text stringlengths 4 2.78M |
|---|
---
abstract: 'We study the localization length $l_c$ of a pair of two attractively bound particles moving in a one-dimensional random potential. We show in which way it depends on the interaction potential between the constituents of this composite particle. For a pair with many bound states $N$ the localization length is proportional to $N$, independently of the form of the two particle interaction. For the case of two bound states we present an exact solution for the corresponding Fokker-Planck equation and demonstrate that $l_c$ depends sensitively on the shape of the interaction potential and the symmetry of the bound state wave functions.'
address: 'Institut f[ü]{}r Theoretische Physik, Technische Universit[ä]{}t Dresden, D-01062, Germany'
author:
- 'M. Turek'
- 'W. John'
title: Localization of a pair of bound particles in a random potential
---
,
Disordered Systems; Localization 72.15.Rn ,61.43.-j
Introduction {#Introduction}
============
Within the single-parameter scaling hypothesis [@Abrahams79] the dimensionless conductance $g(L)$ as a function of system size $L$ determines the transport properties of noninteracting electrons in a random potential (for a review, see Kramer and MacKinnon [@Kramer93]). In one-dimensional disordered systems all states are localized [@Mott61] and the conductance $g(L)$ decreases exponentially [@Anderson80] with increasing sample size $L$. The localization length $l_c$ can be derived from the asymptotic behavior of the conductance $g(L)$ using the relation $$l_c^{-1} = - \lim_{L \to \infty} \frac{d}{dL} \ln g(L).$$ The localization length $l_c$ and the Ljapunov exponent $\gamma \sim l_c^{-1}$ are self-averaging, i.e. non-random quantities [@Lifshitz88] whereas the distribution function of the conductance is approximately log normal [@Abrikosov81]. In the weak disorder limit characterized by a small Fermi wavelength compared to the scattering length $l$ the localization length $l_c$ is given by the mean free path $l$ for backward scattering [@Lifshitz88].
In this paper we study the localization of a composite particle with internal degrees of freedom. As a model we consider a single pair of two particles in a weak one-dimensional white noise potential. We restrict our considerations to the limit of two tightly bound constituents so that the pair does not decay during its motion through the disorder potential. Considering the semiclassical limit the kinetic energy of the center of mass motion is assumed to be larger than the typical energy of the disorder potential. Typical length scales of our model include the mean free paths $l_b$ and $l_f$ of the pair for backward and forward scattering, respectively. The scattering processes cause a transfer of kinetic energy of the center of mass motion to the relative motion within the pair and vice versa. Therefore, the scattering lengths $l_b$ and $l_f$ depend on the structure of the pair, i.e. on the interaction potential between the constituents. Thus one expects that the pair localization length $l_c$ also depends on the shape of the attractive pair potential.
As another but different example for the interplay between interaction and disorder many recent papers study the coherent propagation of two interacting particles (TIP) in a one-dimensional random potential (see [@TIP] and the literature cited there). It is worth mentioning that the TIP model considers the propagation of two repulsing or weakly attracting particles which form a pair due to the scattering by the disordered potential. The size of the pair is of the order of the one particle localization length $l$. As Shepelyansky [@Shepelyansky94] and Imry [@Imry95] argued, attractive as well as repulsive interaction between the particles cause a localization length of the pair which is much larger than the single particle localization length. In the present work, we study two particles with strong interaction which form bound states and have a size that is much smaller than the one particle localization length. For that reason the TIP model and the composite particle model studied in this work consider different limiting cases, which are not directly comparable. Moreover, in the mentioned TIP approach a lattice model of two particles interacting at one lattice site is considered. Generally the coherent motion of the interacting pair is studied only in the middle of the band ($E=0$). However, we study a continuum model of two interacting particles in the semiclassical regime with a Fermi wavelength much smaller than the scattering length ($\lambda_F \ll l$).
A composite particle with $N$ bound states may be considered as an $N$-channel problem for the center of mass motion. On the other hand the motion of a single electron in a thick wire is also a multichannel problem since the transversal motion is quantized. In the thick wire the scattering processes between different channels are random and of the same strength. The localization length $l_c$ of the wire increases with the channel number $N$ as $l_c \sim N l$ because effectively only one of these channels gives rise to coherent backward scattering and produces localization [@Dorokhov83; @Beenakker97]. In our composite particle model the scattering processes between different bound states are also random. However, in contrast to the model of a thick wire, the scattering probabilities depend strongly on the channel numbers. Therefore it is interesting to investigate the localization length in the composite particle model and compare the results for $l_c$ of both different models.
The model of two particles with strong attractive interaction might be relevant for the Coulomb-correlated electron-hole pairs in disordered semiconductors. Recent work [@Brinkmann99] has dealt with the dynamics of the electron-hole pair in a 1D model. Another physical problem which can be related to the model we consider below is the superconductor-isolator transition. In order to study the effects of interaction and disorder, Lages and Shepelyansky [@Lages] investigated numerically the Cooper problem of two quasiparticles with attractive interaction above a frozen Fermi sea in the presence of disorder.
The basic ideas of the model we consider were introduced by Dorokhov [@Dorokhov90]. He studied a pair of particles bound by a harmonic oscillator potential in a weak random disorder potential. For this very special interaction he found a dominant forward scattering in the case of many relevant bound states $N$ ($l_b/l_f \sim \ln N$) employing the specific properties of the harmonic oscillator eigenfunctions. Moreover, the pair scattering length $l_b$ was shown to be on the order of the single particle mean free path $l$ and independent of the strength of the oscillator potential. Using the fact that the forward scattering is the dominant process the pair localization length $l_c = N l_b / 2 \sim N l$ was calculated.
Our results show that Dorokhov’s conclusions are generic for any attractive interaction potential in the case of many bound states $N \gg 1$ regardless of its shape. We extend the method [@Dorokhov90] from the harmonic oscillator potential to an arbitrary attractive interaction potential. This can be accomplished since the excited states of the relative motion within the pair can be described in the semiclassical approximation. We show that the specific properties of the harmonic oscillator wave functions (in [@Dorokhov90] the result was obtained using the recurrence relations of the Hermitian polynomials) are of no importance in the multi-channel case. Using WKB wave functions for the bound states of the pair we estimate the scattering matrix elements and the mean free paths $l_b$ and $l_f$ in the case $N \gg 1$. We find that $l_b \gg l_f$ for any interaction potential. We also show that $l_b$ is qualitatively independent of the interaction. For the localization length $l_c$ we obtain $l_c \sim N l_b / 2$ which implies that $l_c$ is also independent of the interaction.
In addition to the considerations for many bound states $N \gg 1$ we also investigate the first nontrivial case of a small number of bound states, namely $N=2$. For a single particle in a random white noise potential the mean free paths $l_b$ and $l_f$ are equal. We show that already for a pair with only two bound states, the forward scattering is enhanced relative to the backward scattering and that the ratio $l_f/l_b$ depends sensitively on the energy. Finally we derive an exact expression for the localization length of the pair with two accessible bound states. In contrast to earlier treatments [@Dorokhov83] of this problem we do not have to impose any additional restrictions to the scattering matrix elements. Therefore, our solution allows to study the influcence of the two particle interaction on the localization directly. The result indicates that no simple relation exists between $l_c$ and the mean free path of the pair $l_b$ as it was in the limit of many channels $N \gg 1$. As an application of our general result we present numerical calculations showing that the localization length $l_c$ depends sensitively on the interaction potential, the symmetry of the bound state wave functions and the total energy of the pair.
The paper is organized as follows. In Section \[Model\_and\] we describe the model and briefly summarize the transfer matrix approach applied to the composite particle model with arbitrary interaction potential between the two constituents. In Section \[Multi-channel\] we consider a pair with high enough total energy allowing for many bound states. We determine the scattering lengths $l_b$ and $l_f$ as well as the pair localization length $l_c$. In Section \[Two\_channel\] we derive an exact expression for the localization length of a pair with two bound states ($N=2$). Furthermore, we present numerical results for the localization length $l_c$ as well as for the mean free paths $l_b$ and $l_f$ for different interaction potentials.
Model and method of calculations {#Model_and}
================================
We consider two interacting particles in a one-dimensional random potential $V(x)$. The Hamiltonian is given by $$\label{H2}
{\mathcal H} = -\frac{1}{4} \frac{\partial ^2}{\partial x^2} -
\frac{\partial ^2}{\partial y^2}
+ u(y) + V(x+y/2)
+ V(x - y/2),$$ where the units are chosen in a way that $\hbar=1$ and the single electron mass $m=1$. The variable $x$ denotes the center of mass coordinate of the pair while $y$ represents the relative coordinate. The symmetric interaction potential $u(y)$ between the two particles is assumed to be attractive. The bound states $\phi_n(y)$ and the corresponding energy eigenvalues $\epsilon_n$ are given by $$\label{SGL:relat}
\left[ - \frac{\partial^2}{\partial y^2}
+ u(y) \right] \phi_n(y) = \epsilon_n \phi_n(y).$$ In the case of two identical particles the states $\phi_n(y)$ of even parity describe two bosons or a singlet state of fermions while the wave functions of odd parity correspond to a triplet state of fermions. The random potential $V(x)$ is nonzero in the finite interval $0<x<L$. This region of length $L$ is enclosed by two ideal leads without any disorder.
The scattering of the center of mass of the pair with $N$ bound states can be mapped onto an $N$-channel problem of a single free particle. The stationary scattering states of the pair in the leads are given by $$\label{Ansatz2}
\Psi(x,y)=\sum\limits_{n=0}^{N-1} \phi_n(y) \left[ A_n
\frac{\exp{(i k_n x)}}{\sqrt{k_n}} +
B_n \frac{\exp{(- i k_n x)}}{\sqrt{k_n}}
\right].$$ Here, the momentum of the center of mass is denoted by $k_n$. The total energy of the pair $E = k_n^2/4 + \epsilon_n$ determines the number of bound states (channels) $N$ which are included in the sum in Eq. (\[Ansatz2\]). For a given total energy $E$ this number $N$ can be obtained from the relation $\epsilon_{N-1} < E < \epsilon_N$.
As far as the coefficients $A_n$ and $B_n$ are concerned one has to distinguish between the left lead and the right lead. They can be written as vectors $(\vec{A}^L,\vec{B}^L)$ and $(\vec{A}^R,\vec{B}^R)$ for the left and right one, respectively. The transfer matrix $\tau$ describes the scattering in the region between the two leads. It relates the states in the right lead to the states in the left lead [@Beenakker97]: $$\label{Def:tau}
\left( \begin{array}{c}
\vec{A}^L\\ \vec{B}^L
\end{array}
\right) = \tau
\left( \begin{array}{c}
\vec{A}^R\\ \vec{B}^R
\end{array}
\right).$$ The transfer matrix $\tau$ is a $2N \times 2N$ symplectic matrix which can be expressed by the $N \times N$ reflection matrix $r$ and the transmission matrix $t$. These matrices describe the scattering between the channels $n$ and $m$ which correspond to the associated bound states of the pair. The dimensionless conductance $g$ can be expressed [@Imry97] in terms of the $N$ eigenvalues $T_n$ of the Hermitian matrix $t^\dagger t$ as $g=\sum T_n$. Therefore, it is convenient to employ the Hermitian matrix $M\equiv \tau^\dagger \tau$ which can be expressed by the polar decomposition as [@Dorokhov83] $$\label{param.:M}
M = \left( \begin{array}{cc} u & 0 \\ 0 & u^*
\end{array} \right)
\left( \begin{array}{cc} \cosh 2\Gamma &
- \sinh 2\Gamma \\ - \sinh 2\Gamma & \cosh 2\Gamma
\end{array} \right)
\left( \begin{array}{cc} u^\dagger & 0 \\ 0 & u^T
\end{array} \right).$$ In this parameterization $u$ is a $N \times N$ unitary matrix and $\Gamma$ is a $N \times N$ diagonal matrix. The transmission eigenvalues $T_n$ are then related to the matrix elements $\Gamma_n$ by $T_n = 2/[\cosh (2 \Gamma_n) + 1]$.
To evaluate the transmission eigenvalues $T_n(L)$ for a given size $L$ we now consider the matrix $M$ as a function of the sample size $L$. As was shown in [@Dorokhov90] the evolution of $M(L)$ with increasing $L$ can be described by a set of Langevin equations. The random potential $V(L)$ enters these equations as a multiplicative factor. In order to derive these equations the following assumptions were made:
- The potential $V(x)$ is a weak Gaussian white noise potential with the correlation function $$\label{correlation:V}
\langle V(x) V(x') \rangle = 2D \delta (x-x'),$$ where $\langle \dots \rangle$ denotes the ensemble average. The mean free path $l$ of the pair in comparison to the inverse wave numbers $k_n^{-1}$ is the appropriate quantity to describe the strength of the disorder. The semiclassical limit is considered where $k_n l \gg 1$. The mean free path $l$ of a single particle with Fermi energy $E_F$ is given by $l=E_F / D$.
- The mean free path $l$ is large compared to the typical width $b_n$ of the bound states $\phi_n(y)$: $l \gg b_n$. This assumption ensures the concept of mean free path for the center of mass motion.
The localization length $l_c$ of the pair is studied by the asymptotic behavior of the transmission eigenvalues $T_n(L)$ for large system sizes $L \to \infty$. These coefficients $T_n(L)$ decrease exponentially with the system size $L$. The corresponding length scales are defined by $$\label{Def:l_n}
l_n^{-1} \equiv - \lim\limits_{L \to \infty} \frac{d}{dL}
\langle \ln T_n \rangle.$$ The transmission eigenvalues can be arranged in a hierarchical order $T_{N-1} \ll T_{N-2} \ll \dots \ll T_0 \ll 1$. The localization length $l_c$ is then given by the largest length $l_0$. All lengths $l_n$ averaged over the $N$ channels just give the mean free path $l_b$ of the center of mass for backward scattering [@Dorokhov83]: $$\label{avg.l_n}
\frac{1}{N} \sum\limits_{n=0}^{N-1} l_n^{-1} = l_b^{-1}$$ (see Eq. (\[l\_b,f\]) below for the backward scattering length). Taking into account the exponential smallness of all transmission eigenvalues and their different order of magnitude, the Langevin equations for the matrix elements $\Gamma_n$ and $u_{nm}$ are given by $$\begin{aligned}
\label{LG:gamma_n}
\frac{\partial \Gamma_n}{\partial L} &=& i
\left( \beta_u^* - \beta_u \right)_{nn} \cdot V(L) \\
\label{LG:u_nm}
\frac{\partial u_{nm}}{\partial L} &=& -2 i
\sum\limits_{l=0}^{N-1}
\left[ \alpha_{nl} u_{lm} + u_{nl}(\beta_u^*)_{lm}
\Theta(l-m) \right. \nonumber \\
& & \quad \quad \quad \quad
\left. + \; u_{nl}(\beta_u)_{lm} \Theta(m-l) \right]
\cdot V(L)\end{aligned}$$ where the abbreviation $\beta_u$ stands for $\beta_u \equiv u^\dagger \beta u^*$. The value of the step function $\Theta(k)$ for $k=0$ is defined as $\Theta(0) \equiv 1/2$. The matrix elements $\alpha_{nm}$ and $\beta_{nm}$ in Eq. (\[LG:u\_nm\]) describe the forward scattering and the backward scattering of the pair, respectively. They depend on the two particle interaction and can be written as $$\begin{aligned}
\label{W_nm:alpha,beta}
\alpha_{nm} &=& \frac{e^{i(k_n-k_m) L}}{\sqrt{k_n k_m}}
W_{nm}(k_n - k_m) , \nonumber \\
\beta_{nm} &=& \frac{e^{-i(k_n+k_m) L}}{\sqrt{k_n k_m}}
W_{nm}(k_n + k_m) ,\end{aligned}$$ where $W_{nm}(k)$ is $$\label{W_nm}
W_{nm}(k) \equiv 2 \int\limits_{-\infty}^{\infty}
dy \; \phi_n(y) \phi_m(y) \, \cos \left( \frac{ky}{2} \right).$$ Within the Born approximation, the mean free paths $l_{b,f}$ of the pair for backward and forward scattering are given by [@Dorokhov90]: $$\label{l_b,f}
l_b^{-1} = \frac{8 D}{N} \sum\limits_{n,m}^{N-1}
| \beta_{nm} |^2 \quad \text{and} \quad
l_f^{-1} = \frac{8 D}{N} \sum\limits_{n,m}^{N-1}
| \alpha_{nm} |^2.$$ Note that one has to consider bound states of a given parity only. Due to the finite width of the pair we therefore expect that the forward scattering is enhanced relative to the backward scattering although the scattering of a single particle is isotropic. This circumstance allows to solve the many channel problem [@Dorokhov90] with $N\gg 1$ (see section \[Multi-channel\]).
Using the fact that the transport is effectively determined by just one channel ($T_0 \gg T_{n>0}$) Eq. (\[Def:l\_n\]) results in the following expression for the pair localization length [@Dorokhov90] $$\label{l_c:general}
l_c^{-1} = 16D \sum\limits_{n,m=0}^{N-1}
\langle |u_{n0}|^2 |u_{m0}|^2 \rangle (1-\frac{1}{2}\delta_{nm})
|\beta_{nm}|^2.$$ Hence, the localization length $l_c$ depends on the correlation function $\langle |u_{n0}|^2 |u_{m0}|^2 \rangle$ which has to be evaluated using the nonlinear Langevin Eqs. (\[LG:u\_nm\]).
For $N=1$ the unitary matrix $u$ is simply a phase factor with $|u_{00}|=1$. The localization length $l_c$ is then given by the mean free path $l_b$ for backward scattering $$\label{l_c:N=1}
l_c^{-1} = l_b^{-1} = 8D |\beta_{00}|^2$$ and the interaction potential $u(y)$ enters via the ground state wave function in the matrixelement (\[W\_nm\]). In the next section we consider the multi-channel case $N \gg 1$ and show that the localization length depends only weakly on the interaction.
Multi-channel localization length {#Multi-channel}
=================================
In order to evaluate the localization length (\[l\_c:general\]) in the multi-channel case $N \gg 1$ one has to compute the correlation function $\langle |u_{n0}|^2 |u_{m0}|^2 \rangle$ using the nonlinear Langevin equations (\[LG:u\_nm\]). For the harmonic oscillator interaction it was shown in [@Dorokhov90] that the forward scattering dominates in Eq. (\[LG:u\_nm\]). Therefore the nonlinear terms due to the backward scattering processes were neglected in (\[LG:u\_nm\]) and it was found that the correlation function is properly described by the invariant unitary ensemble $$\label{correlation:u}
\langle |u_{n0}|^2 |u_{m0}|^2 \rangle = (1 + \delta_{nm})/N^2.$$ In the following we proof that the same result is valid for any attractive interaction potential. The order of magnitude of forward and backward scattering processes is determined by the corresponding mean free paths $l_f$ and $l_b$ given by (\[l\_b,f\]). These mean free paths depend on the pair interaction potential $u(y)$ via the bound state wave functions $\phi_n(y)$ in Eq. (\[W\_nm\]). We calculate the matrix elements $W_{nm}(k)$ using WKB wave functions and evaluate the integral (\[W\_nm\]) in the saddle point approximation. This approximation is justified by the semiclassical approximation and the assumption $N \gg 1$, which implies that the vast majority of bound states is properly described by the WKB approximation. In this approximation the following expression can be derived from (\[W\_nm\]) $$\begin{aligned}
\label{WKB:W_nm}
W_{nm}(k) = &4& (-1)^{(n \pm m)} \left(
\frac{2 \pi}{{\mathcal T}_n {\mathcal T}_m |u'(y_0) \cdot k|} \right)^{1/2}
\nonumber \\
&\times & \cos \left[ \mp \frac{1}{2} \int\limits_{k_{2,1}}^{|k|}
d\tilde{k} \; y_0(\tilde{k}) - \frac{\pi}{4} \right].\end{aligned}$$ The time ${\mathcal T}_n$ denotes the period of the semiclassical motion of the bound state $n$. The two saddle points $\pm y_0$ are functions of $k$ and follow from the implicite equation $$\label{WKB:u(y_0)}
u(y_0) =
\frac{\epsilon_n+\epsilon_m}{2} - \frac{k^2}{16}
-\frac{(\epsilon_n - \epsilon_m)^2}{k^2}.$$ In Eq. (\[WKB:W\_nm\]) $u'(y_0)$ stands for the derivative of the potential evaluated at position $y_0$. It results from the Gaussian integral at the saddle point. Calculating the matrix elements $W_{nm}(k)$ for backward or forward scattering we set $k=k_n \pm k_m$. In this relation as well as in Eq. (\[WKB:W\_nm\]) the upper sign and $k_2$ characterize the backward scattering while the lower sign and $k_1$ represent the forward scattering. The expression (\[WKB:W\_nm\]) is valid within the interval $k_1 < |k| < k_2$ where Eq. (\[WKB:u(y\_0)\]) has a real solution $y_0(k)$. The limits of the interval are given by $$\label{k_1,2}
k_{2,1} = 2 (\sqrt{\epsilon_n-u_0} \pm \sqrt{\epsilon_m-u_0})$$ with the potential minimum $u_0 \equiv u(y=0)$. Outside this $k$ region the saddle point is shifted to the complex $y$-plane and the matrix element $W_{nm}(k)$ decreases exponentially [@LLBd3]. Calculating the mean free paths (\[l\_b,f\]) we will neglect these exponentially small contributions.
To demonstrate the accuracy of the WKB wave functions together with the saddle point approximation we show an example for a matrix element in Fig. \[fig:w\_22\]. We have calculated the exact function $W_{22}(k)$ for the harmonic oscillator potential $u_{osc}(y) = \xi^{-2} (y/\xi)^2$ using the analytical expression for the third symmetric bound state $\phi^s_2(y)$. This exact result is compared with our WKB result (\[WKB:W\_nm\]). One clearly sees that the oscillatory part is very well described by the approximation. Of course, close to $k=k_{1,2}$ the saddle point approximation fails because $u'(y_0)$ goes to zero and the Gaussian approximation is not valid. Furthermore, the diagonal matrix elements $W_{nn}$ cannot be calculated from Eq. (\[WKB:W\_nm\]) for $k=0$. However, definition (\[W\_nm\]) immediately yields $W_{nn}(k=0) = 2$.
=0.47
The scattering lengths are then evaluated by replacing the sum in (\[l\_b,f\]) by an integral (with $k_n \to k$ and $k_m \to \tilde{k}$). We obtain $$\label{l_b,f:integral}
l_{b,f}^{-1} = \frac{8D}{4 \pi N} \int
\frac{dk \; d\tilde{k}}{(k \pm \tilde{k}) \cdot |u'(y_0)|}.$$ This result may be understood as follows. For $k^2=(k_n \pm k_m)^2$ the saddle point equation (\[WKB:u(y\_0)\]) can be rewritten as $$\label{saddlepoint}
u(y_0) = E - \frac{k_n^2}{4} - \frac{k_m^2}{4} = \epsilon_n
- \frac{k_m^2}{4} = \epsilon_m - \frac{k_n^2}{4}.$$ This means that the kinetic energy of the relative motion $\epsilon_n-u(y_0)$ for the initial channel $n$ transforms into the kinetic energy $k_m^2/4$ of the center of mass in the final channel $m$ and vice versa. The transition rate into the final channel $m$ is therefore proportional to the probability density $\rho_n(k_m)$ of finding the pair in the initial channel $n$ with the appropriate relative momentum. The expression (\[l\_b,f:integral\]) for the mean free paths follows then immediately from $\rho_n(k_m)=4/({\mathcal T}_n |u'(y_0)|)$.
We will now show that Eq. (\[l\_b,f:integral\]) implies a mean free path for the backward scattering that is almost independent of the channel number $N$. Using Eq. (\[saddlepoint\]) we write $u'(y_0)$ formally as $$\label{u'(y_0)}
u'(y_0) = F(u(y_0)) = F\left(
E - \frac{k^2}{4} - \frac{\tilde{k}^2}{4} \right).$$ This is always a unique representation if the interaction potential is assumed to be monotonic. Introducing polar coordinates the angular integration in (\[l\_b,f:integral\]) can be performed and we obtain the expression for the backward scattering length $$\label{WKB:l_b,result}
l_b^{-1} = \sqrt{2}
\ln \left[ \frac{\sqrt{2}+1}{\sqrt{2}-1} \right]
\frac{\int\limits_0^1 dt \,
(1-t)^{-1/2} \, \left[ F(Et) \right]^{-1}}
{\int\limits_0^1 dt \,
(1-t)^{1/2} \, \left[ F(Et) \right]^{-1}} l^{-1}.$$ The denominator in (\[WKB:l\_b,result\]) results from the number $N$ of channels with given parity below the energy $E$. Within the WKB approximation the relation between the number $N$ and the energy $E$ can be expressed by using expression (\[u’(y\_0)\]) once more: $$\begin{aligned}
\label{WKB:N}
N(E) &=& \frac{1}{2 \pi} \int dy \, \sqrt{E - u(y)} \nonumber \\
&=& \frac{1}{\pi} E^{3/2}
\int\limits_0^1 dt \; (1-t^2)^{1/2} \,
\left[ F(Et) \right]^{-1}.\end{aligned}$$ From Eq. (\[WKB:l\_b,result\]) follows that $l_b \sim l$ with a prefactor of order one. This prefactor depends only weakly on the interaction potential $u(y)$ and the number of accessible bound states $N$. To demonstrate this we consider two examples of interaction potentials.
In the case of a scale invariant potential with $u(y)\sim y^\nu$ and $F(u) \sim u^{(\nu - 1)/ \nu}$ one immediately realizes that $l_b / l$ is completely independent of the energy as it was the case for the harmonic oscillator ($\nu=2$). From (\[WKB:l\_b,result\]) follows $$\label{u_si:l_b}
l_b^{-1} = l^{-1} 4 \sqrt{2} \ln \left[
\frac{\sqrt{2}+1}{\sqrt{2}-1} \right] \cdot \left(
\frac{1}{2} + \frac{1}{\nu} \right).$$ This expression includes the result for the harmonic oscillator presented in [@Dorokhov90] which was found by calculating the matrix elements $W_{nm}$ exactly. For any interaction potential $u(y)$ with many bound states we expect only a weak dependence of $l_b$ on the energy $E$ or on the number of channels $N$ which are below this pair breaking energy. As a further example we give the result for the P[ö]{}schel-Teller interaction [@Junker96] $u^a_{PT}(y)=\xi^{-2} (a^2-a(a+1)/\cosh^{2}[y/\xi])$ where $a$ is an integer parameter. In contrast to the scale invariant potentials this interaction has a maximal number $N_{max}$ of bound states given by the parameter $a$. The pair breaking energy is $\sim \xi^{-2}a^2$. The semiclassical limit can be achieved for a large parameter $a$ implying $N_{max} \gg 1$. The backward scattering length (\[WKB:l\_b,result\]) can then be written as $$\label{u_PT:l_b}
l_b^{-1} = l^{-1} \sqrt{2} \ln \left[
\frac{\sqrt{2}+1}{\sqrt{2}-1} \right]
\frac{1}{1-(N/N_{max})^2}.$$ If the total energy of the pair is much smaller than the pair breaking energy then $N \ll N_{max}$ follows and again the ratio $l_b/l$ is only very weakly energy dependent.
Using the same approach for the forward scattering length $l_f$ one realizes that the main contribution in Eq. (\[l\_b,f:integral\]) is due to the scattering between adjacent channels with $k \approx \tilde{k}$. However, the saddle point approximation fails for $k=\tilde{k}$ and the integral in (\[l\_b,f:integral\]) diverges. Therefore, one has to exclude a small strip of order $|k-\tilde{k}| \sim 1/N$ in the integral and calculate the in-channel scattering separately. We find in accordance with the ealier result for the harmonic oscillator interaction [@Dorokhov90] that $$\label{WKB:l_f,result}
l_f^{-1} \sim l^{-1} \; \ln N$$ for $N\gg 1$. From this estimate (\[WKB:l\_f,result\]) follows $l_f \ll l_b$ for the multi-channel case. This means that the forward scattering is dominant and the relation (\[correlation:u\]) can be applied for any interaction potential $u(y)$. The Eqs. (\[WKB:l\_b,result\], \[WKB:l\_f,result\]) are thus the first major result of this work.
Using the correlation function (\[correlation:u\]) we then obtain the localization length $$\label{final_result:l_c}
l_c = \frac{N}{2} l_b$$ from expression (\[l\_c:general\]). Thus we find a delocalization proportional to the number of open channels $N$ compared to the one particle localization length which equals $l$. This shows that the localization effect is qualitatively independent of the shape of the attractive interaction potential $u(y)$. The interaction only gives rise to a weak energy dependence and a numerical prefactor in the backward scattering length of the pair $l_b$ compared to the single particle scattering length $l$.
Our result (\[final\_result:l\_c\]) has the same structure as in the different model of a thick wire [@Beenakker97]. For that case the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation describes the evolution of the distribution function for the transmission eigenvalues $T_n$ (or $\Gamma_n$) with increasing length of the wire. Compared to our model an important simplification there is the isotropy assumption for the scattering between the channels on average. Using this assumption the Fokker-Planck equation for the transmission eigenvalues $T_n$ decouples from the distribution function of the matrix elements $u_{nm}$. At least in the absence of time-reversal symmetry the DMPK equation could be solved exactly [@Beenakker94]. In our model the dominance of the forward scattering processes is the important property which yields the correlation function (\[correlation:u\]) and leads with (\[l\_c:general\]) to the final result (\[final\_result:l\_c\]).
Two channel system {#Two_channel}
==================
As yet another application of the general result (\[l\_c:general\]) we now derive the exact solution for the case of two open channels $N=2$. Here, the matrix $u_{nm}$ is a $2 \times 2$ unitary matrix. Thus only the expression $\zeta \equiv |u_{00}|^2 - |u_{10}|^2$ occurs in Eq. (\[l\_c:general\]) for the localization length. One can rewrite (\[l\_c:general\]) in terms of this variable and obtains $$\label{l_c:N=2,a}
l_c^{-1} = 8D \left[ (\beta^+ - |\beta_{10}|^2)
\langle \zeta^2 \rangle
+ 2 \beta^- \langle \zeta \rangle +
(\beta^+ + |\beta_{10}|^2) \right]$$ with $\beta^\pm = (|\beta_{00}|^2 \pm |\beta_{11}|^2)/4$. This reduces the ensemble average over combinations of matrix elements $u_{nm}$ to the moments $\langle \zeta \rangle$ and $\langle \zeta^2 \rangle$. In order to find the stationary distribution function $P(\zeta)$ we use the parameterization $\zeta=\cos 2 \theta$ and $$\label{param.:u}
u = \left( \begin{array}{cc} e^{-i \phi_0} & 0 \\
0 & e^{-i \phi_1}
\end{array} \right)
\left( \begin{array}{cc} \cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array} \right)
\left( \begin{array}{cc} e^{-i \gamma} & 0 \\
0 & e^{i \gamma}
\end{array} \right)$$ for the unitary matrix $u_{nm}$. The stationary probability distribution can then be obtained from the associated Fokker-Planck equation. Analyzing this equation we find that the exponentials $e^{i \phi_{0,1}}$ and $e^{i \gamma}$ always occur together with the matrix elements $\alpha$ and $\beta$, respectively. If one neglects the rapidly oscillating contributions in this Fokker-Planck equation then only those terms which contain the magnitudes $|\alpha_{nm}|^2$ and $|\beta_{nm}|^2$ remain. For that reason, the stationary probability distribution is independent of the phases $\phi_{1,2}$ and $\gamma$. The solution $P(\zeta)$ of the stationary Fokker-Planck equation follows from the first order differential equation $$\begin{aligned}
\label{DGL:P,N=2}
0 &=& \left[ \frac{\partial}{\partial \zeta}
(1+A \zeta^2) + B \right] P(\zeta) \\
\mbox{with} \quad
A & \equiv& \frac{ |\beta_{10}|^2-\beta^+}{|\alpha_{10}|^2+\beta^+}
\quad \text{and} \quad
B \equiv \frac{2 \beta^-}{|\alpha_{10}|^2+\beta^+}.\end{aligned}$$ It is of the type $P(\zeta) \sim \exp [-B \int^\zeta dx (1+Ax^2)^{-1}]/(1+A \zeta^2)$. Thus we arrive at the following expression for the two channel localization length: $$\begin{aligned}
\label{l_c:N=2}
l_c^{-1} = 8 \, E \, l^{-1} ( |\alpha_{10}|^2 + \beta^+)
& & \left[ 1 +
\frac{|\beta_{10}|^2+\beta^+}{|\alpha_{10}|^2+\beta^+}
\right. \nonumber \\
& & - \left. \frac{|\alpha_{10}|^2 + |\beta_{10}|^2}
{|\alpha_{10}|^2+\beta^+} P^+ \right]\end{aligned}$$ with $P^+ \equiv [P(1)+P(-1)]$. The length scale $l$ is the mean free path of a single particle having the same energy as the pair, i.e. $E=2 E_F$. The probabilities $P(\pm 1)$ for $\zeta = \pm 1$ can easily be obtained from the solution of Eq. (\[DGL:P,N=2\]). It yields $$\begin{aligned}
P^+ &=&
\frac{B}{1+A} \coth \left[ \frac{B}{\sqrt{A}} \arctan \sqrt{A}
\right] \quad \text{for} \quad A>0 \\
P^+ &=&
\frac{B}{1+A} \coth \left[ \frac{B}{\sqrt{-A}} \mbox{artanh} \sqrt{-A}
\right] \quad \text{for} \quad A<0.\end{aligned}$$ Expression (\[l\_c:N=2\]) provides a general solution for the localization length in the two channel regime without any further restrictions concerning the matrix elements $\alpha_{nm}$ and $\beta_{nm}$. It is therefore the main result regarding the two channel case.
Some special cases of (\[l\_c:N=2\]) may be of interest. For isotropic backward scattering $|\beta_{00}|^2=|\beta_{11}|^2 = 2 |\beta_{10}|^2$ follows $A=0$ and $B=0$ in (\[DGL:P,N=2\]) and the distribution function is independent of the forward scattering: $P(\zeta) = 1/2$. This case corresponds to the invariant distribution of an ensemble of unitary $2 \times 2$ matrices. If the two channels are equivalent, so that $|\beta_{00}|^2=|\beta_{11}|^2$ and thus $B=0$, the distribution function $P(\zeta) \sim (1+A \zeta^2)^{-1}$ follows from (\[DGL:P,N=2\]). The special case where $|\beta_{00}|^2 = |\beta_{11}|^2 = |\beta_{10}|^2$ was already discussed in [@Dorokhov83] and is correctly reproduced by our general solution.
We complete our discussion of the $N=2$ case with the presentation of some numerical results for two specific examples of interaction potentials. Using the result (\[l\_c:N=2\]) one can analyze the influence of the interaction potential $u(y)$ on the localization length $l_c$. The requirement of exactly two open channels restricts the energy range to $\epsilon_1 < E < \epsilon_2$ where $\epsilon_0, \epsilon_1, \dots , \epsilon_n$ are the energies corresponding to the bound states with given parity. In order to demonstrate how the localization length depends on the structure of the pair, we present a comparison between the harmonic oscillator interaction $u_{osc}(y)$ and the interaction where the two particles are bound together by a box-like potential $u_{box}(y)$. While $u_{osc}(y)$ is a smooth function of the relative coordinate $y$ the second interaction potential $u_{box}(y)$ is constant for $|y|$ being smaller than some fixed value $y_s$ and jumps to infinity outside this region, see Fig. \[fig:u\_bereich\].
We first discuss the case of bound states with even parity. The size of the box and a total energy shift for the harmonic oscillator potential were adjusted so that the energy levels $\epsilon_{0,1}$ entering (\[l\_c:N=2\]) are at the same position. This approach ensures that all direct energy dependencies of the localization length have the same structure for either interaction type. The energy window for the $N=2$ case is then given by $4.5 < E \xi^2 < 8.5$ ($4.5 < E \xi^2 < 12.5$) for the harmonic oscillator (box-like potential), Fig. \[fig:u\_bereich\].
=0.47
The next step is to evaluate the matrix elements $\alpha$ and $\beta$ as given by (\[W\_nm:alpha,beta\]) and (\[W\_nm\]). Inserting these matrix elements in (\[l\_c:N=2\]) gives the result for the localization length $l_c(E)$ which is presented in Fig. \[fig:l\_c\_symm\]. The interaction between the constituents of the composed particle clearly causes a delocalization compared to the single particle with no internal degrees of freedom. The reason is that the finite size of the pair effectively smoothes out the disorder. However, this delocalization effect sensitively depends on the total energy $E$ and the type of the two particle interaction. The resonance in the case of the box-like potential is clearly more pronounced. If the energy is close to this resonance, i.e. $E\xi^2 \approx 6.6$, the delocalization effect differs by a factor $\sim 5$ due to the different interaction potentials.
=0.47
Furthermore, we have examined the ratio between forward and backward scattering $l_f / l_b$ as a function of the energy. As Fig. \[fig:l\_bf\] indicates, the forward scattering relative to the backward scattering is already for just two channels enhanced. Although the ratio $l_f/l_b$ depends on the energy this effect holds true for the entire energy window accessible for the two channel case. The suppression of the backward scattering is because the finite size of the pair reduces scattering processes with large momentum transfer as can be concluded from Eqs. (\[W\_nm:alpha,beta\], \[W\_nm\]). This effect is even more pronounced if many channels are open, a fact that is of importance for the solution of the multi-channel case, as discussed in Sec. \[Multi-channel\].
=0.47
Using the sum rule (\[avg.l\_n\]) we calculate the length scale $l_1$ of the transmission eigenvalue of the second internal channel. It is given by $l_1 = (2 l_b^{-1} - l_c^{-1})^{-1}$, see Fig. \[fig:l\_c1\_symm\]. We find that the ratio $l_1/l_c$ is smaller than $0.6$ over the entire energy range for either interaction potential. This justifies the assumption that the transmission is mainly due to one internal channel if the system is large, i.e. $L \gg l$, because of the exponential dependence of the transmission amplitudes on the system size.
=0.47
Finally we compute the localization length for the situation where the two bound states are of odd parity. Again, the parameters of the interaction potentials are fixed such that the energies of the two involved energy levels coincide for the different interaction potentials. The resulting localization length is plotted in Fig. \[fig:l\_c\_anti\]. There is a significant difference between the localization lengths in the case of wave functions with even parity compared to odd parity. For the harmonic oscillator potential there is a sharp resonance near the lower limit $\epsilon_1$ of the energy window in the second case (Fig. \[fig:l\_c\_anti\]) while the first case shows a smooth behavior with a weak resonance in the middle of the energy range (Fig. \[fig:l\_c\_symm\]). For the $u_{box}(y)$ interaction the resonance position and height also changes significantly. Since the energy levels are shifted to the same position for either case the differences result only from the different structure of the bound state wave functions. The largest delocalization effect occurs for $u_{box}(y)$ for the states of odd parity and is about $l_c/l \approx 25$ at the resonance.
=0.47
Conclusion {#Conclusion}
==========
In this work we have studied the localization of a pair of two bound particles in a one-dimensional system with weak disorder. The interaction potential within this composite particle is fully taken into account within the semiclassical approximation. Our results are based on a method introduced by Dorokhov for two particles bound by a harmonic oscillator potential. We have generalized the method from this special interaction potential to an arbitrary attractive interaction. This allows us to analyze the dependence of the pair localization on the interaction. We find an enhancement of the pair localization length $l_c$ in comparison to the single particle mean free path $l$. This enhancement is independent of the form of the pair interaction potential in the limit of many bound states $N \gg 1$. It is given by $l_c/l \sim N$ as can be seen from the central result (\[final\_result:l\_c\]) in conjunction with (\[WKB:l\_b,result\]). Furthermore we derived an exact solution for the two channel case $N=2$. For a bound pair with $N=1,2$ we observe a sensitive dependence of the localization length (\[l\_c:N=1\]), (\[l\_c:N=2\]) on the shape of the interaction potential, the parity of the involved bound states and the kinetic energy of the pair. We expect that such a behavior is typical for bound pair with a small number of bound states.
The approach described in this work is valid for two identical particles. Nevertheless, it is possible to extend the method without difficulties to a pair of different particles (e.g. electron-hole pair). In this case, the expression for the matrix elements (\[W\_nm\]) has to be modified taking into account that the random potential and the mass may be different for the two particles. However, the described method cannot be extended to a repulsing interaction because the assumption that the pair size is smaller than the mean free path breaks down.
[99]{} E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Lett. [**42**]{}, 673 (1979) B. Kramer and A. MacKinnon, Rep. Prog. Phys. [**56**]{}, 1469 (1993) N.F. Mott and W.D. Twose, Adv. Phys. [**10**]{}, 107 (1961) P.W. Anderson, D.J. Thouless, E. Abrahams and D.S. Fisher, Phys. Rev. B [**22**]{}, 3519 (1980) I.M. Lifshitz, S.A. Gredeskul and L.A. Pastur, [*Introduction to the theory of disordered systems*]{}, (Wiley, New York, 1988) A.A. Abrikosov, Solid State Comm. [**37**]{}, 997 (1981) O.N. Dorokhov, Zh. Eksp. Teor. Fiz. [**85**]{}, 1040 \[Sov. Phys. JETP [**58**]{}, 606\] (1983) C.W.J. Beenakker, Rev. Mod. Phys. [**69**]{}, 731 (1997) R.A. R[ö]{}mer, M. Schreiber and T. Vojta, Phys. Stat. Sol. (b) [**211**]{}, 681 (1999); O. Halfpap, Ann. Phys. (Leipzig) [**10**]{}, 623 (2001) D.L. Shepelyansky, Phys. Rev. Lett. [**73**]{}, 2607 (1994) Y. Imry, Europhys. Lett. [**30**]{}, 405 (1995) D. Brinkmann, J.E. Golub, S.W. Koch, P. Thomas, K. Maschke and I. Varga, Eur. Phys. J. B [**10**]{}, 145, (1999) J. Lages and D. Shepelyansky, Phys. Rev. B [**62**]{}, 8665, (2000); J. Lages and D. Shepelyansky, Eur. Phys. J. B [**21**]{}, 129, (2001) O.N. Dorokhov, Zh. Eksp. Teor. Fiz. [**98**]{}, 646 \[Sov. Phys. JETP [**71**]{}, 360\] (1990) Y. Imry, [*Introduction to mesoscopic physics*]{}, (Oxford University Press, 1997) G. Junker, [*Supersymmetric Methods in Quantum and Statistical Physics*]{}, (Springer Verlag Berlin Heidelberg, 1996) L.D. Landau and E.M. Lifschitz, [*Quantenmechanik*]{}, (Akademie-Verlag Berlin, 1988) C.W.J. Beenakker and B. Rejaei, Phys. Rev. B [**49**]{}, 7499, (1994)
|
---
abstract: 'The effect of a randomly fluctuating gap, created by a random staggered potential, is studied in a monolayer and a bilayer of graphene. The density of states, the one-particle scattering rate and transport properties (diffusion coefficient and conductivity) are calculated at the neutrality point. All these quantities vanish at a critical value of the average staggered potential, signaling a continuous transition to an insulating behavior. The calculations are based on the self-consistent Born approximation for the one-particle scattering rate and a massless mode of the two-particle Green’s function which is created by spontaneous symmetry breaking. Transport quantities are directly linked to the one-particle scattering rate. Moreover, the effect of disorder is very weak in the case of a monolayer but much stronger in bilayer graphene.'
author:
- 'K. Ziegler'
title: Random gap model for graphene and graphene bilayers
---
Graphene, a sheet of carbon atoms, or bilayer graphene are semimetals with good conducting properties [@novoselov05; @zhang05; @geim07]. In particular, the minimal conductivity at the neutrality point (NP) is very robust and almost unaffected by disorder or thermal fluctuations [@geim07; @tan07; @chen08; @morozov08]. Recent experiments with hydrogenated graphene [@elias08] and biased bilayer graphene [@ohta06; @oostinga08; @gorbachev08] have revealed that a staggered potential (SP) can be created in graphene and bilayer graphene which breaks the sublattice symmetry. This opens a gap at the Fermi energy, leading to an insulating behavior. With this opportunity one enters a new field, where one can switch between a conducting and an insulating regime of a two-dimensional material, either by a chemical process (e.g. oxidation or hydrogenation) or by applying an external electric field [@castro08].
It is clear that the opening of a uniform gap destroys the metallic state immediately. This means that the (minimal) conductivity at the NP drops from a finite value of order $e^2/h$ directly to zero. In a realistic system, however, the gap may not be uniform. This means that locally gaps open, whereas in other regions of the sample there is no gap. The situation can be compared with a classical random network of broken and unbroken bonds. The conductivity of such a network is nonzero as long as there is a percolating cluster of unbroken bonds. In such a system the transition from conducting to insulting behavior is presumably a second order percolation transition [@cheianov07].
Disorder in graphene has been the subject of a number of recent numerical studies [@xiong07; @zhang08]. The results can be summarized by the statement that chiral-symmetry preserving disorder provides delocalized states whereas a chiral-symmetry breaking potential disorder leads to Anderson localization, even at the NP.
Conductivity and other transport properties in graphene can be evaluated by solving the Bethe-Salpeter equation for the average two-particle Green’s function (Cooperon) [@suzuura02; @peres06; @khveshchenko06; @mccann06; @yan08]. Unfortunately, the Bethe-Salpeter equation is usually a complex matrix equation which is difficult to handle. Therefore, a different approach will be employed here that eliminates a part of the complexity by focusing on continuous symmetries and spontaneous symmetry breaking. This allows us to identify a (massless) diffusion mode in the system with a randomly fluctuating gap. Consequently, diffusion can only stop when the spontaneous symmetry breaking vanishes. It will be discussed in this paper that this can happen if the average SP approaches a critical value. Moreover, there is no drop of the conductivity but a continuous decay to zero, depending on the fluctuations of the SP.
[*model:*]{} Quasiparticles in monolayer graphene (MLG) or bilayer graphene (BLG) are described in tight-binding approximation by a nearest-neighbor hopping Hamiltonian =-t\_[<r,r’>]{}c\^\_r c\_[r’]{}+\_r m\_r c\^\_r c\_r +h.c. , \[ham00\] where the underlying structure is either a honeycomb lattice (MLG) or two honeycomb lattices with Bernal stacking (BLG).
The sublattice symmetry of the honeycomb lattice is broken by a staggered potential (SP) $m_r$ which is positive (negative) on sublattice A (B) [@mccann06b; @koshino06]. Such a potential can be the result of chemical absorption of other atoms (e.g. oxygen or hydrogen [@elias08]) or of an external gate voltage applied to the two layers of BLG [@ohta06]. Neither in MLG nor in BLG the potential $m_r$ and, therefore, the gap is uniform, because of fluctuations in the coverage of the MLG by additional non-carbon atoms or by the fact that the graphene sheets are not planar [@morozov06; @meyer07; @castroneto07b]. Deviations from the planar structure in the form of ripples cause fluctuations in the distance of the two sheets in BLG which results in an inhomogeneous potential $m_r$ along each sheet. It is assumed that the gate voltage is adjusted at the NP such that in average the SP is exactly antisymmetric: $\langle m_A\rangle=-\langle m_B\rangle$.
At first glance, the Hamiltonian in Eq. (\[ham00\]) is a standard hopping Hamiltonian with random potential $m$, frequently used to study the generic case of Anderson localization [@anderson58]. The dispersion, however, is special in the case of graphene due to the honeycomb lattice: at low energies it consists of two valleys $K$ and $K'$ [@castroneto07b; @mccann06]. It is assumed that weak disorder scatters only at small momentum such that intervalley scattering is not relevant. Then each valley contributes separately to transport, and the contribution of the two valleys to the conductivity $\sigma$ is additive: $
\sigma=\sigma_K+\sigma_{K'}
$. This allows us to consider for the low-energy properties a Dirac-type Hamiltonian for each valley separately H=h\_1\_1+h\_2\_2+m\_3 \[ham01\] with Pauli matrices $\sigma_j$ and with $h_j$ h\_j=i\_j (MLG), h\_1=\_1\^2-\_2\^2, h\_2=2\_1\_2 (BLG) . \[elements\] $\nabla_j$ is the lattice difference operator in $j$ ($=1,2$) direction. Within this approximation the SP $m_r$ is a random variable with mean value $\langle m_r\rangle_m ={\bar m}$ and variance $\langle (m_r-{\bar m})(m_{r'}-{\bar m})\rangle_m=g\delta_{r,r'}$. The following transport calculations will be based entirely on the Hamiltonian of Eq. (\[ham01\]). In particular, the average Hamiltonian $\langle H\rangle_m$ can be diagonalized by Fourier transformation and is $
%\langle H\rangle_m =
k_1\sigma_1+k_2\sigma_2+{\bar m}\sigma_3
$ for MLG with eigenvalues $E_k=\pm\sqrt{{\bar m}^2+k^2}$. For BGL the average Hamiltonian is $
%\langle H\rangle_m =
(k_1^2-k_2^2)\sigma_1+2k_1k_2\sigma_2+{\bar m}\sigma_3
$ with eigenvalues $E_k=\pm\sqrt{{\bar m}^2+k^4}$.
[*symmetries:*]{} Transport properties are determined by the model properties on large scales. The latter are controlled by the symmetry of the Hamiltonian and of the corresponding one-particle Green’s function $G(i\epsilon)=(H+i\epsilon)^{-1}$. In the absence of sublattice-symmetry breaking (i.e. for $m=0$), the Hamiltonian $H=h_1\sigma_1+h_2\sigma_2$ has a continuous chiral symmetry H e\^[\_3]{} He\^[\_3]{}=H \[contsymmetry\] with a continuous parameter $\alpha$, since $H$ anticommutes with $\sigma_3$. The SP term $m\sigma_3$ breaks the continuous chiral symmetry. However, the behavior under transposition $h_j^T=-h_j$ for MLG and $h_j^T=h_j$ for BLG provides a discrete symmetry: H-\_j H\^T\_j =H , \[discretesymm\] where $j=1$ for MLG and $j=2$ for BLG. This symmetry is broken for the one-particle Green’s function $G(i\epsilon)$ by the $i\epsilon$ term. To see whether or not the symmetry is recovered for $\epsilon\to0$, the difference G(i)+\_jG\^T(i)\_j=G(i)-G(-i)=i(E=0) \[op\] must be evaluated, where $\rho(E=0)\equiv\rho_0$ is the density of states at the NP. Here the limit $\epsilon\to0$ is implicitly assumed. Thus the order parameter for spontaneous symmetry breaking is $\rho_0$.
[*conductivity:*]{} The conductivity can be calculated from the Kubo formula. Here we focus on interband scattering between states of energy $\omega/2$ and $-\omega/2$, which is a major contribution to transport near the NP. The frequency-dependent conductivity then reads [@ziegler08] \_0() =-\^2 \_[-/2]{}|r\_k\^2|\_[/2]{}\_m , \[cond0b\] where $|\Phi_{E}\rangle$ is an eigenstate of $H$ in Eq. (\[ham01\]) with energy $E$. In other words, the conductivity is proportional to a matrix element of the position operator $r_k$ ($k=1,2$) with respect to energy eigenfunctions from the lower and the upper band. The matrix element $\langle\Phi_{\omega/2}|r_k^2|\Phi_{-\omega/2}\rangle$ is identical with the two-particle Green’s function \_[r]{} r\_k\^2 Tr\_2 . \[cond2\] This indicates that transport properties are expressed by the two-particle Green’s function $G(i\epsilon)G(-i\epsilon)$. Each of the two Green’s functions, $G(i\epsilon)$ and $G(-i\epsilon)$, can be considered as a random variable which are correlated due to the common random variable $m_r$. Their distribution is defined by a joint distribution function $P[G(i\epsilon),G(-i\epsilon)]$. In terms of transport theory, both Green’s functions must be included on equal footing. This is possible by introducing the extended Green’s function $${\hat G}(i\epsilon)=\pmatrix{
G(i\epsilon) & 0 \cr
0 & G(-i\epsilon) \cr
} =\pmatrix{
H+i\epsilon & 0 \cr
0 & H-i\epsilon \cr
}^{-1} \ .$$ In the present case one can use the symmetry transformation of $H$ in Eq. (\[discretesymm\]) to write the extended Green’s function as $$\pmatrix{
\sigma_0 & 0 \cr
0 & i\sigma_j \cr
}\pmatrix{
H+i\epsilon & 0 \cr
0 & H^T+i\epsilon \cr
}^{-1} \pmatrix{
\sigma_0 & 0 \cr
0 & i\sigma_j \cr
} \ .$$ This introduces an extended Hamiltonian ${\hat H}=diag(H,H^T)$ which is invariant under a global “rotation” e\^[S]{}[H]{}e\^[S]{}=[H]{} , S= \[symmetry2\] with continuous parameters $\alpha,\alpha'$, since ${\hat H}$ anticommutes with $S$. The $i\epsilon$ term of the Green’s function also breaks this symmetry. According to Eq. (\[op\]), the symmetry is broken spontaneously for $\epsilon\to0$ if the density of states $\rho_0$ is nonzero. Since this is a continuous symmetry, there is a massless mode which describes diffusion [@ziegler97]. Symmetry breaking should be studied for average quantities. Therefore, the average density of states must be evaluated.
[*spontaneous symmetry breaking:*]{} The average one-particle Green’s function can be calculated from the average Hamiltonian $\langle H\rangle_m$ by employing the self-consistent Born approximation (SCBA) [@suzuura02; @peres06; @koshino06] G(i)\_m(H\_m - 2)\^[-1]{} G\_0(i,m\_s) . \[scba1\] The self-energy $\Sigma$ is a $2\times2$ tensor due to the spinor structure of the quasiparticles: $\Sigma=-(i\eta\sigma_0+m_s \sigma_3)/2$. Scattering by the random SP produces an imaginary part of the self-energy $\eta$ (i.e. a one-particle scattering rate) and a shift $m_s$ of the average SP ${\bar m}$ (i.e., ${\bar m}\to m'\equiv {\bar m}+m_s$). $\Sigma$ is determined by the self-consistent equation =-g\_3(H\_m -2)\^[-1]{}\_[rr]{}\_3 . \[spe00\] For simplicity, the dc limit $\omega\sim0$ is considered here. The average density of states at the NP is proportional to the scattering rate: $\rho_0=\eta/2g\pi$. This reflects that scattering by the random SP creates a nonzero density of states at the NP. It should be noticed that the entire calculation of the one-particle scattering rate $\eta$ is based on the average one-particle Green’s function. Therefore, it is unrelated to the continuous symmetry of Eq. (\[symmetry2\]). On the other hand, $\eta>0$ implies spontaneous breaking of this symmetry.
Eq. (\[spe00\]) can also be written in terms of two equations, one for the one-particle scattering rate $\eta$ and another for the shift of the SP $m_s$, as = gI, m\_s=-[|m]{} gI/(1+gI) . \[scba2\] $I$ is a function of ${\bar m}$ and $\eta$ and also depends on the Hamiltonian. For MLG it reads with momentum cutoff $\lambda$ $$I= %\frac{1}{\pi}\int_0^\lambda({\eta}^2+({\bar m}+m_s)^2+k^2)^{-1}kdk =
\frac{1}{2\pi}\ln\left[ 1+\frac{\lambda^2}{{\eta}^2 +({\bar m}+m_s)^2}\right]
%=-{1\over \pi}\ln|\mu|
\label{int1}$$ and for BLG $$I\sim \frac{1}{4\sqrt{{\eta}^2+({\bar m}+m_s)^2}}\ \ \ \ (\lambda\sim\infty) \ .
\label{int2}$$ A nonzero solution $\eta$ requires $gI=1$ in the first part of Eq. (\[scba2\]), such that $m_s=-{\bar m}/2$ from the second part. Since the integrals $I$ are monotonically decreasing functions for large ${\bar m}$, a real solution with $gI=1$ exists only for $|{\bar m}|\le m_c$. For both, MLG and BLG, the solutions read \^2=(m\_c\^2-[|m]{}\^2)(m\_c\^2-[|m]{}\^2)/4 , \[scattrate\] where the model dependence enters only through the critical average SP $m_c$: \~2e\^[-g/]{} (MLG), g/2 (BLG) . \[gap11\] $m_c$ is much bigger for BGL (cf. Fig. 1), a result which indicates that the effect of disorder is much stronger in BLG. This is also reflected by the scattering rate at ${\bar m}=0$ which is $\eta=m_c/2$.
[*diffusion:* ]{} The average two-particle Green’s function $$K^{-1}_{rr'}(i\epsilon)=-\langle Tr_2[G_{rr'}(-i\epsilon)G_{r'r}(i\epsilon)]\rangle_m$$ can be evaluated from an effective field theory [@ziegler97]. If $\eta>0$ the corresponding spontaneous breaking of the symmetry in Eq. (\[symmetry2\]) creates one massless mode, which is related to a diffusion propagator in Fourier space: $$\frac{1}{K_q(i\epsilon)}\sim -\frac{\eta/g}{\epsilon+D q^2}$$ with the diffusion coefficient D= g\_r r\_k\^2 Tr\_2\[G\_[0,r0]{}(-i)G\_[0,0r]{}(i)\] . \[diffcoeff\] Within this approximation the matrix element of the position operator reads \_[/2]{} |r\_k\^2|\_[-/2]{}\_m =|\_[q=0]{} \~-8D . \[matrixelement\] Using the relation between the matrix element and the two-particle Green’s function in Eq. (\[cond2\]), the diffusion coefficient becomes $D=(g\eta/2)\langle\Phi_{i\eta} |r_k^2|\Phi_{-i\eta}\rangle$. Inserting this on the right-hand side of Eq. (\[matrixelement\]) gives a simple relation between the disorder averaged matrix element of $r_k^2$ and the corresponding matrix element without disorder: \_[/2]{} |r\_k\^2|\_[-/2]{}\_m =-\_[i]{} |r\_k\^2|\_[-i]{} . \[mespm\] This is similar to the relation of the average one-particle Green’s function in the SCBA of Eq. (\[scba1\]). Like in the latter case, the averaging process leads to a change of energies $\omega/2\to i\eta$ (i.e. a replacement of the frequency by the scattering rate). Moreover, in the relation of the two-particle Green’s function there is an extra prefactor $-\eta^2/(\omega/2)^2$. It is important for the transport properties, since the average matrix element diverges like $\omega^{-2}$. This indicates that the states $|\Phi_{\pm\omega/2}\rangle$ are delocalized for $\omega=0$ in the presence of weak SP disorder, and localization increases as one goes away from the NP. Such a behavior was also found for bond disorder in analytic [@ziegler08] and in numerical studies [@xiong07].
![ upper panel: dc conductivity in units of $e^2/h$ for BL graphene (upper curve) and ML graphene (lower curve) vs. the average staggered potential ${\bar m}$, calculated from Eq. (\[result\]) for $g=1$ and $\lambda=1$. lower panel: critical average staggered potential as a function of $g$ (variance of the staggered potential fluctuations) for BL graphene (upper curve) and ML graphene (lower curve).[]{data-label="diff"}](rgap2.eps "fig:"){width="7cm" height="6cm"} ![ upper panel: dc conductivity in units of $e^2/h$ for BL graphene (upper curve) and ML graphene (lower curve) vs. the average staggered potential ${\bar m}$, calculated from Eq. (\[result\]) for $g=1$ and $\lambda=1$. lower panel: critical average staggered potential as a function of $g$ (variance of the staggered potential fluctuations) for BL graphene (upper curve) and ML graphene (lower curve).[]{data-label="diff"}](rgap3.eps "fig:"){width="7cm" height="6cm"}
After evaluating $\langle\Phi_{i\eta} |r_k^2|\Phi_{-i\eta}\rangle$, the results for the diffusion coefficient in Eq. (\[diffcoeff\]) and for the conductivity in Eqs. (\[cond0b\]), (\[matrixelement\]) can be summerized in the following expressions D\~ , \_0\~ , \[result\] where $a=1$ ($a=2$) for MLG (BLG). First, this result indicates that the physical relevant quantity is the one-particle scattering rate $\eta$. The difference between MLG and BLG is only due to the parameter $a=1,2$ and due to the ${\bar m}$-dependent scattering rate $\eta$. Second, the result reflects a diffusive behavior as long as the scattering rate $\eta$ does not vanish. Eq. (\[scattrate\]) gives a vanishing scattering rate for ${\bar m}=m_c$, where the critical value $m_c$ is twice the scattering rate at ${\bar m}=0$. Moreover, the average density of states at the NP is proportional to $\eta$. Therefore, a global gap opens only for ${\bar m}>m_c$. Details of the transport properties distinguish between ${\bar m}=0$ and ${\bar m}\ne 0$.
${\bar m}=0$: A fluctuating SP with $g$ not too large has no effect on the conductivity. This can also be understood from the Einstein relation $\sigma_0\propto D \rho$, since the density of states $\rho$ is proportional and the diffusion coefficient $D$ is inversely proportional to the normalized scattering rate $\eta/g$. Such a behavior was also observed in the chiral-invariant case with random bond disorder which is related to ripples [@morozov06; @geim07; @peres06; @ziegler08]. The scattering rate $\eta$ increases with disorder strength $g$ (cf. Fig. 1). Consequently, the density of states at the NP $\rho_0=\eta/2g\pi$ increases with $g$ for MLG, at least for small values of $g$, whereas it is constant for BLG. On the other hand, the diffusion coefficient decreases with $g/\eta$, as a result of increased scattering.
${\bar m}\ne 0$: The conductivity decreases with ${\bar m}$ and eventually goes to zero at ${\bar m}=m_c$. This is due to two effects, namely the reduction of the density of states and the reduction of the diffusion coefficient with ${\bar m}$, caused by a fluctuating gap. Since the product of the two quantities give the conductivity in the Einstein relation, the conductivity also decreases.
The only difference between MLG and BLG in our calculation is the linear (MLG) and the quadratic (BLG) spectrum. This has quantitative consequences for the conductivity: For BLG it is twice as big as for MLG at ${\bar m}=0$ and also decays on a larger scale for $0<{\bar m}\le m_c$, since the critical value is $m_c=g/2$ for BLG, whereas it is $m_c\sim \exp(-\pi/g)$ for MLG. As shown in Fig. 1, the conductivity of MLG vanishes at much lower values of ${\bar m}$. Remarkable is the enormous difference of the scattering rate between the two systems at ${\bar m}=0$. As shown in Fig. 1, $\eta$ is practically zero for a large interval of $g$, whereas it increases linearly with $g$ for BLG. This indicates that disorder has a much stronger effect in the latter.
Our result of the random SP represents a case that is different from random bond disorder (i.e. with chiral symmetry) and random scalar potential (breaks the chiral symmetry but not the sublattice symmetry). The former does not localize states at the NP, whereas the latter has presumably always localized states, with a very large localization length though. In a recent paper, Zhang et al. suggested a Kosterlitz-Thouless (KT) transition for a long-range random potential [@zhang08]. The KT transition is a phase transition that has no spontaneous symmetry breaking but a single massless mode in the ordered phase due to $U(1)$ phase fluctuations. In the case of the random SP the situation is very different: There is spontaneous symmetry breaking in the diffusive phase due to $\eta>0$. Moreover, the symmetry of the fluctuations in Eq. (\[symmetry2\]) has two components rather than one. Therefore, the transition to the insulating behavior due to random SP cannot be linked to the conventional KT transition.
A possible experimental realization of a random gap was recently observed by Adam et al. [@adam08]. It still remains to be seen whether or not the observed transition, which was studied by varying the gate voltage at a fixed gap, can be related to a nonzero average SP. This would require a tuning of the gap fluctuations and measurement of the local density of states.
In conclusion, the one-particle scattering rate, the density of states, the diffusion coefficient, and the conductivity decrease with increasing average SP ${\bar m}$ and vanish at a critical point $m_c$. The latter is exponentially small for MLG but proportional to disorder strength for BLG. Thus the effect of disorder is much stronger in BLG.
This project was supported by a grant from the Deutsche Forschungsgemeinschaft and by the Aspen Center for Physics.
[99]{}
K.S. Novoselov et al., Nature [**438**]{}, 197 (2005)
Y. Zhang et al., Nature [**438**]{}, 201 (2005)
A.K. Geim and K.S. Novoselov, Nature Materials, [**6**]{}, 183 (2007)
Y.-W. Tan et al. Phys. Rev. Lett. [**99**]{}, 246803 (2007)
J.H. Chen et al., Nature Physics [**4**]{}, 377 (2008)
S.V. Morozov et al., Phys. Rev. Lett. [**100**]{}, 016602 (2008)
D.C. Elias et al., arXiv:cond-mat/0810.4706
O. Taisuke et al., Science [**18**]{}, Vol. 313, 951
R.V. Gorbachev et al., Physica E [**40**]{}, 1360 (2008)
J.B. Oostinga et al., Nature Materials [**7**]{}, 151 (2008)
E.V. Castro et al., J. Phys.: Conf. Ser. 129 012002 (2008)
V.V. Cheianov et al., Phys. Rev. Lett [**99**]{}, 176801 (2007)
Y.-Y. Zhang et al., arXiv:cond-mat/0810.1996
S.-J. Xiong and Y. Xiong, Phys. Rev. B [**76**]{}, 214204 (2007)
H. Suzuura and T. Ando, Phys. Rev. Lett. [**89**]{}, 266603 (2002)
N.M.R. Peres, F. Guinea, and A.H. Castro Neto, Phys. Rev. B [**73**]{}, 125411 (2006)
D. Khveshchenko, Phys. Rev. Lett. [**97**]{}, 036802 (2006)
E. McCann et al., Phys. Rev. Lett. [**97**]{}, 146805 (2006)
X.-Z. Yan and C.S. Ting, Phys. Rev. Lett. [**101**]{}, 126801 (2008)
E. McCann and V.I. Fal’ko, Phys. Rev. Lett. [**96**]{}, 086805 (2006); E. McCann, Phys. Rev. B [**74**]{}, 161403(R) (2006)
M. Koshino and T. Ando, Phys. Rev. B [**73**]{}, 245403 (2006)
S.V. Morozov et al., Phys. Rev. Lett. [**97**]{}, 016801 (2006)
A.H. Castro Neto et al., arXiv:cond-mat/0709.1163
J.C. Meyer et al., Nature [**446**]{}, 60 (2007)
P.W. Anderson, Phys. Rev. [**109**]{}, 1492 (1958)
K. Ziegler, Phys. Rev. B [**78**]{}, 125401 (2008)
K. Ziegler, Phys. Rev. B [**55**]{}, 10661 (1997); Phys. Rev. Lett. [**80**]{}, 3113 (1998)
S. Adam, S. Cho, M.S. Fuhrer, and S. Das Sarma, Phys. Rev. Lett. [**101**]{}, 046404 (2008)
|
---
abstract: 'Since the proposal of the AdS/CFT correspondence, made by Maldacena and Witten, there has been some controversy about the definition of conserved Noether charges associated to asymptotic isometries in asymptotically AdS spacetimes, namely, whether they form an anomalous (i.e., a nontrivial central extension) representation of the Lie algebra of the conformal group in odd bulk dimensions or not. In the present work, we shall review the derivation of these charges by using covariant phase space techniques, emphasizing the principle of locality underlying it. We shall also comment on how these issues manifest themselves in the quantum setting.'
author:
- Pedro Lauridsen Ribeiro
title: 'Conserved gravitational charges, locality and the holographic Weyl anomaly - a fresh viewpoint'
---
[^1]
Introduction
============
The issue of defining global conserved charges in General Relativity is quite a delicate one. The difficulties one finds stem from the fact that the dynamical problem posed by Einstein’s equations is a rather *sui generis* one: It does not possess a local dynamics in the usual sense of a “time evolution”, because the very choice of “time” is a local (gauge) symmetry of the system (Hamiltonian constraint), and thus the Noether current corresponding to it, which was supposed to generate the dynamics according to the usual Hamiltonian recipe, actually vanishes “on shell”. This stems from the deep physical fact, put forward by Mach and Einstein, that one needs a material, physical procedure to fix a local notion of time (namely, to fix a coordinate system), in a way that the physical laws ruling these procedures are also local and moreover independent of this choice – namely, one needs a nonvanishing matter energy-momentum tensor. This is the physical content of Einstein’s equations. Another, distinct way of fixing the dynamical interpretation (namely, the improvement terms to be added to the constraints of the theory) is to assume that the metric approaches some fixed, symmetrical background near infinity for which we do know how to fix such an interpretation.\
However, one must be careful with formal computations regarding the on-shell action – its value for the whole spacetime may be infinite. This divergence inserts a new ambiguity in the definition of conserved gravitational charges, this one depending only on the conformal structure of infinity. Whereas the usual definition gives a family of charges whose Poisson algebra is a true representation of the algebra of asymptotic isometries, the potential ambiguity stemming from the above divergence may lead to an obstruction to forming a representation[@sken]. The inception of Maldacena-Witten AdS-CFT correspondence, according to which the asymptotic boundary behaviour of bulk fields corresponds to sources for “dual”, boundary conformal fields, has led on its turn to another, QFT-inspired prescription, according to which nontrivial (with respect to pure AdS) boundary behaviour of the bulk metric acts as a source for the dual energy-momentum tensor, whose renormalization yields a trace anomaly in even boundary dimensions.\
Here, we start from the fact that the variational principle determining dynamics is *inherently local*. Namely, one must keep track of the supports of the functionals – conserved gravitational charges are *quasi-local* quantities, and they must be obtained as suitable limits of local quantities. In this sense, conserved gravitational currents are indeed localized at infinity, since local gravitational charges, as we’ve seen, don’t actually exist (more precisely, they vanish everywhere). The near-boundary behaviour ends up being affected by the same kind of ambiguity one faces in the renormalization of QFT[@scharf]. The result is, however, thatthe anomalous terms are actually independent of the detailed behaviour of the bulk metric (they depend only on the conformal class of the boundary metric)[@him], and must be fixed by additional, physical considerations. The AdS-CFT recipe[@sken] is one possible answer.\
We review the calculations leading to the above ambiguity in Section 2 in an unified way by using the covariant phase space formalism (Peierls bracket[@him]). In Section 3, we propose what we believe to be a QFT counterpart of the action of gravitational charges, *based on first principles* – namely, locality, covariance and causality. By employing a functorial framework in the spirit of [@bfv], we give a prescription, inspired by our previous classical calculations, which seem to capture their essential aspects.
Covariant phase space approach to conserved charges
===================================================
Let $\mathscr{M}$ be a $n$-dimensional manifold, $K\Subset\mathscr{M}$ with regular boundary $\partial K$. Define
$$\label{p1}
S_K[g,\Lambda]=\int_K(R(g)+2\Lambda)\sqrt{-g}d^nx,$$
($\text{Ric}(g)$ and $R(g)$ are resp. the Ricci tensor and the scalar curvature associated with $g$) the Einstein-Hilbert action with cosmological constant $\Lambda$. For variations $\delta g$ supported in the interior of $K$, we have
$$\label{p2}
\delta S_K[g,\Lambda]=\int_K\underbrace{\left(\text{Ric}(g)-\frac{1}{2}R(g)g+
\Lambda g\right)}_{\doteq G^\Lambda_{ab}}\delta g\sqrt{-g}d^n x=0$$
at a $g$ satisfying the Eisntein equations $\text{Ric}(g)-1/2Rg+\Lambda g=0$. Now, consider an *arbitrary* metric (i.e., not necessarily “on shell”), and perform metric variations $\delta g^{ab}=1/2\nabla^{(a}X^{b)}$ coming from (infinitesimal) spacetime diffeomorphisms $X^a$. The Bianchi identities imply that:
$$\label{p3}
G^\Lambda_{ab}\nabla^aX^b=\nabla^b(G^\Lambda_{ab}X^a).$$
If $X$ if timelike, one can see that $\mathscr{C}_a(g,X)\doteq
G^\Lambda_{ab}X^b$ does not contain second derivatives of the metric in the $X$ direction. In other words, the $\mathscr{C}_a(g,X)$’s are *constraints* of the theory, expressing its diffeomorphism invariance. Likewise, these constraints determine the dynamics of any gauge-invariant quantity – instead of showing it in general, we’ll specialize to asymptotically AdS (AAdS) spacetimes, for which the “boundary improvement” procedure mentioned in the Introduction can be implemented rather explicitly.\
Let $(\widehat{\mathscr{M}},\widehat{g})$ be a $n$-dimensional $(n\geq 4)$, AAdS sapcetime with conformal factor $\Omega$, conformal completion $(\mathscr{M},g\doteq\Omega^2\widehat{g})$ and conformal boundary $(\mathscr{I}\doteq\partial\mathscr{M},q\doteq
g\upharpoonright\!_{\mathscr{I}})$, with $q$ lying within the conformal class $[q]$ of the Einstein static universe (ESU). We’ll assume that our AAdS spacetimes satisfy empty space Einstein equations everywhere and are causally simple (i.e., the causal past and future of any compact set are closed), hence stably causal, as in [@ribeiro]. This means they can be foliated by equal-time surfaces (not necessarily Cauchy) by means of a global time function, say $\tau$. We can suppose that $\tau$ can be smoothly extended to $\mathscr{M}$ in such a way that $\tau\upharpoonright\!_{\mathscr{I}}$ is also a global time function.\
We shall consider the following setting: Let $t_1<t_2\in\mathbb{R}$, $\epsilon\in[0,\zeta)$, $\zeta$ such that $d\Omega$ vanishes nowhere in the collar $\mathscr{I}\times[0,\zeta)$, and set $\Sigma_t=
\tau^{-1}(t)$, $\Sigma^\epsilon_t=\Sigma_t\setminus\Omega^{-1}([0,
\epsilon]$. The ingoing and outgoing null hypersurfaces emanating from $\partial\Sigma^\epsilon_{t_i}$, $i=1,2$ cross resp. in $\Delta_i$, $\Delta_o$, forming the edges of the Cauchy surfaces resp. for the regions $\mathscr{O}$, $\mathscr{O}_1$. By choosing $t_1$ and $t_2$ sufficiently close to each other, we can assure that $\Delta_i$ and $\Delta_o$ are smooth (in such a case, $\mathscr{O}$ and $\mathscr{O}_1$ are regular diamonds) and belong to the same Cauchy surface for $\mathscr{O}_1$. We also set $\Omega(\Delta_o)
\subset[\epsilon',\epsilon'']$, $0<\epsilon'<\epsilon''<\epsilon$. Finally, we remark that we can deform the orbits of $\tau$ by suitably redefining the latter, in a way that each orbit of $\Sigma_t\setminus\Sigma^\epsilon_t$ under $\tau$ belong to some level surface of the collar above.\
We’ll compute all quantities we need from the Einstein-Hilbert action $S_{K_{t_1,t_2,\epsilon}}[\widehat{g},\Lambda]$, where $K_{t_1,t_2,\epsilon}
=\cup_{t\in[t_1,t_2]}\Sigma^\epsilon_t\Subset\widehat{\mathscr{M}}$. Given an arbitrary vector field $X^a$, the variation of the action under $X^a$ ($\delta g^{ab}=1/2\nabla^{(a}X^{b)}$) reduces to (we’ll omit the volume elements for simplicity)
$$\delta_X S_{K_{t_1,t_2,\epsilon}}[\widehat{g},\Lambda]=-\int^{t_2}_{t_1}\int_{\partial
\Sigma^\epsilon_t}\mbox{\large(}G^\Lambda_{ab}\omega^a X^b +$$ $$+\omega^a\theta(\widehat{g},\mathfrak{L}_X\widehat{g})_a\mbox{\large)}
+\int_{\Sigma^\epsilon_{t_2}-
\Sigma^\epsilon_{t_1}}\left(G^\Lambda_{ab}\sigma^a X^b+\sigma^a
\theta(\widehat{g},\mathfrak{L}_X\widehat{g})_a\right)\label{p4}$$
($w^a$ and $\sigma^a$ are the unit ingoing spacelike, resp. future directed timelike normals to the timelike, resp. spacelike smooth piecewise components of $\partial K_{t_1,t_2,\epsilon}$). The 1-form $\theta(\widehat{g},\delta\widehat{g})$ is Poincaré dual (with respect to the volume element $\sqrt{-\widehat{g}}$) to the boundary term coming from an arbitrary first variation $\delta\widehat{g}$. Consider now the following, arbitrary second variation, imposing the following conditions on $X^a$:
- $X^a$ vanishes in a neighborhood $N_1$ of $\Sigma_{t_1}$;
- On a neighborhood $N_2$ of $\Sigma_{t_2}$, disjoint from $N_1$, $X^a$ is an *asymptotic Killing field*, i.e., $\mathfrak{L}_X
(\widehat{g})$ decays at least as fast as $\epsilon^{n-2}$ as $\epsilon\rightarrow 0$ (this implies that the integral of the Lie derivative over $\partial\Sigma^\epsilon_t$ has a finite limit as $\epsilon\rightarrow 0$).
This can be done by multiplying an arbitrary asymptotic Killing field $X^a$ by a regularized step function. Now here comes the main point: if $\delta\widehat{g}$ is tangent to a curve of solutions of the Einstein equations, it’s propagated (in a particular gauge for the linearized equations) by convolution with the covariant derivative along the orbits of $\tau$ of some fundamental solution $E_{cdef}$ associated with the globally hyperbolic region $\mathscr{O}_1$, along the timelike component of the boundary (Duhamel’s principle). Let us drop the variation of $\theta$ for now; we shall now exploit the presence of the step function and choose the *retarded* fundamental solution. Putting it all together from (\[p4\]), we get:
$$\begin{aligned}
-\delta\int^{t_2}_{t_1}\int_{\partial
\Sigma^\epsilon_t}G^\Lambda_{ab}\omega^a X^b & = & \int^{t_2}_{t_1}\int_{\partial
\Sigma^\epsilon_t}\omega^a X^b\partial_t\delta g^{ab}.\label{p6}\end{aligned}$$
The calculation of the variation of $\theta$ is more cumbersome and we’ll skip it due to lack of space[@him]. Antisymmetrization of the second variation and repetition of the procedure above leads to the usual expression for the charges, which is no longer dependent neither on the regularized step function, nor on $t$:
$$\label{p7}
Q(X)\varpropto %=-\frac{1}{8\pi}\sqrt{\frac{(n-1)(n-2)}{2(n-3)^2\Lambda}}
\lim_{\epsilon\rightarrow 0}\int_{\partial\Sigma^\epsilon_t}
\Omega^{3-n}C_{abcd}(g)\nabla^a\Omega\nabla^b\Omega X^c\sigma^d,$$
where $C_{abcd}(g)$ is the Weyl tensor of $g$. Notice that the limit is taken by simultaneously taking $t_1 \rightarrow t_2$ and shrinking the regularized step function to the Heaviside step function, so that $\mathscr{O}_1$ remains globally hyperbolic all the way – otherwise, the retarded fundamental solution may cease to exist or no longer be unique. The requirement that the variation of the charges should act as boundary sources for linearized gravity around a solution of Einstein’s equations fix the charges themselves up to dynamically trivial terms. More precisely, it fixes their Peierls bracket with compactly supported, gauge invariant functionals of the on shell metric (local observables). The remaining ambiguity has vanishing Peierls bracket with all local observables and depends on the scaling properties of the local charges w.r.t. $\Omega$ as the limit is taken, as it arises from the ill-defined multiplication of the limit retarded fundamental solution by a Heaviside step function. Depending on the asymptotic scaling degree[@scharf], the needed extension to $\mathscr{I}$ defined by fixing the ambiguity may acquire unavoidable logarithmic terms, violating the expected scaling behaviour. This happens, for instance, for $n$ odd[@sken].
The picture from Local Quantum Physics
======================================
The conceptual advantage of employing the Peierls bracket in the classical calculations above is that it brings the principle of locality to the forefront, in a way akin to QFT. One can emulate the line of reasoning above within local quantum physics (algebraic QFT[@haag]) by means of the functorial formalism proposed in [@bfv]. Let $\underline{\mathscr{M}an}$ is the category of strongly causal, $n$-dimensional spacetimes $(\mathscr{O},g)$, with arrows defined by orientation-preserving isometric embeddings with open, causally convex images, and $\underline{\mathscr{A}lg}$ the category of unital C\*-algebras, whose arrows are unit-preserving C\*-morphisms. A *locally covariant quantum theory* is simply a *covariant functor* $\mathfrak{A}$ between both categories, i.e., the diagram
$$\label{p8}
\xymatrix{(\widehat{\mathscr{M}},\widehat{}g)\ar[d]_{\mathfrak{A}}\ar[r]^\psi &
(\widehat{\mathscr{M}'},\widehat{g'}) \ar[d]_{\mathfrak{A}}\ar[r]^\psi' &
(\widehat{\mathscr{M}''},\widehat{g''}) \ar[d]_{\mathfrak{A}} \nonumber\\
\mathfrak{A}(\widehat{\mathscr{M}},\widehat{g})\ar[r]^{\mathfrak{A}\psi} &
\mathfrak{A}(\widehat{\mathscr{M}'},\widehat{g'})\ar[r]^{\mathfrak{A}\psi'} &
\mathfrak{A}(\widehat{\mathscr{M}''},\widehat{g''})}$$
commutes. We say that $\mathfrak{A}$ is *causal* if the local algebras at spacelike separated regions commute, and *primitively causal* if the embedding of any neighborhood of a Cauchy surface into its Cauchy development induces an isomorphism of the respective algebras. In the latter case, one can define retarded and advanced “scattering morphisms” by suitable metric perturbations. The composition of both gives an automorphism $\beta{g}$ of $\mathfrak{A}(\widehat{\mathscr{M}},\widehat{g})$ (relative Cauchy evolution) whose functional derivative
$$\label{p9}
\frac{i}{2}\langle\Phi,[T^{\mu\nu}(x),\pi_\omega(A)]\Phi\rangle\doteq
\frac{\delta}{\delta g}\langle\Phi,\pi_\omega(\beta_{g}(A))\Phi\rangle$$
acts in the same way (i.e., as a densely defined derivation) as the commutator with the energy-momentum tensor, in the sense of quadratic forms, on the GNS Hilbert space induced by a state satisfying the microlocal spectrum condition, endowed with boundary conditions in the following sense: it approaches an “Rindler-Unruh” type equilibrium state after a sufficiently long relaxation time – for such states, it’s meaningful to speak about the implementation of asymptotic isometries. This derivation on the local algebras shares many properties of the Peierls bracket, and the splitting of the second variation into their retarded and advanced parts is also subject to renormalization ambiguities, in the same sense as above.
I’d like to thank prof. Michael Forger for the enlightening discussions on the principle of locality in classical field theory, which led to the crucial physical insights at the basis of the present work.
[7]{}
R. Brunetti, K. Fredenhagen, R. Verch. Commun. Math. Phys. **193**, 449-470 (1998).
R. Haag. *Local Quantum Physics.* 2nd. ed., Springer-Verlag, 1996.
S. Hollands, A. Ishibashi, D. Marolf. arXiv:hep-th/0503105, and references therein.
H. Friedrich. J. Geom. Phys. **17**, 125-184 (1995).
I. Papadimitriou, K. Skenderis. arXiv:hep-th/0505190, and references therein.
P. L. Ribeiro. arXiv:hep-th/0502096.
G. Scharf. *Finite Quantum Electrodynamics – the Causal Approach.* 2nd. ed., Springer-Verlag, 1995.
[^1]: This is a shortened version of a poster presentation made at the International Conference ”100 Years of Relativity”, held at São Paulo, Brazil, in August 22nd-24th, 2005. This project is supported by FAPESP under grant no. 01/14360-1.
|
---
abstract: 'We study the problem of wave transport in a one-dimensional disordered system, where the scatterers of the chain are $n$ barriers and wells with statistically independent intensities and with a spatial extension $\l_c$ which may contain an arbitrary number $\delta/2\pi$ of wavelengths, where $\delta = k l_c$. We analyze the average Landauer resistance and transmission coefficient of the chain as a function of $n$ and the phase parameter $\delta$. For weak scatterers, we find: i) a regime, to be called I, associated with an exponential behavior of the resistance with $n$, ii) a regime, to be called II, for $\delta$ in the vicinity of $\pi$, where the system is almost transparent and less localized, and iii) right in the middle of regime II, for $\delta$ very close to $\pi$, the formation of a band gap, which becomes ever more conspicuous as $n$ increases. In regime II, both the average Landauer resistance and the transmission coefficient show an oscillatory behavior with $n$ and $\delta$. These characteristics of the system are found analytically, some of them exactly and some others approximately. The agreement between theory and simulations is excellent, which suggests a strong motivation for the experimental study of these systems. We also present a qualitative discussion of the results.'
author:
- 'Marlos Díaz [^1]'
- 'Pier A. Mello'
- Miztli Yépez
- Steven Tomsovic
title: |
Wave Transport in One-Dimensional Disordered Systems\
with Finite-Size Scatterers
---
Introduction {#intro}
============
The problem of wave transport in disordered systems has been extensively studied in the literature, both for uncorrelated disorder (see, e.g., Ref. [@anderson58; @lifshitz; @sheng; @altshuler; @beenakker; @mott; @mello-kumar; @froufe_et_al_2007] and references therein), as well as for the case in which the disordered potential shows correlations [@lifshitz; @dunlap; @bovier92; @flores-hilke; @evangelou93; @izrailev; @moura98; @titov05].
Common features of the problems investigated by our group in Refs. [@mello-kumar; @froufe_et_al_2007] are that i) uncorrelated disordered is contemplated, and ii) the size of the individual scatterers that compose the disordered system is the smallest one occurring in the problem: in particular, it is much smaller than the wavelength of the wave sent along the waveguide, and is thus of no physical relevance. In these models, each individual potential, statistically independent from the others, is modeled by a delta function, and the distance between successive scatterers is subsequently taken to be very small, which allows considering the so-called dense weak-scattering limit (DWSL), an important ingredient in the analysis carried out in those references. Various quantities of physical interest were investigated within this framework, like the conductance, its fluctuations, and the individual transmission coefficients of the disordered system. A particularly attractive property that was found is the insensitivity of the results to details of the individual-scatterer statistical distribution, expressed in the form of a central-limit theorem.
In the present paper we build on previous work [@marlos_et_al] to study the simplest extension of the problems contemplated in Refs. [@mello-kumar; @froufe_et_al_2007]: the problem of wave transport in 1D disordered systems, in which the various scatterers have a finite size. Specifically, we consider a succession of $n$ barriers and wells, to be referred to, generically, as steps, having a finite width. The potential under study is shown schematically in Fig. \[rand\_steps\] below. It contains $n$ steps, assumed to be weak compared with the energy $E$. The steps are characterized by:
i\) A fixed width $l_c$ which may fit an arbitrary number of wavelengths $\delta/2 \pi$, where the parameter $\delta = k l_c$, $k$ being the wave number of the incident wave, will be referred to as the [*phase parameter*]{}.
ii\) Random heights $V_r$ ($r=1,\cdots n$). The $n$ heights $V_r$ are statistically independent of one another; the $n$ distributions are uniform, with zero average, and identical to one another.
The same model has been analyzed later in Ref. [@herrera-izrailev-et-al], using a mapping to a “classical phase space" and iterating that map.
Systems with similar characteristics have been studied in the past and denoted as periodic-on-average systems, and authors would speak of Kronig-Penney-like models (see, e.g., Refs. [@lifshitz; @deych_et_al_1998; @mcgurn_et_al_1993; @freilikher_et_al_1995; @erdos_1982; @gasparian_et_al_1988]). E. g., they study models where all 1D states are localized, but one group shows regular Anderson behavior, and a second group, related to gap states, has non-universal properties [@deych_et_al_1998]. Also, the localization length is found to be very small in the gaps and much larger in the bands [@mcgurn_et_al_1993]. In Ref. [@freilikher_et_al_1995], the surprising result is found that the transmission coefficient for frequencies associated with the gap in the band structure of the periodic system increases with increasing disorder, for sufficiently weak disorder.
In the problem to be studied in the present paper (along the lines of the model outlined above), we elaborate on previous investigations on disordered systems which are periodic on average, and carry on the following analysis.
i\) We also find [*two regimes*]{} with different [*localization properties*]{}, whose “evolution" we study in great detail as function of $\delta$ (for fixed $l_c$, this means as function of the incident momentum $k$) and $n$.
ii\) We study in detail the [*transition*]{} between the two regimes; interestingly, in the transition region the problem exhibits interference fringes that give an oscillatory behavior.
iii\) We can perform such a detailed study thanks to the fact that we are able to provide an [*exact theoretical solution*]{} for the average resistance of the system. We verify this exact solution by means of [*computer simulations*]{}.
iv\) In addition to the exact solution, we also provide a more [*qualitative*]{} analysis, based on:\
a) [*perturbation theory*]{}, that gives a better physical insight, and\
b) the behavior of a finite stretch of a [*periodic Kronig-Penney model*]{}.
To carry on this program, the physical quantities we study are the Landauer resistance of the chain [@landauer] and its Landauer-Büttiker conductance [@buettiker] $(e^2/h)T$ ($R$ and $T$ being the reflection and transmission coefficients of the chain) averaged over an ensemble of realizations, as functions of the number of scatterers $n$ and the phase parameter $\delta$.
The point of view adopted in the present paper is very much oriented towards condensed matter, although the results are actually much more general, as they have to do with wave propagation. We may mention that in the domain of ultracold atoms, Anderson localization has been studied and, more impressively, localized matter waves –in a Bose-Einstein condensate– have been observed (see, e. g., [@billy_et_al_2008; @roati_et_al_2008; @chabe'_et_al_2008; @lugan_et_al2009]). The potential considered is a “speckle potential", an example of a correlated disorder with correlation length $\sigma_R$. It is remarkable that a transition is observed for $k\sigma_R \sim 1$, reminiscent of the transition for $kl_c \approx \pi$ that we observe in our model: it is as if our “steps" could be considered as a potential completely correlated for distances smaller than $l_c$ and completely uncorrelated for distances larger than $l_c$.
The paper is organized as follows. In the next section we describe the theoretical model using the transfer-matrix technique. Section \[av. landauer\] studies the [*exact*]{} theoretical results for the average Landauer resistance $R/T$ of the chain, as well as the results of computer simulations. We first discuss the average Landauer resistance as a function of the number of scatterers $n$ for fixed values of $\delta$, a novel feature of these results being their oscillatory behavior. We develop a perturbation theory for values of $\delta$ not too close to $\pi$, which gives a qualitative understanding of the oscillations. We then discuss the average Landauer resistance as a function of $\delta$ for fixed $n$. The remarkable fact is that we observe the “formation of a gap" very close to $\delta=\pi$ (this region will be designated as $\delta \approx \pi$). In Sec. \[<T>\] we perform a similar study for the average transmission coefficient of the chain. In this case, the theoretical results are subject to a number of approximations and are compared with computer simulations, the agreement between both being excellent. Just as in the case of the resistance, salient features of the results are, on the one hand, their oscillatory behavior and, on the other, the formation of the gap observed for $\delta \approx \pi$. In Sec. \[discussion\] we present a more qualitative explanation of the formation of the gap, based on: i) perturbation theory, and ii) the analogy with a finite stretch of a periodic Kronig-Penney model. We finally conclude in Sec. \[concl\]. A number of appendices are added in order not to interrupt the presentation in the main text.
The theoretical model {#theory}
=====================
In this section we give a theoretical treatment of the 1D system whose potential, represented schematically in Fig. \[rand\_steps\], was described in the Introduction.
The $r$-th scatterer of the chain is shown in Fig. \[barrier\] for the case of a barrier, $V_r>0$; the definitions given below and in the figure also apply to a well, letting $V_r<0$.
In the region of the barrier, the energy $\bar{E}_{r}$ and the wave number $\bar{k}_{r}$ are given by
$$\begin{aligned}
&\hspace{0.37cm}\bar{E}_{r}=E-V_{r},
\label{k_l_j} \\
&\hspace{0.0cm}\left(\bar{k}_{r}\right)^{2}=k^{2}-U_{r},
\label{h_f_d}\end{aligned}$$
\[q\_w\_q\_s\]
where $$U_{r}=\dfrac{2mV_{r}}{\hbar ^{2}},
\hspace{1cm}
k^{2}=\dfrac{2mE}{\hbar ^{2}}.
\label{U,k2}$$ Notice that $k$ is the wavenumber in the absence of barriers. We also introduce the dimensionless parameter $$\hspace{0.8cm}y_{r}=U_{r}l_{c}^{2}=\dfrac{U_{r}}{k^{2}}\left(kl_{c}\right)^{2}\equiv \dfrac{U_{r}}{k^{2}}\delta ^{2},
\label{sogamoso}$$ as a convenient measure of the intensity of the step potential.
The transfer matrix for the $r$-th scatterer has the structure $$\textbf{\textit{M}}_{r}=
\begin{bmatrix}
\alpha_{r} & \beta_{r} \\
\beta_{r}^{*} & \alpha_{r}^{*}
\end{bmatrix},
\label{Mr}$$ with the condition $|\alpha_{r}|^2-|\beta_{r}|^2 =1$, so that it fulfills the properties of flux conservation and time-reversal invariance [@mello-kumar]. For an incident energy $E$ above a barrier ($ 0<y_{r}<\delta ^{2} $), or for arbitrary $E$ in the case of a well, we find
$$\begin{aligned}
&&\alpha_{r}=e^{-i\delta}\left[\cos\left(\sqrt{\delta^{2}-y_{r}}\right)+i\dfrac{2\delta^{2}-y_{r}}{2\delta\sqrt{\delta^{2}-y_{r}}}\sin\left(\sqrt{\delta^{2}-y_{r}}\right)\right]
\equiv \tilde{\alpha}_{r} \;,
\label{alphar} \\
\nonumber \\
&&\hspace{0cm}
\beta_{r}=-ie^{-i(2r-1)\delta}\dfrac{y_{r}}{2\delta\sqrt{\delta^{2}-y_{r}}}\sin\left(\sqrt{\delta^{2}-y_{r}}\right)
\equiv -ie^{-i(2r-1)\delta} \tilde{\beta}_{r} \; ,
\label{betar}\end{aligned}$$
\[alphar,betar\]
where the quantities $\tilde{\alpha}_{r}$ and $\tilde{\beta}_{r}$ are independent of the “running-phase" factor $\exp(-2ir\delta)$. The transfer matrix associated with a chain containing $n$ (non-overlapping) steps will be denoted by (lower indices refer to individual scatterers)
$$\begin{aligned}
\textbf{\textit{M}}^{(n)}
&=& \textbf{\textit{M}}_{n}
\cdots\textbf{\textit{M}}_{r}\cdots\textbf{\textit{M}}_{2}
\textbf{\textit{M}}_{1} \\
&=&
\begin{bmatrix}
\alpha^{(n)} & \beta^{(n)} \\
\left(\beta^{(n)}\right)^{*} & \left(\alpha^{(n)}\right)^{*}
\end{bmatrix}
=
\left[
\begin{array}{cc}
e^{i \varphi^{(n)}} & 0 \\
0 & e^{-i \varphi^{(n)}}
\end{array}
\right]
\left[
\begin{array}{cc}
\sqrt{1+ \lambda^{(n)}} & \sqrt{\lambda^{(n)}} \\
\sqrt{\lambda^{(n)}} & \sqrt{1+ \lambda^{(n)}}
\end{array}
\right]
\left[
\begin{array}{cc}
e^{i \psi^{(n)}} & 0 \\
0 & e^{-i \psi^{(n)}}
\end{array}
\right]
\; .
\nonumber \\\end{aligned}$$
\[M(n)\]
Here, $\varphi^{(n)}$ and $\psi^{(n)}$ are phases, and $\lambda^{(n)}$ is the “radial" parameter in the polar representation of the transfer matrices [@mello-kumar].
Quantities of particular physical interest are the Landauer resistance [@landauer] $\lambda^{(n)}$ of the chain
$$\begin{aligned}
\lambda^{(n)}
&=& |\beta^{(n)}|^2
= \frac{R^{(n)}}{T^{(n)}}
\label{lambda(n)} \end{aligned}$$
and its dimensionless Landauer-Büttiker conductance given by the transmission coefficient $$\begin{aligned}
T^{(n)} &=& \frac{1}{1+\lambda^{(n)}}\; .
\label{T(n)}\end{aligned}$$ \[lambda(n),T(n)\]
The ensemble of chains described in the Introduction is defined by assuming that the $y_{r}$’s ($r=1,\cdots,n$) are statistically independent of one another, each being uniformly distributed in the interval $\left(-y_{0},y_{0}\right)$. This is equivalent to saying that, for fixed $\l_c$, each $U_r$ is uniformly distributed in the interval $\left(-U_{0},U_{0}\right)$, with $y_{0}\equiv U_{0}l_{c}^{2}$. If each chain is represented as in Eq. (\[M(n)\]), the ensemble of chains is described by an [*ensemble of transfer matrices*]{}.
It is relevant here to comment on the dependence of the physical quantities of interest on the parameters that we have introduced. Notice that, although the transfer matrix for a single scatterer depends, in principle, on the three parameters $E, U_r, l_c$, Eqs. (\[alphar,betar\]) show that these parameters occur in the combinations $\delta$ and $y_r$. Thus, for the full chain of $n$ scatterers and a specific realization of disorder, a quantity like the transmission coefficient $T^{(n)}$ depends on the various parameters as $$T^{(n)}
=f(\delta, n, y_1, \cdots, y_n) \; .
\label{T(n) parameters}$$ Its ensemble average is thus given by
$$\begin{aligned}
\langle T^{(n)} \rangle
&=& \int \cdots \int f(\delta, n, y_1, \cdots, y_n)
\; p_{y_0}(y_1) \cdots p_{y_0}(y_n) \; dy_1 \cdots dy_n
\label{<T> a} \\
&=& F(\delta, n, y_0) \; ,
\label{<T> b} \end{aligned}$$
\[<T>\]
which is seen to depend on the three parameters $\delta, n$ and $y_0$ only.
Average Landauer resistance {#av. landauer}
===========================
We assume that the original system of $n$ scatterers is extended with the addition of one scatterer, to be called a [*“building block"*]{} (BB), as shown in Fig. \[rand\_steps\]. The resulting transfer matrix is given by $$\textbf{\textit{M}}^{(n+1)}
=\textbf{\textit{M}}_{n+1} \textbf{\textit{M}}^{(n)}.
\label{M(n+m)=M(m)M(n)}$$ From this combination rule we find the recursion relation for Landauer’s resistance of the chain, averaged over the ensemble, given in App. \[recursion\], Eqs. (\[rec.rel. 1,2\]). Notice that Eqs. (\[rec.rel. 1,2\]) couple the average resistance of the chain, $\langle\vert\beta^{(n)}\vert^{2}\rangle$, to the quantity $\langle\alpha^{(n)}\beta^{(n)}\rangle$. The recursion relations (\[rec.rel. 1,2\]) are exact, and thus take into account all multiple scattering processes occurring in the chain.
Eqs. (\[rec.rel. 1,2\]) can be written as a recursion relation for the quantities
$$\begin{aligned}
A(n)
&=& 1+2\langle\vert\beta^{(n)}\vert^{2}\rangle,
\label{A(n)} \\
b(n)
&=& e^{2in\delta}\langle \alpha^{(n)} \beta^{(n)}\rangle,
\label{b(n)}\end{aligned}$$
\[A(n),b(n)\]
which is given explicitly in Eq. (\[z(n+1)=Mz(n) 1\]). Using the definition $$z(n) = \left[\frac{A(n)}{2}, \frac{i b(n)}{\sqrt{2}},
-\frac{i b^{*}(n)}{\sqrt{2}}\right] ^T \; ,
\label{z}$$ ($T$ meaning transpose), we see that Eq. (\[z(n+1)=Mz(n) 1\]), in turn, has the simple structure $$z(n+1) = \Omega_{y_0}(\delta) z(n) \; .
\label{z(n+1)=Mz(n) 2}$$ We have assumed that all the individual scatterers are equally distributed, so that the various BB averages can be evaluated for the first scatterer. In Eq. (\[z(n+1)=Mz(n) 2\]), $\Omega_{y_0}(\delta)$ is the $3 \times 3$ matrix appearing on the right-hand side of Eq. (\[z(n+1)=Mz(n) 1\]). The matrix $\Omega_{y_0}(\delta)$, which depends on $y_0$ and $\delta$, will be denoted by $\Omega$, for short, when no confusion arises. The various BB averages appearing in $\Omega$ are to be evaluated using the expressions of Eqs. (\[alphar,betar\]).
The matrix $\Omega$ we have defined is [*complex symmetric*]{} and [*independent of $n$*]{}. Thanks to this last property, the solution of Eq. (\[z(n+1)=Mz(n) 2\]) for arbitrary $n$ can be written as
$$\begin{aligned}
z(n) &=& \Omega ^n z(0)
\label{sol. z(n) 1} \\
z(0)&=&[1/2,0,0]^T \;.
\label{z(0)}\end{aligned}$$
\[z(n),z(0)\]
This is done in detail in App. \[diagonalizing Omega\], through the diagonalization of the matrix $\Omega$.
For the average Landauer resistance \[see Eqs. (\[lambda(n)\]) and (\[A(n)\])\] we obtain $$\langle \lambda^{(n)} \rangle
= \frac12[A(n)-1].
\label{lambda(n) 1}$$
Average Landauer resistance in regime I, as function of the number of scatterers $n$ {#<R/T> regime I vs n}
------------------------------------------------------------------------------------
Assume $\delta$ is far from $\pi$. E.g., for $\delta=\pi/2$, the three unperturbed eigenvalues of $\Omega_0$ are $\{\mu_1^{(0)},\mu_2^{(0)},\mu_3^{(0)} \} = \{1, -1, -1\}$. We call [*regime I*]{} the region in which $\{ \mu_2^{(0)},\mu_3^{(0)} \}$ are far away from $\mu_1^{(0)}$, so that they may be considered effectively decoupled when we turn on a weak interaction, $y_0^2 \ll 1$. We then restrict ourselves to the $1\times 1$ block of $\Omega$ in Eq. (\[z(n+1)=Mz(n) 1\]) consisting of the 11 matrix element, and write the solution, Eq. (\[sol. z(n) 1\]), as $$A(n) \approx \Omega_{11}^n A(0)
=\left(1+2
\langle\vert\beta_{1}\vert^{2}\rangle\right)^{n}=e^{2n\frac{1}{2}\ln \left(1+2
\langle\vert\beta_{1}\vert^{2}\rangle\right)}\equiv e^{2nl_{c}/\ell},
\label{A(n) 2}$$ which defines the parameter $\ell$, to be interpreted below. Eq. (\[A(n) 2\]) is the well known exponential behavior found by Landauer [@landauer], where, in the present case, $$\dfrac{l_{c}}{\ell}=\frac{1}{2}\ln \left(1+2
\langle\vert\beta_{1}\vert^{2}\rangle\right),
\label{mfp 1}$$ where $\beta_1$ refers to the first scatterer. In the WSL, $\langle |\beta _{1} |^{2}\rangle =\langle R_{1}/T_{1}\rangle \ll 1$, and we can write
$$\begin{aligned}
\dfrac{l_{c}}{\ell}
&\approx&
\langle\vert\beta_{1}\vert^{2}\rangle =\langle R_{1}/T_{1}\rangle \approx\langle R_{1}\rangle \; ,
\label{lc/l} \end{aligned}$$
so that $$\begin{aligned}
\dfrac{1}{\ell} &\approx& \dfrac{\langle R_{1}\rangle}{l_{c}} \; .
\label{mfp 2}\end{aligned}$$
Thus $1/\ell$ is, approximately, the reflection coefficient per unit length, that we shall identify with the inverse of the [*mean free path*]{} (mfp) [@froufe_et_al_2007], which, in the present 1D problem, is of the order of the localization length.
Explicitly, Landauer’s resistence for the chain consisting of $n$ scatterers in regime I takes the form $$\langle\vert\beta^{(n)}\vert^{2}\rangle =\dfrac{1}{2}\left(e^{2nl_{c}/\ell} -1\right).
\label{landauer(n)}$$ Using Eq. (\[betar\]), we can express $\langle |\beta _{1} |^{2}\rangle$ appearing in (\[mfp 1\]) as function of $\delta $ and $ y_{0}$ as $$\langle |\beta _{1} |^{2}\rangle =\left\langle \dfrac{y_{1}^{2}}{4\delta ^{2}\left(\delta ^{2}-y_{1} \right)}\sin ^{2}\left(\sqrt{\delta ^{2}-y_{1}} \right) \right\rangle .
\label{<beta1^2> 1}$$ Although this average can be computed analytically and expressed in terms of cosine-integral functions, in future calculations it will be more convenient to compute it numerically. However, it is worth noticing that in the WSL it can be expanded in powers of $y_{0}/\delta ^{2}$, giving the rather compact and transparent expression $$\langle |\beta _{1} |^{2}\rangle = \dfrac{l_{c}}{\tilde{\ell}} + O\left(\dfrac{y_{0}}{\delta ^{2}}\right)^{4}, \qquad\qquad \dfrac{l_{c}}{\tilde{\ell}}=\dfrac{y^{2}_{0}}{12\delta^{4}}\sin^{2}\delta .
\label{<beta1^2> 2}$$
Notice that in the present problem [*the mfp depends on the phase parameter*]{} $\delta$.
We now compare the theoretical result of Eq. (\[landauer(n)\]) with numerical simulations. In the WSL we have $y_{0}/\delta ^{2} \ll 1$; we fix $y_0 = 0.09$ and consider $\delta$ in the interval $(1,2.9)$. Figure \[land\_res\_reg\_Ia\] shows the theoretical results and numerical simulations for the average Landauer resistance as functions of the length $n$ of the chain, for various values of $\delta$ in the above interval: the agreement is excellent, indicating that the decoupling leading to the simple equation (\[landauer(n)\]) for the resistance, as well as the expression (\[<beta1\^2> 2\]) for the mfp are very good approximations.
The results indicate the [*tendency of the system to delocalize, with a corresponding increase in the mfp, as the phase parameter $\delta$ increases towards*]{} $\pi$.
Average Landauer resistance in regime II, as function of the number of scatterers $n$ {#<R/T> regime II vs n}
-------------------------------------------------------------------------------------
In the region $2.9 \lesssim \delta \lesssim 3.4$, $\{ \mu_2^{(0)},\mu_3^{(0)} \}$ are not far enough away from $\mu_1^{(0)}$ to be effectively decoupled. We shall see that [*a novel behavior shows up as a consequence of the coupling*]{}.
### The behavior of the average resistance for $\delta = \pi$ {#delta=pi}
For $\delta = \pi$, the three $\mu_a^{(0)}$ are degenerate and equal to 1. In this case, and for weak scattering, i.e., $y_0 \ll 1$, $\Omega$ takes the approximate form given in Eqs. (\[Omega,Omega-red y0<<1\]).
Theoretical results (obtained diagonalizing $\Omega$ of (\[Omega,Omega-red y0<<1\]) numerically) and computer simulations for the average Landauer resistance for $\delta=\pi$ are also shown in Fig. \[land\_res\_reg\_Ia\] as a function of $n$. The excellent agreement between the two results indicates that writing $\Omega$ as in Eqs. (\[Omega,Omega-red y0<<1\]) is a good approximation. What we learn is that the system is [*less delocalized for $\delta = \pi$ than for neighboring values of*]{} $\delta$: i.e., [*the tendency to delocalize as $\delta$ moves towards $\pi$ is reversed for*]{} $\delta = \pi$, where we notice an enhancement of the average resistance.
### Perturbation theory for $\delta$ not too close to $\pi$ {#pert. theory}
For $\delta$ not too close to $\pi$, so that the unperturbed eigenvalues do not become degenerate, we may use perturbation theory (PT) in the parameter $y_0$ to find approximate expressions for the eigenvalues and eigenvectors of the matrix $\Omega$ appearing in the recursion relation (\[z(n+1)=Mz(n) 2\]), as is briefly discussed in App. \[pert. theo.\]. We write $\Omega_{y_0}(\delta) = \Omega_0 (\delta) + \Delta \Omega_{y_0}(\delta)$ as in Eqs. (\[M=M0+DeltaM\]), and consider $\Delta \Omega_{y_0}(\delta)$ as a perturbation; the latter contains the BB expectation values appearing in Eq. (\[z(n+1)=Mz(n) 1\]). The perturbation can be calculated analytically in leading order in $y_0^2$, as we did with $\langle |\beta _{1} |^{2}\rangle$, Eq. (\[<beta1\^2> 2\]). However, just as we mentioned right below Eq. (\[<beta1\^2> 1\]), it is convenient to have an exact expression for these BB quantities, so as to have a better control on the perturbation expansion: they were thus evaluated numerically.
Fig. \[pert\_theo\_land\_res\_d\_3.1380\_y0\_0.09\_0.01\] shows the results of perturbation theory and simulations for the average Landauer resistance as a function of $n$, for four values of $\delta$. A salient novel feature of these results is their [*oscillatory behavior as a function of*]{} $n$; in the case of scatterers with a vanishing size and for a fixed wavelength as in previous studies [@froufe_et_al_2007], oscillations with the present origin were absent. This behavior can be understood as follows. From Eq. (\[A(n) 3\]), $A(n)$ has the structure $$A(n)
\sim
A_1 e^{n \ln (1+ \Delta \mu_1)}
+\left[ A_2 e^{n \ln (e^{2i\delta}+\Delta \mu_2)} + cc \right] \; ,$$ where $A_1$, $A_2$ are constants independent of $n$ and $\Delta \mu_i = \mu_i - \mu_i^{(0)}$. For $\delta = \pi + \epsilon$ and neglecting $\Delta \mu_i$, $$e^{n\ln e^{2i\delta}} = e^{n2i\delta}
= e^{2in(\pi + \epsilon)}
=e^{2in\epsilon} \; .
\label{oscillations}$$ This result oscillates with $n$, with a period $\tau_n$ that satisfies $2 \epsilon \tau_n = 2 \pi$, so that, for $\delta fixed$, we estimate $$\tau_n \sim \frac{\pi}{\epsilon} \; .
\label{tau-n}$$ This estimate for the period $\tau_n$ is independent of $y_0$, it decreases as $\delta$ moves away from $\delta = \pi$, and is consistent with the results of Fig. \[pert\_theo\_land\_res\_d\_3.1380\_y0\_0.09\_0.01\].
### Exact solution for $\delta$ very close to $\pi$ ($\delta \approx \pi$) {#delta very close to pi}
If $\delta$ is very close to $\pi$, perturbation theory fails and $\Omega$ has to be diagonalized exactly. This has been done for a number of cases, shown in Fig. \[land\_res\_d\_pi-3.1380\_y0\_0.01\].
The analytical results are a plot of the solution for the average Landauer resistance given in Eqs. (\[lambda(n) 1\]), in which the matrix $\Omega_{y_0}(\delta)$ was diagonalized numerically. These results, which are essentially exact, have been verified with the aid of computer simulations, also shown in the figure. Notice again the oscillatory behavior of the resistance as a function of $n$: the period $\tau_n$ of the oscillations decreases as $\delta$ goes away from $\pi$, as we already noted in relation with Eq. (\[tau-n\]).
Average Landauer resistance in regimes I and II, as function of $\delta$ for fixed $n$ {#landauer reg I,II vs delta}
--------------------------------------------------------------------------------------
We gain a global picture of the two regimes if we study the behavior of the average resistance $\langle R/T \rangle$ for a fixed length $n$ of the chain, as a function of the phase parameter $\delta$.
Fig. \[land\_res\_reg\_Ia\_6\] shows the analytical results for $n=5000$ scatterers and $1< \delta <4$, covering regimes I and II. We observe in Fig. \[land\_res\_reg\_Ia\_6\]a that the average resistance decreases as $\delta$ moves towards $\pi$, in agreement with the picture we have described of the system becoming more delocalized. The theoretical curve corresponding to regime I ($1< \delta <2.9$ and $\delta >3.4$) was again obtained from Eq. (\[landauer(n)\]), the comparison with the simulation being excellent.
In Regime II, the matrix $\Omega$ was diagonalized as before. These results were verified by computer simulations, also shown in Fig. \[land\_res\_reg\_Ia\_6\]. In agreement with the earlier discussion of Fig. \[land\_res\_reg\_Ia\], we observe that well inside regime II the propensity of the average resistance to decrease as $\delta$ moves towards $\pi$ is reversed, indicating the [*formation of a gap*]{}. A discussion of the physical interpretation of this phenomenon will be given in Sec. \[discussion\].
The inset in panel (a) of Fig. \[land\_res\_reg\_Ia\_6\] exhibits an [*oscillatory behavior of the average Landauer resistance as a function of $\delta$ for fixed $n$*]{}. Again, this effect was not there in earlier studies in which the scatterers had a vanishing size. We can estimate the period from the perturbative result given in Eq. (\[oscillations\]) as $$\tau_{\delta} \sim \frac{\pi}{n} \; ,
\label{tau-delta}$$ if $\delta$ is not too close to $\pi$.
Average transmission coefficient (Landauer-Büttiker conductance) {#<T>}
================================================================
Average transmission coefficient in regime I as function of the number of scatterers $n$ {#<T> regime_I vs n}
----------------------------------------------------------------------------------------
In this section we analyze the average transmission coefficient $\langle T \rangle$ in regime I for the chains that we have been studying. Since for this quantity we have not succeeded in finding a recursion relation of the type obtained in Eq. (\[z(n+1)=Mz(n) 2\]) for the average Landauer resistance, we resort to an approximate treatment.
From Eq. (\[A(n) 2\]), valid in regime I, and treating $n$ approximately as a continuous variable, we write $$\frac{\partial A(n)}{\partial n}
\approx 2 \frac{l_c}{\ell} A(n).
\label{diff eqn A(n)}$$ In terms of the polar representation [@mello-kumar] already employed in previous sections, i.e., $\lambda_r = |\beta_r|^2$ for the $r$-th scatterer and $\lambda^{(n)}=|\beta^{(n)}|^2$ for the chain consisting of $n$ scatterers, Eq. (\[diff eqn A(n)\]) becomes $$\frac{\partial \langle \lambda \rangle_{s}}{\partial s}
= 1+2 \langle \lambda \rangle_{s} \; ,
\label{evol_eqn_for <lambda>_s}$$ where $$s = nl_c/\ell = L/\ell \; .
\label{s}$$ This “evolution" with $s$ of $\langle \lambda \rangle_{s}$ coincides with that found from the evolution equation for the $\lambda$ probability density, $w_{s}(\lambda)$, known as Melnikov’s equation [@mello-kumar; @melnikov] $$\frac{\partial w_{s}(\lambda)}{\partial s}
= \frac{\partial}{\partial \lambda} \left[\lambda(1+\lambda)
\frac{\partial w_{s}(\lambda)}{\partial \lambda}\right].
\label{melnikov}$$ We [*propose*]{} the validity of Melnikov’s equation for regime I and verify the consequences numerically. In particular, from this assumption we can find the statistical properties of the transmission coefficient $T$ which, in terms of $\lambda$, can be written as $$T=\frac{1}{1+\lambda} \; ;
\label{T(lambda)}$$ indeed, from Melnikov’s equation (\[melnikov\]), the expression for the $p$-th moment of $T$ can be reduced to quadratures, with the result [@marlos_et_al] $$\langle T^p \rangle
= \frac{2 {\rm e}^{-\tilde{s}/4}}{\Gamma(p)}
\int_0^{\infty}{\rm e}^{-\tilde{s}t^2}
\left|\Gamma\left(p-\frac12 +it\right)\right|^2 t \; \tanh (\pi t) {\rm d} t \; ,
\label{<Tp>_Melnikov}$$ from which we find the first moment as $$\langle T \rangle
= 2 {\rm e}^{-\tilde{s}/4}
\int_0^{\infty}{\rm e}^{-\tilde{s}t^2}
\pi t [\tanh (\pi t)/\cosh(\pi t)] {\rm d}t \; .
\label{<T>_Melnikov}$$ In Fig. \[<T>\_regime\_I\] we compare result (\[<T>\_Melnikov\]) with numerical simulations obtained for various values of $\delta$ in regime I as a function of the length $n$ of the chain: the agreement is excellent, indicating that the approximation involved in using Melnikov’s equation is reasonable. The localization properties are consistent with what we observed for the resistance in Fig. \[land\_res\_reg\_Ia\]: the transmission [*reduction*]{} shown in panel (b) is also consistent with the resistance [*enhancement*]{} shown in panel (b) of Fig. \[land\_res\_reg\_Ia\].
Average transmission coefficient in regime II as function of the number of scatterers $n$ {#<T> regime_II vs n}
-----------------------------------------------------------------------------------------
In regime II, the theoretical analysis uses the approximation (see Eq. (\[T(n)\])) $$\langle T \rangle
\approx 1 - \langle \lambda \rangle \; ,
\label{T(lambda) 1}$$ since $\langle \lambda \rangle \ll 1$ (see Fig. \[land\_res\_reg\_Ia\_6\]), and $\langle \lambda \rangle$ is obtained from the results of the previous section which make use of the exact recursion relation (\[z(n+1)=Mz(n) 2\]) and diagonalization of the matrix $\Omega$. The results, together with numerical simulations, are shown in Fig. \[<T>\_vs\_n for fixed deltas reg. II\] for $\delta=\pi$ and very close to $\pi$. From the excellent agreement we see that our basic approximation, Eq. (\[T(lambda) 1\]), appears justified.
Again, the oscillations shown in Fig. \[<T>\_vs\_n for fixed deltas reg. II\] are a novel feature of these results, arising from finite-size scatterers. The period $\tau_n$ of the oscillations can be taken over from the footnote to Fig. \[land\_res\_d\_pi-3.1380\_y0\_0.01\] and is consistent with what we observe in Fig. \[<T>\_vs\_n for fixed deltas reg. II\].
Average transmission coefficient in regimes I and II, as function of $\delta$ for fixed $n$ {#<T> reg. I,II vs delta}
-------------------------------------------------------------------------------------------
Just as we did in the case of the resistance in Sec. \[landauer reg I,II vs delta\], we now analyze the behavior of the average conductance $\langle T \rangle$ for a fixed length $n$ of the chain, as a function of the phase parameter $\delta$. Fig. \[<T>\_vs\_delta\] shows the analytical and numerical results for $n=5000$ scatterers and $2.5< \delta <4$, covering regimes I and II. In regime I, the analytical results are obtained from Eq. (\[<T>\_Melnikov\]), which gives an excellent description of the data. In regime II, the analytical results are obtained from Eq. (\[T(lambda) 1\]) and $\langle \lambda \rangle$ is extracted from the results of Sec. \[av. landauer\].
Fig. \[<T>\_vs\_delta\] shows that the average conductance exhibits a “gross-structure" in the form of a “bump". For the case of weak scatterers, the system is almost transparent in regime II, and regime I is more localized. This gross-structure behavior is not entirely surprising. A single barrier with fixed width and strength becomes completely transparent ($T=1$) at the resonance values ${\bar k}l_c= n\pi$, $n=1,2,\cdots$, where $\bar{k}$ is the wave number in the region of the barrier ($\delta \gtrsim \pi$ for low barriers). For a well, $T=1$ at $\delta \lesssim \pi$. For a fixed step width and random strength with zero average, and still for $n=1$, $\langle T \rangle$ reaches a maximum value smaller than unity at $\delta=\pi$. As the number of scatterers $n$ increases, the gross structure seen in $\langle T \rangle$ as a function of $\delta$ is still similar to the above description for one random scatterer, in that [*regime II*]{} ($\delta \sim \pi$) shows [*the system to be almost transparent*]{} and less localized than in regime I.
The behavior of $\langle T \rangle$ for $\delta \approx \pi$ is consistent with that of $\langle R/T \rangle$ shown in Fig. \[land\_res\_reg\_Ia\]: the transmission reduction at $\delta = \pi$ is in agreement with the resistance enhancement in Fig. \[land\_res\_reg\_Ia\]. The physical interpretation of this result will be discussed in the next section.
Discussion of the behavior for $\delta \approx \pi$ {#discussion}
===================================================
The aim of this section is to give a more qualitative and physical explanation of the [*reversal in the trend of the average resistance and transmission coefficient*]{} as $\delta$ approaches $\pi$, a phenomenon which has been described exactly by our mathematical recursion relation.
In Sec. \[pert. theory\] we found that a perturbative approximation in the small parameter $y_0$ can be written down analytically and thus gives a more qualitative description than just the exact numerical solution; indeed, we were able to describe, within this approximate method, the oscillations as a function of $n$.
We can also employ a similar perturbative approach to describe the average resistance as a function of $\delta$, and investigate whether we can find an indication of the reversal in the trend as $\delta$ moves towards $\pi$. Of course, we cannot rely on perturbation theory if $\delta$ gets too close to $\pi$.
Figure \[PT\_and\_sim\_resist\_vs\_delta\] compares computer simulations of the average Landauer resistance with the results of second-order perturbation theory, as a function of the phase parameter $\delta$ and for a fixed number of scatterers, $n=5000$. We observe that the tendency of the average resistance to decrease with increasing $\delta$ and subsequently recover as $\delta$ approaches $\pi$ is reproduced by the approximate, perturbative approach.
The behavior of the system that we have described in the above paragraphs is reminiscent of the incipient formation of a band gap that occurs in a finite stretch of an otherwise infinite, periodic Kronig-Penney model. We now exhibit the similarity of this phenomenon in the the two problems.
The finite stretch of the periodic problem can be formulated by means of a recursion relation in terms of the $2 \times 2$ transfer matrix for the unit cell, assumed to have a length $d$ (see inset in Fig. \[T for finite stretch KP\]), as indicated in Eqs. (\[M(n+m)=M(m)M(n) KP\]) (see, e.g., ref. [@merzbacher], p. 100). Alternatively, the problem can be also formulated by means of a recursion relation in terms of a $3 \times 3$ matrix, again defined for the unit cell, whose structure is similar to that appearing in Eqs. (\[z(n+1)=Mz(n) 2\]), (\[z(n),z(0)\]) and (\[z(n+1)=Mz(n) 1\]) for the disordered problem. With the definitions
$$\begin{aligned}
A(n)
&=& 1+2\vert\beta^{(n)}\vert^{2},
\label{A(n) KP} \\
b(n)
&=& e^{2inkd} (\alpha^{(n)} \beta^{(n)}),
\label{b(n) KP} \\
z(n)&=& \left[A(n)/2, \; (b(n)\sqrt{2})e^{-ikd}, \; (b^{*}(n)\sqrt{2})e^{-ikd}\right]^T \; ,\end{aligned}$$
\[A(n),b(n) KP\]
we rewrite the recursion relations Eqs. (\[M(n+m)=M(m)M(n) KP\]) as $$z(n+1) = \Omega_{y_0}^{KP}(kd) z(n) \; ,
\label{z(n+1)=Omega z(n) KP}$$ which leads to
$$\begin{aligned}
z(n) &=& (\Omega^{KP})^n z(0)
\label{sol. z(n) KP} \\
z(0)&=&[1/2,0,0]^T \;.
\label{z(0) KP}\end{aligned}$$
\[z(n),z(0) KP\]
We thus see that in the ordered problem, [*the quantity $kd$*]{} ($d$ being the size of the unit cell) which appears in the recursion relations (\[A(n),b(n) KP\]) to (\[z(n),z(0) KP\]), [*plays a role similar to $kl_c$*]{} for the disordered problem ($l_c$ being, in this case, the “minimum unit cell"), which enters the recursion relations (\[z(n+1)=Mz(n) 2\]) and (\[z(n),z(0)\]).
We give evidence for the similarity in the response of the two problems by comparing Fig. \[T for finite stretch KP\] with Fig. \[<T>\_vs\_delta\]. In Fig. \[T for finite stretch KP\] we observe the [*incipient formation of a forbidden band*]{}, which manifests itself as a dip in the transmission coefficient $T$ in the vicinity of $kd=\pi$, with interference fringes on each side. We call it “incipient", because $n$ is finite. This is similar to what we observe in Fig. \[<T>\_vs\_delta\] for the disordered case, in the vicinity of $\delta=kl_c=\pi$.
In both cases, i) the dip becomes ever more conspicuous as $n$ increases. This is shown in Fig. \[T for finite stretch KP\] for the ordered case and was verified for the disordered one. As a result, in a scattering experiment carried out in this region, the transmission coefficient in the ordered case, and the average transmission in the disordered one, suffer a reduction, with a peak to valley ratio that increases with $n$. Also, in both cases, ii) the dip becomes wider as the strength of the potential increases (this we verified by changing $y_0$), iii) we see interference fringes at the edges, as seen in the insets of Figs. \[<T>\_vs\_delta\] and \[T for finite stretch KP\]. The above behavior is consistent with the one observed for the average resistance, $\langle R/T \rangle$, described at the beginning of the present section. In the ordered case, the effect discussed above results from the coherent contribution of all the barriers and wells; indeed, it has been described as the [*collective behavior of the poles of the $S$ matrix*]{} for this problem [@gaston]. In the disordered case, we believe it to be a consequence of the barriers and wells having the same width $l_c$, and we [*conjecture*]{} a similar collective behavior.
Conclusions {#concl}
===========
To summarize, we have discussed the problem of wave transport in 1D disordered systems consisting of $n$ weak barriers and wells having a finite, constant width $l_c$, and random strength. For the calculation of the average Landauer resistance, the problem is reduced to the diagonalization of a three-dimensional complex symmetric matrix. Approximate results can be obtained analytically, by truncating the matrix when the phase parameter $\delta=kl_c$ is very far from $\pi$ (regime I). In regime II, the method is improved by using perturbation theory when $\delta$ is not too close to $\pi$. When $\delta \approx \pi$ (well inside regime II), the diagonalization was done numerically, giving essentially exact results. The average conductance was calculated approximately, making use of Melnikov’s equation in regime I and, in regime II, using the results obtained for the resistance. The theoretical results were verified in the two regimes using computer simulations.
In regime I, the average Landauer resistance was found, for a fixed $\delta$, to increase exponentially with $n$. The mfp depends on $\delta$: as $\delta$ increases towards $\pi$, both the average Landauer resistance and the average conductance show that the system becomes more delocalized.
As we enter regime II, a new feature appears, compared with older calculations: the transport properties show an oscillatory behavior as functions of $n$ and/or $\delta$, which we could explain using perturbation theory.
Well inside regime II ($\delta \approx \pi$), a second phenomenon shows up: we found an incipient band gap, or forbidden region, where i) the average conductance suffers a reduction, and ii) the average Landauer resistance increases by various orders of magnitude. In this region, a small change in $\delta$ modifies drastically the transport behavior as a function of $n$. A more qualitative and physical explanation of this behavior is presented in Sec. \[discussion\], i) in terms of an approximate, perturbative approach, and ii) as a reminiscence of the incipient formation of a band gap in a finite stretch or an otherwise infinite, Kronig-Penney problem.
The phenomena we described in the paper and the success of our theoretical analysis in their description suggest the importance of the system’s experimental realization. One possibility we may suggest is in the microwave domain (see, e. g. Refs. [@azi]). One could construct a medium consisting of plastic pieces, all of the same thickness, but with different indeces of refraction. One could then shuffle the plastic pieces and create a different random realization of the sample. The quantity to be measured is the transmission coefficient of each sample. Another possibility is in the domain of elastic waves in metallic bars. This is a problem which, in the last years, has received great attention (see, e.g., Refs. [@flores]). One could construct a bar with indentations and bulges, all of the same length, but with different, random, depths and heights. A collection of such bars would then constitute an approximation to the ensemble we need. MY is grateful to the IFUNAM for its hospitality during the development of this work. PAM acknowledges support from Conacyt, under Contract 79501, and from DGAPA, under Contract PAPIIT IN109014. The authors are greatful to G. García-Calderón for suggesting the analogy with a finite stretch of a Kronig-Penney model.
The recursion relation for the average Landauer resistance {#recursion}
==========================================================
From the combination rule given in Eq. (\[M(n+m)=M(m)M(n)\]) we find a recursion relation for Landauer’s resistance of the chain, averaged over the ensemble, as
$$\begin{aligned}
&&\hspace{-1.0cm}
\Big[1+2\langle\vert\beta^{(n+1)}\vert^{2}\rangle\Big]
-\Big[1+2\langle\vert\beta^{(n)}\vert^{2}\rangle\Big]
=2\langle\vert\beta_{n+1}\vert^{2}\rangle\Big[1+2\langle\vert\beta^{(n)}\vert^{2}\rangle\Big]
\nonumber \\
&&\hspace{6.2cm}
+2\Big[\langle\alpha_{n+1}\beta^{*}_{n+1}\rangle\langle\alpha^{(n)}\beta^{(n)}\rangle +
{\rm c.c.}\Big] \; ,
\label{rec.rel. 1} \\
\nonumber \\
&&\hspace{0.55cm}
\langle\alpha^{(n+1)}\beta^{(n+1)}\rangle-\langle\alpha^{(n)}\beta^{(n)}\rangle=
\langle\alpha_{n+1}\beta_{n+1}\rangle\Big[1+2\langle\vert\beta^{(n)}\vert^{2}\rangle\Big]
\nonumber \\
\nonumber &&\hspace{5.7cm}
+\left(\langle\alpha^{2}_{n+1}\rangle -1\right)\langle\alpha^{(n)}\beta^{(n)}\rangle +\langle\beta^{2}_{n+1}\rangle\langle\alpha^{(n)}\beta^{(n)}\rangle^{*},
\\
\label{rec.rel. 2}\end{aligned}$$
\[rec.rel. 1,2\]
where ${\rm c.c.}$ stands for “complex conjugate". Using the definitions of Eqs. (\[A(n),b(n)\]), Eqs. (\[rec.rel. 1,2\]) take the form $$\begin{aligned}
\left[
\begin{array}{c}
\frac{A(n+1)}{2} \\
\frac{ib(n+1)}{\sqrt{2}} \\
-\frac{ib^{*}(n+1)}{\sqrt{2}}
\end{array}
\right]
&=&\left[
\begin{array}{ccc}
1 + 2\langle |\tilde{\beta}_{1} |^2 \rangle
& \sqrt{2}e^{i\delta}\langle \tilde {\alpha}_{1} \tilde {\beta}_{1}\rangle
& \sqrt{2}e^{-i\delta}\langle \tilde {\alpha}_{1} \tilde {\beta}_{1}\rangle^{*} \\
\sqrt{2}e^{i\delta}\langle \tilde {\alpha}_{1} \tilde {\beta}_{1}\rangle
& e^{2i\delta}\langle \tilde {\alpha}_{1}^2 \rangle
& \langle \tilde {\beta}_{1}^2 \rangle \\
\sqrt{2}e^{-i\delta}\langle \tilde {\alpha}_{1} \tilde {\beta}_{1}\rangle^{*}
& \langle \tilde {\beta}_{1}^2 \rangle
& e^{-2i\delta}\langle \tilde {\alpha}_{1}^2 \rangle^{*}
\end{array}
\right]
\left[
\begin{array}{c}
\frac{A(n)}{2} \\
\frac{ib(n)}{\sqrt{2}} \\
-\frac{ib^{*}(n)}{\sqrt{2}}
\end{array}
\right] ,
\label{z(n+1)=Mz(n) 1} \end{aligned}$$ which can be written in the abbreviated form of Eq. (\[z(n+1)=Mz(n) 2\]). The $3 \times 3$ matrix appearing in Eq. (\[z(n+1)=Mz(n) 1\]) will be designated as $\Omega_{y_0}(\delta)$. It is often useful to write this matrix as
$$\begin{aligned}
\Omega_{y_0}(\delta) &=& \Omega_0 (\delta) + \Delta \Omega_{y_0}(\delta) \; ,
\label{M=M0+DeltaM a} \\
\Omega_0 (\delta) &=&
\left[
\begin{array}{ccc}
\mu_1^{(0)} & 0 & 0 \\
0 & \mu_2^{(0)} & 0 \\
0 & 0 & \mu_3^{(0)}
\end{array}
\right]
=\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & e^{2i\delta} & 0 \\
0 & 0 & e^{-2i\delta}
\end{array}
\right] \; ,
\label{M0}\end{aligned}$$
\[M=M0+DeltaM\]
the unperturbed matrix $\Omega_0(\delta)$ being the limiting value of $\Omega_{y_0}(\delta)$ in the absence of a potential, i.e., for $y_0=0$.
For $\delta = \pi$, the three $\mu_a^{(0)}$ are degenerate and equal to 1. In this case, and for weak scattering, i.e., $y_0 \ll 1$, $\Omega$ takes the approximate form
$$\begin{aligned}
\Omega_{y_0}(\pi) &\approx&I + y_0^2 \Omega_{red}
\label{Omega y0<<1} \\
\Omega_{red} &=&
\frac{1}{12\pi^3}
\left[
\begin{array}{ccc}
0 & -\sqrt{2} & -\sqrt{2} \\
-\sqrt{2} & -(2\pi + i) & 0 \\
-\sqrt{2} & 0 & -(2\pi - i)
\end{array}
\right] \; ,
\label{Omega-red y0<1}\end{aligned}$$
\[Omega,Omega-red y0<<1\]
where $\Omega_{red}$ is approximately (i.e., for $y_0 \ll 1$) independent of $y_0$. In the present case, $\Delta \Omega_{y_0}(\delta)$ of Eq. (\[M=M0+DeltaM a\]) is $\Delta \Omega_{y_0}(\delta) = y_0^2 \Omega_{red}$.
Diagonalization of the matrix $\Omega$, Eq. (\[z(n+1)=Mz(n) 1\]). {#diagonalizing Omega}
=================================================================
The matrix $\Omega$ is complex symmetric; provided it has no double characteristic values, it can be diagonalized by a [*complex orthogonal transformation*]{}: calling $$D
=\left[
\begin{array}{ccc}
\mu_1 & 0 & 0 \\
0 & \mu_2 & 0 \\
0 & 0 & \mu_3
\end{array}
\right]
\label{mu's}$$ the matrix of eigenvalues and $O$ the complex orthogonal matrix whose columns are the eigenvectors of $\Omega$, we have $$\Omega = ODO^T \; .
\label{e-value eqn}$$ The new vector $$z'(n) = O^T z(n)
\label{z'=OTz}$$ has the particularly simple solution
$$\begin{aligned}
z'(n)=D^n z'(0) ,
\label{sol. z'(n)}\end{aligned}$$
with components $$\begin{aligned}
z'_a(n)=(\mu_a)^n z'_a(0) .
\label{z'a(n)}\end{aligned}$$ \[z’\]
The original vector $z(n)$ can thus be expressed as $$z(n) = (O D^n O^T) z(0) \; .
\label{z'=OTz}$$ The first component of this equation gives $A(n)/2$. Using the initial condition (\[z(0)\]) and Eq. (\[z’=OTz\]), we thus find (assuming that $\Omega$ has no double characteristic values) $$A(n) = \sum_{a=1}^3 (O_{1a})^2 (\mu_a)^n \; .
\label{A(n) 1}$$ We notice that only the first component of each of the three eigenvectors enters the expression for $A(n)$.
Perturbation theory {#pert. theo.}
===================
We consider the eigenvalue equation $$\Omega {\bf v}_i = \mu_i {\bf v}_i \; ,
\hspace{1cm} i=1,2,3.
\label{e-value eqn 1}$$ The eigenvectors ${\bf v}_i$ were previously designated as the columns of the matrix $O$ of Eq. (\[e-value eqn\]). The quantity $A(n)$ of Eq. (\[A(n) 1\]) can be written in terms of the above eigenvectors as $$A(n)
= \mu_1^n ({\bf v}_1)_1^2 + \mu_2^n ({\bf v}_2)_1^2 + \mu_3^n ({\bf v}_3)_1^2 \; ,
\label{A(n) 3}$$ where $({\bf v}_i)_1$ designates component 1 of the eigenvector ${\bf v}_i$.
If we express $A(n)$ of Eq. (\[A(n) 3\]) as
$$\begin{aligned}
A(n)&=& \sum_{a=1}^3 A^{(a)}(n)
\label{A(n) 4 a} \\
A^{(a)}(n)
&=& ({\bf v}_a)_1^2 \; \mu_a^n \; ,
\label{Aa(n)}\end{aligned}$$
\[A(n) 4\]
we can write $$\log A^{(a)}(n)
= \log [({\bf v}_a)_1^2]
+ n \log \mu_a.
\label{lnAa(n)}$$ The first term in Eq. (\[lnAa(n)\]) and the coefficient of $n$ are the two parameters of a straight line representing $\log A^{(a)}(n)$ as a function of $n$. If we develop perturbation theory in the eigenvalues and eigenvectors of $\Omega$ so as to give corrections of the same order in $\Delta \Omega$ in both terms of Eq. (\[lnAa(n)\]), we shall be building a [*consistent approximation to the two parameters that define the straight line*]{} that we have just described. A perturbation theory with this criterion is briefly developed in what follows and used in the main text. The theory is taken over, almost [*verbatim*]{}, from the perturbation theory developed in any textbook on Quantum Mechanics, being careful to consider $\Omega$ not as a Hermitean matrix, but as a complex-symmetric matrix.
If we write for the eigenvalues $\mu_i$ of Eq. (\[e-value eqn 1\]) the expansion $$\mu_i = \mu_i^{(0)} + \mu_i^{(1)} + \mu_i^{(2)} + \cdots \; ,
\label{e-values expansion}$$ we find
$$\begin{aligned}
\mu_i^{(0)}
&=& \left\{
\begin{array}{cc}
1 \; , & i=1 \\
{\rm e}^{2i\delta} \; , & i=2 \\
{\rm e}^{-2i\delta} \; , & i=3
\end{array}
\right.
\label{mu(0)} \\
\mu_i^{(1)}
&=& \Delta \Omega _{ii}
\label{mu(1)} \\
\mu_i^{(2)}
&=& \sum_{j (\neq i)}
\frac{\Delta \Omega _{ij} \Delta \Omega _{ji}}{\mu_i^{(0)}- \mu_j^{(0)}}
\label{mu(2)} \\
\cdots \nonumber\end{aligned}$$
\[pert exp mu\]
Similarly, for the eigenvectors ${\bf v}_i$ of $\Omega$ we write the expansion $${\bf v}_i = {\bf v}_i^{(0)} + {\bf v}_i^{(1)} + {\bf v}_i^{(2)} + \cdots \; ,
\label{e-values expansion}$$ and find
$$\begin{aligned}
{\bf v}_i^{(0)}
&=& \left\{
\begin{array}{cc}
(1, 0, 0)^T \; , & i=1 \\
(0, 1, 0)^T \; , & i=2 \\
(0, 0, 1)^T \; , & i=3
\end{array}
\right.
\label{v(0)} \\
{\bf v}_i^{(1)}
&=& \sum_{j (\neq i)} \frac{\Delta \Omega _{ji}}{\mu_i^{(0)}- \mu_j^{(0)}}
{\bf v}_j^{(0)}
\label{v(1)} \\
{\bf v}_i^{(2)}
&=& \sum_{j,k (\neq i)}
\frac{\Delta \Omega _{jk} \Delta \Omega _{ki}}
{(\mu_i^{(0)}- \mu_j^{(0)})(\mu_i^{(0)}- \mu_k^{(0)})}
{\bf v}_j^{(0)}
\nonumber \\
&&-\sum_{j (\neq i)}
\frac{\Delta \Omega _{ii} \Delta \Omega _{ji}}
{(\mu_i^{(0)}- \mu_j^{(0)})^2}
{\bf v}_j^{(0)}
-\frac12
\left[ \sum_{j(\neq i)}
\frac{\Delta \Omega _{ij} \Delta \Omega _{ji}}
{(\mu_i^{(0)}- \mu_j^{(0)})^2} \right]
{\bf v}_i^{(0)}
\label{v(2)} \\
\cdots \nonumber\end{aligned}$$
\[pert exp v\]
Substituting these results in Eq. (\[A(n) 3\]), we can verify the identity $A(0)=1$ up to second order in $\Delta \Omega$.
Recursion relations for a finite stretch of a periodic Kronig Penney model {#recursion KP}
==========================================================================
A finite stretch of a Kronig-Penney problem obeys the recursion relation
$$\begin{aligned}
\textbf{\textit{M}}^{(n+1)}
&=&\textbf{\textit{M}}_{n+1} \textbf{\textit{M}}^{(n)} \\
&=& D^{-1}((n+1)kd) \textbf{\textit{P}}^{n+1} \; , \\
{\rm where}\;\;\;\;\; \textbf{\textit{P}} &=& D(kd) \mathaccent 23 {M}_{1} \\
{\rm and} \;\;\;\;\;
D(kd) &=&
\left[
\begin{array}{cc}
e^{ikd} & 0 \\
0 & e^{-ikd}
\end{array}
\right] \; ,
$$
\[M(n+m)=M(m)M(n) KP\]
written in terms of the $2 \times 2$ transfer matrix for the unit cell, assumed to have a length $d$ (see, e.g., ref. [@merzbacher], p. 100); here, $\mathaccent 23 {M}_{1}$ is the transfer matrix for the unit cell translated to the vicinity of the origin.
Alternatively, with the definitions (\[A(n),b(n) KP\]), we write Eqs. (\[M(n+m)=M(m)M(n) KP\]) as (\[z(n+1)=Omega z(n) KP\]), where $$\begin{aligned}
\Omega_{y_0}^{KP}
&=& \left[
\begin{array}{ccc}
1 + 2 |\mathaccent 23 {\beta}_{1} |^2
& \sqrt{2}e^{ikd} (\mathaccent 23 {\alpha}_{1} \mathaccent 23 {\beta}_{1}^{*})
& \sqrt{2}e^{-ikd} (\mathaccent 23 {\alpha}_{1}
\mathaccent 23 {\beta}_{1}^{*})^{*} \\
\sqrt{2}e^{ikd} (\mathaccent 23 {\alpha}_{1} \mathaccent 23 {\beta}_{1})
& e^{2ikd} \mathaccent 23 {\alpha}_{1}^2
& \mathaccent 23 {\beta}_{1}^2 \\
\sqrt{2}e^{-ikd} (\mathaccent 23 {\alpha}_{1} \mathaccent 23 {\beta}_{1})^{*}
& (\mathaccent 23 {\beta}_{1}^*)^2
& e^{-2i kd} (\mathaccent 23 {\alpha}_{1}^*)^{2}
\end{array}
\right] \; .
\label{Omega KP}\end{aligned}$$ This leads to Eqs. (\[z(n),z(0) KP\]).
Reduction to the results of the dense weak-scattering limit {#steps_vs_dwsl}
===========================================================
In this appendix we briefly investigate the limit in which the results of the present model –consisting of finite-size scatterers– reduce to those obtained in the dense weak-scattering limit (DWSL) of Ref. [@froufe_et_al_2007], consisting of a succession of delta scatterers.
The present model {#present model}
-----------------
A barrier lower than the energy requires (see Eq. (\[sogamoso\])) $y_r < \delta^2$, so that very weak barriers are characterized by $y_0 \ll \delta^2$. We further require the wavelength $\lambda$ to be much larger than the barrier width $l_c$, i.e., $\delta = k l_c \ll 1$. We thus have the joint requirements $$y_0 \ll \delta^2 \ll 1 \; .
\label{weak+long_wl 1}$$ Eq. (\[<beta1\^2> 2\]) for the mfp (designated here by $\ell$) can be written in the equivalent ways
$$\begin{aligned}
\frac{1}{k \ell} &=& \frac{y_0^2}{12 \delta^3} \; ,
\label{1/kl} \\
\eta &\equiv& \frac{1}{\nu \ell}
=\frac{y_0^2}{12 \delta^2} \; ,
\label{eta} \\
\delta &=& \frac{y_0}{\sqrt{12 \eta}} \; ,
\label{delta(y0,eta)} \\
\frac{1}{k \ell} &=& \frac{\eta}{\delta} \; ,
\label{1/kl = eta/delta}\end{aligned}$$
\[1/kl,eta,delta\]
$\nu = 1/l_c$ being the density of scatterers. A problem is thus specified by the three parameters $\eta, \; y_0, \; \delta$, related by one of the above equations, like (\[delta(y0,eta)\]). To satisfy the inequality (\[weak+long\_wl 1\]) we need $$12 \eta \ll y_0 \ll \sqrt{12 \eta} \; .
\label{weak+long_wl 2}$$ We follow the steps:
i\) propose $\eta \ll 1$;
ii\) propose $y_0$ to be consistent with (\[weak+long\_wl 2\]); this is used to set up the numerical barrier model.
iii\) find $\delta$ from (\[delta(y0,eta)\]).
The DWSL model {#dwsl}
--------------
The DWSL model of Ref. [@froufe_et_al_2007] consists of a succession of equally spaced (spacing = $d$) delta potentials, with an rms intensity $u_0$, having units of $k$.
The relation defining the mfp can also be written in various equivalent ways
$$\begin{aligned}
\frac{1}{k \ell}
&=& \frac{u_0^2}{12 k^3 d} \; ,
\label{1/kl-dwsl a} \\
&=& \frac{v_0^2}{3 k d} \; ,
\label{1/kl-dwsl b} \\
\frac{d}{\ell}&=&\frac{v_0^2}{3} \; .
\label{d/l-dwsl}\end{aligned}$$
Here, $d$ is the distance between successive delta potentials and $v_0 = u_0/2k$.
In this model, too, the problem is specified by three parameters: $k \ell, \; kd, \; v_0$, related by one of the above equations, like (\[d/l-dwsl\]).
Connection between the two models {#dwsl}
---------------------------------
We need to connect the two models:
i\) choose $k \ell$ to be the same in the two models
ii\) choose $kd$ of the DWSL delta-potential model to coincide with $kl_c = \delta$ of the finite-size scatterer model. This implies that the fraction of wavelength contained in the interval between the centroids of two successive scatterers is the same in the two models (compare Fig. \[rand\_steps\] of the present paper with Fig. 3 of Ref. [@froufe_et_al_2007]).
iii\) from $kd$ and $k \ell$ we find $d/ \ell$ and hence $v_0$ from Eq. (\[d/l-dwsl\]), which is to be used to set up the numerical delta-potential model.
Fig. \[steps\_vs\_dwsl\] shows computer simulations for the average Landauer resistance for the two models as a function of $L/\ell$, $L$ being the length of the chain, for the parameters indicated in the figure. The agreement is excellent. This figure is similar to Fig. 3 of Ref. [@froufe_et_al_2007]).
[99]{}
P. W. Anderson, [*Phys. Rev*]{}. [**109**]{}, 1492 (1958).
I. M. Lifshitz, S. A. Gredeskul and L. A. Pastur, [*Introduction to the Theory of Disordered Systems*]{}, J. Wiley, N.Y., 1988.
P. Sheng, (ed.)(1990). [*Scattering and localization of classical waves in random media*]{}. World Scientific, Singapore.
B. L. Altshuler, P. A. Lee and R. A. Webb, (ed.)(1991). [*Mesoscopic phenomena in solids*]{}. North-Holland, Amsterdam.
C. W. J. Beenakker and H. van Houten, (1991). [*In Solid state physics*]{} (ed. H. Ehrenreich and D. Turnbull), p. 1. Volume 44. Academic press, New York.
N. F. Mott and W. D. Twose, [*Adv. Phys.*]{}, [**10**]{}, 107 (1961)
P. A. Mello and N. Kumar, [*Quantum Transport in Mesoscopic Systems. Complexity and Statistical Fluctuations*]{} (Oxford University Press, 2010)
L. S. Froufe-Pérez, M. Yépez, P. A. Mello, and J. J. Sáenz, [*Phys. Rev. E*]{}, [**75**]{}, 031113 (2007).
D. H. Dunlap, H-L. Wu, and P. W. Phillips, [*Phys. Rev. Lett*]{}. [**65**]{}, 88 (1990); P. Phillips and H-L. Wu, [*Science*]{} [**252**]{}, 1805 (1991).
A. Bovier, [*J. Phys. A*]{} [**25**]{}, 1021 (1992).
J. C. Flores and M. Hilke, [*J. Phys. A*]{} [**26**]{}, L1255 (1993); M. Hilke, [*J. Phys. A*]{} [**27**]{}, 4773 (1994); M. Hilke and J. C. Flores, [*Phys. Rev. B*]{} [**55**]{}, 10625 (1997).
S. N. Evangelou and E. N. Economou, [*J. Phys. A*]{} [**26**]{}, 2803 (1993).
F. M. Izrailev and A. A. Krokhin, [*Phys. Rev. Lett*]{}. [**82**]{}, 4062 (1999); A. Sánchez, F. Domínguez-Adame, G. Berman and F. Izrailev, [*Phys. Rev. B*]{} [**51**]{}, 6769 (1995).
F. A. B. F. de Moura and M. L. Lyra, [*Phys. Rev. Lett*]{}. [**81**]{}, 3735 (1998).
M. Titov and H. Schomerus, [*Phys. Rev. Lett*]{}. [**95**]{}, 126602 (2005).
M. Díaz, P. A. Mello, M. Yépez and S. Tomsovic, [*Europh. Lett.*]{} [**97**]{}, 54002 (2012).
I. F. Herrera-González, F. M. Izrailev and N. M. Makarov, [*Phys. Rev. E*]{} [**88**]{}, 052108 (2013).
Lev. I. Deych. D. Zavslavsky, and A. A. Lisyansky, Phys. Rev. Lett. [**81**]{}, 5390 (1998).
A. R. Mc Gurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, Phys. Rev. B [**47**]{}, 13120 (1993).
V. D. Freilikher, B. A. Liansky, I. V. Yurkevich, A. A. Maradudin, and A. R. Mc Gurn, P. Erdös and R. C. Herndon, Adv. Phys. [**31**]{}, 65 (1982).
V. M. Gasparian, B. L. Altshuler, A. G. Aronov, and Z. A.Kasamanian, Phys. Lett. A [**132**]{}, 201 (1988).
J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature (London) [**453**]{}, 891 (2008).
G. Roati, C. D’Enrico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, and M. Inguscio, Nature (London) [**453**]{}, 895 (2008).
J. Chabe’, G. Lemarié, B. Grémaud, D. Delande, P. Szriftigiser, and J. C. Garreau, Phys. Rev. Lett. [**101**]{}, 255702 (2008).
P. Lugan, A. Aspect, L. Sanchez-Palencia, D. Delande, B. Grémaud, C. A. Müller, and C. Miniatura, Phys. Rev. A [**80**]{}, 023605 (2009).
R. Landauer, [*Philos. Mag*]{}. [**21**]{}, 863 (1970).
M. Büttiker, [*IBM Res. Dev.*]{} [**32**]{}, 317 (1988).
G. García Calderón, Phys. Rev. B [**56**]{}, 4845 (1997); Phys. Rev. A [**79**]{}, 052103 (2009); Phys. Rev. A [**90**]{}, 062101 (2014), and [*private communication*]{}.
V.I. Mel’nikov, [*Pis’ma Zh. Eksp. Teor. Fiz.,*]{} [**32**]{}, 244 (1980). \[[*JETP Lett.*]{}, [**32**]{}, 225. (1980)\]; [*Fis. Tverd. Tela (Leningrad),*]{} [**23**]{}, 782 (1981). \[[*Sov. Phys. Solid State*]{}, [**23**]{}, 444. (1981)\].
E. Merzbacher, [*Quantum Mechanics*]{}, John Wiley and Sons, New York, Second Edition, 1970.
V. Milner and A. Z. Genack, Phys. Rev. Lett. [**94**]{}, 073901 (2005); J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. [**94**]{}, 113903 (2005); P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. [**96**]{}, 183902 (2006); K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. [**101**]{}, 133901 (2008); A. A. Fernández-Marín, J. A. Méndez-Bermúdez, J. Carbonell, F. Cervera, J. Sánchez-Dehesa, and V. A. Gopar, Phys. Rev. Lett. [**113**]{}, 233901 (2014).
A. Díaz de Anda, J. Flores, L. Gutiérrez, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Jour. Acoust. Soc. Am. [**134**]{}, 4393 (2013); J. Flores, L. Gutiérrez, R. A. Méndez-Sánchez, G. Monsivais, P. Mora, and A. Morales, EPL [**101**]{}, 67002 (2013).
[^1]: Deceased
|
---
author:
- |
[^1],$^a$ Kenta Fujisawa,$^a$ Keiichiro Harada,$^a$ Takumi Nagayama,$^b$ Kousuke Suematsu,$^a$ Koichiro Sugiyama,$^a$ Asao Habe,$^c$ Mareki Honma,$^d$ Noriyuki Kawaguchi,$^d$ Hideyuki Kobayashi,$^d$ Yasuhiro Koyama,$^e$ Yasuhiro Murata,$^f$ Toshihiro Omodaka,$^b$ Kazuo Sorai,$^c$ Hiroshi Sudou,$^g$ Hiroshi Takaba,$^g$ Kazuhiro Takashima,$^h$ and Ken-ichi Wakamatsu$^g$\
Yamaguchi University, Japan\
Kagoshima University, Japan\
Hokkaido University, Japan\
VERA/National Astronomical Observatory, Japan\
National Institute of Information and Communications Technology, Japan\
Japan Aerospace Exploration Agency, Japan\
Gifu University, Japan\
Geographical Survey Institute, Japan\
title: Japanese VLBI Network
---
Introduction
============
The activity of VLBI in Japan has been rapidly growing in this decade. Japan is now one of the most highly populated area in the world in terms of VLBI antennas. Following the idea to connect all Japanese radio telescopes (including the telescopes of VERA; e.g., [@Honma]) into a VLBI-imaging array, which is expected to collaborate with the Korean VLBI Network (KVN; e.g., [@Shon]), other VLBI telescopes in the Asia-Pacific region, and the VSOP-2 (e.g., [@Hirabayashi]) as a ground network, Japanese VLBI Network (JVN; Fujisawa et al. in prep.) was established.
Japanese VLBI Network
=====================
![Japanese VLBI Network.[]{data-label="figure1"}](Fig1.eps){width="66.00000%"}
JVN produced the first fringes in 2004 and the astronomical research has been conducted since 2005. The array consists of ten antennas (Fig. \[figure1\]) that are owned and operated by four research institutes (National Astronomical Observatory of Japan (NAOJ), Japan Aerospace Exploration Agency (JAXA), National Institute of Information and Communications Technology (NICT), and Geographical Survey Institute (GSI)) and four universities (Hokkaido University, Gifu University, Yamaguchi University, and Kagoshima University). These antennas form baselines in the range 50–2560 km across the Japanese islands and provide very dense $u$–$v$ coverage. Three observing bands — 6.7, 8.4, and 22 GHz — are now available. K4/VSOP-terminal system is currently used as a digital back-end with magnetic tapes at 128 Mbps. Correlation processing is performed with the Mitaka FX correlator at NAOJ. The sub-array of five telescopes – Usuda (64 m), Kashima (34 m), Tsukuba (32 m), Yamaguchi (32 m), and Gifu (11 m) – are also connected with optical fibres at 2.4 Gbps. This sub-array has already succeeded in providing real-time fringes on its ten baselines. At the present stage, observations both with JVN and the fibre-connected sub-array are carried out based on the proposals put in by the Japanese VLBI community.
Several observational results
=============================
JVN has carried out scientific observations for more than 400 hours during this 1.5 years since its operations started on May 2005. We have observing sessions throughout the year, $\sim3$ day/month on average. The number of scientific results from JVN steadily increases at all three bands (Fig. \[figure2\]). JVN has demonstrated the capability of Bigradient Phase Referencing (BPR) method, which dramatically improves the phase-reference quality using a weak (i.e., fringe-undetectable) calibrator very close to a target [@Doi_etal.2006a]. Using BPR at 8.4 GHz, we detected radio-quiet quasars (RQQs; Suematsu et al. in prep.) and radio-loud narrow-line Seyfert 1 galaxies (NLS1s) [@Doi_etal.2006b], which had been rarely detected with the VLBI since they belong to AGN classes being very weak radio sources. The giga-hertz peaked spectrum (GPS) source J0111+3906 and other GPS sources were observed at 8.4 GHz in order to study the recurrent activity in their radio jets (Harada et al. in prep.). The spatial and velocity structures of water masers in NML Cygni (Nagayama et al. in prep.), methanol masers in Cep A (Sugiyama et al. to be submitted), and many other cosmic maser sources have recently been obtained with JVN at 22 and 6.7 GHz.
![A few scientific results obtained with JVN.[]{data-label="figure2"}](Fig2.eps){width="63.00000%"}
[99]{} Honma, M., *Parallax measurements of water maser sources beyond 5 kpc with VERA*, in proceedings of *8th European VLBI Network Symposium*, . Sohn, B. W., *Recent progress in Korean VLBI Network (KVN) project*, in proceedings of *8th European VLBI Network Symposium*, . Hirabayashi, H., *VSOP-2 mission*, in proceedings of *8th European VLBI Network Symposium*, . Doi, A. et al., *Bigradient Phase Referencing*, *PASJ*, [**58**]{}, 777 Doi, A., et al., *JVN observations of radio-loud narrow-line Seyfert 1 galaxies*, submitted to *PASJ*
[^1]: E-mail: doi@yamaguchi-u.ac.jp
|
---
abstract: |
We determine the radiative decay amplitudes for decay into $D^*$ and $\bar{D}
\gamma$, or $D^*_s$ and $\bar{D}_s \gamma$ of some of the charmonium like states classified as X,Y,Z resonances, plus some other hidden charm states which are dynamically generated from the interaction of vector mesons with charm. The mass distributions as a function of the $\bar{D} \gamma$ or $\bar{D}_s \gamma$ invariant mass show a peculiar behavior as a consequence of the $D^* \bar{D}^*$ nature of these states. The experimental search of these magnitudes can shed light on the nature of these states.
author:
- |
Wei Hong Liang$^1$, R. Molina$^2$ and E. Oset$^2$\
\
\
\
title: 'Radiative open charm decay of the Y(3940), Z(3930), X(4160) resonances.'
---
Introduction
============
The use of the chiral unitary approach, combining chiral dynamics and unitarity in coupled channels has allowed one to study the interaction of pseudoscalar mesons, and of pseudoscalar mesons with baryons, at higher energies than allowed by perturbation theory [@review]. One of the peculiar findings of the approach is that many resonances appear as poles in the scattering matrix as a consequence of the interaction, which are called dynamically generated resonances, and account for many of the low lying scalar meson and axial vector states, as well as for the low lying baryonic resonances. The success of this theory, providing properties of these states and accurate cross sections in production reactions, has stimulated the extension to the interaction of vector mesons.
A natural extension of the chiral Lagrangians to incorporate vector mesons and their interaction is provided by the hidden local gauge formalism for vector interactions with pseudoscalar mesons, vectors and photons [@hidden1; @hidden2; @hidden3; @Bernard:1988db], which provides a consistent and successful scheme to address many issues of hadron physics. Yet, as was the case with the interaction of pseudoscalar mesons or pseudoscalar mesons with baryons, also here it is the combination of the interaction provided by these Lagrangians with unitary techniques in coupled channels that allows one to obtain a realistic approach to the vector-vector interaction. In this direction, the work of [@raquel; @geng] has allowed to study the vector-vector interaction at intermediate energies, up to about 2000 MeV, where the nonperturbative unitary techniques are essential since many resonances are generated as a consequence of the interaction. In practice one solves a set of coupled channels Bethe Salpeter equations using as kernel the interaction provided by the hidden gauge Lagrangians and regularizing loops with a natural scale [@ollerulf]. The results of [@raquel] show that the $f_0(1370)$ and $f_2(1270)$ mesons are dynamically generated from the $\rho \rho$ interaction. Actually, there are strong experimental arguments to suggest that the $f_0(1370)$ is a $\rho \rho$ molecule [@klempt; @crede].
The work of [@raquel] has been extended to the interaction of all members of the vector nonet with the result that eleven resonances are dynamically generated, most of which can be associated to known resonances, while other ones remain as predictions [@geng].
Some predictions of this approach for physical processes involving these states have readily followed to further support their nature as dynamically generated. In this sense the radiative decay of the $f_0(1370)$ and $f_2(1270)$ mesons into $\gamma \gamma$ [@yamagata], were found in agreement with the experimental data. Similarly the $J/\psi$ decay into $\phi (\omega)$ and one of the $f_2(1270)$, $f'_2(1525)$, $f_0(1710)$ resonances, and into $K^*$ and the $K^*_2(1430)$ [@chinacola], was also found consistent with experiment. In the same line, the $J/\psi$ radiative decay into $\gamma$ and one of these nonstrange resonances was also found in agreement with experimental data [@chinavalgerman]. More recently, the work of [@yamagata] has been extended in [@BranzGeng] to study the $\gamma\gamma$ and $\gamma$-vector meson decays of the eleven dynamically generated resonances of [@geng], with also good agreement with experiment in the cases that there are data.
In [@raquelnaga] the extension to include charm mesons has been done studying the interaction of the $\rho$ meson with $D^*$ mesons, where three states are obtained, one which can be easily associated with the tensor state $D^*_2(2460)$, another one which is very likely to be the $D^*(2640)$ in view of its mass and the natural explanation for the small width compared with that from $D^*_2(2460)$, and a third one which corresponds to a scalar meson, for which no counterpart is yet reported in the PDG [@pdg].
More recently the work has been extended to the interaction of $D^* \bar{D}^*$ in [@xyz], where five resonances are dynamically generated, three of which could be tentatively associated to some X, Y, Z resonances reported recently, concretely the Y(3940), Z(3930), X(4160).
Independently, an alternative approach to the hidden gauge formalism, based on chiral symmetry and heavy quark symmetry, but only in one channel, has been used in [@shilinzhu], where also bound states of the $D^* \bar{D}^*$ systems are found in some cases.
Following the idea of [@shilinzhu; @gutsche] that a Y(3930) and the Y(4140) in [@xyz] could be actually $D^* \bar{D}^*$ and $D^*_s \bar{D}^*_s$ molecules, respectively, an idea was given in [@liuke] that the shape of the spectrum in the radiative decay of these resonances into $D^*\bar{D}\gamma$, or $D^*_s\bar{D}_s\gamma$, respectively, can further test the molecular assignment of these two resonances.
We follow the idea of [@liuke] with a different technical approach, and for the five states dynamically generated in [@xyz]. The work of [@xyz] provides scattering amplitudes for $D^*\bar{D}^*$ and its coupled channels. From there, evaluating the residues at the poles, one determines the coupling of the resonances to the different channels and this is all that one needs to evaluate the radiative decay in the Feynman diagrammatic approach that we follow. This allows us to determine not only shapes of the spectra but also absolute numbers for the radiative decay in terms of the $D^* D
\gamma$ coupling that can be taken from the experiment in the case of $D^{*+}\to D^+ \gamma$ (and $D^{*-}\to D^- \gamma$), and ratios of the $R\to D^*\bar{D}\gamma$ decay width to the radiative width of the $D^*(D^*_s)$ states in general. In [@liuke] a different method based on wave functions of the states is reported and no absolute values are provided. In addition we give arguments on why the $X\to D^*\bar{D}\gamma$ distribution with respect to the $\bar{D} \gamma$ is the observable which connects easier with the dynamically generated nature of the resonances (molecular nature in the wave function picture).
Formalism
=========
In [@xyz] a coupled channel formalism was considered in which one had essentially the hidden charm $D^* \bar{D}^*$, $D^*_s \bar{D}^*_s$ pairs plus all the charmless vector-vector pairs like $\rho \rho$, $\rho \omega$ or $\phi \phi$, which have the same quantum numbers of the states that are investigated and provide the decay width of the XYZ states obtained. Five heavy states were generated, additional to the light ones obtained from the light vector pairs in [@geng], three of which were identified with states observed at Belle and Babar, the Y(3940), Z(3930) and X(4160), and two other states, so far not observed, which were called $Y_p(3945)$ and $Y_p(3912)$. The quantum numbers of the states and their assumed experimental counterparts are summarized in Table \[tab:exp\].
[c|c|c|c|c|c|c]{} $I^G[J^{PC}]$&&\
& Mass \[MeV\] & Width \[MeV\]
------------------------------------------------------------------------
------------------------------------------------------------------------
& Name & Mass \[MeV\] & Width \[MeV\] &$J^{PC}$\
$0^+[0^{++}]$&$3943$&$17$&Y(3940)&$3943\pm 17$&$87\pm 34$&$J^{P+}$\
& & & & $3914.3^{+4.1}_{-3.8}$ & $33^{+12}_{-8}$ &\
$0^-[1^{+-}]$&$3945$&$0$&“$Y_p(3945)$”& & &\
$0^+[2^{++}]$&$3922$&$55$&Z(3930)&$3929\pm 5$& $29\pm 10$&$2^{++}$\
$0^+[2^{++}]$&$4157$&$102$&X(4160)&$4156\pm 29$& $139^{+113}_{-65}$& $J^{P+}$\
$1^-[2^{++}]$&$3912$&$120$&“$Y_p(3912)$”& & &\
[c|cccc]{} & Y(3940)&$Y_p(3945)$&$Z(3930)$&$X(4160)$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
channel&\
$D^* \bar{D}^*$&$18822$&$18489$&$21177$
------------------------------------------------------------------------
------------------------------------------------------------------------
&$1319$\
$D^*_s \bar{D}^*_s$&$8645$&$8763$&$6990$&$19717$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$K^* \bar{K}^*$&$15$&$40$&$44$
------------------------------------------------------------------------
------------------------------------------------------------------------
&$87$\
$\rho \rho$&$52$&$0$&$84$&$73$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\omega \omega$&$1368$&$0$&$2397$
------------------------------------------------------------------------
------------------------------------------------------------------------
&$2441$\
$\phi \phi$&$1011$&$0$&$1999$&$3130$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$J/\psi J/\psi$&$422$&$0$&$1794$&$2841$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\omega J/\psi$&$1445$&$0$&$3433$
------------------------------------------------------------------------
------------------------------------------------------------------------
&$2885$\
$\phi J/\psi$&$910$&$0$&$3062$&$5778$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\omega\phi$&$240$&$0$&$789$&$1828$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
[c|c]{} & $Y_p(3912)$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
channel&$|g_i|$ (MeV)
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$D^* \bar{D}^*$&$20869$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$K^* \bar{K}^*$&$152$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\rho\rho$&$0$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\rho\omega$&$3656$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\rho J/\psi$&$6338$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$\rho\phi$&$2731$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
In [@xyz] the states were identified by observing poles in the vector-vector scattering matrix with certain quantum numbers. The real part of the pole position provides the mass of the resonance and the imaginary part half its width. In addition the residues at the poles provide the product of the couplings of the resonance to the initial and final channels, from where, by looking at the scattering amplitudes in different channels, we can obtain the coupling of the resonance to all channels up to an irrelevant global sign, which is assigned to one particular coupling. In Tables \[tabc1\] and \[tabc2\], the couplings to the channels are also shown. As one can see from these tables, the states obtained correspond to basically bound $D^* \bar{D}^*$ or $D^*_s \bar{D}_s^*$ states, hence the decay into these pairs is forbidden, whereas the light vector-light vector channels provide the width of the states. However, if one looks at the decay channel of the $\bar{D}^*$ into $\bar{D}\gamma$, then the process $X \to D^* \bar{D} \gamma$ is allowed, since the mass of the resonance X, for all the cases listed in Table \[tab:exp\], exceeds the sum of masses of the final state. In Fig. \[fig:fig1\] the corresponding Feynman diagram to the $X\to D^{*+}D^- \gamma$ process is shown. The $D^{*-}$ propagates virtually between the production point of $X \to D^{*+} D^{*-}$ and the decay point of $D^{*-}\to D^-\gamma$. This propagator is the relevant characteristic of the $X\to D^{*+}D^- \gamma$ decay. Thus, this diagram is peculiar to the assumed nature of the resonance X as a molecule of $D^* \bar{D}^*$ and should be largely dominant over other possible processes [@liuke]. The evaluation of this Feynman diagram is easy. All one needs is the coupling of the resonance to $D^{*+} D^{*-}$, together with the corresponding spin projection operator, and the vertex accounting for the decay of $D^{*-}$ into $D^-\gamma$.
![Decay of the X resonance to $D^{*+}D^-\gamma$.[]{data-label="fig:fig1"}](DsDgam.eps "fig:"){width="6cm"}\
The spin projection operators on $J=0,1,2$, evaluated assuming the three momenta of the $D^*$ and $\bar{D}^*$ to be small with respect to the mass of the charmed vector mesons, which is indeed the case here, are given in terms of the polarization vectors by $$\begin{aligned}
{\cal P}^{(0)}&=& \frac{1}{3}{\epsilon}^{(1)}_i {\epsilon}^{(2)}_i {\epsilon}^{(3)}_j {\epsilon}^{(4)}_j\nonumber\\
{\cal P}^{(1)}&=&\frac{1}{2}({\epsilon}_i^{(1)}{\epsilon}_j^{(2)}-{\epsilon}_j^{(1)}{\epsilon}_i^{(2)})\frac{1}{2}({\epsilon}_i^{(3)}{\epsilon}_j^{(4)}-{\epsilon}_j^{(3)}{\epsilon}_i^{(4)})\nonumber\\
{\cal P}^{(2)}&=&\lbrace\frac{1}{2}({\epsilon}_i^{(1)}{\epsilon}_j^{(2)}+{\epsilon}_j^{(1)}{\epsilon}_i^{(2)})-\frac{1}{3}{\epsilon}_l^{(1)}{\epsilon}_l^{(2)}\delta_{ij}\rbrace\nonumber\\
&\times& \lbrace\frac{1}{2}({\epsilon}_i^{(3)}{\epsilon}_j^{(4)}+{\epsilon}_j^{(3)}{\epsilon}_i^{(4)})-\frac{1}{3}{\epsilon}_m^{(3)}{\epsilon}_m^{(4)}\delta_{ij}\rbrace \ .
\label{eq:projectores}\end{aligned}$$ The amplitude obtained after summing all the diagrams included implicitly in the Bethe Salpeter equation, $T=[1-VG]^{-1}\,V$, goes close to a pole, as ${\cal P}^{(k)}\,g_i\, g_j/(s-s_p)$, where $g_{i(j)}$ is the coupling of the resonance to the $i(j)$ channel and ${\cal P}^{(k)}$ are the spin projectors over spin $k=0,1,2$ of Eq. (\[eq:projectores\]), see [@raquel]. This final amplitude is depicted in the diagram of Fig. \[fig:fig2\]. In this way, if we take the case of the Y(3940), with $J=0$, the first vertex in the diagram of Fig. \[fig:fig1\] is $\frac{1}{\sqrt{3}}{\epsilon}^{(1)}_i {\epsilon}^{(2)}_i \,g_{D^*\bar{D}^*}\, \mathrm{F_I}$, where $\mathrm{F_I}$ is the isospin factor needed to change from the isospin basis, where the couplings are evaluated in [@xyz], to the charge basis. In the case of $D^{*+}D^{*-}$, we have $\mathrm{F_I}=\frac{1}{\sqrt{2}}$. In what follows, we will call $\tilde{g}$ the coupling of the resonance to the $VV$ state in isospin basis.
![Representation of the $T$ matrix obtained from the Bethe Salpeter Equation in [@raquelnaga].[]{data-label="fig:fig2"}](DsDgam2.eps "fig:"){width="6cm"}\
On the other hand the anomalous vertex for the $\bar{D}^*$ decay into $\bar{D} \gamma$ is given by $$-it_{\bar D^* \to \bar D \gamma}=-ig_{PV\gamma
}\,\epsilon_{\mu\nu\alpha\beta}\,p^{\mu}\,\epsilon^{\nu}(\bar
D^*)\,k^{\alpha}\,\epsilon^{\beta}(\gamma),
\label{eq:anom}$$ where $p$, $k$ are the momenta of the $D^{*-}$ and $\gamma$ respectively. This amplitude gives rise to a width $$\Gamma_{\bar{ D}^* \to \bar{ D} \gamma}=\frac{1}{48\pi}g^2_{PV\gamma
}\frac{k}{M^2_{\bar{D}^*}}(M^2_{\bar{D}^*}-m^2_{\bar{D}})^2.
\label{eq:radwidth}$$ Unfortunately, only the value for the radiative decay of the $D^{*-} \to D^- \gamma$ and of its positive state partner are known. In this case we will be able to provide an absolute value for the radiative decay width of the XYZ resonances. In the other cases we will give the ratio of the radiative decay of the resonance to that of the $\bar{D}^*$. The value of $g_{PV\gamma}$ for the $D^{*-} \to D^- \gamma$ decay is given by $$g_{PV\gamma}= 1.53 \times10^{-4} MeV^{-1},$$ which can be easily deduced using Eq. (\[eq:radwidth\]) from the experimental value of the width $\Gamma= 1.54 $ KeV.
Let us begin with the decay of the Y(3940). This state has isospin zero and spin zero. According to [@xyz] it couples mostly to $D^* \bar{D}^*$, has a smaller coupling to $D^*_s \bar{D}^*_s$ and very small coupling to pairs of charmless vectors, see Table \[tabc1\]. The couplings in [@xyz] are given in isospin basis. However, we need them now in charge basis, which are readily obtained for the isospin combinations $$\begin{aligned}
|D^*\bar D^*,I=0, I_3=0\rangle &=&\frac{1}{\sqrt{2}}(D^{*+}D^{*-} +
D^{*0}\bar
D^{*0}),\nonumber \\
|D^*\bar D^*,I=1, I_3=0\rangle &=&\frac{1}{\sqrt{2}}(D^{*+}D^{*-} -
D^{*0}\bar
D^{*0}), \\
|D_s\bar D_s,I=0, I_3=0\rangle &=&D_s^{*+}D_s^{*-}.\nonumber\end{aligned}$$ Thus, the couplings of [@xyz] for $D^* \bar{D}^*$ must be multiplied by $1/\sqrt 2$ to get the appropriate coupling for the charged or neutral states (a sign is irrelevant for the width), and do not require an extra factor for the case of $D^*_s \bar{D}^*_s$.
With the previous information we can already write the amplitude for the decay of the Y(3940) into $D^{*+} D^- \gamma$, which is given by $$\begin{aligned}
-it&=&-i\frac{1}{\sqrt{2}}\,\tilde{g}\,\frac{1}{\sqrt{3}}\epsilon_i^{(1)}\epsilon_i^{(2)}\,\frac{i}{p^2-M^2_{D^*}+iM_{D^*}\Gamma_{D^*}}\nonumber \\
&&\times (-i)\,g_{PV\gamma}\,\epsilon_{\mu\nu\alpha\beta}\,p^{\mu}\epsilon^{\nu
(2)}\,k^{\alpha}\,\epsilon^{\beta}(\gamma),\end{aligned}$$ where the indices (1), (2) indicate the $D^{*+}$ and the $D^{*-}$ respectively. The sum over the intermediate $D^{*-}$ polarizations can be readily done as $$\sum\limits_{\lambda}\epsilon_i^{(2)}\epsilon^{\nu
(2)}=-g_i^\nu=-\delta_{i\nu},$$ where we have neglected the three momenta of the intermediate $D^{*-}$ which is in average very small compared with the $D^{*-}$ mass, particularly at large invariant masses of the $D^{-} \gamma$ system which concentrates most of the strength, as we shall see. The sum of $|t|^2$ over the final polarizations of the vector and the photon is readily done and, neglecting again terms of order $\vec{p}\,^2/M_{D^*}^2$, we get the result $$\begin{aligned}
\sum|t|^2&=&\frac{1}{3}\frac{1}{2}\tilde{g}^2g^2_{PV\gamma}\left|\frac{1}{p^2-M^2_{D^*}+iM_{D^*}\Gamma_{D^*}}\right|^2
2(p\cdot k)^2\nonumber \\
&=&\frac{1}{6}\frac{1}{2}\tilde{g}^2g^2_{PV\gamma}\left|\frac{p^2-m^2_D}{p^2-M^2_{D^*}+iM_{D^*}\Gamma_{D^*}}\right|^2.
\label{eq:tdos}
\end{aligned}$$ The differential mass distribution with respect to the invariant mass of the $D^{-} \gamma$ system, $M_{inv}$, with $M_{inv}^2=p^2$, is finally given by $$\frac{d
\Gamma_R}{dM_{inv}}=\frac{1}{4M_R^2}\frac{1}{(2\pi)^3}\,p^*
\tilde{p}_D\sum|t|^2,
\label{eq:dist}$$ where $p^*$ is the momentum of the $D^{*+}$ in the rest frame of the resonance X and $\tilde{p}_D$ is the momentum of the $D^-$ in the rest frame of the final $D^{-} \gamma$ system given by $$\begin{aligned}
p^*&=&\frac{\lambda^{1/2}(M_R^2,M^2_{D^*},M^2_{inv})}{2M_R},\nonumber \\
\tilde{p}_D&=&\frac{M^2_{inv}-m^2_D}{2M_{inv}}.\end{aligned}$$ In the case of the tensor and spin one states we must do extra work since the projector operators are different. In this case we must keep the indices $i, j$ in $t$ and multiply with $t^*$ with the same indices $i, j$ and then perform the sum over the indices $i, j$. This sums over all possible final polarizations but also the initial X polarizations, so in order to take the sum and average over final and initial polarizations, respectively, one must divide the results of the $\sum_{i,j}\,tt^*$ by $(2J+1)$, where $J$ is the spin of the resonance X. The explicit evaluation for the case of the tensor states, $J=2$, of $D^* \bar{D}^*$ proceeds as follows: The $t$ matrix is now written as $$\begin{aligned}
t&=&\frac{1}{\sqrt{2}}\,\tilde{g}\,g_{PV\gamma} \,\left\{
\frac{1}{2}\left(\epsilon_i^{(1)}\epsilon_j^{(2)}+\epsilon_j^{(1)}\epsilon_i^{(2)}
\right)
-\frac{1}{3}\epsilon_l^{(1)}\epsilon_l^{(2)}\delta_{ij}\right\}\nonumber \\
&\times&\frac{1}{p^2-M^2_{D^*}+iM_{D^*}\Gamma_{D^*}}\,\epsilon_{\mu\nu\alpha\beta}\,p^{\mu}\epsilon^{\nu
(2)} k^\alpha\epsilon^\beta(\gamma).\end{aligned}$$ As mentioned above, we must multiply $t_{i,j}$ by $t^*_{i,j}$, recalling that the indices $i,j$ are spatial indices and divide by $(2J+1)$ (5 in this case) in order to obtain the modulus squared of the transition matrix, summed and averaged over the final and initial polarizations. Neglecting again terms that go like $\vec{p}\,^2/m_D^{*2}$ we obtain the same expression as in Eq. (\[eq:tdos\]). It is also easy to see that this is again the case for the $J=1$ states. The normalization of the spin projection operators in Eq. (\[eq:projectores\]) makes this magnitude to be the same in all cases.
Convolution of the $d\Gamma/d M_{inv}$ due to the width of the XYZ states
=========================================================================
Some of the dynamically generated XYZ states have a non negiglible width and, as a consequence, a mass distribution. That means there is a probability of these states to have a mass over the nominal mass and if one consider this fact, the $PV\gamma$ decay width should increase. In order to consider this, we convolute the $d\Gamma/d M_{inv}$ function over the mass distribution of the resonance $R$. We take $\Gamma/2$ to both sides of the peak of the resonance distribution which account for a large fraction of the strength and produces distinct shapes in the $\gamma \bar{D}$ mass distribution. We find:
$$d\Gamma^{\mathrm{conv(\Gamma/2)}}/d M_{inv}=\frac{1}{N}\,\int^{(M_R+\Gamma/2)^2}_{(M_R-\Gamma/2)^2} d\tilde{M}^2\, (-\frac{1}{\pi})\, Im\frac{1}{\tilde{M}^2-M_R^2+i \Gamma M_R}\,d\Gamma/d M_{inv}
\label{eq:conv}$$
with $$N=\int^{(M_R+\Gamma/2)^2}_{(M_R-\Gamma/2)^2}d\tilde{M}^2\, (-\frac{1}{\pi})\, Im\frac{1}{\tilde{M}^2-M_R^2+i \Gamma M_R}\ ,\nonumber$$ As we will see in the next section, the use of Eq. (\[eq:conv\]) leads to an increase of $\Gamma(R\to PV\gamma)$ with respect to the result with the nominal mass $M_R$.
Results
=======
We show here the results for different cases:
The Y(3940): Decay mode $D^{*+} D^- \gamma$
--------------------------------------------
The results are the same reversing the signs of the charges.
In Fig. \[fig:comp\] we show the distribution of Eq. (\[eq:dist\]), together with Eq. (\[eq:tdos\]), between the limits of $M_{inv}$: $m_D$ and $M_R -m_{D^*}$. Also, in order to see the effects produced when one considers the width of the state, we plot in the same figure $d\Gamma^{\mathrm{conv}}/dM_{inv}$, taken from Eq. (\[eq:conv\]).
![The $Y(3940)\to D^{*+}D^-\gamma$: Comparison of $d\Gamma/dM_{inv}$ and $d\Gamma^{\mathrm{conv(\Gamma/2)}}/dM_{inv}$, $Q$ and $Q^{\mathrm{conv(\Gamma/2)}}$ as a function of the $D^-\gamma$ invariant mass.[]{data-label="fig:comp"}](compaa.eps){width="80.00000%"}
We can see a very distinct picture, with most of the strength accumulated at the maximum values of $M_{inv}$. The propagator of the intermediate $D^{*-}$ and the factor $(p.k)^2$ are responsible for that shape. In fact we show superposed in the same figure the result obtained ( normalized to the same area) substituting the propagator by a constant and removing the factor $(p.k)^2$ (or equivalently the factor $(p^2-m_D^2)^2$). We call $Q$ the resulting distribution (or $Q^{\mathrm{conv}}$ when one convolute this function taking into account the width of the $R$ state). We can see that the pictures of $d\Gamma/dM_{inv}$ and $Q$ (or equivalently $d\Gamma^{\mathrm{conv}}/dM_{inv}$ and $Q^{\mathrm{conv}}$) are radically different and the reason is mostly due to the presence of the $D^{*-}$ propagator which carries the memory that the resonance Y(3940) is assumed to be a $D^* \bar{D}^*$ molecule. The effects of considering the convolution are also visible in this picture. Now, $d\Gamma^{\mathrm{conv}}/dM_{inv}$ spreads beyond $M_R-m_{D^*}$, and there is some probability for the state to decay into $PV\gamma$ up to $M_{inv}=M_R+\Gamma/2-m_{D^*}$, where $\Gamma$ is the width of the state. Also in this case, the difference between $d\Gamma^{\mathrm{conv}}/dM_{inv}$ and $Q^{\mathrm{conv}}$ is clearly visible.
For the case of decay into $D^{*0} \bar{D}^0 \gamma$ the matrix element is formally the same except that now we do not know the experimental radiative decay width of the $\bar{D}^{*0}$. In this case we divide the mass distribution of the $D^{*0} \bar{D}^0 \gamma$ decay by the width of the $\bar{D}^{*0}\to \bar{D}^0\gamma$ and plot the magnitude $$\begin{aligned}
\frac{1}{\Gamma_{D^* \to D\gamma}}\frac{d
\Gamma_R}{dM_{inv}}&=&\frac{1}{2}\frac{1}{6}\,\tilde{g}^2g^2_{PV\gamma}
\left|\frac{p^2-m^2_D}{p^2-M^2_{D^*}+iM_{D^*}\Gamma_{D^*}}\right|^2 \nonumber \\
&&\times\frac{48\pi\,
M^2_{D^*}}{k(M^2_{D^*}-m^2_D)^2}\,\frac{1}{4M^2_R}\,\frac{1}{(2\pi)^3}\,p^*\tilde{p}_D,
\label{eq:gammarel}\end{aligned}$$ with $$k=\frac{M^2_{D^*}-m^2_D}{2M_{D^*}}.$$
In Fig. \[fig:comp1\] we show the results of the $d\Gamma_R/dM_{inv}\Gamma_{D^* D\gamma}$ distribution and also we compare with $d\Gamma^{\mathrm{conv}}_R/dM_{inv}\Gamma_{D^*D\gamma}$. We can see that the enlarged range of the mass distribution between the limits $M_{inv}=M_R-m_{D^*}$ and $M_R+\Gamma/2-m_{D^*}$ is responsible for an increase in $\Gamma(Y(3940)\to D^{*0}\bar{D}^0\gamma)$.
![The $Y(3940)\to D^{*0}\bar{D}^0\gamma$: Comparison of $d\Gamma/dM_{inv}\Gamma_{D^* D\gamma}$ and $d\Gamma^{\mathrm{conv(\Gamma/2)}}/dM_{inv}\Gamma_{D^*D\gamma}$.[]{data-label="fig:comp1"}](Y00da.eps){width="80.00000%"}
The $Y_p(3945)$
---------------
This state has zero width, and here we show the difference between $d\Gamma_R/dM_{inv}$ and $Q$ in the case of $Y_p(3945)\to D^{*+}D^{-}\gamma$ in Fig. \[fig:Y01\] to see the effect of the inclusion of the $\bar{D}^*$ propagator in Eq. (\[eq:tdos\]). As one can see, the shapes can be clearly distinguished. Also, in Fig. \[fig:Y01d\] we show the curves for $d\Gamma_R/dM_{inv}\Gamma_{D^*D\gamma}$ for the case of the neutral charm mesons in the final state.
![The $Y_p(3945)\to D^{*+}D^-\gamma$: $d\Gamma/dM_{inv}$ and $Q$ as a function of $M_{inv}$.[]{data-label="fig:Y01"}](Y01.eps){width="80.00000%"}
![The $Y_p(3945)\to D^{*0}\bar{D}^0\gamma$: $d\Gamma/dM_{inv}\Gamma_{D^*D\gamma}$ and $Q$ as a function of $M_{inv}$.[]{data-label="fig:Y01d"}](Y01da.eps){width="80.00000%"}
The Z(3930)
-----------
This state has a larger width compared with the Y(3940) and $Y_p(3945)$ states of $55$ MeV, and for this reason the picture here is very different than in those cases when one takes into account this width. Thus, one can see a big difference between $d\Gamma_R/dM_{inv}$ and $d\Gamma^{\mathrm{conv}}_R/dM_{inv}$, $Q$ and $Q^{\mathrm{conv}}$, as shown in Fig. \[fig:Z02\]. The relatively large width of the resonance taken ($55$ MeV) is responsible for the different shapes compared to Fig. \[fig:Z02\] a). Similar results are obtained for $d\Gamma/dM_{inv}\Gamma_{D^*D\gamma}$ for decay into $D^{*0}\bar{D}^0\gamma$.
![The $Z(3930)\to D^{*+}D^-\gamma$: a) $d\Gamma/dM_{inv}$ and $Q$ as a function of $M_{inv}$. b) $d\Gamma^{\mathrm{conv}}/dM_{inv}$ and $Q^{\mathrm{conv}}$.[]{data-label="fig:Z02"}](Z02a.eps){width="100.00000%"}
The $Y_p(3912)$
---------------
This case is very similar to that of the Z(3930). The shapes of $d\Gamma_R/dM_{inv}$ and $Q$ are very different (also for $d\Gamma^{\mathrm{conv}}_R/dM_{inv}$ and $Q^{\mathrm{conv}}$) as one can see in Fig. \[fig:Y12\]. Now the width is considerably larger compared to that in the previous cases, since $\Gamma=120$ MeV. Similar results are obtained for the case of $Y_p(3912)\to D^{*0}\bar{D}^0\gamma$.
The X(4160)
-----------
In this case the isospin factor is $\mathrm{F_I}=1$ rather than $1/\sqrt{2}$. The formula is the same as before removing a factor $1/2$ in Eq. (\[eq:tdos\]). Once again we do not have the experimental decay rate for the radiative decay of $D_s^{*-}$ and we plot the results for Eq. (\[eq:gammarel\]) in Fig. \[fig:X02\]. In this case the decay into $D^{*+} D^- \gamma$ is also possible. However, the coupling to $D^* \bar{D}^*$ of this resonance (also assumed to be a $D^{*+}_s \bar{D}_s^{*-}$ molecule in [@liuke] and [@gutsche]) is found small in [@xyz], of the order of $17$ times smaller, hence the rate for this channel should be drastically smaller. In order to test the $D^{*+}D^{*-}$ component of this molecule, the allowed strong decay into $D^* \bar{D}^*$ is preferable. This latter measurement is a more efficient tool to get the strength of this coupling and compare with the theoretical predictions.
![The $Y_p(3912)\to D^{*+}D^-\gamma$: a) $d\Gamma/dM_{inv}$ and $Q$ as a function of $M_{inv}$. b) $d\Gamma^{\mathrm{conv}}/dM_{inv}$ and $Q^{\mathrm{conv}}$.[]{data-label="fig:Y12"}](Y12a.eps){width="95.00000%"}
![The $X(4160)\to D^{*+}_sD^-_s\gamma$: a) $d\Gamma_R/dM_{inv}\Gamma_{D_s^*D_s\gamma}$ and $Q$ as a function of $M_{inv}$. b) $d\Gamma^{\mathrm{conv(\Gamma/2)}}_R/dM_{inv}\Gamma_{D_s^*D_s\gamma}$ and $Q^{\mathrm{conv}}$.[]{data-label="fig:X02"}](X02a.eps){width="100.00000%"}
[cccccc]{} State&Decay&$\Gamma$\[keV\]
------------------------------------------------------------------------
------------------------------------------------------------------------
&$\Gamma/\Gamma_{D^{*-}_{(s)}\to D^-_{(s)}\gamma}$&$\Gamma^{\mathrm{conv(\Gamma/2)}}$\[keV\]&$\Gamma^{\mathrm{conv(\Gamma/2)}}/\Gamma_{D^{*-}_{(s)}\to D^-_{(s)}\gamma}$\
Y(3940)&$D^{*+}D^-\gamma$&$2.7\times 10^{-3}$&$1.8\times 10^{-3}$&
------------------------------------------------------------------------
------------------------------------------------------------------------
$2.9\times 10^{-3}$&$1.9\times 10^{-3}$\
$Y_p(3945)$&$D^{*+}D^-\gamma$&$3.1\times 10^{-3}$&$2.0\times 10^{-3}$&$-$ &$-$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
Z(3930)&$D^{*+}D^-\gamma$&$4.1\times 10^{-4}$&$2.6\times 10^{-4}$&$1.0\times 10^{-3}$&$6.7\times 10^{-4}$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$Y_p(3912)$&$D^{*+}D^-\gamma$&$1.0\times 10^{-4}$&$6.7\times 10^{-5}$&$2.7\times 10^{-3}$&$1.8\times 10^{-3}$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
X(4160)&$D_s^{*+}D^-_s\gamma$&$<39.9$&$2.3\times10^{-2}$&$<2.4\times 10^2$&$0.14$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
[cccccc]{} State&Decay&$\Gamma$\[keV\]
------------------------------------------------------------------------
------------------------------------------------------------------------
&$\Gamma/\Gamma_{\bar{D}^{*0}\to \bar{D}^0\gamma}$&$\Gamma^{\mathrm{conv(\Gamma/2)}}$\[keV\]&$\Gamma^{\mathrm{conv(\Gamma/2)}}/\Gamma_{\bar{D}^{*0}\to \bar{D}^0\gamma}$\
Y(3940)&$D^{*0}\bar{D}^0\gamma$&$<2.6$&$3.2\times 10^{-3}$&
------------------------------------------------------------------------
------------------------------------------------------------------------
$<2.7$&$3.4\times 10^{-3}$\
$Y_p(3945)$&$D^{*0}\bar{D}^0\gamma$&$<2.9$&$3.6\times 10^{-3}$&$-$&$-$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
Z(3930)&$D^{*0}\bar{D}^0\gamma$&$<0.48$&$6.0\times 10^{-4}$&$<1.0$&$1.3\times 10^{-3}$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
$Y_p(3912)$&$D^{*0}\bar{D}^0\gamma$&$<0.15$&$1.9\times 10^{-4}$&$<2.4$&$3.0\times 10^{-3}$
------------------------------------------------------------------------
------------------------------------------------------------------------
\
In Tables \[tab:decay1\] and \[tab:decay2\] we show integrated values for $\Gamma(R\to PV\gamma)$ and also rates of $\Gamma(R\to PV\gamma)$ with respect to $\Gamma(D^*_{(s)}\to D_{(s)}\gamma)$. In the case of the decays of the resonance into $D^{*0}\bar{D}^0\gamma$, $D^{*+}_s D^-_s\gamma$, $D^{*0}\bar{D}^0\pi^0$ and $D^{*+}_s D^-_s\pi^0$, we compute $g_{PV\gamma}$ in Eq. (\[eq:radwidth\]) taking $\Gamma(D^{*0})<2.1$ MeV and $\Gamma(D^{*+}_s)<1.9$ MeV. We show in Table \[tab:decay1\], the integrated values for $\Gamma(R\to D^{*+}D^-\gamma)$ which are very small, of the order of $10^{-1}-1$ eV if one does not consider the convolution of the $d\Gamma/dM_{inv}$ distribution. However, when one considers the width of the XYZ resonances given in Table \[tab:exp\], these values become bigger (about one order of magnitude in some cases).
In the case of the X(4160) we can only put a boundary for the $\Gamma(X\to D^{*+}_s D_s^-\gamma)$, which is $39.9$ KeV, but we give rates of $\Gamma(X\to D^{*+}_s D_s^-\gamma)$ respect to $\Gamma(D^{*-}_s \to D^-_s\gamma)$ in Table \[tab:decay1\]. For this observable we get a value of $2.3\times 10^{-2}$ and $0.14$ before and after convolution respectively. When the final state contains neutral charm mesons, we give both amplitudes and rates which can be seen in Table \[tab:decay2\]. In Table \[tab:decay2\], we see that $\Gamma/\Gamma_{\bar{D}^{*0}\to \bar{D}^0\gamma}$ is of the order of $10^{-4}-10^{-3}$ for all the states before the convoluting $d\Gamma_R/dM_{inv}$ and becomes larger when one convolutes this function.
Summary
=======
We have presented results for decay of the heavy dynamically generated states from the vector-vector interaction, with hidden charm, into $D^*$ and $\bar{D}
\gamma$, or $D^*_s$ and $\bar{D}_s \gamma$. We find a very distinctive shape in the $\bar{D} \gamma$ and $\bar{D}_s \gamma$ invariant mass distributions, which is peculiar to the molecular nature of these states as basically bound states of two charmed vector mesons. It was suggested in [@xyz] that some of these states correspond to some of the X,Y,Z states found at the Belle and Babar facilities. We hope the findings of the present paper stimulate experimental work in this direction to further learn about the nature of the X,Y,Z resonances.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is partly supported by DGICYT contract number FIS2006-03438. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU.
[99]{}
J. A. Oller, E. Oset and A. Ramos, Prog. Part. Nucl. Phys. [**45**]{}, 157 (2000) M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. [**54**]{}, 1215 (1985).
M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. [**164**]{}, 217 (1988). M. Harada and K. Yamawaki, Phys. Rept. [**381**]{}, 1 (2003) V. Bernard and U. G. Meissner, Nucl. Phys. A [**489**]{}, 647 (1988). R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D [**78**]{}, 114018 (2008) L. S. Geng and E. Oset, Phys. Rev. D [**79**]{}, 074009 (2009) J. A. Oller and U. G. Meissner, Phys. Lett. B [**500**]{}, 263 (2001) E. Klempt and A. Zaitsev, Phys. Rept. [**454**]{}, 1 (2007)
V. Crede and C. A. Meyer, Prog. Part. Nucl. Phys. [**63**]{}, 74 (2009)
H. Nagahiro, J. Yamagata-Sekihara, E. Oset, S. Hirenzaki and R. Molina, Phys. Rev. D [**79**]{}, 114023 (2009)
A. Martinez Torres, L. S. Geng, L. R. Dai, B. X. Sun, E. Oset and B. S. Zou, Phys. Lett. B [**680**]{}, 310 (2009) L. S. Geng, F. K. Guo, C. Hanhart, R. Molina, E. Oset and B. S. Zou, arXiv:0910.5192 \[hep-ph\]. T. Branz, L. S. Geng and E. Oset, arXiv:0911.0206 \[hep-ph\]. R. Molina, H. Nagahiro, A. Hosaka and E. Oset, Phys. Rev. D [**80**]{}, 014025 (2009)
C. Amsler [*et al.*]{} \[Particle Data Group\], Phys. Lett. B [**667**]{}, 1 (2008).
R. Molina and E. Oset, Phys. Rev. D [**80**]{}, 114013 (2009)
X. Liu, Z. G. Luo, Y. R. Liu and S. L. Zhu, Eur. Phys. J. C [**61**]{}, 411 (2009) T. Branz, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D [**80**]{}, 054019 (2009)
X. Liu and H. W. Ke, Phys. Rev. D [**80**]{}, 034009 (2009)
|
---
abstract: 'The Kerr-Schild version of the Schwarzschild metric contains a Minkowski background which provides a definition of a boosted black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principle null directions. We show that the two corresponding Minkowski backgrounds and their associated boosts have an unexpected difference. We analyze this difference and discuss the implications in the nonlinear regime for the gravitational memory effect resulting from the ejection of massive particles from an isolated system. We show that the nonlinear effect agrees with the linearized result based upon the retarded Green function only if the velocity of the ejected particle corresponds to a boost symmetry of the ingoing Minkowski background. A boost with respect to the outgoing Minkowski background is inconsistent with the absence of ingoing radiation from past null infinity.'
address: |
${}^1$ Núcleo de Astronomía, Facultad de Ingeniería, Universidad Diego Portales, Av. Ejército 441, Santiago, Chile\
${}^{2}$ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\
${}^{3}$ Department of Physics and Astronomy\
University of Pittsburgh, Pittsburgh, PA 15260, USA\
${}^{4}$ Max-Planck-Institut f" ur Gravitationsphysik, Albert-Einstein-Institut,\
14476 Golm, Germany\
author:
- 'Thomas Mädler$^{1,2}$[^1] and Jeffrey Winicour$^{3,4}$'
---
Introduction
============
By considering retarded solutions of the linearized Einstein equation on a Minkowski background, Zeldovich and Polnarev [@zeld] pointed out the existence of a memory effect in the gravitational waves produced by the ejection of massive particles to infinity. Our previous work [@boost] has shown that this effect could also be obtained in linearized theory by considering the transition from an initial state whose exterior was described by a Schwarzschild metric at rest to a final state whose exterior was a boosted Schwarzschild metric. The results were based upon a Kerr-Schild version of the Schwarzschild metric to describe the far field exterior to what we referred to as a Schwarzschild body. For such a body in linearized theory which is initially at rest, then goes through a radiative stage and finally emerges in a boosted state, we showed that the proper retarded solution for the resulting memory effect is described in terms of the ingoing version of the Kerr-Schild metric for both the initial and final states. An outgoing Kerr-Schild or time symmetric Schwarzschild metric would give the wrong result. The result was independent of the details of the intervening radiative period. Because the Kerr-Schild metrics are solutions both in the linearized and nonlinear sense, we extrapolated this result to the nonlinear case.
Here, we investigate this problem from the purely nonlinear perspective. There are two major differences from the linearized view.
The first major difference is that the linearized result in [@boost] was obtained using the boost associated with the Lorentz symmetry of the unperturbed Minkowski background. The Kerr-Schild metrics [@ks1; @ks2] have the form $$g_{\mu\nu} = \eta_{\mu\nu} +H \ell_\mu \ell_\nu
\label{eq:ksk}$$ where $\eta_{\mu\nu}$ is a Minkowski metric, $\ell_\mu$ is a principle null vector field (with respect to both $\eta_{\mu\nu}$ and $g_{\mu\nu}$) and $H$ is a scalar function. In the nonlinear case, there are two natural choices of “Minkowski background” $\eta_{\mu\nu}$ depending on whether the null vector $\ell^\mu$ in the Kerr-Schild metric (\[eq:ksk\]) is chosen to be in the ingoing or outgoing direction. [^2].
The second major difference in the nonlinear case is that there is no analogue of the Green function to construct a retarded solution. Instead, the retarded solution due to the emission of radiation from an accelerated particle is characterized by the absence of ingoing radiation from ${\mathcal I}^-$. A necessary condition that there be no ingoing radiation is that the analogue of the radiation memory at past null infinity ${\mathcal I}^-$ vanishes. In that case the ingoing radiation strain, which forms the free characteristic initial data on ${\mathcal I}^-$, may be set to zero. Otherwise, as explained in Sec. \[sec:mem\], non-vanishing radiation memory at ${\mathcal I}^-$ requires that there must ingoing radiation from ${\mathcal I}^-$.
Consequently, since an initial unboosted Kerr-Schild-Schwarzschild metric has vanishing radiation strain at ${\mathcal I}^-$, the final boosted metric must also have vanishing radiation strain at ${\mathcal I}^-$ if there is no intervening ingoing radiation. This is the case if the boost belongs to the Lorentz subgroup of the BMS group at ${\mathcal I}^-$. This corresponds to the boost symmetry of the Minkowski metric associated with the ingoing version of the Kerr-Schild metric. On the contrary, a boost with respect to the Minkowski metric associated with the outgoing version of the Kerr-Schild metric leads to non-vanishing radiation memory at ${\mathcal I}^-$ so that it is inconsistent with the requirement of vanishing ingoing radiation.
This leads to our main result: the calculation in the nonlinear regime of the memory effect due to the ejection of a massive particle is correctly described by the boost associated with Minkowski background of the ingoing Kerr-Schild metric. The key ingredient is that this boost is a BMS symmetry at ${\mathcal I}^-$ but not at ${\mathcal I}^+$. This leads to vanishing radiation memory at ${\mathcal I}^-$ but to non-zero radiation memory at ${\mathcal I}^+$, which is in precise agreement with the extrapolation from the linearized result based upon the retarded Green function.
In Sec. \[sec:advret\], we discuss unexpected features which result in defining the boost in terms of the Lorentz symmetries of the Minkowski backgrounds of either the ingoing or outgoing versions of the Kerr-Schild metric. This requires considerable notational care, which warrants the formalism presented in Sec. \[sec:advret\]. We show that the boost symmetry of the Minkowski metric associated with the ingoing version of the Kerr-Schild metric corresponds to the boost symmetry of the Bondi-Metzner-Sachs (BMS) [@Sachs_BMS] asymptotic symmetry group at ${\mathcal I}^-$ but is a singular transformation at future null infinity ${\mathcal I}^+$. Conversely, the boost symmetry of the Minkowski metric associated with the outgoing version of the Kerr-Schild metric corresponds to the boost symmetry of the BMS group at ${\mathcal I}^+$ but is a singular transformation at ${\mathcal I}^-$.
The Kerr-Schild metrics have played an important role in the construction of exact solutions; see [@exact]. The most important examples are the Schwarzschild and Kerr black hole metrics. Because their metric form (\[eq:ksk\]) is invariant under the Lorentz symmetry of the Minkowski background $\eta_{\mu\nu}$, the boosted Kerr-Schild versions of the Schwarzschild and Kerr metrics have also played an important role in numerical relativity in prescribing initial data for superimposed black holes in a binary orbit [@ksm1; @ksm2]. The initial data for the numerical simulations are prescribed in terms of the ingoing version of the Kerr-Schild metric, whose coordinatization in terms of advanced time covers the interior of the future event horizon. The initial velocities of the black holes are generated by the boost symmetry of the Minkowski background for the ingoing version of the Kerr-Schild metric. Surprisingly, this boost symmetry, which is a well-behaved BMS transformation at ${\mathcal I}^-$, has singular behavior at ${\mathcal I}^+$. This overlooked asymptotic property of the boost symmetry could possibly introduce spurious asymptotic behavior in the Kerr-Schild construction of binary black hole initial data.
We concentrate here on the boosted Schwarzschild metric. The Kerr case is more complicated because the twist of the principle null direction $\ell_\mu$ does not allow a straightforward construction of well-behaved advanced or retarded null coordinate systems. Although an exact Schwarzschild exterior is unrealistic in a dynamic spacetime it is reasonable to expect that our results are valid if it is a good far field approximation in the neighborhood of null infinity in the limit of both infinite future and infinite past retarded and advanced times. In this respect, our results might also apply to the Kerr case since the metric terms involving the angular momentum parameter fall off faster with $r$ than the terms involving the mass.
Boosts and the Kerr-Schild-Schwarzschild metrics {#sec:advret}
================================================
In time symmetric coordinates $x^\mu = (t,r,x^A)$, with $x^A=(\theta,\phi)$ being standard spherical coordinates, the Schwarzschild metric is $$\begin{aligned}
g_{\mu\nu} &=&-\Big(1-\frac{2M}{r}\Big)t_{,\mu}t_{,\nu}
+\Big(1-\frac{2M}{r}\Big)^{-1}r_{,\mu}r_{,\nu} +r^2q_{\mu\nu}(x^A) .
\label{eq:Schwarz_std}\end{aligned}$$ Here we use standard comma notation to denote partial derivatives, e.g. $f_{,\mu} =\partial_\mu f $, and $q_{\mu\nu}(x^A)$ is the round unit sphere metric defined with respect to the Cartesian coordinates $x^i =r r^i$, $r^i(x^A) = (\sin\theta\cos\phi,\sin\theta\sin\phi, \cos\theta)$ so that $$q_{\mu\nu}dx^\mu dx^\nu = \delta_{ij}r^i_{,A}r^j_{,B}dx^Adx^B
= d\theta^2 +\sin^2\theta d\phi^2.$$ Introduction of the “tortoise” coordinate $r^* = r +2M \ln(\frac{r}{2M}-1)$, with $ r^*_{,\mu} = (rr_{,\mu})(r-2M)^{-1}$, gives $$\begin{aligned}
g_{\mu\nu}
&=&-\Big(1-\frac{2M}{r}\Big)(t_{,\mu}t_{,\nu} -r^*_{,\mu}r^*_{,\nu})
+r^2q_{\mu\nu}(x^A) .
\label{eq:ks0}\end{aligned}$$ In terms of the retarded time $u=t-r^*$, $$g_{\mu\nu} =-\Big(1-\frac{2M}{r}\Big)u_{,\mu}u_{,\nu}
-u_{,\mu}r_{,\nu}- u_{,\nu}r_{,\mu}
+r^2q_{\mu\nu}
\label{eq:ks-}$$ and in terms of the advanced time $v=t+r^*$, $$g_{\mu\nu} =-\Big(1-\frac{2M}{r}\Big)v_{,\mu}v_{,\nu}
+v_{,\mu}r_{,\nu} +v_{,\nu}r_{,\mu}
+r^2q_{\mu\nu}.
\label{eq:ks+}$$
The retarded time version of the Schwarzschild metric (\[eq:ks-\]) has the Kerr-Schild form with Minkowski metric $\eta^{(-)}_{\mu\nu}$, $$g_{\mu\nu} = \eta^{(-)}_{\mu\nu}
+\frac{2M}{r}k_\mu k_\nu , \quad k_\mu = -u_{,\mu}.
\label{eq:mks-}$$ Here, in the associated inertial coordinates $x^{(-)\mu}=(t^{(-)},x^{(-)i})=(t^{(-)},x^{(-)},y^{(-)},z^{(-)})$, with $t^{(-)}=u+r$ and $x^{(-)i} = r r^i(x^A)$, $$\eta^{(-)}_{\mu\nu} dx^{(-)\mu} dx^{(-)\nu}
=-dt^{(-)2} +\delta_{ij}dx^{(-)i} dx^{(-)j} .
\label{eq:outmink}$$
Similarly, the advanced time version (\[eq:ks+\]) has the Kerr-Schild form with the background Minkowski metric $\eta^{(+)}_{\mu\nu}$, $$g_{\mu\nu} = \eta^{(+)}_{\mu\nu}
+\frac{2M}{r} n_\mu n_\nu , \quad n_\mu = -v_{,\mu}
\label{eq:mks+}$$ where in the associated inertial coordinates $x^{(+)\mu}
=(t^{(+)},x^{(+)i})=(t^{(+)},x^{(+)},y^{(+)},z^{(+)})$, with $t^{(+)}=v-r^{(-)}$ and $x^{(+)i} =r r^i(x^A)$, $$\eta^{(+)}_{\mu\nu} dx^{(+)\mu} dx^{(+)\nu}
=-dt^{(+)2}+\delta_{ij}dx^{(+)i} dx^{(+)j} .
\label{eq:inmink}$$ Note that the inertial time coordinates are related by $$\label{eq:inertial_time_relation}
t^{(+)}=t^{(-)}+ 4M \ln\Big(\frac{r}{2M}-1\Big)$$ whereas the inertial spatial coordinates are related by $x^{(+)i}=x^{(-)i}$. As a result, it is unambiguous to write $x^{(+)i}=x^{(-)i}=x^i$ and $r^{(+)}=r^{(-)} =r$, where $r^{(+)2}:=\delta_{ij}x^{(+)i}x^{(+)j}$ and $r^{(-)2}:=\delta_{ij}x^{(-)i}x^{(-)j}$. However, the corresponding directional derivatives are related by $$\partial_{t^{(+)}} = \partial_{t^{(-)}}\, , \quad
\partial_{x^{(+)i}} = \partial_{x^{(-)i}} -\frac{4M}{r-2M} \frac{x^i}{r}\partial_{t^{(-)}}$$ and $$\partial_{r^{(+)}} = \partial_{r^{(-)}} -\frac{4M}{r-2M} \partial_{t^{(-)}} .$$ As will be seen, these transformations have important bearing on the relation between the generators of the BMS group at past and future null infinity.
In [@boost], we showed that the linearized memory effect could be based upon the boosted version of the advanced time Kerr-Schild metric (\[eq:mks+\]). In that linearized treatment, it was assumed that the boost was a Lorentz symmetry of the Minkowski background. However, this cannot be extended unambiguously to the nonlinear case, where there are two distinct Minkowski backgrounds $ \eta^{(-)}_{\alpha\beta}$ and $ \eta^{(+)}_{\alpha\beta}$ defined, respectively, by the retarded and advanced time Kerr-Schild metrics (\[eq:mks-\]) and (\[eq:mks+\]). Since the metrics (\[eq:mks-\]) and (\[eq:mks+\]) are algebraically identical, it cannot be the choice of retarded or advanced metric but the corresponding choice of boost that gives the essential result.
In spherical null coordinates, the Minkowski background metric (\[eq:outmink\]) has the standard retarded time Bondi-Sachs form at ${\mathcal I}^+$, $$\label{eq:BS_mink_ret}
\eta^{(-)}_{\mu\nu}dx^\mu dx^\nu
=-du^2 -2du dr +r^2 q_{AB}dx^Adx^B$$ and (\[eq:inmink\]) has the standard advanced time Bondi-Sachs form at ${\mathcal I}^-$, $$\label{eq:BS_mink_adv}
\eta^{(+)}_{\mu\nu}dx^\mu dx^\nu
=-dv^2 +2dv dr +r^2q_{AB}dx^Adx^B.$$ (See [@bs_scolar] for a review of the Bondi-Sachs formalism.) These Minkowski line elements transform into each other under the Minkowski space relation $u=v-2r$ but not under the Schwarzschild relation between retarded and advanced time $u=v-2r^*$. The retarded and advanced Minkowski metrics and are related by $$\begin{aligned}
\eta^{(+)}_{\mu\nu} &=& \eta^{(-)}_{\mu\nu}
+\frac{2M}{r}\Big( k_{\mu} k_{\nu}
- n_{\mu} n_{\nu}\Big)
\nonumber\\
&=&\eta^{(-)}_{\mu\nu}
-\frac{4M }{r-2M} (u_{,\mu}r_{,\nu}+u_{,\nu}r_{,\mu})
-\frac{8Mr}{(r-2M)^2}r_{,\mu}r_{,\nu} .
\label{eq:eta}\end{aligned}$$ Because of the non-vanishing $ r_{,\mu} r_{,\nu}$ term in (\[eq:eta\]), although $\eta^{(+)}_{\mu\nu} $ has the advanced time Bondi-Sachs form near ${\mathcal I}^-$ it does not have the retarded time Bondi-Sachs form near ${\mathcal I}^+$. The reverse is true of $\eta^{(-)}_{\alpha\beta} $, which has the retarded time Bondi-Sachs form near ${\mathcal I}^+$ but not the advanced time form near ${\mathcal I}^-$. This leads to a non-trivial difference between the boosts $B^{(-)}$ and $B^{(+)}$, with generators $B_{x^{(-)i}}$ and $B_{x^{(+)i}}$, which are symmetries of $\eta^{(-)}_{\alpha\beta}$ and $\eta^{(+)}_{\alpha\beta}$, respectively.
To be specific, consider a boost in the $z^{(-)}$-direction intrinsic to $\eta^{(-)}_{\mu\nu}$ with generator $B_{z^{(-)}}
= z^{(-)} \partial_{t^{(-)}} + t^{(-)}\partial_{z^{(-)} }$. In retarded spherical null coordinates $$\partial_{t^{(-)}} =\partial_u \,, \quad
\partial_{z^{(-)}} = -\cos\theta(\partial_u -\partial_{r^{(-)}})
-\frac{\sin\theta}{r}\partial_\theta.$$ This leads to the retarded time dependence $$B_{z^{(-)}} =-u\cos\theta \partial_u +(u+r)\cos\theta\partial_{r^{(-)}}
-(\frac{u}{r} +1)\sin\theta\partial_\theta,
\label{eq:bm}$$ which has the proper asymptotic behavior to be the generator of a BMS boost symmetry at ${\mathcal I}^+$.
However, expressed in terms of advanced null coordinates, using $u=v-2r^*$, $$\begin{aligned}
B_{z^{(-)}} &=&-(v-2r^*)\cos\theta \partial_v
+(2\partial_r r^*)(v-2r^*+r) \cos\theta \partial_v
\nonumber \\
&+&(v-2r^*+r)\cos\theta\partial_{r^{(+)}} -\left(\frac{v-2r^*}{r} +1\right)\sin\theta\partial_\theta
\nonumber \\
&=&\left\{\frac{(r+2M)[v-4M \ln(\frac{r}{2M} -1)]
-4Mr}{r-2M} \right \} \cos\theta \partial_v \nonumber \\
&+&\left[v-r-4M \ln\left(\frac{r}{2M} -1\right)\right](\cos\theta\partial_{r^{(+)}}
-\frac{1}{r}\sin\theta\partial_\theta).
\label{eq:bzm} \end{aligned}$$ Here, because the $\partial_v$ coefficient goes to infinity as $ \ln(\frac{r}{2M} -1)$ for large $r$, $B_{z^{(-)}}$ generates a singular transformation at ${\mathcal I}^-$.
Now consider the corresponding boost in the $z^{(+)}$-direction intrinsic to $\eta^{(+)}_{\mu\nu}$ with generator $B_{z^{(+)}}= z^{(+)} \partial_{t^{(+)}} + t^{(+)}\partial_{z^{(+)}}$. In advanced null coordinates, $$\partial_{t^{(+)}} =\partial_v , \quad
\partial_{z^{(+)}} = \cos\theta(\partial_v +\partial_{r^{(+)}})
-\frac{\sin\theta}{r}\partial_\theta.$$ This leads to the advanced time dependence $$B_{z^{(+)}}=v\cos\theta \partial_v +(v-r)\cos\theta\partial_{r^{(+)}}
-\left(\frac{v}{r} -1\right)\sin\theta\partial_\theta ,
\label{eq:bp}$$ which has the proper asymptotic behavior to be the generator of a BMS boost symmetry at ${\mathcal I}^-$. However, expressed in terms of retarded null coordinates $$\begin{aligned}
B_{z^{(+)}}
&=&-\left\{\frac{(r+2M)[u+4M \ln(\frac{r}{2M} -1)]
+4Mr}{r-2M} \right \} \cos\theta \partial_u \nonumber \\
&+&\left[u+r+4M \ln\left(\frac{r}{2M} -1\right)\right](\cos\theta\partial_{r^{(-)}}
-\frac{1}{r}\sin\theta\partial_\theta),
\label{eq:bzp} \end{aligned}$$ which generates a singular transformation at ${\mathcal I}^+$. As will be seen in the next section, these unexpected gauge singularities of $B_{z^{(-)}}$ at ${\mathcal I}^-$ and $B_{z^{(+)}}$ at ${\mathcal I}^+$ do not affect calculation of the radiation memory.
Boosts and radiation memory {#sec:mem}
===========================
In retarded null coordinates $x^\mu=(u,r,x^A)$, where the angular coordinates $x^A=(\theta,\phi)$, are constant along the outgoing null rays and $r$ is an areal coordinate which varies along the null rays, the metric takes the Bondi-Sachs form $$\label{BS_metric}
\fl g_{\mu\nu}dx^\mu dx^\nu = -\frac{V}{r}e^{2\beta} du^2-2 e^{2\beta}dudr
+r^2h_{AB}\Big(dx^A-U^Adu\Big)\Big(dx^B-U^Bdu\Big) .$$ The choice of areal coordinate $r$ and the choice $x^A=(\theta,\phi)$ as angular coordinates requires $$\det [h_{AB}] = \det [q_{AB}] =\sin^2 \theta .
\label{eq:det}$$ As a result, the conformal 2-metric $h_{AB}$ has only two degrees of freedom, which encode the two polarization modes of a gravitational wave.
In the neighborhood of ${\mathcal I}^+$, asymptotic flatness allows the construction of inertial coordinates such that the metric approaches the Minkowski metric, $$\frac{V}{r} =1 +O(1/r)\, , \quad
\beta = O(1/r^2) \, , \quad
U^A =-\frac{1}{2r^2}\eth_E c^{EA}+O(1/r^3)$$ and $$h_{AB} = q_{AB} + c_{AB}/r +O(1/r^2).$$ Here $\eth_A$ is the covariant derivative with respect to $q_{AB}$ and the determinant condition (\[eq:det\]) implies that $c_{AB}(u,x^C)$ is traceless, $q^{AB}c_{AB}=0$. Evaluation of the geodesic deviation equation in the linearised limit of the Bondi-Sachs metric shows that $\sigma_{AB} = \frac{1}{2}c_{AB}$ is the $O(1/r)$ strain tensor of the gravitational radiation.
The gravitational wave memory effect is determined by the change in the radiation strain between infinite future and past retarded time, $$\label{def_memory}
\Delta \sigma_{AB}(x^C):
= \sigma_{AB}(u=\infty,\theta,\phi) - \sigma_{AB}(u=-\infty,\theta,\phi).$$ This produces a net displacement in the relative angular position of distant test particles,[^3] $$\label{eq:displacment}
\Delta ( x_2^A - x_1^A) = \frac{1}{r}
( x_2^C - x_1^C)q^{AB} \Delta \sigma_{BC} .$$
A compact way to describe the radiation is in terms of a complex polarization dyad $q_A$ satisfying $$q _{AB} = \frac{1}{2}(q_A \bar q_B +\bar q_A q_B), \quad
q^A \bar q_A =2,
\quad q^A q_A =0.$$ For the standard form of the unit sphere metric in spherical coordinates $x^A=(\theta, \phi)$, we set $q^A\partial_A = \partial_\theta +(i/\sin\theta)\partial_\phi $. In the associated inertial Cartesian coordinates, the dyad $q^A$ has components $q^\mu = r Q^\mu = (0,rQ^i)$, where $$Q^i = r^i_{,A}q^A =
( \cos\theta\cos\phi-i\sin\phi, \cos\theta\sin\phi+i\cos\phi,-\sin\theta)$$ and $\delta_{ij}Q^i \bar Q^j =2$. The dyad decompsition $$\label{ }
\sigma_{AB}=\frac{1}{4}
\Big[(q^Eq^F\sigma_{EF})\bar q_A\bar q_B
+ (\bar q^E\bar q^F\sigma_{EF})q_Aq_B\Big]\;\;,$$ leads to the spin-weight-2 representation of the strain, $$\label{def_rad_mem_spin0}
\sigma:=\frac{1}{2}q^A q^B \sigma_{AB}.$$ Note that $\sigma$ also corresponds to the leading $(r^{-2})$ coefficient of the shear of the null hypersurfaces $u=const$. Its retarded time derivative $N(u,x^A):=\partial_u\sigma(u, x^A)$ is the Bondi news function.
The shear-free property of the Schwarzschild metric in its rest frame implies that $\sigma=0$. For a transition from an initially static Schwarzschild frame to a final boosted state, the resulting spin-weighted radiation memory is then $$\Delta \sigma(x^C) = \sigma(u=\infty, x^C)- \sigma(u=-\infty,x^C) ,
\label{eq:memeff}$$ where $\sigma(u=\infty, x^C)$ is the radiation strain of the final boosted state and initially $\sigma(u=-\infty, x^C) =0$.
Under the retarded time transformation $u\rightarrow u
+\alpha(x^A)$, which corresponds to the supertranslation freedom in the BMS group [@Sachs_BMS], the asymptotic strain has the gauge freedom $$\sigma(u,x^A) \rightarrow \sigma(u,x^A)+\eth^2 \alpha(x^A),
\label{eq:super}$$ where $\eth$ is the Newman-Penrose spin-weight raising operator [@BMS2]. Since the finiteness of the radiative mass loss requires that the news function $N=\partial_u \sigma$ vanish as $u\rightarrow \pm \infty$, the strain $\sigma$ can be gauged to zero either as $u\rightarrow \infty$ or $u\rightarrow -\infty$. The memory effect $\Delta \sigma$ is gauge invariant but determines a supertranslation $\alpha(x^A)$ according to $$\eth^2 \alpha(x^A) =\Delta \sigma(x^A),$$ which relates the strains at $u=\pm \infty$. The energy flux of the radiation is given by the absolute square, $N \bar N $, of the Bondi news function $N$, which is also gauge invariant. If the memory effect (\[eq:memeff\]) is non-zero then there must be intervening radiation.
These attributes of ${\mathcal I}^+$ have corresponding attributes at ${\mathcal I}^-$. In particular, the outgoing radiation strain $\sigma(u,x^A)$ has as its analogue an ingoing radiation strain $\Sigma(v,x^A)$. In analogy with (\[eq:memeff\]), the gravitational wave memory at ${\mathcal I}^-$ due to ingoing radiation is $$\Delta \Sigma(x^C) = \Sigma(v=\infty, x^C)
- \Sigma(v=-\infty,x^C) .
\label{eq:pastmemeff}$$ If there is no ingoing radiation, as required in the linearized case by a retarded solution, then $\Delta \Sigma(x^C) =0$.
Of the BMS transformations, only the supertranslations (\[eq:super\]) affect the radiation strain. As shown in Sec. \[sec:advret\], a $B^{(-)}$ boost is a BMS boost symmetry at ${\mathcal I}^+$ so that it does not introduce outgoing radiation memory $\Delta \sigma$. Conversely, a $B^{(+)}$ boost is a BMS boost symmetry at ${\mathcal I}^-$ so that it does not introduce ingoing radiation memory $\Delta \Sigma$. These results are explicitly demonstrated below.
Effect of a $B^{(-)}$ boost {#sec:bminus}
---------------------------
Consider first the transition from a static Kerr-Schild-Schwarzschild metric to the $B^{(-)}$ boosted version with 4-velocity $v^\mu = \Gamma(1, V^i)$, where $\Gamma = (1 -\delta_{ij}V^iV^j)^{-1/2}$. For a $B^{(-)}$ boost, $\eta^{(-)}_{\mu\nu} \rightarrow \eta^{(-)}_{\mu \nu}$. The boosted version of the static retarded time Kerr-Schild-Schwarzschild metric (\[eq:mks-\]), can be obtained by the further substitutions $$\fl \partial_\mu t^{(-)} \rightarrow -v_\mu, \quad
r^2 \rightarrow R^{(-)2} = x^{(-)}_\mu x^{(-)\mu} +(x^{(-)}_\mu v^\mu)^2, \quad
k_\mu \rightarrow K_\mu = v_\mu+R^{(-)} _\mu,$$ where $$\partial_\mu r \rightarrow R^{(-)} _\mu =
\frac{1}{ R^{(-)}}(x^{(-)}_\mu
+ v_\mu x^{(-)}_\nu v^\nu).$$ The boosted metric is $$g^{(B^-)}_{\mu\nu} = \eta^{(-)}_{\mu\nu}
+\frac{2M}{R^{(-)}}K_\mu K_\nu .$$ This Lorentz covariant transformation reduces to the rest frame expression when $V^i=0$.
In order to calculate the resulting radiation strain, we note that $q^\mu q^\nu \eta^{(-)}_{\mu\nu}=0$ and $q^\mu x_\mu$ =0 so that $$\sigma^{(B^-)}=\frac{1}{4} rq^\mu q^\nu g^{(B^-)}_{\mu\nu} \big|_{\mathcal{I}^+}
=\frac{M r}{2R^{-}}(q^\mu K_\mu)^2\big|_{\mathcal{I}^+},$$ where $q^\mu K_\mu = q^\mu v_\mu (1 + R^{-1} x^{(-)}_\nu v^\nu)$.
For the limit at ${\mathcal I}^+$, in retarded null coordinates $$\begin{aligned}
R^{(-)2} &=& r^2\Big[-\frac{u^2}{r^2}-\frac{2u}{r}
+ \Gamma^2\Big(1-\frac{x _iV^i}{r} + \frac{u}{r}\Big)^2\Big]\;,\\
x^{(-)}_\nu v^\nu &=& \Gamma(-u-r + x_i V^i) \;,\end{aligned}$$ so that $$\begin{aligned}
\lim_{\stackrel{r\rightarrow\infty}{u=const}}\frac{R^{(-)}}{r}
&=&\Gamma(1-\frac{V^i x_i}{r}) \;, \\
\lim_{\stackrel{r\rightarrow\infty}{u=const}}
\frac{ x^{(-)}_\nu v^\nu}{r}&=& -\Gamma(1-\frac{V^i x_i}{r}) \;.\end{aligned}$$ Consequently, $$\lim_{\stackrel{r\rightarrow\infty}{u=const}}
\frac{x^{(-)}_\nu v^\nu}{R^{(-)}} = -1$$ and $$\lim_{\stackrel{r\rightarrow\infty}{u=const}}
q^\mu K_\mu =0.$$ Therefore $\sigma^{(B^-)}(u,x^C)=0$ and in particular $\sigma^{(B^-)}(u=\infty, x^C)=0$. So, as expected from the BMS property of the $B^{(-)}$ boost at ${\mathcal I}^+$, it produces no radiation memory at ${\mathcal I}^+$. Now consider the boosted strain on ${\mathcal I}^-$, $$\Sigma^{(B^-)}
= \frac{r}{4}q^\mu q^\nu g^{(B^-)}_{\mu\nu} |_{{\mathcal I}^-}
=\frac{Mr}{2R^{(-)}}(q^\mu v_\mu )^2
\Big(1+\frac{x^{(-)}_\nu v^\nu}{R^{(-)}}\Big)^2
\Big|_{{\mathcal I}^-} .$$ In order to calculate the limit at ${\mathcal I}^-$, for which $r\rightarrow \infty$ holding $v =t^{(+)} +r$ constant, we must express $\Sigma^{(B^-)}$ as a function of the unboosted advanced coordinates $(v,r,x^A)$. Using , a straightforward calculation gives $$x^{(-)}_\mu v^\mu = r \Gamma\Big[1-\frac{v}{r}
+\frac{4M}{r}\ln(\frac{r}{2M}-1)
+r_i V^i\Big],$$ $$x^{(-)}_\mu x^{(-)\mu}= r[v-4M \ln(\frac{r}{2M}-1)]
[2-\frac{v}{r}+\frac{4M}{r} \ln(\frac{r}{2M}-1)] ,$$ which leads to the limits $$\lim_{\stackrel{r\rightarrow\infty}{v=const}}\frac{R^{(-)}}{r}
=\Gamma(1+V^ir_i) ,$$ $$\lim_{\stackrel{r\rightarrow\infty}{v=const}}
\frac{x^{(-)}_\mu v^\mu }{R^{(-)}}
= \lim_{\stackrel{r\rightarrow\infty}{v=const}}
\big[ \frac{r}{R^{(-)}} \big ]\big [\frac{x^{(-)}_\mu v^\mu }{r} \big]
=1 .$$ We then obtain $$\label{eq:pastboostmem}
\Sigma^{(B^-)} = \frac{2Mr}{R^{(-)}}(q^\mu v_\mu )^2
|_{{\mathcal I}^-}
=\frac{2M\Gamma (q^iV_i)^2}{1+V^ir_i} \;\;.$$ Consequently, for a non zero boost, the resulting radiation memory on ${\mathcal I}^-$ does not vanish, which requires the existence of ingoing radiation.
Thus the $B^{(-)}$ boost is inconsistent with vanishing ingoing radiation and produces zero radiation memory on ${\mathcal I}^+$. Both of these results contradict the linearized result based upon the retarded Green function so that $B^{(-)}$ is not the appropriate boost to model the memory effect.
Effect of a $B^{(+)}$ boost
---------------------------
Consider now the transition from a static to a $B^{(+)}$ boosted version of the Kerr-Schild-Schwarzschild metric with 4-velocity $v^\mu = \Gamma(1, V^i)$. For the $B^{(+)}$ boost, $\eta^{(+)}_{ab} \rightarrow \eta^{(+)}_{ab}$. With respect to the advanced time version of the static Kerr-Schild-Schwarzschild metric (\[eq:mks+\]), the boosted version can be obtained by the further substitutions $$\fl \partial_\mu t^{(+)} \rightarrow -v_\mu, \quad
r^2 \rightarrow R^{(+)2} = x^{(+)}_\mu x^{(+)\mu}
+(x^{(+)}_\mu v^\mu)^2 , \quad
n_\mu \rightarrow N_\mu
= v_\mu-R^{(+)} _\mu,$$ where $$\partial_\mu r \rightarrow R^{(+)} _\mu =
\frac{1}{ R^{(+)}}(x^{(+)}_\mu
+ v_\mu x^{(+)}_\nu v^\nu).$$ The boosted metric is $$g^{(B^+)}_{\mu\nu} = \eta^{(+)}_{\mu\nu}
+\frac{2M}{R^{(+)}}N_\mu N_\nu$$ with the corresponding boosted strain on ${\mathcal I}^-$ given by $$\fl\quad
\Sigma^{(B^+)}
= \frac{r}{4}q^\mu q^\nu g^{(B^+)}_{\mu\nu}\Big|_{{\mathcal I}^-}
= \frac{M r}{ 2R^{(+)}}(q^\mu N_\mu )^2 \Big|_{{\mathcal I}^-}
=\frac{Mr}{2R^{(+)}}(q^\mu v_\mu )^2
\Big(1-\frac{x^{(+)}_\nu v^\nu}{R^{(+)}}\Big)^2
\Big|_{{\mathcal I}^-}. \quad$$ The calculation of the limit proceeds in a time reversed sense as in Sec. \[sec:bminus\].
In advanced null coordinates $$\begin{aligned}
R^{(+)2} &=& r^2\Big[-\frac{v^2}{r^2}+\frac{2v}{r}
+ \Gamma^2\Big(1+\frac{x _iV^i}{r} - \frac{v}{r}\Big)^2\Big]\;,\quad \\
x^{(+)}_\nu v^\nu &=& \Gamma(-v+r + x_i V^i) \;,\end{aligned}$$ so that $$\begin{aligned}
\lim_{\stackrel{r\rightarrow\infty}{v=const}}\frac{R^{(+)}}{r}
&=&\Gamma(1+\frac{V^i x_i}{r}) \;,\qquad \\
\lim_{\stackrel{r\rightarrow\infty}{v=const}}
\frac{ x^{(+)}_\nu v^\nu}{r}&=& \Gamma(1+\frac{V^i x_i}{r}) \;.\end{aligned}$$ Consequently, $$\lim_{\stackrel{r\rightarrow\infty}{v=const}}
\frac{x^{(+)}_\nu v^\nu}{R^{(+)}}
= \lim_{\stackrel{r\rightarrow\infty}{v=const}}
\big[\frac{r}{R^{(+)}} \bi]\big[ \frac{x^{(+)}_\nu v^\nu}{r}\big]
= 1 \qquad$$ and therefore $\Sigma^{(B^+)}(v,x^C)=0$. So, as expected from the BMS property of the $B^{(+)}$ boost at ${\mathcal I}^-$, there is no radiation memory at ${\mathcal I}^-$. It is thus consistent to set the free characteristic initial data $\Sigma$ to zero on ${\mathcal I}^-$ so that there is no ingoing radiation.
Now consider the boosted strain on ${\mathcal I}^+$, $$\sigma^{(B^+)}
= \frac{r}{4}q^\mu q^\nu g^{(B^+)}_{\mu\nu}
|_{{\mathcal I}^+}
=\frac{Mr}{2R^{(+)}}(q^\mu v_\mu )^2
\Big(1-\frac{x^{(+)}_\nu v^\nu}{R^{(+)}}\Big)^2
\Big|_{{\mathcal I}^+} .$$ In order to calculate the limit at ${\mathcal I}^+$, for which $r\rightarrow \infty$ holding $u =t^{(-)} -r$ constant, we must express $\sigma^{(B^+)}$ as a function of the unboosted retarded coordinates $(u,r,x^A)$. A straightforward calculation gives $$x^{(+)}_\mu v^\mu = -r \Gamma\Big[1+\frac{u}{r}
+\frac{4M}{r}\ln(\frac{r}{2M}-1)
-r_i V^i\Big],$$ $$x^{(+)}_\mu x^{(+)\mu}= -r[u+4M \ln(\frac{r}{2M}-1)]
[2+\frac{u}{r}+\frac{4M}{r} \ln(\frac{r}{2M}-1)] ,$$ which leads to the limits $$\lim_{\stackrel{r\rightarrow\infty}{u=const}}\frac{R^{(+)}}{r}
=\Gamma(1-V^ir_i) ,$$ $$\lim_{\stackrel{r\rightarrow\infty}{u=const}}
\frac{x_\nu v^\nu}{R^{(+)}}
= \lim_{\stackrel{r\rightarrow\infty}{u=const}}
\frac{-\Gamma r\Big[1-V^i r_i +\frac{u}{r}
+\frac{4M}{r}\ln(\frac{r}{2M}-1)\big]}{R^{(+)}} =-1 .$$ We then obtain $$\sigma^{(B^+)} =\frac{2Mr}{R^{(+)}}(q^\mu v_\mu )^2
|_{{\mathcal I}^+} = \frac{2M\Gamma}{(1-r_i V^i)}(q^i V_i )^2.$$ The resulting radiation memory due to the ejection of a Schwarzschild body is $$\Delta \sigma^{(B^+)} = \frac{2M\Gamma}{1-r_i V^i}(q^i V_i )^2.$$ This is in exact agreement with the linearized result.
Discussion {#sec:discuss}
==========
We have shown that the boost symmetry $B^{(+)}$ of the Minkowski background $\eta^{(+)}_{\mu\nu} $ of the ingoing Kerr-Schild version of the Schwarzschild metric leads to a nonlinear model for determining the memory effect due to the ejection of a massive particle. An initially stationary Kerr-Schild-Schwarzschild metric followed by an accelerating interval which produces radiation and leads to a final $B^{(+)}$ boosted state is consistent with the absence of ingoing radiation and produces outgoing radiation in agreement with the linearized memory effect obtained from a retarded solution. The corresponding results for a $B^{(-)}$ boost of the Minkowski background $\eta^{(-)}_{\mu\nu}$ produces results expected in the linearized limit from the use of an advanced Green function.
In [@boost], we have given an analysis of how radiation memory affects angular momentum conservation. In a non-radiative regime, a preferred Poincar[é]{} subgroup can be picked out from the BMS group. This difference $\Delta \sigma$ between initial and final radiation strains induces the supertranslation shift (\[eq:super\]) between the preferred Poincar[é]{} groups at $u=\pm \infty$. The rotation subgroups associated with the initial and final Poincar[é]{} groups differ by a supertranslation. As a result, the corresponding components of angular momentum intrinsic to the initial and final states differ by supermomenta. This complicates the interpretation of angular momentum flux conservation laws. There might be a distinctly general relativistic mechanism for angular momentum loss. This is a ripe area for numerical investigation.
In prescribing initial data for the numerical simulation of binary black holes using superimposed Kerr-Schild metrics [@ks1; @ks2], $B^{(+)}$ is used to induce the orbital motion. Although $B^{(+)}$ has a logarithmic singularity (\[eq:bzp\]) at ${\mathcal I}^+$, this is a pure gauge effect which does not show up in the memory effect measured by the change in asymptotic strain $\Delta \sigma$ but it could introduce spurious effects in the prescription of binary black hole initial data. Whether this adversely affects the asymptotic gauge behavior of the data deserves further study.
The model presented here provides a scheme for studying these issues. Although our example of a transition from a asymptotically stationary to boosted state is highly idealized, the chief criterion for the model is that, to an asymptotic approximation, the far field behavior of the initial and final states consist of the Kerr-Schild superposition of distant Schwarzschild bodies. The model is also applicable to an initial state whose far field is a superposition of boosted Schwarzschild bodies which, after some dynamic, radiative process, coalesce to form a boosted Kerr black hole. Of course, the intermediate radiative epoch, which determines the final mass and velocity, must be treated by numerical methods. The Kerr-Schild model offers a framework for interpreting such results.
We thank the AEI in Golm for hospitality during this project. T.M. appreciates support from C. Malone and the University of Cambridge. J.W. was supported by NSF grant PHY-1505965 to the University of Pittsburgh.
References {#references .unnumbered}
==========
[10]{}
Zeldovich Y and Polnarev A G 1974 Radiation of gravitational waves by a cluster of superdense stars [*Sov. Astron.*]{} [**18**]{} 17
M[ä]{}dler T and Winicour J 2017 Radiation memory, boosted Schwarzschild spacetimes and supertranslations [*Class. Quantum Grav.*]{} 34 115009
Kerr R P and Schild A 1965 Some algebraically degenerate solutions of Einstein’s gravitational field equations [*Proc. Symp. Appl. Math.*]{} [**17**]{} 199
Kerr R P and Schild A 1965 A new class of vacuum solutions of the Einstein field equations [*Atti degli Convegno Sulla Relativita Generale*]{} p. 222 (Firenze)\
Kerr R P and Schild A 2009 Gen. Relativ. Gravit. 41 2485 (republication) Sachs R 1962 Asymptotic symmetries in gravitational theory [*Phys. Rev.*]{} [**128**]{} 2851
Kramer D, Stephani H, MacCallum M and Herlt E 1980 [*Exact solutions of Einstein’s field equations*]{}, ed. E. Schmutzer (Cambridge University Press, Cambridge)
Matzner R A, Huq M F and Shoemaker D 1998 Initial data and coordinates for multiple black hole systems [*Phys. Rev. D*]{} [**59**]{} 024015
Bonning E, Marronetti P, Neilson D, and Matzner R A 2003 Physics and initial data for multiple black hole spacetimes [*Phys. Rev. D*]{} [**68**]{} 044019
Mädler T and Winicour J 2016 Bondi – Sachs Formalism [*Scholarpedia*]{} [**11**]{}(12) 33528
Newman E and Penrose R 1966 Note on the Bondi-Metzner-Sachs group [*J. Math. Phys.*]{} [**7**]{}, 863
Mädler T and Winicour J 2016 The sky pattern of the linearized gravitational memory effect [*Class. Quantum Grav.* ]{} [**33**]{} 175006
[^1]: Email:thomas.maedler@mail.udp.cl
[^2]: Note that a time symmetric version of the Schwarzschild metric, see , does not single out such a preferred Minkowski background in the nonlinear case.
[^3]: Note corrects a missing $1/r$ factor in the corresponding equation in [@sky_pattern].
|
---
abstract: 'The presence, nature, and impact of chemical short-range order in the multi-principal element alloy CrCoNi are all topics of current interest and debate. On the basis of first-principles calculations, we present a theory of exchange interaction-driven atomic ordering in this system centered on the elimination of like-spin Cr-Cr neighbors, with significant contributions from certain magnetically aligned Co-Cr and Cr-Cr atoms. Together, these effects can explain anomalous magnetic measurements across a range of compositions and provide implications for related high-entropy alloys.'
author:
- Flynn Walsh
- 'Robert O. Ritchie'
- Mark Asta
bibliography:
- 'main.bib'
title: |
Interdependence of Magnetic and Chemical Short-Range Order\
in the CrCoNi Multi-Principal Element Alloy
---
Multi-principal element alloys (MPEAs), often referred to as high-entropy alloys, have become intensely investigated in recent years as they offer a practically limitless design space that, in the small portion thus far explored, has already yielded several promising materials [@yeh2004; @miracle2017; @george2019; @george2020]. In particular, a large body of research has been devoted to face-centered cubic (*fcc*) systems composed of 3$d$ transition metals, namely the equimolar (Cantor) alloy CrMnFeCoNi [@cantor2004] and its derivatives. These ostensibly disordered *fcc* MPEAs display highly desirable combinations of mechanical properties that are attributable to deformation mechanisms [@george2019; @george2020] that can be tuned through careful control of alloy parameters such as chemistry [@li2017b] and even magnetic structure [@wu2020]. Another potentially important, if enigmatic, factor in the engineering of this class of materials is the presence of atomic-scale short-range order (SRO). In this regard, particular attention has been given to the equiatomic CrCoNi alloy, a representative system that is noteworthy for its cryogenic damage tolerance and general mechanical superiority to the five-component CrMnFeCoNi [@wu2014; @gludovatz2016; @laplanche2017]. Several computational [@tamm2015; @ding2018; @pei2020] and experimental [@zhang2017; @zhang2020] studies have provided evidence for both the presence and possibly significant impact of SRO in this material; however, its microscopic origin has largely escaped scrutiny.
In the present work, we employ density-functional theory (DFT) calculations to explain the large energetic driving force ($\sim40$ meV/atom) for SRO that has been found in this system by previous first-principles investigations [@tamm2015; @ding2018]. Our results reveal the fundamental importance of exchange interactions among Cr-Cr and Cr-Co neighbors and the role of SRO in reducing magnetic frustration inherent in random solid solutions of these elements. While previous studies have identified, in various terms, the importance of Cr antiferromagnetism in the ordering of a related CrFeCoNi alloy [@tamm2015; @niu2015], the role of magnetism in the chemical SRO of CrCoNi has not, to the best of our knowledge, been carefully examined. Newly identified ordering principles are extended to non-stoichiometric Cr-Co-Ni compositions, offering an explanation for previously noted discrepancies between experimental measurements of low-temperature magnetization and DFT calculations assuming complete compositional disorder [@sales2016; @sales2017].
------- ------- ------- -------------------------------
Cr-Cr 0.42 0.40 ${{\alpha}_{\text{CrCr}}}$
Co-Cr -0.16 -0.25 -${{\alpha}_{\text{CrCr}}}$/2
Ni-Cr -0.27 -0.15 -${{\alpha}_{\text{CrCr}}}$/2
Ni-Co 0.15 0.19 ${{\alpha}_{\text{CrCr}}}$/2
Co-Co 0.01 0.06 0.0
Ni-Ni 0.12 -0.04 0.0
------- ------- ------- -------------------------------
: WC SRO parameters (see Eq. (\[eq:WC\])) reported by two previous DFT-MC studies (columns two and three), compared to our simple structural model (column four) that is intended to relieve frustration of magnetic interactions by eliminating Cr-Cr bonds. []{data-label="table:WC"}
{width="\textwidth"}
SRO in CrCoNi was theoretically first examined by Tamm *et al.* [@tamm2015] through Monte Carlo (MC) optimization of on-lattice density functional theory (DFT) calculations. Their results are summarized in Table \[table:WC\] in terms of nearest neighbor Warren-Cowley (WC) SRO parameters [@cowley1950], a measure of the frequency of a chemical pairing relative to that expected in a random solution. For adjacent species $i$ and $j$, the latter with concentration $c_j$, the WC SRO parameter is $$\label{eq:WC}
\alpha_{ij} = 1 - \frac{P(j \mid i)}{c_j}
= 1 - \frac{P(j \cap i)}{P(j)P(i)}.$$ Negative values indicate more $ij$-type neighbors than in a random alloy (corresponding to $\alpha_{ij} = 0$ for all neighbor types) and positive the inverse. Ding *et al.* [@ding2018] later performed similar simulations, finding comparable order parameters that are also summarized in Table \[table:WC\]. Both computational datasets are supported by analysis of x-ray absorption fine structure from Ref. [@zhang2017] and share a pronounced positive value of ${{\alpha}_{\text{CrCr}}}$ that, despite very short simulation times, approaches the maximum of 0.5 for ${x_{\text{Cr}}}=\frac{1}{3}$ on an *fcc* lattice.
We build upon these prior computational studies by investigating in detail the link between magnetic exchange interactions and bonding preferences in CrCoNi solid solutions. To begin, we note that a random distribution of Cr atoms on a third of the sites of an *fcc* lattice will average four Cr nearest neighbors per Cr atom. Given the expected antiferromagnetic interactions between Cr atoms, this configuration will give rise to frustration among neighboring Cr moments [@tamm2015; @sales2016; @niu2018; @li2019b]. We examine the precise nature of this frustration—and later the energetics of the system—by employing collinearly spin-polarized DFT calculations, using the Vienna *Ab initio* Simulation Package (VASP) [@kresse1993; @kresse1996; @kresse1996a] with Perdew, Burke, and Ernzerhof’s (PBE) parametrization of the generalized gradient approximation (GGA) [@perdew1996] in tandem with projector-augmented wave (PAW) potentials [@kresse1999]. All simulation cells contained 108 atoms and were structurally relaxed, while electronic states were sampled in reciprocal space with a $3\times3\times3$ Monkhorst-Pack grid and 420 eV plane wave cutoff.
For the quasirandom [@zunger1990] supercell depicted in Fig. \[fig:frustration\](a)(i), the calculated local moments of Cr atoms are shown in Fig. \[fig:frustration\](a)(iii) with nearest neighbor Cr-Cr bonds identified. Even by inspection, it is clear that Cr atoms bonded to several other Cr display suppressed local moments, while those with fewer Cr bonds resolve into a network of alternating polarizations. Simplistically assigning Cr atoms “up" and “down" states from the sign of their local moment enables the calculation of a WC value for same-spin Cr (denoted ${\alpha_{\rm{Cr_{\uparrow \uparrow}}}}$) as $0.65 \pm 0.04$, while alternate spin pairs are commensurately more likely with ${\alpha_{\rm{Cr_{\uparrow \downarrow}}}}= -0.37 \pm 0.08$, as graphed in Fig. \[fig:frustration\](a)(ii). Although not accounting for moment magnitudes, the importance of which is emphasized below, these numbers highlight the unfavorability of same-spin nearest neighbors. In principle, noncollinear arrangements of moments could offer a pathway to resolving magnetic frustration, but this behavior is rarely observed in Cr clusters [@ruiz-diaz2010] and, consistent with Niu *et al.* [@niu2018], we could only converge our DFT calculations to collinear solutions.
Alternatively, we find local chemical ordering can reduce the mean number of same-species nearest neighbors in the solution to as low as two ($\alpha_\text{CrCr} = 0.5$), which should offer significant relief for frustration. To investigate this effect further, we develop supercells following a simple structural model in which $\alpha_\text{CrCr}$ is the dominant ordering term and other values are non-zero only by conservation of probability. Order parameters corresponding to this picture are presented in the last column of Table \[table:WC\] for arbitrary values of $\alpha_\text{CrCr}$. For ${{\alpha}_{\text{CrCr}}}=0.3,0.4,0.45,0.5$ (plus the quasirandom case of ${{\alpha}_{\text{CrCr}}}=0$), twenty 108 atom supercells were configured according to the model. Energies and net moments for these configurations, calculated as previously described, are plotted in Fig. \[fig:energies\], alongside those of supercells matching the nearest neighbor WC parameters of Tamm *et al.* and Ding *et al*. reproduced in Table \[table:WC\]. It should be emphasized that these are not exact replicas of those studies’ configurations. Indeed, Tamm *et al.* report a formation energy of 43.7 meV/atom, lower than the $62.2 \pm 2.7$ meV/atom recalculated presently—the discrepancy between these values suggests the importance of additional correlations, which we will address shortly.
![\[fig:energies\] Formation energy and magnetization for configurations representing various models of SRO in CrCoNi (twenty per datum) as a function of the Cr-Cr nearest neighbor WC SRO parameter (see Eq. \[eq:WC\]). ](energies2.pdf){width="\linewidth" height="5in"}
Nevertheless, the results displayed in Fig. \[fig:energies\] indicate that, within the margin of error (standard deviation of twenty supercells), the energy and magnetization of all configurations linearly follow ${{\alpha}_{\text{CrCr}}}$. In the extreme case of ${{\alpha}_{\text{CrCr}}}= 0.5$, formation energy and net moment are $52.0 \pm 3.5$ meV/atom and $ 0.054 \pm 0.04$ ${\rm{\mu_B}}$/atom, respectively, reduced from $88.0 \pm 3.3$ meV/atom and $0.28 \pm 0.04$ ${\rm{\mu_B}}$/atom for a quasirandom solution. As reflected in parts (ii) and (iii) of Fig. \[fig:frustration\](b), we can attribute reduction in net moments and energies to the minimization of Cr nearest neighbors with aligned local moments; indeed, ${\alpha_{\rm{Cr_{\uparrow \uparrow}}}}$ approaches unity (i.e., zero like-spin bonds). Identification of Cr-Cr bonds as the dominant neighbor interaction fulfills the purpose of the simple structural model. However, the large variation in atomic moments obtained from supercells with the same value of ${{\alpha}_{\text{CrCr}}}$ implies that other types of interactions are also relevant. Indeed, for configurations with the same ${{\alpha}_{\text{CrCr}}}$ there exists a strong correlation between moment and energy: those with lower net magnetizations, which arise from local moments closer to integer ${\rm{\mu_B}}$ values ($\pm2$ for Cr, 1 for Co), are also more energetically favorable.
![\[fig:moments\] Atomic moments of Co and Cr atoms from configurations minimizing adjacent Cr in the simple structural model (${{\alpha}_{\text{CrCr}}}= 0.5$). Co and Cr moments are plotted against the cumulative moments of their nearest and second-nearest neighbor Cr atoms, respectively. ](moments2.pdf){width="\linewidth" height="7in"}
After analyzing many aspects of these configurations, we have identified two additional ordering factors clearly affecting energy and magnetization. First, Co is found to favor moments that are antiparallel to those of neighboring Cr atoms, as shown in Fig. \[fig:moments\] considering the the ${{\alpha}_{\text{CrCr}}}= 0.5$ configurations. While Ni atoms possess negligible local moments under all degrees of order, most Co align ferromagnetically, the direction of which will define a reference spin “up" state to which Cr atoms are either aligned (${\text{Cr}_{\uparrow}}$) or opposed (${\text{Cr}_{\downarrow}}$). The preferred anti-alignment of Cr and Co moments is reflected in Figs. \[fig:frustration\] and \[fig:moments\] (as well as Tamm *et al.*), where ${\text{Cr}_{\downarrow}}$ outnumber ${\text{Cr}_{\uparrow}}$ 3:1. There exists a simple explanation for this ratio: $x_{{\text{Cr}_{\downarrow}}} = \frac{1}{4}$ is maximum possible concentration that can exist on an *fcc* lattice with no nearest neighbors, while $x_{{\text{Cr}_{\uparrow}}} = \frac{1}{12}$ represent the remaining Cr. The antiferromagnetic alignment of Cr-Cr and Cr-Co pairs is consistent with Anderson-type theory of magnetic interactions, as is the suppression of Ni moment in the vicinity of these elements [@moriya1965].
The second ordering principle is that the magnitude of a ${\text{Cr}_{\downarrow}}$ atom’s moment correlates strongly with the sum of the local moments of its Cr second-nearest neighbors, as also shown in Fig. \[fig:moments\]. In particular, these data indicate that second-nearest neighbor sublattices of ${\text{Cr}_{\downarrow}}$ (i.e., ${\text{Cr}_{\downarrow}}$ with six ${\text{Cr}_{\downarrow}}$ second-nearest neighbors) consistently display -2 ${\rm{\mu_B}}$ local moments. Furthermore, configurations containing the most ${\text{Cr}_{\downarrow}}$-${\text{Cr}_{\downarrow}}$ second-nearest neighbors possess the lowest ${\text{Cr}_{\downarrow}}$ moments, overall magnetization, and formation energy. This interaction may be mediated by some form of indirect exchange through Ni or Co atoms, although identification of the exact mechanism would require analysis beyond the scope of the present work. Interestingly, the magnitudes of Cr$_\uparrow$ local moments follow no such trend.
From the above results, we develop the hypothesis that configurations giving rise to the aforementioned near-integer ${\rm{\mu_B}}$ local magnetic moments are lower in energy, a simple principle we use to motivate a further set of configurations that we will denote as “spin-ordered." Specifically, we prescribe moments to maximize the alignment of second-nearest neighbor ${\text{Cr}_{\downarrow}}$ atoms so that they form a simple cubic sublattice, as depicted in Fig. \[fig:frustration\](c). These represent $\frac{3}{4}$ of Cr; the remaining $\frac{1}{4}$ Cr$_\uparrow$ atoms are randomly distributed while avoiding nearest neighbors. Ni and Co are assigned to remaining sites to minimize unfavorable Co-${\text{Cr}_{\uparrow}}$ pairs. The average formation energy and net moment for twenty of these supercells are $38.9 \pm 2.0$ meV/atom and $0.015 \pm 0.01$ ${\rm{\mu_B}}$/atom, respectively. These values, included in Fig. \[fig:energies\], are not only substantially lower than in the simple structural model, but also display minimal spread. Although we do not necessarily claim this precise ordering state is present in any material, we believe it captures the physics driving local chemical rearrangement in the nominally disordered CrCoNi system. Interestingly, these ordering principles appear similarly applicable to hexagonal close-packed lattices, which DFT consistently predicts to be lower energy for all degrees of order.
![\[fig:compositions\] Zero-field and temperature magnetization predictions for CrCoNi calculated with several different approaches, compared to 5K experimental measurements [@sales2016] for a range of Cr concentrations where $x_{\text{Co}} = x_{\text{Ni}} = (1 - {x_{\text{Cr}}})/2$. Supercell data represent the average of five distinct configurations. ](compositions.pdf){width="\linewidth" height="5in"}
We turn next to comparing how computations accounting for SRO compare to the low-temperature magnetic measurements reported in Refs. [@sales2016; @sales2017]. In these experimental studies, zero-field magnetization was measured at 5K for samples of *fcc* Cr$_x$Co$_{(1-x)/2}$Ni$_{(1-x)/2}$ with $0.2 \leq x \leq 0.355$. Values from Ref. [@sales2016] are reproduced in Fig. 4, alongside computational results they obtained using the KKR-CPA method [@khan2016], which assumes complete compositional disorder. The KKR-CPA results agree well with our own calculations for quasirandom configurations, as shown in Fig. 4, but both of these methods predict net moments significantly larger than experimental measurements. This discrepancy was discussed in Ref. [@sales2017] as reflecting effects not captured by DFT, but we propose that the presence of chemical and magnetic SRO in the experimental samples provides an alternative explanation, as explored below.
Included in Fig. 4 are two further sets of computational results, each representing a model of SRO. The first dataset (gold squares) extends the simple structural model of maximally eliminating Cr nearest neighbors to additional compositions. In a similar manner, the second model (magenta diamonds) applies the chemical and magnetic ordering of the spin-ordered state. These results indicate that, if the experimental samples contained SRO following, to varying extents, the principles described above, then the magnetic measurements of Ref. [@sales2016] are largely consistent with DFT. Of course, this intepretation requires further experimental analysis. Refs. [@sales2016; @sales2017] report no annealing that would specifically induce SRO, but neither do they describe rapid quenching that would indisputably suppress any local ordering. We note that Ref. [@sales2017] includes high-resolution electron microscopy results that rule out appreciable local clustering or compositional heterogeneity in the samples, but we do not believe they eliminate the possibility of atomic-scale chemical SRO.
The identification of SRO in CrCoNi as a predominantly magnetic phenomenon leads to several significant implications. Of immediate concern is the modeling of the system through classical interatomic potentials, which typically omit explicit magnetic interactions. For example, our results confirm magnetism as the reason that a recently developed embedded-atom method potential, a purely chemical formalism, predicts a hierarchy of neighbor interactions [@li2019a] differing from the DFT-MC studies represented in Table \[table:WC\]. Additionally, we emphasize the importance of carefully characterizing an alloy’s degree of order for both theoretical predictions and experimental interpretations concerning CrCoNi or related materials.
More broadly, our results imply a generally underappreciated role of magnetism in the properties of alloys containing multiple principal 3$d$ elements, supporting an emerging trend connecting the conventionally disparate domains of structural and magnetic properties [@niu2015; @niu2018; @wu2020]. For instance, the energy cost associated with Cr agglomeration should hinder the formation of a Cr-rich *bcc* (DFT) ground state via a martensitic mechanism, perhaps buoying the alloy’s metastability. We further postulate that the ability of SRO to minimize frustration among Cr atoms could be determining the *fcc* CrCoNi solution’s compositional range of stability. Indeed, the above results raise many questions; e.g., can the energetics of stacking faults in CrCoNi be connected to the presence of like-spin Cr bonds to the same degree as we showed for bulk energy? As a potential corollary, will Cr spins reorient as their local environment is changed by the transmission of dislocations—and how will the dynamics of this process affect slip properties? Given the more prevalent role of magnetism at low temperatures, we wonder if it could be connected to the material’s superior mechanical performance at cryogenic temperatures.
Nonetheless, we believe we have identified the magnetic interactions dictating chemical ordering behavior in *fcc* CrCoNi. In summary, this is primarily the unfavorability of like-spin Cr-Cr bonds, leading to frustration in a random solid solution. Complementarily, Cr opposing Co prefer to maximize their second-nearest neighbor alignment, which increases their local moments to negate those of opposite-spin Cr and Co, matching experimental measurements of minimal net magnetization. Together with a reduction in magnetically aligned Co-Cr pairs, these principles lead to a “spin-ordered" state we predict to further reduce energy while remaining largely disordered in chemical terms. This investigation highlights not only the dominant role of magnetism in the SRO of a representative MPEA, but also the need to analyze the ordering of these materials in spin-polarized rather than purely chemical terms.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract No. DE-AC02-05CH11231 within the Damage-Tolerance in Structural Materials (KC13) program. Simulations were performed using resources provided by the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under the same contract number. FW was supported by the Department of Defense through the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program and credits Anirudh R. Natarajan and Anton van der Ven for spurring his interest in the magnetic interactions of this system.
|
---
abstract: 'Studying fundamental symmetries of Nature has proven fruitful in particle physics. I argue that recent results at the LHC, and the naturalness problem highlighted by them, provide a renewed motivation for tests of CPT symmetry as a probe for physics beyond quantum field theory. I also discuss prospects for antihydrogen CPT tests with sensitivities to Planck scale suppressed effects.'
address: |
TRIUMF National Laboratory for Particle and Nuclear Physics\
Vancouver, British Columbia, V6T 2A3, Canada\
and\
Department of Physics and Astronomy, University of Calgary\
Calgary, Alberta, T2N 1N4, Canada\
E-mail: Makoto.Fujiwara@triumf.ca
author:
- 'MAKOTO C. FUJIWARA'
title: 'ANTIHYDROGEN, CPT, AND NATURALNESS[^1] '
---
Introduction
============
It is clear that testing CPT invariance at highest possible precision is a worthwhile effort, given its fundamental importance in modern physics.[@SME] Comparisons of antihydrogen atoms ($\rm {\overline{H}}$) with their well-studied matter counterpart, atomic hydrogen ($\rm {H}$), could provide competitive tests of CPT. (See, e.g., Ref. ). In this short paper, I will try to put $\rm {\overline{H}}$ studies in the larger context of current particle physics, and argue that the recent LHC results provide enhanced motivations for symmetry tests with $\rm {\overline{H}}$.
Naturalness and CPT
===================
Before going into antihydrogen, let us digress and ask a rather basic question: what is particle physics? According to Grossman[@Grossman], particle physics is about asking a simple question: $$\mathcal{L} = ?
\label{eq1}$$ To this simple question, we seem to have a simple answer: the Standard Model (SM), including the recently discovered final piece, a Higgs boson. However, as is well known, the SM has a number of open issues, not least of which is the (technical) naturalness problem. Loosely speaking, the naturalness (also called the hierarchy, or the fine-tuning) problem in this context refers to the lightness of the observed Higgs mass compared to the Planck (or the grand unification) scale, which implies fine tuning of the Higgs parameters by some $O(30)$ within the SM.
In the past decades, the issue of naturalness has been a central guiding principle in particle physics. A number of solutions to the problem have been proposed and studied extensively, the most popular scenario being supersymmetry. These solutions usually require that new phenomena appear at the energy scale near the electroweak scale, leading to the expectation that we would observe new physics beyond the SM at the LHC. The lack of such observations thus far has ruled out many of the most attractive new physics scenarios, and it seems to be putting many particle physicists in the state of ‘soul searching.’ Of course, there is still room for discoveries as the LHC energy is increased, which may solve the naturalness problem, but most of the surviving scenarios appear rather contrived. In addition to the light Higgs mass, the cosmological constant presents an even greater challenge to our belief in naturalness, naively requiring a fine-tuned cancelation at the $O(120)$ level.
This apparently disparate situation has lead to the increasing popularity of the anthropic principle, which states certain parameters in physics are fine tuned (possibly in a landscape of the ‘multiverse’) to allow the existence of the observer. Before accepting this controversial (but logically possible) option, however, I wish to step back and ask: are we asking the right question? When the answer we get (i.e., incredible degrees of fine tuning) does not make sense, it is possible that we are asking a wrong question.
In fact, implicit in the question in Eq. [(\[eq1\])]{} is the validity of quantum field theory (QFT), at least as the low energy effective description of Nature. However, it is the effective QFT framework itself that gives rise to the fine-tuning problems in the first place. The question we should be asking may be about the validity of the framework itself, rather than the ingredients in it. The possibility of such an option, although admittedly speculative, motivates putting the QFT framework to stringent experimental tests.
How does one test the validity of QFT? One possibility is to improve the precision of measurements of physical quantities that can be predicted precisely. The electron $g-2$ factor is a leading candidate. However, as a test of QED, it is currently limited by independent knowledge of the fine structure constant. Another approach is to search for a violation of symmetries guaranteed in QFT, such as CPT. $\rm {\overline{H}}$-$\rm {H}$ comparisons fall into the latter category. By directly comparing matter and antimatter, we would be free from the uncertainties due to theory or the constants.
Search for violation of CPT and/or Lorentz invariance has been the subject of considerable recent activities. A general effective field theory framework by Kostelecký , known as the Standard-Model Extension (SME), has provided an extremely powerful framework in which to test these symmetries.[@SME] It has been well known that the CPT theorem is guaranteed in QFT under general assumptions including Lorentz invariance, locality, and unitarity. A notable finding in the past decade is that within the SME (and presumably in any local QFT), CPT violation is always accompanied by Lorentz violation. This implies that limits on Lorentz violation in matter-only experiments also provide constraints on CPT violation. Given that extremely sensitive limits of the SME parameters are being placed via matter-only experiments[@SME], is there need for antimatter experiments?
Matter-antimatter comparison experiments confront the entire framework of (effective) QFT by testing CPT-odd, but Lorentz-even interactions, which are forbidden in local QFT. In other words, any violation of CPT, e.g., in comparison of $\rm {\overline{H}}$ and $\rm {H}$ atomic spectra, would indicate physics beyond QFT, forcing a fundamental change in our understanding of Nature.
Of course, it is by no means guaranteed that any potential modifications of QFT would lead to an observable CPT violating effect in $\rm {\overline{H}}$ experiments. However, antihydrogen should serve at least as a ‘lamp post’ case, because of the potentially very high sensitivities it could offer.[@AIP]
Since the CPT’13 meeting, I learned that there are in fact attempts to explain the fine-tuning problem using (mildly) nonlocal theory beyond the conventional QFT, e.g., by considering effects of wormholes in the multiverse.[@non-local] Whether this class of models would have observable implications at low energies is an open question.
Antihydrogen experiments at the CERN AD
=======================================
Several experiments are ongoing or under construction at the CERN Antiproton Decelerator, with the goal of testing fundamental symmetries with $\rm {\overline{H}}$. Here, I briefly discuss one of the experiments, ALPHA (Antihydrogen Laser PHysics Apparatus). Since the CPT’10 meeting, ALPHA has made significant progress. Indications of trapped $\rm {\overline{H}}$, which I reported at CPT’10,[@mcf] has now been unambiguously confirmed.[@trapping] The confinement times have been extended to as long as 1000 seconds[@longtime]. First proof-of-principle demonstration of spectroscopic measurement on $\rm {\overline{H}}$ has been performed by driving hyperfine transitions via microwaves[@microwaves]. An entirely new trap (ALPHA-2) has been constructed and is being commissioned to allow laser and improved microwave spectroscopy.
Laser spectroscopy of the 1s-2s level is the golden mode for a CPT test with $\rm {\overline{H}}$, since the same transition in $\rm {H}$ is measured to parts in 10$^{-15}$. Hyperfine spectroscopy offers a complementary test.[@Widmann; @AIP] In this respect, recent developments in measurements of the bare antiproton $g$-factor are encouraging. While the latter probes the long-distance magnetic property of the antiproton, $\rm {\overline{H}}$ hyperfine splitting can probe the antiproton’s internal structure via the contact interaction of the positron and the antiproton.[@Widmann] Recall that while the proton electric charge is very well known, there is a puzzle in its charge distribution. The gravitational interaction of antimatter is another increasingly active subject. See, e.g., Ref. .
A benchmark for CPT tests may be the sensitivity to Planck scale suppressed effects, $\Delta E \sim m_{\rm proton}^2/M_{\rm Planck} \sim 10^{-18}$ GeV.[@AIP] In frequency units, this corresponds to the precision of $\Delta f \sim 100$ kHz. This is within the reach of current antihydrogen experiments.
Acknowledgments {#acknowledgments .unnumbered}
===============
I wish to thank Alan Kostelecký and the organizing committee for a stimulating meeting. I thank the members of ALPHA for fruitful collaboration, Art Olin and Dave Gill for a critical reading of the manuscript. This work is supported in part by Canada’s NSERC, and TRIUMF.
[xx]{}
V.A. Kostelecký and N. Russell, 2013 edition, arXiv:0801.0287v6.
M.C. Fujiwara , AIP Conf. Proc. [**1037**]{}, 208 (2008), arXiv:0805.4082.
Y. Grossman, Introduction to flavor physics, arXiv:1006.3534.
D.L. Bennet, C.D. Froggatt, and H.B. Nielsen, arXiv:hep-ph/9504294; C.D. Froggatt and H.B. Nielsen, Phys. Lett. B [**368**]{}, 96 (1996); H. Kawai and T. Okada, Prog. Theor. Phys. [**127**]{}, 689 (2012).
M.C. Fujiwara , in V.A. Kostelecký, ed., [*Proceedings of the Fifth Meeting on CPT and Lorentz Symmetry,*]{} World Scientific, Singapore, 2010, arXiv:1104.4661.
G.B. Andresen , Nature [**468**]{}, 673 (2010).
G.B. Andresen , Nature Physics [**7**]{}, 558 (2011).
C. Amole , Nature [**483**]{}, 439 (2012).
E. Widmann , nucl-ex/0102002.
C. Amole , Nature Comm.[**4**]{}, 1785 (2013).
[^1]: Based on an invited talk at CPT’13 – the Sixth Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, June 17-21, 2013.
|
---
abstract: 'Superintegrable $d$ - dimensional quantum mechanical systems with spin, which admit a generalized Laplace-Runge-Lenz vector are presented. The systems with spins 0, $\frac12$ and 1 are considered in detail. All these systems are exactly solvable for arbitrary $d$, and their solutions are presented explicitly.'
---
[**Laplace-Runge-Lenz vector with spin in any dimension**]{} [^1]
.5cm
Introduction\[int\]
===================
Exactly solvable systems present perfect tools for understanding of grounds of quantum mechanics. A very attractive property of the majority of such systems is that they admit extended symmetries which are nice and important signs of their solvability.
Maybe the most important exactly solvable system is the Hydrogen atom. In addition to transparent invariance with respect to the rotation group SO(3), this system admits a hidden (Fock) symmetry with respect to group SO(4) [@Fock]. In other words, the Hydrogen atom is a superintegrable system. In addition, it admits a supersymmetry, since its potential is shape invariant with respect to a special Darboux transform [@khare].
There are numerous reasons to search for other superintegrable and supersymmetric quantum mechanical systems. In particular, that is a promissing way to discover new exactly solvable systems. Integrable and superintegrable systems for $d=2$ have been first classified in papers [@wint1] and [@BM]. 3d systems with second order integrals of motion are described in ([@ev1]) and ([@ev2]). A fresh survey of the classic and contemporary results in this field can be found in [@wint01].
Supersymmetric systems with scalar potentials belong to a well developed and in some aspects almost completed research field, see survey [@khare]. However, the recent discovery of an infinite family of new shape invariant systems generated by the exceptional polynomials [@quesne] opens new interesting ways in this area.
A much less studied field includes shape invariant systems with matrix potentials. A systematic search for such systems was started with papers [@N3] and [@N4], also particular examples of shape invariant matrix potentials where discussed in many papers, see, e.g., [@fu] and [@nb]. The completed list of additive shape invariant $2\times2$ matrix potentials is presented in paper [@N4].
Matrix potentials appears naturally in all problems including quantum mechanical particles with spin. Superintegrable systems with spin which include the spin-orbit interaction were studied in papers [@w6], [@w7] and [@w8].
But there is one more type of spin interaction which has perfect grounds in quantum mechanics. It is the Pauli interaction which can be represented, e.g., by the Stern-Gerlach term $~{\bf S}\cdot{\bf B}$ where $\bf S$ and ${\bf B}$ are a spin vector and a vector of magnetic field strength correspondingly.
A perfect example of an exactly solvable system with Pauli interaction, which is both supersymmetric and superintegrable, was discovered some time ago by Pron’ko and Stroganov [@Pron]. This system is effectively planar and simulates a neutral particle with a non-trivial magnetic or electric dipole momentum (e.g., neutron). Like the Hydrogen atom, the Pron’ko-Stroganov (PS) system admits a vector integral of motion which is a $2d$ quantum analogue of the Laplace-Runge-Lenz (LRL) vector. Moreover, this system describes a particle with spin $\frac12$. A relativistic analogue of the PS system was obtained in [@N10], the other new 2d and 3d superintegrable systems with spin $\frac12$ were discussed in [@N11].
Only over thirty years a generalization of the PS system to the case of arbitrary spin was obtained [@Pron2]. Like the initial PS model, the systems with arbitrary spin are superintegrable and supersymmetric. They have been integrated in [@N5] using shape invariant superpotentials classified in [@N3].
A new exactly solvable $3d$ system with Fock symmetry was discussed in [@N1]. That is a generalization of the PS system to the tree-dimensional case. The systems proposed in [@N1] are supersymmetric and admits a hidden symmetry with respect to group $SO(4)$. In other words, they keep all symmetries admitted by the Hydrogen atom, and so can be solved exactly [@N1]. But, in contrast with the non-relativistic Hydrogen atom, these systems include orbital particles with spin $\frac12$.
The next natural step was to search for $3d$ systems with arbitrary spin, which keep the hidden symmetry with respect to group $SO(4)$. Such systems were presented in paper [@N6]. In this way a generalization of the LRL vector for systems including orbital particles with arbitrary spin has been constructed.
In the present paper the discussion of superintegrable systems with spin is extended to the case of arbitrary dimensional space. More exactly, the $d$ dimensional systems which admit a LRL vector are presented. The systems with spin 0, $\frac12$ and 1 are considered in detail. All these systems appears to be exactly solvable for arbitrary $d$, and their solutions are presented explicitly.
Schrödinger equations in d-dimensional space
==============================================
General analysis
----------------
Consider a $d$-dimensional stationary Schrödinger equation with a matrix potential $$\begin{gathered}
H\Psi\equiv\left(\frac{p^2}{2m} +V({\bf x})\right)\Psi=E\Psi{\label}{se}\end{gathered}$$ where $p^2=p_1^2+p_2^2+...+p_d^2$, $p_1=-{{\mathrm i}}\frac{{\partial}}{{\partial}x_1}$, and $V$ is a matrix potential dependent on ${\bf x}=\left(x_1, x_2, ..., x_d\right)$.
We suppose Hamiltonian $H$ be invariant with respect to the rotation group SO($d$) whose generators can be chosen in the standard form: $$\begin{gathered}
J_{\mu\nu}=x_\mu p_\nu-x_\nu p_\mu+S_{\mu\nu}{\label}{Jmn}\end{gathered}$$ where indices $\mu$ and $\nu$ run over the values $1, 2, ..., d$, and $S_{\mu\nu}$ are matrices satisfying the familiar so($d$) commutation relations: $$\begin{gathered}
{\label}{Smn}[S_{\mu\nu},S_{\lambda\sigma}]=
{{\mathrm i}}(\delta_{\mu\lambda}S_{\nu\sigma}+\delta_{\nu\sigma}S_{\mu\lambda}-
\delta_{\mu\sigma}S_{\nu\lambda}-\delta_{\nu\lambda}
S_{\mu\sigma})\end{gathered}$$ where $\delta_{\mu\lambda}$ is the Kronecker symbol. By definition the matrix potential $V({\bf x})$ should commute with generators (\[Jmn\]): $$\begin{gathered}
=0.{\label}{c1}\end{gathered}$$
Let us search for such equations (\[se\]) which admit additional integrals of motion $K_\mu, \ \mu=1,2,...,d$ of the following generic form: $$\begin{gathered}
{\label}{lrl}K_\mu=\frac1{2m}\left(p_\nu J_{\mu\nu}+
J_{\mu\nu}p_\nu\right)+x_\mu V.\end{gathered}$$ By definition, $K_\mu$ should commute with $H$. This condition generates the following equations for potential: $$\begin{gathered}
{ x_\nu}\nabla_\nu V+V=0,{\label}{c2}\\
S_{\mu\nu}\nabla_\nu V+\nabla_\nu VS_{\mu\nu}
=0{\label}{c3}\end{gathered}$$ where $\nabla_\nu=\frac{{\partial}}{{\partial}{ x_\nu}}$ and summation from 1 to $d$ is imposed over the repeating index $\nu$.
Equations (\[c1\]), (\[c2\]) and (\[c3\]) present the necessary and sufficient conditions of the commutativity of hamiltonian $H$ with operators $J_{\mu\nu}$ and $K_\nu$. For $d=2$ and $d=3$ these equations can be reduced to the commutativity conditions obtained in [@Pron2] and [@N1] for two- and three-dimensional systems respectively.
If conditions (\[c1\]), (\[c2\]) and (\[c3\]) are fulfilled then operators $J_{\mu\nu}$ and $K_\mu$ satisfy the following relations: $$\begin{gathered}
{\label}{a2}[J_{\mu\nu}, H]=[ K_\mu, H]=0,
\\{\label}{a3}\begin{split}&[ K_\mu,J_{\nu\lambda}]=i(\delta_{\mu\lambda} K_\nu-\delta_{\mu\nu}\hat K_\lambda),\\&[ K_\mu, K_\nu]=-\frac{2{{\mathrm i}}}mJ_{\mu\nu}H,\end{split}\\
[J_{\mu\nu},J_{\lambda\sigma}]={{\mathrm i}}(\delta_{\mu\lambda}J_{\nu\sigma}+
\delta_{\nu\sigma}J_{\mu\lambda}-
\delta_{\mu\sigma}J_{\nu\lambda}-\delta_{\nu\lambda}J_{\mu\sigma}).{\label}{a4}\end{gathered}$$
If $ H$ is changed by its eigenvalue $E$ then relations (\[a3\]) and (\[a4\]) define a Lie algebra isomorphic to so($d+1$) if $E<0$ and to so($1,d$) for $E$ positive. In the special case $E=0$ we obtain the Lie algebra of the Euclidean group in $d$ dimensions. More exactly, for $E\neq0$ operators $J_{\mu\nu}$ and $$\begin{gathered}
{\label}{JJ}J_{d+1 \nu}=\sqrt{-\frac{m}{2E}}{ K}_\nu\end{gathered}$$ satisfy the following commutation relations: $$\begin{gathered}
{\label}{a44}[J_{\mu\nu},J_{\lambda\sigma}]={{\mathrm i}}(g_{\mu\lambda}J_{\nu\sigma}+
g_{\nu\sigma}J_{\mu\lambda}-
g_{\mu\sigma}J_{\nu\lambda}-g_{\nu\lambda}J_{\mu\sigma})\end{gathered}$$ where all subindices run over values 1, 2, ..., $d+1$ and non-zero entries of tensor $g_{\mu\nu}$ are $$g_{11}=g_{22}=...=g_{dd}=-\text{sign}(E)g_{d+1\ d+1}=-1.$$
In other words, all systems whose potentials satisfy conditions (\[c1\]), (\[c2\]) and (\[c3\]) admit a hidden symmetry of Fock type. The corresponding integral of motion (\[lrl\]) is an analogue of the LRL vector for $d-$dimensional space.
This hidden symmetry makes it possible to impose an additional condition on $\Psi$, which fixes eigenvalue $\omega$ of the second order Casimir operator $C_2=\frac12J_{\mu\nu}J_{\mu\nu}$: $$\begin{gathered}
{\label}{acon}\frac12J_{\mu\nu}J_{\mu\nu}\Psi=\omega\Psi, \quad \mu, \nu=1,2,...d+1\end{gathered}$$ Using (\[Jmn\]), (\[lrl\]) and (\[c2\]) this condition can be rewritten in the following form: $$\begin{gathered}
{\label}{AK}\begin{split}&\left(-\frac12S_{\mu\nu}S_{\mu\nu}p^2+
S_{\lambda\mu}S_{\lambda\nu}p_\mu p_\nu-\frac{ m}2\left(S_{\mu\nu}J_{\mu\nu}V+VS_{\mu\nu}J_{\mu\nu}
\right)+ m^2r^2V^2\right)\Psi\\&=-2mE\left(\frac{(d-1)^2}4+\omega\right)\Psi\end{split}\end{gathered}$$
Thus equation (\[se\]) can be supplemented by additional equation (\[AK\]) which should be compatible with (\[se\]). We will see that the compatibility condition for this system presents effective tools for solving the initial equation (\[se\]).
Scalar systems
--------------
Let us start with a very particular case when matrices $S_{\mu\nu}$ in (\[Jmn\]) are trivial. The corresponding system (\[se\]) is reduced to a single Schrödinger equation with a scalar potential. In accordance with (\[c1\]) this potential should be a rotational scalar, i.e., a function of $r^2=x_1^2 +x_2^2+...+x_d^2$. In addition, it has to satisfy one more condition, i.e., (\[c2\]) (since (\[c3\]) turns to identity). The general solution of (\[c2\]) for $V=V(r)$ is: $$\begin{gathered}
{\label}{kp}V=-\frac\alpha{r}\end{gathered}$$ where $\alpha$ is a constant. The corresponding Schrödinger equation (\[se\]) is superintegrable and admits a $d-$dimensional analogue of the LRL vector given by equation (\[lrl\]), where $$\begin{gathered}
{\label}{lmn}J_{\mu\nu}\to L_{\mu\nu}=x_\mu p_\nu-x_\nu p_\mu\end{gathered}$$ and $V$ is the $d$-dimensional Coulomb potential (\[kp\]).
Thus we recover a well known result [@Sud] concerning the generalization of the LRL vector in $d$ dimensions. Moreover, we also present a formal proof that the only scalar potential which is compatible with the $d$-dimensional LRL vector is the one given by equation (\[kp\]).
In the following we restrict ourselves to attractive potentials with $\alpha>0$.
Systems with spin $\frac12$
---------------------------
Consider now a more complicated case when matrices $S_{\mu\nu}$ in (\[Jmn\]) are non-trivial. The corresponding eigenvalue problem (\[se\]) includes a system of coupled Schrödinger equations in $d$ dimensional space.
Let us restrict ourselves to the cases when matrices $S_{\mu\nu}$ realize irreducible representation $D(\frac12,\frac12,...,\frac12)$ of algebra so($d$) for even $d$ or representation $D(\frac12,\frac12,...,-\frac12)$ for $d$ odd. Here the symbols in brackets are the Gelfand-Tsetlin numbers [@Gelf]. Making reduction of these representations on subalgebra so(3) we obtain a direct sum of representations D($\frac12$). Thus it is possible to interpret the corresponding equations (\[se\]) as a model of a particle with spin $\frac12$.
The considered matrices $S_{\mu\nu}$ admit the following uniform representation $$\begin{gathered}
{\label}{a45}S_{\mu\nu}=\frac14\left(\gamma_\mu\gamma_\nu-
\gamma_\nu\gamma_\mu\right)\end{gathered}$$ where $\gamma_\mu$ are basis elements of the Clifford algebra satisfying the following relations $$\begin{gathered}
{\label}{a5}\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu= 2\delta_{\mu\nu}.\end{gathered}$$
The dimension of irreducible matrices (\[a45\]) is equal to $2^{\left[\frac{d}2\right]}$ where $\left[\frac{d}2\right]$ is the entire part of $\frac{d}2$.
Let us search for potentials $V$ which satisfy relations (\[c1\]), (\[c2\]) and (\[c3\]) together with matrices (\[a45\]). The generic form of potential satisfying (\[c1\]) is given by the following equations: $$\begin{gathered}
{\label}{a6} V= f_1(r) +f_2(r)\gamma_\nu x_\nu\end{gathered}$$ for $d$ odd, and $$\begin{gathered}
{\label}{a7} V= f_3(r)+f_4(r)\gamma_\nu x_\nu+f_5(r)\gamma_{d+1}\gamma_\nu x_\nu\end{gathered}$$ for $d$ even.
Here $f_1(r), ..., f_5(r)$ are arbitrary functions. Condition (\[c2\]) specifies these functions: $f_\nu=\frac{\alpha_\nu}{r^2}$ where $\nu=1, 2, ..., 5$ and $\alpha_\nu$ are constants. The remaining condition (\[c3\]) reduces potentials (\[a6\]) and (\[a7\]) to the following unified form: $$\begin{gathered}
{\label}{a8}\hat V=\frac{\alpha}r\gamma_\nu n_\nu\end{gathered}$$ where $n_\nu=\frac{x_\nu}r$.
Thus we specify a $d$-dimensional system with spin $\frac12$ which is invariant with respect to group SO(d) and admits the generalized LRL vector. The corresponding hamiltonian (\[se\]) includes potential (\[a8\]), i.e., has the following form: $$\begin{gathered}
{\label}{a9} H=\frac{p^2}{2m}+\frac{\alpha}{r^2}\gamma_\nu x_\nu.\end{gathered}$$ For $d$=2 and $d=3$ this operator is equivalent to hamiltonians discussed in papers [@Pron] and [@N1] respectively.
The spectra of hamiltonians (\[a9\]) and solutions of the corresponding equations (\[se\]) are presented in section 3.
Systems with spin 1
-------------------
Consider a bosonic $d$-dimensional system admitting generalized LRL vector (\[lrl\]). We suppose the corresponding matrices $S_{\mu\nu}$ are irreducible and realize representation D(1,0,0,...0) of algebra so($d$), where the symbols in brackets are the Gelfand-Tsetlin numbers. Up to equivalence, their entries $\left(S_{\mu\nu}\right)_{ab}$ can be represented in the following form: $$\begin{gathered}
{\label}{s2}\left(S_{\mu\nu}\right)_{ab}={{\mathrm i}}(\delta_{\mu a}\delta_{\nu b}-\delta_{\nu a}\delta_{\mu b}),\quad \mu, \nu,\ a, b=1,2,...d.\end{gathered}$$ Being reduced to its subalgebra so(3), algebra of matrices (\[s2\]) is decomposed to a direct sum of representations $D(1)\oplus D(0)\oplus D(0)...$ which includes a spin-one (vector) representation $D(1)$ and $d-3$ scalar representations $D(0)$.
There are three basic scalars commuting with the corresponding total orbital momentum (\[Jmn\]): functions of module of the $d$-dimensional radius-vector $r$ multiplied by the unit matrix and such functions multiplied by the scalar matrix $S_{\mu\nu}x_\nu S_{\mu\lambda}x_\lambda$. Thus we can search for potentials of the following form: $$\begin{gathered}
{\label}{v3}V=\frac1r\left(c_1 +c_2\frac{S_{\mu\nu}x_\nu S_{\mu\lambda}x_\lambda}{r^2}\right)\end{gathered}$$ where $c_1$ and $c_2$ are constants.
Such potentials automatically satisfy conditions (\[c1\]) and (\[c2\]). Substituting (\[v3\]) and (\[s2\]) into (\[c3\]) we obtain the following solution: $$\begin{gathered}
{\label}{v4}V=\frac\alpha{(d-2)r}\left({(d-1)(d-4)}+ {2}S_{\mu\nu}n_\nu S_{\mu\lambda}n_\lambda\right),\quad d\neq2.\end{gathered}$$ Using realization (\[s2\]) for matrices $S_{\mu\nu}$ it is possible to find entries $V_{\mu\nu}$ of matrix potential (\[v4\]) in the following form: $$\begin{gathered}
V_{\mu\nu}=\frac\alpha{2r}\left({(d-3)}\delta_{\mu\nu}+2n_\mu n_\nu\right). {\label}{v5}\end{gathered}$$
It is interesting to note that, in contrast with representation (\[v4\]), formula (\[v5\]) is valid for the case $d=2$ also. Moreover, in the cases $d=2$ and $d=3$ potential (\[v5\]) is equivalent to potentials discussed in [@N5] and [@N1] respectively.
Potential (\[v5\]) admits one more matrix representation alternative to (\[v4\]). Indeed, using matrices which realize the representation D(1,0,0,...,0) of algebra so($d+1$), we can construct the following form: $$\begin{gathered}
{\label}{v6}\hat V =\frac\alpha{2r}\left(d-3+2(S_{d+1 \mu}n_\mu)^2\right)=\begin{pmatrix}\frac{(d-1)\alpha}{2r}&0\\0^\dag&V
\end{pmatrix}\end{gathered}$$where 0 is the $d\times 1$ zero matrix and $V$ is a $d\times d$ matrix whose entries are given by equation (\[v5\]). In other words, it is a direct sum of the Coulomb potential and potential (\[v5\]). For $d=2$ potential (\[v6\]) is equivalent to potential for spin 1 discussed in [@Pron2].
Hamiltonian (\[se\]) with potentials (\[v4\]) – (\[v6\]) commutes with the total orbital momentum (\[Jmn\]) and the LRL vector (\[lrl\]) where $S_{\mu\nu}$ are matrices (\[s2\]). It means that the corresponding Schrödinger equation (\[se\]) is possessed of Fock symmetry.
Exact solutions
===============
Solutions for scalar equations
------------------------------
Thanks to their rotation invariance all equations presented above admit solutions in separated variables. For scalar equations such solutions are well known and they will be a starting point in our discussion.
To separate variables in equation (\[se\]) with potential (\[kp\]) it is possible to use the hyper-spherical variables which are related to the Cartesian variables via the following relations: $$\begin{gathered}
{\label}{rv}\begin{split}&
x_d = r \cos \theta_{d-1},\\&
x_{d-1} = r \sin \theta_{d-1} \cos \theta_{d-2},\\&
x_{d-2} = r \sin \theta_{d-1} \sin \theta_{d-2} \cos \theta_{d-3},\\&
...\\&
x_2 = r \sin \theta_{d-1} \sin \theta_{d-2} . . . \sin \theta_{2} \cos \theta_{1},\\&
x_1 = r \sin \theta_{d-1} \sin \theta_{d-2} . . . \sin \theta_{2} \sin \theta_1.\end{split}\end{gathered}$$ In addition, wave function $\psi$ should be expressed via hyper-spherical harmonics $Y^l_\lambda$ $$\begin{gathered}
{\label}{hsh}\Psi=r^{\frac{1-d}2}
\psi_{l\lambda}Y^l_\lambda\end{gathered}$$ where $Y^l_\lambda$ satisfy the following condition: $$\begin{gathered}
{\label}{L2}\frac12L_{\mu\nu}L_{\mu\nu}Y^l_\lambda=
l(l+d-2)Y^l_\lambda,\end{gathered}$$ and $\lambda$ is the multiindex enumerating eigenvalues of other Casimir operators of algebra so($d$) whose generators $L_{\mu\nu}$ are given by equation (\[lmn\]).
Substituting (\[hsh\]) into (\[se\]) we come to the following equation for the radial wave function: $$\begin{gathered}
{\label}{re} H_\mu\psi_{l\lambda}(r)\equiv\left(-\frac{{\partial}^2}{{\partial}r^2}+\frac{\mu(\mu+1)}{r^2}+\hat V\right)\psi_{l\lambda}(r)=\epsilon\psi_{l\lambda}(r)\end{gathered}$$ where $\hat V=2mV=-\frac{2m\alpha}r$, $\epsilon=2mE$ and $$\begin{gathered}
\mu=l+\frac{d-3}2, \quad l=0, 1, 2, ...\end{gathered}$$
Notice that admissible values of $E$ can be found algebraically, without solving equation (\[re\]). Indeed, we have one more constraint, i.e., equation (\[AK\]). For trivial matrices $S_{\mu\nu}$ this equation is reduced to the algebraic condition $$\left(\frac12(d-1)^2mE+m^2\alpha^2\right)\Psi=-2mE\omega\Psi$$ provided equation (\[se\]) is satisfied. Thus eigenvalues $E$ can be expressed via the spectral parameter $\omega=n(n+d-1), \ n=0, 1, 2,...$ whose values can be found within the representation theory of group so($d+1$) [@Kir]. Thus $$\begin{gathered}
{\label}{spectr}E=-\frac{m\alpha^2}{2N^2}\end{gathered}$$ where $$\begin{gathered}
{\label}{N} N=n+l+\frac{d-1}2,\quad n=0, 1, 2, ...\end{gathered}$$ For $d=3$ equation (\[spectr\]) is reduced to the familiar Balmer formula.
Up to the meaning of quantum number $\mu$ equation (\[re\]) coincides with the radial equation for the 3$d$ Hydrogen atom. Its solutions can be found using tools of supersymmetric quantum mechanics. Indeed, hamiltonian $H_\mu$ can be factorized: $$\begin{gathered}
H_\mu=a^+_\mu a_\mu+c_\mu{\label}{fac}\end{gathered}$$ where $$\begin{gathered}
a_\mu=\frac{{\partial}}{{\partial}r}+W_\mu{\label}{amu}\end{gathered}$$ with $\quad W_\mu=
-\frac{m\alpha}{\mu+1}+\frac{\mu+1}r$ and $c_\mu=\frac{(m\alpha)^2}{(\mu+1)^2}.$ Moreover, the following intertwining relations are satisfied: $$\begin{gathered}
{\label}{ir}a^+_{\mu+1}H_{\mu+1}=H_\mu a_\mu^+.\end{gathered}$$
The ground state vector $\psi^0_\mu$ should solve the first order equation $a_\mu^+\psi^0_\mu=0$ and so has the following form $$\begin{gathered}
{\label}{gs}\psi^0_\mu=C_{0\mu}r^{\mu+1}\exp\left(\frac{-\alpha r m}{2(\mu+1)^2}\right).\end{gathered}$$ Then vectors of exited states $\psi^n_\mu$ and the corresponding eigenvalues $\epsilon_n$ are easily found using the following relation: $$\begin{gathered}
\label{es}\psi^n_{\mu}=
a_{\mu}^+a_{\mu+1}^+ \cdots
a_{\mu+n-1}^+\psi^0_{\mu+n,k},\quad\epsilon_n=-\frac{(m\alpha)^2}{4(\mu+n+1)^2}.\end{gathered}$$
In this way we obtain $$\begin{gathered}
{\label}{phi1}\psi^n_\mu=C_{n\mu} z^{\mu+1}\exp(-z){\cal F}(-n,2\mu+2,2z)\end{gathered}$$ where $\cal F(.,.,.)$ is the confluent hypergeometric function, $z=\frac{m\alpha r}{n+\mu+1}$. The corresponding energy levels are $E=-\frac{\epsilon_n}{2m}$, i.e., coincide with (\[spectr\]).
Solutions for spinor systems
----------------------------
Consider now equations (\[se\]) with matrix hamiltonian (\[a9\]). To separate variables it is sufficient to use hyper-spherical variables (\[rv\]) and expand solutions as: $$\begin{gathered}
{\label}{psi}\Psi({\bf r})=r^{\frac{d-1}2}
\psi_{j\varrho\lambda}(r)\Omega^j_{\varrho\lambda}\end{gathered}$$ where $\Omega^j_{\varrho\lambda}$ are hyper-spherical spinors satisfying the conditions $$\begin{gathered}
\begin{split}&\frac12J_{\mu\nu}J_{\mu\nu}
\Omega^j_{\varrho\lambda}=
\left(j(j+d-2)+\frac18(d-2)(d-3)\right)\Omega^j_{\varrho\lambda},
\\&D\Omega^j_{\varrho\lambda}\equiv\left(\frac12 \gamma_\mu\gamma_\nu L_{\mu\nu}+\frac{d-1}2\right)
\Omega^j_{\varrho\lambda}=\varrho\Omega^j_{\varrho\lambda}.\end{split}{\label}{casim}\end{gathered}$$ Here $j$ and $\varrho$ are quantum numbers which take the following values $$j=\frac12, \frac32, ...,\quad \varrho=\pm\left(j+\frac{d-2}2\right).$$
Substituting (\[a8\]) and (\[psi\]) into (\[se\]) we come to the following equation for radial wave functions (see Appendix A): $$\begin{gathered}
{\label}{req}H_\varrho\phi\equiv\left(-\frac{{\partial}^2}{{\partial}r^2}+\frac{\varrho^2+\sigma_3\varrho}{r^2}+
\frac\omega{r}\sigma_1\right)\phi=\varepsilon\phi,\qquad \phi=\begin{pmatrix}\psi_{j|\varrho|\lambda}(r)\\
\psi_{j-|\varrho|\lambda}(r)\end{pmatrix}\end{gathered}$$ where $\varepsilon=2mE,\ \omega=2m\alpha$, $\sigma_1$ and $\sigma_3$ are Pauli matrices.
Equation (\[req\]) includes effective potential $V_\varrho=\frac{\varrho^2+\sigma_3\varrho}{r^2}+
\frac\omega{r}\sigma_1$ which belongs to the list of shape invariant potentials presented in paper [@N3], see equation (5.11) for $\mu=\varrho-\frac12$ and $\kappa=\frac12$ therein. The general solutions of this equation for negative eigenvalues $E$ has been obtained in [@N3] using tools of SUSY quantum mechanics.
Let us apply the generic results presented in [@N3] to our particular system. First we note that Hamiltonian (\[req\]) can be factorized, i.e., represented in the form (\[req\]) where $$a_\varrho=-\frac{{\partial}}{{\partial}\varrho}+W_\varrho,\quad W_\varrho=\frac{\sigma_3-2\varrho-1}{2r}+\frac{\omega\sigma_1}{2\varrho+1}.$$ The ground state vector $\phi^0_\varrho$ is a solution of the first order equation $a\varrho^+\psi^0_\varrho=0$ whose explicit form is given by the following equation: $$\begin{gathered}
{\label}{bst}\phi^0_\varrho=Cy^{\varrho+1}\begin{pmatrix}
K_1(y)\\K_0(y)\end{pmatrix}\end{gathered}$$ where $K_0(y)$ and $K_1(y)$ are the modified Bessel functions, $y=\frac{\omega r}{2\varrho+1}$, and $C$ is the integration constant: $$\begin{gathered}
C=2^{-2\varrho}\left(G^{22}_{00}\left(1|^{-1,0}_{\varrho-\frac12,
\varrho+\frac12}\right)+G^{22}_{00}\left(1|^{0,0}_{\varrho+\frac12,
\varrho+\frac12}\right)\right)^{-1}\end{gathered}$$ with $G^{22}_{00}(.|^{..}_{..})$ being the Meijer functions.
The exited states vectors $\phi^n_\varrho$ are defined as $$\begin{gathered}
{\label}{bst2}\phi^n_{\varrho}=
a_{\varrho}^+a_{\varrho+1}^+ \cdots
a_{\varrho+n-1}^+\psi^0_{\varrho+n}\end{gathered}$$ while the corresponding eigenvalues $E$ are given by equation (\[spectr\]) where $$\begin{gathered}
{\label}{NN}N=\varrho+n=j+n+\frac{d-1}2,\quad n=0,1,2,...\end{gathered}$$
All vectors (\[bst\]) and (\[bst2\]) are square integrable and vanish at $r=0$. In particular cases $d=2$ and $d=3$ they are reduced to solutions discussed in [@Hau], [@N5] and [@N1].
Consider also the additional condition (\[AK\]) which fixes the eigenvalue of the second order Casimir operator of the hidden symmetry algebra so($d$+1). Substituting (\[a4\]) and (\[a8\]) into (\[AK\]) and using the found eigenvalues $E$ we come to following algebraic relation $$\begin{gathered}
2E(\omega+\frac{d(d-1)}8+\alpha^2m=0\end{gathered}$$ or $$\begin{gathered}
\omega={\hat j}({\hat j}+d-1)+\frac{(d-1)(d-2)}8{\label}{ome}\end{gathered}$$ where $\hat j=j+n=\frac12, \frac32,...$ Notice that eigenvalues (\[ome\]) which correspond to algebra so($d$+1) and eigenvalues in the first line of equation (\[casim\]) which correspond to algebra so($d$) are connected via the excepted relation $d\to d-1$.
Solutions for vector systems
----------------------------
Finally, let us discuss equation (\[se\]) with matrix potential whose entries are given by equation (\[v5\]). Solutions of this equation are $d$-component vectors $\Psi=$column$(\Psi_1, \Psi_2, ..., \Psi_d)$ which form the space of irreducible representation $D(1,0,0,...,0)$ of group SO($d$).
We will consider (\[se\]) together with the supplementary condition (\[AK\]). Using equation (\[se\]) and the following identities $$\begin{gathered}
\begin{split}&S_{\mu\nu}L_{\mu\nu}V+VS_{\mu\nu}L_{\mu\nu}=
\frac{\alpha}r\left((d-2)S_{\mu\nu}L_{\mu\nu}+1+
\frac{d(d-3)}2\right)+(d-2)V,\\
&\frac12S_{\mu\nu}S_{\mu\nu}=d-1,\quad S_{\lambda\mu}S_{\lambda\nu}p_\mu p_\nu=p^2+(d-2)P\end{split}\end{gathered}$$ where $P$ is a matrix with entries $P_{\mu\nu}=p_\mu p_\nu$, we reduce (\[AK\]) to the following form: $$\begin{gathered}
{\label}{AC}\begin{split}&\left(p_\mu p_\nu-\frac{\alpha m}x(L_{\mu \nu}+\delta_{\mu\nu}-n_\mu n_\nu)+m^2\alpha^2n_\mu n_\nu+\varepsilon\right)\Psi_\nu=0\end{split}\end{gathered}$$ where $$\begin{gathered}
{\label}{var}\varepsilon=\frac1{4(d-2)}\left(2mE
\left({(d-3)^2}+
4\omega\right)+{m^2\alpha^2(d-3)^2}\right). \end{gathered}$$
Equations (\[se\]) and (\[AC\]) where $V$ is potential (\[v5\]) admit solutions in separated variables. To separate variables we represent entries of the wave function as linear combinations of the following linearly independent terms: $$\begin{gathered}
{\label}{vh3}\Psi_\mu=\varphi^1_{l\lambda}(r)\Phi^1_\mu +\varphi^2_{l\lambda}(r)\Phi^2_\mu+\Phi^3_\mu\end{gathered}$$ where $$\begin{gathered}
{\label}{vh}\Psi^1_\mu=\frac{x_\mu}r Y^l_\lambda,\quad \Psi^2_\mu =r\nabla_\mu Y^l_\lambda,\end{gathered}$$ $Y^l_\lambda$ are hyper-spherical harmonics discussed in section 3.1, and $\Phi^3_\mu$ is a vector satisfying the following relations: $$\begin{gathered}
{\label}{vh2} x_\mu\Phi^3_\mu=0,\quad \quad \nabla_\mu\Phi^3_\mu=0.\end{gathered}$$
For $d=3$ vector $\Phi^3_\mu$ can be represented as $\Phi^3_\mu=\frac12\varepsilon_{\mu\nu\sigma}L_{\nu\sigma}
\varphi^3_{l\lambda}(r)Y^l_\lambda$ were $\varepsilon_{\mu\nu\sigma}$ is the Levi-Civita tensor. In this case $\Psi^1_\mu, \Psi^2_\mu$ and $\Psi^1_\mu$ are nothing but the familiar vector harmonics. We will not specify $\Phi^3_\mu$ for arbitrary $d$ since it will not be present in the final solutions.
Introducing hyper-spherical variables (\[rv\]) and substituting (\[v5\]) and (\[vh3\]) into (\[se\]) we recognize that the latter system is decoupled to two subsystems. One of them involves only $\Phi^3_\mu$ and has the following form: $$\begin{gathered}
{\label}{ph3}\left(p^2+\frac{\tilde\alpha}r\right)\Phi^3_\mu
=2mE\Phi^3_\mu\end{gathered}$$ where $\tilde\alpha=m(d-3)\alpha$. The other subsystem includes equations for radial functions $\varphi_1=\varphi_1^{l\lambda}$ and $\varphi_2=\varphi_2^{l\lambda}$ and can be written as: $$\begin{gathered}
{\label}{vph1}\begin{split}&-\varphi_1''-\frac{d-1}r\varphi_1'+
\frac1{r^2}((l(l+d-2)+d-1)\varphi_1-2l(l+d-2))\varphi_2+
\frac{m\alpha(d-1)}r\varphi_1\\&=2mE\varphi_1,\end{split}\\{\label}{vph2}
-\varphi_2''-\frac{d-1}r\varphi_1'+
\frac1{r^2}((l(l+d-2)-d+3)\varphi_2-2\varphi_1)+
\frac{m\alpha(d-3)}r\varphi_2=2mE\varphi_2\end{gathered}$$ where $\varphi_1'=\frac{{\partial}\varphi_1}{{\partial}r}$, etc.
The system (\[AC\]) can be decoupled too. Namely, substituting (\[vh3\]) into (\[AC\]) and using (\[vph1\]) we obtain: $$\begin{gathered}
\varepsilon\Phi_\mu=0,{\label}{AC3}\\{\label}{AC1}
l(l+d-2)(\varphi_1-(r\varphi_2)'+\alpha mr\varphi_2)-r^2(m^2\alpha^2+2mE+\varepsilon)\varphi_1=0,\\
(d-2){\varphi_1}+{(r\varphi_1)'}-l(l+d-2)
{\varphi_2}+{m\alpha}r\left(\varphi_1-
\frac{d-3}2\varphi_2\right)-r^2\varepsilon\varphi_2=0.
{\label}{AC2}\end{gathered}$$
Thus we have two separated systems of equations. The first of them describes functions $\Phi^3_\mu$ and includes equations (\[ph3\]) and (\[AC3\]). The other one involves equations (\[vph1\]), (\[vph2\]) and (\[AC1\]), (\[AC2\]) for variables $\varphi_1$ and $\varphi_2$.
Let us start with the latter equations, i.e., the first order system (\[AC1\]), (\[AC2\]) and the second order system (\[vph1\]), (\[vph2\]). The compatibility condition of these systems reads: $$\begin{gathered}
{\label}{CC}2mE+\varepsilon+\alpha^2m^2=0.
\end{gathered}$$ Indeed, differentiating equations (\[AC1\]) and (\[AC2\]) and expressing the second order derivatives in accordance with (\[vph1\]) and (\[vph2\]) we immediately come to (\[CC\]), otherwise $\varphi_1$ and $\varphi_2$ should be trivial.
Equations (\[CC\]) and (\[var\]) make it possible to express energy levels $E$ via eigenvalues $\omega$ of the second order Casimir operator of algebra so($d+1$): $$\begin{gathered}
{\label}{E1} E=-\frac{m\alpha^2(d-1)^2}{2(d-1)^2+8\omega}. \end{gathered}$$
The last term in the left hand side of equation (\[AC1\]) is equal to zero, therefore $$\begin{gathered}
{\label}{AC4}\varphi_1=(r\varphi_2)'-\alpha m r\varphi_2.\end{gathered}$$ Substituting that into (\[vph2\]) and setting $\varphi_2=r^{-\frac{d+1}2}\phi$ we obtain the following equation: $$\begin{gathered}
-\phi''+\left(\frac{\mu(\mu+1)}{r^2}-
\frac\kappa{r}\right)\phi=2mE\phi{\label}{se2}\end{gathered}$$ where $\kappa=\alpha(d-1)m$ and $\mu=l+\frac{d-3}2$. In other words, we again recognize equation (\[re\]) where, however, $\alpha\to\frac12(d-1)\alpha$. Thus the admissible eigenvalues $E$ and the corresponding solutions $\phi$ of (\[se2\]) are given by formulas (\[spectr\]) and (\[phi1\]) with $\alpha\to\frac12(d-1)\alpha$. The related functions $\varphi_1$ are easy calculated using equation (\[AC4\]). Thus $$\begin{gathered}
{\label}{spectra}
E=-\frac{m\alpha^2}{2k^2}\end{gathered}$$ where $k=\frac{2n+2l+d-1}{d-1}$.
Comparing energy levels (\[spectra\]) and (\[E1\]) we obtain the spectrum of the second order Casimir operator $C_2$: $$\omega=l'(l'+d-1) \quad \text{where} \quad l'=l+n=0, 1, 2, ...$$
The radial wave functions are: $$\begin{gathered}
{\label}{ei1}\begin{split}&\varphi_2=C_{ln}z^{l+d-1}\exp(-z){\cal F}(-n,l+d-1,2z),\\&\varphi_1=C_{ln}\left(z^{l+d-1}\exp(-z)\left((l+d+(k-1)z){\cal F}(-n,l+d-1,2z)\right.\right.\\&\left.\left.-\frac{2kz}{l+d-1}{\cal F}(-n+1,l+d,2z)\right)\right)\end{split}\end{gathered}$$ where $z=\frac{m\alpha r}{k}.$ They are normalizable and vanish at $z=0$.
Consider now equations (\[ph3\]) and (\[AC3\]). The first of them fixes admissible energy values in the following form (see definition (\[var\])): $$\begin{gathered}
{\label}{E2} E=-\frac{m\alpha^2(d-3)^2}{2(d-3)^2+8\omega}. \end{gathered}$$ This relation is incompatible with (\[E1\]). Thus solutions with non-trivial $\Psi_\mu^3$ correspond to trivial $\varphi_1$ and $\varphi_2$ and vise versa. It happens that in fact $\Psi_\mu^3$ should be trivial (see Appendix B) while $\varphi_1$ and $\varphi_2$ are presented by equations (\[ei1\]).
Discussion
==========
Thus we extend the field of superintegrable systems admitting LRL vector. Originally this vector (both classical and quantum mechanical) was specified for three dimensional space. Then it was generalized to the case of $d$ dimensions [@Sud], but it was done only for scalar QM systems.
The first example of LRL vector with spin was proposed apparently in paper [@Pron], where a 2$d$ superintegrable system was discovered. LRL vectors with spin in three dimensional space were discussed in [@N1] and [@N6].
In the present paper the results of papers [@N1] and [@N6] are extended to the case of arbitrary dimension. Effectively, the number of presented systems is infinite, but countable. All of them are exactly solvable, and this property is used to obtain their exact solutions in explicit forms.
We restrict ourselves to discussion of systems with spins 0, 1/2 and 1, which correspond to the trivial representation of algebra so($d$) and also irreducible representations $D(\frac12,\frac12,...,\frac12)$, $D(\frac12,\frac12,...,-\frac12)$ and $D(1,0, ...,0)$ of this algebra. Starting with other representations and using the determining equations (\[c1\]), (\[c2\]) and (\[c2\]) it is possible to construct additional superintegrable systems with spin, admitting the LRL vector. However, in contrast with the 2d and 3d cases [@N5], [@N6], it is seemed to be impossible to construct explicitly the generic model for arbitrary irreducible representation of algebra so($d$).
The $d$ dimensional systems with spin admitting LRL keep all basic properties of the systems with $d$=3. First, they are superintegrable. Their extended symmetries present effective tools for finding exact solutions. In particular, these symmetries make it possible to impose the additional condition (\[AK\]) on solutions of the Schrödinger equation. For systems with spin 0 and $\frac12$ this condition is reduced to algebraic equations for the Hamiltonians eigenvalues while for the case of spin one we have a system of first order equations (\[AC1\]) and (\[AC2\]).
The very existence of the first order system compatible with the initial (second order) equations can be interpreted as a conditional symmetry, see [@Rena] for exact definitions. But in contrast with the generic conditional symmetry which gives rise to existence of particular exact solutions, its particular case generated by the LRL vector makes it possible to find the [*general*]{} solution of equation (\[se\]).
The discussed systems with spin $\frac12$ are supersymmetric, i.e., the corresponding radial equations are shape invariant. Moreover, they belong to the list of shape invariant systems classified in [@N3] and [@N4]. Effectively, in the present paper a countable set of systems with shape invariant matrix potentials is constructed. As it was shown in Section 3.2, all these systems can be successfully solved using tools of SUSY quantum mechanics. The same is true for the scalar systems whose general solutions are presented in section 3.1.
The considered systems with spin 1 are neither supersymmetric nor shape invariant. However, they are superintegrable and exactly solvable, see section 3.3 for the explicit solutions. Supersymmetric systems in $d$ dimensions were discussed in paper [@Kir].
Some results of the present paper had been announced in the conference proceedings published in [@NNN].
Appendix A. Separation of variables in spinor equations
=======================================================
A standard way to separate variables in equation invariant w.r.t. algebra so($d$) is to expand solutions via eigenvectors of the complete set of $d-1$ commuting integrals of motion, which includes the Casimir operators of algebra so($d$) (whose generators are given by equations (\[Jmn\]) and (\[a4\])) and also operators $J_{12}, J_{34}, ...$ and change Cartesian coordinates to radial and hyper-spherical variables. Let present a more simple and straightforward way to obtain the radial equations discussed in section 3.2.
First we define the following operator $$\begin{gathered}
D=\frac12 \gamma_\mu\gamma_\nu L_{\mu\nu}+\frac{d-1}2.{\label}{di}\end{gathered}$$ Its important properties are:
- $D$ anticommutes with operators $\gamma_\mu p_\mu$ and $\gamma_\mu x_\mu$ $$\begin{gathered}
{\label}{ak}D\gamma_\mu p_\mu=-\gamma_\mu p_\mu D,\quad \gamma_\mu x_\mu=-\gamma_\mu x_\mu D;\end{gathered}$$
- In a space of eigenvectors of the second order Casimir operator $C_2=\frac12J_{\mu\nu}J_{\mu\nu}$ the square of $D_d$ is proportional to the unit matrix, since $$\begin{gathered}
{\label}{dsr}(D)^2=\frac12J_{\mu\nu}J_{\mu\nu}+
\frac18(d-1)(d-2);\end{gathered}$$ Moreover, eigenvalues of operators $D$ and $C_2$ are given by formulae (\[casim\]).
- If $d$ is even then multiplying $D$ by $\gamma_{d+1}=\frac1{d!}\varepsilon_{\mu_1\mu_2...\mu_d}\gamma_{\mu_1}\gamma_{\mu_2}... \gamma_{\mu_d}$ where $\varepsilon_{\mu_1\mu_2...\mu_d}$ is the absolutely antisymmetric unit tensor, we obtain an integral of motion for hamiltonian (\[a9\]). For $d=3$ operator $\gamma_{d+1}D$ is reduced to Dirac constant of motion for the relativistic Hydrogen atom.
To deduce the equation for radial functions we use the identities $$\begin{gathered}
{\label}{p2}p^2=\left(-{{\mathrm i}}\gamma_\mu {\partial}_\mu\right)^2,\quad (-{{\mathrm i}}\gamma_\nu n_\nu)^2=1, \quad \gamma_\mu\gamma_\nu=-\delta_{\mu\nu}-2{{\mathrm i}}S_{\mu\nu}\end{gathered}$$ and evaluate $\gamma_\mu {\partial}_\mu$ in the following way: $$\begin{gathered}
{\label}{gmpm}\begin{split}&\gamma_\mu {\partial}_\mu\equiv (-{{\mathrm i}}\gamma_\nu n_\nu)^2\gamma_\mu {\partial}_\mu=
-{{\mathrm i}}\gamma_\nu n_\nu\left(n_\mu p_\mu+{{\mathrm i}}\frac{d-1}{2r}-\frac{i}r D\right)\\&=-{{\mathrm i}}\gamma_\nu n_\nu\left(-i\frac{{\partial}}{{\partial}r}+{{\mathrm i}}\frac{d-1}{2r}-\frac{i}r D\right).\end{split}\end{gathered}$$ Substituting (\[gmpm\]) into (\[p2\]) and using (\[ak\]) we obtain the following convenient representation of the $d$ dimensional Laplace operator: $$\begin{gathered}
{\label}{la}p^2=-\Delta_d=-\frac{{\partial}^2}{{\partial}r^2}-\frac{d-1}r\frac{{\partial}}{{\partial}r}-\frac1{4r^2}(d-1)(d-3)+\frac1{r^2}D(D+1)\end{gathered}$$ which generates equation (\[req\]) for radial wave function.
Appendix B. Some calculation details for vector systems
=======================================================
Considering equations (\[ph3\]) for wave functions $\Psi_\mu^3$ we conclude that this system is completely decoupled. Moreover, for any fixed $\mu$ we have an equation which, up to the change $\alpha\to\tilde\alpha=\frac12(d-3)\alpha$, coincides with the scalar equation solved in section 3.1. Thus we know both eigenvectors (which will be discussed later) and the corresponding eigenvalues: $$\begin{gathered}
{\label}{E3}E=-\frac{m\alpha^2(d-3)^2}
{2\left(2l+2n+d-1\right)^2}.\end{gathered}$$
The compatibility condition for equations (\[E2\]) and (\[E3\]) can be written as $$\begin{gathered}
{\label}{om}\omega=\tilde l(\tilde l+\tilde d-2)\end{gathered}$$ where $\tilde l=l+n+1=1,2,..., \ \tilde d=d-1.$ Let us show that this relation is not realizable.
Equation (\[om\]) should define the spectrum of the second order Casimir operator of algebra so($d+1$). This algebra includes subalgebra so($d$) whose basis elements are given by equations (\[Jmn\]) and (\[s2\]). The spectrum of the corresponding second order Casimir operator (\[acon\]) with $\mu, \nu=1,2,...,d$ is given by the following relation: $$\begin{gathered}
{\label}{sp}\omega=j(j+d-2),\quad j=0, 1, 2, ...\end{gathered}$$ However, spectra (\[om\]) and (\[sp\]) are not compatible. More exactly, representations of subalgebra so($d$) which correspond to spectrum (\[sp\]) cannot be obtained by reduction of hypothetical representations of algebra so($d+1$) corresponding to spectrum (\[sp\]), since such reduction should be attended by [*decreasing*]{} of quantum number $d$, while in our case it [*increases*]{}. Thus eigenvalues (\[om\]) for the Casimir operator and the corresponding eigenvalues (\[E3\]) for the Hamiltonian are forbidden, and so vectors $\Phi^3_\mu$ in expansion (\[vh3\]) should be trivial.
[99]{} Fock V 1935 Zur Theorie des Wasserstoffatoms, [*Z. Phys.*]{} [**98**]{} 145
Cooper F Khare A and Sukhatme U 1995 Supersymmetry and Quantum Mechanics [*Physics Reports*]{} [**251**]{} 267-385 (arXiv:9405029v2.)
Winternitz P, Smorodinsky J, Uhlir M and Fris I 1966 Symmetry groups in classical and quantum mechanics, [*Yad. Fiz.*]{} [**4**]{} 625-35\
Winternitz P, Smorodinsky J, Uhlir M and Fris I 1967 Symmetry groups in classical and quantum mechanics [*Sov. J. Nucl. Phys.*]{} [**4**]{} 444-50 (Engl. transl.)
Makarov A, Smorodinsky J, Valiev Kh and Winternitz P 1967 A systematic search for non-relativistic systems with dynamical symmetries [*Nuovo Cim. A*]{} [**52**]{} 1061-1084
Evans N 1991 Group theory of the Smorodinsky-Winternitz system [*J. Math. Phys.*]{} [**32**]{} 3369–-3375
Evans N W 1990 Super-integrability of the Winternitz system [*Phys. Lett.*]{} [**147**]{} 483–486
Miller W, Jr, Post S and Winternitz P 2013 Classical and Quantum Superintegrability with Applications [*J. Phys. A: Math. Theor.*]{} [**46**]{} 423001 (arXiv:1309.2694)
Quesne C 2009 Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics [*SIGMA*]{} [**5**]{} 084
Nikitin A G and Karadzhov Yu 2011 Matrix superpotentials [*J. Phys. A: Math. Theor.*]{} [**44**]{} 305204 (arXiv:1101.4129)
Nikitin A G and Karadzhov Yu 2011 Enhanced classification of matrix superpotentials [*J. Phys. A: Math. Theor.*]{} [**44**]{} 445202 (arXiv:1107.2525)
Fukui T 1993 Shape-invariant potentials for systems with multi-component wave functions [*Phys. Lett. A*]{} [**178**]{} 1–6
Beckers J Debergh N and Nikitin A G 1992 More on supersymmetries of the Schrödinger equation, [*Modern Phys. Lett. A* ]{} [**7**]{} 1609–16
Winternitz P and Yurdusen I Integrable and superintegrable systems with spin 2006 [*J. Math. Phys.*]{} [**47**]{} 103509 (arXiv:0604050)
Winternitz P and Yurdusen I 2009 Integrable and superintegrable systems with spin in three-dimensional euclidean space [*J.Phys. A: Math. Theor.*]{}[**42**]{} 38523 (arXiv:0906.1021v2)
Désilets J -F, Winternitz P and Yurdusen I 2012 Superintegrable systems with spin and second-order integrals of motion [*Phys. A: Math. Theor.*]{} [**45**]{} 475201 (arXiv:1208.2886v1)
Pron’ko G P and Stroganov YG 1977 New example of quantum mechanical problem with hidden symmetry, [*Sov. J. JETP* ]{} [**45**]{} 1075–77
Ferraro E, Messina N and Nikitin A G 2010 Exactly solvable relativistic model with the anomalous interaction [*Phys. Rev. A*]{} [**81**]{} 042108 (arXiv:0909.5543)
Nikitin A G 2012 Integrability and supersymmetry of Schrodinger-Pauli equations for neutral particles [*J. Math. Phys.*]{} [**53**]{} 122103 (arXiv:1204.5902)
Pronko G P 2007 Quantum superintegrable systems for arbitrary spin [*J. Phys. A: Math. Theor.*]{} [**40**]{} 13331 (arXiv:0706.0310v2)
Nikitin A G Matrix superpotentials and superintegrable systems for arbitrary spin 2012 [*J. Phys. A: Math. Theor.*]{} [**45**]{} 225205 (arXiv:1201.4929)
Nikitin A G 2012 New exactly solvable systems with Fock symmetry [*J. Phys. A: Math. Theor.*]{} [**45**]{} 485204 (arXiv:1205.3094)
Nikitin A G 2013 Laplace-Runge-Lenz vector for arbitrary spin [*J. Math. Phys.*]{} [**54**]{} 123506 (arXiv:1308.4279)
Sudarshan E C G, Mukunda N and O’Raifeartaigh L 1965 Group theory of the Kepler problem [*Phys. Lett.*]{} [**19**]{} 322–25
Gel’fand I M Minlos A P and Shapiro Z Ja 1963 Representations of the Rotation and Lorentz Groups and their applications (Vol. 35. Oxford: Pergamon Press)
Kirchberg A Lange J D Pisani P A G and Wipf A 2003 Algebraic Solution of the Supersymmetric Hydrogen Atom in $d$ Dimensions [*Annals of Physics*]{} [**303**]{} 359–-88 (arXiv:0208228v2)
Hau L V, Golovchenko G A and Burns M M 1995 Supersymmetry and the binding of a magnetic atom to a filamentary current [*Phys. Rev. Lett.*]{} [**74**]{} 3138
Zhdanov R Z, Tsyfra I M and Popovych R O 1999 A precise definition of reduction of partial differential equations [*J. Math. Anal. Appl.*]{} [**238**]{} 101–23 (arxiv:0207023)
Nikitin A G Superintegrable systems with arbitrary spin (2013) [*Ukr. J. Phys.*]{} [**58**]{} 1046–1054
[^1]: E-mail: [nikitin@imath.kiev.ua]{}
|
---
abstract: 'We study a microscopic model of a thermocouple device with two connected correlated quantum wires driven by a constant electric field. In such isolated system we follow the time– and position–dependence of the entropy density using the concept of the reduced density matrix. At weak driving, the initial changes of the entropy at the junctions can be described by the linear Peltier response. At longer times the quasiequilibrium situation is reached with well defined local temperatures which increase due to an overall Joule heating. On the other hand, strong electric field induces nontrivial nonlinear thermoelectric response, e.g. the Bloch oscillations of the energy current. Moreover, we show for the doped Mott insulators that strong driving can reverse the Peltier effect.'
author:
- 'M. Mierzejewski'
- 'D. Crivelli'
- 'P. Prelovšek'
bibliography:
- 'bibliography.bib'
title: Peltier effect in strongly driven quantum wires
---
Significant progress has recently been achieved in understanding the properties of strongly driven quantum many–body systems. The physics beyond the linear response (LR) regime is interesting for basic research and important for future applications. The underlying phenomena have become accessible to novel experimental techniques like ultrafast time–resolved spectroscopy of solids [@matsuda1994; @dalconte12; @rettig12; @novelli12; @gadermaier10; @okamoto2010; @cortes2011; @kim12] or measurements of relaxation processes in ultracold atoms driven far from equilibrium. Most of theoretical studies on transport beyond LR focus on charge currents driven by strong electromagnetic fields [@oka2003; @jim2006; @hasegawa2007; @sugimoto2008; @takahashi2008; @my1; @my2; @my3; @lev2011; @lev2011_1; @eckstein2011; @aron2012; @amaricci11; @einhellinger12] or heat/spin transport in electric insulators subject to a large temperature gradient [@Tomaz2011]. The thermoelectric phenomena beyond LR while important for power generation or cooling applications remain mainly unexplored, except for the specific case of non–interacting particles[@prosen2013]. First efforts in filling this gap have recently been reported in [@leijnse; @kirchner] and [@sanchez2013] for quantum dots and mesoscopic systems, respectively.
A thermoelectric couple (TEC) is the circuit build out of two different wires and is the basic device for heat–to–current conversion or heat pumping. In this Letter we explore the behavior of a simple quantum model of an isolated TEC device connecting two wires of different materials with charge carriers being electrons and holes, respectively. In a closed circuit the either weak or strong electric field can we introduce via induction. We follow the real–time evolution of the TEC by solving the time–dependent Schrödinger equation. Since the system is isolated (decoupled from any thermal bath) the essential tool to investigate the local thermal properties is the concept of reduced density matrix (DM) of small subsystems. The latter allows to study how the entropy density increases/decreases in different parts of the TEC. It allows also to specify the limits of the local equilibrium (LoE) regime. Although the Joule heating is the dominating non–linear effect it does not immediately break the LoE. On the contrary, the time– and position–dependent temperature consistent with a canonical ensemble can be introduced also for moderate drivings far beyond the LR. We find that LoE persist up to much stronger fields, when the energy current starts to undergo the Bloch oscillations.
We choose as the simple model for TEC the one-dimensional (1D) ring with $L$ sites and spinless but interacting fermions where different materials are modeled by site-dependent local potentials $\varepsilon_i$. Steadily increasing magnetic flux $\phi(t)$ induces an electric field $F=-\dot{\phi}(t)/L$, as described by the time–dependent Hamiltonian $$\begin{aligned}
H(t)&=&-t_0 \sum_i \left\{ {\mathrm e}^{i \phi(t)/L}\; c^{\dagger}_{i+1}c_i +{\mathrm h.c.} \right\} + \sum_i
\varepsilon_i n_i \nonumber \\
& &+ V \sum_i \tilde{n}_i \tilde{n}_{i+1} + W \sum_i \tilde{n}_i \tilde n_{i+2},
\label{ham}\end{aligned}$$ where $n_i= c^{\dagger}_{i}c_i$ and $\tilde n_i=n_i -1/2$, $t_0$ is the hopping integral and periodic boundary conditions are used. $V$ and $W$ are repulsive interactions on nearest neighbors and next to nearest neighbors, respectively. The reason behind introducing $W$ is to stay away from the integrable case ($W=0$, $\varepsilon_i=\mathrm{const}$), which shows anomalous relaxation [@gge; @Eckstein2012; @Cassidy2011; @my3] and charge transport [@u2; @my1; @my2; @Tomaz2011; @Marko2011; @Sirker2009; @Robin2011]. We model different wires assuming a symmetric situation shown in Fig. \[fig1\]a, i.e. $ \varepsilon_i= -\varepsilon_0 $ and $\varepsilon_0$ for $i \in [1,L/2]$ and $i \in [L/2+1,L]$, respectively, while overall system is half-filled, i.e. the number of electrons $N_e=L/2$. Such a choice means that carriers in both wires are of opposite character, i.e. they are electrons and holes, respectively.
The dynamics of TEC is studied within a procedure described in Refs. [@my1; @my2]. Initially $F=0$ and we generate a microcanonical state $|\Psi(0) \rangle$ for the target energy $E_0= \langle \Psi(0)| H(0) |\Psi(0) \rangle$ and small energy uncertainty $\delta^2 E_0=\langle \Psi(0)| [H(0)-E_0]^2|\Psi(0) \rangle $. Then the driving is switched on and the time evolution $ |\Psi (0) \rangle \rightarrow |\Psi (t) \rangle$ is calculated by the Lanczos propagation method [@lantime] applied to small time intervals $(t,t+\delta t)$. We use units in which $\hbar=k_B=t_0=1$.
![(Color online) a): sketch of TEC. b): entropy difference between hot and cold junctions $\Delta S^{hc}$ for $\varepsilon_0=1.2$ calculated for $M$–site subsystems in comparison with Heikes Eq. (\[ds\]). c) and d) show $s_i(t)-s_i(0)$ for $F=0.2$ and $F=0.4$, respectively, for $M=4$, $\varepsilon_0=1.6$, with line denoting $s_i(t)=s_i(0)$. []{data-label="fig1"}](fig1){width="45.00000%"}
Since TEC is a composite object, its microscopic model may include several free parameters. With our choice of filling $N_e=L/2$ and $\varepsilon_i=\pm \varepsilon_0$ two parts of the TEC can be transformed to each other by a particle–hole transformation (see Fig. \[fig1\]). As a result, the concentration of fermions on one side of each junction is the same as the concentration of holes on other side $\langle n_{i} \rangle = 1 - \langle n_{L+1-i} \rangle$. It holds at any time provided that the same holds for $|\Psi(0)\rangle$. In an isolated TEC the chemical potential $\mu$ is irrelevant, still within the grand canonical ensemble the considered $N_e=L/2$ case would correspond to $\mu=0$.
The basic characteristics of the driven TEC are obtained from charge current $j^N_i =\langle J^N_i \rangle$ and energy current $j^E_i=\langle J^E_i \rangle$ defined from Eq. (\[ham\]) by the relations [@Naef97] $$\nabla J^N_i \equiv J^N_{i+1}-J^N_{i}=i[n_i,H], \quad
\nabla J^E_i = i[h_i,H],
\label{jne}$$ where $H= \sum_i h_i $. So defined currents fulfill the continuity relations [@my1] $$\frac{{\mathrm d} }{{\mathrm d} t} \langle n_i \rangle + \nabla j^N_i = 0 \label {c1}, \quad \quad
\frac{{\mathrm d} }{{\mathrm d} t} \langle h_i \rangle + \nabla j^E_i = F(t) j^N_i \label {c2}.$$ Due to the imposed particle–hole symmetry in the chosen model, the charge currents are the same on both sides of the junctions $j^N_i=j^N_{L+1-i}$, while the energy currents flow in the opposite directions $j^E_i=- j^E_{L+1-i}$ (see Fig. \[fig1\]a). The latter property implies that magnitude of $\nabla j^E_i$ is particularly large at the junctions, what is the essence of the Peltier heating or cooling. The energy density changes also due to the Joule heating, as represented by the source term on the rhs. of Eq. (\[c2\]). However, the heating is of the order of at least $F^2$ while $\nabla j^E_i \propto F$.
Since the initial state is a pure state with the corresponding DM $\rho(t=0)= |\Psi (0) \rangle \langle \Psi (0)|$ and the TEC is isolated from the surroundings, it stays in a pure state $|\Psi (t) \rangle \langle \Psi (t)|$ and the von Neumann entropy is identically zero. However, employing the concept of local reduced DM [@our2013] the entropy density can be obtained from DM of small subsystems of the TEC. For subsystems of $M$ consecutive lattice sites we calculate $\rho=\mathrm{Tr}_{L-M} |\Psi (t) \rangle \langle \Psi (t)|$ where the partial trace is taken over the remaining $L-M$ sites. Then, $S_i(t)=-\mathrm{Tr}_M(\rho \log \rho)$ is the local entropy and $s_i(t)=S_i(t)/M$ corresponding entropy density where $i$ labels the position of the subsystem within TEC. $s$ is thermodynamically relevant intensive quantity [@our2013; @rigolast] except for the low–energy regime where typically $s \propto M^{-1} $ according to the area laws [@Eisert2010]. Hence, we choose in this study the initial microcanonical states corresponding to high temperatures, i.e. initial $\beta(0)\simeq 0.3$. Furtheron we also set the size to largest available within our numerical approach, $L=26$.
In order to identify the hallmarks of LoE we focus on the weak–field regime. We consider metallic regime $V=1.4, W=1$ where the linear response functions are featureless [@u2]. Figs. \[fig1\]c and \[fig1\]d show $s_i(t)$ for the TEC driven by $F=$const. Major changes of $s_i(t)$ are clearly visible at the junctions, i.e. at $i=$13 and 26. For short times $t<10$, $s_{13}(t)$ strongly decreases (we dub it the cold junction) while $s_{26}(t)$ strongly increases (hot junction). Due to particle–hole symmetry, driving does not affect the average concentration of fermions in subsystems covering the junctions. Therefore, the change of the entropy at the junctions must be due to genuine heating/cooling. Further support for this interpretation follows from Fig. \[fig1\]b, which shows the difference of the total entropies of subsystems which cover the hot and the cold junctions. Initially, the results are independent of $M$, indicating that entropy is gained/lost mostly at the junctions consistently with Peltier heating $\dot{Q} = T \dot{S} =
2 \Pi j^N$. At high $T$ we can employ the Heikes formula for each wire $ \Pi \simeq -\mu \sim \pm \varepsilon_0 $. The estimate is then $$\Delta S^{hc} \equiv S^{\mathrm hot}(t)-S^{\mathrm cold}(t) \simeq 4 \beta(0) \int_0^t
{\mathrm d t'} \varepsilon_0 j^N(t'), \label{ds}$$ In the investigated regime the particle currents are determined by LR [@my1; @my2]. Hence, the rate of the entropy gain/loss at the junctions is roughly proportional to $F$ as it is as shown in Fig. \[fig2\]a, well consistent with Eq. (\[ds\]).
![(Color online) Results for $M=4$ and $\varepsilon_0=1.2$. Difference of the entropy-densities $\Delta s^{hc}$ is shown vs. a) $Ft$ and b) $F^2t$. c) shows $j^N$ and $j^E$ in the middle of the left wire for $F=0.2$; d) the same but for $F$ switched off at $t=15$. []{data-label="fig2"}](fig2){width="45.00000%"}
Next we discuss the long–time regime shown in Fig. \[fig2\]b. Here $\Delta s^{hc}(t)=\Delta S^{hc}(t)/M$ decays approximately as $\exp(-a F^2 t)$, where $a$ is independent of $F$. The same time–dependence has been found for particle current (see Fig. \[fig2\]c and Refs. [@my1; @my2]) and explained as a result of the Joule heating. It has also been recognized as a hallmark of the quasiequilibrium (QE) evolution when $\rho$ is determined only by the instantaneous energy[@our2013]. Contrary to the case of homogeneous systems [@my1; @my2; @our2013], the QE regime of TEC cannot be characterized by a single time–dependent $\beta(t)$.
An important property of the long time regime can be inferred from Fig. \[fig2\]c that shows $j^N$ and $j^E$ in the middle of the left part of TEC (far from the junctions). Initially, both currents show similar time–dependence, however $j^E$ vanishes for $t>10$ while $j^N$ remains large. In order to explain this result we recall that the in LoE regime both currents are driven by two independent forces: $F$ and $\nabla \beta$. A particular combination of these forces may cause vanishing of $j^N$ (Seebeck effect) or $j^E$ (present case). In order to explicitly show that vanishing of $j^E$ originates from compensation of two forces we instantaneously switch off one of them: the electric field. As shown in Fig. \[fig2\]d, the remaining force drives $j^E$ in the opposite direction. The magnitude of the resulting energy current is comparable with its values during the initial evolution under $F \ne 0$. Below we demonstrate that $\nabla \beta_i(t)$ is indeed the second driving force.
![(Color online) Results at $F=0.2$, $\varepsilon_0=1.2$ for: a) $\beta_{i}(t)$ for $i=6$ (away from junctions), b) $s_{i}(t)-s_{i}(0)$ for $i=6,M=4$ determined directly from $\rho$ and from $\int {\mathrm d} \epsilon_6 \beta_6 $, c) $\beta_i(t)$ for $t=10,20,30$, and d) $\langle n_i(t) \rangle $ for times $t=0$ and $t=30$ (points) compared with the HTE result (lines).[]{data-label="fig3"}](fig3){width="45.00000%"}
It has been shown for a driven homogeneous wire that $\rho$ is block–diagonal with respect to the number of particles in the subsystem. In the QE regime $\rho \propto \exp[ -\beta(t) H_{eff}]$ within each block [@our2013] and the spectrum $\{E_m \}$ of the effective Hamiltonian $H_{eff}$ is independent of $\beta$. Although for small subsystems $H_{eff}$ may significantly differ from $H$, one may still estimate $\beta(t)$ without specifying explicit form of $H_{eff}$. For the initial microcanonical state with known inverse temperature $\beta(0)$ we determine the eigenvalues $\tilde{\lambda}_{m}$ of the largest block of $\rho$. Then a similar spectrum $\lambda_m$ is determined for a driven system in a QE. Assuming the same $\{E_m\}$ one can then estimate $\beta(t)/\beta(0) =\log(\lambda_m/\lambda_1)/ \log(\tilde{\lambda}_m/\tilde{\lambda}_{1})$. Fig. \[fig3\]a shows the resulting $\beta_i(t)$ (averaged over $m \ne 1$) for the subsystem in the middle between hot and cold junctions. Being almost independent of $M$, $\beta$ is a well defined intensive quantity. Finally, we demonstrate that $\beta$ is consistent with the 2nd law of thermodynamics. In Fig. \[fig3\]b we compare $s_i(t)-s_i(0)$ determined directly from $\rho$ with the integral $\int_0^t {\mathrm d} \epsilon_i(t') \beta_i(t')$ where $\epsilon_i(t) =\langle h_i(t) \rangle$ is the energy density in the subsystem. Both quantities are very close to each other. Therefore, we conclude that in the QE regime one may introduce $\beta_i(t)$ consistent with the canonical ensemble as well as with equilibrium thermodynamics. This consistency breaks down only for subsystem covering one of the junctions. In fig. \[fig3\]c we show snapshots of the temperature profiles $T_i$ for various $t$ in the QE regime. The temperature gradient is clearly visible, however there exists also an asymmetry between the change of $T_i$ at hot and cold junctions due to the heating effects.
In an inhomogeneous system $j^N \neq 0$ causes a redistribution of particles within the TEC. This in turn may be another (in addition to $F$) driving force for the transport of particles. In the investigated TEC this effect should be insignificant at least within the QE regime because of the particle–hole symmetry. In order to confirm this expectation we plot in Fig. \[fig3\]d the spatial distribution of particles $\langle n_i (t) \rangle$ and compare it with the equilibrium high–temperature expansion (HTE), $n_i^{HTE}(t)= \bar{\varepsilon}_i \beta_i(t)/4 $, where $\varepsilon_i$ are averaged over all sites of the subsystem. Indeed, the changes of $\langle n_i (t) \rangle$ can be reasonably explained as originating only from the time–dependence of $\beta_i(t)$.
![(Color online) a) and b) show parametric plots $s_i(t)$ vs. $\epsilon_i(t)$ for cold and hot junctions, respectively. c) and d) show $\Delta s^{hc}(t)$ for various $V$ and $F$, respectively. Again, $M=4$ and $\varepsilon_0=1.2$.[]{data-label="fig4"}](fig4){width="45.00000%"}
Next we concentrate on nonequilibrium phenomena related with the operation of the TEC under strong $F$. The first one concerns the magnitude of $F$ which destroys the LoE. Since the TEC is spatially inhomogeneous LoE can be destroyed in certain parts of TEC while persisting in the other parts. In the LoE regime, intensive quantities including $s_i(t)$ and $\epsilon_i(t)$, are uniquely determined by $\beta_i(t)$. Such a universal relation is confirmed for $F \leq 0.4$ in Figs. \[fig4\]a (cold junction) and \[fig4\]b (hot junction). In the former case the curves for weak $F$ merge during the entire evolutions, while in the latter case it happens only in the long–time regime after the nonequilibrium transient. Results for $s_i(t)$ within the wires (not shown) are intermediate to the cases shown in Figs. \[fig4\]a and \[fig4\]b. Hence, one can observe that the LoE regime is broken first at the hot junction. For large $F$, $\epsilon_i(t)$ starts to oscillate, while oscillations of $s_i(t)$ are rather limited. Therefore, the equilibrium relation between $\epsilon_i$ and $s_i$ is broken when the [*energy current*]{} $j^E_i(t)$ starts to undergo the Bloch oscillations. It is indicative to compare this result with recent finding for the driven homogeneous systems [@my1; @eckstein2011] when the Bloch oscillations of the [*particle current* ]{} $j^N_i(t)$ mark the onset of the nonequilibrium evolution.
Finally we test the nonequilibrium response of TEC build out of two doped Mott insulators. Fig. \[fig4\]c shows the operation of TEC when the interaction $V$ is tuned from small (metallic) $V<2$ to large values $V \gg 2$ corresponding, close to half–filling, to lightly doped Mott insulators. Such tuning reverses the dc flow of entropy (at longer $t$) and effectively interchanges the role of junctions (hot and cold, respectively). This effect is not unexpected being the result of changing the charge carriers close to half–filling from electrons in metallic regime to holes in the Mott-insulating regime. In contrast, results in Fig. \[fig4\]d are even more surprising. One can see that under strong driving $F > 0.5$, the Mott-insulating TEC operates in the same way as expected for generic metals, i.e. the current is again carried by electrons. Breaking of the Mott insulator [*ground state*]{} by strong $F$ has intensively been investigated during the last decade [@oka2003; @oka2005; @hasegawa2007; @sugimoto2008; @takahashi2008; @eckstein2010; @zala2012; @zala2012a] and explained mostly as a kind the Landau–Zener transitions from the dispersionless ground state to a dispersionful excited state. However in the present case, the breakdown concerns a [*doped*]{} Mott insulator and involves only excited states with rather high energy, so a proper explanation remains a challenge.
In conclusion, we have studied a simple model of driven isolated TEC that can offer a useful and novel insight into several aspects of thermoelectric and nonequilibrium phenomena. Here, the concept of reduced (subsystem) DM is crucial for the discussion of increasing /decreasing entropy density, local temperature and local equilibrium. Starting with an equilibrium state, we have shown that the onset of driving field $F$ first leads to local Peltier heating/cooling at junctions according to LR theory. Concerning the long–time regime of weakly/moderately driven TEC the behavior can be dubbed as “local quasiequilibrium”. Similarly to the standard LoE one may introduce well defined $\beta_i(t)$. However, the changes of $\beta_i(t)$ originate not only from the energy and particle currents flowing within TEC, but also from the Joule heating due to external driving in analogy to QE in homogeneous systems [@my1] with homogeneous $\beta(t)$.
The presented method also allows to find the regions of evident departures from LoE. In the metallic regime of the model, strong $F$ leads to the breakdown of the relation between local temperature $T_i(t)$ and local energy $\epsilon_i(t)$ which is incompatible with the notion of LoE. Even more dramatic are the effects in the regime of doped Mott insulator where the charge carriers (within the equilibrium LR response) change the electron/hole character. Such systems are promising for the thermoelectric applications [@zlatic2013]. Here we find that large $F$ can even reverse the thermoelectric response.
Authors acknowledge stimulating discussions with Veljko Zlatič. This work has been carried out within the NCN project “Nonequilibrium dynamics of correlated quantum systems”. P.P. acknowledges the support by the Program P1-0044 and project J1-4244 of the Slovenian Research Agency.
|
.1.5cm
[Non-normal and Stochastic Amplification of Magnetic 0.1cm]{}
[Energy in the Turbulent Dynamo: Subcritical Case]{}
1.0cm
[Sergei Fedotov]{}$^{1}$
1.0cm
$^1$ Department of Mathematics, UMIST - University of Manchester
Institute of Science and Technology, Manchester, M60 1QD UK,
e-mail: Sergei.Fedotov@umist.ac.uk
Web-page: http://www.ma.umist.ac.uk/sf/index.html
0.5cm
Submitted to Phys. Rev. Lett.
1.0cm
**Abstract**0.3cm
Our attention focuses on the stochastic dynamo equation with non-normal operator that gives an insight into the role of stochastics and non-normality in galactic magnetic field generation. The main point of this Letter is a discussion of the generation of a large-scale magnetic field that cannot be explained by traditional linear eigenvalue analysis. We present a simple stochastic model for the thin-disk axisymmetric $\alpha
\Omega $ dynamo involving three factors: (a) non-normality generated by differential rotation, (b) nonlinearity reflecting how the magnetic field affects the turbulent dynamo coefficients, and (c) stochastic perturbations. We show that even for the *subcritical case,* there are three possible mechanisms for the generation of magnetic field. The first mechanism is a deterministic one that describes an interplay between transient growth and nonlinear saturation of the turbulent $\alpha -$effect and diffusivity. It turns out that the trivial state is nonlinearly unstable to small but finite initial perturbations. The second and third are stochastic mechanisms that account for the interaction of non-normal effect generated by differential rotation with random additive and multiplicative fluctuations. In particular, we show that in the *subcritical case* the average magnetic energy can grow exponentially with time due to the multiplicative noise associated with the $\alpha -$effect.
The generation and maintenance of large scale magnetic fields in stars and galaxies has attracted enormous attention in past years [@Mof]-[@RShS] (see also a recent review [@Widrow]). The main candidate to explain the process of conversion of the kinetic energy of turbulent flow into magnetic energy is the mean field dynamo theory [@KrR]. The standard dynamo equation for the large scale magnetic field $\mathbf{B}(t,%
\mathbf{x})$ reads $\partial \mathbf{B/}\partial t=$ curl$(\alpha \mathbf{B}%
)+\beta \Delta \mathbf{B}+$curl$(\mathbf{u}\times \mathbf{B}),$ where $%
\mathbf{u}\ $is the mean velocity field, $\alpha \ $is the coefficient of the $\alpha $-effect and $\beta $ is the turbulent magnetic diffusivity. This equation has been widely used for analyzing the generation of the large-scale magnetic field. Traditionally the mathematical procedure consists of looking for exponentially growing solutions of the dynamo equation with appropriate boundary conditions. While this approach has been quite successful in the prediction of large scale magnetic field generation, it fails to predict the *subcritical* onset of a large-scale magnetic field for some turbulent flow. Although the trivial solution $\mathbf{B}=0$ is linearly stable for the *subcritical* case, the non-normality of the linear operator in the dynamo equation for some turbulent flow configurations leads to the transient growth of initial perturbations [@nonnormal]. It turns out that the non-linear interactions and random fluctuations might amplify this transient growth further. Thus, instead of the generation of the large scale magnetic field being a consequence of the linear instability of trivial state $\mathbf{B}=0$, it results from the interaction of transient amplifications due to the non-normality with nonlinearities and stochastic perturbations. The importance of the transient growth of magnetic field for the induction equation has been discussed recently in [@FI1; @Proctor]. Comprehensive reviews of *subcritical* transition in hydrodynamics due to the non-normality of the linearized Navier-Stokes equation, and the resulting onset of shear flow turbulence, can be found in [@Grossmann; @SH].
The main purpose of this Letter is to study the non-normal and stochastic amplification of the magnetic field in galaxies. Our intention is to discuss the generation of the large-scale magnetic field that cannot be explained by traditional linear eigenvalue analysis. It is known that non-normal dynamical systems have an extraordinary sensitivity to stochastic perturbations that leads to great amplifications of the average energy of the dynamical system [@F0]. Although the literature discussing the mean field dynamo equation is massive, the effects of non-normality and random fluctuations are relatively unexplored. Several attempts have been made to understand the role of random fluctuations in magnetic field generation. The motivation was the observation of rich variability of large scale magnetic fields in stars and galaxies. Small scale fluctuations parameterized by stochastic forcing were the subject of recent research by Farrell and Ioannou [@FI1]. They examined the mechanism of stochastic field generation due to the transient growths for the induction equation. They did not use the standard closure involving $\alpha $ and $\ \beta $ parameterization. Hoyon with his colleagues has studied the effect of random alpha-fluctuations on the solution of the kinematic mean-field dynamo[@Hoyng]. However they did not discuss the non-normality of the dynamo equation and the possibility of stochastic transient growth of magnetic energy. Both attempts have involved only the linear stochastic theory. Numerical simulations of magnetoconvection equations with noise and non-normal transient growth have been performed in [@Proctor]
It is the purpose of this Letter to present a simple stochastic dynamo model for the thin-disk axisymmetric $\alpha \Omega $ dynamo involving three factors: non-normality, non-linearity and stochastic perturbations. Recently it has been found [@Fedotov] that the interactions of these factors leads to noise-induced phase transitions in a “toy” model mimicking a laminar-to-turbulent transition. In this Letter we discuss three possible mechanisms for the generation of a magnetic field that are not based on standard linear eigenvalue analysis of the dynamo equation. The first mechanism is a deterministic one that describes an interplay between linear transient growth and nonlinear saturation of both turbulent parameters: $%
\alpha $ and $\beta $. The second and third are stochastic mechanisms that account for the interaction of the non-normal effect generated by differential rotation with random additive and multiplicative fluctuations.
Here we study the nonnormality and stochastic perturbation effects on the growth of galactic magnetic field by using a Moss’s “no-z” model for galaxies [@Beck]. Despite its simplicity the “no-z” model proves to be very robust and gives reasonable results compared with real observations. We consider a thin turbulent disk of conducting fluid of uniform thickness $\
2h\ $and radius $\ R\ $($R\gg h$), which rotates with angular velocity $\
\Omega (r)$ [@ZRS; @RShS]. We consider the case of $\ \alpha \Omega -$dynamo for which the differential rotation dominates over the $\alpha $-effect. Neglecting the radial derivatives one can write the stochastic equations for the azimuthal, $B_{\varphi }\left( t\right) ,\ $and radial,$\
B_{r}\left( t\right) ,$ components of the axisymmetric magnetic field $$\frac{dB_{r}}{dt}=-\frac{\alpha (|\mathbf{B}|,\xi _{\alpha }(t))}{h}%
B_{\varphi }-\frac{\pi ^{2}\beta (|\mathbf{B}|)}{4h^{2}}B_{r}+\xi _{f}(t),$$ $$\frac{dB_{\varphi }}{dt}=gB_{r}-\frac{\pi ^{2}\beta (|\mathbf{B}|)}{4h^{2}}%
B_{\varphi },\ \label{governing}$$ where $\alpha (|\mathbf{B}|,\xi _{\alpha }(t))$ is the random non-linear function describing the $\alpha -$effect, $\beta (|\mathbf{B}|)$ is the turbulent magnetic diffusivity, $g=rd\Omega /dr\ $is the measure of differential rotation (usually $rd\Omega /dr<0).$
Nonlinearity of the functions $\alpha (|\mathbf{B}|,\xi _{\alpha }(t))$ and $%
\beta (|\mathbf{B}|)$ reflects how the growing magnetic field $\mathbf{B}$ affects the turbulent dynamo coefficients. This nonlinear stage of dynamo theory is a topic of great current interest, and, numerical simulations of the non-linear magneto-hydrodynamic equations are necessary to understand it. There is an uncertainty about how the dynamo coefficients are suppressed by the mean field and current theories seem to disagree about the exact form of this suppression [@backreaction]. Here we describe the dynamo saturation by using the simplified forms [@Widrow] $$\alpha (|\mathbf{B}|,\xi _{\alpha }(t)=(\alpha _{0}+\xi _{\alpha
}(t))\varphi _{\alpha }(|\mathbf{B}|),\;\;\;\beta (|\mathbf{B}|)=\beta
_{0}\varphi _{\beta }(|\mathbf{B}|), \label{nonlinear}$$ where $\varphi _{\alpha ,\beta }(|\mathbf{B}|)$ is a decaying function such that $\varphi _{\alpha ,\beta }(0)=1.$ In what follows we use [@Widrow] $$\varphi _{\alpha }(|\mathbf{B}|)=\left( 1+k_{\alpha }(B_{\varphi
}/B_{eq})^{2}\right) ^{-1},\;\;\;\varphi _{\beta }(|\mathbf{B}|)=\left( 1+%
\frac{k_{\beta }}{1+(B_{eq}/B_{\varphi })^{2}}\right) ^{-1},
\label{backreaction}$$ where $k_{\alpha }$ and $k_{\beta }$ are constants of order one, and $%
B_{eq}$ is the equipartition strength. It should be noted that for the $%
\alpha \Omega -$dynamo the azimuthal component $B_{\varphi }\left( t\right) $ is much larger$\ $than the radial field$\ B_{r}\left( t\right) ,$ therefore, $\mathbf{B}^{2}\simeq B_{\varphi }^{2}.$ We did not include the strong dependence of $\alpha $ and $\beta $ on the magnetic Reynolds number $R_{m}$. The back reaction of the magnetic field on the differential rotation is also ignored.
The multiplicative noise $\xi _{\alpha }(t)$ describes the effect of rapid random fluctuations of $\alpha .$ We assume that they are more important than the random fluctuations of the turbulent magnetic diffusivity $\beta $ [@Hoyng]. The additive noise $\xi _{f}(t)$ represents the stochastic forcing of unresolved scales [@FI1]. Both noises are independent Gaussian random processes with zero means $<\xi _{\alpha }(t)>=0,$ $<\xi
_{f}(t)>=0$ and correlations: $$<\xi _{\alpha }(t)\xi _{\alpha }(s)>=2D_{\alpha }\delta (t-s),\;\;\;<\xi
_{f}(t)\xi _{f}(s)>=2D_{f}\delta (t-s). \label{noise}$$ The intensity of the noises is measured by the parameters $D_{\alpha }$ and $%
D_{f}$. One can show [@Fedotov] that the additive noise in the second equation in (\[governing\]) is less important.
The governing equations (\[governing\]) can be nondimensionalized by using an equipartition field strength $B_{eq},$ a length $h$, and a time $\Omega
_{0}^{-1},$ where $\Omega _{0}$ is the typical value of angular velocity. By using the dimensionless parameters $$g\rightarrow -\Omega _{0}|g|,\;\;\;\delta =\frac{R_{\alpha }}{R_{\omega }}%
,\;\;\;\varepsilon =\frac{\pi ^{2}}{4R_{\omega }},\;\;\;R_{\alpha }=\frac{%
\alpha _{0}h}{\beta },\;\;\;R_{\omega }=\frac{\Omega _{0}h^{2}}{\beta },$$ we can write the stochastic dynamo equations in the form of SDE’s $$dB_{r}=-(\delta \varphi _{\alpha }(B_{\varphi })B_{\varphi }+\varepsilon
\varphi _{\beta }(B_{\varphi })B_{r})dt-\sqrt{2\sigma _{1}}\varphi _{\alpha
}(B_{\varphi })B_{\varphi }dW_{1}+\sqrt{2\sigma _{2}}dW_{2},$$ $$dB_{\varphi }=-(|g|B_{r}+\varepsilon \varphi _{\beta }(B_{\varphi
})B_{\varphi })dt\ , \label{basic}$$ where $W_{1}$ and $W_{2}$ are independent standard Wiener processes. The dynamical system (\[basic\]) is subjected to the multiplicative and additive noises with the corresponding intensities: $$\sigma _{1}=\frac{D_{\alpha }}{h^{2}\Omega _{0}},\;\;\;\sigma _{2}=\frac{%
D_{f}}{B_{eq}^{2}\Omega _{0}}. \label{intensity}$$ It is well-known that the presence of noise can dramatically change the properties of a dynamical system [@LH]. Since the differential rotation dominates over the $\alpha $-effect ($R_{\alpha }\ll |R_{w}|),$ the system (\[basic\]) involves two small parameters $\delta =$ $R_{\alpha }/R_{\omega
}$ and $\varepsilon =1/R_{\omega }$ whose typical values are $0.01-0.1$ ($%
R_{\omega }=10-100,$ $\ R_{\alpha }=0.1-1).$ These parameters play very important roles in what follows. For small values $\delta $ and $\varepsilon
$ , the linear operator in (\[basic\]) is a highly non-normal one $($ $%
|g|\sim 1).$ This can lead to a large transient growth of the azimuthal component $B_{\varphi }\left( t\right) $ in a *subcritical case.* We then expect a high sensitivity to stochastic perturbations. Similar deterministic low-dimensional models have been proposed to explain the *subcritical* transition in the Navier-Stokes equations (see, for example, [@Trefethen; @GS]). The main difference is that the nonlinear terms in (\[governing\]) are not energy conserving.
The probability density function $p(t,B_{r},B_{\varphi })$ obeys the Fokker-Planck equation associated with (\[basic\]) [@Gardiner] $$\frac{\partial p}{\partial t}=-\frac{\partial }{\partial B_{r}}\left[ \left(
\delta \varphi (B_{\varphi })B_{\varphi }+\varepsilon \varphi (B_{\varphi
})B_{r}\right) p\right] -\frac{\partial }{\partial B_{\varphi }}\left[
\left( |g|B_{r}+\varepsilon \varphi (B_{\varphi })B_{\varphi }\right) p%
\right] +$$ $$(\sigma _{1}\varphi ^{2}(B_{\varphi })B_{\varphi }^{2}+\sigma _{2})\frac{%
\partial ^{2}p}{\partial B_{r}^{2}}.\$$ Using this equation in the linear case one can find a closed system of ordinary differential equations for the moments $<B_{r}^{2}>,$ $%
<B_{r}B_{\varphi }>,$ and $\ <B_{\varphi }^{2}>$ $$\frac{d}{dt}\left(
\begin{array}{c}
<B_{r}^{2}> \\
<B_{r}B_{\varphi }> \\
<B_{\varphi }^{2}>
\end{array}
\right) =\left(
\begin{array}{ccc}
-2\varepsilon & -2\delta & \sigma _{1} \\
-|g| & -2\varepsilon & -\delta \\
0 & -2|g| & -2\varepsilon
\end{array}
\right) \left(
\begin{array}{c}
<B_{r}^{2}> \\
<B_{r}B_{\varphi }> \\
<B_{\varphi }^{2}>
\end{array}
\right) +\left(
\begin{array}{c}
\sigma _{2} \\
0 \\
0
\end{array}
\right) . \label{moments}$$ The linear system of equations (\[moments\]) allows us to determine the initial evolution of the average magnetic energy $E(t)=<B_{r}^{2}>+<B_{r}B_{%
\varphi }>+<B_{\varphi }^{2}>.$ Similar equations emerge in a variety of physical situations, such as models of stochastic parametric instability that explain why the linear oscillator subjected to multiplicative noise can be unstable [@BF].
Now we are in a position to discuss three possible scenarios for the *subcritical* generation of galactic magnetic field.
**Deterministic subcritical generation.** Let us examine the deterministic transient growth of the magnetic field in the *subcritical case.*To illustrate the non-normality effect consider first the linear case without noise terms. The dynamical system (\[basic\]) takes the form $$\frac{d}{dt}\left(
\begin{array}{c}
B_{r} \\
B_{\varphi }
\end{array}
\right) =\left(
\begin{array}{cc}
-\varepsilon & -\delta \\
-|g| & -\varepsilon
\end{array}
\right) \left(
\begin{array}{c}
B_{r} \\
B_{\varphi }
\end{array}
\right) . \label{linear}$$ Since $\delta <<1$, $\varepsilon <<1$ and $|g|\sim 1$, this system involves a highly non-normal matrix. Even in the* subcritical case (*$%
0<\delta <\varepsilon ^{2}/|g|$ see below*)* when all eigenvalues are negative, $B_{\varphi }$ exhibits a large degree of transient growth before the exponential decay. Assuming that $B_{r}(t)=e^{\gamma t}$and $B_{\varphi
}(0)=b\ e^{\gamma t}$we find two eigenvalues $\gamma _{1.2}=-\varepsilon \pm
\sqrt{\delta |g|}$ (the corresponding eigenvectors are almost parallel). The *supercritical* excitation condition $\gamma _{1}>0$ can be written as $\sqrt{\delta |g|}>\varepsilon $ or $\sqrt{R_{\alpha }R_{\omega }|g|}>\pi
^{2}/4$ [@RShS]. Consider the *subcritical* case when $0<\delta
<\varepsilon ^{2}/|g|.$ The solution of the system (\[linear\]) with the initial conditions $B_{r}(0)=-2c\sqrt{\delta /|g|},$ $B_{\varphi }(0)=0\ $ is $$B_{r}(t)=-c\sqrt{\frac{\delta }{|g|}}(e^{\gamma _{1}t}+e^{\gamma
_{2}t}),\;\;\;B_{\varphi }(t)=c(e^{\gamma _{1}t}-e^{\gamma _{2}t}).\$$ Thus $B_{\varphi }(t)$ exhibits large transient growth over a timescale of order $1/\varepsilon $ before decaying exponentially. In Fig. 1 we plot the azimuthal component $B_{\varphi }$ as a function of time for $|g|=1,$ $%
\delta =10^{-4}$ and $\varepsilon =2\cdot 10^{-2}$ and different initial values of $B_{r}$ ($B_{\varphi }(0)=0$).
=0.6
Of course without nonlinear terms any initial perturbation decays. However if we take into account the back reaction suppressing the effective dissipation ($\varphi _{\beta }(|\mathbf{B}|)$ is a decaying function), one can expect an entirely different global behaviour. In the deterministic case there can be three stationary solutions to (\[basic\]). In Fig. 2 we illustrate the role of transient growth and nonlinearity in the transition to a non-trivial state using (\[backreaction\]) with $k_{\alpha }=0.5$ and $k_{\beta }=3$. We plot the azimuthal component $B_{\varphi }$ as a function of time with the initial condition $B_{\varphi }(0)=0.$ We use the same values of parameters $|g|,$ $\delta $ and $\varepsilon $ and three initial values of $B_{r}(0)$ as in Fig. 1. One can see from Fig. 2 that the trivial solution $B_{\varphi }=B_{r}=0$ is nonlinearly unstable to small but finite initial perturbations of $B_{r},$ such as, $B_{r}(0)=-0.03$. For fixed values of the parameters in nonlinear system (\[basic\]), there exists a threshold amplitude for the initial perturbation, above which $B_{\varphi
}(t)$ grows and below which it eventually decays.
=0.6
**Stochastic subcritical generation due to additive noise.** This scenario has been already discussed in the literature [@FI1] (see also [@F0] for hydrodynamics). The physical idea is that the average magnetic energy is maintained by additive Gaussian random forcing representing unresolved scales. It is clear that the non-zero additive noise ($\sigma
_{2}\neq 0$) ensures the stationary solution to (\[moments\]). If we assume for simplicity $\sigma _{1}=0$ and $\delta =0$ then the dominant stationary moment is $$<B_{\varphi }^{2}>_{st}=\frac{g^{2}\sigma _{2}}{4\varepsilon ^{3}}.
\label{stationary}$$ We can see that due to the non-normality of the system (\[linear\]) the average stationary magnetic energy $E_{st}\sim <B_{\varphi }^{2}>_{st}$ exhibits a high degree of sensitivity with respect to the small parameter $%
\varepsilon :E_{st}\sim \varepsilon ^{-3}$ [@F0; @Fedotov].
**Stochastic subcritical generation due to multiplicative noise.** Here we discuss the divergence of the average magnetic energy $%
E(t)=<B_{r}^{2}>+<B_{r}B_{\varphi }>+<B_{\varphi }^{2}>$ with time $t$ due to the random fluctuations of the $\alpha -$parameter. Although the first moments tend to zero in the *subcritical case,* the average energy $%
E(t)$ grows as $e^{\lambda t}$ when the intensity of noise $\sigma _{1}$ exceeds a critical value. The growth rate $\lambda $ is the positive real root of the characteristic equation for the system (\[moments\]) $$(\lambda +2\varepsilon )^{3}-4\delta |g|(\lambda +2\varepsilon )-2\sigma
_{1}|g|=0. \label{ch}$$ For $\delta =0,$ the growth rate is $\lambda _{0}=-2\varepsilon +(2\sigma
_{1}|g|)^{1/3}$ as long as it is positive, and the excitation condition can be written as $\sigma _{1}>\sigma _{cr}=4\varepsilon ^{3}/|g|.$ It means that the generation of average magnetic energy occurs for $\alpha _{0}=0$ ! It is interesting to compare this criterion with the classical *supercritical* excitation condition: $\delta |g|>\varepsilon ^{2}$[@RShS]. To assess the significance of this parametric instability it is useful to estimate the magnitude of the critical noise intensity $\sigma _{cr}.$ First let us estimate the parameter $\varepsilon =\pi ^{2}\beta /(4\Omega
_{0}h^{2}).$ The turbulent magnetic diffusivity is given by $\beta \simeq
lv/3,$ where $v$ is the typical velocity of turbulent eddy $v\simeq 10$ km s$%
^{-1},$ and $l$ is the turbulent scale, $l\simeq 100$ pc. For spiral galaxies, the typical values of the thickness, $h,$ and the angular velocity, $\Omega _{0},$ are $h\simeq 800$ pc and $\Omega _{0}\simeq
10^{-15}$ s$^{-1};$ $|g|\simeq 1$[@RShS]. It gives an estimate for $%
\varepsilon \simeq 3.2\times 10^{-2}$ , that is, $\sigma _{cr}\simeq
1.\,\allowbreak 3\times 10^{-4}.$ In general $\lambda (\delta )$ $=\lambda
_{0}+$ $(4/3|g|(2\sigma _{1}|g|)^{-1/3})\delta +$ $o(\delta ).$ This analysis predicts an amplification of the average magnetic energy in a system (\[basic\]) where no such amplification is observed in the absence of noise. The value of the critical noise intensity parameter $\sigma _{cr},$ above which the instability occurs, is proportional to $\varepsilon ^{3}$, that is, very small indeed. To some extent, the amplification process exhibits features similar to those observed in the linear oscillator submitted to parametric noise [@BF]. To avoid the divergence of the average magnetic energy, it is necessary to go beyond the kinematic regime and consider the effect of nonlinear saturations.
In summary, we have discussed galactic magnetic field generation that cannot be explained by traditional linear eigenvalue analysis of dynamo equation. We have presented a simple stochastic model for the $\alpha \Omega $ dynamo involving three factors: (a) non-normality due to differential rotation, (b) nonlinearity of the turbulent dynamo $\alpha $ effect and diffusivity $\beta
$, and (c) additive and multiplicative noises. We have shown that even for the *subcritical case,* there are three possible scenarios for the generation of large scale magnetic field. The first mechanism is a deterministic one that describes an interplay between transient growth and nonlinear saturation of the turbulent $\alpha -$effect and diffusivity. We have shown that the trivial state $\mathbf{B}=0$ can be nonlinearly unstable with respect to small but finite initial perturbations. The second and third are stochastic mechanisms that account for the interaction of non-normal effect generated by differential rotation with random additive and multiplicative fluctuations. We have shown that multiplicative noise associated with the $\alpha -$effect leads to exponential growth of the average magnetic energy even in the *subcritical case.*
**Acknowledgements** I am grateful to Anvar Shukurov for constructive discussions.
[99]{} H. K. Moffatt, *Magnetic Field Generation in Electrically Conducting Fluids* (Cambridge University Press, New York, 1978).
F. Krause and K. H. Radler, *Mean-Field Magnetohydrodynamics and Dynamo Theory* (Academic-Verlag, Berlin, 1980).
Ya. B. Zeldovich, A.A. Ruzmaikin and D. D. Sokoloff, *Magnetic Fields in Astrophysics* (Gordon and Breach Science Publishers, New York, 1983).
A.A. Ruzmaikin, A. M. Shukurov and D. D. Sokoloff, *Magnetic Fields in Galaxies* (Kluwer Academic Publishers, Dordrecht, 1988).
L. Widrow, Rew. Mod. Phys. **74**, 775 (2002).
An operator is said to be non-normal if it does not commute with its adjoint in the corresponding scalar product. Typical examples of transient growth can be represented by the functions: $%
f_{1}(t)=t\exp (-t)$ and $f_{2}(t)=$ $\exp (-t)$ $-\exp (-2t)$ for $t\geq 0.$
B. Farrel, and P. Ioannou, 1999, ApJ, **522**, 1088.
J. R. Gog, I. Opera, M. R. E. Proctor, and A. M. Rucklidge, Proc. R. Soc. Lond. A , **455**, 4205 (1999).
S. Grossmann, Rev. Mod. Physics, **72**, 603 (2000).
P. J. Schmid, D. S. Henningson, *Stability and Transition in Shear Flows,* (Springer, Berlin, 2001).
B. F. Farrell, and P. J. Ioannou, Phys. Fluids, **5**, 2600 (1993); Phys. Rev. Lett. **72**, 1188 (1994); J. Atmos. Sci., **53**, 2025 (1996).
P. Hoyng, D. Schmitt, and L. J. W. Teuben, Astron. Astrophys. **289**, 265 (1994).
S. Fedotov, I. Bashkirtseva, and L. Ryashko, Phys. Rev. E., **66**, 066310 (2002).
R. Beck, A. Brandenburg, D. Moss, A. Shukurov, and D. Sokoloff, Ann. Rev. Astron. Astrophys. **34**, 155 (1996).
L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscol, Science **261**, 578 (1993).
T. Gebhardt and S. Grossmann, Phys. Rev. E **50**, 3705 (1994); J. S. Baggett, T. A. Driscoll, and L. N. Trefethen, Phys. Fluids **7**, 833 (1995); J. S. Baggett and L. N. Trefethen, Phys. Fluids **9**, 1043 (1997).
C. W. Gardiner, *Handbook of Stochastic Methods*, 2nd ed. (Springer, New York, 1996).
W. Horsthemke and R. Lefever, *Noise-Induced Transitions* (Springer, Berlin, 1984).
S. I. Vainshtein and F. Cattaneo, ApJ 393, **165** (1992); F. Cattaneo, and D. W. Hughes, Phys. Rev. E **54**, R4532 (1996); A. V. Gruzinov and P. H. Diamond, Phys. Rev. Lett. **72**, 1651 (1994); A. Brandenburg, ApJ **550**, 824 (2001); I. Rogachevskii and N. Kleeorin Phys. Rev. E **64**, 056307 (2001); E. G. Blackman, and G. B. Field, Phys. Rev. Lett. **89**, 265007 (2002).
R. Bourret, Physica **54**, 623 (1971); U. Frisch, and A. Pouquet, *ibid,* **65**, 303 (1973).
|
---
abstract: 'We study the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full description of the long- time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by solitons-solitons and solitons-radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou [@DZ03] revisited by the $\overline{\partial}$-analysis of Dieng-McLaughlin [@DM08] and complemented by the recent work of Borghese-Jenkins-McLaughlin [@BJM16] on soliton resolution for the focusing nonlinear Schrödinger equation.'
address:
- 'Department of Mathematics, University of Arizona, Tucson, Arizona 85721–0089'
- 'Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506–0027'
- ' Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506–0027'
- 'Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada '
author:
- Robert Jenkins
- Jiaqi Liu
- Peter Perry
- Catherine Sulem
title: 'Global Well-posedness and soliton resolution for the Derivative Nonlinear Schrödinger equation'
---
[^1] [^2]
[^1]: P. Perry supported in part by a Simons Research and Travel Grant.
[^2]: C. Sulem supported in part by NSERC Grant 46179-13
|
---
abstract: 'This poster paper illustrates the color–magnitude diagrams discussed by Piotto in the preceding paper. We present CMDs for 13 clusters; and we emphasize the discovery of additional blue horizontal branches in two metal-rich clusters, and the four-mode HB of NGC 2808.'
author:
- |
C.SOSIN$^1$, G.PIOTTO$^2$, S.G.DJORGOVSKI$^3$,\
I.R.KING$^1$, R.M.RICH$^4$, B.DORMAN$^5$,\
J.LIEBERT$^6$,
- ' A.RENZINI$^7$'
title: 'Globular-Cluster Color–Magnitude Diagrams with HST[^1]'
---
\#1[\_[[\#1]{}]{}]{} =cmr9 \#1[[[ \#1]{}]{}]{}
@scaling[.95]{} \#1[@scaling[\#1]{}]{}
\#1[ =@scaling]{}
\#1\#2[ =.45 =.45]{}
psfig.tex
Introduction
============
The factors that determine the morphology of the horizontal branch in globular clusters are still not well understood. Metal abundance plays an important role, with the most metal-rich clusters usually having the reddest HBs. However, a number of “second parameters,” such as age, have been proposed as explanations of the clusters that deviate from this rule.
Here we present preliminary results from a [*Hubble Space Telescope*]{} program that aims to explore connections between stellar evolution and cluster dynamics, and to investigate the CMD morphology of some clusters that are difficult to observe from the ground. The central regions of ten clusters were observed with the WFPC2 on [*HST*]{}. Some of the clusters chosen had a high central density and/or concentration; others had ultraviolet flux of unknown origin detected by [*IUE*]{}. We also include here three additional CMDs that we have derived from archival data originally taken by Yanny in another program.
We present two results: the discovery of additional blue horizontal branches in two metal-rich globular clusters, and the intriguing “clumpy” nature of the blue horizontal-branch tail of NGC 2808.
Blue horizontal branches in metal-rich clusters
===============================================
An exciting result from this program has been the discovery of additional [*blue*]{} horizontal branches in two [*metal-rich*]{} clusters, NGC 6388 and NGC 6441. Both are extremely crowded from the ground, and their HBs have not been seen previously.
Both clusters were also detected in the ultraviolet by [*IUE*]{}, and it was suggested that their UV brightness arose from blue HB stars. While a handful of blue HB stars have indeed been detected in metal-rich populations, blue populations as large as those discovered here have not previously been seen. (For discussion of these points, and references, see the preceding paper by Piotto .) What is puzzling, however, is that some other clusters with nearly the same metal abundance have no blue HB stars (see our CMDs of 47 Tuc and NGC 5927). It appears that another “second-parameter” problem may be at hand!
The Long, Clumpy HB of NGC 2808
===============================
The horizontal branch of NGC 2808 has provided another fascinating result. Already known to be separated from the red stub, its blue end is now seen to extend many magnitudes fainter in $V$ than previously known, down to $V \simeq 21$.
A CMD taken with an ultraviolet filter (F218W, $\lambda_{\rm eff} =
2189 {\rm \AA}$) spreads out the HB tail in color, allowing better separation of stars of different envelope mass. A histogram of this “blue vertical branch” shows that it is made up of at least two distinct groups, and possibly a third at its extreme blue end (in addition to the red stub that does not appear in the UV diagram). No mass-loss mechanism is known that could lead to such well-defined HB clumps.
Other clusters
==============
The analysis of these data continues. Color–magnitude diagrams for our other clusters are shown in Figure 3; a number of interesting features in them are being investigated.
[^1]: Based on observations with the NASA/ESA [*Hubble Space Telescope*]{}, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555
|
[**Glueball-Induced Partonic Energy Loss in Quark-Gluon Plasma** ]{}\
Dong-Pil Min$^a$ [^1] and Nikolai Kochelev$^{a,b}$[^2],\
[(a) *Department of Physics and Astronomy, Center for Theoretical Physics,\
Seoul National University, Seoul 151-747, Korea*]{}\
1ex [(b) *Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia*]{} 1ex
0.5cm
**Abstract**
We discuss the energy loss of energetic parton jets in quark-gluon plasma above the deconfinement temperature $T_c$ by the interaction with scalar and pseudoscalar glueballs. It is shown that the loss by this mechanism is quite important and may play the important role of the observed jet-quenching. 0.3cm
The investigation on jets in the relativistic heavy ion collisions (RHIC) provides us insights to understand the properties of quark-gluon plasma (QGP) [@STAR; @PHENIX; @BRAHMS; @PHOBOS]. One of the important RHIC discoveries is the jet quenching phenomenon coming from the partonic energy loss in QGP. In the conventional approach to the jet quenching the perturbative (pQCD) type of energy loss is taken into account by the channels, elastic and radiative, of one-gluon exchange between the jet and the massless gluons and quarks (see recent discussion in [@radiative]). However, the large quark-gluon rapidity density $dN_{qg}/dy\approx 2000$ which is needed to describe the RHIC jet quenching data within this approach, seems to be in contradiction with the restriction $dN_{qg}/dy\leq 1/4dS/dy\approx 1300$ coming from the measured final entropy density $dS/dy\approx 5000$ [@muller]. Furthermore, the lattice calculations show that even at very high temperature gluons and quarks still interact strongly in QGP [@lattice1].
Recently, it was suggested that the glueballs, the bound states of gluons, can exist above deconfinement temperature and may play an important role in the dynamic of strongly interacting QGP [@vento; @kochmin]. In particular, in [@kochmin] it is suggested that a very light pseudoscalar glueball can exist in QGP and might be responsible for the residual strong interaction between gluons. The lattice results showing a change of sign of the gluon condensate [@lattice1] and a small value of the topological susceptibility [@lattice2] above $T_c$ can be explained in the glueball picture as well. Furthermore, one expects that the suppression of the mixing between glueballs and quarkonium states in the QGP leads to a smaller width for former as compared to the vacuum [@vento]. This property opens the possibility for a clear separation of the glueball and the quark states in heavy ion collisions. Such separation is rather difficult in other hadron reactions due to existence of strong glueball-quarkonium mixing in the vacuum.
In this communication we report our study on the contribution of glueballs to the energy loss by high energy partons propagated in QGP. We will argue that significant partonic energy loss can result from strong glueball-gluon coupling.
Our starting point is the effective pseudoscalar glueball-gluon vertex employed in Refs. [@kochmin; @soni] $$\begin{aligned}
{\cal L}_{Ggg}=\frac{1}{f_S}(\alpha_sG^a_{\mu\nu}{G}^a_{\mu\nu}S+
\xi\alpha_sG^a_{\mu\nu}\widetilde{G}^a_{\mu\nu}P), \label{lag}\end{aligned}$$ where $G^a_{\mu\nu}$ is gluon field strength, $\widetilde{G}^a_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}{G}^a_{\alpha\beta}/2$, $ S $ and $P$ are scalar and pseudoscalar glueball fields, respectively, $\xi\approx 1$, and $f_S\approx 0.35$ GeV which is fixed by low energy theorem [@kochmin]. In Fig.1 the diagrams contributing to the energy loss in high energy limit $s>> (-t,M^2_{S,P})$ are illustrated.
The elastic energy loss is given by Bjorken’s formula [@bjorken] $$\frac{dE}{dx}(T)=\int d^3kn(k,T)[Flux factor]\int
dt\frac{d\sigma}{dt}\nu, \label{bj}$$ where $\nu=E^\prime-E$ energy difference between fast incoming and outcoming partons, $ [Flux factor]=(1-cos\theta)$, $\theta $ is the laboratory angle between the incident partons, $d\sigma/dt$ is partonic cross section and $n(k,T)$ is density of target parton in QGP at the temperature $T$.
The result of the calculation of diagrams in Fig.1 is $$\begin{aligned}
\frac{d\sigma_a}{dt}\approx\frac{15\alpha_s^3}{f_S^2|t|}F(t), \ \
\frac{d\sigma_b}{dt}\approx\frac{15\alpha_s^3}{4f_S^2|t|}F(t),\nonumber\\
\frac{d\sigma_c}{dt}\approx\frac{16\alpha_s^3}{3f_S^2|t|}F(t), \ \
\frac{d\sigma_d}{dt}\approx\frac{\alpha_s^3}{3f_S^2|t|}F(t),
\label{cross}\end{aligned}$$ where $|t|=2k\nu(1-cos\theta)$, and the form factor in gluon-glueball vertex reads [@kochmin] $$F(t)=e^{-\Lambda^2|t|}, \label{cut}$$ with $\Lambda\approx 0.6$ GeV$^{-1}$. In the high energy limit we will neglect the small effect coming from finite masses of the produced glueballs but we will take into account the finite value of their masses in the densities, Eq.\[dens\]. Furthermore, our consideration here is restricted by calculation to the leading order in $\alpha_s$. So the possible thermal gluon mass effects, $m_g\propto \alpha_sT$, are not considered. Therefore, in the case of energetic parton it is enough to keep only leading energy independent terms of the partonic cross sections shown in Eq.\[cross\].
The final result for energy loss for gluon and quark jets reads $$\begin{aligned}
\frac{dE_{g}}{dx}(T)&=& \frac{15\alpha_s^3}{2f_S^2\Lambda^2}\int
\frac{d^3k}{k}[n_S(k,T)+n_P(k,T)+\frac{n_g(k,T)}{4}+\frac{n_q(k,T)}{45}]\nonumber\\
\frac{dE_{q}}{dx}(T)&=&\frac{8\alpha_s^3}{3f_S^2\Lambda^2}\int
\frac{d^3k}{k}[n_S(k,T)+n_P(k,T)+\frac{n_g(k,T)}{8}],
\label{loss2}\end{aligned}$$ For estimation we will assume in QGP gluons, quarks and glueballs are in thermodynamical equilibrium and will use the gas approximation for gluon and glueball densities $$n_{i}(k,T)=\frac{N_i}{(2\pi)^3(exp{(\sqrt{k^2+M^2_i}/T)}\pm
1)},
\label{dens}$$ where the plus (minus) sign is for fermions (bosons) and numbers of degrees of freedom are $N_S=1$ for scalar and $N_P=1$ for pseudoscalar glueballs, respectively, $N_g=16$ for gluons and $N_q=12 $ for number of light quark flavors $N_F=2$. In our previous paper [@kochmin] it was argued that in the model with Lagrangian Eq. \[lag\] the behaviour of the masses of pseudoscalar and scalar glueballs above $T_c$ is very different. Indeed, it was shown that the scalar glueball remain to be massive, $M_S\approx 1.5$ GeV, but pseudoscalar glueball is very light, $M_P\approx 0$, above deconfinement temperature. Within such approximation we obtain $$\frac{dE_{g,q}}{dx}(T)\approx
C^G_{g,q}\frac{\alpha_s^3T^2}{f_S^2\Lambda^2},
\label{loss3}$$ where $C^G_g=79/24 $ is the coefficient for the glueball contribution to gluon energy loss and $C^G_q=2/3$ is the correspondent coefficient for quark energy loss. [^3]
The numerical result presented in Figs.(2,3) of the temperature dependence of elastic energy loss due to interaction with glueballs is to be compared with the recent re-analysis of perturbative QCD elastic contribution in the range of temperatures $T_c<T<2T_c$, which is accessible at RHIC experiments. The pQCD elastic contribution is as following [@pQCD] $$\frac{dE^{pQCD}_{g,q}}{dx}(T)=
C^{p}_{g,q}\frac{8\pi^2\alpha_sT^2}{b_0}(1+\frac{N_F}{6}),
\label{lossp}$$ where $C^p_g=3/2, C^p_q=2/3$ are coefficients for the perturbative gluon and quark energy loss, respectively, and $b_0=11/3N_c-2/3N_F$. We take $T_c=170$ MeV for $N_F=2$ [@lattice] and $\alpha_s\approx 0.6$ at $T_c<T<2T_c$ [@latticealpha; @alpha2] for the estimation of energy loss in gluon-glueball plasma. It follows from Figs. 2,3 that glueball-induced energy loss is large for both gluon and quark jets. In particular, for the gluon jet such contribution is about of few GeV/fm and approximately twice larger than the perturbative elastic loss [@pQCD]. In spite of the fact that for the quark jet the glueball contribution is smaller than perturbative elastic loss, it can not be neglected in comparison with latter one. It is evident that the origin of such large contribution is in strong glueball-gluon coupling in Eq.\[lag\]. We should point out that more than one half of contribution to the gluon energy loss comes from interaction of gluon with the light pseudoscalar glueball in QGP. Therefore, existence of such light bound state of gluons above $T_c$ is crucial for the understanding of the large observed partonic energy loss in QGP.
In summary, we made the estimation of the energy loss induced by interaction of an energetic parton, which was produced in the hard scattering of two heavy ion’s partons, with glueballs in hot quark-gluon plasma. It is shown that such contribution leads to a significant energy loss. We conclude that not only pQCD type of energy loss but also glueball-induced loss, arising from existence of scalar and pseudoscalar glueballs in QGP, are important for the understanding of the RHIC results such as the jet quenching. We should emphasize that the main goal of our paper is to show the significance of the glueball-induced energy loss and we left the detailed comparison with experiment, which should include also the consideration of the effects of both elastic and radiative pQCD losses, for a forthcoming publication.\
[**Acknowledgments**]{}\
We would like to thank V.Vento for useful discussions. This work was supported by Brain Pool program of Korea Research Foundation through KOFST, grant 042T-1-1. NK is very grateful to the School of Physics and Astronomy of Seoul National University for their warm hospitality during this work.
[99]{}
STAR Collaboration: J. Adams, et al., Nucl. Phys. [**A757**]{} (2005) 102 .
PHENIX Collaboration: K. Adcox, et al., Nucl. Phys. [**A757**]{} (2005) 184 .
BRAHMS Collaboration: I.Arsene, et al., Nucl. Phys. [**A757**]{} (2005) 1 .
PHOBOS Collaboration: B. B. Back, et al., Nucl. Phys. [**A757**]{}(2005) 28 .
S. Wicks and M. Gyulassy, arXiv:nucl-th/0701088.
B. Muller and J. L. Nagle, Ann. Rev. Nucl. Part. Sci. [**56**]{}, 93 (2006)
D. E. Miller, Phys. Rept. [**443**]{} (2007) 55 .
V. Vento, Phys. Rev [**D75**]{} (2007) 055012 ; ibid. [**D73**]{} (2006) 054006 .
N.Kochelev and D.-P. Min, Phys. Lett. [**B650**]{} (2007) 239.
B. Alles, M. D’Elia and A. Di Giacomo, Nucl. Phys. [**B494**]{} (1997) 281; Erratum: ibid [**B679**]{} (2004) 397 ; B. Alles and M. D’Elia, hep-lat/0602032.
J. M. Cornwall, A. Soni, Phys. Rev. [**D29**]{} (1984) 1424.
J. D. Bjorken, “Energy Loss Of Energetic Partons In Quark - Gluon Plasma: Possible Extinction Of High P(T) Jets In Hadron - Hadron Collisions,” Preprint FERMILAB-PUB-82-059-THY.
A. Peshier, Phys. Rev. Lett. [**97**]{} (2006) 212301 .
F. Karsch and E. Laermann, arXiv:hep-lat/0305025.
O. Kaczmarek and F. Zantow, Phys. Rev. [**D71**]{} (2005) 114510
A. Peshier, arXiv:hep-ph/0601119.
[^1]: dpmin@snu.ac.kr
[^2]: kochelev@theor.jinr.ru
[^3]: We neglect the small contribution arising from the interaction of energetic parton with massive scalar glueball in QGP due to small value of its density $n_S<<n_P$.
|
---
abstract: 'We review and extend the recently proposed model of combinatorial quantum gravity. Contrary to previous discrete approaches, this model is defined on (regular) random graphs and is driven by a purely combinatorial version of Ricci curvature, the Ollivier curvature, defined on generic metric spaces equipped with a Markov chain. It dispenses thus of notions such as simplicial complexes and Regge calculus and is ideally suited to extend quantum gravity to combinatorial structures which have a priori nothing to do with geometry. Indeed, our results show that geometry and general relativity emerge from random structures in a second-order phase transition due to the condensation of cycles on random graphs, a critical point that defines quantum gravity non-perturbatively according to asymptotic safety. In combinatorial quantum gravity the entropy area law emerges naturally as a consequence of infinite-dimensional critical behaviour on networks rather than on lattices. We propose thus that the entropy area law is a signature of the random graph nature of space-(time) on the smallest scales.'
author:
- 'C.Kelly'
- 'C.A.Trugenberger'
title: 'Combinatorial Quantum Gravity: Emergence of Geometric Space form Random Graphs'
---
Introduction
============
Ultraviolet (UV) fixed points of statistical mechanics models define renormalizable quantum field theories via the Wilson renormalization group [@zinn]. Here we review the evidence, first presented in [@tru], that quantum gravity is defined by an UV fixed point for a graph model [@graphrev]. The asymptotic safety scenario [@safety] would thus be realized on networks, rather than traditional statistical mechanics models.
In the traditional discrete approach to quantum gravity [@triang] a smooth background is assumed, which is then approximated by piecewise flat geometries on which curvature is computed by Regge calculus [@regge]. In [@tru], instead, one of us first posited that the fundamental structures on Planckian scales are not smooth but, rather, graphs, on which even notions like Regge curvature are lost. Random graphs are generic metric spaces. When equipped with a Markov chain, like a probability measure, a purely combinatorial notion of Ricci curvature, first introduced by Ollivier [@olli1; @olli2; @olli3], can be defined on such structures. This was used in [@tru] to define a model of purely combinatorial quantum gravity. This approach was subsequently pursued in [@loll], where a modified version of the Ollivier curvature was introduced.
Albeit in a simplified model, we will provide strong evidence that geometric space emerges from random graphs at a second-order phase transition driven by a combinatorial version of the Einstein-Hilbert action and corresponding to the condensation of elementary loops on the graphs. In the geometric phase the combinatorial Einstein-Hilbert action becomes its standard continuum version. One notable result is that, in this model, the entropy of quantum space automatically follows an area law. The posited critical point on graphs could thus be the origin of the famed area law for the entropy in quantum gravity. Note that a relation between geometry and the density of triangles (the so-called clustering coefficient [@graphrev]) has been also noted in the network literature [@krioukov]. In the simplified model considered here we will be dealing with squares but the general case can also be treated [@kelly].
The Ollivier curvature and the combinatorial Einstein-Hilbert action
====================================================================
The continuum Ricci curvature is associated with two (infinitely) close points on a manifold, defining a tangent vector. It can be thought of as a measure of how much (infinitesimal) spheres around these points are, on average, closer (positive Ricci curvature) or more distant (negative Ricci curvature) than the two points at their centres. Its combinatorial version, the Ollivier curvature [@olli1; @olli2; @olli3], is a discrete version of the same measure. Consider two vertices $i$ and $j=i+e_{ij}$ separated by edge $e_{ij}$ on a graph. The Ollivier curvature compares the Wasserstein (or earth-mover) distance $W\left( \mu_i, \mu_j \right)$ between two uniform probability measures $\mu_{i,j}$ on the unit spheres around $i$ and $j$ to the distance $d(i,j)$ on the graph and is defined as $$\kappa (i,j)= 1- {W\left( \mu_i, \mu_j \right) \over d(i,j)} \ .
\label{olli}$$ The Wasserstein distance between two probability measures $\mu_1$ and $\mu_2$ on the graph is defined as $$W\left( \mu_1, \mu_2 \right) = {\rm inf} \sum_{i,j} \xi(i,j)d(i,j) \ ,
\label{wasser}$$ where the infimum has to be taken over all couplings (or transference plans) $\xi(i,j)$ i.e. over all plans on how to transport a unit mass distributed according to $\mu_1$ around $i$ to the same mass distributed according to $\mu_2$ around $j$ without losses, $$\sum_j \xi (i,j) = \mu_1(i) \ , \qquad
\sum_i \xi (i,j) = \mu_2(j) \ .
\label{transplan}$$
The Ollivier curvature is very intuitive but, in general not easy to compute and work with. Fortunately, it becomes much simpler for bipartite graphs [@olli3], which have no odd cycles. Since the Ollivier curvature of an edge depends only on the triangles, squares and pentagons supported on that edge (a discrete form of locality) [@olli2] and there are no triangles and pentagons on bipartite graphs, one can use for all practical purposes the simpler version of the Ollivier curvature for bipartite regular graphs [@olli3]: $$\begin{aligned}
\kappa (i,j) &&= -{1\over d} \Big[ (2d-2) -|N_1(j)|
\nonumber \\
&&+ \sum_{a} \left( |L_a(j)|-|U_a(i)| \right) \times {\bf 1}_{\left\{ |U_a(i)| < |L_a(j)|\right\} } \Big] _+\ ,
\label{riccibipartite}\end{aligned}$$ where $N_1(i)$ denotes the set of neighbours of $i$ which are on a 4-cycle supported on $(ij)$, ${\bf 1}$ denotes the indicator function (1 if the corresponding condition is satisfied, 0 otherwise) and the undescript “+" denotes $z_+ = Max(z,0)$ so that the Ollivier Ricci curvature for bipartite graphs is always zero or negative. The definition of $U$ and $L$ is as follows: suppose that $R(i,j)$ is the subgraph induced by $N_1(i) \cup N_1(j)$ and $R_1(i,j)$...$R_q(i,j)$are the connected components of $R(i,j)$. Then $U_a(i) = R_a(i,j) \cap N_1(i)$ and $L_a(j) = R_a(i,j) \cap N_1(j)$ for $a=1 \dots q$.
Equipped with a combinatorial version of Ricci curvature we can now formulate a purely combinatorial version of the Einstein Hilbert action. First we define the combinatorial Ricci scalar as $$\kappa (i) = \sum_{j \sim i} \kappa (i,j) \ ,
\label{orscalar}$$ where $\sim$ denotes the neighbours of $i$ on the graph. Then we obtain the combinatorial Einstein-Hilbert action simply as $$S_{\rm EH} = - {1\over g} \sum_i \kappa(i) \ ,
\label{cehaction}$$ where the sum runs over all the vertices of the graph and $g$ is a coupling constant with dimension 1/action. This expression for the Ollivier curvature still looks forbidding. However, as we will show in a moment, it will become extremely simple on the physical configuration space.
Configuration space and combinatorial quantum gravity
=====================================================
To fully specify a combinatorial quantum gravity model we need, in addition to the action, a configuration space over which to sum in the partition function. The configuration space in our simplified model consists of all random bipartite graphs. There is, however a further restriction that must be taken into account. As mentioned above we will model the emergence of geometry by the condensation of elementary loops on the graphs. We have thus to remind ourselves that even the Bose condensation of point particles is not well defined in absence of interactions, because of the infinite compressibility of the condensate. In exactly the same way, the condensation of “non-interacting" loops is unstable, since it leads to crumpling and disconnected graphs (baby universes). We will thus follow the same route as for point particles and introduce, as the simplest stability mechanism, a hard-core condition for elementary loops. However, while for point particles the meaning of a hard core condition is unequivocal, for loops we must define what exactly we have in mind. The definition we will use is that elementary squares on the graphs will be allowed to share one edge but not more. Note that two squares can share two edges without being identical: it is exactly these configurations that we exclude.
When the hard-core condition is implemented, the Ollivier combinatorial curvature becomes really simple. Indeed, it is easy to convince oneself that the second term in (\[riccibipartite\]), involving the sum of connected components of a subgraph, only contributes for squares that share 2 edges. Indeed, for an isolated square $|N_1|=1$ for all vertices on the square. If an edge supports $N_s$ squares which do not share another edge, then $|N_1(i)| = |N_1(j)|= N_s$ and $|U_a(i)|=|L_a(j)|$ since all the vertices within $N_1(i)$ and $N_1(j)$ are disconnected because of the absence of triangles in a bipartite graph and all the vertices of $N_1(i)$ are disconnected from those in $N_1(j)$ since, by assumption, the edge does not support two different squares. The Ollivier combinatorial curvature reduces thus simply to $$\kappa (i,j) = -{1\over d} \big[ (2d-2) -N_s(ij) \big]_+ \ ,
\label{riccireduced}$$ where $N_s(ij)$ is the total number of squares supported on edge $(ij)$. The full model of combinatorial quantum gravity can thus be specified as $${\cal Z} = \sum_{CF} {\rm exp} \left( - {1\over g\hbar} \sum_i \kappa(i) \right) \ ,
\label{combqg}$$ where CF denotes the configuration space of random regular bipartite graphs with squares satisfying the hard-core condition and $\kappa (i) $ given by (\[orscalar\]) and (\[riccireduced\]).
The classical limit and the mean field action
=============================================
The classical limit $\hbar \to 0$ corresponds to the weak coupling limit of small $g$. In the quantum regime $\hbar g \gg 1$ the Boltzmann probability becomes uniform over all configuration space of random regular bipartite graphs. Random regular bipartite graphs are locally tree-like, with very sparse short cycles governed by a Poisson distribution with mean $(2d-1)^l/$ for cycles of length $l$ on $2d$-regular graphs [@wormald]. The quantity $(2d-2)$, instead, is the number of squares supported on an edge in a ${\mathbb Z}^d$ lattice. The combinatorial Einstein-Hilbert action (\[cehaction\]) thus favours the formation of squares on the graph until the amount corresponding to a ${\mathbb Z}^d$ lattice is reached, after which it vanishes. The number of squares based on an edge, however, can be larger than $(2d-2)$ even for graphs with hard core squares, indeed it can reach up to $(2d-1)$. This is the maximum that the “mean field" version of the action (\[riccireduced\]) without subscript $+$ [@tru] would favour in the classical limit. It can be shown, however, that configurations attaining this maximum split into large quantities of disconnected baby universes while the classical limit of the exact action avoids this fate and approaches a regular ${\mathbb Z}^d$ lattice [@kelly]. The hard core condition is thus sufficient to stabilize space.
On a ${\mathbb Z}^d$ lattice the optimal coupling for the Wasserstein distance between the two vertices at the extremities of an edge is the translation along the lattice links connecting the unit balls around the two vertices. As derived in [@olli1], the Ollivier curvature then becomes the average of the sectional curvatures of the planes defined by the original edge and each of the links defining the unit ball around one of the vertices. If we assign a length $\ell$ to each link of the lattice and scale this as $\ell = \ell_0 N^{-1/d}$, with $\ell_0$ a renormalization constant related to the Planck length, we obtain the formal continuum limit $${1\over \hbar g} \sum_i \kappa(i) \to {1\over 2(d+2) \ell_0^{d-2}} {N^{1-2/d}\over \hbar g} \int d{\rm Vol} \ R \ .
\label{ren}$$ Both sides do of course vanish since the ${\mathbb Z}^d$ lattice is Ollivier flat and the Euclidean space ${\mathbb R}^d$ it approximates is Ricci flat. However, this formal continuum limit shows, first, that the combinatorial Einstein-Hilbert action goes over into the continuum Einstein-Hilbert action and, secondly, that this requires a scaling of $g\hbar \sim N^{1-2/d}$. We will return to this all-important scaling below. For the moment let us retain that the the combinatorial quantum gravity model has the correct formal continuum limit.
Having established that the hard-core condition is sufficient to stabilize space and obtain the correct (formal) continuum limit we can adopt the simpler mean field action expressed in terms of the total number of squares provided we explicitly exclude configurations with more than $(2d-2)$ squares per edge. To this end we compute $$\begin{aligned}
\sum_i \kappa (i) &&= -(4d-4) N + {1\over d} \sum_i \sum_{e_i} N_s\left( e_i \right)
\nonumber \\
&&= {-8\over d} \left[ {d(d-1)\over 2} N-N_s \right] \ ,
\label{totcursca}\end{aligned}$$ which gives the final result $$S_{EH}^{mf}= {4d-4\over g} N \big[ 1 -\zeta \big] \ ,
\label{meanfield}$$ where $\zeta = 2 N_s/(d(d-1)N)$ ($0\le \zeta \le 1$) is the density of squares.
A continuous network phase transition and the entropy area law
==============================================================
Let us consider the free energy of the model (divided by the “temperature" $\hbar g$) $$F= {4d-4\over \hbar g} N \big[ 1 -\zeta \big] - S\left( N \right) \ ,
\label{freeenergy}$$ where $S(N)$ is the entropy of the graphs. In traditional statistical mechanics models the degrees of freedom, typically (but not necessarily) living on the vertices of a lattice, interact with a fixed number of their neighbours. As a consequence, both the energy and the entropy are extensive quantities, scaling like the volume $N$ (number of vertices of the lattice). Phase transitions, thus, show up when the external intensive parameter temperature $T$ (or coupling constant in case of quantum phase transitions) crosses a critical value $T_c$ where energy and entropy exactly compensate. This is not so in the statistical mechanics of networks [@newman]. On networks, interactions are represented by edges. Each vertex can thus interact with a number of other vertices that diverges in the limit $N\to \infty$: contrary to traditional statistical mechanics models on lattices, network are infinite-dimensional. Moreover, there is no a priori notion of locality on networks. On random graphs, e.g., there can be an edge between any two vertices. As explained in detail in [@graphrev], the infinite-dimensionality of networks has the consequence that the phase structure of networks is determined by [*critical functions*]{} of $N$ rather than critical values, e.g. $$\begin{aligned}
{\rm phase \ 1} \ \ {\rm if} \ \ &&{\rm lim}_{N\to \infty} \left({T(N)\over T_c(N)}\right) = 0 \ ,
\nonumber \\
{\rm phase \ 2} \ \ {\rm if} \ \ &&{\rm lim}_{N\to \infty} \left({T(N)\over T_c(N)}\right) = \infty \ .
\label{phasesnet}\end{aligned}$$ When the temperature (or the coupling in the quantum case) is chosen to scale exactly as the critical function, $${\rm lim}_{N\to \infty} \left({T(N)\over T_c(N)}\right) = t \ ,
\label{crfct}$$ the phase transition appears typically as a traditional one in terms of the rescaled coupling $t$, one phase appearing for $t<t_c$ and the other for $t>t_c$ with the difference that energy, entropy and free energy need not be extensive quantities anymore but can have a different scaling in terms of the volume $N$ (see e.g. the mean field solution of the two-star model in [@newman]).
This is exactly what seems to be realized in the present model of combinatorial quantum gravity. The formal continuum limit (\[ren\]) is well-defined only if the coupling $\hbar g$ scales as $\hbar g \sim N^{1-2/d}$. And remarkably, exactly when this scaling is chosen, the order parameter $\zeta = 2N_s/(d(d-1)N)$ representing the density of squares collapses onto an $N$-independent function suggestive of a traditional second-order phase transition, as shown in Fig. 1 for the case $d=4$.
![\[fig:Fig. 1\] Monte Carlo simulation of the average number of squares for $d=4$ and $N=300$, $N=400$ and $N=500$ as a function of the rescaled coupling $\hbar g/\sqrt{N}$. Random regular graphs with sparse squares $N_s \sim {\rm Poisson\ } (600.25)$ and logarithmic distance scaling at large values of the coupling constant turn into $\mathbb{Z}^4$ lattices with the maximum number of squares $N_s=6N$ and power-law distances when gravitation becomes weak.The horizontal lines correspond to the expected number of squares for random regular graphs of the corresponding volume $N$.](scalingdice.eps){width="8cm"}
Second order phase transitions are characterized by the divergence of the correlation length at the critical point. We can define a correlation length also on graphs. To this end we define the local order parameter $\zeta(i) = 2N_s(i)/(d(d-1))$ characterizing the density of squares at each vertex $i$ and we compute the correlation function $$C\left(d (ij) \right) = {<(\zeta (i) - \bar \zeta) (\zeta(j)-\bar \zeta) >\over \sigma (i) \sigma (j) } \ ,
\label{correl}$$ as a function of the graph distance $d(ij)$ between vertices. Here $\bar \zeta$ denotes the expectation value whereas $\sigma(i)$ is the standard deviation of $\zeta (i)$. Expressing, as usual, this correlation function as $$C\left(d (ij) \right) = {\rm exp} \left( - {d(ij)\over \xi} \right) \ ,
\label{corle}$$ defines the correlation length $\xi$. The correlation length in units of the graph diameter, averaged over different graph sizes, is plotted as a function of the rescaled coupling $ G= \hbar g$ in Fig. 2 for the case $d=3$.
![\[fig:Fig. 1\] The correlation length in units of the graph diameter, averaged over different graph sizes as a function of the rescaled coupling $ G= \hbar g$ in Fig. 2 for the case $d=3$. ](CorrelationLength.eps){width="8cm"}
Remarkably, this plot indeed shows the typical behaviour of the correlation length at a second order phase transition, the divergence being of course cut-off by finite size effects. This second order phase transition, if confirmed by further studies, marks the emergence of geometric space, in the form of flat tori locally isomorphic to ${\mathbb Z}^d$ from random regular graphs as a consequence of the condensation of the shortest cycles, squares. This critical point would provide a non-perturbative definition of quantum gravity according to the asymptotic safety scenario [@safety], in which the critical value $G_c \simeq 400 \ (d=4) $ (see Fig.1) of the rescaled coupling $\hbar g/N^{1-2/d}$ defines Newton’s gravitational constant via the relation $G_{\rm Newton} = (3/4\pi) G_c \ell_0^2/\hbar$.
A further important consequence of this second-order phase transition is derived from the expression (\[freeenergy\]) of the free energy. As explained above, the scaling $\hbar g \sim N^{1-2/d}$ is exactly the critical function of the graph model. The existence of a second-order phase transition with this scaling would imply the existence of a balance critical value between energy and entropy, which would further imply that, with this scaling, energy and entropy have themselves the same scaling behaviour with the volume $N$. This immediately leads to the scaling behaviour $S(N) \sim N^{2/d}$ for the entropy of the graphs, in both phases of course. Since $N$ represents the volume, this scaling law is a combinatorial version of the entropy area law in quantum gravity [@area].
[9]{} Zinn-Justin J 1989 [*Quantum Field Theory and Critical Phenomena*]{} (Oxford: Clarendon Press) Trugenberger C A 2017 [*JHEP*]{} [**1709**]{} 45 Albert R and Barabasi L 2002 [*Rev. Mod. Phys.*]{} [**74**]{} 47 Eichhorn A arXiv:1709.03696 Ambjorn J, Görlich A, Jurkiewicz J and Loll R 2012 [*Phy. Rep.*]{} [**58**]{} 127 Regge T 1961 [*Nuovo Cim.*]{} [**19**]{} 558; Williams R M and Tuckey P A 1992 [*Class. Quant. Gravity*]{} [**9**]{} 1409 Ollivier, Y 2009 [*J. Funct. Anal.*]{} [**256**]{} 810; Ollivier, Y 2010 [*Adv. Stud. Pure Math.*]{} [**57**]{} 343; Linn, Y, Lu L and Yau, S T 2011 [*Tohoku Math. J.*]{} [**63**]{} 605; Loisel B and Romon P 2014 [*Axioms*]{} [**3**]{} 119; Jost J and Liu S 2014 [*Discrete Comput. Geom.*]{} [**51**]{} 300. Bhattacharya B B and Mukherjee S 2015 [*Discrete Mathematics*]{} [**338**]{} 23. Klitgaard N and Loll R 2018 [*Phys. Rev*]{} D [**97**]{} 046008; arXiv:1802.10524. Dall J and Christensen M 2002 [*Phys. Rev.* ]{} E [**66**]{} 016121; Krioukov D 2016 [*Phys. Rev. Lett.*]{} [**116**]{} 208302 Kelly C, Trugenberger C A and Biancalana F 2018, in preparation. O’Neil P E 1969 [*Bull. Am. Math. Soc.*]{} [**75**]{} 1276; Wormald N C 1999 [*Surveys in Combinatorics*]{}, LMS Lectures Note Series [**267**]{}, Lamb J D and Pierce D A eds. 239. Park J and Newman M J 2004 [**Phys. Rev.**]{} [**E70**]{} 066117-1. BekensteinJ D 1973[*Phys. Rev.* ]{} [**D7**]{} 2333; Jacobson T 1995 [*Phys. Rev. Lett.*]{} [**75**]{} 1260, Jacobson T 2016 [*Phys. Rev. Lett.*]{} [**116**]{} 201101.
|
---
abstract: 'We report the detection of repeat bursts from the source of FRB171019, one of the brightest fast radio bursts (FRBs) detected in the Australian Square Kilometre Array Pathfinder (ASKAP) fly’s eye survey. Two bursts from the source were detected with the Green Bank Telescope in observations centered at 820 MHz. The repetitions are a factor of $\sim$590 fainter than the ASKAP-discovered burst. All three bursts from this source show no evidence of scattering and have consistent pulse widths. The pulse spectra show modulation that could be evidence for either steep spectra or patchy emission. The two repetitions were the only ones found in an observing campaign for this FRB totaling $1000$ hr, which also included ASKAP and the 64-m Parkes radio telescope, over a range of frequencies (720–2000MHz) at epochs spanning two years. The inferred scaling of repetition rate with fluence of this source agrees with the other repeating source, FRB121102. The detection of faint pulses from FRB171019 shows that at least some FRBs selected from bright samples will repeat if follow-up observations are conducted with more sensitive telescopes.'
author:
- Pravir Kumar
- 'R. M. Shannon'
- 'Stefan Os[ł]{}owski'
- Hao Qiu
- Shivani Bhandari
- Wael Farah
- Chris Flynn
- Matthew Kerr
- 'D.R. Lorimer'
- 'J.-P. Macquart'
- Cherry Ng
- 'C. J. Phillips'
- 'Danny C. Price'
- Renée Spiewak
bibliography:
- 'references.bib'
title: Faint Repetitions from a Bright Fast Radio Burst Source
---
Introduction {#sec:intro}
============
We are now starting to unravel the enigmatic astrophysical phenomenon of fast radio bursts (FRBs), millisecond-duration transient events first discovered over a decade ago [@lorimerburst]. The observed dispersion measures (DMs) of FRBs significantly exceed the expected contribution from the Milky Way [@thornton_frbs], suggesting extragalactic origins. The localization of several bursts sources (@r1_localization_chatterjee [@askap_localized; @ravi_localization]) unequivocally places them at cosmological distances; nevertheless, their physical origin has yet to be determined.
[ccccccccr]{} ASKAP & PAF & 0.1 & 50 & 1297.5 & 336 & 60 & 51.8 & 986.6\
Parkes & Multibeam & 0.7 & 23 & 1382 & 340 & 14 & 1.10 & 12.4\
GBT & Prime Focus 1 & 2.0 & 20 & 820 & 200 & 15 & 0.27 & 9.7\
GBT & L-band & 2.0 & 20 & 1500 & 800 & 9 & 0.13 & 0.9
There are currently about 100 FRB sources published [FRBCAT[^1]; @frbcat], most of which have only been detected once. The repeat bursts from FRB121102 [@repeater_1] enabled precise localization of the burst source and the identification of its host galaxy [@r1_localization_chatterjee; @r1_localization_tendulkar]. The existence of repetitions ruled out cataclysmic progenitor scenarios for the origin of its emission. Since its discovery, more than 100 bursts (@zhang_frbs_r1 [@Hessels19]) have been detected from this source in a broad range of frequencies, from as high as 8 GHz [@r1_highestfreq_gajjar] to as low as 600 MHz [@r1_lowestfreq_josephy]. The discovery of second repeating source, FRB180814 [@repeater_2], with properties similar to FRB121102, strengthened evidence for the existence of a substantial population of repeating FRB sources. Recently the Canadian Hydrogen Intensity Mapping Experiment (CHIME) telescope reported detection of eight new repeating FRB sources [@chime_mega_repeaters].
The localization of the ostensibly one-off (single pulse detection, which has not been shown to repeat) FRB180924 to a position 4 kiloparsecs from the center of a luminous galaxy at a redshift of $z=0.32$ [@askap_localized], enabled the first comparison of burst host galaxies. The massive ($\sim 10^{10} M_\odot$) host galaxy of FRB 180924 is in stark contrast with the low-mass ($\sim 10^8 M_\odot$), low-metallicity dwarf galaxy of the repeating source FRB121102 [@r1_localization_tendulkar], thus raising questions whether there are multiple FRB formation channels. Recently, another burst (FRB190523) has also been localized to $10''\times2''$ uncertainty, and associated with a massive ($\sim 10^{11} M_\odot$) host galaxy [@ravi_localization], partially based on the agreement between the burst DM ($760.8$${{\rm pc~cm^{-3}}}$) and the galaxy redshift ($z=0.66$). One of the most exciting open questions is the relationship between the repeating and one-off FRB sources. It is not clear whether all FRBs repeat. Are there two (or more) classes of FRBs, or are the one-off FRBs simply the most energetic bursts from repeating sources? The absence of repeat bursts even after hundreds of hours of follow-up (@ravi_carina [@ravi_2016]) and the diversity in properties (e.g., temporal structure and polarization) of one-off FRBs could be evidence for multiple populations of FRBs (@caleb_repeating [@james_limits]). However, in a recent analysis, [@ravi_repeating] has suggested that the volumetric rate of one-off FRBs is inconsistent with the rate of all possible cataclysmic FRB progenitors and concludes that most FRBs are repeating sources.
Among the strongest constraints on FRB repetition so far come from [@askap_nature] with the discovery of 20 FRBs in the first Commensal Real-time ASKAP Fast Transient [CRAFT[^2]; @craft_start] survey. The survey was conducted using a “fly’s eye” configuration to maximize sky coverage at a Galactic latitude of $|b| = 50 \pm 5~\deg$ and a central frequency of 1.3 GHz. The survey produced a well-sampled population of FRBs and established a relationship between burst dispersion and observed luminosity. The mean spectral index for these bursts ($\alpha \approx -1.5$, where $E_{\nu} \propto \nu^\alpha$) is found to be similar to that of the normal pulsar population [@jp_askapfrbs]. A key feature of the survey was that it revisited the same positions hundreds of times over its duration, producing $ \sim$ 12,000hr (@askap_nature [@craft_performance_james]) of (self) follow-up observations, which included times before and after bursts were detected. No repeat bursts from detected FRBs were found in the survey.
One possible reason for the lack of repeat detections is that ASKAP is insufficiently sensitive to faint repetitions from the bursts. Conducting follow-up observations with more sensitive instruments will be more effective [@cordes_chatterjee]; for example, Parkes has a repeat detection rate $\sim 10^4$ times greater than ASKAP, assuming the luminosity distribution follows a power-law where, above some luminosity $\mathcal{L}$, the number of detections $N (> \mathcal{L}) \propto \mathcal{L}^{\alpha}$ assuming $\alpha = -2$ [@connor_petroff_repetitions]. To complement the ASKAP self follow-up, we have also been conducting sensitive monitoring campaigns of ASKAP detections with the 64-m Parkes radio telescope and the 110-m Robert C. Byrd Green Bank Telescope (GBT). The arcminute localization of FRBs, made possible by the multi-beam detection [@bannister_askap] using ASKAP’s phased-array feed (PAF) enabled the follow-up of FRB fields with large aperture telescopes.
 \[fig:timeline\]
In this Letter, we report[^3] the discovery of repetitions from FRB171019, one of the brightest bursts found in the ASKAP fly’s-eye survey. The burst was $\sim 5$ms wide with a measured fluence of $220$ Jyms [@askap_nature]. The observed DM was 460 ${{\rm pc~cm^{-3}}}$, a factor of 11 in excess of the NE2001 model [@cordes2001model] prediction along that line of sight. In Section \[sec:observations\], we describe the observational campaigns for this FRB. In Section \[sec:repeats\], we present the properties of the repeat pulses. In Section \[sec:discussion\], we discuss the implications for the FRB population as a whole.
Observations and Data Processing {#sec:observations}
================================
We searched for repeat pulses from FRB171019 using ASKAP, Parkes, and the GBT. The observing details for all three telescopes used are summarized in Table \[tab:followupobs\]. Each telescope was pointed at the position of FRB171019 reported in [@askap_nature], i.e., R.A. = $22^{\rm{h}}17^{\rm{m}}32^{\rm{s}}$ and decl. = $-08\arcdeg39\arcmin32\arcsec$ (J2000.0 epoch). This position was obtained with $10'\times10'$ uncertainty (90% confidence) as described in [@bannister_askap]. As such, the positional uncertainty was well within the full-width at half maximum (FWHM) of the follow-up telescopes. Figure \[fig:timeline\] shows a timeline of the radio observations of FRB171019.
ASKAP Searches
--------------
ASKAP follow-up was conducted in fly’s eye configuration with each antenna pointing at a different position in the sky, and the survey regularly revisiting the same positions [@askap_nature]. FRB searches are performed in near-real-time using [FREDDA]{} [@fredda_ascl], a GPU-based implementation of the fast dispersion measure transform algorithm [FDMT; @fdmt]. For a description of the detection methods and search pipeline, see [@bannister_askap]. We found no other astrophysical events at similar DMs of FRB171019 exceeding a threshold signal-to-noise ratio (S/N) of 9.5 (which corresponds to a fluence sensitivity of 52 Jy ms for a pulse duration of 5 ms) in 987 hr of observations.
Parkes Searches
---------------
At Parkes, we used the 20-cm multibeam receiver to search for bursts from FRB171019, using the Berkeley-Parkes Swinburne Recorder (BPSR) mode of the HI-Pulsar system to record full-stokes spectra with 64 $\upmu$s time and 390kHz frequency resolution [@StaveleySmith:1996; @Price:2016]. The search process [@ppta_oslowski] was similar to that of the SUrvey for Pulsars and Extragalactic Radio Bursts project “Fast” pipeline (SUPERB; details in @parkes_multibeam_keith [@parkes_multibeam_keane]). The online pipeline stored the 8-bit data stream from all 13 beams in a ring buffer over the bandwidth of 340 MHz centered at 1382 MHz. The data were then searched using [Heimdall]{} [@barsdell12] up to a maximum DM of 4096 ${{\rm pc~cm^{-3}}}$ with a tolerance (S/N loss tolerance between each DM trial) of 20 %. The transient pipeline sorts candidate FRB events from radio interference using the methods detailed in [@parkes_multibeam_bhandari]. The pipeline searched for bursts above a threshold S/N of 10, thus sensitive up to a fluence of $1.1$ Jyms for a burst of width similar to FRB171019. No bursts were found in all the 12.4 hr of observations at the dispersion measure of FRB171019.
-------------------------------------- -------------------------------------- -----------------------------------
  
-------------------------------------- -------------------------------------- -----------------------------------
[cccccccc]{} 0 & ASKAP & 58045.56061371(2) & 219 $\pm$ 5 & 5.4 $\pm$ 0.3 & 24.8 & $-$12.6 $\pm$ 1.4 & 461 $\pm$ 1\
& & & 388 $\pm$ 12 & 5.2 $\pm$ 0.2 & 32.4 & $-$9.9 $\pm$ 2.0 &\
1 & GBT & 58319.356770492(1) & 0.60 $\pm$ 0.04 & 4.0 $\pm$ 0.3 & 15.2 & $-$7.8 $\pm$ 1.2 & 456.1 $\pm$ 0.4\
& & & 1.11 $\pm$ 0.07 & 4.2 $\pm$ 0.3 & 16.7 & $-$0.9 $\pm$ 1.8 &\
2 & GBT & 58643.321088777(1) & 0.37 $\pm$ 0.05 & 5.2 $\pm$ 0.8 & 7.9 & $-$13.2 $\pm$ 2.8 & 457 $\pm$ 1\
& & & 0.61 $\pm$ 0.07 & 3.7 $\pm$ 0.5 & 9.1 & $-$9.6 $\pm$ 3.3 &\
GBT Searches
------------
The GBT observations were obtained with the Prime Focus 1 (centered at 820MHz) and L-band receivers (details in Table \[tab:followupobs\]), and data recorded with the Green Bank Ultimate Pulsar Processing Instrument [GUPPI; @2008SPIE.7019E..1DD]. Each pointing was sampled with a time resolution of 81.92 $\upmu$s and 2048 frequency channels (512 channels for the L-band receiver), and written to a PSRFITS format file with full-Stokes parameters.
To search the GBT data for bursts, we first converted the PSRFITS data to total intensity SIGPROC[^4] filterbank format. The dynamic spectra were then normalized to remove the receiver bandpass by scaling each channel to a mean of zero and standard deviation of unity. Using the PRESTO[^5] [@ransom_presto] tool [rfifind]{} and the median absolute deviation statistics, we identified bad channels affected by radio frequency interference (RFI). The resulting data were then searched using [Heimdall]{} for dispersed pulses. We performed two searches: a narrow search within the DM range of 446 to 474 ${{\rm pc~cm^{-3}}}$ over 220 trials using a tolerance of 1% and then a wider search in a DM range of 0 to 2000 ${{\rm pc~cm^{-3}}}$ with a tolerance of 5%. Candidates satisfying the following criteria were retained for further analysis: S/N $\geq 6.5 $ (7.5 for the wider search), pulse width $\leq 41.94 $ ms and members[^6] $\geq 2 $. For the L-band data, we also apply a minimum threshold for pulse width (0.65ms) to mitigate false-positives produced by spurious narrow-band short-duration candidates.
------------------------------------------------- ------------------------------------------------- ----------------------------------------------
  
  
------------------------------------------------- ------------------------------------------------- ----------------------------------------------
We used deep neural network trained models, as developed by [@fetch][^7] to perform the FRB/RFI binary classification of the candidates. Following their prescription, we created dedispersed frequency-time and DM-time image data for each candidate, which were then classified using [keras]{} [@keras] with the [TensorFlow]{} [@tenserflow] backend. We took the union of all the 11 model predictions and visually inspected each one of the resulting FRB candidates to identify astrophysical pulses. We found two bursts (hereafter GBT-1 and GBT-2) at similar DM to that of FRB171019 in the observations.
GBT Periodicity Searches
------------------------
We also conducted a search for periodicity in the GBT data using Fourier domain searching with the PRESTO routine [accelsearch]{}, as well as time domain searching using the Fast Folding Algorithm (FFA[^8]) package [riptide]{}. Before searching, frequency channels and time blocks significantly affected by RFI were identified using [rfifind]{} and masked. The data were corrected for dispersion over 240 trial DMs evenly spaced from 400 to 520 ${{\rm pc~cm^{-3}}}$, generating a time series at each trial. We used [dedisp]{} [@barsdell12], a GPU-accelerated package, to create time series. The FFA-based periodicity search was carried out to find long-period signals, where we searched periods ranging from 0.2 to 10s. We detected no significant periodic astrophysical signal in the data above a S/N threshold of 10 (chosen to minimize the number of false-positive candidates).
The Repeat bursts {#sec:repeats}
=================
The two repeat bursts were detected in 820 MHz GBT observations 9 and 20 months after the initial ASKAP detection, and are marked with red circles in Figure \[fig:timeline\]. The dynamic spectra of the bursts are shown in Figure \[fig:repeaterplots\], along with the original detection at ASKAP. To measure the width of the bursts, we fit the frequency-averaged pulse profile with a Gaussian model and report the FWHM[^9] ; both bursts are approximately 4.5ms in duration. The residual after subtracting the best-fit model from pulse profiles appears to be white, thus there is no underlying temporal sub-structure in the dynamic spectrum of either repeat burst. The burst durations are well in excess of the maximum DM smearing across a channel for the GBT data, which is 1.0 ms. For reference, we also calculate the properties of the ASKAP detection. The time resolution for ASKAP data is 1.26 ms with a maximum DM smearing of 2.66 ms present within a channel. All three bursts are visible in the lower half of the band but not detected in the top half. Thus, the lower sub-band fluences are larger than the full-band averaged values. The burst properties obtained from the full band as well as from the lower half of the band are listed in Table \[tab:burstsproperties\]. The spectral structures of the bursts are described in Section \[sec:substructure\].
Scattering and dispersion analysis
----------------------------------
To obtain scattering timescales and burst DMs, we perform multi sub-band modeling of the burst pulse profiles using the nested sampling method `Dynesty` [@dynesty] implemented in the parameter estimation code `Bilby` [@bilby]. We model each of the pulse profiles to be a Gaussian convolved with an exponential pulse-broadening function. The broadening time $\tau$ is assumed to vary with frequency with a fixed index, $\tau \propto \nu^{-4}$. We model both interchannel dispersion delay (which causes the pulse to arrive at different times in different sub-bands) and intrachannel dispersion smearing (which increases the pulse width in quadrature with an intrinsic width). Based on the ratio of Bayesian evidence between models with and without scattering, we conclude that the data do not support presence of scattering. For the ASKAP pulse, we limit the scattering timescale to be $ < 3.52$ ms, at a reference frequency of 1GHz. For the repeat bursts, we group the lower half band of the data into four sub-bands to perform the analysis; we limit the scattering time scales (referenced to 1 GHz) to be $< 0.79$ ms and $< 1.77$ ms for GBT-1 and GBT-2 respectively. In contrast, the optimized DMs of the bursts shown in Table \[tab:burstsproperties\] suggest that the repetitions have a different apparent DM than the higher-frequency ASKAP detection.
Polarization Properties
-----------------------
We extracted the GBT/GUPPI data for the detected repeat bursts using `dspsr` [@dspsr] producing a full-Stokes archive file. We found no evidence for linear or circular polarization in the pulse data. It is possible that the non-detection of linear polarization is the result of Faraday rotation of the burst through magnetized plasma. We searched for Faraday rotation using the PSRCHIVE (@psrchive [@psrchive2]) `rmfit` routine in the range ${\rm |RM|} \leq 3 \times 10^4 \,{\rm rad\, m^{-2}}$ (this is the rotation measure (RM) at which the polarization position angle rotates by one radian in one frequency channel at the center of the band), but no significant RM was found. We note that no polarization calibration procedures were conducted during GBT observations. For the ASKAP burst, only the total intensity data were retained; hence, no polarimetric properties could be derived from this burst.
Spectral Properties {#sec:substructure}
-------------------
The spectrum for each burst shown in Figure \[fig:spectrabursts\] is formed by integrating the signal over the time samples within twice the measured FWHM of the frequency-averaged pulse. The amplitude of each spectrum was then scaled to fluence, using the system equivalent flux density (SEFD) and the radiometer equation. Modest changes to the window do not significantly affect estimates of fluence. All three bursts show lower fluences at higher frequencies. One possibility is that the bursts have steep spectra. We characterize this by fitting a power-law model $ E_{\nu} \propto \nu^{\alpha} $. Spectral indices, $ \alpha $ obtained from the fits to individual spectra are in Table \[tab:burstsproperties\]. All three bursts show steep spectra in the observed bands with $ \alpha $ ranging from $-$13 to $-$8. While both the ASKAP burst and the GBT-2 spectra is extremely steep in the lower half band as well, the GBT-1 spectrum is nearly flat.
Off-axis attenuation is unlikely to significantly change the fluences or spectral indices of the repetitions. Based on the posterior distribution from the ASKAP multi-beam localization in [@askap_nature], the median correction to the fluence results in an increase of 8%, and is $< 24\%$ with 90% confidence. The median spectral index correction is $-0.07$, and with 95% confidence is less than $< - 0.2$. This analysis assumes the GBT beam can be modeled as a Gaussian with an FWHM (beamwidth) of $15\arcmin$ at 820 MHz (Table \[tab:followupobs\]). We therefore rule out any primary beam offset as the cause of the observed steep spectra for the GBT pulses.
The spectral modulation in the bursts could be intrinsic to the emission or due to the propagation effects. To characterize this, we calculate the autocorrelation function (ACF) of the burst spectra [@wael_frb] as shown in Figure \[fig:spectrabursts\]. We fit the ACF with Gaussian component models using a non-linear optimization approach [@lmfit] to find the frequency scales of characteristic modulation in spectra. We detect two characteristic frequency scales in the ASKAP spectrum of band extent 13 and 147 MHz. For the GBT-1 spectrum, the ACF can be best described with a single component (100 MHz), which is the total bandwidth over which the pulse is visible. We observe a bright spike in the spectrum (at $\sim 776$ MHz), but its width is comparable to the channel width. It is unclear if this is astrophysical or RFI. For the GBT-2, apart from the frequency scale of 82 MHz, we also see marginal evidence for a second component (7 MHz wide). However, because the second component is not present in an analysis of the lower half of the band where the burst is bright, it is most likely due to RFI or noise fluctuations. We also estimate the amplitude of the spectra variability using the square of the modulation index $ m^2$, by computing the mean-normalized spectral autocovariance [@jp_askapfrbs] from the spectrum of bursts. The estimated values of $ m^2$ for the three bursts are 2.4, 1.1, and 1.9 respectively.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
![Posterior distributions for burst rate parameters. Panels A, C, and F show the one-dimensional marginalized distributions for $R_0$, $\gamma$, and $\beta$, with peak probability densities normalized to unity. Panels B, D, and E show the two-dimensional distributions (normalized again such that peak probability density are unity), with grayscale shown in panel G. The rate $R_0$ has been scaled to ASKAP sensitivities and frequencies ($52$Jyms; see Table \[tab:followupobs\]). []{data-label="fig:rate_corner"}](corner-eps-converted-to.pdf "fig:")
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
Inferring the repetition rate
-----------------------------
We use Bayesian methodology to characterize the repetition statistics of FRB171019, given the detection of pulses with ASKAP at 1.3 GHz and the GBT at 820 MHz, and the non-detections with the GBT at 1.5 GHz and Parkes at 1.3 GHz. We assume that the cumulative burst rate above a fiducial fluence $S$ at a frequency $\nu$ is $$R(>S, \nu) = R_0 \left(\frac{S}{S_0(\nu)} \right)^\gamma,$$ where $R_0$ is the rate of bursts above fluence $$S_0(\nu) = S_0 \left(\frac{\nu}{\nu_0}\right)^{\beta}$$ at a frequency $\nu_0$.
We assume that the burst event rate in a survey $i$ of total integration time $T_i$ will follow Poisson statistics with rate parameter $\lambda_i = T_i R(>S_i, \nu_i)$, where $\nu_i$ is the observing frequency of the survey and $S_i$ is the survey sensitivity. In this case we can infer the parameters in the survey $R_0$, $\beta$, and $\gamma$ using the likelihood $$L = \prod_{i=1}^{N_s} \frac{1}{n_i!} e^{- \lambda_i} \left( \lambda_i \right)^{n_i},$$ where $n_i$ is the number of bursts found in survey $i=1$ to $N_s$.
We sample the posterior distribution using the `multinest` algorithm [@2009MNRAS.398.1601F] assuming uniform priors on $\beta$ and $\gamma$ ($-10 < \beta, \gamma < 10$), and logarithmic priors on $R_0$ between $10^{-6}$ and $1$hr$^{-1}$, where the reference frequency $\nu_0=1.3$ GHz and sensitivity $S_0=52$Jyms. We do not take into account the spectral index obtained for bursts (Table \[tab:burstsproperties\]) in this repetition analysis, which allows for an independent estimation of the spectral index. The posterior distribution is shown in Figure \[fig:rate\_corner\]. We find that the slope of the burst intensity distribution is consistent with a power-law distribution with an index between $-1.5 \lesssim \gamma \lesssim 0$. The value depends strongly on the spectral dependence of the burst emission rate $\beta$. The inferred steep values of $\beta$ ($\beta \ll -1.5 $, with the lower prior acceptable) are consistent with the observed spectra (in the case, the spectrum is attributed to a steep power-law process), but inconsistent with the ASKAP population overall [@jp_askapfrbs]. The observed shallow values of $\gamma$ are consistent with observations of the first repeating FRB121102 [@law_repeating].
Discussion and Conclusions {#sec:discussion}
==========================
The bursts in FRB171019 extend over the range of 219 Jyms to 0.37 Jyms, a fluence range of $\sim$ 590. At a similar frequency range, this is a factor of $\sim$ 3 larger than what has been observed in FRB121102 (@gourdji_lower_limit [@Hessels19]) and an order of magnitude larger than any other repeating FRB source [@chime_mega_repeaters]. The wide range in observed fluences shows that, like Galactic pulsars and magnetars, repeating FRB sources can emit pulses with a wide range of luminosities, and that repeating sources can emit bright pulses like the initial ASKAP detection. The inferred isotropic peak luminosity of bursts ranges from $L\sim6\times 10^{43}$ erg s$^{-1}$ to $L\sim6\times 10^{40}$ erg s$^{-1}$, nearly 3 orders in magnitude. Models for burst emission need to account for this wide range.
We find evidence for variations in the apparent DMs of the pulses. It is unclear whether the difference is genuine DM variation or due to non-dispersive effects as has been observed in FRB121102 [@Hessels19]. We note that this discrepancy in apparent DM can also be due to the different volumes of the medium being probed by the ASKAP and the GBT. All three bursts are temporally resolved with similar widths. We note that the pulse width of the GBT-2 is less reliable when measured in the whole band due to the presence of RFI in the upper half of the band. However, taking the DM smearing and sampling time into account, the intrinsic width of all bursts are consistent within uncertainties. We find no evidence for sub-structure in the pulse profile as seen in other FRBs (@wael_frb [@Hessels19]). We would be insensitive to any sub-structure narrower than $\sim$ 1 ms in GBT detections.
The band extent of spectral features differs between ASKAP and GBT pulses and is inconsistent with diffractive scintillation. The burst exhibited a large degree of spectral modulation in the original ASKAP detection. It was not clear whether the bright structures were intrinsic to the burst or due to propagation effects [@jp_askapfrbs]. If the spectral structures observed in the ASKAP detection were the result of diffractive scintillation, we would expect the band extent of the structures present in the GBT pulses to be factor of $(\nu_{\rm GBT}/\nu_{\rm ASKAP})^{-4} \approx 6 $ smaller. The widest structures in the ASKAP burst (width approximately half the band) would be observed to be $\sim 25$MHz wide in the GBT spectrum. However, we only see evidence for structures much wider than this in the GBT observations. We do not find any conclusive evidence of diffractive scintillation in repeat bursts.
All of the bursts from FRB171019 are only visible in lower half of their respective bands, which could be evidence of an extremely steep spectrum. This argument is also consistent with the non-detection of repetitions at Parkes and GBT L-band receivers. If we assume this steep spectrum ($ \sim -9$) to be the case, it provides a very natural way to understand the detection of repetitions from this source in the context of all the non-detections (@askap_nature [@james_limits]) from other ASKAP FRBs (assuming a non-negligible fraction are repeaters). It would make the repeat bursts at least a factor of $(\nu_{\rm ASKAP}/\nu_{\rm GBT})^{9} \approx 60 $ fainter at the center frequency of ASKAP. In that scenario, the fluence discrepancy between ASKAP and the GBT detection is actually $ > 10^4 $, assuming a constant spectral index that makes FRB 171019 special within the ASKAP population of flatter-spectrum FRBs [@jp_askapfrbs]. However, we are cautious not to over-interpret this result, as there are not many physical mechanisms to produce such a steep spectrum. It is quite possible that the spectrum is similar to patchy emission, seen in the other repeater FRB sources (@michilli_repeater1 [@chime_mega_repeaters]). This is evident as the GBT-1 spectrum is nearly flat in the lower half of the band. Also, the ASKAP detection has a large spectral modulation that can not be explained by scintillation. In this scenario, the power-law model might not be the correct approach for the spectral index measurement [@mwa_steep].
Another possibility is FRBs having stochastic patchy or modulated emission in different parts of the frequency band for an individual burst but, when ensemble-averaged, produce steep spectra as observed in ASKAP one-off FRBs sample (@askap_nature [@jp_askapfrbs]). This would be tested with further detections.
The other published repeating burst sources (@repeater_2 [@Hessels19]) share common features such as spectra variability, sub-structures in their dynamic spectrum, and sub-components in pulse profile. We do not observe any of these features in all three bursts. A coherently dedispersed detection from FRB171019 with high time resolution will provide more information on these distinctions. The bursts from the FRB171019 source are fainter at higher frequencies, which is not the case with many of the bursts from the FRB121102 source, where bursts have been reported brighter at higher frequencies [@gourdji_lower_limit]. The first detection of FRB171019 comes from a different sample of bright FRBs [@askap_nature] than the CHIME detections [@chime_frbs] and FRB 121102. The host galaxy of a localized burst [FRB180924; @askap_localized] from the ASKAP population also originates from a galaxy significantly different to that of FRB121102. It will be interesting to see if all repeating FRBs have similar environments as of FRB121102. If not, it could be indicative of a different channel for producing repeat burst sources. The detection of further repetitions from this source[^10] and localization to a host galaxy will be key to understanding the nature of FRB171019 and its relation to other repeating burst sources.
We thank C. James for useful discussions. P.K., R.M.S., S.O., and R.S. acknowledge support through Australian Research Council (ARC) grant FL150100148. R.M.S. and J.P.M. acknowledge support through ARC grant DP180100857. R.M.S. also acknowledges support through ARC grant CE170100004. D.R.L. was supported by the National Science Foundation through the award number OIA-1458952. This work was performed on the OzSTAR national facility at Swinburne University of Technology. OzSTAR is funded by Swinburne University of Technology and the National Collaborative Research Infrastructure Strategy (NCRIS). Work at NRL is supported by NASA. The Green Bank Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Parkes radio telescope is part of the Australia Telescope National Facility which is funded by the Australian Government for operation as a National Facility managed by CSIRO. The Australian SKA Pathfinder is part of the Australia Telescope National Facility which is managed by CSIRO. Operation of ASKAP is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy. ASKAP uses the resources of the Pawsey Supercomputing Centre. Establishment of ASKAP, the Murchison Radio-astronomy Observatory and the Pawsey Supercomputing Centre are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry Endowment Fund. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site.
[^1]: <http://www.frbcat.org>; visited 2019 August 26.
[^2]: <https://astronomy.curtin.edu.au/research/craft/>
[^3]: Analysis of the entire campaign is ongoing and will be reported elsewhere.
[^4]: <http://sigproc.sourceforge.net>
[^5]: <https://github.com/scottransom/presto>
[^6]: Number of individual boxcar/DM trials clustered into a candidate.
[^7]: All 11 trained models are taken from <https://github.com/devanshkv/fetch>
[^8]: Based on <https://bitbucket.org/vmorello/riptide>
[^9]: The measured FWHM values are consistent with the W50 estimates (width at 50% of pulse peak).
[^10]: We note that CHIME has recently detected a repeat burst from this source [@chime_atel].
|
---
abstract: |
For a spectrally positive strictly stable process with index in (1,2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval’s lower end point before jumping over its upper end point, and ii) the joint distribution of the time, the undershoot, and the jump of the process when it makes first exit the other way around. For i), the density of the time of first exit is expressed as an infinite sum of functions, each the product of a polynomial and an exponential function, with all coefficients determined by the roots of a function. For ii), conditional on the undershoot, the time and the jump of first exit are independent, and the marginal conditional densities of the time has similar features as i).
*Keywords and phrases.* Two-sided exit problem; process; stable; spectrally positive;
2000 Mathematics Subject Classifications: Primary 60G51; Secondary 60E07.
---
**Law of two-sided exit by a spectrally positive strictly stable process[^1]**\
Zhiyi Chi\
Department of Statistics\
University of Connecticut\
Storrs, CT 06269, USA,\
E-mail: zhiyi.chi@uconn.edu\
Introduction
============
The so-called exit problems, which concern the random event that a stochastic process gets out of a set for the first time, occupy a prominent place in the study of processes. For spectrally positive processes, years of intensive research have revealed many remarkable facts about first exit from a bounded closed interval [@bertoin:96:cup; @kyprianou:14:sh; @doney:07:sg-b]. An essential tool for many of the results is the scale function. Since the function can be analytically extended to $\Coms$ ([@kyprianou:14:sh], Lemma 8.3), it is amenable to treatments by complex analysis. By combining the scale function and residual calculus, this paper obtains series expressions of the distribution of first exit of a spectrally positive strictly stable process with index in $(1,2)$.
Henceforth, without loss of generality, let $X$ be a process with $$\begin{aligned}
\label{e:X}
\mean [e^{-q X_t}] = e^{t q^\alpha}, \quad t>0,\ q>0,\ \alpha\in
(1,2).\end{aligned}$$ Given $b,c\in (0, \infty)$, first exit by $X$ from interval $[-b,c]$ consists of two possibilities: either the process makes a continuous downward passage of $-b$ before it makes an upward jump across $c$ or the other way around. The probability of each possibility is well-known (cf. [@bertoin:96:cup], Theorem VII.8). Meanwhile, the scale functions of the first exit have been known [@kyprianou:14:sh]. On the other hand, not much is known about the explicit joint probability density function () for the first exit.
In [@chi:18:tr], by using Laplace transform, the distribution of first upward passage of a fixed level by $X$ is obtained. It turns out that the method used there can be extended to first exit from an interval. Section \[s:lower\] considers the time of first exit from $[-b,c]$ at $-b$. It will be shown that the probability density function () of the time has an expression in terms of the residuals of a certain function at the roots of a function, and as a result, is of the form $f(t) = \sum_\root p_\root(t)
e^{t\root}$, where the sum runs over the roots and for each root $\root$, $p_\root(t)$ is a polynomial in $t$ whose coefficients are determined by $\root$ and several functions. For all but a finite number of $\root$, $p_\root(t)$ is of order 0. The result provides a connection to some known results on first exit of a standard Brownian motion. It also highlights the importance of gaining more information on the roots of functions [@popov:13:jmathsci]. Section \[s:upper\] considers the joint distribution of the time, the undershoot, and the jump of $X$ when its first exit from $[-b,c]$ occurs at $c$. It will be shown that conditional on the undershoot, the time and the jump are independent. This allows the joint distribution to be factorized into the marginal of the undershoot, and the marginal conditional ’s of the time and the jump, respectively. The expression of the marginal conditional of the time has similar features as the one of first exit at the lower end. This is in contrast to the power series expression of the time of first upward passage of $c$ [@chi:18:tr; @bernyk:08:ap; @simon:11:sto; @peskir:08:ecp], even though the first passage can be regarded as the limit of first exit from $[-b,c]$ as $b\toi$. In both sections, the asymptotics of the time of first exit as $t\to 0$ or $\infty$ are also considered.
The rest of the section fixes notation and collects some background information for later use.
#### Integral transforms.
For $f\in L^1(\Reals)$, denote its Laplace transform and Fourier transform, respectively, by $$\begin{gathered}
\LT f(z) = \int e^{-t z} f(t)\,\dd t
\quad\text{and}\quad
\FT f(\theta) = \LT f(-\iunit \theta), \quad \theta\in\Reals.\end{gathered}$$ The domain of $\LT f$ is $\{z\in\Coms: e^{-t z} f(t)\in L^1(\dd t)\}$. Similarly, for a finite measure $\mu$ on $\Reals$, denote its Laplace transform and Fourier transform, respectively, by $$\begin{gathered}
\LT\mu(z) = \int e^{-t z} \mu(\dd t)
\quad\text{and}\quad
\FT\mu(\theta) = \LT\mu(-\iunit\theta),\quad \theta\in\Reals.\end{gathered}$$ The domain of $\LT\mu$ is $\{z\in\Coms: e^{-t z} \in L^1(\dd t)\}$.
Let $z_0\in\Coms$. Denote $U_r(z_0) = \{z\in\Coms: |z-z_0|<r\}$. If function $g$ is analytic in $U_r(z_0) \setminus\{z_0\}$ for some $r>0$ and has $z_0$ as a pole, possibly removable, then the residual of $g$ at $z_0$ is $$\begin{aligned}
\res(g(z), z_0) = c_1 = \frac1{2\pi\iunit} \oint_\gamma g(z)\,\dd z,\end{aligned}$$ where $\gamma$ is any counterclockwise simple closed contour in $U_r(z_0)\setminus \{z_0\}$ ([@rudin:87:mcgraw], p. 224).
#### Some properties of a function.
A function with parameters $a>0$ and $b\in\Coms$ is an entire function defined as $$\begin{aligned}
E_{a, b}(z) = \sumzi n \frac{z^n}{\Gamma(a n + b)},
\quad z\in\Coms.\end{aligned}$$ Let $\Scr Z_{a, b} = \{z\in\Coms: E_{a,b}(z) = 0\}$. The focus will be mostly on $E_{\alpha, \alpha}(z)$ with $\alpha\in (1,2)$. By [@popov:13:jmathsci], Theorems 1.2.1, 1.4.2, and 1.5.1, $$\begin{aligned}
\label{e:asym-infty}
E_{\alpha, \alpha}(z)
=
\alpha^{-1} z^{\ralph-1} \exp(z^\ralph)
-
\frac{(\alpha - 1)\alpha z^{-2}}{\Gamma(2-\alpha)}
+ O(|z|^{-3}), \quad |z|\toi,\end{aligned}$$ where the $O(\cdot)$ term is uniform in $\arg z$ and the principle branch of $z$ is used so that $\arg(e^{\iunit\theta}) = \theta -
2k\pi$ for any $\theta\in \Reals$ with $k$ the unique integer satisfying $2k-1 < \theta/\pi \le 2k+1$. Furthermore, from Theorems 2.1.1 and 4.2.1, and Chapter 6 in [@popov:13:jmathsci],
1. $\Scr Z_{\alpha,\alpha}$ has infinitely many elements and all those with large enough modulus are simple roots of $E_{\alpha,
\alpha}(z)$ and can be enumerated as $\root_{\pm n}$, $n\ge N$, for some large $N$, such that $$\begin{aligned}
\label{e:zero-asym}
\root_{\pm n}
=
[1+o(1)](2\pi n)^\alpha e^{\pm\iunit\alpha\pi/2},
\quad n\toi;
\end{aligned}$$
2. $\Scr Z_{\alpha,\alpha}\subset \{z: |\arg(z)| >
\alpha\pi/2\}$; and
3. $E_{\alpha, \alpha}(z)$ has a finite positive number of real roots, all being negative.
#### First passage time and hitting time.
For $c>0$ and $x\in\Reals$, denote $T_c = \inf\{t>0: X_t>c\}$ and $\tau_x = \inf\{t>0:\ X_t = x\}$. Under the law of $X$, $T_c <
\tau_c < \infty$ and $X_{T_c} > c > X_{T_c-}$ a.s. [@simon:11:sto], both $T_c$ and $\tau_x$ have ’s [@bertoin:96:cup; @bernyk:08:ap; @peskir:08:ecp; @simon:11:sto], and given $b>0$, as downward movement is continuous and 0 is regular for $(-\infty, 0)$, $\tau_{-b} = \inf\{t: X_t < -b\}$ ([@doney:07:sg-b], Theorem 5.17), so the time of first exit from $[-b,c]$ is $\min(\tau_{-b}, T_c)$. When $\tau_{-b} < T_c$ (resp. $\tau_{-b} > T_c$), $X$ is said to make first exit from $[-b,c]$ at the lower (resp. upper) end. Denote by $g_t$ the of $X_t$ and by $f_x$ the of $\tau_x$. The distribution of $\tau_x$ is classical for $x<0$ ([@bertoin:96:cup], Theorem VII.1) and is also known for $x>0$ [@simon:11:sto].
We will rely on the scale function $W\Sp q$ of $-X$ for the derivation ([@kyprianou:14:sh], Chapter 8; also cf. [@bertoin:96:cup; @doney:07:sg-b]). For the spectrally negative strictly stable process $-X$, $$\begin{aligned}
\label{e:ss-scale}
W\Sp q(x) = x^{\alpha-1}_+ E_{\alpha,\alpha}(q x^\alpha_+), \quad
q\ge 0,\end{aligned}$$ where $x_+ = \max(x,0)$ ([@kyprianou:14:sh], p. 250).
Distribution of first exit time at lower end {#s:lower}
============================================
Given $c>0$ and $x<c$, denote by $$\begin{aligned}
k_{x,c}(t)
=
\frac{\pr\{\tau_x\in\dd t,\, X_s<c\,\forall s\le t\}}{\dd t}
=
\frac{\pr\{\tau_x\in\dd t,\, T_c>\tau_x\}}{\dd t}.\end{aligned}$$ Since $\tau_x$ has a , $k_{x,c}(t)$ is well-defined, and since its integral over $t$ is $\pr\{T_c > \tau_x\}<1$, it is a sub-rather than a proper one. For $b>0$ and $c>0$, it is well-known that ([@kyprianou:14:sh], Theorem 8.1) $$\begin{aligned}
\label{e:LT-k}
\LT{k_{-b,c}}(q) = \frac{W\Sp q(c)}{W\Sp q(b+c)}
=
\frac{c^{\alpha-1}}{(b+c)^{\alpha-1}}
\frac{
E_{\alpha, \alpha}(c^\alpha q)
}{
E_{\alpha, \alpha}((b+c)^\alpha q)
}.\end{aligned}$$
\[p:exit-lower\] Given $b>0$ and $c>0$, put $s = c^\alpha/(b+c)^\alpha$. Then $$\begin{aligned}
\label{e:k-psi}
k_{-b,c}(t)
=
\frac{c^{\alpha-1}}{(b+c)^{2\alpha-1}}
\psi_s\Grp{\frac{t}{(b+c)^\alpha}},
\end{aligned}$$ where $$\begin{aligned}
\label{e:residual}
\psi_s(t)
=
\sum_{\root\in\Scr Z_{\alpha,\alpha}} \res(H_s(z) e^{t z},
\root), \quad t>0,
\end{aligned}$$ is a concentrated on $[0,\infty)$ and for $s\in [0,1)$ $$\begin{aligned}
\label{e:H_s(q)}
H_s(z) = \frac{
E_{\alpha, \alpha}(s z)
}{
E_{\alpha, \alpha}(z)
}, \quad z\in\Coms.
\end{aligned}$$ Furthermore, $\psi_s\in C^\infty(\Reals)$ such that for all $n\ge
1$, $\psi\Sp n_s(x)\to 0$ as $x\dto 0$ or $x\toi$.
Since $\pr\{\tau_{-b} < T_c\} = \pr\{\tau_{-b} < \tau_c\} =
\lfrac{c^{\alpha-1}}{(b+c)^{\alpha-1}}$ ([@bertoin:96:cup], Theorem VII.8), by , conditional on the event that $X$ makes first exit from $[-b,c]$ by hitting $-b$, the scaled exit time $(b + c)^\alpha\tau_{-b}$ has $\psi_s$ with $s
=c^\alpha/(b+c)^\alpha$.
The main feature of Proposition \[p:exit-lower\] is that it expresses the of the time of first exit in terms of the roots of the function $E_{\alpha, \alpha}(z)$. As one may suspect, the expression results from residual calculus for as a meromorphic function on $\Coms$. However, since currently there is little precise knowledge on the values of the roots of $E_{\alpha,\alpha}$, the contour involved in the calculation has to be chosen carefully. For each term in the sum , if $\root\in \Scr Z_{\alpha, \alpha}$ has multiplicity $n$, then for $t>0$, $$\begin{aligned}
\res(H_s(z) e^{t z}, \root)
=
\nth{(n-1)!}
\lim_{z\to\root}
\frac{\dd^{n-1}}{\dd z^{n-1}}\Sbr{
\frac{E_{\alpha, \alpha}(s z) e^{t z}}
{E_{\alpha, \alpha}(z)/(z-\root)^n}
},\end{aligned}$$ which has the form $\sum^{n-1}_{k=0} c_k(\root) t^{n-1-k} e^{t
\root}$. Moreover, if $\root$ has large enough modulus, then it is a simple root ([@popov:13:jmathsci], Theorem 2.1.1), giving $$\begin{aligned}
\res(H_s(z) e^{t z}, \root)
=
\frac{E_{\alpha, \alpha}(s\root) e^{t\root}}
{E'_{\alpha,\alpha}(\root)}. \end{aligned}$$
Proposition \[p:exit-lower\] is an extension of a similar result on first exit of a standard Brownian motion. If $\alpha=2$, then by $\mean[e^{-q X_t}] = e^{t q^2}$ for $q>0$, $X_t = B_{2t}$ with $B_t$ a standard Brownian motion. By $E_{2,2}(z) = \sinh(\sqrt z)/\sqrt z$, $\Scr Z_{2,2} = \{-k^2\pi^2, k\in\Nats\}$ and $E'_{2,2}(z) =
[\cosh(\sqrt z)- E_{2,2}(z)]/(2 z)$. Since $E'_{2,2}(-k^2 \pi^2) =
(-1)^{k-1}/(2k^2\pi^2)$, each root is simple. Then from the above display with $s = c^2/(b+c)^2$ and Proposition \[p:exit-lower\], $$\begin{aligned}
\frac{\pr\{\tau_{-b} \in \dd t,\, \tau_c>\tau_{-b}\}}{\dd t}
=
\frac{2\pi}{(b+c)^2} \sumoi k (-1)^{k-1} k \sin\Grp{
\frac{k\pi c}{b+c}
}
\exp\Cbr{-\frac{k^2\pi^2 t}{(b+c)^2}}.\end{aligned}$$ The series expansion is different from the one in [@borodin:02:bvb] (p. 212, 3.0.6). However, it can be proved using the heat equation method ([@morters:10:cup], section 7.4); see for example [@chi:16:tr] for a derivation.
Following a heuristic for a standard Brownian motion (cf. [@morters:10:cup], p. 217), one can get a different expression of $k_{-b,c}$ analogous to the one for the standard Brownian motion in [@borodin:02:bvb] (p. 212, 3.0.6): $$\begin{aligned}
\nonumber
k_{-b,c}
&=
f_{-b} - f_c * f_{-b-c} + f_{-b} * f_{b+c} * f_{-b-c} -
f_c * f_{-b-c} * f_{b+c} * f_{-b-c} + \cdots
\\\label{e:k-alt}
&=
\sumzi n f_{-b}* (\delta - f_c*f_{-c}) * (f_{b+c} * f_{-b-c})^{*n},\end{aligned}$$ where all the terms involved are taken as ’s of time, $\delta$ is the degenerate distribution at 0, and $p^{*0}:=\delta$ for any $p$. Indeed, thinking of $f_{-b}(t)$ as the probability that $X$ hits $-b$ at time $t$ for the first time, and $k_{-b,c}(t)$ as the one that $X$ does so before it ever hits $c$, $f_{-b}(t) - k_{-b,c}(t)$ is the probability that $X$ does so after it hits $c$, so by strong Markov property, $$\begin{aligned}
\label{e:k-k}
k_{-b,c}(t) = f_{-b}(t) - (k_{c,-b} * f_{-b-c})(t),\end{aligned}$$ where we have used the extended definition $k_{x,c}(t)=\lfrac{\pr\{\tau_x \in \dd t,\, \tau_c > \tau_x\}}{\dd
t}$ for any $x$, $c\in\Reals$. Likewise, $k_{c,-b}(t) = f_c(t) -
(k_{-b,c} * f_{b+c})(t)$. Then iterating the two identities gives . A rigorous proof of will be considered in Section \[ss:exit-lower-alt\].
Properties of scaled first exit time at lower end
-------------------------------------------------
This subsection considers some properties of $H_s(z)$ as defined in . Along the way it proves the smoothness of $\psi_s$ stated at the end of Proposition \[p:exit-lower\].
From and scaling, it follows that for any $s\in
(0,1)$, $H_s(q)$ is the Laplace transform of the probability distribution $$\begin{aligned}
\mu_s(\dd t)
=
\pr\{\tau_{s^\ralph-1}
\in\dd t\gv \tau_{s^\ralph - 1} < T_{s^\ralph}\}, \quad
t>0,\end{aligned}$$ i.e., for $q>0$, $$\begin{aligned}
\label{e:mu_s}
\LT{\mu_s}(q) = H_s(q)
=
\frac{E_{\alpha,\alpha}(sq)}{E_{\alpha,\alpha}(q)}.\end{aligned}$$ It is seen that the identity holds for all $q\in\Coms$ with $\Re(q)
\ge0$. By , given $s\in (0,1)$, $H_s(q)$ is completely monotone in $q\ge 0$, and for $z\in\Coms$ with $\Re(z)\ge0$, $|H_s(z)|\le1$, and so $|E_{\alpha, \alpha}(s z)|\le |E_{\alpha,
\alpha} (z)|$. Thus for any $\theta\in [-\pi/2,\pi/2]$, $|E_{\alpha, \alpha}(e^{\iunit \theta} r)|$ is increasing in $r\ge0$, so letting $r\to0+$, $$\begin{aligned}
|E_{\alpha, \alpha}(z)|\ge E_{\alpha, \alpha}(0)
=
\frac1{\Gamma(\alpha)},
\quad \Re(z)\ge 0.\end{aligned}$$
Fix $s\in (0,1)$. For $\theta\in\Reals$, $$\begin{aligned}
\label{e:FT}
\FT{\mu_s}(\theta) = H_s(-\iunit \theta)
=
\frac{E_{\alpha, \alpha}(-\iunit s\theta)}
{E_{\alpha, \alpha}(-\iunit \theta)}.\end{aligned}$$ On the other hand, $|e^{(-\iunit \theta)^\ralph}| = e^{\lambda
|\theta|^\ralph}$ with $\lambda =\cos(\alpha^{-1} \pi/2)>0$, so by , $$\begin{aligned}
\label{e:asym-infty2}
|E_{\alpha, \alpha}(-\iunit\theta)| \sim \alpha^{-1}
|\theta|^{\ralph-1} e^{\lambda|\theta|^\ralph}, \quad
\theta\to\pm\infty\end{aligned}$$ where $x\sim y$ means $x/y\to1$. Then from , $|\FT{\mu_s}(\theta)| \sim s^{\ralph-1} e^{\lambda(s^\ralph-1)
|\theta|^\ralph}$. As a result, $\int |\FT{\mu_s}(\theta)|
|\theta|^n \,\dd \theta < \infty$ for all $n\ge 0$, so $\mu_s$ has a in $C^\infty(\Reals)$ with vanishing derivative of any order at $\pm\infty$ ([@sato:99:cup], Proposition 28.1). By , the is exactly $\psi_s$ in Proposition \[p:exit-lower\]. Since $\psi_s$ is supported on $[0,\infty)$, it is seen that $\psi\Sp n_s(x)\to 0$ as $x\to0+$.
From and the Continuity Theorem of characteristic functions (cf. [@breiman:92:siam], Theorem 8.28), as $s\to 0+$, $\mu_s$ weakly converges to a probability distribution $\mu_0$ with $$\begin{aligned}
\FT{\mu_0}(\theta) =
H_0(-\iunit\theta) =
\frac1{\Gamma(\alpha)E_{\alpha, \alpha}(-\iunit\theta)},
\quad\theta\in \Reals.\end{aligned}$$ Similar to $\mu_s$ with $s\in (0,1)$, $\mu_0$ has a $\psi_0\in
C^\infty(-\infty, \infty)$ with support on $[0,\infty)$ such that all its derivatives $\psi\Sp n_0(x)$ vanish as $x\to 0+$, $\infty$. Consequently, for each $s\in [0,1)$, $\psi_s$ cannot be analytically extended to a neighborhood of 0, otherwise, $\psi_s$ would be constant 0. On the other hand, by Fourier inversion ([@rudin:87:mcgraw], p. 185), $$\begin{aligned}
\label{e:exit-lower-ift}
\psi_s(t)
=
\nth{2\pi} \intii \FT{\mu_s}(\theta) e^{-\iunit\theta t}\,\dd\theta
=
\nth{2\pi} \lim_{M\toi} \int^M_{-M}
\FT{\mu_s}(\theta) e^{-\iunit\theta t}\,\dd\theta,\end{aligned}$$ and from , $\mu_s$ has finite moment of any order, with the $n\th$ moment equal to $(-1)^n H\Sp n_s(0)$.
Contour integration
-------------------
This subsection completes the proof of Proposition \[p:exit-lower\]. Because directly follows from , it only remains to show .
Define function $$\begin{aligned}
\sigma(\theta) = \nth{|\sin(\theta/\alpha)|}.
\end{aligned}$$ Since $\alpha>1$, $\sigma(\theta)$ is bounded on $[-\pi, -\pi/2]\cup
[\pi/2, \pi]$. Fix $\beta\in (\pi/2, \pi/\alpha)$. For $R>0$, let $C_R$ be the contour that travels along the curve $$\begin{aligned}
\label{e:contour}
\{[R \sigma(\theta)]^\alpha e^{\iunit\theta}:\,
\pi/2\le |\theta|\le \pi\}
\end{aligned}$$ from $\iunit (R\sigma_0)^\alpha$ to $-\iunit (R\sigma_0)^\alpha$, where $\sigma_0 = \sigma(\pi/2)$. The contour is smooth except at its intersection with $(-\infty,0)$ and its length is proportional to $R^\alpha$. Fix $\beta \in (\pi/2, \alpha\pi/2)$. Let $C_{R,1}$ denote the part of $C_R$ in the section $\pi/2\le |\arg z|\le
\beta$, and $C_{R,2}$ the part in the section $\beta \le |\arg z|
\le \pi$.
For $z = re^{\iunit\theta}$ with $\theta = \arg z$, $|\exp(z^\ralph)| = \exp\{r^\ralph \cos(\theta/\alpha)\}$. If $z\in
C_{R,1}$, then $|\theta/\alpha|\le \beta/\alpha < \pi/2$, and so $\cos(\theta/\alpha)\ge \lambda:=\cos(\beta/\alpha)>0$. As a result, for $z\in C_{R,1}$, $$\begin{aligned}
\label{e:modulus-exp}
|\exp(z^\ralph)| \ge \exp(\lambda |z|^\ralph).
\end{aligned}$$ Then by , given $s\in (0,1)$, as $R\toi$, $$\begin{aligned}
H_s(z)
= \frac{E_{\alpha, \alpha}(s z)}{E_{\alpha, \alpha}(z)}
&=
(1+o(1))
\frac{(s z)^{\ralph-1} \exp((s z)^\ralph)}
{z^{\ralph-1} \exp(z^\ralph)}
\\
&=
(1+o(1)) s^{\ralph-1} \exp\{(s^\ralph-1) z^\ralph\}, \quad
z\in C_{R,1},
\end{aligned}$$ where the $o(1)$ term is uniform in $z\in C_{R,1}$. Since $|z|\ge
R^\alpha$, from , $$\begin{aligned}
\label{e:contour1}
\sup_{z\in C_{R,1}} |H_s(z)| = O(\exp\{-\lambda(1-s^\ralph) R\}).
\end{aligned}$$
We also need a bound for $|H_s(z)| = |E_{\alpha,\alpha}(s z) /
E_{\alpha, \alpha}(z)|$ on $C_{R,2}$. However, since $E_{\alpha,
\alpha}(z)$ has infinitely many roots in the section $\beta \le
|\arg z|\le \pi$, $R$ cannot be any large positive number but has to be selected appropriately. We need the following result.
\[l:contour\] Let $R_n = 2\pi n$, $n=1,2,\ldots$. Then given any $A\in
\Reals\setminus\{0\}$, $$\begin{aligned}
\label{e:dist}
\Linf_{n\toi} \inf_{z\in C_{R_n,2}}
|z^{\ralph+1} \exp(z^\ralph) - A|>0.
\end{aligned}$$
Assuming the lemma is true for now, let $A = \alpha^2(\alpha-1)/
\Gamma(2-\alpha)$. By , $$\begin{aligned}
E_{\alpha, \alpha}(z)
=
\alpha^{-1} z^{-2} [z^{\ralph+1} \exp(z^\ralph) - A] +
O(|z|^{-3}).
\end{aligned}$$ Then by Lemma \[l:contour\], there is $\rx>0$, such that for all large $n$ and $z\in C_{R_n,2}$, $E_{\alpha, \alpha}(z) \ge\rx
|z|^{-2}$. Let $m_0 = \sup_{\pi/2\le |\theta|\le\pi}
\sigma(\theta)$. Then by $|z|\le (m_0 R_n)^\alpha$, $$\begin{aligned}
E_{\alpha, \alpha}(z) \ge\rx m^{-2\alpha}_0 R^{-2\alpha}_n.
\end{aligned}$$ On the other hand, again by , $$\begin{aligned}
|E_{\alpha,
\alpha}(s z)|\le
E_{\alpha, \alpha}(|z|) \le E_{\alpha, \alpha}(m^\alpha_0
R^\alpha_n) = O(R^{1-\alpha}_n \exp(m_0 R_n)).
\end{aligned}$$ Combining with the lower bound, this implies $$\begin{aligned}
\label{e:contour2}
\sup_{z\in C_{R_n,2}} |H_s(z)|
= O(R^{1+\alpha}_n e^{m_0 R_n}), \quad n\toi.
\end{aligned}$$
Let $D_R$ be the domain bounded by $C_R$ and $\{\iunit\theta:
|\theta| \le (R\sigma_0)^\alpha\}$. Let $t>0$. If $C_R\cap \Scr
Z_{\alpha,\alpha} = \emptyset$, then by and residual theorem, $$\begin{aligned}
\nonumber
\nth{2\pi}
\int^{(R\sigma_0)^\alpha}_{-(R\sigma_0)^\alpha}
\FT{\mu_s}(\theta) e^{-\iunit\theta t}\,\dd\theta
&=
\nth{2\pi\iunit}
\int^{\iunit (R\sigma_0)^\alpha}_{-\iunit (R\sigma_0)^\alpha}
H_s(z) e^{t z}\,\dd z
\\\label{e:ift}
&=
\nth{2\pi\iunit}
\int_{C_R} H_s(z) e^{t z}\,\dd z -
\sum_{\root\in D_R\cap \Scr Z_{\alpha,\alpha}}
\res(H_s(z) e^{t z}, \root).
\end{aligned}$$ Consider the contour integral along $C_R$. For $z = r
e^{\iunit\theta}\in C_R$ with $\theta = \arg z$, by $\pi/2\le
|\theta|\le \pi$, $|e^{t z}|= e^{r t \cos\theta} \le 1$. Then by , $$\begin{aligned}
\Abs{\int_{C_{R,1}} H_s(z) e^{t z}\,\dd z}
&\le
\text{Length}(C_{R,1})\times O(e^{-\lambda(1-s^\ralph) R})
\\
&=
O(1) R^\alpha e^{-\lambda(1-s^\ralph) R}.
\end{aligned}$$ Furthermore, if $z\in C_{R,2}$, then by $\beta\le |\theta| \le\pi$ and $r\ge R^\alpha$, $|e^{t z}| \le e^{- b_0 R^\alpha t}$, where $b_0 = - \cos\beta>0$. Then by , $$\begin{aligned}
\nonumber
\Abs{\int_{C_{R_n,2}} H_s(z) e^{t z}\,\dd z}
&\le
\text{Length}(C_{R_n,2}) \times O(R^{1+\alpha}_n e^{m_0 R_n - b_0
R^\alpha_n t})
\\\label{e:contour3}
&=
O(1) R^{1+2\alpha}_n e^{m_0 R_n - b_0 R^\alpha_n t}.
\end{aligned}$$ By $\alpha>1$, combining the above two bounds yields $$\begin{aligned}
\int_{C_{R_n}} H_s(z) e^{t z}\,\dd z\to0.
\end{aligned}$$ Then by the Fourier inversion and , $$\begin{aligned}
\psi_s(t) =
\lim_{n\toi}
\sum_{\root\in D_{R_n}\cap \Scr Z_{\alpha,\alpha}}
\res(H_s(z) e^{t z}, \root), \quad t>0.
\end{aligned}$$
The above formula is proved for $s\in (0,1)$. For $s=0$, the formula can be similarly proved. To complete the proof, it only remains to show that the series on the of converges absolutely. Since in any bounded domain there are only a finite number of roots of $E_{\alpha,
\alpha}(z)$, it suffices to show that for a large enough $M>0$, $$\begin{aligned}
\sum_{\root\in \Scr Z_{\alpha,\alpha}\setminus U_M(0)}
|\res(H_s(z) e^{t z}, \root)|<\infty.
\end{aligned}$$
By Theorems 2.1.1 and Chapter 6 of [@popov:13:jmathsci], $M>0$ can be chosen such that all elements in $\Scr Z_{\alpha,\alpha}
\setminus U_M(0)$ are not real and are simple roots of $E_{\alpha,
\alpha}$. Then for each $\root\in \Scr Z_{\alpha,
\alpha}\setminus U_M(0)$, $$\begin{aligned}
\res(H_s(z) e^{t z}, \root)
=
\res\Grp{
\frac{E_{\alpha, \alpha}(s z) e^{t z}}{E_{\alpha, \alpha}(z)},
\root
}
=
\frac{E_{\alpha, \alpha}(s\root) e^{t\root}}
{E'_{\alpha,\alpha}(\root)}.
\end{aligned}$$ Letting $z = \root$ in the following identity (, p. 333) $$\begin{aligned}
\alpha z E'_{\alpha, \alpha}(z) + (\alpha-1) E_{\alpha, \alpha}(z)
=
E_{\alpha, \alpha-1}(z),
\end{aligned}$$ one gets $E'_{\alpha, \alpha}(\root) = E_{\alpha,
\alpha-1}(\root)/(\alpha\root)$ and hence $$\begin{aligned}
\label{e:ML-recursive}
\res(H_s(z) e^{t z}, \root)
=
\frac{\alpha \root E_{\alpha, \alpha}(s\root)e^{t\root}}
{E_{\alpha,\alpha-1}(\root)}.
\end{aligned}$$ By $E_{\alpha,\alpha}(\root)=0$ and , $$\begin{aligned}
\alpha^{-1} \root^{\ralph-1} \exp(\root^\ralph)
=
\frac{\alpha(\alpha-1) \root^{-2}}{\Gamma(2-\alpha)}
+ O(|\root|^{-3}).
\end{aligned}$$ On the other hand, by Theorems 1.2.1, 1.4.2, and 1.5.1 in [@popov:13:jmathsci], $$\begin{aligned}
E_{\alpha,\alpha-1}(\root)
=
\alpha^{-1} \root^{2/\alpha-1} \exp(\root^\ralph) +
\frac{\alpha(\alpha^2-1)\root^{-2}}{\Gamma(2-\alpha)}
+ O(|\root|^{-3}).
\end{aligned}$$ Combining the two displays yields $E_{\alpha,\alpha-1}(\root) \asymp
\root^{\ralph-2}$. Then by , $$\begin{aligned}
|\res(H_s(z) e^{t z},\root)|
=
O(1) |\root^{3-\ralph} E_{\alpha, \alpha}(s \root) e^{t\root}|.
\end{aligned}$$ From , as $|\root|\toi$, $$\begin{aligned}
|E_{\alpha,\alpha}(s
\root)| \le E_{\alpha, \alpha}(|\root|) = O(1) |\root|^{\ralph-1}
\exp(|\root|^\ralph).
\end{aligned}$$ On the other hand, let $\root = r e^{\iunit \theta}$ with $\theta =
\arg\root$. By $\alpha\pi/2 < |\theta|\le \pi$, $|e^{\root t}| =
e^{t r\cos\theta} \le \exp(-\lambda |\root| t)$, where $\lambda =
-\cos(\alpha\pi/2)>0$. Then $$\begin{aligned}
|\res(H_s(z) e^{t z},\root)|
=
O(1) |\root|^2 e^{|\root|^\ralph - \lambda t |\root|}
=
O(1) e^{-\eta|\root|}
\end{aligned}$$ for some $\eta = \eta(t)>0$. By , for $M>0$ large enough, all $\root\in \Scr Z_{\alpha,\alpha} \setminus U_M(0)$ can be enumerated as $\root_{\pm n}$, $n=N$, $N+1$, …, with $N\ge1$ being some large integer, such that $|\root_{\pm n}| \asymp
n^\alpha$. This combined with the above display then yields the desired absolute convergence.
For $z = [R \sigma(\theta)]^\alpha e^{\iunit\theta}\in C_{R,2}$ with $\theta = \arg z$, $$\begin{aligned}
z^{\ralph+1} \exp(z^\ralph)
&=
[R \sigma(\theta)]^{1+\alpha} e^{\iunit(\ralph+1)\theta}
\exp\{R \sigma(\theta) e^{\iunit\theta/\alpha}\}
\\
&=
[R \sigma(\theta)]^{1+\alpha} e^{R \sigma(\theta)
\cos(\theta/\alpha)}
e^{\iunit[(\ralph+1)\theta + R \sigma(\theta) \sin(\theta/\alpha)]}
\\
&=
[R \sigma(\theta)]^{1+\alpha} e^{R \sigma(\theta)
\cos(\theta/\alpha)}
e^{\iunit[(\ralph+1)\theta + R\text{sign}(\theta)]}.
\end{aligned}$$ Put $a(\theta, R)=[R \sigma(\theta)]^{1+\alpha} e^{R \sigma(\theta)
\cos(\theta/\alpha)}$. Then for $z\in C_{R_n,2}$, by $R_n = 2\pi
n$, $z^{\ralph+1} \exp(z^\ralph) = a(\theta, R_n) e^{\iunit(\ralph
+ 1)\theta}$. If there were $z_n = [R \sigma(\theta_n)]^\alpha
e^{\iunit \theta_n} \in C_{R_n,2}$ such that $z^{\ralph + 1}_n
\exp(z^\ralph_n) \to A$, then taking modulus, $a(\theta_n, R_n) =
[R_n \sigma(\theta_n)]^{1 + \alpha} e^{R_n \sigma(\theta_n)
\cos(\theta_n/\alpha)} \to |A|>0$. By $|R_n \sigma(\theta_n)|
\toi$, it follows that $\cos(\theta_n/\alpha)\to 0$, as any sequence $n$ with $R_n \sigma(\theta_n) \cos(\theta_n/\alpha)\toi$ (resp. $-\infty$) has $a(\theta_n, R_n)\toi$ (resp. 0). Because $|\theta_n|/\alpha \in (\pi/(2\alpha), \pi/\alpha]$, this implies $\theta_n/\alpha = k_n\pi/2 + \rx_n$ with $k_n = \pm1$ and $\rx_n\to0$. But then $$\begin{aligned}
z^{\ralph+1}_n \exp(z^\ralph_n) =
a(\theta_n, R_n) e^{\iunit (\ralph + 1) \theta_n} = |A|e^{\iunit(1 +
\alpha) k_n\pi/2} + o(1)\not\to A,
\end{aligned}$$ a contradiction.
Alternative expression and asymptotics {#ss:exit-lower-alt}
--------------------------------------
We first consider . Let $$\begin{aligned}
M(t) = \sumzi n f_{-b}*(\delta + f_c
*f_{-c})*(f_{b+c}*f_{-b-c})^{*n}(t).\end{aligned}$$ Given $q>0$, by Fubini’s theorem, $\LT M(q) =\LT{f_{-b}}(q) [1 +
\LT{f_c}(q) \LT{f_{-c}}(q)] \sumzi n \LT{f_{b+c}}(q)^n \LT{f_{-b
-c}}(q)^n$, which is finite due to $\LT{f_{-b-c}}(q) < 1$. Then $M(t)<\infty$ a.e. As a result, the of converges a.e. Denote it by $F(t)$ for now. By dominated convergence and the formula for $\LT M(q)$, $$\begin{aligned}
\LT F(q)
=
\frac{\LT{f_{-b}}(q) [1- \LT{f_c}(q) \LT{f_{-c}}(q)]}
{1-\LT{f_{b+c}}(q) \LT{f_{-b-c}}(q)}.\end{aligned}$$ On the other hand, it is known that [@doney:91:jlms] (also cf.[@kyprianou:14:sh], p. 253) $$\begin{aligned}
\LT{f_x}(q) = e^{x q^\ralph} - \alpha x^{\alpha-1}_+ q^{1 - \ralph}
E_{\alpha, \alpha}(x^\alpha_+ q), \quad x\in\Reals,\ q>0.\end{aligned}$$ Then for $x\ge0$, $\LT f_{-x}(q) = e^{-x q^\ralph}$ and $1 - \LT f_x(q) \LT f_{-x}(q) = \alpha x^{\alpha-1} q^{1-\ralph}
E_{\alpha,\alpha}(x^\alpha q) e^{-x q^\ralph}$. Plugging the identities into the display, it is seen that $\LT F(q)$ equals the of , giving $F(t) = k_{-b,c}(t)$.
Based on the alternative expression, it is quite easy to get that as $t\dto0$, $$\begin{aligned}
\label{e:asym-k-0}
k_{-b,c}(t) \sim f_{-b}(t),\end{aligned}$$ in particular, by Eq. (14.35) in [@sato:99:cup], $\ln k_{-b,c}(t)
\sim -C b^{\alpha/(\alpha-1)} t^{-1/(\alpha-1)}$, where $C>0$ is constant. First, by , $0<f_{-b}(t) - k_{-b,c}(t) =
(k_{c,-b}*f_{-b - c})(t) < (f_c*f_{-b-c})(t) = (u*f_{-b})(t)$, where $u = f_c*f_{-c}$ is a and we have used $k_{c,-b} < f_c$ and $f_{-b-c} = f_{-b} * f_{-c}$. Next, $$\begin{aligned}
(u*f_{-b})(t)
=
\int^t_0 u(s) f_{-b}(t-s)\,\dd s
\le
\sup_{s\le t} f_{-b} \times \int^t_0 u
=
o(1) \sup_{s\le t} f_{-b}, \quad t\dto 0.\end{aligned}$$ Since $f_{-b}$ is unimodal ([@sato:99:cup], p. 416), for all $t>0$ small enough, $\sup_{s\le t} f_{-b} = f_{-b}(t)$, implying .
Although Proposition \[p:exit-lower\] gives a series expression of $\mu_s$, it does not provide the radius of convergence of $\LT{\mu_s}$, defined as $\sup\{r>0: e_z\in L^1(\mu_s)\,\forall z\in
U_r(0)\}$, where $e_z$ is the function $t\mapsto e^{-t z}$. This is also related to the tail of $k_{-b,c}(t)$ as $t\toi$. We have the following.
\[p:root\] Let $-\varrho$ be the largest real root of $E_{\alpha,
\alpha}(z)$, where $\varrho>0$. Then given $s\in [0,1)$, $\LT{\mu_s}(x) < \infty$ for $x\in(-\varrho, \infty)$ and $\LT{\mu_s}(x)\uto \LT{\mu_s}(-\varrho) = \infty$ as $x\dto
-\varrho$. In particular, the radius of convergence of $\LT{\mu_s}$ is $\varrho$ and $\Scr Z_{\alpha,\alpha} \subset \{z: |\arg z| >
\alpha\pi/2,\, \Re(z)\le -\varrho\}$.
Recall that for a measure $\nu$ on $[0,\infty)$ with finite total mass, the domain of $\LT{\mu_s}$ contains $\{z: \Re(z)\ge 0\}$, and if $\LT\nu(z)$ can be analytically extended to $U_r(0)\cap \{z:
\Re(z) < 0\}$ for some $r>0$, then the domain of $\LT\nu$ contains $\{z: \Re(z)>-r\}$ and $\LT\nu$ is analytic in the region.
Let $\root_0$ be a root of $E_{\alpha,\alpha}(z)$ with the largest real part. Put $a = -\Re(\root_0)$. Clearly, $\varrho \ge a$. From $\Scr Z_{\alpha,\alpha} \subset \{z: |\arg(z)| >
\alpha\pi/2\}$ ([@popov:13:jmathsci], Theorem 4.2.1), $a>0$. Since $E_{\alpha, \alpha}(z) \ne0$ for any $z$ with $\Re(z)>-a$, $H_s(z)$ is analytic in $U_a(0)$. Since, by and , $\LT{\mu_s}(z) = H_s(z)$ for $z$ with $\Re(z) \ge0$, from the above remark, the domain of $\LT{\mu_s}$ contains $\{z: \Re(z) >
-a\}$. Let $z\to\root_0$ along the ray from 0 to $\root_0$. Then $|\LT{\mu_s}(z)| = |H_s(z)| \to |H_s(\root_0)|= \infty$. By $|\LT{\mu_s}(z)| \le \LT{\mu_s}(\Re(z))$, it follows that if $x\in
\Reals$ and $x\dto -a$, then $\LT{\mu_s}(x) = H_s(x)\toi$. Since $E_{\alpha,\alpha}(x) \to E_{\alpha,\alpha}(-sa)>0$, then $E_{\alpha, \alpha}(-a)=0$, and so $a = \varrho$. Thus, $\Scr
Z_{\alpha, \alpha} \subset \{z: \Re(z)\le -\varrho\}$. The proof is complete.
Combining Propositions \[p:exit-lower\] and \[p:root\], if the multiplicity of $-\varrho$ is $n\ge 1$, then $k_{-b,c}(t)$ decreases exponentially fast with $$\begin{aligned}
\limsup_{t\toi}
\frac{\ln k_{-b,c}(t)}{t} = -\frac{\varrho}{(b+c)^\alpha}.\end{aligned}$$ However, the exact asymptotic of $k_{-b,c}(t)$ depends on the multiplicity of $-\varrho$ as well as other roots of $E_{\alpha, \alpha}(z)$ on the line $\Re(z) =
-\varrho$, if there are any.
Distribution of first exit at upper end {#s:upper}
=======================================
The main result of this section is Theorem \[t:exit-upper\] below. It provides a factorization of the joint sub-of the time $T_c$, the undershoot $X_{T_c-}$, and the jump $\Delta_{T_c} = X_{T_c}
- X_{T_c-}$ when $X$ makes first exit from $[-b,c]$ by jumping upward across $c$ before hitting $-b$. The following function plays an important role. For $x\in (-b,c)$ and $t>0$, define $$\begin{aligned}
\label{e:no-exit}
l_{x,-b,c}(t)
=
\frac{\pr\{X_t\in\dd x,\, X_s\in (-b,c)\,\forall s\le t\}}{\dd x}.\end{aligned}$$ While the function can be defined for any process that has a at any time point, in the case of a spectrally positive strictly stable process, it has an explicit representation.
To start with, it is known that for $q\ge 0$ ([@kyprianou:14:sh], Theorem 8.7) $$\begin{aligned}
\nonumber \LT{l_{x, -b, c}}(q)
&=
\frac{W\Sp q(c) W\Sp q(b+x)}{W\Sp q(b+c)} - W\Sp q(x_+)
\\\label{e:LT-l}
&=
\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
\frac{
E_{\alpha, \alpha}(c^\alpha q)
E_{\alpha, \alpha}((b+x)^\alpha q)
}{
E_{\alpha, \alpha}((b+c)^\alpha q)
}
- x^{\alpha-1}_+ E_{\alpha,\alpha}(x^\alpha_+ q).\end{aligned}$$
\[t:exit-upper\] Fix $b>0$ and $c>0$. For $x\in \Reals$, $$\begin{aligned}
\label{e:undershoot}
\pr\{T_c<\tau_{-b},\, X_{T_c-} \in\dd x\}
=
\cf{x\in (-b,c)}
\frac{|\sin(\alpha\pi)|}\pi
\Sbr{
\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
- x^{\alpha-1}_+
} \frac{\dd x}{(c-x)^\alpha},
\end{aligned}$$ and conditional on $T_c < \tau_{-b}$ and $X_{T_c-} = x\in (-b,c)$, $\Delta_{T_c}$ and $T_c$ are independent, such that $\Delta_{T_c}$ follows the Pareto distribution with $$\begin{aligned}
\pi(u)
=
\alpha (c-x)^\alpha u^{-\alpha-1} \cf{u>c-x}
\end{aligned}$$ and $T_c$ has $$\begin{aligned}
\label{e:exit-time-upper}
p(t) = \Gamma(\alpha)\Sbr{
\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
- x^{\alpha-1}_+
}^{-1} l_{x, -b,c}(t)
\end{aligned}$$ with $l_{x,-b,c}(t)$ having the following expression $$\begin{aligned}
\label{e:residual2}
l_{x,-b,c}(t)
=
\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
\sum_{\root\in \Scr Z_{\alpha,\alpha}}
\res\Grp{
\frac{
E_{\alpha,\alpha}(c^\alpha z) E_{\alpha,\alpha}((b+x)^\alpha
z)
}{
E_{\alpha,\alpha}((b+c)^\alpha z)
} e^{t z},\, \frac\root{(b+c)^\alpha}
}.
\end{aligned}$$ Furthermore, given $t>0$, the mapping $x\to l_{x,-b,c}(t)$ has an analytic extension from $(-b,c)$ to $\Coms\setminus (-\infty, -b]$.
From , one may suspect that can be obtained by residual calculus. However, when $x>0$, the function in the residuals of have the term $x^{\alpha-1}
E_{\alpha, \alpha}(x^\alpha q)$ missing. A direct inversion of the Laplace transform when $x>0$ seems to be involved. Instead, we will first show for $x<0$ by residual calculus, and then establish for $x\ge0$ through analytic extension.
Factorization and conditional independence {#ss:factor}
------------------------------------------
The factorization in Theorem \[t:exit-upper\] follows from the next result, which actually holds under much more general assumptions on a process.
\[l:first-exit\] Given $b>0$ and $c>0$, for $t>0$, $$\begin{gathered}
\label{e:joint-exit-up}
\pr\{T_c<\tau_{-b},\, T_c\in\dd t,\, X_{T_c-}\in\dd x,\,
\Delta_{T_c}\in \dd u\}
\\
=
\cf{x>-b,\, u>c-x>0}
\dd t \, \pr\{X_t\in\dd x,\, X_s \in (-b,c)\,\forall s\le t\}
\,\Pi(\dd u).
\end{gathered}$$
The proof follows the one on p. 76 of [@bertoin:96:cup]. As already noted, $\pr\{X_{T_c}>c > X_{T_c-}>-b\}=1$. By Rogozin’s criterion (cf. [@doney:07:sg-b], Theorem 5.17), under the law of $X$, 0 is regular for $(0,\infty)$ as well as for $(-\infty, 0)$, so $X_t<c$ for all $t<T_c$ and $X_t>-b$ for all $t<\tau_{-b}$. Therefore, almost surely, for any bounded function $f(t,x,u)\ge 0$, $$\begin{aligned}
&
f(T_c,\, X_{T_c-}, \Delta_{T_c})\cf{T_c < \tau_{-b}}
\\
&=
\sum_t f(t,\, X_{t-},\, \Delta_t)\cf{\Delta_t > c-X_{t-} > 0,\,
X_{t-}>-b, \, -b<X_s<c\,\forall s\le t}.
\end{aligned}$$ The sum on the is well-defined as it runs over the set of $t$’s where $X$ has a jump, which is countable. The rest of the proof then applies the compensation formula to show that the expectation of the random sum is an integral of $f(t, x, u)$ with respect to the measure on the of . Since the argument has become standard, it is omitted for brevity.
Since the measure of $X$ is $$\begin{aligned}
\Pi(\dd x) = \frac{\alpha(\alpha-1) \cf{x>0}}
{\Gamma(2-\alpha)} \frac{\dd x}{x^{\alpha+1}}.\end{aligned}$$ by and Lemma \[l:first-exit\], $$\begin{aligned}
\nonumber
&
\pr\{T_c<\tau_{-b},\, T_c\in\dd t,\, X_{T_c-}\in\dd x,\,
\Delta_{T_c}\in \dd u\}
\\\label{e:l-factor}
&=
\cf{c>x>-b}
l_{-x,b,c}(t)\,\dd t\,\dd x\,
\frac{\cf{u>c-x}\alpha(\alpha-1)\dd
u}{\Gamma(2-\alpha)u^{\alpha+1}}.\end{aligned}$$ Now let $q=0$ in . Then $$\begin{aligned}
\intzi l_{x,-b,c}(t)\,\dd t
=
\LT{l_{x,-b,c}(0)}
=
\frac1{\Gamma(\alpha)}
\Sbr{\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
- x^{\alpha-1}_+
}.\end{aligned}$$ Then by , for $x\in (-b,c)$, $$\begin{aligned}
\pr\{T_c<\tau_{-b},\, X_{T_c-}\in\dd x\}
&=
\dd x\intzi l_{x,-b,c}(t)\,\dd t \int^\infty_{c-x}
\frac{\alpha(\alpha-1)}
{\Gamma(2-\alpha)} \frac{\dd u}{u^{\alpha+1}}
\\
&=
\Sbr{
\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
- x^{\alpha-1}_+
}
\frac{(\alpha-1)}{\Gamma(\alpha)\Gamma(2-\alpha)}
\frac{\dd x}{(c-x)^\alpha}.\end{aligned}$$ Since $\pr\{X_{T_c-}\in (-b,c)\}=1$ by Lemma \[l:first-exit\], then the sub-formula in follows. On the other hand, given $x\in (-b,c)$, $$\begin{aligned}
\pr\{T_c\in\dd t,\, \Delta_{T_c}\in \dd u \gv T_c<\tau_{-b}, \,
X_{T_c-}\in\dd x\}
=
C l_{-x,b,c}(t) \,\dd t \times \frac{\cf{u>c-x}\dd u}{u^{\alpha+1}},\end{aligned}$$ for some constant $C = C(x)$. It follows that conditional on $T_c <
\tau_{-b}$ and $X_{T_c-} = x$, $T_c$ and $\Delta_{T_c}$ are independent, with $\Delta_{T_c}$ following a Pareto distribution and $T_c$ having a of the form $C' l_{-x,b,c}(t)$ with $C'$ another constant, as claimed in Theorem \[t:exit-upper\].
Contour integration
-------------------
The main step is to get the expression of $l_{x,-b,c}(t)$ for given $x\in (-b,c)$. As marked after Theorem \[t:exit-upper\], we need to show that $l_{x,-b,c}(t)$ as a function of $x$ can be analytically extended from $x\in (-b,0)$ to the entire $(-b,c)$. To this end, define $$\begin{aligned}
\label{e:no-overshoot}
h_{x,c}(t)
=
\frac{\pr\{X_t\in \dd x,\, X_s < c\,\forall s\le t\}}{\dd x}.\end{aligned}$$ Given $c>0$, $h_{x,c}(t)$ can be either regarded as a function of $t$ or a function of $x$. It already plays a critical role in [@chi:18:tr] in the derivation of the distribution of the triple $(T_c, X_{T_c-}, X_{T_c})$, known as Gerber-Shiu distribution ([@kyprianou:14:sh], Chapter 10).
\[l:lhk-renewal\] Given $b>0$, $c>0$, and $x\in (-b,c)$, $l_{x,-b,c}(t) = h_{x,c}(t) -
(k_{-b,c}*h_{b+x, b+c})(t)$.
Let $f\ge 0$ be a function with support in $(-b,c)$. Then for any $t>0$, $$\begin{aligned}
\mean [f(X_t) \cf{X_s\in (-b,c)\forall s\le t}]
=
\int^c_{-b} f(x) l_{x,-b,c}(t)\,\dd x.
\end{aligned}$$ On the other hand, the expectation can be decomposed as $$\begin{aligned}
\mean [f(X_t) \cf{X_s<c\forall s\le t}]
-
\mean [f(X_t) \cf{\tau_{-b}\le t,\, X_s<c\forall s\le t}].
\end{aligned}$$ The first expectation in the display is equal to $$\begin{aligned}
\int^c_{-b} f(x) h_{x,c}(t)\,\dd x.
\end{aligned}$$ By strong Markov property of $X$, the second expectation is equal to $$\begin{aligned}
&
\int^c_{-b} \int^t_0 f(x) \pr\{X_t\in\dd x,\, \tau_{-b}\in\dd u,\,
X_s < c\forall s\le t\}
\\
&=
\int^c_{-b} \int^t_0 f(x) \pr\{X_t\in\dd x+b,\, X_s<b+c\forall
s\le t-u\} \pr\{\tau_{-b}\in\dd u,\, X_s < c\forall s\le u\}
\\
&=
\int^c_{-b} f(x) \Sbr{
\int^t_0 h_{x+b,b+c}(t-u) k_{-b,c}(u)\,\dd u
}\,\dd x.
\end{aligned}$$ Comparing the integrals and by $f$ being arbitrary, the claimed identity follows.
\[e:l-extend\] Given $b>0$, $c>0$, and $t>0$, the mapping $x\to l_{x,-b,c}(t)$ has an analytic extension from $(-b,c)$ to $\{z-b: z\in\Omega\}
\cap\{c-z: z\in\Omega\}$, where $\Omega = \{z\in\Coms: |\arg z| <
\kappa^{-1}\pi/2\}$ with $1/\kappa = 1 - \ralph$.
It is shown in [@chi:18:tr] that given $c>0$ and $t>0$, the mapping $x\to h_{x,c}(t)$ has an analytic extension from $(-\infty,
c)$ to $\{c-z: z\in \Omega\}$. Put $a = b+c$. Then by Lemma \[l:lhk-renewal\], it suffices to show that $x\to (k_{-b,c}*h_{x,
a})(t)$ has an analytic extension from $(0, a)$ to $\Omega\cap
\{a-z: z\in\Omega\}$.
For $x < a$, by p. 4/10 of [@michna:15:ecp], $$\begin{aligned}
h_{x,a}(t) = g_t(x) - \phi_t(x) > 0,
\end{aligned}$$ with $$\begin{aligned}
\phi_t(x) = \int^t_0 f_{x-a}(t-s) g_s(a)\,\dd s>0.
\end{aligned}$$ Then $(k_{-b,c}*h_{x,a})(t) = G(x) - \Phi(x)$, where $$\begin{gathered}
G(x) = \int^t_0 k_{-b,c}(t-s) g_s(x)\,\dd s
\intertext{and}
\Phi(x) = \int^t_0 k_{-b,c}(t-s) \phi_s(x)\,\dd s.
\end{gathered}$$ We shall show that $G(x)$ has an analytic extension from $(0,
\infty)$ to $\Omega$ and $\Phi(x)$ has an analytic extension from $(-\infty, a)$ to $\{a-z: z\in\Omega\}$. This will finish the proof.
First consider $\Phi(x)$. In [@chi:18:tr], the proof of its Lemma 9 establishes that $\phi_t$ has an analytic extension from $(-\infty, a)$ to $\{a-z: z\in\Omega\}$ such that, given $0<r_1
< r_2 < \infty$ and $0< \beta_0 < \kappa^{-1}\pi/2$, for $z =
re^{\iunit\beta}\in \Omega$ with $r\in [r_1, r_2]$ and $-\beta_0
\le\beta = \arg z\le \beta_0$, letting $x_0 = r_1 d(\beta_0)>0$, where $d(\beta) = \cos(\kappa\beta)^{1/\kappa}$, $$\begin{aligned}
|\phi_t(a-z)|
\le
(r_2/x_0)^{\alpha/(\alpha-1)} g_t(a-x_0)<\infty.
\end{aligned}$$ By scaling, $g_t(a-x_0) \le t^{-\ralph} \sup g_1$. By , $\sup k_{-b,c}<\infty$. Then by $$\begin{aligned}
\int^t_0 k_{-b,c}(t-s) |\phi_s(a-z)|\,\dd s
\le
\sup k_{-b,c} \int^t_0 (r_2/x_0)^{\alpha/(\alpha-1)}
s^{-\ralph} \sup g_1\,\dd s,
\end{aligned}$$ the is uniformly bounded for $z$ in any compact subset of $\Omega$. By dominated convergence, $\Phi(a-z)$ is continuous in $\Omega$. Then by Fubini’s theorem followed by Morera’s theorem (cf. [@rudin:87:mcgraw], p. 208), $\Phi(a-z)$ is analytic in $\Omega$, as desired.
Next consider $G(x)$. It is known that $x\to g_t(x)$ has an analytic extension from $\Reals$ to $\Coms$ ([@sato:99:cup], p. 88). Then, as above, it suffices to show $\int^t_0 k_{-b,c}(t-s)|g_s(z)|
\,\dd s$, or more simply, $\int^t_0 |g_s(z)|\,\dd s$ is uniformly bounded for $z$ in any compact subset of $\Omega$. By , for $x>0$, $$\begin{aligned}
g_t(x)
=
\frac{\alpha\pi}{\alpha-1} (x/t)^{\kappa/\alpha}
\int^{\pi/2}_{\pi(\ralph-1/2)} a(\theta)
\exp\{-x^\kappa t^{-\kappa/\alpha}
a(\theta)\}\,\dd\theta,
\end{aligned}$$ where $a(\theta)>0$ is a continuous function in the open interval of the integral. Since $z\to z^\kappa$ maps $\Omega$ to $\{z:
\Re(z)>0\}$, the integral on the can be analytically extended to $\Omega$, and hence the identity can be extended to all $z\in\Omega$. If $z = re^{\iunit\beta}$ with $\beta = \arg z$, then by $|\beta|< \kappa^{-1}\pi / 2$, $\Re(z^\kappa) = [r
d(\beta)]^\kappa>0$, where $d(\beta)$ is defined above. Then $$\begin{aligned}
|g_t(z)|
&\le
\frac{\alpha\pi}{\alpha-1} (r/t)^{\kappa/\alpha}
\int^{\pi/2}_{\pi(1/\alpha-1/2)} a(\theta)
\exp\{-[r d(\beta)]^{\kappa} t^{-\kappa/\alpha}
a(\theta)\}\,\dd\theta
\\
&=
\frac{g_t(r d(\beta))}{d(\beta)^{\kappa/\alpha}}
\le
\frac{t^{-\ralph} \sup g_1}{d(\beta)^{\kappa/\alpha}},
\end{aligned}$$ where the equality on the second line follows by comparing the integral with the previous display. Since any compact subset of $\Omega$ is contained in a section $\{|\arg z|\le \beta_0\}$ for some $\beta_0<\kappa^{-1}\pi/2$ and since $d(\beta)>0$ is decreasing in $|\beta|$ in the section, it is then easy that $|g_t(z)| \le
t^{-\ralph} \sup g_1/d(\beta_0)^{\kappa/\alpha}$, yielding the desired uniform boundedness.
We finally can prove the expression for $l_{x,-b,c}(t)$ and that it can analytically extended to $\Coms\setminus (-\infty, b)$, which then finishes the proof of Theorem \[t:exit-upper\].
First, suppose $x\in (-b,0)$. Then $$\begin{aligned}
\LT{l_{x,-b,c}}(q)
=
\frac{c^{\alpha-1} (b+x)^{\alpha-1}}{(b+c)^{\alpha-1}}
\frac{
E_{\alpha, \alpha}(c^\alpha q)
E_{\alpha, \alpha}((b+x)^\alpha q)
}{
E_{\alpha, \alpha}((b+c)^\alpha q)
}
\end{aligned}$$ and will follow once it is proved that $\FT
{l_{x,-b,c}}(\theta) = \LT {l_{x,-b,c}}(-\iunit\theta)$ is in $L^1(\dd\theta)$ and that $$\begin{aligned}
\nth{2\pi\iunit} \int^{\iunit\infty}_{-\iunit\infty}
\frac{
E_{\alpha, \alpha}(c^\alpha z)
E_{\alpha, \alpha}((b+x)^\alpha z)
}{
E_{\alpha, \alpha}((b+c)^\alpha z)
} e^{z t}\,\dd z
\end{aligned}$$ is equal to the sum on the of . The argument is similar to that for Proposition \[p:exit-lower\]. Let $$\begin{aligned}
\label{e:sv}
s = c^\alpha/(b+c)^\alpha, \quad
v = (b+x)^\alpha/(b+c)^\alpha.
\end{aligned}$$ Then by making change of variables $z' = (b+c)^\alpha z$ and $t' =
t/(b+c)^\alpha$, and using the function $H_s(z)$ defined in , it boils down to showing that $$\begin{aligned}
\label{e:l-ft}
\intii |H_s(-\iunit\theta) E_{\alpha,\alpha}(-\iunit
v\theta)|\,\dd\theta <\infty
\end{aligned}$$ and for any $t>0$, $$\begin{aligned}
\label{e:l-contour}
\int_{C_{R_n}}
H_s(z) E_{\alpha,\alpha}(v z) e^{t z}\,\dd z\to0, \quad n\toi,
\end{aligned}$$ where the contour $C_R$ and the numbers $R_n$ are defined in the proof of Proposition \[p:exit-lower\].
By , with $\lambda = \cos(\alpha^{-1}\pi/2)>0$, $$\begin{aligned}
|H_s(-\iunit\theta) E_{\alpha,\alpha}(-\iunit
v\theta)| \sim
\alpha^{-1} |s v\theta|^{\ralph-1} e^{\lambda
(s^\ralph + v^\ralph-1) |\theta|^\ralph}, \quad |\theta|\toi.
\end{aligned}$$ Then easily follows by noting $$\begin{aligned}
\label{e:negative}
s^\ralph + v^\ralph - 1 = x/(b+c)<0.
\end{aligned}$$ Next, as in the proof of Proposition \[p:exit-lower\], divide $C_R$ into $C_{R,1}$ and $C_{R,2}$. As $R\toi$, uniformly for $z\in
C_{R,1}$, $H_s(z) = O(1) \exp\{(s^\ralph-1) z^\ralph\}$ and $E_{\alpha,\alpha}(v z) = O(1) (v z)^{\ralph-1} \exp\{v^\ralph
z^\ralph\}$; see the derivation of . From again, there is $\eta = \eta(x)>0$, such that $\sup_{z\in C_{R,1}} |H_s(z) E_{\alpha, \alpha}(v z)| = O(e^{-\eta
R})$. Meanwhile, $|e^{t z}|\le 1$ for $z\in C_{R,1}$ and Length$(C_{R,1}) = O(R^\alpha)$. Then $$\begin{aligned}
\int_{C_{R,1}} H_s(z) E_{\alpha,\alpha}(v z) e^{t z}\,\dd z
= O(R^\alpha e^{-\eta R}) \to0,
\quad R\toi.
\end{aligned}$$ On the other hand, according to derivation of , for some $m_0>0$, $$\begin{aligned}
\sup_{z\in C_{R_n,2}} |H_s(z) E_{\alpha,\alpha}(v z)|
=
O(R^{1+\alpha}_n e^{m_0 R_n})\cdot O(R^{1-\alpha}_n e^{m_0 R})
= O(R^2_n e^{2 m_0 R_n}).
\end{aligned}$$ Meanwhile, according to the argument leading to , for some $b_0>0$, $|e^{t z}| \le e^{-b_0 R^\alpha t}$ for $t\in
C_{R,2}$. Then by $\alpha>1$ and $t>0$, $$\begin{aligned}
\int_{C_{R_n,2}} H_s(z) E_{\alpha,\alpha}(v z) e^{t z}\,\dd z
=
O(R^{2+\alpha}_n e^{2m_0 R_n - b_0 R^\alpha_n t}),
\quad n\toi.
\end{aligned}$$ The desired convergence in then follows and hence is proved in the case $x\in (-b,0)$.
It only remains to show that the of as a function of $x$ has an analytic extension to $\Coms\setminus
(-\infty, -b]$ for given $t>0$. Once this is done, since by Lemma \[e:l-extend\], $x\to l_{x,-b,c}(t)$ has an analytic extension to a domain containing $(-b,c)$ and since it was just proved that the two functions are equal on $(-b,0)$, then they must be equal on $(-b,c)$ and $x\to l_{x,-b,c}(t)$ can actually be extended to $\Coms\setminus (-\infty, -b]$, finishing the proof.
Thus, let $$\begin{aligned}
w_\root(x) =
\res\Grp{
\frac{
E_{\alpha,\alpha}(c^\alpha z) E_{\alpha,\alpha}((b+x)^\alpha
z)
}{
E_{\alpha,\alpha}((b+c)^\alpha z)
} e^{t z}, \frac\root{(b+c)^\alpha}
}.
\end{aligned}$$ It is easy to see that $w_\root$ has an analytic extension from $(-b,c)$ to $\Coms \setminus (-\infty, -b]$. All $\root\in\Scr
Z_{\alpha, \alpha}$ with large enough modulus are simple roots of $E_{\alpha, \alpha}(z)$ and have $|\arg\root|$ arbitrarily close but strictly greater than $\alpha\pi/2$. For each such $\root$ and each $z\in \Coms\setminus (-\infty,0]$, $$\begin{aligned}
w_\root(z-b) =
\frac{E_{\alpha,\alpha}(s\root)e^{t'\root}}{(b+c)^\alpha
E'_{\alpha,\alpha}(\root)} \times
E_{\alpha, \alpha}\Grp{\frac{z^\alpha \root}{(b+c)^\alpha}},
\end{aligned}$$ where $s$ is defined as in and $t' = t/(b+c)^\alpha$. From the last part of the proof for Proposition \[p:exit-lower\], there is $c' = c'(t')>0$, such that $$\begin{aligned}
\Abs{\frac{E_{\alpha,\alpha}(s\root) e^{t'\root}}
{(b+c)^\alpha E'_{\alpha,\alpha}(\root)}
} = O(1) e^{-c'|\root|}.
\end{aligned}$$ On the other hand, by , there is a constant $C>0$ such that $$\begin{aligned}
\Abs{
E_{\alpha,\alpha}\Grp{\frac{z^\alpha\root}{(b+c)^\alpha}}
}
\le E_{\alpha,\alpha}\Grp{
\frac{(|z|\vee 1)^\alpha |\root|}{(b+c)^\alpha}
}= O(1)
\exp\Cbr{C(|z|\vee 1) |\root|^\ralph}.
\end{aligned}$$ Combining the two bounds then yields a bound on $|w_\root(z-b)|$. Following the last part of the proof of Proposition \[p:exit-lower\], $\sum_{\root\in \Scr Z_{\alpha, \alpha}}|w_\root(z -
b)|$ converges uniformly for $z$ in any compact subset of $\Coms
\setminus(-\infty, 0]$. Then $z\mapsto\sum_{\root\in \Scr
Z_{\alpha, \alpha}} w_\root(z-b)$ is continuous on $\Coms
\setminus(-\infty, 0]$. Then by Fubini’s theorem followed by Morera’s theorem, the of as a function of $x$ has an analytic extension to $\Coms\setminus(-\infty, -b]$.
Asymptotics
-----------
We consider the asymptotics of $l_{x,-b,c}(t)$ as $t\dto0$ or $\toi$. First, we have
\[p:asym-l-0\] Given $b>0$, $c>0$, and $x\in (-b,c)$, as $t\dto0$, $l_{x,-b,c}(t)
\sim g_t(x)$.
It is clear that $l_{x,-b,c}(t) < g_t(x)$. On the other hand, $$\begin{aligned}
g_t(x) - l_{x,-b,c}(t)
&\le
\frac{\pr\{X_t\in \dd x,\, \tau_{-b}<t\}}{\dd x}
+
\frac{\pr\{X_t\in \dd x,\, T_c<t\}}{\dd x}
\\
&=
\frac{\pr\{X_t\in \dd x,\, \tau_{-b}<t\}}{\dd x}
+
\frac{\pr\{X_t\in \dd x,\, \tau_{c}<t\}}{\dd x}
\\
&=
\frac{\pr\{X_t\in \dd x,\, \tau_{-b}<t\}}{\dd x}
+
\frac{\pr\{X_t\in \dd x,\, \tau_{-(c-x)}<t\}}{\dd x},
\end{aligned}$$ where the third line follows by considering $\sup\{s<t: X_s=c\}$ as well as time reversal. Note that both $b$ and $c-x$ are greater than $(-x)_+$. Then it suffices to show that for any $x\in\Reals$ and $b>(-x)_+$, $j(t):=\pr\{X_t\in\dd x,\, \tau_{-b}<t\}/\dd x =
o(g_t(x))$ as $t\to 0$.
By strong Markov property and $\sup g_s = O(s^{-1/\alpha})$, $$\begin{aligned}
j(t) = \int^t_0 f_{-b}(t-s) g_s(x+b)\,\dd s
= O(1) t^{1-1/\alpha} \sup_{s\le t} f_{-b}(s).
\end{aligned}$$ Since $f_{-b}$ is unimodal ([@sato:99:cup], p. 416), then for small $t>0$, $j(t) = O(1) f_{-b}(t)$. On the other hand, $$\begin{aligned}
g_t(x) \asymp
\begin{cases}
t & \text{if } x>0,\\
t^{-1/\alpha} & \text{if } x=0,\\
t f_x(t)/(-x) & \text{if } x<0
\end{cases}
\quad\text{as } t\dto 0,
\end{aligned}$$ where the last line is by Corollary VII.3 in [@bertoin:96:cup]. It is then clear that if $x\ge0$, then $j(t) = o(g_t(x))$. On the other hand, if $x<0$, since $b>|x|$, then $f_{-b}(t) =
o(f_{-x}(t))$, again yielding $j(t) = o(g_t(x))$.
Finally, by the same argument for the tail of $k_{-b,c}(t)$, as $t\toi$, $l_{x,-b,c}(t)$ decreases exponentially fast with $$\begin{aligned}
\limsup_{t\toi}
\frac{\ln l_{x,-b,c}(t)}{t} = -\frac{\varrho}{(b+c)^\alpha},\end{aligned}$$ where $-\varrho < 0$ is the largest real root of $E_{\alpha,\alpha}$ and $n$ its multiplicity. Again, the exact asymptotic of $l_{x,-b,c}(t)$ depends on more detail of the roots along the line $\Re(z) = -\varrho$.
, [Dalang, R. C.]{}, [and]{} [Peskir, G.]{} (2008). The law of the supremum of a stable [L]{}évy process with no negative jumps. **36**, 5, 1777–1789.
(1996). . Cambridge Tracts in Mathematics, Vol. **121**. Cambridge University Press, Cambridge.
(2002). , Second ed. Probability and its Applications. Birkhäuser Verlag, Basel.
(1992). . Classics in Applied Mathematics, Vol. **7**. Society for Industrial and Applied Mathematics, Philadelphia, PA. Corrected reprint of the 1968 original.
(2016). Exact sampling of first passage event of certain symmetric [L]{}évy processes with unbounded variation. Preprint. Available as arXiv:1606.06411.
(2018). Law of the first passage triple of a spectrally positive strictly stable process. Preprint. Available as arXiv:1801.06891.
(1991). Hitting probabilities for spectrally positive [L]{}évy processes. **44**, 3, 566–576.
(2007). . Lecture Notes in Mathematics, Vol. **1897**. Springer, Berlin.
(2014). , Second ed. Universitext. Springer, Heidelberg. Introductory lectures.
, [Palmowski, Z.]{}, [and]{} [Pistorius, M.]{} (2015). The distribution of the supremum for spectrally asymmetric [L]{}évy processes. [*20*]{}, no. 24, 10.
(2010). . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. With an appendix by Oded Schramm and Wendelin Werner.
(2008). The law of the hitting times to points by a stable [L]{}évy process with no negative jumps. [*13*]{}, 653–659.
(2013). Distribution of roots of [M]{}ittag-[L]{}effler functions. **190**, 2, 209–409. First published in Russian in [*Sovrem. Math. Fundam. Napravl.*]{}, Vol 40, pp. 3–171, 2011.
(1987). , Third ed. McGraw-Hill, New York.
(1999). . Cambridge Studies in Advanced Mathematics, Vol. **68**. Cambridge University Press, Cambridge. Translated from the 1990 Japanese original, revised by the author.
(2011). Hitting densities for spectrally positive stable processes. **83**, 2, 203–214.
(1966). On the representation of stable laws by integrals. [*6*]{}, 84–88. First published in Russian in 1964.
[^1]: Research partially supported by NSF Grant DMS 1720218.
|
---
abstract: 'We study a scale invariant two measures theory where a dilaton field $\phi$ has no explicit potentials. The scale transformations include a translation of a dilaton $\phi\rightarrow\phi +const$. The theory demonstrates a new mechanism for generation of the exponential potential: in the conformal Einstein frame (CEF), after SSB of scale invariance, the theory develops the exponential potential and, in general, non-linear kinetic term is generated as well. The scale symmetry does not allow the appearance of terms breaking the exponential shape of the potential that solves the problem of the flatness of the scalar field potential in the context of quintessential scenarios. As examples, two different possibilities for the choice of the dimensionless parameters are presented where the theory permits to get interesting cosmological results. For the first choice , the theory has standard scaling solutions for $\phi$ usually used in the context of the quintessential scenario. For the second choice, the theory allows three different solutions one of which is a scaling solution with equation of state $p_{\phi}=w\rho_{\phi}$ where $w$ is predicted to be restricted by $-1<w<-0.82$. The regime where the fermionic matter dominates (as compared to the dilatonic contribution) is analyzed. There it is found that starting from a single fermionic field we obtain exactly three different types of spin $1/2$ particles in CEF that appears to suggest a new approach to the family problem of particle physics. It is automatically achieved that for two of them, fermion masses are constants, the energy-momentum tensor is canonical and the “fifth force” is absent. For the third type of particles, a fermionic self-interaction appears as a result of SSB of scale invariance.'
address: 'Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel'
author:
- 'E. I. Guendelman [^1] and A. B. Kaganovich [^2]'
title: |
SSB of scale symmetry, fermion families\
and quintessence without the long-range force problem
---
PACS number(s): 98.80.Cq, 98.62.Gq, 11.30.Qc, 12.15.Pf
Introduction
============
Recent observations imply that the Universe now is undergoing era of acceleration[@P]. This is most naturally explained by the existence of a vacuum energy which can be of the form of an explicit cosmological constant. Alternatively, there may be a slow rolling scalar field, whose potential (assumed to have zero asymptotic value) provides the negative pressure required for accelerating the Universe. This is the basic idea of the quintessence[@Wett1988NP668]. Some of the problems of the quintessence scenario connected to the field theoretic grounds of this idea, are: i) what is the origin of the quintessence potential; ii) why the asymptotic value of the potential vanishes (this is actually the “old” cosmological constant problem[@CCP] ); iii) the needed flatness of the potential[@KLyth]; iv) without the symmetry $\phi\rightarrow\phi +const$ it is very hard to explain the absence of the long-range force if no fine tuning is made[@Carroll; @Dolgov], but such a translation-like symmetry is usually incompatible with a nontrivial potential.
One of the main aims of this paper is to show how the above problems can be solved in the context of the two measures theories (TMT)[@GK1; @GK2; @GK3; @GK4; @G; @G1; @G2; @K]. These kind of models are based on the observation that in a generally covariant formulation of the action principle one has to integrate using an invariant volume element, which is not obliged to be dependent of the metric. In GR, the volume element $\sqrt{-g}d^{4}x$ is indeed generally coordinate invariant, but nothing forbids us from considering the invariant volume element $\Phi d^{4}x$ where $\Phi$ is a scalar density that could be independent of the metric[@GK1].
If the measure $\Phi$ is allowed, we have seen in a number of models[@GK2; @GK3; @GK4; @G] that, in the conformal Einstein frame (CEF), the equations of motion have the canonical GR structure, but the scalar field potential produced in the CEF is such that zero vacuum energy for the ground state of the theory is obtained without fine tuning, that is the “old” cosmological constant problem can be solved[@GK4].
If both measures ($\sqrt{-g}$ and $\Phi$) are allowed, this opens new possibilities concerning scale invariance[@G; @G1; @G2; @K]. In this context we study here a theory which is invariant under scale transformations including also a translation-like symmetry for a dilaton field of the form $\phi\rightarrow\phi +const$ discussed by Carroll[@Carroll]. For the case when the original action does not contain dilaton potentials at all, it is found that the integration of the equation of motion corresponding to the measure $\Phi$ degrees of freedom, spontaneously breaks the scale symmetry and the generation of a dilaton potential is a consequence of this spontaneous symmetry breaking (SSB). When studying the theory in the CEF, it is demonstrated in Sec. III that the spontaneously induced dilaton potential has the exponential form and in addition, also non-linear kinetic terms appear in general.
In Sec. IV we discuss possible cosmological applications of the theory when the dilaton field is the dominant fraction of the matter: it is found that quintessential solutions are possible.
In Sec. V we show that in the presence of fermions, the theory displays a successful fermionic mass generation after the spontaneous symmetries break (SSB), and this is actually the second main aim of this paper. In the regime when the fermionic density is of the order typical for the normal particle physics (which in the laboratory conditions is always much higher than the dilaton density ), there are constant fermion masses, gravitational equations are canonical and the “fifth force” is absent, - all this without any additional restrictions on the parameters of the theory. A possible way for explanation to the “family puzzle” of particle physics also appears naturally in the context of this model. For one of the families, a fermion self-interaction appears as a result of the SSB of scale symmetry.
Two Measures Theory (TMT)
=========================
The main idea of these kind of theories[@GK1; @GK2; @GK3; @GK4] is to reconsider the basic structure of generally relativistic actions, which are usually taken to be of the form $$S = \int d^{4}x\sqrt{-g}L
\label{SE}$$ where $L$ is a scalar and $g=\det(g_{\mu\nu})$. The volume element $d^{4}x\sqrt{-g}$ is an invariant entity. It is however possible to build a different invariant volume element if another density, that is an object having the same transformation properties as $\sqrt{-g}$, is introduced. For example, given four scalar fields $\varphi_{a}$, $a=1,2,3,4$ we can build the density $$\Phi
=\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\partial_{\mu}\varphi_{a}
\partial_{\nu}\varphi_{b}\partial_{\alpha}\varphi_{c}\partial_{\beta}\varphi_{d}
\label{Phi}$$ and then $\Phi d^{4}x$ is also an invariant object. Notice also that $\Phi$ is a total derivative since $$\Phi
=\partial_{\mu}(\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\varphi_{a}
\partial_{\nu}\varphi_{b}\partial_{\alpha}\varphi_{c}\partial_{\beta}
\varphi_{d})
\label{Phideriv}$$ Therefore if we consider possible actions which use both $\Phi$ and $\sqrt{-g}$ we are lead to TMT $$S = \int L_{1}\Phi d^{4}x +\int L_{2}\sqrt{-g}d^{4}x
\label{S}$$
Since $\Phi$ is a total derivative, a shift of $L_{1}$ by a constant, $L_{1}\rightarrow L_{1}+const$, has the effect of adding to S the integral of a total derivative , which does not change equations of motion. Such a feature is not present in the second piece of Eq. (\[S\]) since $\sqrt{-g}$ is not a total derivative. It is clear then that the introduction of a new volume element has consequences on the way we think about the cosmological constant problem, since the vacuum energy is related to the coupling of the volume element with the Lagrangian. How this relation is modified when a new volume element is introduced, was discussed in [@GK2; @GK3; @GK4].
It has been shown that a wide class of TMT models[@GK4], containing among others a scalar field, can be formulated which are free of the “old” cosmological constant problem. An important feature of those models consists in the use of the “first order formalism” where the connection coefficients $\Gamma^{\lambda}_{\mu\nu}$, metric $g_{\mu\nu}$ and in our case also $\varphi_{a}$ and any matter fields that may exist are treated as independent dynamical variables. Any relations that they satisfy are a result of the equations of motion. The models allow the use of the so called conformal Einstein frame (CEF) where the equations of motion have canonical GR form and the effective potential has an absolute minimum at zero value of the effective energy density without fine tuning. This was verified to be the case in all examples studied in Ref. [@GK4], provided the action form (\[S\]) is preserved, where $L_{1}$ and $L_{2}$ are $\varphi_{a}$-independent. If this is so, an infinite symmetry appears[@GK4]: $\varphi_{a}\rightarrow\varphi_{a}+f_{a}(L_{1})$, where $f_{a}(L_{1})$ is an arbitrary function of $L_{1}$.
Scale invariant model with spontaneous symmetry\
breaking giving rise to a potential
================================================
If we believe that there are no fundamental scales in physics, we are lead to the notion of scale invariance. In the context of TMT, to implement global scale invariance one has to introduce a “dilaton” field[@G; @G1]. In this case the measure $\Phi$ degrees of freedom also can participate in the scale transformation[@G; @G1]. In [@G; @G1], explicit potentials (of exponential form) which respect the symmetry were introduced. Fundamental theories however, like string theories, etc. give most naturally only massless particles, which means that only kinetic terms and no explicit potentials appear from the beginning naturally. Let us therefore explore a similar situation in the context of a scale invariant TMT model. We postulate then the form of the action $$S=\int d^{4}x\Phi e^{\alpha\phi/M_{p}}\left[
-\frac{1}{\kappa}R(\Gamma ,g)+\frac{1}{2}g^{\mu\nu}\phi_{,\mu}
\phi_{,\nu}\right]
+\int d^{4}x\sqrt{-g}e^{\alpha\phi/M_{p}}\left[
-\frac{b_{g}}{\kappa}R(\Gamma ,g)+\frac{b_{k}}{2}g^{\mu\nu}\phi_{,\mu}
\phi_{,\nu}\right]
\label{totac}$$ where we proceed in the first order formalism and $R(\Gamma,g)=g^{\mu\nu}R_{\mu\nu}(\Gamma)$, $R_{\mu\nu}(\Gamma)=R^{\alpha}_{\mu\nu\alpha}(\Gamma)$ and $R^{\lambda}_{\mu\nu\sigma}(\Gamma)\equiv
\Gamma^{\lambda}_{\mu\nu,\sigma}+
\Gamma^{\lambda}_{\alpha\sigma}\Gamma^{\alpha}_{\mu\nu}-
(\nu\leftrightarrow\sigma)$. By means of a redefinition of factors of $\phi$ and of $\Phi$ one can always normalize the kinetic term of $\phi$ and the $R$-term that go together with $\Phi$ as done in (\[totac\]). Once this is done, this freedom however is not present any more concerning the second part of the action going together with $\sqrt{-g}$. The appearance of the constants $b_{g}$ and $b_{k}$ is a result of this.
The action (\[totac\]) is invariant under the scale transformations: $$\begin{aligned}
&& g_{\mu\nu}\rightarrow
e^{\theta}g_{\mu\nu}, \quad
\phi\rightarrow \phi-\frac{M_{p}}{\alpha}\theta ,\quad
\Gamma^{\sigma}_{\mu\nu}\rightarrow \Gamma^{\sigma}_{\mu\nu},
\nonumber\\
&&\varphi_{a}\rightarrow \lambda_{a}\varphi_{a},\quad a=1,2,3,4
\quad
where \quad \Pi\lambda_{a}=e^{2\theta}.
\label{st} \end{aligned}$$
Notice that (\[totac\]) is the most general action of TMT invariant under the scale transformations (\[st\]) where the Lagrangian densities $L_{1}$ and $L_{2}$ are linear in the scalar curvature and quadratic in the space-time derivatives of the dilaton but [*without explicit potentials*]{}. In Refs. [@G; @G1], actions of such type were discussed, but with explicit potentials and without kinetic term going with $\sqrt{-g}$. A different definition of the metric have been used also in [@G; @G1] ($g^{\mu\nu}$ in [@G; @G1] instead of the combination $e^{\alpha\phi/M_{p}}g^{\mu\nu}$ here) so that no factor $e^{\alpha\phi/M_{p}}$ appeared multiplying $\Phi$ in Ref.[@G; @G1]. Also it is possible to formulate a consistent scale invariant model keeping only the simplest structure (namely, only the measure $\Phi$ is used), provided $L_{1}$ contains 4-index field strengths and an exponential potential for the dilaton[@G2]. Then SSB of the scale invariance can lead to a quintessential potential[@G2]. Another type of the field theory models with explicitly broken scale symmetry have been studied in Ref.[@K] where it is shown that the quintessential inflation[@PV] type models can be obtained without fine tuning.
We examine now the equations of motion that arise from (\[totac\]). Varying the measure fields $\varphi_{a}$, we get $$A^{\mu}_{a}\partial_{\mu}\left[
e^{\alpha\phi/M_{p}}\left(-\frac{1}{\kappa}R(\Gamma
,g)+
\frac{1}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}\right)\right]=0
\label{7}$$ $$A^{\mu}_{a}=\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}
\partial_{\nu}\varphi_{b}\partial_{\alpha}\varphi_{c}
\partial_{\beta}\varphi_{d}.
\label{8}$$
Since $Det (A^{\mu}_{a})
= \frac{4^{-4}}{4!}\Phi^{3}$ it follows that if $\Phi\neq 0$, $$e^{\alpha\phi/M_{p}}\left[ -\frac{1}{\kappa}R(\Gamma,g)+
\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]=sM^{4}=const,
\label{varphi}$$ where $s=\pm 1$ and $M$ is a constant with the dimension of mass. It can be noticed that the appearance of a nonzero integration constant $sM^4$ spontaneously breaks the scale invariance (\[st\]).
The variation of $S$ with respect to $g^{\mu\nu}$ yields $$-\frac{1}{\kappa}R_{\mu\nu}(\Gamma)(\Phi +b_{g}\sqrt{-g})+
\frac{1}{2}\phi_{,\mu}\phi_{,\nu}
(\Phi +b_{k}\sqrt{-g})-
\frac{1}{2}\sqrt{-g}g_{\mu\nu}
\left[-\frac{b_{g}}{\kappa}R(\Gamma ,g)+
\frac{b_{k}}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}\right]=0
\label{varg}$$
Contracting Eq. (\[varg\]) with $g^{\mu\nu}$, solving for $R(\Gamma ,g)$ and inserting into Eq. (\[varphi\]) we obtain the constraint $$M^{4}(\zeta -b_{g})e^{-\alpha\phi/M_{p}}+
\frac{\Delta}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}
=0,
\label{con1}$$ where the scalar $\zeta$ is the ratio of two measures $$\zeta \equiv\frac{\Phi}{\sqrt{-g}}
\label{zeta}$$ and $\Delta =b_{g}-b_{k}$. It is very interesting that [*the geometrical quantity $\zeta$ is defined by a constraint where neither Newton constant nor curvature enter*]{}.
Varying the action with respect to $\phi$ and using Eq. (\[varphi\]) we get $$(-g)^{-1/2}\partial_{\mu}\left[(\zeta +b_{k})e^{\alpha\phi/M_{p}}
\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)\right]-
\frac{\alpha}{M_{p}}\left[M^{4}(\zeta +b_{g})
-\frac{\Delta}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}
e^{\alpha\phi/M_{p}}\right]=0
\label{se}$$
Considering the term containing connection $\Gamma^{\lambda}_{\mu\nu}$, that is $R(\Gamma ,g)$, we see that it can be written as $$S_{\Gamma}=-\frac{1}{\kappa}\int \sqrt{-g}e^{\alpha\phi/M_{p}}(\zeta +b_{g})
g^{\mu\nu}R_{\mu\nu}(\Gamma) =
-\frac{1}{\kappa}\int \sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}R_{\mu\nu}(\Gamma),
\label{Gamac}$$ where $\tilde{g}_{\mu\nu}$ is determined by the conformal transformation $$\tilde{g}_{\mu\nu}=e^{\alpha\phi/M_{p}}(\zeta +b_{g})g_{\mu\nu}
\label{ct}$$
It is clear then that the variation of $S_{\Gamma}$ with respect to $\Gamma$ will give the same result expressed in terms of $\tilde{g}_{\mu\nu}$ as in the similar GR problem in Palatini formulation. Therefore, if $\Gamma^{\lambda}_{\mu\nu}$ is taken to be symmetric in $\mu , \nu$, then in terms of the metric $\tilde{g}_{\mu\nu}$, the connection coefficients $\Gamma^{\lambda}_{\mu\nu}$ are Christoffel’s connection coefficients of the Riemannian space-time with the metric $\tilde{g}_{\mu\nu}$: $$\Gamma^{\lambda}_{\mu\nu}=
\{
^{\lambda}_{\mu\nu}\}|_{\tilde{g}_{\mu\nu}}=\frac{1}{2}\tilde{g}^{\lambda\alpha}
(\partial_{\nu}\tilde{g}_{\alpha\mu}+\partial_{\mu}\tilde{g}_{\alpha\nu}-
\partial_{\alpha}
\tilde{g}_{\mu\nu}).
\label{Gama}$$
So, it appears that working with $\tilde{g}_{\mu\nu}$, we recover a Riemannian structure for space-time. We will refer to this as the conformal Einstein frame (CEF). Notice that $\tilde{g}_{\mu\nu}$ is invariant under the scale transformations (\[st\]) and therefore the spontaneous breaking of the global scale symmetry (see Eq. (\[varphi\]) and discussion after it) is reduced, in CEF, to the spontaneous breaking of the shift symmetry $\phi\rightarrow\phi +const$ for the dilaton field. In this context, it is interesting to notice that Carroll pointed to the possible role of the shift symmetry for a scalar field in the resolution of the long range force problem of the quintessential scenario.
Equations (\[varg\]) and (\[se\]) in CEF take the following form: $$G_{\mu\nu}(\tilde{g}_{\alpha\beta})=\frac{\kappa}{2}T_{\mu\nu}^{eff}
\label{gef}$$ $$T_{\mu\nu}^{eff}=\frac{1}{2}\left(1+\frac{b_{k}}{b_{g}}\right)
\left(\phi_{,\mu}\phi_{,\nu}-K\tilde{g}_{\mu\nu}\right)-
\frac{\Delta^{2}Ke^{2\alpha\phi/M_{p}}}{2b_{g}M^{4}}
\left(\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}K\tilde{g}_{\mu\nu}\right)+
\tilde{g}_{\mu\nu}\frac{sM^{4}}{4b_{g}}e^{-2\alpha\phi/M_{p}}
\label{Tmn}$$ $$N\left[(-\tilde{g})^{-1/2}\partial_{\mu}
(\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi ) +
\tilde{g}^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}
\ln N\right]
+\frac{\alpha\Delta^{2}}{M_{p}M^{4}}K^{2}e^{2\alpha\phi/M_{p}}
-\frac{\alpha M^{4}}{M_{p}}e^{-2\alpha\phi/M_{p}}=0
\label{phief}$$
Here $$K\equiv\frac{1}{2}\tilde{g}^{\alpha\beta}\phi_{,\alpha}
\phi_{,\beta}, \quad N\equiv
b_{g}+b_{k}-\frac{\Delta^{2}}{M^{4}}Ke^{2\alpha\phi/M_{p}},
\label{not}$$ $G_{\mu\nu}(\tilde{g}_{\alpha\beta})$ is the Einstein tensor in the Riemannian space-time with metric $\tilde{g}_{\mu\nu}$ and the constraint (\[con1\]) have been used which in CEF takes the form $$\zeta =b_{g}\frac{M^{4}-\Delta Ke^{2\alpha\phi/M_{p}}}
{M^{4}+\Delta Ke^{2\alpha\phi/M_{p}}}
\label{con2}$$
Notice that in $T_{\mu\nu}^{eff}$ we can recognize an effective potential $$V_{eff}=\frac{sM^{4}}{4b_{g}}e^{-2\alpha\phi/M_{p}}
\label{Veff}$$ which appears in spite of the fact that no explicit potential term was introduced in the original action (\[totac\]). As we see, the existence of $V_{eff}$ is associated with the constant $sM^{4}$, appearance of which spontaneously breaks the scale invariance. This is actually a new mechanism for generating the exponential potential[^3].
Notice also that if $b_{g}\neq b_{k}$, the effective energy-momentum $T_{\mu\nu}^{eff}$ as well as the dilaton equation of motion contain the non-canonical terms nonlinear[^4] in gradients of the dilaton $\phi$. It will be very important that the non-canonical in $\phi_{,\alpha}$ terms are multiplied by a very specific exponential of $\phi$. As we will see, these non-canonical terms may be responsible for the most interesting scaling solutions. In the context of FRW cosmology, this structure provides conditions for quintessential solutions if $s=1$.
Scaling solutions
=================
In the context of a spatially flat FRW cosmology with a metric $ds^{2}_{eff}=
\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu} =dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$, the equations (\[gef\])-(\[phief\]), with the choice $s=+1$, become: $$H^{2}=\frac{1}{3M_{p}^{2}}\rho_{eff}(\phi)
\label{FRWgr}$$ $$\begin{aligned}
&&\left(b_{g}+b_{k}-
\frac{\Delta^{2}}{2M^{4}}\dot{\phi}^{2}e^{2\alpha\phi/M_{p}}\right)
\left[\ddot{\phi}+3H\dot{\phi}+
\dot{\phi}\partial_{t}\ln{\Big |}
b_{g}+b_{k}-
\frac{\Delta^{2}}{2M^{4}}\dot{\phi}^{2}e^{2\alpha\phi/M_{p}}{\Big |}
\right]
\nonumber\\
&&+\frac{\alpha\Delta^{2}}{4M^{4}M_{p}}\dot{\phi}^{4}e^{2\alpha\phi/M_{p}}
-\frac{\alpha M^{4}}{M_{p}}e^{-2\alpha\phi/M_{p}}=0
\label{FRWphi}\end{aligned}$$ where the energy density of the dilaton field is $$\rho_{eff}(\phi)=\frac{1}{4}\left(1+\frac{b_{k}}{b_{g}}\right)\dot{\phi}^{2}
-\frac{3\Delta^{2}}{16b_{g}M^{4}}\dot{\phi}^{4}e^{2\alpha\phi/M_{p}}+
\frac{M^{4}}{4b_{g}}e^{-2\alpha\phi/M_{p}}
\label{rho}$$ and the pressure $$p_{eff}(\phi)=\frac{1}{4}\left(1+\frac{b_{k}}{b_{g}}\right)\dot{\phi}^{2}
-\frac{\Delta^{2}}{16b_{g}M^{4}}\dot{\phi}^{4}e^{2\alpha\phi/M_{p}}-
\frac{M^{4}}{4b_{g}}e^{-2\alpha\phi/M_{p}}
\label{p}$$
One can see that Eqs. (\[FRWgr\])-(\[rho\]) allow solutions of a familiar quintessential form[@Wett1988NP668; @FJ] $$\phi(t)=\frac{M_{p}}{2\alpha}\phi_{0}+\frac{M_{p}}{\alpha}\ln (M_{p}t)
\label{phiq}$$ $$a(t)=t^{\gamma}
\label{aq}$$ which provides scaling behaviors of the dilaton energy density $$\rho_{eff}(\phi)\propto 1/a^{n}.
\label{scalrho}$$ The important role for possibility of such solutions belongs to the remarkable feature of the nonlinear terms in Eqs. (\[FRWgr\])-(\[rho\]) that appear only in the combination $\dot{\phi}^{2}e^{2\alpha\phi/M_{p}}$ which remains constant for the solutions (\[phiq\]) and (\[aq\]): $$\dot{\phi}^{2}e^{2\alpha\phi/M_{p}}=const
\label{constnonlinear}$$
Eqs. (\[phiq\])-(\[scalrho\]) describe solutions of Eqs. (\[FRWgr\])-(\[rho\]) with $n=\frac{2}{\gamma}$ if $$\gamma =\frac{b_{g}+b_{k}-y}{4b_{g}\alpha^{2}}
\label{gam}$$ where $$y\equiv\frac{\Delta^{2}M_{p}^{4}e^{\phi_{0}}}{2M^{4}\alpha^{2}}
\label{y}$$ is a solution of the cubic equation $$y^{3}-2(b_{g}+b_{k}-b_{g}\alpha^{2})y^{2}+(b_{g}+b_{k})(b_{g}+b_{k}-
\frac{4}{3}b_{g}\alpha^{2})y-\frac{2}{3}b_{g}\alpha^{2}\Delta^{2}=0.
\label{eqy}$$
Up to now we did not make any assumptions about parameters of the theory. We will now suppose that $b_{g}$ and $b_{k}$ are positive and consider two particular cases.
[*The case I*]{}. If $$b_{k}=b_{g}=b
\label{keqb}$$ then one can immediately see that Eqs. (\[FRWgr\])-(\[p\]) describe the FRW cosmological model in the context of the standard GR when the minimally coupled scalar field $\phi$ with the potential $\frac{M^{4}}{4b}e^{-2\alpha\phi/M_{p}}$ is the only source of gravity. In this case the scaling solution (\[phiq\]), (\[aq\]) coincides with the standard one[@FJ] where $$\gamma =\frac{1}{2\alpha^{2}}, \qquad n=4\alpha^{2}.
\label{qstand}$$
[*The case II*]{}. Another interesting possibility consists of the assumption that $$b_{k}\ll b_{g}
\label{bkey}$$ Then ignoring corrections of the order of $b_{k}/b_{g}$, the solutions of Eq. (\[eqy\]) are $$y_{1}=b_{g}
\label{soly1}$$ $$y_{2}=\frac{b_{g}}{2}\left[1-2\alpha^{2}+\sqrt{4\alpha^{4}-
\frac{20}{3}\alpha^{2}+1}\right]
\label{soly2}$$ $$y_{3}=\frac{b_{g}}{2}\left[1-2\alpha^{2}-\sqrt{4\alpha^{4}-
\frac{20}{3}\alpha^{2}+1}\right]
\label{soly3}$$
The solution $y_{1}$ corresponds to the static universe ($\gamma =0$ and $a(t)=const$) supported by the slow rolling scalar field $\phi$, Eq. (\[phiq\]). However, taking into account corrections of the order $b_{k}/b_{g}$ to $y_{1}$ we will get $\gamma\propto {\cal O}(b_{k}/b_{g})$.
Solutions $y_{2}$ and $y_{3}$ exist and are positive (see the definition (\[y\])) only if $$\alpha^{2}\leq \frac{1}{6}
\label{alpha}$$
The solution $y_2$ corresponds to the values of the parameter $\gamma$ monotonically varying from $\gamma_{min}=2/3$ up to $\gamma =1$ as $\alpha^2$ changes from 0 up to $1/6$.
The most interesting solution is given by $y_{3}$ that provides the values of the parameter $\gamma$ monotonically varying from $\gamma_{min}= 1$ up to $\infty$ as $\alpha^2$ changes from $1/6$ up to zero. In this case, Eqs. (\[phiq\])-(\[aq\]) describe an accelerated universe for all permissible values of $\alpha^2$ and the energy density of the dilaton field scales as in Eq. (\[scalrho\]) with monotonically varying $n$, $2\geq n\geq 0$ as $\alpha^2$ changes from $1/6$ up to zero. For the dilatonic matter equation-of-state $p=w\rho$ we get $$-1\leq w\leq -32/39\approx -0.82
\label{w}$$
In the conclusion of this section let us revert to one of the problems of the quintessence discussed in Introduction, namely to the flatness problem[@KLyth]. This is a question of the field theoretic basis for the choice of the flat enough potential. In fact, Kolda and Lyth noted [@KLyth] that an extreme fine tuning is needed in order to prevent the contribution from another possible terms breaking the flatness of the potential (see also for a review by Binetruy in Ref.[@CCP]). In the theory we study here, there is a symmetry (scale symmetry (\[st\])) which forbids the appearance of such dangerous contributions into $V_{eff}$, at least on the classical level. One can hope that the soft breaking of the scale symmetry guaranties that the symmetry breaking quantum corrections to the classical effective potential (\[Veff\]) will be small.
Here we have to make a note concerning quantization of the dilaton field. If $\Delta\neq 0$ then one can see from Eq. (\[rho\]) that there is a possibility of negative energy contribution from the space-time derivatives of the dilaton. This raises of course the suspicion that the quantum theory may contain ghosts. Let us see that this problem does not appear when considering small perturbations around the background determined by the studied above scaling solutions. To see this, let us calculate the canonically conjugate momenta to $\phi$, starting from the original action (\[totac\]) and expressing it in terms of the variables defined in CEF, Eq. (\[ct\]): $$\pi_{\phi}=\frac{1}{2b_{g}}\left(b_{g}+b_{k}-
\frac{\Delta^{2}}{sM^{4}}Ke^{2\alpha\phi/M_{p}}\right)\sqrt{-\tilde{g}}
\tilde{g}^{00}\dot{\phi}
\label{canconj}$$
As we have seen, the cosmological scaling solutions provide backgrounds where $Ke^{2\alpha\phi/M_{p}}=const$. Moreover, it is easy to see that for the scaling solutions $$\pi_{\phi}=\frac{1}{2b_{g}}\left(b_{g}+b_{k}-y\right)a^{3}\dot{\phi}=
2\alpha^{2}\gamma a^{3}\dot{\phi},
\label{canconjscaling}$$ where $\gamma$ and $y$ are defined by Eqs. (\[gam\]) and (\[y\]). We have seen also that for studied scaling solutions, $\gamma$ gets positive values. Therefore we conclude that in such backgrounds $\pi_{\phi}$ and $\dot{\phi}$ have the same sign, that guaranties a ghost-free quantization. The only exclusion is the particular case when $b_{k}=0$, $y=b_{g}$. As we have seen, such solution describes a static universe. In this case the canonically conjugate momenta $\pi_{\phi}=0$ and therefore it appears that in this vacuum there are no particles associated with the scalar field $\phi$.
Scale invariant fermion-dilaton coupling\
without the long-range force problem
=========================================
In general scalar-tensor theories, particle masses depend on time, when the theory is studied in the frame where Newton’s constant is really a constant. However, for all the fermionic matter observed in the universe, the cosmological variation of particle masses (including those of electrons) is highly constrained. We want to show now how the theory presented in this paper avoids this problem and also the so called fifth force problem, in spite of the need to include exponential couplings of the dilaton field to fermionic matter in order to ensure global scale invariance.
To describe fermions, normally one uses the vierbein ($e_{a}^{\mu}$) and spin-connection ($\omega_{\mu}^{ab}$) formalism where the metric is given by $g^{\mu\nu}=e^{\mu}_{a}e^{\nu}_{b}\eta^{ab}$ and the scalar curvature is $R(\omega ,e) =e^{a\mu}e^{b\nu}R_{\mu\nu ab}(\omega)$ where $$R_{\mu\nu ab}(\omega)=\partial_{\mu}\omega_{\nu ab}
+\omega_{\mu a}^{c}\omega_{\nu cb}
-(\mu\leftrightarrow\nu).
\label{B}$$
Following the general idea of the model, we now treat the geometrical objects $e_{a}^{\mu}$, $\omega_{\mu}^{ab}$, the measure fields $\varphi_{a}$, as well as the dilaton $\phi$ and the fermionic fields as independent variables. In this formalism, the natural generalization of the action (\[totac\]) keeping the general structure (\[S\]), when a fermion field $\Psi$ is also present and which also respect scale invariance is the following: $$\begin{aligned}
&S=& \int d^{4}x e^{\alpha\phi /M_{p}}
(\Phi +b\sqrt{-g})\left[-\frac{1}{\kappa}R(\omega ,e)
+\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]
\nonumber\\
&+&\int d^{4}x e^{\alpha\phi /M_{p}}
\left[(\Phi +k\sqrt{-g})\frac{i}{2}\overline{\Psi}
\left(\gamma^{a}e_{a}^{\mu}\overrightarrow{\nabla}_{\mu}
-\overleftarrow{\nabla}_{\mu}\gamma^{a}e_{a}^{\mu}\right)\Psi
-(\Phi +h\sqrt{-g})e^{\frac{1}{2}\alpha\phi/M_{p}}m\overline{\Psi}\Psi
\right]
\label{totaction}\end{aligned}$$ where $\overrightarrow{\nabla}_{\mu}=\overrightarrow{\partial}_{\mu}+
\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd}$ and $\overleftarrow{\nabla}_{\mu}=\overleftarrow{\partial}_{\mu}-
\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd}$.
The action (\[totaction\]) is invariant under the global scale transformations $$\begin{aligned}
e_{\mu}^{a}\rightarrow e^{\theta /2}e_{\mu}^{a}, \quad
\omega^{\mu}_{ab}\rightarrow \omega^{\mu}_{ab}, \quad
\varphi_{a}\rightarrow \lambda_{a}\varphi_{a}\quad
where \quad \Pi\lambda_{a}=e^{2\theta}
\nonumber
\\
\phi\rightarrow \phi-\frac{M_{p}}{\alpha}\theta ,\quad
\Psi\rightarrow e^{-\theta /4}\Psi, \quad
\overline{\Psi}\rightarrow e^{-\theta /4} \overline{\Psi}.
\label{stferm} \end{aligned}$$
In (\[totaction\]) two types of fermionic “kinetic-like terms” (as well as “mass-like terms”) which respect scale invariance have been introduced: they are coupled to the measure $\Phi$ and to the measure $\sqrt{-g}$ respectively. As we have discussed in the previous section, the quantum theory may in general contain ghosts if $b_{g}\neq b_{k}$. Taking this into account and also for the sake of a simplification of the presentation of the results we have chosen $b_{g}= b_{k}= b$. Notice however that in the framework of the classical theory, all conclusions will be made below are true also if $b_{g}\neq b_{k}$. Except for this, Eq.(\[totaction\]) describes the most general action[^5] satisfying the formulated above symmetries.
We can immediately obtain the equations of motion. From these going through similar steps to those performed in Sec. III, a constraint follows again which replaces (\[con1\]) and which contains now a contribution from the fermions. The spin-connection can be found by the variation of $\omega^{\mu}_{ab}$.
Similar to what we learned from the treatment of Sec.III, we can consider the theory in the CEF which in this case involves also a transformation of the fermionic fields: $$\begin{aligned}
\tilde{g}_{\mu\nu}=e^{\alpha\phi/M_{p}}(\zeta +b)g_{\mu\nu}, \quad
\tilde{e}_{a\mu}=e^{\frac{1}{2}\alpha\phi/M_{p}}(\zeta
+b)^{1/2}e_{a\mu},
\nonumber
\\
\Psi^{\prime}=e^{-\frac{1}{4}\alpha\phi/M_{p}}
\frac{(\zeta +k)^{1/2}}{(\zeta +b)^{3/4}}\Psi
\label{ctferm}\end{aligned}$$
In terms of these variables, the transformed spin-connections $\tilde{\omega}_{\mu }^{cd}$ turns out to be that of the Einstein-Cartan space-time and, besides, the new variables $\tilde{g}_{\mu\nu}$, $\tilde{e}_{a\mu}$, $\Psi^{\prime}$ and $\overline{\Psi}^{\prime}$ are invariant under the scale transformations (\[stferm\]). In the CEF the only field which still has a non trivial transformation property is the dilaton $\phi$ which gets shifted (according to (\[stferm\])). Thus, the presence of fermions does not change a conclusion made in Sec.III after Eq.(\[Gama\]): the spontaneous breaking of the scale symmetry is reduced, in the CEF, to the spontaneous breaking of the shift symmetry $\phi\rightarrow\phi +const$ for the dilaton field.
In terms of $\tilde{e}_{a\mu}$, $\Psi^{\prime}$, $\overline{\Psi}^{\prime}$ and $\phi$, the constraint (again arising as a self-consistency condition of equations of motion) which now replaces (\[con2\]) and which contains now a contribution from the fermions is $$(\zeta -b)M^{4}e^{-2\alpha\phi/M_{p}}+
F(\zeta)(\zeta +b)^{2}
m\overline{\Psi}^{\prime}\Psi^{\prime}=0.
\label{confermEin}$$ where we have chosen $s=+1$ for definiteness and the function $F(\zeta)$ is defined by $$F(\zeta)\equiv
\frac{1}{2(\zeta +k)^{2}(\zeta +b)^{1/2}}
[\zeta^{2}+(3h-k)\zeta +2b(h-k)+kh]
\label{F}$$
The dilaton field equation is $$(-\tilde{g})^{-1/2}\partial_{\mu}
\left(\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi \right)
-\frac{\alpha M^{4}}{M_{p}(\zeta +b)}e^{-2\alpha\phi/M_{p}}+
\frac{\alpha m}{M_{p}}F(\zeta)\overline{\Psi}^{\prime}\Psi^{\prime}
=0.
\label{phief+ferm}$$
The fermionic equation of motion in terms of the variables (\[ctferm\]) takes the standard structure of that in the Einstein-Cartan space-time[@Hehl] where a fermion field is the only source of a non-riemannian part of the connection. The only novelty of the fermionic equation consists of the form of the $\zeta$- depending fermion “mass” $m^{(eff)}(\zeta)$: $$m^{(eff)}(\zeta)=
\frac{m(\zeta +h)}{(\zeta +k)(\zeta +b)^{1/2}}
\label{muferm}$$
The gravitational equations are of the standard form (\[gef\]) with $$T_{\mu\nu}^{eff}=
\phi_{,\mu}\phi_{,\nu}-K\tilde{g}_{\mu\nu}
+\frac{b_{g}M^{4}}{(\zeta +b)^{2}}
e^{-2\alpha\phi/M_{p}}\tilde{g}_{\mu\nu}
+T_{\mu\nu}^{(f,canonical)}
-mF(\zeta)
\overline{\Psi}^{\prime}\Psi^{\prime}\tilde{g}_{\mu\nu},
\label{Tmn+f}$$ where $$T_{\mu\nu}^{(f,canonical)}=
\frac{i}{2}[\overline{\Psi}^{\prime}\gamma^{a}e_{a(\mu }^{\prime}
\nabla_{\nu )}\Psi^{\prime}-(\nabla_{(\mu}\overline{\Psi}^{\prime})
\gamma^{a}e_{\nu )a}^{\prime}\Psi^{\prime}]
\label{Tmnfcanon}$$ is the canonical energy-momentum tensor for the fermionic field in the curved space-time[@Birrel] and $\nabla_{\mu}\Psi^{\prime}
=\left(\partial_{\mu}+
\frac{1}{2}\tilde{\omega}_{\mu }^{cd}\sigma_{cd}\right)\Psi^{\prime}$ and $\nabla_{\mu}\overline{\Psi}^{\prime}=
\partial_{\mu}\overline{\Psi}^{\prime}-
\frac{1}{2}\tilde{\omega}_{\mu}^{cd}\overline{\Psi}^{\prime}\sigma_{cd}$.
The scalar field $\zeta$ is defined by the constraint (\[confermEin\]) in terms of the dilaton and fermion fields as a solution of the seventh degree algebraic equation that makes finding $\zeta$ in general a very complicated question. However there are two physically most interesting limiting cases when solving (\[confermEin\]) is simple enough.
Let us first analyze the constraint (\[confermEin\]) when the fermionic density (proportional to $\overline{\Psi}^{\prime}\Psi^{\prime}$) is very low as compared to the contributions of the dilaton potential ($\propto M^{4}e^{-2\alpha\phi/M_{p}}$). In this limiting case, the constraint gives again the expression (\[con2\]) for $\zeta$ where we have to take now $\Delta =0$, that is constraint yields the constant value[^6] $\zeta =b$. Inserting this value of $\zeta$ into (\[muferm\]) we see that the mass of a “test” fermion (that is when we ignore the effect of the fermion itself on the dilatonic background) is constant.
An opposite regime is realized when the contribution of the fermionic density to the constraint (\[confermEin\]) is very high as compared to the contribution of the dilaton potential. In the context of the present day universe, this regime corresponds in particular to the normal laboratory conditions in particle physics. Then according to the constraint (\[confermEin\]), one of the possibilities for this to be realized consists in the condition $$F(\zeta)= 0
\label{F=0}$$ from which we find two possible [*constant*]{} values for $\zeta$ $$\zeta_{1,2}=\frac{1}{2}\left[k-3h\pm\sqrt{(k-3h)^{2}+
8b(k-h)
-4kh}\,\right]
\label{zeta12}$$ These solutions, i.e. values $\zeta_{1}$ and $\zeta_{2}$, are real and different for very broad range of the parameters $b$, $k$ and $h$. These conditions have to be considered together with the obvious requirement $\zeta +b >0$ (see transformations (\[ctferm\])). For instance, for $h>0$, all these conditions are satisfied provided that parameters are situated in the broad region defined by the system of inequalities $(b-h)(b-k)>0$ and $(k-h)[k-h+8(b-h)]>0$.
We see from (\[muferm\]) that two different constants $\zeta$ given by (\[zeta12\]) define in general [*two specific masses*]{} for the fermion. We will assume that these two fermionic states should be identified with the first two fermionic generations.
The separate possibility relevant to the high fermionic density (again, as compared to the contributions of the dilaton potential) is the case when $$\zeta +b\approx 0
\label{zeta=0}$$ is a solution. However, the solution $\zeta +b=0$ is singular one as we see from equations of motion. This means that one can not neglect the first term in the constraint (\[confermEin\]) and instead of $\zeta +b=0$ we have to take the solution $\zeta_{3} \approx -b$ by solving $\zeta +b$ in terms of the dilaton field and the primordial fermion field itself. Then it follows from (\[confermEin\]) and (\[F\]) that $$\frac{1}{\sqrt{\zeta_{3} +b}}\approx
\left[\frac{m(h-b)}{4M^{4}b(k-b)}
\overline{\Psi}^{\prime}\Psi^{\prime}e^{2\alpha\phi /M_{p}}\right]^{1/3}.
\label{srtzeta}$$ Therefore, instead of constant masses, as it was for $\zeta_{1}$ and $\zeta_{2}$ (i.e. in the case $F(\zeta)=0$), this leads to higher fermion self-interaction which can be represented by the following term in the effective fermion Lagrangian in the dilatonic background $\phi =\bar{\phi}$: $$L^{ferm}_{selfint}=
3\left[\frac{1}{b}\left(\frac{m(h-b)}{4M(k-b)}
\overline{\Psi}^{\prime}\Psi^{\prime}\right)^{4}
e^{2\alpha\bar{\phi} /M_{p}}\right]^{1/3}.
\label{selfint}$$ The coupling constant of this self-interaction depends on the dilaton $\phi$. The condition (\[zeta=0\]) is realized, for example, as the classical cosmological background value $\phi =\bar{\phi}(t)\rightarrow\infty$ that corresponds to the late universe in the quintessence scenario. A full treatment of the case with $\zeta =\zeta_{3}$, which we assume corresponds to the third fermion generation, requires the study of quantum corrections. We expect that after $\overline{\Psi}^{\prime}\Psi^{\prime}$ develops an expectation value, the fermion condensate will give the third family appropriate masses similar to what we know in NJL model [@NJL] (for recent progress in this subject see e. g. Ref. [@Cv]). It is interesting to note that appearance of the higher fermion self-interaction here is related to the SSB of the scale invariance. In fact, the appearance of the integration constant $M$ in Eqs. (\[srtzeta\]) and (\[selfint\]) tells us that without SSB of scale invariance such interaction is not defined.
Concluding this analysis of equations when the fermionic density is of the order typical for the normal particle physics (which in the laboratory conditions is always much higher than the dilaton density ) we see that starting from a single primordial fermionic field we obtain exactly three different types of spin $1/2$ particles in the CEF. This appears to be a new approach to the family problem in particle physics and it will be subject of a detail study in another publication.
Coming back to the first two fermion families generated in the regime of fermion dominance as $F(\zeta)=0$ we note that surprisingly the same factor $F(\zeta)$ appears in the last terms of Eqs. (\[phief+ferm\]) and (\[Tmn+f\]). Therefore, in the regime where [*regular*]{} fermionic matter (i.e. $u$ and $d$ quarks, $e^{-}$ and $\nu_{e}$) is a dominant fraction, the last terms of Eqs. (\[phief+ferm\]) and (\[Tmn+f\]) [*automatically*]{} vanish. In Eq. (\[phief+ferm\]), this means that the fermion density $\overline{\Psi}^{\prime}\Psi^{\prime}$ is not a source for the dilaton and thus the long-range force disappears automatically. Notice that there is no need to require no interactions of the dilaton with regular matter at all to have agreement with observations but it is rather enough that these interactions vanish in the regime where regular fermionic matter dominates over other matter fields. In Eq. (\[Tmn+f\]), the condition (\[F=0\]) means that in the region where the regular fermionic matter dominates, the fermion energy-momentum tensor becomes equal to the canonical energy-momentum tensor of a fermion field in GR. [^7]
Discussion and conclusions
==========================
In this paper the possibility of a spontaneously generating exponential potential for the dilaton field in the context of TMT with spontaneously broken global scale symmetry was studied. The symmetry transformations formulated in terms of the original variables (\[st\]) (or (\[stferm\]) in the presence of fermions) include the global scale transformations of the metric, of the scalar fields $\varphi_{a}$ related to the measure $\Phi$ (and of the fermion fields) and in addition the dilaton field $\phi$ undergoes a global shift. In the CEF (see Eqs. (\[ct\]) or (\[ctferm\]) where the theory is formulated in the Riemannian (or Einstein-Cartan) space-time), all dynamical variables are invariant under the transformations (\[st\]) (or (\[stferm\])) except for the dilaton field which still gets shifted by a constant. Thus, SSB of the scale symmetry that appears firstly in (\[varphi\]) when solving Eq. (\[7\]), is reduced, in the CEF, to SSB of the shift symmetry $\phi\rightarrow\phi
+const$.
The original action does not includes potentials but in the CEF, the exponential potential appears as a result of SSB of the scale symmetry. In the generic case $\Delta
=b_{g}-b_{k}\neq
0$, the process of SSB also produces terms with higher powers in derivatives of the dilaton field.
Cosmological scaling solutions of the theory were studied. The flatness of the potential $V_{eff}$ which is associated here with the exponential form, is protected by the scale symmetry. Quintessence solutions (corresponding to accelerating universe) were found possible for a broad range of parameters.
Finally, the behavior of fermions in such type of models was investigated. Scale invariant fermion mass-like terms can be introduced in two different ways since they can appear coupled to each of the two different measures of the theory. Although an exponential of the dilaton field $\phi$ couples to the fermion in both of these terms, it is found that when the fermions are treated as a test particles in the scaling background, their masses in the CEF are constants.
Even more surprising is the behavior of the fermions in the limit of high fermion density as compared to the dilaton density. This approximation is regarded as more realistic if we are interested in the regular particle physics behavior of these fermions under normal laboratory conditions. It is found then that in the CEF, a given fermion can behave in three different ways according to the three different solutions of the fundamental constraint (\[confermEin\]). Two of the solutions correspond to fermions with constant masses and the other - to a higher fermion self-interaction which, we expect, can generate mass on the quantum level in a manner similar to a NJL model[@NJL]. From one primordial fermion three are obtained for free. This suggests a new approach to the “family problem” in particle physics.
In addition to this, for the two mentioned above solutions (\[zeta12\]) corresponding to constant fermion masses, the fermion-dilaton coupling in the CEF (proportional to $F(\zeta)$, Eq.(\[F\])) disappears automatically. If one of these types of fermions is associated to the first family (regular matter, i.e., $u$ and $d$ quarks, $e^{-}$ and $\nu_{e}$), we obtain that normal matter decouples from the dilaton.
All what has been done here concerning fermions is in the context of a toy model without Higgs fields, gauge bosons and the associated $SU(2)\times U(1)\times SU(3)$ gauge symmetry of the standard model. As we have seen in other models (see [@GK4], the second reference of [@G1] and [@K]), it is possible to incorporate the two measure ideas with the gauge symmetry and Higgs mechanism. Now the differences consist of: i) the presence of global scale symmetry, ii) the most general TMT structure for gravitation and dilaton sector. The complete discussion of the standard model in the context of such TMT structure will be presented in a separate publication[@prep]. Here we want only to explain shortly the main ideas that provides us the possibility to implement this program. It is important that in a simple way gauge fields can be incorporated so that they will not appear in the fundamental constraint[^8] in contrast to the fermions (see for comparison Eq. (\[confermEin\])). We can also work without significant changes in the discussion of the fermionic sector if instead of explicit mass-like terms we will work with similar terms where the coupling constants with the dimensionality of the mass are replaced by gauge invariant Yukawa couplings to the Higgs field. Proceeding in the spontaneously broken $SU(2)\times U(1)$ gauge theory[@prep] and starting from one correspondent primordial fermions family we observe again[@prep] the effect of generation of three fermion families, as was above in the toy model. Generating mass of two of them is automatic as in the previous discussion. For the third we need again some quantum effect that gives rise to a fermion condensate.
The analysis of the constraint (\[confermEin\]) provides in general seven solutions for $\zeta$. It could be that among of them there is a solution corresponding to a fermionic state responsible for dark matter. For example, the solution (\[srtzeta\]) after inserting into $00$-component of the energy-momentum tensor (\[Tmn+f\]) makes the last three terms of (\[Tmn+f\]) to be dependent in the same manner only on the combination $\overline{\Psi}^{\prime}\Psi^{\prime}
e^{\frac{1}{2}\alpha\phi /M_{p}}$ and they appear to be of the same order of magnitude. This implies that fermion contributions to the energy and those of the scalar field are of the same order that provides then a possible explanation of the “cosmic coincidence” problem. A consistent study of these cosmological questions will become possible after we will explore in detail[@prep] the field theoretic aspects of the displayed here “families birth effect”.
Acknowledgments
===============
We are grateful to J. Alfaro, S. de Alwis, M. Banados, J. Bekenstein, Z. Bern, R. Brustein, K. Bronnikov, V. Burdyuzha, L. Cabral, S. del Campo, C. Castro, A. Chernin, A. Davidson, G. Dvali, M. Eides, P. Gaete, H. Goldberg, M. Giovannini, A. Guth, F. Hehl, V. Ivashchuk, V. Kiselev, A. Linde, L. Horwitz, M. Loewe, A. Mayo, Y. Ne’eman, Y. Jack Ng, M. Pavsic, M.B.Paranjape, L. Parker, J. Portnoy, V.A. Rubakov, E. Spallucci, Y. Verbin, A. Vilenkin, M. Visser, K. Wali, C. Wetterich, P. Wesson and J.Zanelli for discussions on different aspects of this paper.
[99]{}
See, for example, N. Bahcall, J.P. Ostriker, S.J. Perlmutter and P.J. Steinhardt, Science [**284**]{}, 1481 (1999) and references therein.
C. Wetterich, Nucl. Phys. [**B302**]{}, 668 (1988); B. Ratra and P.J.E. Peebles , Phys. Rev. [**D37**]{}, 3406 (1988); P.J.E. Peebles and B. Ratra, Astrophys. J. [**325**]{}, L17 (1988); R. Caldwell, R. Dave and P. Steinhardt, Phys. Rev. Lett. [**80**]{}, 1582 (1998); N. Weiss, Phys. Lett. [**B197**]{}, 42 (1987); Y. Fujii and T. Nishioka, Phys. Rev. [**D42**]{}, 361 (1990); M.S. Turner and M. White, Phys. Rev. [**D56**]{}, R4439 (1997); E. Copeland, A. Liddle and D. Wands, Phys. Rev. [D57]{}, 4686 (1998); C.T. Hill, D.N. Schramm and J.N. Fry, Comments Nucl. Part. Phys. [**19**]{}, 25 (1989); J. Frieman, C. Hill and R. Watkins, Phys. Rev. [**D46**]{}, 1226 (1992); J. Frieman, C. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett. [**75**]{}, 2077 (1995); C. Wetterich, Nucl. Phys. [**B302**]{}, 645 (1988); Astron. Astrophys. [**301**]{}, 321 (1995); P. Ferreira and M.Joyce, Phys. Rev. Lett. [**79**]{}, 4740 (1997); I. Zlatev, L. Wang and P. Steinhardt, Phys. Rev. Lett. [**82**]{}, 896 (1999); P. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. [**D59**]{}, 123504 (1999).
P. Ferreira and M.Joyce, Phys. Rev. [**D58**]{}, 023503 (1998).
I. Novikov, [*Evolution of the Universe,*]{} Cambridge University Press, 1983. S.Weinberg, Rev. Mod. Phys. [**61**]{}, 1 (1989) and [*The Cosmological Constant Problem*]{}, astro-ph/0004075; Y.J. Ng, Int. J. Mod. Phys. [**D1**]{}, 145, (1992); S.M. Carroll, W.H. Press and E.L. Turner, Ann. Rev. Astron. and Astrophys. [**30**]{},499 (1992); V. Sahni and A. Starobinsky, “The Case for a Positive Cosmological $\Lambda$-term”, astro-ph/9904398. S.M. Carroll, [*The cosmological constant*]{}, astro-ph/0004075; P. Binetruy, [*Cosmological constant vs. quintessence*]{}, hep-ph/0005037; J. Garriga and A. Vilenkin, [*Solutions to the cosmological constant problems*]{}, hep-th/0011262.
C. Kolda and D. Lyth, Phys. Lett [B458]{}, 197 (1999).
S.M. Carroll, Phys. Rev. Lett. [**81**]{}, 3067 (1998).
A.D. Dolgov, Phys. Reports [**320**]{}, 1 (1999).
E.I. Guendelman and A.B. Kaganovich, Phys. Rev. [**D53**]{}, 7020 (1996); Mod. Phys. Lett. [**A12**]{}, 2421 (1997); Phys. Rev. [**D55**]{}, 5970 (1997).
E.I. Guendelman and A.B. Kaganovich, Phys. Rev.[**D56**]{}, 3548 (1997).
E.I. Guendelman and A.B. Kaganovich, Phys. Rev.[**D57**]{}, 7200 (1998); Mod. Phys. Lett. [**A13**]{}, 1583 (1998)
E.I. Guendelman and A.B. Kaganovich, Phys. Rev. [**D60**]{}, 065004 (1999).
E.I. Guendelman, Mod. Phys. Lett. [**A14**]{}, 1043 (1999); Class. Quant. Grav. [**17**]{}, 361 (2000); gr-qc/0004011.
E.I. Guendelman, Mod. Phys. Lett. [**A14**]{}, 1397 (1999); gr-qc/9901067; hep-th/0106085.
E.I. Guendelman, hep-th/0011049, to appear in Foundations of Physics.
A.B. Kaganovich, Phys. Rev.[**D63**]{}, 025022 (2001).
P.J.E. Peebles and A. Vilenkin, Phys. Rev. [**D59**]{}, 063505 (1999).
C. Wetterrich, Nucl. Phys. [**B252**]{}, 309 (1985); J. Halliwell, ibid. [**B266**]{}, 228 (1986); Q. Shafi and C. Wetterrich, Phys. Lett. [**152B**]{}, 51 (1985); Q. Shafi and C. Wetterrich, Nucl. Phys. [**B289**]{}, 787 (1987).
E. Cremer et al., Phys. Lett. [**133B**]{}, 61 (1983); J. Ellis et al., ibid. [**134B**]{}, 429 (1984); H. Nishino and E. Sezgin, ibid [**144B**]{}, 187 (1984); E. Witten, ibid. [**155B**]{}, 151 (1985); M. Dine et al., ibid. [**156B**]{}, 55 (1985); J. Halliwell, ibid. [**185B**]{}, 341 (1987). J. Yokoyama and M. Maeda, ibid. [**207B**]{}, 31 (1988).
J. Barrow and S. Cotsakis, Phys. Lett., [**214B**]{}, 515 (1988); S. Cotsakis, Phys. Rev. [**D47**]{}, 1437 (1993).
D. Gross and E. Witten, Nucl. Phys. [**B277**]{}, 1 (1986); J. Polchinski, [*Superstrings*]{}, Vol.II (Cambridge U. Press, Cambridge, 1998), Ch. 12.
F.H. Hehl and B.K.Datta, J.Math.Phys. [**12**]{}, 1334 (1971); V. de Sabbata and M. Gasperini, [*Introduction to Gravity*]{} (World Scientific, Singapore, 1985).
N.D. Birrell and P.C.W. Davies, [*Quantum fields in curved space*]{} (Cambridge University Press, 1982).
Y. Nambu and G. Jona-Lasinio, Phys. Rev. [**122**]{}, 345.
G. Cvetic, Rev. Mod. Phys. [**71**]{}, 513 (1999).
E.I. Guendelman and A.B. Kaganovich, in preparation.
[^1]: guendel@bgumail.bgu.ac.il
[^2]: alexk@bgumail.bgu.ac.il
[^3]: See for comparison Refs. [@exppot1; @exppot2; @exppot3] and a general discussion in Ref. [@FJ]
[^4]: Other possible origin for the non-linear kinetic terms, known in the literature[@WP], are higher order gravitational corrections in string and supergravity theories.
[^5]: Recall that in this paper we restrict ourselves to the models without explicit dilaton potentials in the original action
[^6]: Notice that if we had chosen $b_{g}\neq b_{k}$ and assumed the quintessential cosmological solution of Sec.IV where $Ke^{2\alpha\phi/M_{p}}=const$, we would again get a constant value of $\zeta$.
[^7]: The decoupling of the dilaton in the CEF in the case of high fermion density was discussed also in a simpler spontaneously broken scale invariant model (with $b=k=0$ and explicit exponential potentials) in Ref. [@G1]. In the framework of other TMT model[@K] (with small [*explicit*]{} breaking of the scale invariance) in the context of the quintessential scenario, a special tuning of the parameters is needed to achieve the dilaton-fermion decoupling.
[^8]: This may be done by making the gauge field kinetic terms coupled only to $\sqrt{-g}$ which is dictated by local scale invariance of that part of the action.
|
---
abstract: 'An approximate equation of motion is proposed for screw and edge dislocations, which accounts for retardation and for relativistic effects in the subsonic range. Good quantitative agreement is found, in accelerated or in decelerated regimes, with numerical results of a more fundamental nature.'
author:
- 'L. Pillon'
- 'C. Denoual'
- 'Y.-P. Pellegrini'
title: Equation of motion for dislocations with inertial effects
---
Introduction
============
Dislocation behavior in solids under dynamic conditions (e.g. shock loading [@HORN62; @CLIF81; @TANG03]) has recently attracted renewed attention, [@TANG03; @HIRT98; @GUMB99; @MARI04; @DENO04; @PILL06; @DENO07] partly due to new insights provided by molecular dynamics studies. [@GUMB99; @TANG03; @MARI04] Whereas theoretical investigations mainly focused on the stationary velocities that regular or twinning dislocations can attain as a function of the applied stress (possibly intersonic or even supersonic with respect to the longitudinal wave speed $c_\text{L}$),[@WEER69; @GUMB99; @ROSA01] one other major concern is to establish an equation of motion [@ESHE53; @CLIF81; @ROSA01; @HIRT98] (EoM) suitable to instationary dislocation motions towards or from such high velocities, and which is computationally cheap. This would be an important step towards extending dislocation dynamics (DD) simulations [@DEVI97; @BULA06; @WANG06] to the domain of high strain rates, in order to better understand hardening processes in such conditions.
The key to instationary motion of dislocations lies in the inertia arising from changes in their long-ranged displacement field, which accompany the motion. These retarded rearrangements take place at finite speed, through wave emission and propagation from the dislocation. As a consequence, dislocations possess an effective inertial mass,[@ESHE53] which has bearings on the process of overcoming dynamically obstacles such as dipoles, etc. [@PILL06; @WANG06; @BITZ05] Inertial effects are non-local in time, and are related to effective “viscous” losses. For small velocities where the EoM is linear,[@ESHE53] this relation takes the form of the Kramers-Krönig relations between the reactive and dissipative parts of the causal mass kernel.[@NABA51; @ESHE62; @ALSH71] One major ingredient of the EoM should thus be the effective visco-inertial force exerted on the dislocation by its own wave emission.[@ESHE53; @CLIF81] An EoM results from balancing it by the applied stress, and by drags of various origins.[@ALSH86] EoMs with effective masses, but which ignore retardation (e.g., Ref.), cannot truly capture visco-inertial effects. Previous works on these questions having mainly been confined to the linear regime, their influence in the relativistic domain remains largely unexplored in spite of analytical progresses, partly due to the complexity of the formalism (especially for edge dislocations).
Hereafter, Eshelby’s EoM for screws with a rigid core,[@ESHE53] valid at small velocities, is first re-examined, and cast under a simple form which suggests a straightforward regularization procedure for finite core effects. This allows us to appeal to previous results for point dislocations valid at high velocities.[@CLIF81] We then build in an heuristic way an EoM for accelerated or decelerated screw and edge dislocations in the drag-dominated subsonic regime, that consistently accounts for saturation effects at velocities comparable to the sound speed. Results from the equation are compared to quasi-exact calculations from a numerical method of the phase-field type. Having in mind applications to DD simulations, the scope of the study is limited to continuum theory, so that dispersion effects due to the atomic lattice,[@ISHI73] or to the influence of the Peierls potential,[@ALSH71] are not explicitly considered.
Eshelby’s force
===============
Within the Peierls-Nabarro model in isotropic elasticity,[@PEIE40; @NABA47] and with the usual $\arctan$ ansatz for the relative displacement $u(x,t)$ of the atoms on both sides of the glide plane, Eshelby computed the visco-inertial force $F$ experienced by a screw dislocation of Burgers vector $b$, centered on position $x(t)$ at time $t$, moving with a velocity $v=\dot{x}$ small compared to the shear wave speed $c_\text{S}$:[@ESHE53] $$\begin{aligned}
\label{eq:eshforce}&&F(t)=m_0\int_{-\infty}^t{\rm d}\!\tau
\frac{\dot{v}(\tau)}{\left[(t-\tau)^2+t_{\rm S}^2\right]^{1/2}}\\
&&{}+m_0\int_{-\infty}^t \hspace{-1em}{\rm d}\!\tau \frac{t_{\rm
S}^2}{\left[(t-\tau)^2+t_{\rm S}^2\right]^{3/2}} \frac{\rm d}{{\rm
d}\tau}\left(\frac{x(t)-x(\tau)}{t-\tau}\right)\nonumber.\end{aligned}$$ The dislocation is assumed to have a *rigid* core of half-width $\zeta_0$. Then $t_{\rm S}=2\zeta_0/c_\text{S}$ is the time of shear wave propagation over the core width. The mass per unit dislocation length $m_0=\mu b^2/(4\pi
c_\text{S}^2)$ depends on the shear modulus $\mu$. In Ref. (and in Ref. as well), an incorrect factor $1/2$ is present in front of the second integral, and has been removed here. This factor is of no important physical consequence, save for different values of the linear response kernels; see below.
That (\[eq:eshforce\]) is correct can be verified as follows. Starting from Eshelby’s expression of the force as a double integral in Eq. (26) of Ref., and expanding it to linear order in the velocity $v(\tau)$ or in $x(t)-x(\tau)$, the following expression is easily obtained: $$\begin{aligned}
\label{eq:flin1}
F(t) &=& 2 m_0\int_{-\infty}^t {\rm
d}\tau\,\left\{\frac{\dot{v}(\tau)}{\sqrt{\Delta
t^2+t_{\rm S}^2}}\right.\nonumber\\
&&\left. {}-\frac{2 t_{\rm S}^2-\Delta t^2}{(\Delta t^2+t_{\rm
S}^2)^{5/2}}[x(t)-x(\tau)]\right\},\end{aligned}$$ where $\Delta t=t-\tau$. Using integrations by parts over $\tau$, each of Eq. (\[eq:eshforce\]) and (\[eq:flin1\]) can be put under the following irreducible form: $$\label{eq:canonical} \frac{F(t)}{m_0}=2\frac{v(t)}{t_{\rm
S}}-2\frac{x(t)}{t_{\rm S}^2}+2\int_{-\infty}^t {\rm d}\tau\,\frac{
x(\tau)}{\left[(t-\tau)^2+t_{\rm S}^2\right]^{3/2}},$$ which shows them to coincide.
By the same token, we check that (\[eq:eshforce\]) can be further simplified as: $$\label{eq:flin2} F(t)=\int_{-\infty}^t{\rm d}\!\tau \frac{2
m_0}{\left[(t-\tau)^2+t_{\rm S}^2\right]^{1/2}}\frac{\rm d}{{\rm
d}\tau}\left(\frac{x(t)-x(\tau)}{t-\tau}\right).$$ By Fourier transforming $F(t)$ \[under the form (\[eq:canonical\])\] and by writing $$F(\omega)\equiv [-\omega^2 m(\omega)-i\omega \eta(\omega)]\,x(\omega),$$ we identify effective mass $m(\omega)$ and viscosity $\eta(\omega)$ kernels. [@NABA51] Their expression in closed form involves the modified Bessel and Struve functions $K_1$, $I_1$ and $\mathcal{L}_{-1}$:
\[eq:masskercf\] $$\begin{aligned}
\label{meshcf} \frac{m(\omega)}{m_0}&=&2\frac{1-t_{\rm S}|\omega|K_1(t_{\rm S}|\omega|)}{(t_{\rm S}|\omega|)^2}\\
\label{veshcf}\frac{\eta(\omega)}{m_0}&=&\frac{2}{t_{\rm
S}}\left\{1+\frac{\pi}{2}\bigl[I_1(t_{\rm S}|\omega|)-\mathcal{L}_{-1}(t_{\rm
S}|\omega|)\bigr]\right\}\end{aligned}$$
To leading orders in the pulsation $\omega$,
\[eq:massker\] $$\begin{aligned}
\label{mesh} m(\omega)/m_0&=&\left(\frac{1}{2}+\ln\frac{2e^{-\gamma}}{t_{\rm
S}|\omega| }
\right)+O\left((t_{\rm S}\omega)^2\ln t_{\rm S}\omega\right)\\
\label{vesh} \eta(\omega)/m_0&=&\frac{\pi}{2}|\omega|+O(t_{\rm S}|\omega|^2)\end{aligned}$$
where $\gamma$ is Euler’s constant. Moreover, we observe that $$\label{eq:viscoinfty} \eta(|\omega|\to\infty)/m_0=2/t_{\rm S}.$$ Result (\[eq:massker\]) coincides to leading order with Eshelby’s, [@ESHE53] as $\omega\to 0$. The mass increase with wavelength as $\omega\to
0$ implies very different behaviors for, e.g., quasi-static and shock loading modes, since the latter involves a wider frequency range. We note that $\eta(\omega)\to 0$ as $\omega\to 0$, since losses should be absent from the model in the stationary subsonic regime. [@ESHE53] The non-analytical behavior of the kernels at $\omega=0$ (due to $|\omega|$), and its associated non-locality in time has been emphasized in Ref. .
The finite “instantaneous” viscosity (\[eq:viscoinfty\]) stems from the first term in the R.H.S. of (\[eq:canonical\]), and is responsible for a velocity jump $\Delta v$ undergone by the dislocation when subjected to a jump $\Delta F$ in the applied force.[@ESHE53; @CLIF81] From (\[eq:viscoinfty\]) we deduce: $$\label{eq:vjump} \Delta v=\frac{\Delta F}{\eta(\infty)}=\frac{t_{\rm S}\Delta
F}{2 m_0}=4\pi\frac{\zeta_0 c_\text{S}\Delta F}{\mu b^2}.$$ The velocity jump (\[eq:vjump\]) increases with core width. It was first predicted by Eshelby from his equation,[@ESHE53] and can be understood as follows for a screw dislocation along the $z$ axis: the force jump $\Delta F$ is due to a shear stress jump $\Delta\sigma_{yz}=\Delta F/b$ attaining simultaneously all the points of the whole glide plane (e.g., as the result of shear loading applied on faces of the system containing the plane, parallel to the latter). Neglecting material inertia of the atoms on both sides of the dislocation plane, the medium undergoes an elastic strain jump $\Delta
\sigma_{yz}/\mu=\Delta \varepsilon_{yz}\sim\Delta v_m/c_\text{S}$, determined by a material velocity jump $\Delta v_m$. The latter is equilibrated through outward emission of a shear wave with velocity $c_\text{S}$. On the other hand, the slope of the displacement function near the core is $(\partial u/\partial
x)\sim b/(2\zeta_0)$, so that $\Delta v_m$ is related to the dislocation velocity jump $\Delta v$ by $\Delta v_m\sim \Delta v\, b/(2\zeta_0)$. Combining these relationships yields (\[eq:vjump\]), up to a numerical constant factor. The same argument applies to other types of dislocations. In case of several relaxation waves (e.g., longitudinal and shear waves for an edge dislocation), that of lowest celerity controls the amplitude of the velocity jump. It should be borne in mind, however, that accounting for material inertia from the atoms on both sides of the glide plane results in an instantaneous inertial force of order $F_i=2m_0 \ddot{x}$ to be added to (\[eq:eshforce\]).[@ESHE53] By balancing the forces, it is seen that this force should spread the velocity jump over a short rise time $$\label{eq:risetime} \Delta t \sim t_{\rm S}.$$
Equation of motion
==================
No expression analogous to (\[eq:eshforce\]) is available for edge dislocations. However, Clifton and Markenscoff computed the force acting on a *point* screw or edge dislocation moving with any subsonic velocity in an isotropic medium, that jumps instantaneously at instant $t=\tau$ from rest to a constant velocity $v$.[@CLIF81] A generalization to anisotropic media is available.[@WU02] To maintain its velocity constant, this dislocation must be subjected, at time $t>\tau$, to the time-decaying force $$\label{eq:fcm} F^{\rm CM}(t-\tau,v)=\frac{g\bigl(v\bigr)}{t-\tau},$$ where the function $g$ depends on its character and on anisotropy. [@ESHE53; @CLIF81] We now construct heuristically a force for accelerated motion by interpreting such a motion as a succession of infinitesimal velocity jumps. Assuming that, for instationary motion, $v$ in (\[eq:fcm\]) can be interpreted as $v(\tau)$, the elementary force that would arise from the elementary jump $\delta v(\tau)$ at $t=\tau$ is: $ \delta F=[\partial F^{\rm
CM}\bigl(t-\tau,v(\tau)\bigr)/\partial v(\tau)]\delta v(\tau)$ $=$ $g'\bigl(v(\tau)\bigr)\delta v(\tau)/(t-\tau)$. Then, the total force experienced by the dislocation results from integrating such elementary forces over past history: $$\label{eq:newforce0} F(t)=\int_{-\infty}^t {\rm d}\!\tau\, \frac
{g'\bigl(v(\tau)\bigr)}{t-\tau}\dot{v}(\tau).$$ Comparing (\[eq:newforce0\]) to (\[eq:flin2\]) shows, firstly, that the relevant “accelerations” at linear order are different. However, we remark that $2({\rm d}/{\rm d}t)\{[x(t)-x(\tau)]/(t-\tau)\}\to \dot{v}(\tau)$ as $t\to\tau$, and moreover that for a screw dislocation, $g'(v\simeq 0)=m_0$. [@CLIF81] Hence, since we interpret $v$ in (\[eq:fcm\]) as $v(\tau)$, the numerator of the integrand in (\[eq:newforce0\]) is correct at least for small velocities and for small times $t\to\tau$. Its relevance for large velocities is demonstrated below through comparisons to full-field calculations. Next, integral (\[eq:newforce0\]) is singular at $\tau=t$, due the point-dislocation hypothesis at the root of (\[eq:fcm\]). However, using (\[eq:flin2\]) as a physical motivation, we propose a regularization consisting in replacing the kernel $1/t$ in (\[eq:newforce0\]) by $1/[t^2+t_0^2]^{1/2}$ where $t_0$, the counterpart of $t_{\rm S}$ in (\[eq:eshforce\]), is some time characteristic of sound propagation over a core diameter. In Sec. \[sec:appli\], $t_0$ is chosen alternatively proportional to $t_{\rm S}=2\zeta_0/c_\text{S}$ and to $t_{\rm
L}=2\zeta_0/c_\text{L}$ in the case of edge dislocations for illustrative purposes, whereas $t_0$ is proportional to $t_{\rm S}$ for screws. The proportionality factor, 1/2 in all cases, is justified below. From a physical point of view, inertia is controlled by the slowest wave so that better results are expected using $c_\text{S}$ for all types of dislocations. Given Eshelby’s rigid-core hypothesis in (\[eq:flin2\]), and the approximations made, it would be pointless to refine this treatment. Another kind of regularization is used in Ref. (p. 195), which consists in replacing the upper bound $t$ of integral (\[eq:newforce0\]) by $t-t_0$ (in Ref. , the integrand assumes that $v\simeq 0$).
With the above regularization the force eventually reads: $$\label{eq:newforce1} F_{\text{reg}}(t)=\int_{-\infty}^t {\rm d}\!\tau\, \frac
{g'\bigl(v(\tau)\bigr)}{[(t-\tau)^2+t_0^2]^{1/2}}\dot{v}(\tau).$$ Its Fourier transform for small velocities where $g'(v)\simeq g'(0)$ yields, in terms of modified Bessel and Struve functions of order 0,
\[eq:masskercfapprox\] $$\begin{aligned}
\label{meshcfapprox} m(\omega)/g'(0)&=&K_0(t_0|\omega|)\\
&=&\ln\frac{2e^{-\gamma}}{t_{\rm S}|\omega|}
+O\left((t_{\rm S}\omega)^2\ln t_{\rm S}\omega\right),\nonumber\\
\label{veshcfapprox}\eta(\omega)/g'(0)&=&\frac{\pi}{2}|\omega|
\bigl[I_0(t_0|\omega|)-\mathcal{L}_{0}(t_0|\omega|)\bigr]\nonumber\\
&=&\frac{\pi}{2}|\omega|+O(t_{\rm S}|\omega|^2)\\
\label{veshcfapproxinf}\eta(|\omega|\to\infty)/g'(0)&=&1/t_0.\end{aligned}$$
The approximation therefore preserves the logarithmic character of the mass, and the viscosity, to leading order. The mass is slightly decreased, the constant $m_0/2$ in (\[mesh\]) being absent. This difference is insignificant given the approximations made. In the limit of small velocity for a screw dislocation, our approximation amounts to retaining in (\[eq:eshforce\]) the first integral only. In order to recover a correct velocity jump for screws, we must take $t_0\simeq t_{\rm S}/2$ since the instantaneous viscosity (\[veshcfapproxinf\]) is different from (\[eq:viscoinfty\]). This “calibration” is used in the next section for screws and (somewhat arbitrarily) for edges as well.
In the stationary limit, the visco-inertial force (\[eq:newforce1\]) vanishes. For $v\ll c_{\rm S}$, the asymptotic velocity should be determined by a viscous drag force, mainly of phonon origin,[@ALSH86] $F_{\rm
drag\,0}=\eta_0 v$, where $\eta_0$ is the viscosity. This force is modified (in the context of the Peierls-Nabarro model) by the relativistic contraction of the core, into $F_{\rm drag}(v)=\eta(v) v$. For subsonic velocities, $\eta(v)=\eta_0 D(0)/D(v)$, where:[@ROSA01] $$\label{eq:contract}
D(v)=\left[A^2(v)+\alpha^2 (v/c_{\rm S})^2 \right]^{1/2},$$ with $\alpha=\eta_0\zeta_0/(2 m_0 c_{\rm S})$, is an effective viscosity-dependent core contraction factor, such that the core length in the laboratory frame reads: $\zeta(v)=\zeta_0 D(v)/D(0)$. The purely relativistic contraction factor $A(v)$ is, with $\gamma_{\rm L,S}=\left(1-v^2/c_{\rm
L,S}^2\right)^{1/2}$:[@WEER61; @ESHE49; @ROSA01] $$A(v)= \left\{
\begin{array}{c}
\frac{1}{2}(c_{\rm S}/v)^2\left(4\gamma_{\rm L}
-\gamma_{\rm S}^{-1}-2\gamma_{\rm S}-\gamma_{\rm S}^{3}\right)\mbox{ for edges},\\
\vphantom{\Bigl(}\frac{1}{2}\gamma_{\rm S}\hspace{12em}\mbox{ for screws}.
\end{array}
\right.\nonumber$$ With this drag, and introducing the applied stress $\sigma_a$, the EoM finally reads: $$\label{eq:pdpeq} \frac{\mu b^2}{2\pi}\int_{-\infty}^t{\rm d}\!\tau\,
\frac{\widetilde{g}'\bigl(v(\tau)\bigr)\dot{v}(\tau)}{[(t-\tau)^2+t_0^2]^{1/2}}+F_{\rm
drag}(v(t))=b\sigma_a,$$ where $g(v)\equiv 2 m_0\, \widetilde{g}(v)$, and where:[@CLIF81] $$\begin{aligned}
\widetilde{g}(v)&=&(\gamma_{\rm S}^{-1}-1)/v,\hspace{3.5em}\text{for screw dislocations},\\
&=&(8\gamma_{\rm L}+4\gamma_{\rm L}^{-1}-7\gamma_{\rm S}-6\gamma_{\rm S}^{-1}+\gamma_{\rm S}^{-3})c_{\rm S}^2/v^3\nonumber\\
&&-2[1-(c_{\rm S}^2/c_{\rm L}^2)^2]/v,\quad\text{for edge dislocations}.\end{aligned}$$ This is our main result. By construction, it reproduces the asymptotic velocities of Ref. .
We checked numerically that the replacement of $\zeta_0$ by $\zeta(v)$ in $t_0$ does not change by more than a few percent the overall results described in the following section. Since this change in $t_0$ would bring in nothing useful, we choose to use $\zeta_0$ in $t_0$ in the following section.
![\[fig:rosakis\] Upper: Relationship between dimensionless applied stress $\sigma/\mu$ and asymptotic velocity $v/c_\text{S}$ provided by the PNG code, [@DENO07] for an accelerated edge dislocation in the stationary regime of an accelerated dislocation (dots), compared to that predicted by Rosakis’s Model I[@ROSA01] (lines) for different viscosity parameters $\alpha$, for a screw dislocation in the subsonic regime. Lower: Normalized velocity-dependent core width $\zeta(v)/\zeta(0)$ measured under same conditions. $c_\text{R}$ is the Rayleigh velocity.](PillonEtAl_Rosakis.eps){width="8cm"}
Applications {#sec:appli}
============
Setting $v(t)=\sum_i \Delta v_i \theta(t-t_i)$, Eq. (\[eq:pdpeq\]) is solved numerically for edge and screw dislocations, in an implicit way with a time step $\Delta t=t_{i+1}-t_i$ small enough. Results are compared with numerical points obtained with the *Peierls-Nabarro-Galerkin* (PNG) approach [@DENO04; @DENO07] used here as a benchmark. This method is less noisy than molecular dynamics, allows for full-field dynamic calculations of the displacement and stress fields in the whole system, accounting for wave propagation effects, and allows for better flexibility. We can thus, e.g., control boundary conditions by applying analytically computed forces, so as to prevent image dislocations from perturbing the simulation window.
Firstly, to check the accuracy of the benchmark, asymptotic velocities of screw and edge dislocations were compared to the stationary predictions of Rosakis’ Model 1.[@ROSA01] In the PNG method, the permanent lattice displacement field (which is part of the full atomic displacement, $u$) is relaxed by means of a Landau-Ginzburg equation, with viscosity parameter $\eta_\text{PNG}$. An exact correspondence holds between this viscosity and Rosakis’s viscosity parameter $\alpha$, namely $\eta_\text{PNG}=\alpha\mu/c_\text{S}$, as can be shown by specializing to one dimension the general field equations of Ref.. A $\gamma$-potential $\gamma(u)=(1/2)\gamma_0\sin^2(\pi
u/b)$, with $\gamma_0=(2/3)C_{44}b/\pi^2$, is used. The material is an elastically cubic material, with elastic moduli taken such that $C_{44}=C_{12}=C_{11}/3$ to insure isotropy. Due to the elastic correction made to the $\gamma$-surface potential in order to remove its quadratic elastic part,[@DENO04; @DENO07] the core at rest is a bit larger in the PNG results than in the Peierls-Nabarro solution. The time dependent core width $\zeta\bigl(v(t)\bigr)$ is measured from the numerical simulations by using $b^2/(2\pi\zeta)\equiv \int {\rm d}x\, [u'(x)]^2$ (the value corresponding to a core of the arctan type). Two-dimensional calculations are carried out using a simulation box of size $300\times 30$ $b^2$, with a unique horizontal glide plane along $Ox$. Eight nodes per Burgers vector are used in both directions. Forces are applied on the top and bottom sides so as to induce shear on the unique glide plane. Free boundary conditions are used on sides normal to the $Ox$ axis. Measurements are done near the center of the box, where the mirror attracting forces these sides generate on the dislocation, are negligible. The box is wide enough so that the dislocation accelerates and reaches its terminal velocity. Comparisons between PNG results and Rosakis’ model are displayed in Fig. \[fig:rosakis\] for different viscosities $\alpha$, in the case of an edge dislocation. The core scaling factor $D(v)$ and the asymptotic velocity $v/c_\text{S}$, are directly measured from simulations under different applied stresses $\sigma$, and compared to theory. [@ROSA01] The PNG asymptotic velocities were found to be $5\%$ systematically lower than the theoretical results. This correction is accounted for in the figure. The overall agreement is excellent. It is emphasized that core contraction effects in the viscous drag \[Eq. (\[eq:contract\])\] are required in order to obtain a good match.
![\[fig:step\_vis\]Velocities vs. time for accelerated screw dislocations: white dots, PNG code; solid, Ref. ; dots (in left curve only), Eq. (\[eq:eshforce\]); dash-dots, linear approximation to (\[eq:pdpeq\]); dashes, fully relativistic equation (\[eq:pdpeq\]).](PillonEtAl_step_vis.eps){width="8.5cm"}
![\[fig:step\_coin\]Velocities vs. time for accelerated edge dislocations: white dots, PNG code; solid, Ref. ; dash-dots, linear approximation to (\[eq:pdpeq\]); dashes, fully relativistic equation (\[eq:pdpeq\]). Curves obtained from (\[eq:pdpeq\]) are duplicated, using either $t_0$ computed with $c_\text{S}$ (upper), or with $c_\text{L}$ (lower), see text.](PillonEtAl_step_coin.eps){width="8.5cm"}
Next, comparisons in the accelerated regime are made with Eq. (\[eq:pdpeq\]) and with other models. Fig. \[fig:step\_vis\] displays, as a function of time, the velocity of a screw dislocation accelerated from rest by a constant shear stress $\sigma_a$ applied at $t=0$. Low and high shear stresses are examined. These stresses lead to terminal asymptotic velocities $v_t=v(t=\infty)=0.007\, c_{S}$ and $0.75\, c_{S}$, computed from (\[eq:pdpeq\]). The results displayed are obtained: (i) with the PNG approach (white dots); (ii) with Eq. (\[eq:pdpeq\]) using fully “relativistic” expressions of $\widetilde{g}(v)$ and $D(v)$ (dashes); (iii) with linear small-velocity approximations of $\widetilde{g}(v)$, but with the full expression of $D(v)$, in order to emphasize the importance of relativistic effects in the retarded force (dash-dots, for the case $v_\text{T}=0.75
c_\text{S}$); (iv) with a previous EoM,[@HIRT98] using a typical cut-off radius $R=500$ nm in the logarithmic core term (solid) corresponding to a typical dislocation density of $10^{12}$/m${}^2$. The result arising from using (\[eq:eshforce\]) in the EoM is also displayed for the lowest speed (dots, left figure only).
Figure \[fig:step\_coin\] presents similar curves for an edge dislocation. For the latter, $t_0$ is taken either as $t_{\rm S}/2$ or as $t_{\rm L}/2$, $t_{\rm
L}=2\zeta_0/c_\text{L}$, thus providing two limiting curves. The curves with $t_{\rm S}$ provide the best matches, consistently with the above observation that the wave of lowest velocity $c_{\rm S}<c_{\rm L}$ should provide the main contribution to inertia. At low and high speeds, good agreement is obtained between PNG points and Eq. (\[eq:pdpeq\]), provided that fully “relativistic” expressions are used for $g(v)$ (especially for edge dislocations); otherwise, inertia is strongly underestimated. In all the curves, the relativistic expression of the non-linear viscous terms was used. Moreover, variations of the core width with velocity,[@ROSA01] implicitly present in PNG calculations, and ignored in the expression of $t_0$ used in the visco-inertial term of (\[eq:pdpeq\]), are not crucial to accelerated or to decelerated motion (see Fig. \[fig:deceler\]); still, the core width shrinks by 20% during the acceleration towards $v_t=0.75\, c_{L}$. On the other hand, retardation effects in the effective mass are crucial: curves with non-local inertial forces are markedly different from the solid ones using the masses of Ref. , computed at constant velocity. The version of the PNG code used here does not include the above-mentioned effects of material inertia in the glide plane, so that the full-field velocity curves indeed display what resembles a velocity jump, like the EoM. Owing to (\[eq:risetime\]), this lack of accuracy solely concerns the time interval between the time origin and the first data point: hence we can consider that the velocity jump is a genuine effect, and not an artefact, at least from the point of view of full-field calculations in continuum mechanics. However, we should add that, to our knowledge, this effect has not been reported so far in molecular dynamics simulations.
![\[fig:deceler\]Velocities vs. time for a decelerated screw dislocation. Comparison between the PNG method (white dots)[@DENO04] and equation (\[eq:pdpeq\]) with (dashes) or without (dot-dash) fully relativistic expressions.](PillonEtAl_deceleration.eps){width="8.5cm"}
Figure \[fig:deceler\] displays the velocity of screw and edge dislocations decelerated from the initial velocity $v_i$. Comparisons between EoMs and PNG calculations are then harder to make than in the accelerated case. Indeed, the non-relativistic (resp. relativistic) theoretical curves from Eq.(\[eq:pdpeq\]) (dashed-dot) \[resp. (dashed)\] are obtained by assuming that an applied stress abruptly vanishes at $t=0$. This induces a negative velocity jump in the curves. This jump is larger if non-relativistic expressions are used, which demonstrates in passing the higher inertia (i.e. “mass”) provided by relativistic expressions. The same loading was tried in the PNG calculations as well, but led to non-exploitable results due to multiple wave-propagation and reflection phenomena. Therefore, PNG curves for decelerated motion were obtained instead using a somewhat artificial loading: the medium was split in a zone of constant stress, separated from a zone of zero stress by an immobile and sharp boundary. The dislocation is then made to accelerate in the zone of constant stress. Due to the finite core width, the boundary is crossed in a finite time $\simeq \zeta / v_i$, which explains the smoothed decay of the velocity in the PNG data points. This type of loading cannot be realistically implemented within the framework of Eq.(\[eq:pdpeq\]) because the dislocation core is not spatially resolved. Hence, though the curves strongly suggest that relativistic effects are as important in deceleration as in acceleration, and that (\[eq:pdpeq\]) reproduces well the PNG points, the comparison between the latter and theoretical curves should be taken here with a grain of salt. On the other hand, the EoM of Ref. (solid) is once again clearly imprecise. As a final remark, we expect our neglecting of retardation effects in the nonlinear viscous term of (\[eq:pdpeq\]) to induce an underestimation of damping effects. This may explain why the PNG curves decay faster than that from Eq. (\[eq:pdpeq\]).
Concluding remarks
==================
An empirical relativistic equation of motion for screw and edge dislocations, accounting for retardation effects in inertia, Eq. (\[eq:pdpeq\]), has been proposed. We compared it, together with another available approximate EoM, to a quasi-exact numerical solution of a dynamical extension of the Peierls-Nabarro model, provided by the *Peierls-Nabarro Galerkin* code.[@DENO07] The latter was beforehand shown to reproduce very well the asymptotic velocities of Rosakis’s model 1[@ROSA01] in the subsonic regime. The best matches with full-field results were found with our EoM, both for accelerated and for decelerated motion, thus illustrating quantitatively the importance of retardation and of relativistic effects in the dynamic motion of dislocations. To these effects, our EoM provides for the first time a satisfactory approximation for high velocities in the subsonic range. Our comparisons rule out the use of masses computed at constant velocity. One of the restrictions put forward by Eshelby to his EoM was its limitation to weakly accelerated motion, mainly due to the rigid core assumption.[@ESHE53] Ours makes no attempt to explicitly overcome this simplification. However, comparisons with full-field calculations, where the core structure is not imposed from the start, but emerges as the result of solving the evolution equation for the displacement field, shows that this rigid-core assumption is acceptable on a quantitative basis as far as inertia is concerned, at least for velocities high, but not too close to $c_{\rm S}$.
The authors thank B. Devincre for stimulating discussions, and F. Bellencontre for his help during preliminary calculations with the PNG code.
[99]{} E. Hornbogen, Acta Metall. [**10**]{}, 978 (1962). R.J. Clifton and X. Markenscoff, J. Mech. Phys. Solids. [**29**]{}, 227 (1981). D. Tanguy, M. Mareschal, P.S. Lomdahl, T.C. Germann, B.L. Holian and R. Ravelo, Phys. Rev. B [**68**]{}, 144111 (2003). J.P. Hirth, H.M. Zbib and J.P. Lothe, Modelling Simul.Mater. Sci. Eng. [**6**]{}, 165 (1998). P. Gumbsch and H. Gao, Science [**283**]{}, 965 (1999). J. Marian, W. Cai and V.V. Bulatov, Nature Materials [**3**]{}, 158 (2004). C. Denoual, Phys. Rev. B [**70**]{}, 024 106 (2004). L. Pillon, C. Denoual, R. Madec and Y.-P. Pellegrini, J. Phys. IV (France) [**134**]{}, 49 (2006). C. Denoual, Comput. Methods Appl. Mech. Engrg. [**196**]{}, 1915 (2007). J. Weertman in *Mathematical theory of dislocations*, edited by T. Mura (ASME, New York, 1969), p. 178. P. Rosakis, Phys. Rev. Lett. [**86**]{}, 95 (2001). J.D. Eshelby, Phys. Rev. [**90**]{}, 248 (1953). B. Devincre and L.P. Kubin, Mater. Sci. Eng. A [**234**]{}-[**236**]{}, 8 (1997). V.V. Bulatov, L.L. Hsiung, M. Tang, A. Arsenlis, M.C. Bartelt, W. Cai, J.N. Florando, M. Hiratani, M. Rhee, G. Hommes, T.G. Pierce and T. Diaz de la Rubia, Nature [**440**]{}, 1174 (2006). Z.Q. Wang, I.J. Beyerlein and R. LeSar, Los Alamos preprint LA-UR-06-3602 (unpublished). E. Bitzek and P. Gumbsch Mater. Sci. Eng. A [**400**]{}, 40 (2005). J.D. Eshelby, Proc. R. Soc. London, Ser. A [**266**]{}, 222 (1962). V.I. Al’shitz, V.L. Indenbom and A.A. Shtol’berg, Zh. Eksp. Teor. Fiz. [**60**]{}, 2308 (1971) \[Sov. Phys. JETP [**33**]{}, 1240 (1971)\]. F.R.N. Nabarro, Proc. R. Soc. London, Ser. A [**209**]{}, 278 (1951). V.I. Alshits and V.L. Indenbom, in *Dislocations in Solids* vol. 7, edited by F.R.N. Nabarro (North-Holland, Amsterdam, 1986). S. Ishioka, J. Phys. Soc. Japan [**34**]{}, 462 (1973). R. Peierls, Proc. Phys. Soc. [**52**]{}, 34 (1940). F.R.N. Nabarro, Proc. Phys. Soc. [**59**]{}, 256 (1947). K.C. Wu, Acta Mechanica [**158**]{}, 85 (2002). J.P. Hirth and J.P. Lothe, *Theory of crystal dislocations* (Krieger, New York, 1982). J. Weertman, in *Response of materials to high velocity deformations*, edited by P.G. Shewmon and V.F. Zackay (Interscience, New York,1961), p. 205. J.D. Eshelby, Proc. Phys. Soc. A [**62**]{}, 307 (1949).
|
---
abstract: 'We present the first implementation of Active Galactic Nuclei (AGN) feedback in the form of momentum driven jets in an Adaptive Mesh Refinement (AMR) cosmological resimulation of a galaxy cluster. The jets are powered by gas accretion onto Super Massive Black Holes (SMBHs) which also grow by mergers. Throughout its formation, the cluster experiences different dynamical states: both a morphologically perturbed epoch at early times and a relaxed state at late times allowing us to study the different modes of BH growth and associated AGN jet feedback. BHs accrete gas efficiently at high redshift ($z > 2$), significantly pre-heating proto-cluster halos. Gas-rich mergers at high redshift also fuel strong, episodic jet activity, which transports gas from the proto-cluster core to its outer regions. At later times, while the cluster relaxes, the supply of cold gas onto the BHs is reduced leading to lower jet activity. Although the cluster is still heated by this activity as sound waves propagate from the core to the virial radius, the jets inefficiently redistribute gas outwards and a small cooling flow develops, along with low-pressure cavities similar to those detected in X-ray observations. Overall, our jet implementation of AGN feedback quenches star formation quite efficiently, reducing the stellar content of the central cluster galaxy by a factor 3 compared to the no AGN case. It also dramatically alters the shape of the gas density profile, bringing it in close agreement with the $\beta$ model favoured by observations, producing quite an isothermal galaxy cluster for gigayears in the process. However, it still falls short in matching the lower than Universal baryon fractions which seem to be commonplace in observed galaxy clusters.'
bibliography:
- 'author.bib'
date: 'Accepted 2010 July 09. Received 2010 June 04; in original form 2010 April 11'
title: 'Jet-regulated cooling catastrophe'
---
\[firstpage\]
galaxies: clusters: general – galaxies: active – galaxies: jets – methods: numerical
Introduction
============
The well-known over–cooling problem in galaxy formation is encountered in numerical simulations for a range of galaxy sizes, spanning dwarves to red ellipticals. Stellar feedback mechanisms, such as winds from young stars and supernovae explosions are potentially good candidates for expelling large amounts of cold gas from galaxies. Unfortunately, these mechanisms are inadequate for the most massive galaxies, where the energy liberated by star formation activity is too low to unbind material from their gravitational potential wells.
Different theories find ways to partially or totally prevent the cooling catastrophe in the most massive structures: galaxy groups and clusters. Propagating heat with a Spitzer conductivity from the outskirts to the central parts of a cluster, thermal conduction could solve the cooling problem for the more massive clusters . However, due to the propagation of electrons along magnetic field lines, the conduction is essentially anisotropic leading to instabilities (such as the Heat flux Buoyancy Instability, HBI, see ) in the cluster core. These can reorient the magnetic field lines and stop the net inflow of heat towards the centre [@parrish09; @bogdanovic09]. Pre-heating of the gas destined to fall into a cluster potential well during proto-galactic stages has also been proposed as a means to empty gas on galactic scales in these massive halos [@babuletal02].
However, in most models of massive galaxy formation, AGN play a crucial role in regulating their gas content. Red and dead galaxies commonly exhibit the signatures of SMBHs which are thought to power high velocity jets into the hot surroundings of the galaxies. Observational evidence for strong AGN activity in groups and clusters is plenty [@arnaudetal84; @carillietal94; @mcnamaraetal01; @mcnamaraetal05; @fabianetal02; @birzanetal04; @formanetal07]. When spatially resolved, this activity often takes the form of radio lobes or cavities [@boehringeretal93; @owenetal00; @birzanetal04; @mcnamaraetal05; @fabianetal06; @tayloretal06; @dongetal10; @dunnetal10], or thin extended jets [@bridleetal94]. The radio lobes or cavities are associated with low AGN activity, i.e. the radio mode, and jets with the quasar modes.
AGN feedback is invoked to efficiently suppress star formation in the most massive galaxies either by ejecting gas from their Interstellar Medium (ISM) into the Intra-Cluster Medium (ICM), or by preventing the ICM gas from collapsing into galactic discs . The powerful ejection of gas by AGN is also supposed to suppress the formation of cool cores in some fraction of galaxy clusters, and in turn to make the ICM turbulent [@duboisetal09].
Many of the previous numerical studies that attempt to simulate the growth of BHs and their associated AGN feedback have been performed using the Smoothed Particle Hydrodynamics (SPH) technique . As shown by [@mitchelletal09], SPH codes suffer from underestimating the true entropy[^1] profile in cluster cores because of their trouble resolving Kelvin-Helmoltz instabilities in regions of strong density contrast [@agertzetal07]. As it is generally assumed that BHs accrete gas at a Bondi rate which is related to the local entropy $\dot M_{\rm BH}\propto K^{-3/2}$, a poor estimate of this entropy leads to an incorrect calculation of the accretion rate onto a BH and, thus, its energy release. To circumvent this issue, we use a sink particle approach to follow the growth and AGN feedback of BHs within an Adaptive Mesh Refinement (AMR) code.
Moreover, in previous cosmological simulations, AGN feedback is modeled with a thermal input of energy. Meanwhile, theoretical work and observations suggest that AGN feedback is mostly mechanical, not thermal. Numerous numerical simulations have implemented and tested the formation and propagation of AGN jets on cluster scales and their impact on the ICM using either idealized simulations , or cosmological simulations [@heinzetal06; @morsonyetal10], but none so far have followed the BH growth self-consistently (i.e. resolving both the BH growth and jet-AGN feedback in cosmological simulation over a Hubble time). As the AGN feedback is tightly linked to its BH growth history, it is of crucial importance to model both the BH evolution through time and its jet-energy release. In this paper, we propose to bridge this gap by performing the first cosmological simulation including a self-consistent treatment of BH evolution and its associated AGN jet energy release.
The paper is organized as follows. In section \[physics\], we describe the physical ingredients of our simulation. We start by presenting our cooling and star formation prescriptions and then we introduce the scheme for following the formation and mergers of BHs, gas accretion onto them, and their energy release in the form of jets. In section \[IC\], we describe our initial condition set-up and our simulation run. In section \[BHgrowth\], we show how the BH growth is linked to the galaxy cluster formation history, and what drives the different modes of AGN feedback (radio versus quasar). In section \[Cooling\_catastrophe\], we demonstrate that this type of anisotropic mechanical AGN feedback is able to suppress the cooling catastrophe occuring in the galaxy cluster. Finally, in section \[discussion\], we discuss our results.
Modeling the physics of galaxy formation {#physics}
========================================
Modeling star formation {#sec:sf}
-----------------------
Gas in our simulation is allowed to radiate energy by atomic collisions in a H/He primordial gas so that it can collapse into dark matter potential wells to form galaxies [@silk77]. To model reionization from $z=8.5$, heating from a UV background is followed with the prescriptions from . Star formation occurs in high density regions $\rho> \rho_0$ ($\rho_0=0.1\, \rm H.cm^{-3}$). When the density threshold is surpassed, a random Poisson process spawns star cluster particles according to a Schmidt law $\dot \rho_*= \epsilon \rho/t_{\rm ff}$, where $t_{\rm ff}$ is the gas free-fall time and $\epsilon$ is the star formation efficiency, taken to be $\epsilon=0.02$ in order to reproduce the observational surface density laws [@kennicutt98]. The reader can consult and for more information on the star formation implementation.
In this work, we do not model supernova feedback. Several authors have argued that supernovae can only have a dynamical impact on low mass galaxies . Thus, as a first order approximation, we assume that supernova feedback has very little effect on the growth of BHs in massive galaxies as they alone appear incapable of removing a substantial fraction of gas from the ISM. Whilst this simplification allows us to properly isolate the effect of AGN feedback on the surrounding gas from any other galactic feedback mechanism, it does not allow us to study the role of metal enrichment on cooling in the centre of massive halos. However, we will see that the main features of a cooling catastrophe are already captured with zero metallicity cooling.
Finally it is possible to view the modification of the temperature at high density $\rho> \rho_0$ by a polytropic equation of state (EoS) that we introduce for numerical reasons in section \[sec:accretion\] as a way to take into account the thermal effect of the heating of the ISM by supernovae. As a matter of fact, a similar EoS approach is used by other authors (e.g. ) as a simple model for the unresolved multiphase structure of the ISM in cosmological simulations. More specifically, the minimum temperature in dense regions becomes $T_{\rm min}=T_0 (\rho/\rho_0)^{n-1}$, with $T_0=10^4K$, and $n=4/3$ which leads to a constant Jeans mass $M_{\rm J}=1.3\, 10^9 \, \rm M_{\odot}$. Such a value of the polytropic index $n$ roughly compares with the complex functional form of the EoS obtained by analytical considerations on the multiphase structure of the ISM in .
SMBHs as sink particles {#sink_algorithm}
-----------------------
Sink particles were first introduced by [@bateetal95] in a SPH code. Sinks are massive particles that capture gas particles in their surroundings. They mimic the formation of unresolved compact objects, e.g. proto-stellar cores in the ISM, black holes in the ISM, central super-massive black holes in galaxies, etc. Due to the very Lagrangian nature of the sink particle technique, it had been extensively and exclusively used in SPH codes until [@krumholzetal04] extended its use to grid codes. The version in [ramses]{} [@teyssier02] is strongly inspired by the [@krumholzetal04] numerical implementation.
Sink particles are created in regions where the Jeans criterion is violated, i.e. in regions where the maximum level of refinement is reached and where the gas density is large enough to potentially produce a numerical instability, in other words where: $${\Delta x\over 4} > \lambda_{\rm J}=\sqrt{\pi c_s^2\over G \rho}\, .
\label{L_Jeans}$$ Here $\Delta x$ is the size of the smallest cell, $\lambda_{\rm J}$ the Jeans length, $c_s$ the sound speed and $\rho$ the gas density. According to [@trueloveetal97], the numerical stability of a gravitationally bound object is ensured if it is resolved with at least 4 cells. With a mixed composition of matter (dark matter, gas, stars), Jeans stability is not trivial anymore, but we can reasonably assume that gas is the dominant source of gravitational potential inside dense collapsed objects, like galaxies, in our case.
For numerical stability, each time that the Jeans criterion is violated we should spawn a sink particle with a mass corresponding to the depleted mass. However, in cosmological simulations this leads to excessively large sink masses. The reason is that the gas is concentrated in structures (galaxies) that are poorly resolved (kpc scale). As a result an entire galactic disk can be defined by only a few Jeans-violating cells leading to massive sinks. To form sufficiently small seed BHs in the centres of the galaxies, we prefer to choose their initial mass, $M_{\rm seed}$, thereby introducing a free parameter. We set $M_{\rm seed}=10^5 \, \rm M_{\odot}$ in agreement with previous cosmological simulations (e.g. ). However, BHs are still spawned only in cells belonging to the maximum level of refinement and that verify equation (\[L\_Jeans\]) . One consequence of this self-controlled formation of the sink BHs is that they are not allowed to accrete gas when the Jeans criterion is violated. The only way for them to accrete gas is to do so by a reasonable physical process such as Bondi accretion. With this prescription for initializing the mass of the seed black hole, it is conceivable that gas could be numerically violently Jeans unstable, but this issue is partially solved by the consumption of gas in the star forming process that temporarily restores gravitational stability.
To get only one BH per massive galaxy, a halo finder is usually run on-the-fly during the simulation to check if candidate galaxies already host a BH . We prefer a simpler, more direct, and computationally cheaper approach. To avoid creating multiple sinks inside the same galaxy, we ensure that each time a cell could potentially produce a sink (i.e. it verifies eq. (\[L\_Jeans\])), it is farther than a minimum radius $r_{\rm min}$ from all other pre-existing sinks. This distance has to be larger than the typical size of galactic discs and smaller than the typical average inter-galactic distance. Test runs suggest that the choice $r_{\rm min}=100$ kpc produces very satisfactory results.
To avoid formation of sink particles in low density regions that are Jeans-unstable, we set a minimum threshold for the density $\rho>\rho_0$ of gas that can create a new sink, where $\rho_0$ is the same density threshold that we use for star formation. In order to avoid the creation of sink black holes before the formation of the very first stars, we check that the star density $\rho_*$ verifies $$f_*={\rho_*\over \rho_*+\rho} > 0.25 \, ,
\label{fraction_star}$$ before a sink is spawned, where $\rho$ is the gas density.
When a sink particle is finally created it is split into several cloud particles with equal mass. Cloud particles are spread over a $4 \Delta x$ radius sphere and positioned every $0.5 \Delta x$ in (x,y,z). The exact number of cloud particles in this configuration is therefore $n_{\rm cloud}=2109$ per sink. This splitting process is useful in many ways. First, it keeps a heavy sink particle from becoming the dominant gravitational contribution in one single cell. The latter could catapult particles far from their host galaxy by two body encounters. Second, it provides a simple canvas over which to compute averaged quantities around the sink, for example we use it to determine the Bondi accretion rate.
Sinks are also allowed to merge together if they lie at a distance closer than $4\Delta x$ from each other. Mass is conserved in this process and momentum vectors of the old sinks are simply added to compute the momentum of the new sink.
Finally we insist on the fact that sink positions and velocities are updated in the classical way used to update standard particles such as DM particles. No correction on their positions and velocities is done to force them to stay near their host galaxy (as could be done with the Halo finder approach). Thus, weakly bound BHs, such as BHs in satellite galaxies of large groups and clusters, may easily be stripped from their host galaxy. These BHs behave like star particles that tidal forces compel to populate the stellar halo of massive galaxies.
Accretion rate onto SMBHs {#sec:accretion}
-------------------------
Since we fail to resolve the accretion disk around our SMBHs, whose size is sub-parsec even for the most massive ones ($\sim 10^{-3}$ pc according to [@morganetal10] from micro-lensing estimates), we use the common prescription that these BHs accrete gas at a Bondi-Hoyle-Lyttleton rate [@bondi52] $$\dot M_{\rm BH}={4\pi \alpha G^2 M_{\rm BH}^2 \bar \rho \over (\bar c_s^2+v^2) ^{3/2}}
\label{dMBH}$$ where $\alpha$ is a dimensionless boost factor ($\alpha\ge 1$), $M_{\rm BH}$ is the black hole mass, $\bar \rho$ is the average gas density, $\bar c_s$ is the average sound speed, and $v$ is the gas velocity relative to the BH velocity. One of the major difficulties encountered with the computation of the relative gas velocity is that in cosmological runs, the ISM is poorly resolved and leads to a very thin scale height for galaxies (compared to the resolution). Moreover, due to poor sampling of the gravitational force in the galactic disc, BHs can slightly oscillates in their host galaxy. For this reason a BH close to the centre of a galaxy can feel the infalling material coming from the halo at a relative velocity much higher than the typical velocity inside the bulge. As $v$ is not a reliable quantity, we do not compute $v$ as the gas velocity relative to the sink velocity but we prefer to set it to the average gas velocity dispersion in the ISM which is assumed constant and equal to $\sigma=10\, \rm km.s^{-1}$ [@dibetal06; @agertzetal09].
The average density $\bar \rho$ and sound speed $\bar c_s$ are computed around the BH using the cloud particles for this operation, as mentioned in section \[sink\_algorithm\]. To compute the averages, the cell in which each particle sits is assigned a weight given by a kernel function $w$, similar to the one used in [@krumholzetal04]: $$w\propto\exp \left( -r^2/r_K^2\right )\, ,$$ where $r$ is the distance from the cloud particle to the sink particle and $r_K$ is the radius defined as $$r_K =
\left\{
\begin{array}{lr}
\Delta x/4 & \,\, r_{\rm BH}<\Delta x /4\, ,\\
r_{\rm BH}&\,\, \Delta x /4 \le r_{\rm BH} \le 2 \Delta x\, , \\
2\Delta x & \,\, r_{\rm BH}>2\Delta x\, .
\end{array}
\right.$$ The Bondi-Hoyle radius $r_{\rm BH}$ is given by: $$r_{\rm BH}={GM_{\rm BH} \over c_s^2}\, ,$$ where $c_s$ is the exact sound speed in the cell where the sink lies.
The true accretion rate onto the sink is finally limited by its Eddington rate $$\dot M_{\rm Edd}={4\pi G M_{\rm BH}m_{\rm p} \over \epsilon_{\rm r} \sigma_{\rm T} c}\, ,
\label{dMEdd}$$ where $\sigma_{\rm T}$ is the Thompson cross-section, $c$ is the speed of light, $m_P$ is the proton mass, and $\epsilon_{\rm r}$ is the radiative efficiency, assumed to be equal to $0.1$ for the accretion onto a Schwarzschild BH.
The accretion rate is computed at each time step and a fraction $\dot M_{\rm BH} \Delta t / n_{\rm cloud}$ of gas mass is depleted from the cell where the cloud particle lies and is added to that cloud particle, and its sink mass is updated accordingly. At each coarse time step cloud particles are re-scattered with equal-mass $M_{\rm BH}/n_{\rm cloud}$. As the timestep does not depend on the accretion speed onto BHs and as low density cells can be close to high density cells, a BH might remove more mass than is acceptable. To avoid negative densities and numerical instabilities arising from this, we do not allow a cloud particle to deplete more than 25% of the gas content in a cell.
In such large scale cluster simulations, it is impossible to resolve the scale and the clumpiness of the ISM. To prevent the collapse of the gas from numerical instabilities and to take into account the mixing of the different phases in the ISM (cold and warm components), we use the polytropic EoS described in section \[sec:sf\]. Applying this EoS means that it is impossible to know the “true” density and the “true” sound speed in the ISM, thus the accretion rate onto the BHs must be modified. Previous work modeling the accretion rate onto BHs with such a polytropic EoS set the $\alpha$ parameter to a constant 100 [@springeletal05; @sijackietal07; @dimatteoetal08]. Here we follow the prescription from who show that $\alpha = (\rho/\rho_0)^2$ is the best parametric choice to match observational laws.
We stress that this polytropic EoS has important consequences on the accretion rate onto BHs in gas rich systems: equation (\[dMBH\]) turns into $\dot M_{\rm BH}\propto{M_{\rm BH}^2 \rho^{5/2}}$, and the temperature dependence is removed. On the other hand, as soon as the cold gas component has been evaporated by star formation or feedback mechanisms from massive galaxies, the accretion rate of the black hole is, by definition, the proper Bondi accretion rate. This $\alpha$ boost of the accretion rate is an artificial way of modeling the very fast accretion of gas within cold and gas-rich galaxies at early epochs, where the clumpiness of the ISM is unresolved in large-scale cosmological simulations.
AGN feedback modeling
---------------------
Following [@ommaetal04], we assume that the primary source of AGN feedback is a sub-relativistic, momentum-imparting bipolar outflow. As discussed in detail by these authors, there are many considerations which support this hypothesis, the most blatant one being that bipolar outflows are observed around virtually all accreting objects in the Universe: stars, black holes and galaxies. Along with these authors, we further assume that the advection dominated inflow-outflow solution (ADIOS) developed by to explain the low luminosities of AGNs compared to their estimated Bondi-Hoyle accretion rates is correct. The most important feature of the ADIOS model, as far as we are concerned, is that the bulk of the accretion energy (released as plasma falls into the SMBH) drives a (sub-relativistic) wind from the surface of the accretion disk. We emphasize that, as pointed out in [@ommaetal04], this bipolar wind is distinct from observed relativistic synchrotron jets which are probably powered by the spin of the SMBH itself, although both jets can simultaneously be present. However the synchrotron jet is very likely irrelevant in terms of AGN feedback since the mechanical luminosity of the sub-relativistic outflow is much higher than the synchrotron luminosity. Finally, we note that in very dense environments it is probable that much of the accretion energy is radiated away, driving outflows with velocities $\approx 0.1 c$ in objects with photon luminosities also on the order of the Eddington luminosity . We argue that as more and more shock-heated rarefied gas fills the central parts of the growing host halo, more and more energy will come out as mechanical energy. More specifically, we assume that a fraction $\epsilon_{\rm f}$ of the radiated energy is imparted to the ambient gas $$\dot E_{\rm AGN}=\epsilon_{\rm f} L_r=\epsilon_{\rm f} \epsilon_{\rm r} \dot M_{\rm BH}c^2\, ,
\label{E_BH}$$ in the form of a kinetic jet.
Such an implementation has the advantage of continuously releasing energy without radiating it away by cooling on the scale of a hydro timestep. Indeed this problem plagues supernovae feedback modelling where energy is generally injected in thermal form and leads to the feedback having no dynamical impact on the surrounding gas. To bypass this issue, some authors release AGN thermal energy only after a sufficient amount of gas has been accreted onto the BH so as to more severely impact the ambient medium . In contrast, we model the AGN feedback as a jet-like structure with the same momentum profile defined in [@ommaetal04]. This model has already been used to follow the self-consistent BH growth and its energy release in an isolated galaxy cluster . Mass, momentum and energy are spread over a small cylinder of radius $r_{\rm J}$ and height 2$h_{\rm J}$ ($h_{\rm J}$ for one side of the jet) multiplied with a kernel window function $$\psi \left (r_{\rm cyl} \right)={1 \over 2\pi r_{\rm J}^2} \exp \left( -{r_{\rm cyl}^2\over 2 r_{\rm J}^2 } \right) \, ,$$ where $r_{\rm cyl}$ is the distance to the axis of the cylinder. The mass deposition follows $$\dot M_{\rm J} \left (r_{\rm cyl} \right)={\psi \left (r_{\rm cyl} \right) \over \| \psi \|} \eta \dot M_{\rm BH} \, ,$$ where $\| \psi \|$ is the integrated value of $\psi$ over the whole cylinder, and $\eta=100$ is an arbitrary value that represents the mass loading factor of the jet on unresolved scales. Mass is transferred from the central cell (where the sink lies) to all the enclosed cells within the jet. Momentum $\mathbf{q_{\rm J}}$ is deposited in opposite directions from the centre along the jet axis, according to $$\| \mathbf {\dot q_{\rm J} }\| \left (r_{\rm cyl} \right)
= \dot M_{\rm J} \left (r_{\rm cyl} \right)\| \mathbf{u_{\rm J}\|}
={\psi \left (r_{\rm cyl} \right) \over \| \psi \|} \dot M_{\rm BH} \sqrt{2 \epsilon_{\rm f}\epsilon_{\rm r}\eta} c {\mathbf{j}.{\rm d}\mathbf{r}\over \| {\rm d}\mathbf{r} \| } \, ,
\label{mom_input}$$ where $\| \mathbf{u_{\rm J}} \| =(2\epsilon_{\rm f}\epsilon_{\rm r}/\eta)^{1/2} c$ is the velocity of the jet ($\| \mathbf{u_{\rm J}} \| \simeq 10^4 \, \rm km.s^{-1}$ for $\epsilon_{\rm f}=1$), $\mathbf{j}$ is the unit spin vector of the BH which defines the jet axis, and ${\rm d}\mathbf{r}$ is the distance vector from the centre of the black hole. $\mathbf{j}$ is computed by adding the different contributions from the neighbouring cells (sampled with the cloud particles) to the total angular momentum $$\mathbf{J}=\sum_{i=1}^{n_{\rm clouds}} m_{i} {\rm d}\mathbf{r_{i}}\times \mathbf{u_{i}} \, ,$$ where $m_i$ and $\mathbf{u_i}$ are the mass and velocity of the gas in the cell harbouring the cloud particle $i$, so that $\mathbf{j}=\mathbf{J}/\| \mathbf{J}\|$. Finally the kinetic energy deposited within a cell is $$\dot E_{\rm J}\left (r_{\rm cyl} \right)
={\mathbf {\dot q_{\rm J} }^2 \left (r_{\rm cyl} \right) \over 2 \dot M_{\rm J}\left (r_{\rm cyl} \right)}
={\psi \left (r_{\rm cyl} \right) \over \| \psi \|} \dot E_{\rm AGN} \, .$$ Integrating this energy deposition over all the cells within the jet, we recover $\dot E_{\rm AGN}$.
We point out that our jet has no opening angle and should therefore propagate along a straight line as it is perfectly collimated. [@ommaetal04] have shown that this kind of jet stays collimated over quite long distances ($100$ kpc) compared to its initial broadness and length ($1$ kpc). It broadens as it reaches equilibrium with the surrounding hot ambient medium. The same behavior is also pointed out by and [@duboisetal09], but with the difference that when strong turbulent motions begin to develop in the cluster core due to the formation of a cooling flow, the jet is more quickly mixed with the ICM. The choice of the jet velocity input $\| \mathbf{u_{\rm J}} \| \simeq 10^4 \, \rm km.s^{-1}$ (or equivalently the mass loading factor $\eta=100$) is particularly arbitrary but based on earlier works from [@ommaetal04], and [@duboisetal09]. The same simulation performed with $\| \mathbf{u_{\rm J}} \| \simeq 3.10^4 \, \rm km.s^{-1}$ produces results in very strong agreements with the ones presented here (mass of the most massive BH and stellar mass of the central galaxy agrees within 1%) suggesting that even strong variations of $\| \mathbf{u_{\rm J}} \|$ keep our results unchanged.
We set $r_{\rm J}$ and $h_{\rm J}$ equal to $\Delta x$, and the energy efficiency $\epsilon_{\rm f} = 1$ so as to reproduce the $M_{\rm BH}$–$M_*$ and $M_{\rm BH}$–$\sigma_*$ observational relations. Larger values of $r_{\rm J}$ and $h_{\rm J}$ have been tested at the same resolution $\Delta x \simeq 1$ kpc in a cosmological simulation (as opposed to a resimulation like the one presented in this paper) and they produce BHs which are too massive with respect to their host galaxy. Note that our total efficiency $\epsilon_{\rm r}\epsilon_{\rm f} = 0.1$ is also in good agreement with the average value obtained by general relativistic magneto-hydrodynamics numerical simulations of the accretion-ejection mechanism in accretion discs around spinning BHs (e.g. [@devilliersetal05], , or and references therein). Lower $\epsilon_{\rm f}$ values again cause black holes to become more massive, overshooting the $M_{\rm BH}$–$M_*$ observational relation.
Simulation Set-up {#IC}
=================
The simulations are run with the Adaptive Mesh Refinement (AMR) code [ramses]{} [@teyssier02]. The evolution of the gas is followed using a second-order unsplit Godunov scheme for the Euler equations. The Riemann solver used to compute the flux at a cell interface is the acoustic solver using a first-order MinMod Total variation diminishing scheme to reconstruct the interpolated variables from their cell-centered values. Collisonless particles (dark matter, stars and sink particles) are evolved using a particle-mesh solver with Cloud-in-Cell (CIC) interpolation.
We assume a flat $\Lambda$CDM cosmology with total matter density $\Omega_{m}=0.3$, baryon density $\Omega_b=0.045$, dark energy density $\Omega_{\Lambda}=0.7$, fluctuation amplitude at $8 \, h^{-1}.\rm Mpc$ $\sigma_8=0.90$ and Hubble constant $H_0=70\, \rm km.s^{-1}.Mpc^{-1}$ that corresponds to the Wilkinson Microwave Anisotropies Probe (WMAP) 1 year best-fitting cosmology [@spergeletal03]. The simulations are performed using a resimulation (zoom) technique: the coarse region is a $128^3$ grid with $M_{\rm DM}=2.9\times10^{10} \, \rm M_{\odot}$ DM resolution in a $80\,\rm h^{-1} Mpc$ simulation box. This region contains a smaller $256^3$ equivalent grid in a sphere of radius $20\,\rm h^{-1} Mpc$ with $M_{\rm DM}=3.6\times10^9 \, \rm M_{\odot}$ DM resolution, which in turn encloses the final high resolution sphere with radius $6\,\rm h^{-1} Mpc$, $512^3$ equivalent grid and $M_{\rm DM}=4.5\times10^8 \, \rm M_{\odot}$ DM resolution. Figure \[nice\_dm\_stars\] shows the distribution of DM and the distribution of stars in the zoom region and in the galaxy cluster at $z=0$.
The smallest region is the resimulation zone where cells may be refined up to $\ell_{\rm max}=16$ levels of refinement, reaching $1.19 \, \rm h^{-1}.kpc$, following a quasi–Lagragian criterion: if more than 8 dark matter particles lie in a cell, or if the baryon mass exceeds 8 times the initial dark matter mass resolution, the cell is refined. This strategy allows AMR codes, such as [ramses]{}, which use CIC interpolation in their gravity solver, to avoid propagating discreetness noise from small scales [@romeoetal08]. A Jeans length criterion is also added to ensure the numerical stability of the scheme on all levels $\ell < \ell_{\rm max}$ [@trueloveetal97], and where $\delta \rho=\rho/\bar \rho > 10^5$: the cells fulfilling these conditions must sample the local Jeans length with more than 4 cells. We point out that the $\ell_{\rm max}=16$ level of refinement is only reached at $a_{\rm exp}=(1+z)^{-1}=0.8$, and that the actual maximum level of refinement for a given redshift is increased as the expansion factor grows with time, i.e. $\ell_{\rm max}=15$ at $a_{\rm exp}=0.4$, $\ell_{\rm max}=14$ at $a_{\rm exp}=0.2$, etc. This allows us to resolve the smallest scales with a roughly constant physical size ($0.95<\Delta x<1.9\, \rm h^{-1}$.kpc), rather than a constant comoving size.
The resimulated region tracks the formation of a galaxy cluster with a 1:1 major merger occurring at $z=0.8$ . Indeed throughout its formation, the cluster chosen for resimulation passes through different dynamical stages: both a morphologically perturbed epoch occurring at half the age of the Universe, and a relaxed state at late times, which permits us to study the different associated states of the BH self-regulated growth. Figure \[mergertree\_DM\] shows the dark matter halo merger tree history for this cluster. Halos and sub-halos are identified and followed using the Most massive Sub-node Method (MSM) algorithm described in [@tweedetal09], which isolates bound substructures. The cluster experiences a major halo merger at $z\simeq 1.7$ and two proto-clusters progenitors merge together earlier at $z\simeq 3.1$. These mergers end at $z \sim 0.8$ and $z \sim 1.7$ respectively, when the central galaxies hosted in the (sub)halos merge. Central BHs hosted by these galaxies merge later, as shown in Fig \[mbhvsredshift\]. At later times, most of the mass growth of the cluster occurs through diffuse accretion or minor mergers. In the following, we discuss how such events might trigger or halt the AGN activity of the central (most massive) BH.
Growth of SMBHs and their activity {#BHgrowth}
==================================
The growth of a BH is tightly linked to the accretion history of its host halo (c.f. coeval growth scenario advocated by [@milleretal06] and [@hopkinsetal07]). In principle, cold gas which flows directly into the central nucleus in a free-fall time will very efficiently grow BHs. However this rapid growth might be substantially reduced by AGN activity that could expel both energy and material in the vicinity of the BH. In this case, self-regulation of the BH growth possibly drives the relations observed between BH mass and their host galaxy properties .
At high redshift ($z>2$), most galaxies seem to harbor a massive cold gas disc component, both in cosmological hydrodynamics simulations [@ocvirketal08] and in the observations [@shapiroetal08], which can be tapped to fuel rapid BH growth. Accordingly, in the cosmological re-simulation of a cluster presented here, the central BH reaches a few tenths of its final mass when the Universe is less than 2 Gyr old, accreting at a rate above $\sim$1 % of its Eddington limit (figure \[mbhvsredshift\]). As the initial seed mass of the black hole is $10^{5}\, \rm M_{\odot}$, this means that its mass increases by a factor $10^{4}$ in a tenth of the Hubble time. On Fig. \[mbhvsredshift\] the growth of two of its most massive BH progenitors (indicated by red dashed and blue dot-dashed lines on the figure) is also displayed until they merge with the final BH ($M_{\rm BH}=1.7\,10^{10}\, \rm M_{\odot}$ at $z=0$). These mergers (vertical dotted lines in upper panel of Fig \[mbhvsredshift\]) coincide with two important halo mergers in the history of the cluster (branches labelled 2 and 3 on Fig. \[mergertree\_DM\]). At high redshift ($z>2$), these BH progenitors behave like the main one: they accrete gas at a fraction of their Eddington rate and this fraction steadily decreases with time from $z=4$ to $z=2$. We know that for the most massive halos (those with masses $M_{\rm stream} \geq 6 \times 10^7 (1+z)^8 \,\rm M_{\odot}$ at $z \geq 2$), cold accretion of gas from the IGM is efficiently thermalized at a few virial radii by an accretion shock, and as a result we expect the accretion rate in the centre of the halo to drop. From Fig. \[mergertree\_DM\] one can see that the DM halos of branches 2 and 3 have masses $ \approx 10^{13}\,\rm M_{\odot}$ at redshift $z=3$ and therefore satisfy the $M_{\rm stream}$ criterion. On that account we claim that this explains, in part, the decrease of the BH accretion rate relative to its Eddington rate.
Mergers are a non-negligible growing mode at intermediate and lower redshift, as there is less cold gas to feed the BH in the massive cluster. Indeed fig. \[mbhvsredshift\] shows that the BH doubles its mass at redshift $\simeq1.6$ when two BHs of comparable mass ($M_{\rm BH}=3.1 \, 10^{9}\,\rm M_{\odot}$ and $M_{\rm BH}=8.2 \, 10^{8}\,\rm M_{\odot}$) coalesce. The extra amount of mass ($\simeq 2 \, 10^9 \, \rm M_\odot$) comes from the fast accretion of material brought in by the galaxy major merger ($M_{*}=1.3 \, 10^{12}\,\rm M_{\odot}$ and $M_{*}=6.1 \, 10^{11}\,\rm M_{\odot}$). This merger appears in the DM merger tree (figure \[mergertree\_DM\]) when branches 1 and 2 join at redshift $1.6$. It is interesting to note that the first BH that forms is not necessarily the most massive one at late times (in our case the BH hosted by branch 2 halo forms first), as already pointed out by [@dimatteoetal08].
Fig. \[nice\_jet\], \[nice\_merger\] and \[nice\_radio\] show three different episodes of the formation and evolution of the cluster, respectively a high-redshift major merger between two gas-rich galaxies (Fig. \[nice\_jet\]), the major merger of the two clusters (Fig. \[nice\_merger\]), and the relaxation of the cluster at late times (Fig \[nice\_radio\]).
The halo merger between branches 1 and 2 at $z=3.1$ results in a cataclysmic episode for its host galaxies at $z\simeq 1.6$: a large amount of gas is expelled far from the core of the halo, reaching the virial radius, and the resulting disc of gas from the two merging galaxies is almost completely disrupted. This sequence of events is illustrated in figure \[nice\_jet\]. On the left panel, we observe the encounter of the two gas-rich galaxies before they merge. Shortly after they merge (fig. \[nice\_jet\] middle panel), their respective BHs do so as well which results in a strong jet that disrupts most of the cold baryon content in the galaxy and shock heats the ambient medium to high temperature. The jet propagates supersonically at Mach $3$ ($u_{\rm jet}\simeq 3000\, \rm km.s^{-1}$ and $c_{\rm s}\simeq 1000 \, \rm km.s^{-1}$) before being stopped by the intergalactic medium at $r\simeq1.2$ Mpc, which corresponds to about $3 \, r_{\rm vir}$ at this redshift (fig \[nice\_jet\] right panel).
The disruption of cold material by AGN feedback has already been noted by [@dimatteoetal05] in idealized simulations of a gas rich merger. It is comforting to confirm their results within a cosmological setting. [@khalatyanetal08] have also pointed out that mergers could trigger a high level of AGN activity during the formation of a small galaxy group. Finally, we see two hotspots during the jet propagation, that look like radio lobes (fig. \[nice\_jet\] right panel). Such events (strong jets following a merger) become rarer as time goes on since the combined action of star formation and early AGN activity strongly diminishes the cold and dense gas content in massive halos.
To check how this striking result is affected by our limited spatial resolution, we performed the same simulation with one more level of refinement ($\Delta x=0.6\, \rm h^{-1}.kpc$). The same burst appears at the same redshift but its power is slightly lower, because more gas has been pre-heated by a strong AGN activity in a previous merger taking place at higher redshift $z\simeq4$. This is not a very surprising effect: with more resolution, the density contrast is more pronounced especially in poorly resolved high-redshift galaxies, which, in turn, leads to a faster accretion rate at early times. As in our standard run, the cold gas in the core of the halo is strongly disrupted by the AGN activity triggered by the wet merger occurring at $z=1.6$, so that BH growth and AGN luminosity are suppressed for 3 Gyrs (see figure \[lagnvsredshift\] from $z=1.6$ to $z=0.6$). Therefore we can reasonably claim that most of the important features describing the BH growth are already captured by our standard resolution run.
Figure \[lagnvsredshift\] also shows the energy that would be released by supernova feedback if it were implemented in our simulation. To estimate this energy release, we assume that stars are distributed according to a [@salpeter55] Initial Mass Function, for which each massive star ($M_{\star} > 8 \, \rm M_{\odot}$) deposits $10^{51}$ erg per $10\, \rm M_{\odot}$ into the ISM. We see that the energy from this form of feedback is always lower than that from the AGN at all times, except during the post-merger phase from $z=1.6$ to $z=0.6$. However, in this post-merger phase the high level of supernova feedback is an artifact of the way star formation is computed: we trace back the star formation history (SFH) of the central galaxy using all the stars which belong to it at $z=0$, so that the star formation and hence the supernova rate we derive from it includes that of its accreted satellites. In this redshift range ($0.6<z<1.6$), the star formation rate is dominated by the galaxy progenitor that has not undergone the cataclysmic quasar phase (hosted by the branch 3 halo on Fig \[mergertree\_DM\]), whilst the star formation activity in the quasar galaxy progenitor (hosted by the branch 1 halo) is completely suppressed. In light of this, it is a fair approximation to neglect the feedback from supernovae on the evolution of this galaxy cluster.
At $z\simeq 1.7$, another halo major merger (1:1) occurs (branches 1 and 3 in the merger tree (figure \[mergertree\_DM\])). However, the most massive ($M_{\rm BH}=5.9 \, 10^{9}\,\rm M_{\odot}$) BH only merges with the ($M_{\rm BH}=4.3 \, 10^{9}\,\rm M_{\odot}$) BH of its cluster companion at a much later epoch ($z=0.58$ right panel of Fig \[nice\_merger\]). Note that, in this case, the BH merger also takes place quite a long time after the central galaxies hosting the BHs merge together. As a matter of fact, the $z=0.8$ major galaxy merger drives the cluster gas to temperatures twice the virial temperature thanks to a violent shock wave (middle panel of Fig \[nice\_merger\]). However, during this galaxy merger phase, the accretion rate onto the most massive BH drops to negligible values ($\sim 10^{-5}\dot M_{\rm Edd}$ fig. \[mbhvsredshift\] solid black line), and only its companion continues to accrete at a moderate rate (a few $\sim 10^{-3}\dot M_{\rm Edd}$, fig. \[mbhvsredshift\] red dashed line). The resulting AGN activity, even when boosted by the final BH merger at $z=0.58$ does not seriously impact the highly pressurized ICM gas which confines the jet energy to the very central parts of the cluster (right panel of Fig \[nice\_merger\]). After the major BH merger, the accretion rate onto the central BH becomes extremely small ($10^{-4} \dot M_{\rm Edd}$) due to the complete evaporation of leftover cold material by the final outburst of AGN activity. Subsequently the BH stays in this almost-dead phase for 2 Gyr.
It is striking that the less massive BH progenitor of the latter BH merger (at $z=0.58$) accretes gas at $0.01\,\dot M_{\rm Edd}$ before the merger, whereas accretion onto the most massive BH is negligible. The reason for this different behavior can be understood by looking at the temperature maps of both cluster progenitors (left panel of figure \[nice\_merger\]): the most massive cluster (branch 1, red circle) is slightly warmer than its companion (branch 3, blue circle), as it has been pre-heated by important quasar activity at earlier redshifts (z$\sim1.56$). On the other hand, the less massive progenitor did not experience such strong pre-heating, and as a result has a lower gas temperature, and therefore a higher accretion rate during the pre-merger phase.
Finally, the cluster relaxes and the inner halo cold gas reservoir gets replenished, fueling a faint accretion onto the BH. This translates into low AGN activity. Episodically, stronger jets are produced by the AGN in this phase which yield small perturbations of the ICM temperature in the form of sound waves (figure \[nice\_radio\]). These jets are roughly sonic ($u_{\rm jet}\simeq 1000\, \rm km/s$ and $c_s \simeq 1300 \, \rm km.s^{-1}$), and are efficiently thermalized by the ICM. As a result they do not propagate farther than a few $10$ kpc.
Nevertheless, the part of the jet energy which is carried by these sound waves limits the cooling flow in the cluster core. However the cooling time in the core ($<100$ kpc) is extremely short (smaller than a Gyr, see figure \[ctimevsr\]), so the cooling flow eventually develops again and feeds the BH afresh. As a result, the BH accretion rate increases from $10^{-4}\, \dot M_{\rm Edd}$ to a few $10^{-2}\, \dot M_{\rm Edd}$ at z$\sim0$, giving rise to late-time AGN activity.
Figure \[jet\_slice\] shows different physical properties in a slice of gas cut through the AGN jet at $z=0$. They show that the jet is at low-density and high temperature in good agreement with high resolution simulations of jet-formation (see [@heinzetal06; @simionescuetal09] for example). A remarkable feature is the formation of two under-dense cavities filled with sonic-jet material. We have computed a simulated X-ray map of these cavities in 3 different temperature bands (see figure \[nice\_Xray\]). These cavities are reminiscent of the ones observed in Perseus A by [@fabianetal06] in which a strong cooling core is also visible. In our simulation, an extended cooling-core (a few $10$ kpc across) is absent, but the cooling flow which gives rise to late-time AGN activity is clearly present. As in the [@fabianetal06] observations, we interpret the ripples induced by our jet modelling as sound waves. This can be seen in figure \[profiles\_pencil\] where radial velocities are always sub-sonic. These sound waves, provided one can dissipate them viscously [@fabianetal03; @ruszkowskietal04b] can reheat the ICM at distances larger than the scalelength of the jet. In our simulation, no explicit viscosity is included, but these spherical sound waves do not appear at radii larger than $r_{500\rho_{\rm c}}$, suggesting that they have been dissipated by numerical viscosity on these scales.
Regulation of the cooling catastrophe {#Cooling_catastrophe}
=====================================
In the absence of strong feedback processes to offset the cooling of gas in the potential wells of massive DM halos, too many massive galaxies are formed both in CDM cosmological hydrodynamical simulations and semi-analytic models of galaxy formation and evolution. In our cluster zoom simulation, when no AGN feedback is considered, the final mass of stars in the central cD galaxy is very high, $M_{*}\simeq1.7\, 10^{13} \, \rm M_{\odot}$, for a $M_{\rm 500}=2.4\, 10^{14}\, \rm M_{\odot}$ ($M_{\rm 200}=2.9\, 10^{14}\, \rm M_{\odot}$) dark matter halo with radius $r_{\rm 500}=940$ kpc (resp. $r_{\rm 200}=1370$ kpc, see figure \[nice\_dm\_stars\])[^2]. In the presence of stirring from AGN feedback, the total stellar mass is reduced to $M_{*}\simeq5.6 \,10^{12} \, \rm M_{\odot}$, i.e. by more than a factor of 3. To compute the stellar mass content, we use the same MSM algorithm [@tweedetal09] as for the dark matter but with different parameters, since stars are more clustered than DM particles. This tool efficiently separates one galaxy from another, especially the central galaxy from its satellite galaxies (see figure \[nice\_dm\_stars\]). However, the algorithm used in this method attributes all the stars present in the dark matter halo and not part of satellite galaxies to the central one. As a result, a non-negligible part of the stellar mass of the central galaxy resides in the intra-cluster stellar halo (composed of all the stars stripped from satellite galaxies of the cluster), which has a very large extent (up to $\sim 400$ kpc). This caveat must be borne in mind when comparing the stellar mass of the central object with observations: our estimate only provides an upper limit of the stellar mass content of the central galaxy.
Figure \[mgalvsredshift\] compares the stellar mass evolution as a function of redshift for the most massive galaxy at $z=0$ (solid line) and its two most massive progenitors (dashed and dot-dashed lines, central galaxies of DM halos identified as branch 1 and 3 in figure \[mergertree\_DM\]) in the AGN and no-AGN runs (red and black sets of curves respectively). We observe that the reduction of the stellar mass is a continuous process which begins at an early stage (around $z\sim5$) but gets amplified as time goes on to reach a factor 3 at $z\sim1$. Both progenitors seems to follow the same reduction of their stellar content which suggests that even the branch 3 cluster, which does not exhibit any strong quasar activity at high redshift, is able to prevent some gas from falling onto the central galaxy. The process is more quiescent as can be seen in its BH accretion rate (red dashed curve in figure \[mbhvsredshift\]), but even this continuous and moderate AGN activity can efficiently reduce the star formation.
We evaluate the bulge to disc mass ratio of the central galaxy by using the kinematics of its star particles. First, we identify the rotation axis of the galaxy to define the correct cylindrical reference frame in which we project the velocity components of each star particle. A particle belongs to the bulge if its circular velocity is lower than half its total velocity. With that definition, the bulge-to-total mass ratio of the central galaxy is $0.75$ for the simulation with AGN, and $0.80$ for the simulation without AGN. Thus it appears that although AGN dramatically change the SFH, they have a much less significant impact on the morphology of a galaxy.
Figure \[sfr\_mygal\] shows the star formation rate for the central galaxy as a function of time for the two runs. To compute the time evolution of the SFR, we simply have identified the stars belonging to the central galaxy at $z=0$, and traced them back in time. Thus it is the SFH summed over all the stars of all the satellite galaxies that have been accreted onto the central galaxy throughout its evolution. The SFR continuously decreases with time due to early AGN activity (z$\sim4$), but the dramatic decline in SFR occurs around $z=0.6$ when the BHs hosted by the central galaxy merger remnant of branch 1 and 3 halos finally merge. The vast majority of the cold gas is heated up by the AGN during this violent merger. In contrast, the cold gas in the no-AGN case is simply compressed in the galaxy merger which results in a double small star formation peak around the same redshift ($z\sim0.6$). The latter effect is a well-known property of merging galaxies without AGN . However, mergers of galaxies containing BHs boost the accretion of gas onto the BH fueling strong AGN activity and produce a dip in the SFR by reducing the cold-gas content. These features are clearly seen at redshift $z=1.6$ and $z=0.6$ in figure \[sfr\_mygal\]. Such a behavior has already been analyzed in detail in idealized (as opposed to cosmological) simulations of galaxy merger [@springeletal05; @dimatteoetal05]. Our work confirms that it also occurs in more realistic cosmological configurations.
We have measured the cumulative mass profiles of the different components (gas, stars, dark matter) as a function of radius for the cluster at $z=0$ (figure \[massvsr\]). In the absence of feedback, most of the cold baryons are concentrated within the galaxy (in the inner 10 kpc): i.e. the cooling catastrophe has occurred. There is a strong difference in the cumulative profiles between the runs with and without AGN activity, especially in the central region of the cluster. They differ by a factor 5 in total mass at $r=10$ kpc and by a factor 1.5 at $r=100$ kpc. Without AGN, the gravitational potential is steeper in the centre of the cluster, baryons accumulate and gravitationally pull DM along with them. This is predicted to have severe consequences when simulating the gravitational lensing effect of such structures [@peiranietal08; @meneghettietal10].
Baryon fraction as a function of radius provides us with a good benchmark to quantify the influence of feedback processes. We have computed the cumulative fractions of stars, gas and dark matter for both runs (figure \[mfracvsr\]). We observe a very small difference in the total baryon fraction at $r_{200}$ and $r_{500}$ which means that this quantity is relatively immune to the presence of feedback. In contrast, the baryon fraction has decreased by a factor 2 at $r=0.1 r_{500}$. This means that half of the baryons that were concentrated in the inner parts of the halo have been redistributed by the AGN feedback at larger radii. However, as the baryon fraction seems relatively independent of the presence of the AGN at very large radii ($>r_{500}$), we can conclude that only a modest fraction of the gas is expelled by the AGN outside of the cluster.
Stellar and gas fractions yield more clues as to the impact of AGN feedback on the cluster history. There is a strong decrease in the stellar fraction, even at large radii ($r>r_{500}$), meaning that star formation has been efficiently suppressed by the AGN activity. We see that the stellar fraction at $r_{500}$ has been lowered by more than a factor 2, which is comparable with the results obtained by [@duffyetal10]. Table \[tabfrac\] show the stellar fractions $f_{500}^{\rm star}$, gas fractions $f_{500}^{\rm gas}$ and baryon fractions (gas and stars) $f_{500}^{\rm b}$ measured at $r_{500}$ in our simulations and compared to X-ray or near infrared measurements made by various groups [@linetal03; @gastaldelloetal06; @vikhlininetal06; @arnaudetal07; @gonzalezetal07; @giodinietal09; @sunetal09]. Observational data values are obtained using the best fit these authors provide for $f_{500}^{\rm star}$, $f_{500}^{\rm gas}$ and $f_{500}^{\rm b}$ as a function of $M_{500}$. From this comparison with observations, the simulation which does not include AGN feedback is clearly ruled out, as the stellar fraction is too high by a factor $\sim 3-4$ and the gas fraction too low by 25% in the most favorable case. This clearly indicates that the ICM has undergone a cooling catastrophe: too much gas has been depleted and transformed into stars. On the other hand, the simulation with AGN feedback shows reasonable agreement with the stellar fraction estimated from X-ray measurements by [@gonzalezetal07] and [@giodinietal09], but still overestimates the stellar fraction from near infrared data by [@linetal03] by about a factor 2. Gas fraction in the simulation with AGN, is also within the range of values inferred from X-ray gas emission, albeit on the high side. The major flaw of our simulations is their inability to match the lower than Universal baryon fraction observed in small galaxy clusters ($M_{500}< 10^{15} \, \rm M_{\odot}$). We note that the discrepancy would be even more blatant had we used WMAP 5 year parameters [@dunkleyetal09] since the Universal baryon fraction goes up to 18% for this cosmology. In the case of our AGN simulation, the cluster baryon fraction is close to the Universal baryon fraction $\Omega_{\rm b}/\Omega_{\rm m}=0.15$, but we fail to push gas far enough out of the cluster potential well to match lower observational values sitting around $\sim0.13$.
no AGN AGN L03$^a$ G07$^b$ G09$^c$
---------------------- -------- ------- --------- --------- ---------
$f_{500}^{\rm star}$ 0.090 0.036 0.019 0.023 0.032
$f_{500}^{\rm gas}$ 0.073 0.116 0.117 0.101 0.090
$f_{500}^{\rm b}$ 0.163 0.152 0.136 0.124 0.122
: Comparison of the stellar, gas and baryon fractions at $r_{500}$ in our simulations with the observational data.[]{data-label="tabfrac"}
$^a$ Observational data from [@linetal03]\
$^b$ Observational data from [@gonzalezetal07]. Their gas fraction is the best fit to data from [@vikhlininetal06] and [@gastaldelloetal06].\
$^c$ Observational data from [@giodinietal09]. Their stellar fraction are the best fit of their data combined with data from [@linetal03]. Their gas fraction is the best fit to data from [@vikhlininetal06], [@arnaudetal07] and [@sunetal09]
The gas fraction behavior is somewhat counter-intuitive: it is larger in the run with AGN feedback, whatever the radius is. This is explained by the fact that AGN removes gas from the central regions of the cluster to replenish its outer parts. AGN feedback thus transforms cold gas contained in the central galactic disc into hot and diffuse halo gas. Moreover, the gas fraction has a remarkable feature in the form of a pronounced dip at intermediate radius (15 kpc for the no-AGN and 100 kpc for the AGN case) which marks the transition between the cold/dense phase ($n >0.1 \, \rm cm^{-3}$), and the hot/diffuse component. Such dips in the gas fraction also are commonplace in X-ray cluster surveys [@vikhlininetal06]. Another interesting result from Fig. \[mfracvsr\] is that the dark matter to total mass fraction in the cluster core ($r < 10$kpc) is higher in the case of AGN feedback, even though the total amount of dark matter is lower in this case. We attribute this to the domination of the mass budget by the stellar component which pulls DM along with it through adiabatic contraction [@blumenthaletal86].
Finally, we compare the thermodynamical properties of the gas in the two runs in figure \[gasprofiles\]. We have fitted the gas density profile in the AGN run with a $\beta$ profile of the form $$\rho=\rho_{\rm s} \left (1+(r/r_{\rm c})^2\right)^{-3\beta/2}\, ,$$ where $\rho_{\rm s}=0.5\, \rm cm^{-3}$, $r_{\rm c}=10$ kpc and $\beta=0.6$. This profile matches the density profile of the relaxed cluster at different times ($z=0$, $z=0.04$ and $z=0.09$, which are separated by $500$ Myr) in the intermediate radius range $0.05$-$1$ $r_{500}$. When the cluster is relaxed, the same analytic profile extends to the core of the cluster, but as soon as a cooling flow develops it fails to describe the numerical gas density accurately. Indeed, the core of the cluster shrinks as gas flows in, so $r_{\rm c}$ drops, but the $\beta$ index stays identical as the profile remains unaffected by the cooling flow on large scales. By contrast, fitting the gas density profile with a $\beta$ profile for the simulation where no AGN is present turns out to be an impossible task, since the core radius becomes smaller than the spatial resolution in that case.
Surprisingly, the simulation without the AGN, which has endured a cooling catastrophe for Gyrs, exhibits a very hot gas core (temperature in excess of 9 keV) with a very steep profile (see second panel from the top on Fig \[gasprofiles\]). Actually a massive central cold disc component is also present in this run and would appear on Fig \[gasprofiles\] if we were measuring mass-weighted instead of volume-weighted quantities, but the properties of the gas at the center of the cluster would then reflect the properties of the ISM instead of the diffuse ICM. This very hot thermal part, in the no-AGN run, arises from the cluster need for more thermal energy to support the gas against the extra gravitational compression generated by adiabatic contraction. Due to the very high temperature and a lack of diffuse gas around the post-merger galaxy this energy is not easily radiated away (the cooling time is greater than 2 Gyr as shown on Fig \[ctimevsr\]). However we can clearly see in figure \[velprofiles\] that the gas, close to the galactic disc ($r< 50$ kpc), still collapses onto that galaxy due to the lack of pressure support (figure \[gasprofiles\]), which explains the depletion of the gas component at $r\simeq100$ kpc.
On the other hand, when the AGN is active the temperature profile is stabilized and looks quasi-isothermal in the range $0.05$-$1$ $r_{500}$ (second panel of Fig \[gasprofiles\]). Before $z=0$, the temperature is a factor 2 to 4 higher in the inner 10 kpc, due to heating from the jet, which remains confined in that region. As the gas radiates away the jet energy, its temperature drops and a cooling flow develops because of a lack of pressure support in the core (second panel from the bottom on Fig \[gasprofiles\]). Small variations of temperature with radius in the form of wiggles can be observed in Fig \[gasprofiles\] (second panel from the top, solid red curve). These correspond to the propagation of sound waves into the intra-cluster medium. These waves contribute to reheating the cooling plasma in the cluster as a whole by propagating and isotropising the energy injected by the AGN jet. They manage to offset the extremely short cooling time within $r< 0.1\, r_{500}$ which is at least one order of magnitude shorter than the time elapsed since the last major merger (see figure \[ctimevsr\]), and thus prevent most of the gas from collapsing onto the central galaxy.
The pressure cavities seen on figure \[jet\_slice\] and in the X-ray map (figure \[nice\_Xray\]), are visible in the pressure profile of figure \[gasprofiles\] (second panel from the bottom): at $z=0$, there is a small depression in the pressure profile around $r\simeq 15$-$30$ kpc, which does not appear at earlier times when the AGN is not active enough to form these cavities. This feature is also detectable in the radial velocity profile of the gas on figure \[velprofiles\] (bottom panel): there is a net radial gas outflow at $r \simeq 15-30$ kpc whose maximum corresponds to the maximum extent of the jet and whose outwardly decreasing profile reflects the pre-shocked cocoon region. The volume-averaged velocity which we plot on this figure is however under-estimated, because the sonic outflowing component of the jet is mixed with the quasi-steady flow or inflowing regions. The velocity inside the jet is much faster, around $1000 \, \rm km/s$.
Finally, entropy profiles (bottom panel of Fig \[gasprofiles\]) shows a plateau in the cluster core (at $r<20$ kpc for the AGN run, and $r< 300$ kpc for the no-AGN run) with a strong departure from the scaling power law $K\propto r^{1.2}$. This indicates the level of turbulent mixing in the gas as explained in detail by [@mitchelletal09] and is nicely illustrated by comparing entropy profiles on figure \[gasprofiles\] (bottom panel) with radial velocity dispersion profiles on figure \[velprofiles\] (top panel). Such a comparison clearly shows that the stronger the turbulence level (or equivalently the radial velocity dispersion), the higher the entropy.
Discussion and conclusion {#discussion}
=========================
We have numerically studied how a self-regulating model of supermassive black hole growth and AGN feedback impacts the formation history of a large cluster of galaxies. Using a resimulation technique to explicitly account for the cosmological context ($\Lambda$CDM Universe) which drives the cluster growth and an AMR technique to solve the equations of hydrodynamics without running into entropy issues, we find that:
1. BHs accrete gas efficiently at high redshift ($z>2$), significantly pre-heating proto-cluster halos in the process.
2. some, but not all, wet (gas-rich) mergers fuel strong episodic jet activity which transport gas from the cluster core to its intermediate/outer regions.
3. reduced infall of cold gas during the more secular phase evolution of clusters produces smaller outbursts from the central AGN which contribute to heat the whole cluster via sound waves but are inefficient at redistributing the gas outwards.
4. late-time AGN activity forms two large cavities correlated with the emergence of a small cooling flow.
Whilst our model for accretion onto the black hole is commonly used in simulations in the literature , this is the first time, to the best of our knowledge, that a momentum driven jet is implemented as AGN feedback in cosmological simulations and followed in a self-consistent way. We argue that this is one step in the correct direction since powerful jets are observed in the Universe on scales well resolved by any of these simulations (e.g. [@bridleetal94]). Other authors have adopted a more phenomenological approach where energy is either accumulated by the BH before being released as a thermal pulse , or simply used as a continuous heat source [@springeletal05; @dimatteoetal05; @dimatteoetal08], or used both heating modes [@sijackietal07]. Injection of non-thermal relativistic protons in rising buoyant AGN bubbles has also been explored as an alternative feedback mechanism [@sijackietal08]. It is worth noting the recrudescence of efforts done to improve models of AGN feedback in idealised (non-cosmological) simulations of galaxy evolution, where kinetic energy is deposited either isotropically [@debuhretal10; @poweretal10] or – as in the present paper – as a collimated jet . These models probably capture more of the relevant jet physics that what we achieve here, and we believe that as cosmological simulation spatial resolution increases, more physical insight into the impact of AGN feedback on the ICM will be gained by coupling them to such models. While we believe that most of the results we get are similar to those obtained with the continuous heating model and that our feedback seems less efficient at stopping the cooling catastrophe than the accumulated heating prescription, the devil certainly is in the details and we defer a more detailed comparison to a future paper (Dubois et al in prep). It is however interesting to note that all these models, including ours, are calibrated to provide an acceptable $M_{\rm BH}$ vs $M_{\star}$ (or $M_{\rm BH}$ vs $\sigma_{\star}$) relation so that it is very unlikely that this relation can be used to constrain feedback mechanisms.
From both semi-analytic prescriptions [@boweretal06; @cattaneoetal06] and recent numerical simulations , it has become clear that AGN feedback must play a major role in reducing the central galaxy stellar mass in groups and clusters by a large amount. Our implementation of feedback succeeds at least partially in reaching this goal: the stellar mass of the central object is reduced by more than a factor 3 in the run where AGN feedback stirs the gas whose properties (density profile, temperature, radial velocity) also are in better agreement with observations. Another success of our AGN modeling is its capacity to reproduce double cavities separated by a cold component as seen in the X-ray emission of observed clusters (e.g. in Perseus, [@fabianetal06]). Our experiments with a simple isotropic thermal input [@teyssieretal10] suggest that these cannot be reproduced.
One drawback of our simulation is that no supernova feedback is included. However we do not expect this feedback to be energetically relevant as a simple estimate based on the star formation history of our galaxies shows that it is always an order of magnitude lower than the AGN energy input (figure \[lagnvsredshift\]), On the other hand, supernovae feedback releases metals into the ICM. As the cluster gas is heated to very high temperatures ($\sim 3$ keV), cooling in the ICM is primarily due to free-free collisions, thus we do not expect metals to strongly alter the cooling rate of the plasma. Indeed, at $T=3.5$ keV, the relative difference in the net cooling rate between a zero and a one third of solar metallicity plasma is $0.2$ according to . This means that, assuming a Z$=0.3\, \rm Z_{\odot}$ metallicity in the ICM, our simulation underestimates the gas cooling rate by 20 %.
As always when analysing numerical simulations, one has to worry about spatial and mass resolution related issues. Indeed, accretion onto a SMBH happens on (sub)parsec scales, well below the kpc size of the smallest cell in our cluster re-simulation. Even though we use the empirically well motivated subgrid model of to account for this lack of resolution, one would like to test its validity by performing direct simulations. Whilst this is beyond the reach of the current generation of supercomputers, we have tried to assess both the robustness of our subgrid and our jet implementations by performing a sub-kpc run and conclude that our results are by-and-large unchanged at this increased spatial resolution. This is also true for a modest increase in mass resolution. We are therefore quite confident that the conclusions we draw in this paper are robust vis-a-vis resolution issues and defer a more thorough resolution study to future work.
Finally, other physical mechanisms, which we do not model here, could potentially play a significant role in preventing the development of massive cooling flows. These mechanisms involve tapping into the heat reservoir provided by the outer regions of galaxy clusters to raise the gas temperature in their core. In particular, recent efforts have been made to investigate the importance of anisotropic thermal conduction in idealized galaxy clusters . These studies have shown that the HBI can reorient the magnetic field lines in the cluster core in an azimuthal configuration that stops the inward heat flux. The most recent of these simulations [@parrishetal10] demonstrated that if some small turbulence is brought to break this magnetic field topology, heating is able to proceed. One has to wonder, in the context of anisotropic thermal conduction, whether the turbulence induced by small-scale and large-scale motions is able to reorder magnetic fields in cosmological simulations of galaxy clusters , and, thus, break the shelf-shielding of the heat-flux. In particular, the presence of AGN stirring can help to disturb this magnetic equilibrium [@duboisetal09], so we believe that this problem needs to be addressed with MHD cosmological simulations including both AGN stirring and anisotropic thermal conduction.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Taysun Kimm and Stas Shabala for useful comments and discussion. YD is supported by an STFC Postdoctoral Fellowship. The simulations presented here were run on the TITANE cluster at the Centre de Calcul Recherche et Technologie in CEA Saclay on allocated resources from the GENCI grant c2009046197.
\[lastpage\]
[^1]: Here and later, the entropy is defined as $K=T \rho^{-2/3}$, where $T$ is the gas temperature and $\rho$ is the gas mass density
[^2]: All quantities with subscripts 200 or 500 refer to regions with overdensities 200 or 500 times larger than critical ($\rho_c=3H^2/(8\pi G)$).
|
---
abstract: 'We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in particular, to the setting in which the two finite element spaces consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure $O(h^\gamma)$, where $\gamma$ is a nonnegative scalar and $h$ is the mesh spacing. The projector may be, for example, the orthogonal projector with respect to the $L^2$- or $H^1$-inner product. In these and other circumstances, the bounds are superconvergent under a few mild regularity assumptions. That is, under mesh refinement, the two projections differ in norm by an amount that decays to zero at a faster rate than the amounts by which each projection differs from $u$. We present numerical examples to illustrate these superconvergent estimates and verify the necessity of the regularity assumptions on $u$.'
address: 'Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA; '
author:
- 'Evan S. Gawlik'
- 'Adrian J. Lew'
bibliography:
- 'projections.bib'
date:
title: Supercloseness of Orthogonal Projections onto Nearby Finite Element Spaces
---
Introduction {#sec:intro}
============
One of the hallmarks of the finite element method is its geometric flexibility: it permits the construction of numerical approximations to solutions of partial differential equations using meshes that are designed according to the practitioner’s discretion. When two meshes are used to solve the same problem, the norm of the difference between the corresponding numerical solutions is, of course, no larger than the sum of the norms of the differences between each numerical solution and the exact solution. This paper addresses the question of whether or not a sharper estimate holds in the event that the two meshes coincide over a large fraction of the domain.
Beyond its inherent mathematical appeal, the question raised above has important consequences in the study of numerical solutions to time-dependent PDEs on meshes that change abruptly in time. Notable examples are remeshing during finite element simulations of problems with moving boundaries, and adaptive refinement during finite element simulations of problems on fixed (or moving) domains. The relevance of the aforementioned question in this setting is elucidated in [@Gawlik2012b], where it is shown that if a parabolic PDE is discretized in space with finite elements and the solution is transferred finitely many times between meshes using a suitable projector, then it is possible to derive an upper bound on the error in the numerical solution at a fixed time $T>0$ that involves the norms of the jumps in $r_h u(t)$ across the remeshing times, where $r_h u(t)$ denotes an elliptic projection of the exact solution $u(t)$ onto the current finite element space. These jumps are precisely the differences between the finite element solutions of an elliptic PDE on two different meshes.
#### Intuition.
It is perhaps not surprising that two finite element solutions associated with nearly identical meshes should differ by an amount that is small relative to their individual differences with the exact solution, under suitable conditions on the finite element spaces and the PDE under consideration. To develop some intuition, it is instructive to first consider the similarity between the *interpolants* of a smooth function $u$ onto two finite element spaces associated with nearby meshes.
To this end, consider two families of shape-regular, quasi-uniform meshes $\{\mathcal{T}_h\}_{h \le h_0}$ and $\{\mathcal{T}_h^+\}_{h \le h_0}$ of an open, bounded, Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \ge 1$. Assume that the two families are parametrized by a scalar $h$ that equals the maximum diameter of an element among all elements of $\mathcal{T}_h$ and $\mathcal{T}_h^+$ for every $h \le h_0$, where $h_0$ is a positive scalar. Let $\mathcal{V}_h$ and $\mathcal{V}_h^+$ be finite element spaces consisting of, for definiteness, continuous functions that are elementwise polynomials of degree at most $r-1$ over $\mathcal{T}_h$ and $\mathcal{T}_h^+$, respectively, where $r>1$ is an integer.
For $s \ge 0$ and $p \in [1,\infty]$, we denote by $W^{s,p}(\Omega)$ the Sobolev space of differentiability $s$ and integrability $p$, equipped with the norm $\|\cdot\|_{s,p}$ and semi-norm $|\cdot|_{s,p}$. We sometimes write $\|\cdot\|_{s,p,\Omega}$ and $|\cdot|_{s,p,\Omega}$ to emphasize the domain under consideration. We denote $H^s(\Omega) = W^{s,2}(\Omega)$ for every $s \ge 1$ and $L^p(\Omega) = W^{0,p}(\Omega)$ for every $p \in [1,\infty]$.
For finite element spaces of the aforementioned type, the nodal interpolants $i_h u \in \mathcal{V}_h$ and $i_h^+ u \in \mathcal{V}_h^+$ of a function $u \in W^{r,\eta}(\Omega) \cap C^0(\overline{\Omega})$ onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively, satisfy the standard interpolation estimate $$\label{standard_interp}
\|i_h^+ u - u\|_{s,\eta} + \|i_h u - u\|_{s,\eta} \le C h^{r-s} |u|_{r,\eta}$$ for any $s \in \{0,1\}$, any $\eta \in [2,\infty]$, and every $h \le h_0$ [@Ern2004]. Here and throughout this paper, the letter $C$ denotes a constant that is not necessarily the same at each occurrence and is independent of $h$.
Using the triangle inequality and (\[standard\_interp\]) with $\eta=2$ gives an immediate upper bound on the $L^2$- and $H^1$-norms of the difference between $i_h^+ u$ and $i_h u$. Namely, $$\label{naive_estimate}
\|i_h^+ u - i_h u \|_{s,2} \le C h^{r-s} |u|_{r,2}$$ for any $s \in \{0,1\}$ and every $h \le h_0$.
Suppose, however, that $\mathcal{T}_h$ and $\mathcal{T}_h^+$ are nearby in the following sense: the two meshes coincide except on a region of measure $O(h^\gamma)$ for some scalar $\gamma \ge 0$. In this scenario, $i_h u$ and $i_h^+ u$ agree everywhere except in the region over which the meshes differ. Hence, by an application of Holder’s inequality (cf. Lemma \[lemma:supp\]), the triangle inequality, and (\[standard\_interp\]), $$\begin{aligned}
\|i_h^+ u - i_h u\|_{s,2}
&\le C h^{\gamma(1/2-1/\eta)} \|i_h^+ u - i_h u\|_{s,\eta} \nonumber \\
&\le C h^{\gamma(1/2-1/\eta)} \left( \|i_h^+ u - u\|_{s,\eta} + \|u - i_h u\|_{s,\eta} \right) \nonumber \\
&\le C h^{r-s+\gamma(1/2-1/\eta)} |u|_{r,\eta} \label{interp_superclose}\end{aligned}$$ for any $s \in \{0,1\}$, any $\eta \in [2,\infty]$, and every $h \le h_0$.
A comparison of (\[interp\_superclose\]) with the naive estimate (\[naive\_estimate\]) reveals that $i_h u$ and $i_h^+ u$ are *superclose* in the $L^2$- and $H^1$-norms when the corresponding meshes are nearby. The primary goal of this paper is to prove an analogous superconvergent estimate when $i_h u$ and $i_h^+ u$ are replaced by the orthogonal projections $r_h u$ and $r_h^+ u$ of $u$ onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively, with respect to a coercive, continuous bilinear form $a : \mathcal{V} \times \mathcal{V} \rightarrow \mathbb{R}$, where $\mathcal{V} \subseteq H^s(\Omega)$ and $s$ is a nonnegative integer. As special cases, our results apply to $L^2$-projections (the case $s=0$) and elliptic projections (the case $s=1$) onto piecewise polynomial finite element spaces. Another applicable case of interest is that in which the bilinear form $a$ is of the form $$a(u,w) = \int_\Omega \nabla u \cdot \nabla w \, dx - \int_\Omega (v \cdot \nabla u) w \, dx + \kappa \int_\Omega u w \, dx$$ with a constant $\kappa>0$ and a vector field $v : \Omega \rightarrow \mathbb{R}^d$. This bilinear form appears in the analysis of finite element methods for the diffusion equation on a moving domain [@Gawlik2012b], with $v$ playing the role of the velocity of a moving mesh and $\kappa$ an auxiliary constant introduced to ensure coercivity.
It is not obvious that superconvergent estimates of the form (\[interp\_superclose\]) should hold in these settings, since the projections of $u$ onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$ need not agree on the region over which the meshes coincide. Nevertheless, Corollaries \[corollary1\] and \[corollary2\] provide such estimates under suitable assumptions on the finite element spaces $\mathcal{V}_h$ and $\mathcal{V}_h^+$ and the bilinear form $a$. The proof uses the observation that, loosely speaking, $a(r_h^+ u - r_h u, r_h^+ u - r_h u)$ is small if $r_h^+ u - r_h u$ is well-approximated by an element of $\mathcal{V}_h^+ \cap \mathcal{V}_h$, since $$a(r_h^+ u - r_h u, w_h) = a(r_h^+ u - u, w_h) + a(u - r_h u, w_h) = 0$$ for any $w_h \in \mathcal{V}_h^+ \cap \mathcal{V}_h$. In particular, if $\|r_h^+ u - r_h u - w_h\|_{s,2}$ decays to zero more rapidly as $h \rightarrow 0$ than do $\|r_h^+ u - u\|_{s,2}$ and $\|r_h u - u\|_{s,2}$, then a superconvergent estimate for $\|r_h^+ u - r_h u\|_{s,2}$ follows from the relation $$a(r_h^+ u - r_h u, r_h^+ u - r_h u) = a(r_h^+ u - r_h u, r_h^+ u - r_h u - w_h)$$ together with the coercivity and continuity of $a$. We in fact prove a more general result that applies to the case in which the projectors $r_h$ and $r_h^+$ are associated not only with different subspaces $\mathcal{V}_h$ and $\mathcal{V}_h^+$, but also with different bilinear forms $a_h$ and $a_h^+$ that may depend on $h$.
#### Organization.
This paper is organized as follows. In Section \[sec:summary\], we summarize our main results. We begin with an abstract estimate (Theorem \[thm:abstract\_estimate\]) for the $H^s$-norm of $r_h^+ u - r_h u$. We then apply Theorem \[thm:abstract\_estimate\] to the setting of finite element spaces with nontrivial intersection in Theorem \[thm:Hsestimate\]. Under some additional assumptions on the finite element spaces, the bilinear forms, and the regularity of $u$, we deduce in Corollary \[corollary1\] a superconvergent estimate for $\|r_h^+ u - r_h u\|_{s,2}$ that parallels (\[interp\_superclose\]). Next, we specialize to the case in which $s=1$ and $a_h$ and $a_h^+$ are bilinear forms associated with elliptic operators that possess smoothing properties. We use a duality argument to prove a superconvergent estimate (Theorem \[L2estimate\] and Corollary \[corollary2\]) for the $L^2$-norm of $r_h^+ u - r_h u$ that is up to one order higher than the corresponding estimate in the $H^1$-norm given by Corollary \[corollary1\].
In Section \[sec:proofs\], we present proofs of the preceding results and provide a few remarks along the way.
In Section \[sec:regularity\], we demonstrate the necessity of the regularity assumptions on $u$ that are imposed in the theorems by exhibiting an example of a pair of projectors $r_h$ and $r_h^+$ and a function $u$ whose insufficient regularity leads to a reduction in the rates of convergence of $\|r_h^+u - r_h u\|_{1,2}$ and $\|r_h^+u - r_h u\|_{0,2}$.
Finally, we give numerical examples to illustrate our positive theoretical results in Section \[sec:numerical\].
#### Related work.
The results presented in this paper bear resemblance to the well-studied phenomenon of superconvergence in finite element theory, where the functions under comparison are typically the solution to a PDE and the numerical solution to a finite element discretization of the same problem. The phenomenon often manifests itself as an exceptional rate of convergence of the finite element solution to the exact solution at isolated points in the domain, as in [@Barlow1976; @Kvrivzek1987; @Babuska1996; @Goodsell1994; @Schatz1996; @Wahlbin1995]. Related results involve exceptional rates of convergence of the finite element solution to a discrete representative of the exact solution, such as its interpolant [@Oganesyan1969; @Li2004; @Huang2008; @Liu2012; @Bank2003; @Andreev2005; @Brandts2003]. Finally, post-processing techniques can lead to modifications of a finite element solution that converge more rapidly to the exact solution than the unprocessed finite element solution [@Zienkiewicz1992; @Babuska1984; @Bank2003; @Kvrivzek1984; @Kvrivzek1987; @Goodsell1989; @Cockburn2003]. To our knowledge, however, little attention has been paid to the supercloseness of finite element solutions associated with differing meshes.
Statement of Results {#sec:summary}
====================
#### Notation.
Fixing a nonnegative integer $s$ and an open, bounded, Lipschitz domain $\Omega \subset \mathbb{R}^d$, let $\mathcal{V}$ be a closed subspace of $H^s(\Omega)$. Let $a_h : \mathcal{V} \times \mathcal{V} \rightarrow \mathbb{R}$ and $a_h^+ : \mathcal{V} \times \mathcal{V} \rightarrow \mathbb{R}$ be bilinear forms that may depend on a parameter $h \le h_0$, where $h_0$ is a positive scalar. We assume that $a_h$ and $a_h^+$ are continuous and coercive uniformly in $h$. In other words, for every $h \le h_0$ and every $u,w \in \mathcal{V}$, the inequalities $$\begin{aligned}
a_h(u,u) &\ge \alpha\|u\|_{s,2}^2, \\
a_h(u,w) &\le M\|u\|_{s,2} \|w\|_{s,2}\end{aligned}$$ hold with constants $\alpha$ and $M$ independent of $h$, and similarly for $a_h^+$ (with the same constants $\alpha$ and $M$).
Let $\{\mathcal{V}_h\}_{0<h\le h_0}$ and $\{\mathcal{V}_h^+\}_{0<h\le h_0}$ be two families of finite element subspaces of $\mathcal{V}$. It is a consequence of the Lax-Milgram theorem that the maps $r_h : \mathcal{V} \rightarrow \mathcal{V}_h$ and $r_h^+ : \mathcal{V} \rightarrow \mathcal{V}_h^+$ defined by the relations $$a_h(r_h u - u, w_h) = 0 \;\;\; \forall w_h \in \mathcal{V}_h$$ and $$a_h^+(r_h^+ u - u, w_h^+) = 0 \;\;\; \forall w_h^+ \in \mathcal{V}_h^+,$$ respectively, are well-defined linear projectors.
For intuition, it is useful to think of $\mathcal{V}_h$ and $\mathcal{V}_h^+$ as finite element spaces associated with a pair of meshes $\mathcal{T}_h$ and $\mathcal{T}_h^+$ of $\Omega$, with the parameter $h$ denoting the maximum diameter of an element among all elements of $\mathcal{T}_h$ and $\mathcal{T}_h^+$. This level of concreteness, however, is not needed for a presentation of the results that follow.
#### Abstract estimate.
Our first result is an abstract estimate for the $H^s$-norm of $r_h^+ u - r_h u$. It provides an alternative to the obvious upper bound $$\|r_h^+ u - r_h u\|_{s,2} \le \|r_h^+ u - u\|_{s,2} + \|u - r_h u\|_{s,2}$$ that one obtains from the triangle inequality. Its utility will be made apparent shortly.
\[thm:abstract\_estimate\] Let $a_h^+$ and $a_h$ be uniformly coercive and continuous bilinear forms on $\mathcal{V} \times \mathcal{V}$. Then for every $u \in \mathcal{V}$ and every $h \le h_0$, $$\label{abstract_estimate}
\begin{split}
\| r_h^+ u - r_h u \|_{s,2} \le \inf_{\substack{e_h \in \mathcal{V}_h \\ e_h^+ \in \mathcal{V}_h^+}}
\Big[ \frac{M}{\alpha} &\left\| r_h^+ u - r_h u - (e_h + e_h^+) \right\|_{s,2} \\
+ &\frac{1}{\sqrt{\alpha}} \big( \left|a_h^+(r_h^+u - u, e_h)\right|^{1/2} + \left|a_h(r_h u - u, e_h^+)\right|^{1/2} \\
&+ \left|a_h^+(r_h u - u, e_h + e_h^+) - a_h(r_h u - u, e_h + e_h^+)\right|^{1/2} \big) \Big].
\end{split}$$
The preceding theorem provides a heuristic for estimating the $H^s$-norm of $r_h^+ u - r_h u$. Namely, one seeks functions $e_h \in \mathcal{V}_h$ and $e_h^+ \in \mathcal{V}_h^+$ that are nearly (right-) orthogonal to $r_h^+ u - u$ and $r_h u - u$ with respect to $a_h^+(\cdot,\cdot)$ and $a_h(\cdot,\cdot)$, respectively, but whose sum is close to $r_h^+ u - r_h u$. In general, near orthogonality and closeness to $r_h^+ u - r_h u$ are competing interests. Exact orthogonality holds for $e_h,e_h^+ \in \mathcal{V}_h^+ \cap \mathcal{V}_h$, whereas $e_h + e_h^+$ can be made equal to $r_h^+ u - r_h u$ by choosing, for instance, $e_h^+ = r_h^+u$ and $e_h = -r_h u$. If a suitable choice of $e_h$ and $e_h^+$ leads to adequate satisfaction of both interests simultaneously, and if $a_h^+$ is close to $a_h$ (in the sense that the last term in (\[abstract\_estimate\]) is small), then the prospects of producing a superconvergent bound on $\|r_h^+ u - r_h u\|_{s,2}$ are favorable.
#### Finite element spaces with nontrivial intersection.
We now apply Theorem \[thm:abstract\_estimate\] to the case in which the finite element spaces $\mathcal{V}_h^+$ and $\mathcal{V}_h$ intersect nontrivially. The setting that we have in mind is that in which $\mathcal{V}_h$ and $\mathcal{V}_h^+$ consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes of $\Omega$ that coincide except on a region of measure $O(h^\gamma)$ for some constant $\gamma \ge 0$. To allow for more generality, we state the assumptions on $\mathcal{V}_h^+$ and $\mathcal{V}_h$ abstractly, and we refer the reader to Appendix \[sec:appendix\_Pk\] for a proof of their satisfaction in the aforementioned setting.
In particular, we assume the existence of a constant $\eta \in [2,\infty]$ such that the following properties hold:
1. \[Wseta\] For every $h \le h_0$, $\mathcal{V}_h,\mathcal{V}_h^+ \subset W^{s,\eta}(\Omega) \cap \mathcal{V}$.
2. \[inverse\] There exists $C>0$ independent of $h$ such that the inverse estimate $$\|w_h\|_{m,\eta} \le Ch^{-m}\|w_h\|_{0,\eta}$$ holds for every $m=0,1,\dots,s$, every $w_h \in \mathcal{V}_h^+ \cap \mathcal{V}_h$, and every $h \le h_0$.
3. \[assumption2b\] There exist constants $\gamma \ge 0$ and $C>0$ independent of $h$ and a map $\pi_h : \mathcal{V}_h^+ + \mathcal{V}_h \rightarrow \mathcal{V}_h^+ \cap \mathcal{V}_h$ such that $$\|\pi_h w_h\|_{0,\eta} \le C \|w_h\|_{0,\eta}$$ and $$|\mathrm{supp}(\pi_h w_h - w_h)| \le C h^\gamma$$ for every $w_h \in \mathcal{V}_h^+ + \mathcal{V}_h$ and every $h \le h_0$.
In the context of finite element spaces consisting of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes of $\Omega$, a befitting choice for $\pi_h$ in (\[assumption2b\]) is the nodal interpolant onto $\mathcal{V}_h^+ \cap \mathcal{V}_h$; see Appendix \[sec:appendix\_Pk\]. In that setting, the constant $\gamma$ appearing in (\[assumption2b\]) may take on any real value between $0$ and $d$, unless the two meshes coincide entirely (in which case $\gamma$ may be chosen arbitrarily large). To realize a pair of meshes $\mathcal{T}_h$ and $\mathcal{T}_h^+$ fulfilling (\[assumption2b\]) with $\gamma \in [0,d]$, one may, for instance, consider a shape-regular, quasi-uniform mesh $\mathcal{T}_h$ of $\Omega$ and perturb the positions of $O(h^{-d+\gamma})$ of its nodes by a sufficiently small amount to define $\mathcal{T}_h^+$.
The following theorem results from applying Theorem \[thm:abstract\_estimate\] to the setting delineated in conditions (\[Wseta\]-\[assumption2b\]), with the choice $e_h=\pi_h(r_h^+ u - r_h u)$ and $e_h^+=0$ in (\[abstract\_estimate\]).
\[thm:Hsestimate\] Suppose the conditions of Theorem \[thm:abstract\_estimate\] hold and the finite element spaces $\mathcal{V}_h^+$ and $\mathcal{V}_h$ satisfy conditions (\[Wseta\]-\[assumption2b\]). Suppose further that there exist constants $C_1>0$, $\delta \ge 0$, $1 \le q \le \eta$, and $\mu,\nu \in \{0,1,\dots,s\}$ independent of $h$ such that $$\label{assumption1}
|a_h^+(v,w) - a_h(v,w)| \le C_1 h^\delta \|v\|_{\mu,\eta} \|w\|_{\nu,q}$$ for every $v,w \in W^{s,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$. Then there exists $C>0$ independent of $h$ such that for any $h \le h_0$ and any $u \in W^{s,\eta}(\Omega) \cap \mathcal{V}$, $$\begin{split}
\|r_h^+ u - r_h u\|_{s,2} \le C h^{\sigma-s} \Big[ &h^s\|r_h^+ u - u\|_{s,\eta} + h^s\|r_h u - u\|_{s,\eta} + \|r_h^+ u - u\|_{0,\eta} + \|r_h u - u\|_{0,\eta} \\
&+ \left(h^\mu \|r_h u - u\|_{\mu,\eta}\right)^{1/2} \left(\|r_h^+ u - u\|_{0,\eta} + \|r_h u - u\|_{0,\eta} \right)^{1/2} \Big]
\end{split}$$ with $$\label{sigma}
\sigma = \min\left\{\gamma\left(\frac{1}{2}-\frac{1}{\eta}\right), \frac{\delta+2s-\mu-\nu}{2} \right\}.$$
The meaning of Theorem \[thm:Hsestimate\] is clearest when the quantities $h^m \|r_h u - u\|_{m,p}$ and $h^m \|r_h^+ u - u\|_{m,p}$, $m = 0,1,\dots,s$, $p=2,\eta$, all decay at the same rate with respect to $h$ as $h \rightarrow 0$. In such a setting, the theorem states that $\|r_h^+ u - r_h u\|_{s,2}$ tends to zero faster than $\|r_h u - u\|_{s,2} + \|r_h^+ u - u\|_{s,2}$ by a factor $O(h^\sigma)$, where the order of superconvergence $\sigma$ depends primarily upon two features: (1) the extent to which the finite element spaces $\mathcal{V}_h$ and $\mathcal{V}_h^+$ coincide, as measured by the constant $\gamma$ in (\[assumption2b\]), and (2) the difference between the bilinear forms $a_h$ and $a_h^+$, as measured by the constants $\delta$, $\mu$, and $\nu$ in (\[assumption1\]). The regularity of $u$ also plays a role in the estimate via the constant $\eta$, which is in the best case equal to $\infty$.
To be more concrete, let us point out that in many contexts (which we detail in Appendix \[sec:appendix\_Linf\]), the quantities $r_h u - u$ and $r_h^+ u - u$ satisfy estimates of the form $$\begin{aligned}
\|r_h u - u\|_{0,\eta} + \|r_h^+ u - u\|_{0,\eta} &\le C \ell(h) h^r |u|_{r,\eta} \label{ep0eta}, \\
\|r_h u - u\|_{m,\eta} + \|r_h^+ u - u\|_{m,\eta} &\le C h^{r-m} |u|_{r,\eta}, \;\;\; m=1,2,\dots,s, \label{epmeta}\end{aligned}$$ for every $u \in W^{r,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$, where $r>s$ is an integer and $\ell(h)$ is either identically unity or equal to $\log(h^{-1})$. Note that (\[epmeta\]) is vacuous when $s=0$. When such estimates hold, the following corollary to Theorem \[thm:Hsestimate\] is immediate.
\[corollary1\] Suppose that the conditions of Theorem \[thm:Hsestimate\] are satisfied and that both $r_h$ and $r_h^+$ satisfy estimates of the form (\[ep0eta\]-\[epmeta\]) for an integer $r>s$. Then there exists $C>0$ independent of $h$ such that $$\|r_h^+ u - r_h u\|_{s,2} \le C \ell(h) h^{r-s+\sigma} |u|_{r,\eta}$$ for every $u \in W^{r,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$, with $\sigma$ given by (\[sigma\]).
In particular, if $a_h=a_h^+$, then $$\|r_h^+ u - r_h u\|_{s,2} \le C \ell(h) h^{r-s+\gamma(1/2-1/\eta)} |u|_{r,\eta}$$ for every $u \in W^{r,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$.
Note that to deduce the preceding corollary, the case $a_h=a_h^+$ is handled by taking $\delta = \infty$ and choosing any admissible $\mu$, $\nu$ and $q$ in (\[assumption1\]).
#### $L^2$ estimates for elliptic projections.
Finally, we restrict our attention to the case $s=1$ with $\mathcal{V}=H^1_0(\Omega)$, so that $a_h$ and $a_h^+$ are coercive, continuous bilinear forms on $H^1_0(\Omega) \times H^1_0(\Omega)$, uniformly in $h$. Here, $H^1_0(\Omega)$ denotes the space of functions in $H^1(\Omega)$ with vanishing trace on $\partial \Omega$. Our aim is to provide an estimate for the $L^2$-norm of $r_h^+ u - r_h u$ that parallels the estimate in the $H^1$-norm provided by Corollary \[corollary1\] but is of a higher order by up to one power of $h$.
In addition to the assumptions stated in Theorem \[thm:Hsestimate\], we make the following assumptions on the bilinear forms $a_h$ and $a_h^+$.
1. \[cond:smoothing\] The bilinear forms $a_h$ and $a_h^+$ are associated with elliptic operators whose adjoints possess *smoothing properties* (cf. [@Ern2004 Definition 3.14]), uniformly in $h$. Precisely, let $f \in L^2(\Omega)$ and consider the following problem: Find $w \in \mathcal{V}$ such that $$\label{adjoint_problem}
a_h(y,w) = (f,y) \;\;\; \forall y \in \mathcal{V},$$ where $(f,y) := \int_\Omega fy$. Then $a_h$ is said to have smoothing properties (uniformly in $h$) if there exists a constant $C>0$ independent of $h$ such that for every $f \in L^2(\Omega)$ and every $h \le h_0$, there exists a unique solution $w$ to (\[adjoint\_problem\]) satisfying the elliptic regularity estimate $$\|w\|_{2,2} \le C\|f\|_{0,2}.$$
2. \[cond:subdomains\] There exists $C>0$ such that for any $h \le h_0$, any subdomain $R \subseteq \Omega$, and any $v,w \in \mathcal{V}$ with $\mathrm{supp}(w) \subseteq R$, $$|a_h(v,w)| \le C \|v\|_{1,2,R} \|w\|_{1,2,R},$$ where the constant $C$ is independent of $h$ and $R$, and similarly for $a_h^+$.
3. \[qrestriction\] The constant $q$ appearing in the bound (\[assumption1\]) satisfies the additional restriction $$\begin{cases}
q < \infty &\mbox{ if } d=4-2\nu, \\
q \le \frac{2d}{d-4+2\nu} &\mbox{ if } d > 4-2\nu.
\end{cases}$$
Condition (\[qrestriction\]) guarantees the validity of the Sobolev emdedding $H^2(\Omega) \subset W^{\nu,q}(\Omega)$. Note that it places no additional restriction on $q$ if $d < 4-2\nu$.
Furthermore, we assume the existence of interpolation operators $i_h : \bar{\mathcal{V}} \rightarrow \mathcal{V}_h$ and $i_h^+ : \bar{\mathcal{V}} \rightarrow \mathcal{V}_h^+$ defined on a space $H^2(\Omega) \cap \mathcal{V} \subseteq \bar{\mathcal{V}} \subseteq \mathcal{V}$ that satisfy the following properties.
1. \[cond:interp\_stability\] There exists $C>0$ independent of $h$ such that $$\|i_h w\|_{\nu,q} + \|i_h^+ w\|_{\nu,q} \le C\|w\|_{\nu,q}$$ for every $w \in H^2(\Omega) \cap \mathcal{V}$ and every $h \le h_0$.
2. \[cond:interp\] There exists $C>0$ independent of $h$ such that $$\|i_h w - w\|_{1,2} + \|i_h^+ w - w\|_{1,2} \le C h |w|_{2,2}$$ for every $w \in H^2(\Omega) \cap \mathcal{V}$ and every $h \le h_0$.
3. \[cond:interp\_coincide\] For every $w \in H^2(\Omega) \cap \mathcal{V}$ and every $h \le h_0$, $$\mathrm{supp}(i_h^+ w - i_h w) \subseteq \mathcal{R}_h,$$ where $$\mathcal{R}_h := \bigcup_{w_h \in \mathcal{V}_h + \mathcal{V}_h^+} \mathrm{supp}(w_h - \pi_h w_h)$$ and $\pi_h$ is the map introduced in (\[assumption2b\]).
Our estimate for the $L^2$-norm of $r_h^+ u - r_h u$, whose proof employs a duality argument, is as follows.
\[L2estimate\] Suppose the conditions of Theorem \[thm:Hsestimate\] hold with $s=1$. Assume further that conditions (\[cond:smoothing\]-\[cond:interp\_coincide\]) hold. Then there exists $C>0$ independent of $h$ such that for every $u \in W^{1,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$, $$\begin{split}
\|r_h^+ u - r_h u\|_{0,2} \le C h^{\sigma'} \, \Big[ &h\|r_h^+ u - u\|_{1,\eta} + h\|r_h u - u\|_{1,\eta} + \|r_h^+ u - u\|_{0,\eta} + \|r_h u - u\|_{0,\eta} \\
&+ \left(h^\mu \|r_h u - u\|_{\mu,\eta}\right)^{1/2} \left(\|r_h^+ u - u\|_{0,\eta} + \|r_h u - u\|_{0,\eta} \right)^{1/2} \\
&+ h^\mu \|r_h u - u\|_{\mu,\eta} \Big],
\end{split}$$ with $$\label{sigmaprime}
\sigma' = \min\left\{\gamma\left(\frac{1}{2}-\frac{1}{\eta}\right), \frac{\delta+2-\mu-\nu}{2}, \delta - \mu \right\}.$$
Just as in Theorem \[thm:Hsestimate\], the meaning of Theorem \[L2estimate\] is clearest when the quantities $h^m \|r_h u - u\|_{m,p}$ and $h^m \|r_h^+ u - u\|_{m,p}$, $m = 0,1,\dots,s$, $p=2,\eta$, all decay at the same rate with respect to $h$ as $h \rightarrow 0$. In such a setting, Theorem \[L2estimate\] states that $\|r_h^+ u - r_h u\|_{0,2}$ tends to zero faster than $\|r_h u - u\|_{0,2} + \|r_h^+ u - u\|_{0,2}$ by a factor $O(h^{\sigma'})$, where the order of superconvergence $\sigma'$ is given by (\[sigmaprime\]). Note that $\sigma' \le \sigma$, where $\sigma$ is the order of superconvergence of the $H^1$-norm of $r_h^+ u - r_h u$ that was provided in Theorem \[thm:Hsestimate\].
Concretely, when estimates of the form (\[ep0eta\]-\[epmeta\]) hold for $u \in W^{r,\eta}(\Omega) \cap \mathcal{V}$ with an integer $r>1$, we arrive immediately at the following corollary to Theorem \[L2estimate\].
\[corollary2\] Suppose that the conditions of Theorem \[thm:Hsestimate\] are satisfied and that both $r_h$ and $r_h^+$ satisfy estimates of the form (\[ep0eta\]-\[epmeta\]) for an integer $r>1$. Then there exists $C>0$ independent of $h$ such that $$\|r_h^+ u - r_h u\|_{0,2} \le C \ell(h) h^{r+\sigma'} |u|_{r,\eta}$$ for every $u \in W^{r,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$, with $\sigma'$ given by (\[sigmaprime\]).
In particular, if $a_h = a_h^+$, then $$\|r_h^+ u - r_h u\|_{0,2} \le C \ell(h) h^{r+\gamma(1/2-1/\eta)} |u|_{r,\eta}$$ for every $u \in W^{r,\eta}(\Omega) \cap \mathcal{V}$ and every $h \le h_0$.
Note that to deduce the preceding corollary, the case $a_h=a_h^+$ is again handled by taking $\delta = \infty$ and choosing any admissible $\mu$, $\nu$ and $q$ in (\[assumption1\]).
Proofs {#sec:proofs}
======
This section presents proofs of Theorems \[thm:abstract\_estimate\], \[thm:Hsestimate\], and \[L2estimate\].
Let $e_h \in \mathcal{V}_h$ and $e_h^+ \in \mathcal{V}_h^+$, and write $$\begin{split}
a_h^+(r_h^+ u - r_h u, r_h^+ u - r_h u) =&\; a_h^+\left(r_h^+ u - r_h u, r_h^+ u - r_h u - (e_h + e_h^+)\right) \\
&+ a_h^+(r_h^+ u - r_h u, e_h + e_h^+).
\end{split}$$ The uniform coercivity and continuity of $a_h^+$ imply $$\| r_h^+ u - r_h u \|_{s,2}^2 \le \frac{1}{\alpha} \left( M \| r_h^+ u - r_h u \|_{s,2} \|r_h^+ u - r_h u - (e_h + e_h^+)\|_{s,2} + |a_h^+(r_h^+ u - r_h u, e_h + e_h^+)| \right).$$ Using the fact that for real numbers $x,a,b \ge 0$, $$x^2 \le a x + b \implies x \le a + \sqrt{b},$$ we deduce that $$\| r_h^+ u - r_h u \|_{s,2} \le \frac{M}{\alpha} \|r_h^+ u - r_h u - (e_h + e_h^+)\|_{s,2} + \frac{1}{\sqrt{\alpha}} |a_h^+(r_h^+ u - r_h u, e_h + e_h^+)|^{1/2}$$ The result will then follow from the identity $$\label{term2expansion}
\begin{split}
a_h^+(r_h^+ u - r_h u, e_h + e_h^+)
&= a_h^+(r_h^+ u - u, e_h) + a_h(u - r_h u, e_h^+) \\
&\;\;\;+ a_h^+(u - r_h u, e_h + e_h^+) - a_h(u - r_h u, e_h + e_h^+)
\end{split}$$ together with the subadditivity of the square root operator.
To prove (\[term2expansion\]), use the decomposition $r_h^+ u - r_h u = (r_h^+ u - u) + (u - r_h u)$ to write $$a_h^+(r_h^+ u - r_h u, e_h + e_h^+) = a_h^+(r_h^+ u - u, e_h + e_h^+) + a_h^+(u - r_h u, e_h + e_h^+).$$ Now add and subtract $a_h(u - r_h u, e_h + e_h^+)$ to obtain $$\begin{split}
a_h^+(r_h^+ u - r_h u, e_h + e_h^+)
&= a_h^+(r_h^+ u - u, e_h + e_h^+) + a_h(u - r_h u, e_h + e_h^+) \\
&\;\;\;+ a_h^+(u - r_h u, e_h + e_h^+) - a_h(u - r_h u, e_h + e_h^+).
\end{split}$$ Finally, use the definitions of $r_h^+$ and $r_h$ to simplify the first two terms, giving (\[term2expansion\]).
We remark that while the estimate (\[abstract\_estimate\]) is not symmetric in the “+” variables and their unadorned counterparts, it can easily be made symmetric by exchanging the roles of $r_h^+$ and $a_h^+$ with $r_h$ and $a_h$, respectively, and averaging the resulting estimates. The same holds true for the estimates in Theorems \[thm:Hsestimate\] and \[L2estimate\].
We now turn to the proof of Theorem \[thm:Hsestimate\]. We begin with a lemma concerning the relationship between a function’s support and its Sobolev norms.
\[lemma:supp\] Let $f \in W^{k,p}(\Omega)$, $k \ge 0$, $p \in [1,\infty]$. Then for any $1 \le t \le p$, $$\|f\|_{k,t} \le |\mathrm{supp}(f)|^{1/t-1/p} \|f\|_{k,p}.$$
Let $\chi : \Omega \rightarrow \{0,1\}$ denote the indicator function for $\mathrm{supp}(f)$. We have $$\begin{aligned}
\|f\|_{k,t}
&= \sum_{|\alpha| \le k} \|\partial^\alpha f\|_{0,t} \\
&= \sum_{|\alpha| \le k} \|\chi \partial^\alpha f\|_{0,t}.\end{aligned}$$ Now let $\tilde{p} \in [1,\infty]$ be such that $\frac{1}{\tilde{p}}+\frac{1}{p}=\frac{1}{t}$. By Holder’s inequality, $$\begin{aligned}
\|f\|_{k,t}
&\le \sum_{|\alpha| \le k} \|\chi\|_{0,\tilde{p}} \|\partial^\alpha f\|_{0,p} \\
&= |\mathrm{supp}(f)|^{1/\tilde{p}} \sum_{|\alpha| \le k} \|\partial^\alpha f\|_{0,p} \\
&= |\mathrm{supp}(f)|^{1/t-1/p} \|f\|_{k,p}.\end{aligned}$$
The proof of Theorem \[thm:Hsestimate\] is as follows.
Choose $e_h^+=0$ and $e_h = \pi_h (r_h^+ u - r_h u)$ in (\[abstract\_estimate\]). By the stability assumption in (\[assumption2b\]), $$\begin{aligned}
\|e_h\|_{0,\eta}
&\le C\|r_h^+ u - r_h u\|_{0,\eta} \\
&\le C \left( \|r_h^+ u - u\|_{0,\eta} + \|u - r_h u\|_{0,\eta} \right).\end{aligned}$$ Thus, for any $m=0,1,\dots,s$, $$\label{ehmeta}
\| e_h \|_{m,\eta} \le C h^{-m} \left( \|r_h^+ u - u\|_{0,\eta} + \|u - r_h u\|_{0,\eta} \right)$$ by (\[inverse\]). It follows that $$\begin{aligned}
\|r_h^+ u - r_h u - (e_h + e_h^+) \|_{s,\eta}
&\le \|r_h^+ u - u\|_{s,\eta} + \|u - r_h u\|_{s,\eta} + \|e_h\|_{s,\eta} + \|e_h^+\|_{s,\eta} \\
&\le C \big( \|r_h^+ u - u\|_{s,\eta} + \|u - r_h u\|_{s,\eta} \\
&\hspace{0.4in}+ h^{-s}\|r_h^+ u - u\|_{0,\eta} + h^{-s} \|u - r_h u\|_{0,\eta} \big).\end{aligned}$$ Now note that $r_h^+ u - r_h u - (e_h + e_h^+)$ has support of measure $O(h^\gamma)$ by (\[assumption2b\]). Consequently, by Lemma \[lemma:supp\], $$\begin{aligned}
\|r_h^+ u - r_h u - (e_h + e_h^+) \|_{s,2}
&\le C h^{\gamma(1/2-1/\eta)} \|r_h^+ u - r_h u - (e_h + e_h^+) \|_{s,\eta} \nonumber \\
&\le C h^{\gamma(1/2-1/\eta)} \big( \|r_h^+ u - u\|_{s,\eta} + \|r_h u - u\|_{s,\eta} \nonumber \\
&\hspace{0.7in} + h^{-s}\|r_h^+ u - u\|_{0,\eta} + h^{-s} \|r_h u - u\|_{0,\eta}\big). \label{term1}\end{aligned}$$ To estimate the remaining terms that appear in (\[abstract\_estimate\]), note that $$a_h^+(r_h^+ u - u, e_h) = 0$$ since $e_h \in \mathcal{V}_h^+ \cap \mathcal{V}_h \subseteq \mathcal{V}_h^+$, and $$a_h(r_h u - u, e_h^+) = 0$$ since $e_h^+ = 0$. Finally, using (\[ehmeta\]) with $m=\nu$ together with (\[assumption1\]) shows that $$\begin{aligned}
\big|a_h^+(r_h u - u, e_h + &e_h^+) - a_h(r_h u - u, e_h + e_h^+)\big| \\
&\le C h^{\delta} \|r_h u - u\|_{\mu,\eta} \|e_h\|_{\nu,q} \\
&\le C h^{\delta} \|r_h u - u\|_{\mu,\eta} \|e_h\|_{\nu,\eta} \\
&\le C h^{\delta-\nu} \|r_h u - u\|_{\mu,\eta} \left(\|r_h^+ u - u\|_{0,\eta} + \|u - r_h u\|_{0,\eta}\right).\end{aligned}$$ Taking the square root and adding (\[term1\]) proves the claim.
Note that the preceding proof treats the estimate (\[assumption1\]) wastefully when $q<\eta$, in the sense that the ultimate bound on $\|r_h^+ u - r_h u\|_{s,2}$ is unchanged if $q$ is replaced by $\eta$. The importance of considering scenarios in which $q$ may be chosen less than $\eta$ is made apparent in Theorem \[L2estimate\], where the restriction (\[qrestriction\]) is enforced.
With this in mind, we now prove Theorem \[L2estimate\].
Define $w \in \mathcal{V}$ as the solution to the dual problem $$\label{dual_problem}
a_h^+(y,w) = (r_h^+ u - r_h u, y) \;\;\; \forall y \in \mathcal{V}.$$ Note that $w \in H^2(\Omega) \cap \mathcal{V}$ by (\[cond:smoothing\]).
For any $w_h^+ \in \mathcal{V}_h^+$, $w_h \in \mathcal{V}_h$, we have $$\begin{aligned}
\|r_h^+ u - r_h u\|_{0,2}^2
&= a_h^+(r_h^+ u - r_h u, w) \\
&= a_h^+(r_h^+ u - r_h u, w - w_h^+) + a_h^+(r_h^+ u - r_h u, w_h^+) \\
&= a_h^+(r_h^+ u - r_h u, w - w_h^+) + a_h^+(u - r_h u, w_h^+) \\
&= a_h^+(r_h^+ u - r_h u, w - w_h^+) + a_h^+(u - r_h u, w_h^+ - w_h) \\
&\;\;\; + a_h^+(u - r_h u, w_h) - a_h(u-r_h u, w_h) \\
&=: T_1 + T_2 + T_3,\end{aligned}$$ where $$\begin{aligned}
T_1 &= a_h^+(r_h^+ u - r_h u, w - w_h^+), \\
T_2 &= a_h^+(u - r_h u, w_h^+ - w_h), \\
T_3 &= a_h^+(u - r_h u, w_h) - a_h(u-r_h u, w_h).\end{aligned}$$
Now choose $w_h^+ = i_h^+ w$ and $w_h = i_h w$ and bound each term separately. By the continuity of $a_h^+$ and (\[cond:interp\]), $$\begin{aligned}
|T_1|
&\le C \| r_h^+ u - r_h u\|_{1,2} \| w - w_h^+ \|_{1,2} \\
&\le C h \| r_h^+ u - r_h u\|_{1,2} |w|_{2,2}. \\\end{aligned}$$ To bound $T_2$, note that $\mathrm{supp}(w_h^+ - w_h) \subseteq \mathcal{R}_h$ has measure $O(h^\gamma)$ by (\[cond:interp\_coincide\]) and (\[assumption2b\]). Thus, $$\begin{aligned}
|T_2|
&\le C \|u-r_h u\|_{1,2,\mathcal{R}_h} \|w_h^+ - w_h\|_{1,2,\mathcal{R}_h} \\
&\le C h^{\gamma(1/2-1/\eta)} \|u-r_h u\|_{1,\eta} \left(\|w_h^+ - w\|_{1,2,\mathcal{R}_h} + \|w - w_h\|_{1,2,\mathcal{R}_h} \right) \\
&\le C h^{\gamma(1/2-1/\eta)+1} \|u-r_h u\|_{1,\eta} |w|_{2,2}\end{aligned}$$ by (\[cond:subdomains\]), Lemma \[lemma:supp\], and (\[cond:interp\]). For $T_3$, we have by (\[assumption1\]) that $$|T_3| \le C h^\delta \|u-r_h u\|_{\mu,\eta} \|w_h\|_{\nu,q}.$$ Using (\[cond:interp\_stability\]) together with the Sobolev embedding $H^2(\Omega) \subset W^{\nu,q}(\Omega)$ ensured by (\[qrestriction\]) gives $$|T_3| \le C h^\delta \|u-r_h u\|_{\mu,\eta} \|w\|_{2,2}.$$ Combining results and invoking the regularity estimate (\[cond:smoothing\]) leads to $$\begin{split}
\|r_h^+ u - r_h u\|_{0,2} \le C \Big[ h &\|r_h^+ u - r_h u\|_{1,2} \\&+ h^{\min\{\gamma(1/2-1/\eta),\delta-\mu\}} \left( h \|u-r_h u\|_{1,\eta} + h^\mu \|u-r_h u\|_{\mu,\eta} \right) \Big].
\end{split}$$ Conclude using Theorem \[thm:Hsestimate\].
The Need for Regularity {#sec:regularity}
=======================
When $a_h^+=a_h$ and $\gamma$ is fixed, the estimates of Corollaries \[corollary1\] and \[corollary2\] are of the highest order in $h$ when $\eta = \infty$, but in this case they demand that $u \in W^{r,\infty}(\Omega) \cap \mathcal{V}$. If the regularity requirement $u \in W^{r,\infty}(\Omega) \cap \mathcal{V}$ is relaxed, the rates of convergence of $\|r_h^+ u - r_h u\|_{0,2}$ and $\|r_h^+ u - r_h u\|_{1,2}$ as $h \rightarrow 0$ may deteriorate.
Indeed, consider the case in which $\mathcal{V}_h$ is the space of piecewise affine functions on a grid $(0,h,2h,3h,\dots,1)$ of the unit interval in one dimension that vanish at 0 and 1. Let $\mathcal{V}_h^+$ be the space of piecewise affine functions on the nearby grid $(0,3h/2,2h,3h,\dots,1)$ that vanish at 0 and 1. Let $$a_h^+(u,w)=a_h(u,w) = \int_0^1 \frac{\partial u}{\partial x} \frac{\partial w}{\partial x} \, dx,$$ so that the projectors $r_h$ and $r_h^+$ coincide with the nodal interpolants onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively [@Ern2004 Remark 3.25(i)]. In this setting, the conditions of Corollaries \[corollary1\] and \[corollary2\] hold with $\eta=\infty$, $\gamma=1$, $r=2$, and $\ell(h) \equiv 1$, leading to the estimates $$\begin{aligned}
\| r_h^+ u - r_h u \|_{0,2} &\le C h^{5/2} |u|_{r,\infty}, \\
\| r_h^+ u - r_h u \|_{1,2} &\le C h^{3/2} |u|_{r,\infty}\end{aligned}$$ for $u \in W^{2,\infty}(0,1) \cap H^1_0(0,1)$.
However, consider the function $$u(x) = x^{2-1/p} - x$$ with $2 < p < \infty$, so that $u \in W^{2,p-\varepsilon}(0,1) \cap H^1_0(0,1)$ for any $\varepsilon>0$. Then a direct calculation renders that $$\begin{aligned}
\| r_h^+ u - r_h u \|_{0,2} &\ge C h^{5/2-1/p}, \\
\| r_h^+ u - r_h u \|_{1,2} &\ge C h^{3/2-1/p},\end{aligned}$$ which are of a lower order than the rates $h^{5/2}$ and $h^{3/2}$, respectively, obtainable for a function in $W^{2,\infty}(0,1) \cap H^1_0(0,1)$. In fact, by letting $p \rightarrow 2$, these rates can be made arbitrarily close to the quadratic and linear rates that hold in the $L^2$- and $H^1$-norms, respectively, on a pair of unrelated meshes.
Numerical Examples {#sec:numerical}
==================
In this section, we numerically illustrate the superconvergent estimates of Corollaries \[corollary1\] and \[corollary2\] on test cases in one and two dimensions.
[295pt]{}[r|cc|cc]{} & &\
$h_0/h$ & & & &\
1 & 3.2150e-03 & & 1.2843e-04 &\
2 & 5.6505e-04 & 2.5084 & 1.0676e-05 & 3.5886\
4 & 9.9837e-05 & 2.5007 & 9.1277e-07 & 3.5480\
8 & 1.7645e-05 & 2.5003 & 7.9301e-08 & 3.5248\
16 & 3.1189e-06 & 2.5002 & 6.9484e-09 & 3.5126\
32 & 5.5132e-07 & 2.5001 & 6.1146e-10 & 3.5063\
[295pt]{}[r|cc|cc]{} & &\
$h_0/h$ & & & &\
1 & 1.4451e-01 & & 7.4390e-03 &\
2 & 5.1203e-02 & 1.4968 & 1.2835e-03 & 2.5351\
4 & 1.8081e-02 & 1.5017 & 2.2408e-04 & 2.5180\
8 & 6.3851e-03 & 1.5017 & 3.9364e-05 & 2.5090\
16 & 2.2558e-03 & 1.5011 & 6.9369e-06 & 2.5045\
32 & 7.9723e-04 & 1.5006 & 1.2243e-06 & 2.5023\
[295pt]{}[r|cc|cc]{} & &\
$h_0/h$ & & & &\
1 & 3.4546e-03 & & 1.7770e-04 &\
2 & 6.1937e-04 & 2.4796 & 1.5493e-05 & 3.5198\
4 & 1.1019e-04 & 2.4908 & 1.3576e-06 & 3.5124\
8 & 1.9537e-05 & 2.4957 & 1.1943e-07 & 3.5069\
16 & 3.4587e-06 & 2.4979 & 1.0530e-08 & 3.5036\
32 & 6.1186e-07 & 2.4990 & 9.2955e-10 & 3.5018\
#### One dimension.
Consider the case in which $\mathcal{V}_h$ is the space of piecewise polynomial functions of degree at most $r-1$ on a grid $(0,h,2h,3h,\dots,1)$ of the unit interval in one dimension that vanish at 0 and 1. Let $\mathcal{V}_h^+$ be the space of piecewise polynomial functions of the same degree that vanish at 0 and 1, on the same grid but with the node nearest to $x=1/4$ perturbed by $h/4$ in the positive direction. In this scenario, assumption (\[assumption2b\]) is satisfied with $\gamma=1$. Let $u(x) = \sin(\pi x)$ and let $$a_h^+(u,w)=a_h(u,w) = \int_0^1 uw \, dx,$$ so that $r_h$ and $r_h^+$ are the $L^2$-projectors onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively.
Table \[tab:L2proj1d\] shows the $L^2$-norm of the difference $r_h^+ u - r_h u$ for several values of $h$, beginning with $h = 1/8 =: h_0$. The table illustrates the predictions of Corollary \[corollary1\], namely $$\|r_h^+ u - r_h u\|_{0,2} \le
\begin{cases}
C h^{5/2} |u|_{2,\infty} &\mbox{if } r=2, \\
C h^{7/2} |u|_{3,\infty} &\mbox{if } r = 3.
\end{cases}$$
Next, consider the same setup as above, but with $$a_h^+(u,w)=a_h(u,w) = \int_0^1 \frac{\partial u}{\partial x} \frac{\partial w}{\partial x} \, dx,$$ so that $r_h$ and $r_h^+$ are the standard elliptic projectors onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively. Table \[tab:ellipticproj1dH1\] shows the $H^1$ norm of the difference $r_h^+ u - r_h u$ for the sequence of grids described above. The table illustrates the predictions of Corollary \[corollary1\], namely $$\|r_h^+ u - r_h u\|_{1,2} \le
\begin{cases}
C h^{3/2} \log(h^{-1}) |u|_{2,\infty} &\mbox{if } r=2, \\
C h^{5/2} |u|_{3,\infty} &\mbox{if } r = 3.
\end{cases}$$ Table \[tab:ellipticproj1dL2\] shows the $L^2$-norm of the difference $r_h^+ u - r_h u$ for the same sequence of grids. The table illustrates the predictions of Corollary \[corollary2\], namely $$\|r_h^+ u - r_h u\|_{0,2} \le
\begin{cases}
C h^{5/2} \log(h^{-1}) |u|_{2,\infty} &\mbox{if } r=2, \\
C h^{7/2} |u|_{3,\infty} &\mbox{if } r = 3.
\end{cases}$$ Note that we have not attempted to detect the presence of the factor $\log(h^{-1})$ in these numerical experiments.
[165pt]{}[r|cc]{} &\
$h_0/h$ & &\
1 & 6.3533e-03 &\
2 & 7.5614e-04 & 3.0708\
4 & 8.8718e-05 & 3.0914\
8 & 1.1020e-05 & 3.0091\
16 & 1.3781e-06 & 2.9993\
\[tab:ellipticproj2d\]
[295pt]{}[r|cc|cc]{} &\
$h_0/h$ & & & &\
1 & 2.1441e-01 & & 6.6386e-03 &\
2 & 4.7374e-02 & 2.1782 & 7.8678e-04 & 3.0768\
4 & 1.1359e-02 & 2.0603 & 9.6370e-05 & 3.0293\
8 & 2.8114e-03 & 2.0144 & 1.2033e-05 & 3.0016\
16 & 7.0176e-04 & 2.0023 & 1.5106e-06 & 2.9937\
#### Two dimensions.
Consider now the case in which $\mathcal{V}_h \subset H^1_0((0,1) \times (0,1))$ is the space of piecewise affine functions on a mesh of the unit square in two dimensions consisting of equally sized isosceles right triangles, as in Fig. \[fig:squaremesh\](a). Let $\mathcal{V}_h^+ \subset H^1_0((0,1) \times (0,1))$ be the space of piecewise affine functions on the same mesh, but with the node nearest to $(x,y)=(1/4,1/4)$ perturbed by $h/4$ in the positive $x$ direction, as in Fig. \[fig:squaremesh\](b). In this scenario, assumption (\[assumption2b\]) is satisfied with $\gamma=2$. Let $u(x) = \sin(\pi x)\sin(\pi y)$ and let $$a_h^+(u,w)=a_h(u,w) = \int_0^1\int_0^1 uw \, dx dy,$$ so that $r_h$ and $r_h^+$ are the $L^2$-projectors onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively.
Table \[tab:L2proj2d\] shows the $L^2$-norm of the difference $r_h^+ u - r_h u$ for several values of $h$, beginning with $h = \sqrt{2}/4 =: h_0$. The table illustrates the predictions of Corollary \[corollary1\], namely $$\label{est1}
\|r_h^+ u - r_h u\|_{0,2} \le C h^3 |u|_{2,\infty}.$$
Next, consider the same setup as above, but with $$a_h^+(u,w)=a_h(u,w) = \int_0^1\int_0^1 \left( \frac{\partial u}{\partial x} \frac{\partial w}{\partial x} + \frac{\partial u}{\partial y} \frac{\partial w}{\partial y} \right) \, dx dy,$$ so that $r_h$ and $r_h^+$ are the elliptic projectors onto $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively. Table \[tab:ellipticproj2d\] shows the $H^1$- and $L^2$-norms of the difference $r_h^+ u - r_h u$ for the sequence of meshes described above. The table illustrates the predictions of Corollaries \[corollary1\] and \[corollary2\], namely $$\label{est2}
\|r_h^+ u - r_h u\|_{m,2} \le
\begin{cases}
C h^2 \log(h^{-1}) |u|_{2,\infty} &\mbox{if } m=0, \\
C h^3 \log(h^{-1}) |u|_{2,\infty} &\mbox{if } m=1.
\end{cases}$$ Again, we have not attempted to detect the presence of the factor $\log(h^{-1})$.
[165pt]{}[r|cc]{} &\
$h_0/h$ & &\
1 & 2.2504e-02 &\
2 & 4.8445e-03 & 2.2158\
4 & 1.0019e-03 & 2.2736\
8 & 1.9159e-04 & 2.3866\
16 & 3.5132e-05 & 2.4472\
32 & 6.3195e-06 & 2.4749\
\[tab:ellipticproj2dgamma1\]
[295pt]{}[r|cc|cc]{} &\
$h_0/h$ & & & &\
1 & 5.4318e-01 & & 1.9864e-02 &\
2 & 2.8504e-01 & 0.9303 & 4.8794e-03 & 2.0254\
4 & 1.2522e-01 & 1.1867 & 1.0528e-03 & 2.2125\
8 & 4.8674e-02 & 1.3632 & 1.9842e-04 & 2.4075\
16 & 1.7931e-02 & 1.4407 & 3.5671e-05 & 2.4758\
32 & 6.4595e-03 & 1.4730 & 6.3290e-06 & 2.4947\
#### More substantial mesh perturbation in two dimensions.
Finally, consider the same two-dimensional tests as above, but with the mesh of Fig. \[fig:squaremesh\](b) replaced by a different perturbation of the uniform mesh. Namely, consider perturbing all nodes whose distance from the boundary of the unit square is equal to $h/\sqrt{2}$ (the length of the shortest edge of each triangle) via a translation by $h/4$ in the positive $x$ direction, as in Fig. \[fig:squaremesh\](c).
In this scenario, assumption (\[assumption2b\]) is satisfied with $\gamma=1$, so that the estimates (\[est1\]) and (\[est2\]) no longer apply. Their analogues in this case read $$\|r_h^+ u - r_h u\|_{0,2} \le C h^{5/2} |u|_{2,\infty}.$$ and $$\|r_h^+ u - r_h u\|_{m,2} \le
\begin{cases}
C h^{3/2} \log(h^{-1}) |u|_{2,\infty} &\mbox{if } m=0, \\
C h^{5/2} \log(h^{-1}) |u|_{2,\infty} &\mbox{if } m=1,
\end{cases}$$ respectively. Tables \[tab:L2proj2dgamma1\]-\[tab:ellipticproj2dgamma1\] illustrate these predictions. Again, we have not attempted to detect the presence of the factor $\log(h^{-1})$.
Summary {#sec:conclusion}
=======
We have derived estimates for the difference between the orthogonal projections $r_h u$ and $r_h^+ u$ of a smooth function $u$ onto nearby finite element spaces $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively, with respect to bilinear forms $a_h, a_h^+ : \mathcal{V} \times \mathcal{V} \rightarrow \mathbb{R}$, respectively, where $\mathcal{V}$ is a closed subspace of $H^s(\Omega)$. When $s \in \{0,1\}$ and $\mathcal{V}_h$ and $\mathcal{V}_h^+$ consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure $O(h^\gamma)$ for a constant $\gamma \ge 0$, the estimates for $\|r_h^+ u - r_h u\|_{s,2}$ are superconvergent by $O(h^{\gamma/2})$, provided that $u \in W^{s,\infty}(\Omega)$ and $a_h$ and $a_h^+$ are sufficiently close. In addition, when $s=1$ and a few more mild assumptions (namely (\[cond:smoothing\]-\[cond:interp\_coincide\])) are satisfied, an $O(h^{\gamma/2})$-superconvergent estimate for $\|r_h^+ u - r_h u\|_{0,2}$ holds. Numerical experiments illustrated these estimates and verified the necessity of the regularity assumptions on $u$.
Acknowledgments
===============
This research was supported by the U.S. Department of Energy, grant number DE-FG02-97ER25308; Department of the Army Research Grant, grant number: W911NF-07- 2-0027; NSF Career Award, grant number: CMMI-0747089; and NSF, grant number CMMI-1301396.
Properties of Piecewise Polynomial Finite Element Spaces {#sec:appendix_Pk}
========================================================
In this section, we verify conditions (\[Wseta\]-\[assumption2b\]) for piecewise polynomial finite element spaces on nearby meshes for the cases $s=0$ and $s=1$.
As in Section \[sec:intro\], consider two families of shape-regular, quasi-uniform meshes $\{\mathcal{T}_h\}_{h \le h_0}$ and $\{\mathcal{T}_h^+\}_{h \le h_0}$ of an open, bounded, Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \ge 1$. Assume that the two families are parametrized by a scalar $h$ that equals the maximum diameter of an element among all elements of $\mathcal{T}_h$ and $\mathcal{T}_h^+$ for every $h \le h_0$. Let $\mathcal{V}_h$ and $\mathcal{V}_h^+$ be finite element spaces consisting of continuous functions that are elementwise polynomials of degree at most $r-1$ over $\mathcal{T}_h$ and $\mathcal{T}_h^+$, respectively, where $r>1$ is an integer.
In this setting, condition (\[Wseta\]) is automatic for any $\eta \in [2,\infty]$, $s \in \{0,1\}$. Condition (\[inverse\]) is trivial for $s=0$ and is satisfied for $s=1$ and any $\eta \in [2,\infty]$ [@Ern2004].
Condition (\[assumption2b\]) holds for any $\eta \in [2,\infty]$ when $\mathcal{T}_h$ and $\mathcal{T}_h^+$ coincide except on a region of measure $O(h^\gamma)$. To prove this, let $\{N_a\}_{a=1}^A \subset \mathcal{V}_h$ and $\{N_a^+\}_{a=1}^{A^+} \subset \mathcal{V}_h^+$ be the standard Lagrange shape functions that form bases for $\mathcal{V}_h$ and $\mathcal{V}_h^+$, respectively. Our assumptions on $\mathcal{T}_h$ and $\mathcal{T}_h^+$ imply the existence of an integer $I$ such that $N_a = N_a^+$ for every $1 \le a \le I$ and such that $$\left| \left( \displaystyle\bigcup_{a=I+1}^A \mathrm{supp}(N_a) \right) \cup \left( \displaystyle\bigcup_{a=I+1}^{A^+} \mathrm{supp}(N_a^+) \right) \right| \le C h^\gamma$$ for every $h \le h_0$.
Define $\pi_h : \mathcal{V}_h^+ + \mathcal{V}_h \rightarrow \mathcal{V}_h^+ \cap \mathcal{V}_h$ as follows: For any $$\label{wh_expansion}
w_h = \sum_{a=1}^I c_a N_a + \sum_{a=I+1}^A c_a N_a + \sum_{a=I+1}^{A^+} c_a^+ N_a^+$$ belonging to $\mathcal{V}_h^+ + \mathcal{V}_h$, set $$\label{pih_def}
\pi_h w_h := \sum_{a=1}^I c_a N_a.$$
Clearly, $$|\mathrm{supp}(\pi_h w_h - w_h)| \le C h^\gamma$$ for every $w_h \in \mathcal{V}_h^+ + \mathcal{V}_h$ and every $h \le h_0$. To prove that $$\label{pih_stability}
\| \pi_h w_h \|_{0,\eta} \le C \|w_h\|_{0,\eta}$$ for every $w_h \in \mathcal{V}_h^+ + \mathcal{V}_h$ and every $h \le h_0$, there are two cases to consider: $\eta = \infty$ and $2 \le \eta < \infty$.
For $\eta = \infty$, it is enough to note that for each of the two finite element spaces, every shape function is bounded uniformly in $h$ in the maximum norm, the number of shape functions whose support intersects any given element is bounded uniformly in $h$, and the coefficients $c_a$, $1 \le a \le I$, in the expansion (\[wh\_expansion\]) of $w_h$ are bounded by $\|w_h\|_{0,\infty}$. Indeed, the standard degrees of freedom $\sigma_a$, $1 \le a \le I$, for the Lagrange shape functions $N_a (=N_a^+)$, $1 \le a \le I$, satisfy $$\sigma_a(N_b) = \delta_{ab}, \;\;\; 1 \le b \le A$$ and $$\sigma_a(N_b^+) = \delta_{ab}, \;\;\; 1 \le b \le A^+,$$ where $\delta_{ab}$ denotes the Kronecker delta. Hence, for any $1 \le a \le I$, $$|c_a| = |\sigma_a(w_h)| \le \|w_h\|_{0,\infty}.$$
For $2 \le \eta < \infty$, the proof of (\[pih\_stability\]) relies on the following lemma.
\[lemma:norms\] Let $\{\mathcal{T}_h\}_{h \le h_0}$ be a shape-regular, quasi-uniform family of meshes of an open, bounded, Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \ge 1$, with $h$ denoting the maximum diameter of an element $K \in \mathcal{T}_h$. Let $r>1$ be an integer. For any $K \in \mathcal{T}_h$, let $\theta_1,\theta_2,\dots,\theta_{n_{sh}}$ denote the local shape functions for the Lagrange finite element of degree at most $r-1$ on $K$. Then for any $2 \le \eta < \infty$, there exist $C_1,C_2 > 0$ independent of $h$ such that for every $h \le h_0$, every $K \in \mathcal{T}_h$, and every $v = \sum_{i=1}^{n_{sh}} d_i \theta_i$, $$C_1 h^d \sum_{i=1}^{n_{sh}} |d_i|^\eta \le \|v\|_{0,\eta,K}^\eta \le C_2 h^d \sum_{i=1}^{n_{sh}} |d_i|^\eta.$$
A proof of this fact when $\eta=2$ is given in [@Ern2004 Lemma 9.7]. The case $2 < \eta < \infty$ is a trivial modification thereof.
Now let $w_h$ and $\pi_h w_h$ be as in (\[wh\_expansion\]) and (\[pih\_def\]), respectively. Note that the support of $\pi_h w_h$ is contained within the region $Q_h \subseteq \Omega$ over which $\mathcal{T}_h$ and $\mathcal{T}_h^+$ coincide. On any $K \in \mathcal{T}_h$ with $K \subseteq Q_h$, we can write $$\left.w_h\right|_K = \sum_{i=1}^{n_{sh}} d_i \theta_i$$ and $$\left.\pi_h w_h\right|_K = \sum_{i=1}^{n_{sh}} \bar{d}_i \theta_i,$$ with scalars $d_i \in \mathbb{R}$ and $\bar{d}_i \in \{0,d_i\}$ for every $i$. By Lemma \[lemma:norms\], $$\begin{aligned}
\| \pi_h w_h \|_{0,\eta,K}^\eta
&\le C_2 h^d \sum_{i=1}^{n_{sh}} |\bar{d}_i|^\eta \\
&\le C_2 h^d \sum_{i=1}^{n_{sh}} |d_i|^\eta \\
&\le C_2 C_1^{-1} \| w_h \|_{0,\eta,K}^\eta\end{aligned}$$ on every such $K$. Summing over all $K \in \mathcal{T}_h$ with $K \subseteq Q_h$ proves (\[pih\_stability\]) for $2 \le \eta < \infty$.
Estimates for the $L^2$-Projection and Elliptic Projections {#sec:appendix_Linf}
===========================================================
Two exemplary cases in which estimates of the form (\[ep0eta\]-\[epmeta\]) are known to hold are the following. Suppose that $\mathcal{V} = H^s(\Omega) \cap H^1_0(\Omega)$ and $\mathcal{V}_h$ is the space of continuous functions in $\mathcal{V}$ that are elementwise polynomials of degree at most $r-1$ on a shape-regular, quasi-uniform family of meshes $\{\mathcal{T}_h\}_{h \le h_0}$ whose maximum element diameter is $h$. Then:
1. If $s=0$, $d \in \{1,2\}$, and $$a_h(u,w) = \int_\Omega uw \, dx$$ so that $r_h$ is the $L^2$-projector onto $\mathcal{V}_h$, then (\[ep0eta\]) holds with $\ell(h) \equiv 1$ for any $\eta \in [2,\infty]$ [@Crouzeix1987]. Note that the estimate (\[epmeta\]) is vacuous in this case, since $s=0$.
2. If $s=1$, $d \in \{2,3\}$, and $$a_h(u,w) = \int_{\Omega} \left(\sum_{i,j=1}^d a_{ij}(x) \frac{\partial u}{\partial x_i} \frac{\partial w}{\partial x_j} + \sum_{j=1}^d b_j(x) \frac{\partial u}{\partial x_j} w + b_0(x) u w\right) dx$$ with $h$-independent coefficients $a_{ij}$, $i,j=1,2,\dots,d$ and $b_j$, $j=0,1,\dots,d$, then (\[ep0eta\]-\[epmeta\]) hold [@Ern2004] with $\ell(h) \equiv 1$ for any $2 \le \eta < \infty$ (if $r=2$) and any $\eta \in [2,\infty]$ (if $r>2$), provided that
- The coefficients satisfy $b_j \in L^\infty(\Omega)$, $j=0,1,\dots,d$, and $a_{ij} \in L^\infty(\Omega) \cap W^{1,p}(\Omega)$, $i,j=1,2,\dots,d$, with $p>2$ if $d=2$ and $p \ge 12/15$ if $d=3$.
- The coefficients $a_{ij}$ are coercive pointwise, i.e. there exists $c>0$ independent of $x$ such that $$\label{pointwise_coercive}
\sum_{i,j=1}^d a_{ij}(x) \xi_i \xi_j \ge c |\xi|^2$$ for every $0 \ne \xi \in \mathbb{R}^d$ and a.e. $x \in \Omega$.
- There exists $C>0$, $q_0 > d$ such that the continuous Dirichlet problem $$a_h(u,w) = \int_\Omega f w \, dx \;\;\; \forall w \in \mathcal{V}$$ has a unique solution satisfying $$\label{dirichlet_bound}
\|u\|_{2,q} \le C \|f\|_{0,q}$$ for every $f \in L^p(\Omega)$ and every $1 < q < q_0$.
Under the same conditions as above but with $r=2$ and $\eta=\infty$, the estimates (\[ep0eta\]-\[epmeta\]) hold with $\ell(h) = \log(h^{-1})$ in dimension $d=2$ [@Ern2004].
|
---
abstract: 'To understand the factors that encourage the deployment of a new networking technology, we must be able to model how such technology gets deployed. We investigate how network structure influences deployment with a simple deployment model and different network models through computer simulations. The results indicate that a realistic model of networking technology deployment should take network structure into account.'
author:
- Yoo Chung
bibliography:
- 'strings.bib'
- 'articles.bib'
- 'web.bib'
- 'proceedings.bib'
title: Modeling network technology deployment rates with different network models
---
Introduction {#sec:introduction}
============
On January 31, 2011, the Internet Assigned Numbers Authority allocated the last free blocks of IPv4 addresses to the five Regional Internet Registries [@w:ipv4-free-pool-depleted]. It will not be long before even the Regional Internet Registries run out of their own IPv4 addresses to give out. This is the sort of news which gives urgency to the need for IPv6 to replace IPv4 for its greatly expanded address space. And yet IPv6 deployment rates, while they are steadily climbing, remain extremely low [@karpilovsky:pam2009; @colitti:pam2010].
Looking to the distant future, clean-slate design approaches are being proposed to solve problems that seem to be insurmountable or too messy to solve with the design of the Internet today [@feldmann:sigcomm2007; @roberts:aot2009]. Perhaps the technical community may go further than just using a clean-slate design approach for generating new ideas and converge on a clean-slate design for a new Internet that can solve all of the problems with the present Internet. But if IPv6 can hardly get deployed even when there is a clear perceived need for it today, how can we expect this clean-slate Internet to replace the by-then obsolete Internet?
In order to figure out how to design new networking technologies so that they are quickly adopted, we need to understand the factors that are behind the spread of such technologies. To understand these factors, we need to be able to model the deployment of these technologies to a reasonable degree of accuracy.
This work does not attempt to define a comprehensive model for the deployment of new networking technologies, nor does it try to suggest what policies or design principles may promote the deployment of such technologies. Instead, this work focuses on how different *network structure* may influence the spread of a new networking technology throughout a network, an understanding of which may be important in the design of a realistic model. Using computer simulations, we investigate how different network models influence deployment rates. Other work has focused on the deployment of specific networking technologies [@chan:sigcomm2006] or on the role of “converters” which provide compatibility with an old technology in the deployment of a new technology [@joseph:conext2007; @sen:ton2010].
The rest of the paper is structured as follows. Section \[sec:infection\] describes the deployment model used in the computer simulations, which models how each node in the network decides to make the transition to the new networking technology. Section \[sec:networks\] describes the different network models we looked at and shows the growth curves of how a new networking technology spreads with each network model. We end the paper in section \[sec:discussion\], where we discuss how the models might apply to a real world system, the limitations of the modeling done here, and some concluding remarks.
Deployment model {#sec:infection}
================
There are a myriad of factors that affect whether a given node in a network decides to adopt a new networking technology, which makes modeling its spread a very difficult problem. Our focus here is on how different network models can influence the spread of a new networking technology, however, so we use a very simple deployment model as applied to networks of fixed size.
We also assume that once a node decides to adopt a new networking technology, it will not want to revert to the older technology. In fact, we assume that there is only one old networking technology and one new networking technology. To avoid dealing with the influence of converters which provide compatibility between the old and new technologies, we assume a dual-use situation, where nodes with the new technology can continue to use the old technology.
We model the spreading deployment in a network on a fixed graph $G = (V, E)$ representing a network, where $V$ represents the set of nodes and $E$ represents the set of edges between the nodes. The state of deployment throughout the network changes in one discrete time step, where each node that has not already transitioned to the new networking technology decides whether to transition or not. We define $D(t)$ to be the set of nodes that have transitioned to the new networking technology so far at time step $t$.
How does a node decide whether to transition or not? A node would be more inclined to transition if more of its neighbors have already transitioned, since this would allow it to take increasing advantage of the new networking technology to communicate with others. Contrast this to the case when none of its neighbors have transitioned, in which case the old networking technology would have to be used to communicate with any other node in the network. A concrete example in the real world would be a single host supporting IPv6 inside an IPv4 subnet, in which case the node cannot use IPv6 to communicate with nodes outside the subnet, ignoring tunneling to simplify matters.
A node would also more inclined to transition if lots of nodes in the network as a whole have already transitioned, sort of not wanting to “fall behind the times”. It is unclear how this combines with the previous effect to influence the decision of a node on whether to transition or not, but we will model it as multiplicative rather than additive. This would have the effect of a node being much more inclined to transition if its neighborhood has already transitioned, given the same general transition rate for the entire network.
On the other hand, transitioning to a new technology is never free, and the cost associated with a transition would discourage a node from making the transition. We model the cost differently for each network model in section \[sec:networks\] to reflect what the nodes and edges represent in the corresponding model, although we do make the simplifying assumption that the costs do not change over time.
Given these factors, we define a utility value $u(v,t)$ which represents how inclined a vertex $v$ is towards making a transition at time step $t$: $$u(v,t) = \frac{1}{\alpha} \left( h_G(t) h(v, t) - c(v) \right)$$ where $h_G(t)$ is the total number of transitioned nodes in the network, $h(v,t)$ is the number of nodes adjacent to $v$ that have already transitioned, $c(v)$ is a cost function that is specific to each network model, and $\alpha$ is a scaling factor that is used later: $$h_G(t) = |D(t)|$$ $$h(v, t) =| \{ v' \;|\; (v,v') \in E \ \wedge \ v' \in D(t) \} |$$
![The logistic function, a type of sigmoid function.[]{data-label="fig:sigmoid"}](sigmoid){width="8cm"}
However, a node does not decide whether to transition or not based on whether the utility value is positive or not. We take advantage of the logistic function, which is graphed out in \[fig:sigmoid\], to represent the probability that a vertex $v$ will transition or not at time step $t$: $$p(v,t) = \frac{1}{1+e^{-u(v,t) + \beta}}$$
Taking advantage of the logistic function ensures that $p(v,t)$ lies between 0 and 1 while having increasing value with increasing utility value. $\beta$ is a baseline that shifts the logistic function to the right, so that a zero utility value will have low probability. The scaling factor $\alpha$ in the definition of the utility value is used to ensure that most of the probabilities lie on the center slope rather than near 0 or 1. Obviously, $u(v,t)$ and $p(v,t)$ do not apply to nodes that have already transitioned, since we assume that nodes do not revert once they make the transition.
The deployment model described in this section is a very simple model of how nodes in a network decide to adopt a new networking technology, but hopefully it captures enough that we can glean insights to how new networking technologies spread given different network models.
Network models {#sec:networks}
==============
In this section, we investigate how different network models influence the deployment of a new networking technology. The deployment model of section \[sec:infection\] is applied to different network models using computer simulations, where we plot out the number of nodes which have made the transition as time goes by.
In the rest of the section, all networks have exactly 10000 nodes. All of the cases start out with exactly one node having already made the transition, and in most cases we run the simulation until 99% of the nodes have made the transition. We fixed $\beta=3$, but the scaling factor $\alpha$ was adjusted for each model to prevent the transition probabilities $p(v,t)$ from clustering around $\frac{1}{1+e^\beta} \approx 0.047$, which would collapse a growth curve to the uninteresting case of section \[sec:independent\].
Because the constants used in the utility functions and transition probabilities are somewhat arbitrary, the number of steps that a network model uses to attain a certain deployment rate is rather arbitrary as well. This is why we will ignore the absolute speed at which different network models saturate, and instead focus on the features of each deployment growth curve.
Independent {#sec:independent}
-----------
Before investigating the other network models, we take a look at how a new networking technology would spread if every node decided to make the transition independent of whatever the state of the rest of the network might be. In other words, we make the exception for the transition probability in the deployment model so that: $$p(v,t) = \gamma$$
![Deployment growth with independent deployment.[]{data-label="fig:independent-chart"}](independent-chart){width="10cm"}
With $\gamma = 0.05$, the growth is shown in \[fig:independent-chart\], which is unsurprisingly an exponential curve upside down. This case is included as a comparison for what happens when “network effects”, the influence of the rest of the network on a single node, are ignored.
Clique {#sec:clique}
------
As an example of a simple network model that still includes “network effects”, we model the network as a clique, where every node is connected to every other node. Because the number of transitioned nodes in the entire network is identical to the number of transitioned nodes adjacent to a node that has yet to make the transition, this is basically the case where all of the perceived benefit is due to how many nodes have made the transition globally. Because every node is basically the same as every other node, we use a constant cost model:
$$c(v) = \gamma$$
![Deployment growth with clique model.[]{data-label="fig:clique-chart"}](clique-chart){width="10cm"}
With $\alpha = \gamma = 1.25 \times 10^7$, the growth for the clique model is shown in \[fig:clique-chart\]. Deployment steadily grows until it rises exponentially and quickly saturates the network. This roughly replicates the “S curve” that would be expected from technology deployment growth.
Random graph {#sec:random}
------------
As a slightly more realistic network model than the clique model of section \[sec:clique\], we apply the deployment model of section \[sec:infection\] to the Erdős-Rényi model [@gilbert:aoms1959] of random graphs. In this model, every potential edge between nodes in a graph are actualized with probability $p$, which should not be confused with the transition probability $p(v,t)$. Basically, nodes in the network are randomly connected to each other.
As for the cost model, we assume that it is more expensive to make the transition for nodes that are linked to many nodes. In addition, we assume that there is a fixed cost for making the transition itself. Thus we model the cost as follows, where $d(v)$ is the number of edges adjacent to $v$:
$$c(v) = \gamma (1 + d(v))$$
![Deployment growth with random graph model.[]{data-label="fig:random-chart"}](random-chart){width="10cm"}
With $p = 0.001$, $\alpha = 10000$, and $\gamma = 1562.5$, the growth for the random graph model is shown in \[fig:random-chart\]. An alternate way to interpret the Erdős-Rényi model is that edges are randomly removed from a clique, so it should be no surprise that \[fig:random-chart\] looks like \[fig:clique-chart\] for the clique model, despite the difference in the cost model. As in the clique model, deployment steadily grows until it jumps exponentially and quickly saturates the network.
Preferential attachment {#sec:preferential}
-----------------------
One possible scenario for how new networking technologies are adopted is that an organization upgrades everything at once, rather than replacing old equipment and software incrementally. In such a scenario, an organization may be more inclined to make the transition to new networking technology if other organizations it directly communicates with have also made the transition. The organization would also be more inclined to make the transition if the rest of the world has taken up the new technology. For example, an ISP may be more inclined to take up IPv6 if most of its peer ISPs have already transitioned to IPv6, and even more so if the rest of the Internet has also taken up IPv6.
In such a scenario, the basic entity that makes a transition would be an organization, and each organization would be influenced by other organizations it has relationships with. In other words, the network model would be a social network model where vertexes are organizations and edges are relationships. We will use the Barabási-Albert model [@barabasi:science1999] of preferential attachment to model this sort of social network. In this model, nodes with larger number of neighbors are more likely to collect even more neighbors.
A major organization that has more relationships with other organizations is also likely to have more equipment and personnel, which would raise the cost of making a transition. Thus we model the transition cost as follows, which also includes a fixed cost:
$$c(v) = \gamma (1 + d(v))$$
![Deployment growth with preferential attachment model.[]{data-label="fig:preferential-chart"}](preferential-chart){width="10cm"}
With an initial network of 100 nodes connected as a ring, $\alpha = 3333$, and $\gamma = 2500$, the deployment growth for the preferential attachment model is shown in \[fig:preferential-chart\]. Superficially, the curve looks like those for the clique and random graph models as shown in \[fig:clique-chart\] and \[fig:random-chart\], respectively.
A closer look reveals a significant difference, however. Whereas deployment increases initially with a steady growth rate in \[fig:clique-chart\] and \[fig:random-chart\], growth slightly *flattens* in \[fig:preferential-chart\] before it skyrockets. The deployment rate in a preferential attachment model does not quite reflect the traditional “S curve” ascribed to technology adoption. It is not certain why there is a flattening of the growth rate until a critical mass is reached, but it does suggest that caution should be exercised if one attempts to estimate how fast a new networking technology is being widely deployed by fitting a partial growth curve to an “S curve”.
Binary tree {#sec:hierarchical}
-----------
Instead of an organization deciding to make the transition to a new networking technology all at once, another scenario would be each piece of equipment being replaced or upgraded one by one. The decision to upgrade a piece of networking equipment would be influenced by whether neighboring networking equipment has already made the transition, not to mention an industry-wide trend of adopting the new technology. In this scenario, actual networking hardware would be the vertexes and the communication links between them would be the edges. A concrete example would be the router-level graph of the Internet.
The router-level graph of the Internet is built up to maximize their utility within the constraints enforced by the hardware technology that is available, which results in a network model that is very unlike a scale-free network constructed through preferential attachment [@alderson:ton2005]. One of the features which makes them very different is that the routers that form the core of the Internet are typically very high bandwidth, low-degree nodes in the network, since supporting very high bandwidth links makes it extremely difficult to support more than a few links, while the routers with high degree are typically located at the fringe of the network, connecting low-bandwidth end hosts to the rest of the Internet.
While we do not try to realistically model the router-level graph of the Internet here, we will try to model one aspect of the graph: routers near the core are typically a lot more expensive than those at the fringe since they must handle much higher bandwidths. The network model itself that we use is a very simple binary tree, while the cost to transition a node grows exponentially the nearer it is to the root of the tree. The cost $c(v)$ is defined as follows, where $l(v)$ is the depth of vertex $v$ in the binary tree:
$$c(v) = \gamma \, 2^{-l(v)}$$
![Deployment growth with a binary tree model.[]{data-label="fig:hierarchical-chart"}](hierarchical-chart){width="10cm"}
With $\alpha = 312.5$ and $\gamma =2000000$, the growth for the binary tree model is shown in \[fig:hierarchical-chart\]. Deployment spreads exponentially at first and then slows down, saturating the network slowly. This looks more like the traditional “S curve” compared to the other models. The slow saturation at the end seems to be an innate feature of the binary tree model, not an artifact of the constants that we used in the simulation. In fact, a larger scaling factor $\alpha$ shifts the curve generally to the left, which is basically slowing down the saturation even more.
![Deployment growth with a binary tree model using a smaller scale factor.[]{data-label="fig:hierarchical-chart-scale128"}](hierarchical-chart-scale128){width="10cm"}
Even more interesting behavior occurs when we use smaller scaling factors. When we set the scaling factor to $\alpha = 78$, we get growth as shown in \[fig:hierarchical-chart-scale128\]. Instead of a single S-like curve, we get a double-S curve, almost as if there are two growth spurts occuring sequentially. The smaller scaling factor increases the sensitivity of the deployment model to benefits and costs, so it may be the case that the high transition cost of nodes near the root may delay the deployment in a large subtree in the network, and once the root of this subtree makes its own transition, the rest of the subtree quickly follows. This could explain the double bursts of growth seen in \[fig:hierarchical-chart-scale128\].
![Deployment growth with a binary tree model using a very small scale factor.[]{data-label="fig:hierarchical-chart-scale256"}](hierarchical-chart-scale256){width="10cm"}
Yet smaller scaling factors, which is equivalent to an increased cost-benefit sensitivity, gives rise to even more complex growth curves. When we set the scaling factor to $\alpha = 39$, we get the growth curve shown in \[fig:hierarchical-chart-scale256\], where we can see many growth bursts occuring throughout time. The curve shows characteristics of a self-similar curve, which is made clearer when we zoom into a small portion of the curve as in \[fig:hierarchical-chart-scale256-zoom\].
![A zoomed view of \[fig:hierarchical-chart-scale256\].[]{data-label="fig:hierarchical-chart-scale256-zoom"}](hierarchical-chart-scale256-zoom){width="10cm"}
While the binary tree model is still an unrealistic model for real networks, the unexpected behavior in the growth curve shows how different network and cost models can result in qualitatively different behavior in the spread of new networking technologies.
Discussion {#sec:discussion}
==========
Applicability to real networks {#sec:applicability}
------------------------------
One obvious real-world network we may want to compare the network models in section \[sec:networks\] with is the spread of IPv6 in a world of IPv4. Unfortunately, this is difficult not only because IPv6 deployment is still in its initial stages, but also because actually measuring how widely IPv6 is deployed is a very hard problem. Still, we will briefly discuss how the models compare with the growth of IPv6 address block allocations to the Regional Internet Registries, which can be found in 2 of [@karpilovsky:pam2009].
If one were to look at 2 of [@karpilovsky:pam2009], one might notice that the growth curves for RIPENCC and APNIC are vaguely similar to the growth curve for the preferential attachment model in \[fig:preferential-chart\]. They start off with the growth rate slowly flattening until they begin to rise again. The growth for the other regions look more like the other models, although there is not enough data to see if the growth curves look more like one model rather than another.
For RIPENCC and APNIC, the similarity of the growth curves to \[fig:preferential-chart\] might suggest that organizations are transitioning all at once to IPv6 rather than incrementally, at least in terms of allocating IPv6 address blocks, and that they are influenced by other organizations having allocated IPv6 address blocks of their own. Or perhaps the similarity is just a coincidence, the initial growth spurt being due to a clarification of allocation policy which gradually tapered off as suggested in [@karpilovsky:pam2009], and the later rise in growth rates being due to the impending depletion of IPv4 addresses.
Regardless, the similarities are a tantalizing hint that network structure can indeed influence the spread of a new networking technology in a real world situation, and in fact, may be an indication of the decision processes by which organizations determine their plans concerning the deployment of IPv6. This may be an issue that deserves a closer look.
Limitations {#sec:limitations}
-----------
Both the deployment model in section \[sec:infection\] and network models in section \[sec:networks\] are very limited in their realism. In the following, we discuss what those limitations are, which also point to directions for further research.
1. Networks grow. We have assumed networks of fixed size, which does not reflect the reality that even incumbent networks such as IPv4 are growing. Similarly, the way we model deployment would be like embedding an IPv6 network inside a fixed IPv4 network, and technically, an IPv6 network is entirely separate from an IPv4 network. Dual-stack IPv6/IPv4 deployments may be the rule while IPv4 dominates, but this is unlikely to continue to be the case once IPv6 becomes widespread.
2. The decision process for whether a node will make the transition or not should be modeled more realistically. There will be many more factors that influence the decision process. For example, the increasing scarcity of IPv4 addresses along with growth would presumably encourage the deployment of IPv6. In addition, constants used in an ideal deployment should be based on measurable values that are dependent on the underlying factors, rather than being derived by fitting to a curve.
3. The network models should be more realistic. For example, we may look at actual peering relationships between ISPs to model the relationships between organization. We could also use more realistic models of communication networks such as heuristically optimal topologies [@alderson:ton2005].
4. A look at how new networking technologies spread throughout a network. We have only looked at the total number of deployments in a network, and seeing which nodes actually make the transition as time passes would undoubtedly give rise to new insights. Visualizing this in a large network is likely to be a challenging problem.
5. Actual data to validate models would be very useful. As can be seen in the measurement of IPv6 deployment rates, defining what exactly a deployment means can be ambiguous, and actually measuring the deployment rate can be very difficult even when it is defined.
Concluding remarks {#sec:conclude}
------------------
With a simple model for how new networking technologies get deployed throughout a network and applying it to different network models, we have seen how different network structures and cost models can change the qualitative behavior of how a new networking technology spreads. While the research is in its infancy such that we are not able to make predictions or derive insights from existing data with any certainty, we can make the observation that any realistic model of how new network technologies get deployed will have to account for network structure and related cost models.
|
---
abstract: 'In two recent papers the mesoscale model Meso-NH, joint with the Astro-Meso-NH package, has been validated at Dome C, Antarctica, for the characterization of the optical turbulence. It has been shown that the meteorological parameters (temperature and wind speed, from which the optical turbulence depends on) as well as the 2 profiles above Dome C were correctly statistically reproduced. The three most important derived parameters that characterize the optical turbulence above the internal antarctic plateau: the surface layer thickness, the seeing in the free-atmosphere and in the total atmosphere showed to be in a very good agreement with observations. Validation of 2 has been performed using all the measurements of the optical turbulence vertical distribution obtained in winter so far. In this paper, in order to investigate the ability of the model to discriminate between different turbulence conditions for site testing, we extend the study to two other potential astronomical sites in Antarctica: Dome A and South Pole, which we expect to be characterized by different turbulence conditions. The optical turbulence has been calculated above these two sites for the same 15 nights studied for Dome C and a comparison between the three sites has been performed.'
author:
- |
F. Lascaux,$^{1}$[^1] E. Masciadri$^1$, and S. Hagelin$^{1, 2}$\
$^1$INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, I-501 25 Florence, Italy\
$^2$Uppsala Universitet, Department of Earth Sciences, Villavägen 16, S-752 36 Uppsala, Sweden
date: 'Accepted 2010 ??? ??, Received 2010 ??? ??; in original form 2010 ??? ??'
title: 'Mesoscale optical turbulence simulations above Dome C, Dome A and South Pole'
---
2
\[firstpage\]
site testing – atmospheric effects – turbulence
Introduction
============
The Internal Antarctic Plateau represents a potential interesting location for astronomical applications. For almost a decade astronomers have shown more and more interest towards this region of the Earth thanks to its peculiar atmospheric conditions. The extreme cold temperature, the dry atmosphere, the fact that the plateau is at more than 2500 m above the sea level, that the turbulence seems to develop mainly in a thin surface layer of the order of 30-40 m on the top of summits and that the seeing above this surface layer assumes values comparable to those obtained at mid-latitude sites get this region of the earth very appealing for astronomers. South Pole has been the first site equipped with an Observatory in the Internal Antarctic Plateau in which measurements of the optical turbulence have been done (Marks et al. 1996, Marks et al. 1999). Fifteen balloons have been launched in the winter period and it has been observed that the seeing above a surface layer of $\sim$ 220 m was very good (0.37 arcsec). Measurements of the optical turbulence at Dome C are more recent. After the first observations done in 2004 with a MASS (Lawrence et al. 2004), a series of studies done with different instrumentation have been published aiming to provide the assessment of the integrated seeing (Aristidi et al. 2005, Aristidi et al. 2009) and the vertical distribution of the optical turbulence (Trinquet et al. 2008).
-------------- ---------- ---------- --------------------- -------------------- ---------- --------------------- --------------------- ---------- ---------------------
h$_{sl}$ $\sigma$ $\sigma$/$\sqrt{N}$ $\varepsilon_{FA}$ $\sigma$ $\sigma$/$\sqrt{N}$ $\varepsilon_{TOT}$ $\sigma$ $\sigma$/$\sqrt{N}$
(m) (arcsec) (arcsec)
Observations 35.3 19.9 5.1 0.30 0.70 0.20 1.60 0.70 0.20
Model 44.2 24.6 6.6 0.30 0.67 0.17 1.70 0.77 0.21
-------------- ---------- ---------- --------------------- -------------------- ---------- --------------------- --------------------- ---------- ---------------------
\[tab\_sum\]
[|l|r|r|c|c|]{} SITE & LATITUDE & LONGITUDE & MESO-NH & MEASURED\
& & & ALTITUDE (m) & ALTITUDE (m)\
Dome A$^*$ & 80$^{\circ}$22’00“S &077$^{\circ}$21’11”E & 4089 & 4093\
Dome C$^{**}$ & 75$^{\circ}$06’04“S &123$^{\circ}$20’48”E & 3230 & 3233\
South Pole & 90$^{\circ}$00’00“S &000$^{\circ}$00’00”E & 2746 & 2835\
\
\
\[tab0\]
This paper deals with a different approach to the site assessment. In this context we are interested in investigating the abilities of a mesoscale model (Meso-NH) in reconstructing correct optical turbulence features above different sites of the Internal Antarctic Plateau and its abilities in discriminating the optical turbulence properties of different sites. Meso-NH (Lafore et al., 1998) is a non-hydrostatic mesoscale research model developed jointly by the Centre National des Recherches Météorologiques (CNRM) and the Laboratoire d’Aérologie de Toulouse, France. The Astro-Meso-NH package (Masciadri et al. 1999a) was first proven to be able to reconstruct realistic $\CN2$ profiles above astronomical sites by Masciadri et al. (1999b) and Masciadri et al. (2001) and statistically validated later on (Masciadri & Jabouille, 2001, Masciadri et al. 2004, Masciadri & Egner 2006). In the Astro-Meso-NH package all the main integrated astroclimatic parameters such as the isoplanatic angle, the wavefront coherence time, the scintillation rate, the spatial coherence outer scale are coded in the model (Masciadri et al. 1999a). The model is also coded to calculate the astroclimatic parameters in finite vertical slabs (h$_{min}$, h$_{max}$) in the troposphere (Masciadri et al. 1999a, Masciadri & Garfias 2001, Lascaux et al. 2010b). It can be therefore a useful tool for adaptive optics applications in classical as well as GLAO and/or MCAO configurations because we can produce OT vertical distribution in whatever vertical slab we wish and with the suited vertical resolution. More recently, Meso-NH has been statistically validated above Dome C by Lascaux et al. (2009, 2010a). The most important results obtained in these two last papers are summarized in Table \[tab\_sum\]. Briefly, the observations at Dome C, for a set of 15 winter nights (all the available nights for which is known the optical turbulence vertical distribution), gave a mean surface layer thickness $h_{sl,obs}$ = 35.3 $\pm$ 5.1 m. The simulated surface layer thickness obtained with the Meso-NH model ($h_{sl,mnh}$ = 44.2 $\pm$ 6.6 m) is well correlated to measurements. The statistical error is of the order of 5-6 m but the standard deviation ($\sigma$) is of the order of 20-25 m. This indicates that the statistic fluctuation of this parameter is intrinsically quite important. The median simulated free-atmosphere seeing ($\varepsilon_{mnh,FA}$ = 0.30 $\pm$ 0.17 arcsec) as well as the median total seeing ($\varepsilon_{mnh,TOT}$ = 1.70 $\pm$ 0.21 arcsec) are well correlated to observations, respectively $\varepsilon_{obs,FA}$ = 0.3 $\pm$ 0.2 arcsec and $\varepsilon_{obs,TOT}$ = 1.6 $\pm$ 0.2 arcsec.
In the context of this paper we consider that the Meso-NH model is calibrated as shown in Lascaux et al. (2010a) i.e. it produces optical turbulence features in agreement with observations. We therefore apply the Meso-NH model with the same configuration to other two sites of the plateau: South Pole and Dome A (Table \[tab0\]).
---------- ----------- ---------------- ------------------
Domain $\Delta$X Grid Points Surface
(km) (km$\times$km)
Domain 1 25 120$\times$120 3000$\times$3000
Domain 2 5 80$\times$80 400$\times$400
Domain 3 1 80$\times$80 80$\times$80
---------- ----------- ---------------- ------------------
: Meso-NH model configuration. In the second column the horizontal resolution $\Delta$X, in the third column the number of grid points and in the fourth column the horizontal surface covered by the model domain.
\[tab2\]
Why these sites ? Dome A is an almost uncontaminated site of the plateau. It is the highest summit of the plateau and, for this reason, it is expected to be among the best astronomical sites for astronomical applications. The high altitude reduces the whole atmospheric path for light coming from space and above the summit the katabatic wind speed is reduced to minima values. Dome A has been proved to have the strongest thermal stability (Hagelin et al. 2008) in proximity of the ground due to the coldest temperature. Dome A is a chinese base. In the last few years the chinese astronomers gave a great impulse to the site characterization showing a great interest for building astronomical facilities in this site. Optical turbulence measurements during the winter time are not yet available but site testing programs are on-going (Ashley et al. 2010). South Pole is interesting in our study because measurements of optical turbulence are available and, at the same time, the site is not located on a summit but on a gently slope. From the preliminary measurements done in the past we expect a surface turbulent layer that is thicker than the surface layer developed above the other two sites (Dome C and Dome A) due to the ground slope and the consequent katabatic winds in proximity of the surface. The three sites form therefore a perfect sample for a benchmark test on the model behavior and the model abilities.
In Section \[num\] the numerical set-up of the model is presented. In Section \[opt\] results of the complete analysis of the three major parameters that characterize the optical turbulence features: surface layer thickness, seeing in the free atmosphere i.e. calculated above the surface layer and total seeing are reported. Two different criteria to define the surface layer are used with consequent double treatment. Finally, in Section \[concl\] the results of this study are summarized.
Numerical set-up {#num}
================
Meso-NH [@laf] can simulate the temporal evolution of the three-dimensional atmospheric flow over any part the globe. The prognostic variables forecasted by this model are the three cartesian components of the wind $u$, $v$, $w$, the dry potential temperature $\Theta$, the pressure $P$, the turbulent kinetic energy $TKE$.
The system of equation is based upon an anelastic formulation allowing for an effective filtering of acoustic waves. A Gal-Chen and Sommerville@gcs coordinate on the vertical and a C-grid in the formulation of Arakawa and Messinger@am for the spatial digitalization is used. The temporal scheme is an explicit three-time-level leap-frog scheme with a time filter [@as]. The turbulent scheme is a one-dimensional 1.5 closure scheme [@cux] with the Bougeault and Lacarrère@bl mixing length. The surface exchanges are computed in an externalized surface scheme (SURFEX) including different physical packages, among which ISBA [@np] for vegetation. Masciadri et al. (1999a,b) implemented the optical turbulence package to be able to forecast also the optical turbulence ($C_N^2$ 3D maps) and all the astroclimatic parameters deduced from the $C_N^2$. We will refer to the ’Astro-Meso-NH code’ to indicate this package. The integrated astroclimatic parameters are calculated integrating the $C_N^2$ with respect to the zenith in the Astro-Meso-NH code. We list here the main characteristics of the numerical configuration used in this study:
- The interactive grid-nesting technique [@st] is used, with three imbricated domains of increased horizontal mesh-sizes ($\Delta$X=25 km, 5 km and 1 km, Table \[tab2\]). Such a method is used to permit us to achieve the best resolution on a small surface but keeping the volumetric domain in which the simulation is done in thermodynamic equilibrium with the atmospherical circulation that evolves at large spatial scale on larger domains. We shown [@lf09] that the simulations results are sensitive to the chosen horizontal resolution. To achieve a good correlation between model outputs and observations, a grid-nesting configuration with a high horizontal resolution (at least $\Delta$X=1 km) is mandatory.
- The vertical grid is the same for all the domains reported in Table \[tab2\]. The first vertical grid point is at 2 m above ground level (a.g.l.). A logarithmic stretched grid up to 3500 m a.g.l. (with 12 points in the first hundred of meters) is employed. Above 3500 m a.g.l., the vertical resolution is constant ($\Delta$H $\sim$ 600 m). The maximum altitude achieved is around 20 km a.g.l.. The first point at only 2 m above the ground (and with 12 points in the first hundred of meters) is necessary to forecast the typical very thin surface layer observed in the Antartic Plateau.
- All simulations are initialized and forced every 6 hours at synoptic times (00:00, 06:00, 12:00, 18:00) UTC by analyses from the European Center for Medium-range Weather Forecasts (ECMWF)[^2]. The simulations run for 18 hours. Note that the time at which the simulation starts (UTC) differs for Dome A, Dome C and South Pole. This is done so to be able to compare optical turbulence profiles simulated in the same temporal interval with respect to the local time (LT). For each night, a mean vertical profile of 2 is computed between the time interval (20:00 - 00:00) LT as done in Lascaux et al. (2009, 2010a). This range is centered on the time at which the balloons were typically launched at Dome C. In this way we obtain the most representative simulated 2 profile for each night[^3]. In Table \[tab1\] are reported, for each site, the time at which the simulation starts and the duration $\Delta$T of the simulation with respect to the local time.
- An optimized version of the externalized surface scheme ISBA (Interaction Soil Biosphere Atmosphere) for antarctic conditions is employed [@lem; @lem10]. Such a scheme has been used in Lascaux et al. (2010a) and it contributed to provide a realistic reconstruction of the optical turbulence near the surface (optical turbulence strength and turbulence layer thickness). It is indeed obvious that the most critical part of an atmospherical model for this kind of simulations is the scheme that controls the air/ground turbulent fluxes budget. Our ability in well reconstructing the surface temperature T$_{s}$ is related to the ability in reconstructing the sensible heat flux H that is responsible of the buoyancy-driven turbulence in the surface layer.
- The Astro-Meso-NH package (Masciadri et al. 1999a) implemented in the most recent version of Meso-NH has been used to calculate the optical turbulence and derived astroclimatic parameters.
Dome A Dome C South Pole
-------------------- ---------------------- ---------------------- ----------------------
Starting time 06:00 UTC / 11:00 LT 00:00 UTC / 08:00 LT 12:00 UTC / 12:00 LT
Time interval
for 2 computations 15:00 - 19:00 UTC 12:00 - 16:00 UTC 20:00 - 00:00 UTC
(20:00 - 00:00 LT)
\[tab1\]
As shown in [@lf10a], the best choice for the description of the orography is the RAMP (Radarsat Antarctic Mapping Project) Digital Elevation Model (DEM) presented in [@liu], instead of the GTOPO30 DEM from the U.S. Geological Survey used in [@lf09]. For this study, therefore, the RAMP Digital Elevation Model has been used. The orography of each area of interest in this study (Dome C, Dome A, South Pole) is displayed on Fig. \[fig1\]. All the grid-nested domains, from low horizontal resolution (larger mesh-size) to high horizontal resolution (smaller mesh-size) are reported. As can be seen in Fig. \[fig1\] (c,f,i) the orography around Dome C and Dome A is more detailed than the orography in proximity of the South Pole. This is due to the fact that the procedure to obtain a DEM integrates data from many different sources (satellite radar altimetry, airborne surveys, GPS surveys, station-based radar sounding...). However the resolution of some areas (typically those that can hardly receive information from the satellites) remain poorer than others. The region included in the inner circular polar region (and therefore South Pole) fits with this condition and this is the reason why the orography is somehow less detailed than the rest of the Internal Antarctic Plateau. Nevertheless, this is a region with no peaks or mountains and with just a regular and gently slope. We can therefore reasonably expect that the poorer accuracy in the orography has little or minor influence on the results of the numerical simulations done with a mesoscale model such as Meso-NH.\
The same set of 15 winter nights used by [@lf09; @lf10a] to validate the model above Dome C is investigated in this study for the three antarctic sites Dome C, Dome A and South Pole.
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"}
[\[fig1\]]{}
Optical Turbulence above Dome C, Dome A and South Pole {#opt}
======================================================
In this section we investigate and compare the values obtained above the three sites (Dome C, Dome A and South Pole) of three parameters that characterize the optical turbulence features above the antarctic plateau:
- surface layer thickness;
- free atmosphere seeing from the surface layer thickness (h$_{sl}$) up to the top of the atmosphere;
- total seeing from the ground up to the top of the atmosphere. We note that this corresponds to $\sim$10 km because the balloons explode at this altitude due to the high pressure and the strong wind speed.
In a numerical mesoscale model the great challenge and difficulty is related to the parameterization of the optical turbulence. The critical issue is related to the ability of the model in reconstructing the vertical distribution of the optical turbulence (i.e. the $\CN2$). This is the reason why we selected and studied these three fundamental parameters. The other integrated astroclimatic parameters are obtained calculating the integral of the $\CN2$ and wind speed vertical profiles along the troposphere. A forth-coming paper will be dedicated to the analysis of the integrated astroclimatic parameters.
Optical turbulence surface layer thickness
------------------------------------------
To compute the surface layer thickness for each night, the same method employed in [@tr] and [@lf10a] is first used. The thickness $h_{sl}$ is defined as the vertical slab containing 90 per cent of the optical turbulence developed inside the first kilometer above the ground: $$\label{eq:bl1}
\frac{ \int_{8m}^{h_{sl}} C_N^2(h)dh }{ \int_{8m}^{1km} C_N^2(h)dh } < 0.90$$ where $C_N^2$ is the refractive index structure parameter. We remind here that the selection of this criterium (that we call criterium A) is motivated by the fact that we intend to compare our calculations with measurements done by Trinquet et al. (2008). This criterium has been selected by Trinquet et al. (2008) because the typical optical turbulence features above the internal antarctic plateau is characterized by a major bump at the surface and a consistent decreasing of the optical turbulence strength in the first tens of meters. The selection of the percentage is obviously absolutely arbitrary and, in this context, is mainly useful to check the correlation with measurements and to compare predictions on different sites (in relative terms therefore). The choice of the inferior limit of the integral (8 m) is motivated by the fact that Trinquet et al. (2008) intended to compare results obtained with balloons with those provided by the DIMM placed at 8 m from the ground.
[|c|r|r|r|]{} Date & Dome A & Dome C & South Pole\
04/07/05 & 65.0 & 30.4 & 117.6\
07/07/05 & 529.4 & 35.4 & 262.9\
11/07/05 & 28.6 & 80.0 & 131.9\
18/07/05 & 27.7 & 49.7 & 224.0\
21/07/05 & 17.6 & 66.7 & 136.3\
25/07/05 & 15.7 & 27.4 & 298.6\
01/08/05 & 25.5 & 22.6 & 185.2\
08/08/05 & 53.1 & 34.2 & 104.4\
12/08/05 & 19.4 & 16.7 & 59.0\
29/08/05 & 17.4 & 91.4 & 251.3\
02/09/05 & 16.4 & 70.9 & 164.9\
05/09/05 & 125.2 & 338.4 & 128.0\
07/09/05 & 59.8 & 52.5 & 103.6\
16/09/05 & 38.8 & 19.4 & 158.7\
21/09/05 & 20.1 & 21.0 & 148.0\
Mean & 37.9\* & 44.2\* & 165.0\
$\sigma$ & 30.2\* & 24.6\* & 67.3\
$\sigma$/$\sqrt{N}$ & 8.1\* & 6.6\* & 17.4\
\
\
\
\[tab3\]
Table \[tab3\] reports the computed values of the surface layer thickness for each night at the three sites, as well as the mean, the standard deviation ($\sigma$) and the statistical error ($\sigma/\sqrt N$) for the 15 nights. For each night, the surface layer thickness is computed from a computed 2 profile averaged between 20 LT and 00 LT (see Table \[tab1\] for hours in UT) as done in Lascaux et al. (2010a). The calculated mean surface layer thickness above South Pole is h$_{sl}$$=$165 m $\pm$ 17.4 m, at Dome C h$_{sl}$$=$44.2 m $\pm$ 6.6 m and at Dome A h$_{sl}$$=$37.9 m $\pm$ 8.1 m. In this paper we are not forced anymore to use the same inferior limit of the integral in Eq. \[eq:bl1\] (8 m) than Trinquet et al. (2008), and we can compute the surface layer thickness starting the integral at the ground. Under this assumption the calculated mean surface layer thickness at South Pole is h$_{sl}$$=$158.7 m $\pm$ 16.2 m, at Dome C h$_{sl}$$=$45.0 m $\pm$ 7.1 m and at Dome A h$_{sl}$$=$34.9 m $\pm$ 7.9 m. We conclude that at South Pole, h$_{sl}$ is more than three time larger than at Dome C or Dome A in both cases. This difference is well correlated with previous observations done above South Pole. More precisely, observations related to 15 balloons launched during the period (20/6/1995 - 18/8/1995) indicated h$_{sl}$$=$220 m [@ma3]. Measurements in that paper are done in winter but in a different year and for different nights. It is not surprising therefore that the matching between calculations and measurements is not perfect. Unfortunately the precise dates of nights studied in the paper from Marks et al. (1999) are not known. It is therefore not possible to provide a more careful estimate. It is however remarkable that the h$_{sl}$ above South Pole is substantially larger than the h$_{sl}$ above Dome C and Dome A. Also we note that the typical thickness calculated above South Pole with a statistical sample of three months by [@seg] was h$_{sl}$$=$102 m. The authors used however a different definition of turbulent layer thickness. More precisely, they defined h$_{sl}$ as the elevation (starting from the lowest model level) at which the turbulent kinetic energy contains 1 per cent of the turbulent kinetic energy of the lowest model layer. A comparison of this result with our calculations and with measurements is therefore meaningless. The same conclusion is valid for the estimates of h$_{sl}$ given at Dome C as already explained in Lascaux et al. (2009, 2010a). In conclusion, looking at Table \[tab3\], individuals values for each nights show a $h_{sl,SP}$ almost always higher than 100 m, with a maximum close to 300 m (2005 July 25), whereas $h_{sl,DC}$ and $h_{sl,DA}$ are always below 100 m. Dome C and Dome A have a comparable surface layer thickness. For this sample of 15 nights, $h_{sl,DA}$ is 6.3 m smaller than $h_{sl,DC}$. We note also that the number of nights for which $h_{sl}$ is very small (inferior at 30 m) is more important at Dome A (nine instead of six at Dome C). This difference is however not really statistically reliable considering the number of the nights in the sample. For a more detailed discrimination between the h$_{sl}$ value at Dome C and Dome A we need a larger statistic. This analysis is planned for a forthcoming paper.
Looking at the results obtained night by night we can note some specific features observed in specific cases. Two nights (September 5 at Dome C ($h_{sl}$ = 338.4 m) and July 7 at Dome A ($h_{sl}$ = 529.4 m)) present similar characteristics: the surface layer thickness h$_{sl}$ is well larger than the observed one. In these two cases, however, as already explained in [@lf09; @lf10a] for the case of Dome C, the large value of h$_{sl}$ does not mean that a thicker and more developed turbulence is present near the ground but it simply means that, in the first kilometer from the ground, 90 per cent of the turbulence develops in the (0, h$_{sl}$) range. On September 5, at Dome C, the model reconstructs the total seeing on the whole 20 km much weaker than what has been observed and more uniformly distributed and, consequently, the criterium (Eq.\[eq:bl1\]) provides us a much larger value of h$_{sl}$. In both cases (on September 5 at Dome C and on July 7 at Dome A), when we look at the vertical distribution of the 2 calculated by the model, we observe that the turbulence is concentrated well below 20 m in a very thin surface layer with a very weak total seeing (see next section). The case of 5 September at Dome A, is however a case in which the model reconstructed a surface turbulent layer thicker than what has been observed.
It is known that mesoscale model provide a temporal variability of the turbulence in the high part of the atmosphere that is smoother than what observed with vertical profiler. This is due to the fact that a mesoscale model is more active in the low part of the atmosphere where the orographic effects are mainly present. We recently obtained [@lf09] very encouraging results showing that the 2 in the free atmosphere has a temporal variability even on a small dynamic range (-18, -16.5 in logarithmic scale). This is a signature of the improvement of the model activity in the high part of the atmosphere. At present, however, it presents a hazard to quantifying the typical time-scale for temporal variability of all the parameters related to the optical turbulence reconstructed by a mesoscale model. Nevertheless we can describe the temporal variability of the morphology of these parameters such as, for example the thickness of the surface layer.
In Appendix A, we report the temporal evolution of the calculated 2 for all the nights above the three sites. Looking at these pictures we can give a description of the morphology of the temporal variation of the surface layer. Above Dome C and Dome A, the thickness of the surface layer remains mostly stable during the night, even though we have night to night variations as shown in Table \[tab3\]. This fits with preliminary results shown in Ashley et al. (2010) above Dome A. Above South Pole, the thickness varies in a much more important way during the night with oscillations that can reach 50 to 100 m. The larger variability of the typical turbulent surface layer thickness is confirmed also by the larger value of $\sigma$ observed above South Pole (Table \[tab3\], \[tab\_TKE\], \[tab\_TKEbis\]).
In order to compare our calculations and results with those obtained by [@seg] we applied also a different criterium (criterium B) based on the analysis of the vertical profile of turbulent kinetic energy (TKE) instead of the vertical profile of 2. The TKE is certainly an ingredient from which the optical turbulence depends on and it represents the dynamic turbulent energy. However, it is known (Masciadri & Jabouille 2001) that the $\CN2$ depends also on the gradient of the potential temperature and moreover, the selection of the value of percentage of the turbulent kinetic energy (1$\%$, 10$\%$, other...) used as a threshold is absolutely arbitrary. This method is therefore not useful to quantify the absolute value of h$_{sl}$ to be compared to measurements provided by Trinquet et al. (2008) and Marks et al. (1999). It can possibly be useful for relative comparisons between different sites or to compare our calculations with calculations provided by Swain & Gallee (2006).
Using this method (Table \[tab\_TKE\]), the surface layer height is determined as the elevation at which the TKE is X% of the lowest elevation value. We calculated the h$_{sl}$ for X = 1 (Table \[tab\_TKE\]) and X = 10 (Table \[tab\_TKEbis\]). X=1 is the case treated by [@seg]. For each simulation, we first compute the average of the TKE profile for the night between 20 LT and 00 LT. While the average of the $\CN2$ profile is calculated with a 2 minutes rate sample, the average of the TKE is calculated with 5 profiles, available at each hour (20, 21, 22, 23, 00) LT. This gives us an averaged vertical profile of TKE characteristic of the considered night.
The computation of the surface layer thickness is then performed using this averaged TKE profile. It has been observed that, when the night presents only low dynamic turbulence (with a very low averaged TKE at the lowest elevation level), it is very hard to retrieve a surface layer height using this criterium. This means that the turbulence is so weak that we are at the limit of necessary turbulent kinetic energy to resolve the turbulence itself. For these nights (indicated with an asterisk in Table \[tab\_TKE\]) it could happen that we calculated the average on a number of estimates smaller than 5 (as for all the other cases). The results are reported in Table \[tab\_TKE\].
[|c|r|r|r|]{} Date & Dome A & Dome C & South Pole\
04/07/05 & 78\* & 32 & 112\
07/07/05 & 6\* & 32 & 112\*\
11/07/05 & 40 & 76 & 174\
18/07/05 & 32 & 48 & 242\
21/07/05 & 22 & 56 & 144\
25/07/05 & 22 & 12\* & 148\
01/08/05 & 32 & 22 & 186\
08/08/05 & 56 & 32 & 112\
12/08/05 & 22 & 60 & 58\
29/08/05 & 22 & 82 & 250\
02/09/05 & 20 & 60 & 188\
05/09/05 & 136 & 30\* & 146\
07/09/05 & 72 & 74 & 250\
16/09/05 & 56 & 22 & 192\
21/09/05 & 26 & 22 & 170\
Mean & 42.8& 44& 165.6\
$\sigma$ & 33.0& 22.6 & 55.5\
$\sigma$/$\sqrt{N}$ & 8.5& 5.8& 14.3\
\
\
\[tab\_TKE\]
Table \[tab\_TKE\] shows that results obtained with the criterium of the TKE are similar to those obtained with the criterium described in Eq.\[eq:bl1\]. Table \[tab\_TKEbis\] provides smaller values of h$_{sl}$ above all the three sites. We treat the case (X = 10) to show that, tuning the value of the percentage, it is possible to find different values of h$_{sl}$. This means that h$_{sl}$ estimates are useful only if they are compared to measurements using the same criteria.
Date Dome A Dome C South Pole
--------------------- -------- -------- ------------
04/07/05 58 22 56
07/07/05 2 24 62
11/07/05 28 52 64
18/07/05 22 32 186
21/07/05 14 40 110
25/07/05 12 6 102
01/08/05 22 16 126
08/08/05 40 24 102
12/08/05 16 8 40
29/08/05 14 68 192
02/09/05 14 46 112
05/09/05 106 30 88
07/09/05 50 52 68
16/09/05 40 12 150
21/09/05 18 14 116
Mean 30.4 27 104.9
$\sigma$ 26.1 15.6 45.2
$\sigma$/$\sqrt{N}$ 6.7 4 11.7
: Mean surface layer thicknesses $h_{sl}$ computed for the 3 sites, for the same set of nights shown in Table \[tab3\], but computed with a different criterion. The surface layer height is determined as the elevation at which the averaged TKE between 20 LT and 00 LT for each nigh is 10% of the averaged lowest elevation value. Units in meter (m). The mean values are also reported with the associated statistical error $\sigma$/$\sqrt{N}$.
\[tab\_TKEbis\]
To conclude, both criteria (A and B with X=1) give similar mean $h_{sl}$ values for all the 3 sites for this limited set of nights. Evaluating the surface layer thickness over a more extended set of nights should be the next step. It would permit us to compute more reliable and robust statistical estimates for $h_{sl}$ over the 3 antarctic sites and possibly to identify discrimination between the $h_{sl}$ at Dome A and Dome C. This also means that our estimate of h$_{sl}$=165 m above South Pole is better correlated to measurements (h$_{sl}$=220 m) than the estimate (h$_{sl}$=102 m) obtained by [@seg] at the same site.
Seeing in the free atmosphere and seeing in the whole atmosphere
----------------------------------------------------------------
The seeing in the free atmosphere and in the whole atmosphere for $\lambda$$=$0.5$\times$10$^{-6}$m is: $$\varepsilon_{FA}=5.41 \cdot \lambda^{-1/5} \cdot \left( \int_{h_{sl}}^{h_{top}} C_N^2(h) \cdot dh \right) ^{3/5}$$ $$\varepsilon_{TOT}=5.41 \cdot \lambda^{-1/5} \cdot \left( \int_{8m}^{h_{top}} C_N^2(h) \cdot dh \right) ^{3/5}$$ with h$_{top}$ $\sim$ 13 km from the sea level i.e. where the balloons explode and we have no more their signal. Table \[tab4\] shows the simulated total seeing ($\varepsilon_{TOT}$) and free-atmosphere seeing ($\varepsilon_{FA}$) for each night and each sites (Dome C, Dome A and South Pole). We define the free atmosphere as the portion of the atmosphere extended from the mean $h_{sl}$ reported in Table \[tab3\] up to $h_{top}$. The median values of the seeing as well as the standard deviation ($\sigma$) and the statistical error ($\sigma/\sqrt N$) are reported. As expected the total seeing is stronger at Dome A ($\varepsilon_{TOT,DA}$ = 2.37 $\pm$ 0.27 arcsec) than at Dome C ($\varepsilon_{TOT,DC}$ = 1.70 $\pm$ 0.21 arcsec) or South Pole ($\varepsilon_{TOT,SP}$ = 1.82 $\pm$ 0.23). The total seeing is very well correlated with measurements at Dome C (Lascaux et al. 2010a - $\varepsilon_{TOT,obs}$ = 1.6 arcsec) and at South Pole (Marks et al. 1999 - $\varepsilon_{TOT,obs}$ = 1.86 arsec) getting the estimate at Dome A highly reliable. The minimum median free-atmosphere seeing is found at Dome A ($\varepsilon_{FA,DA}$ = 0.23 $\pm$ 0.28 arcsec). The medain free-atmosphere seeing at Dome C is $\varepsilon_{FA,DC}$ = 0.30 $\pm$ 0.17 arcsec and at South Pole, $\varepsilon_{FA,SP}$ = 0.36 $\pm$ 0.11 arcsec. The seeing in the free atmosphere is very well correlated with measurements at Dome C (Lascaux et al. 2010a - $\varepsilon_{FA,obs}$ = 0.30 arcsec) and at South Pole (Marks et al. 1999 - $\varepsilon_{FA,obs}$ = 0.37 arsec) getting again very reliable the method (Meso-NH model) as well as the estimates at Dome A. What is remarkable is that, even if $h_{sl,DA}$ $<$ $h_{sl,DC}$ $<$ $h_{sl,SP}$, Dome A is the site with the lowest free-atmosphere seeing $\varepsilon_{FA}$. This means that at Dome A as well as at Dome C the turbulence is concentrated inside the first tens of meters from the ground. Moreover, the turbulence in the surface layer is stronger at Dome A than at Dome C. This can be explained with the stronger thermal stability of Dome A near the ground. Our results match, therefore, with predictions we did in Hagelin et al. (2008) studying only features of the meteorological parameters.
DOME A DOME C SOUTH POLE
--------------------- ---------------------------------------- ---------------------------------------- -----------------------------------------
Date $\varepsilon_{FA}$/$\varepsilon_{TOT}$ $\varepsilon_{FA}$/$\varepsilon_{TOT}$ $\varepsilon_{FA}$/ $\varepsilon_{TOT}$
[($h_{sl}$=37.9m)]{} [($h_{sl}$=44.2m)]{} [($h_{sl}$=165m)]{}
04/07/05 2.55 / 3.37 0.22 / 2.28 0.40 / 1.67
07/07/05 0.20 / 0.24 0.28 / 1.91 0.31 / 0.70
11/07/05 0.23 / 2.78 1.61 / 1.81 0.47 / 1.96
18/07/05 0.21 / 2.73 0.80 / 1.94 1.46 / 2.28
21/07/05 0.21 / 1.95 0.86 / 1.27 0.31 / 1.71
25/07/05 0.22 / 1.55 0.25 / 0.85 0.32 / 0.76
01/08/05 0.22 / 1.78 0.22 / 2.27 0.52 / 1.78
08/08/05 1.45 / 2.42 0.35 / 1.70 0.28 / 1.69
12/08/05 0.23 / 2.37 0.23 / 0.99 0.29 / 1.82
29/08/05 0.23 / 1.83 2.29 / 2.47 1.55 / 2.11
02/09/05 0.22 / 1.76 1.16 / 1.54 0.81 / 3.56
05/09/05 3.21 / 3.36 0.30 / 0.52 0.31 / 2.98
07/09/05 2.43 / 3.49 1.69 / 3.73 0.31 / 1.41
16/09/05 1.11 / 4.60 0.21 / 1.57 0.99 / 3.96
21/09/05 0.20 / 2.30 0.26 / 1.63 0.36 / 2.32
Median 0.23 / 2.37 0.30 / 1.70 0.36 / 1.82
$\sigma$ 1.08 / 1.03 0.67 / 0.77 0.43 / 0.90
$\sigma$/$\sqrt{N}$ 0.28 / 0.27 0.17 / 0.21 0.11 / 0.23
: Total seeing $\varepsilon_{TOT}$$=$$\varepsilon_{[8m,h_{top}]}$ and seeing in the free atmosphere $\varepsilon_{FA}$$=$$\varepsilon_{[h_{sl},h_{top}]}$ calculated for the 15 nights and averaged in the temporal range 20-00 LT. See the text for the definition of h$_{sl}$ and h$_{top}$.
\[tab4\]
At South Pole, however, the 2 vertical distribution decreases in a less abrupt way because the thermal stability near the ground is less important. The 2 vertical distribution is spread over hundreds of meters from the ground, instead of tens of meters like for Dome A or Dome C. As a consequence the total seeing is also weaker than above Dome C and Dome A.
Such a behavior is evidenced in Figure \[fig2\], which displays the median vertical 2 profiles over the 3 sites.
Looking at Table \[tab4\], we note that the values of $\sigma$ for the total seeing above the three sites is mostly comparable with no significant differences even if Dome C seems a little smaller (0.77) than Dome A (1.03) and South Pole (0.90). This indicates a comparable variability of the turbulence above the three sites. For what concerns the seeing in the free atmosphere, the value of $\sigma$ above Dome A is almost double (1.08) than above Dome C (0.67) and South Pole (0.43).
{width="\textwidth"}
----------------------- ---------- ---------- --------------------- -------------------- ---------- --------------------- ------------------- ---------- ---------------------
h$_{sl}$ $\sigma$ $\sigma$/$\sqrt{N}$ $\varepsilon_{FA}$ $\sigma$ $\sigma$/$\sqrt{N}$ $\varepsilon_{TOT $\sigma$ $\sigma$/$\sqrt{N}$
}$
(m) (arcsec) (arcsec)
Observations - Dome C 35.3 19.9 5.1 0.30 0.70 0.20 1.60 0.70 0.20
Meso-NH - Dome C 44.2 24.6 6.6 0.30 0.67 0.17 1.70 0.77 0.21
Meso-NH - Dome A 37.9 30.2 8.1 0.23 1.08 0.28 2.37 1.03 0.27
Meso-NH - South Pole 165.0 67.3 17.4 0.36 0.43 0.11 1.82 0.90 0.23
----------------------- ---------- ---------- --------------------- -------------------- ---------- --------------------- ------------------- ---------- ---------------------
\[tab\_sum2\]
Conclusion {#concl}
==========
In this study the mesoscale model Meso-NH was used to perform forecasts of optical turbulence (evolutions of 2 profiles) for 15 winter nights at three different antarctic sites: Dome A, Dome C and South Pole. The model has been used with the same configuration previously validated at Dome C (Lascaux et al. 2010a) and simulations of the same 15 nights have been performed above the three sites. The idea behind our approach is that once validated above Dome C, the model can be used above two other sites of the internal antarctic plateau to discriminate optical turbulence features typical of other sites. This should show the potentiality of the numerical tool in the context of the site selection and characterization in astronomy. South Pole has been chosen because in the past some measurements of the optical turbulence have been done and this can represent a useful constraint for the model itself. For Dome A there are not at present time measurements of the optical turbulence and this study provides therefore the first estimates ever done of the optical turbulence above this site. We test this approach above the antarctic plateau because this region is particularly simple from the topographic point of view and certainly simpler than typical mid-latitude astronomical sites. No major mountain chains are present and the local surface circulations is mainly addressed by the energy budget air/ground transfer, the polar vortex circulation at synoptic scale and the katabatic winds generated by gravity effects on gently slopes due to the cold temperature of the iced surface. The main results we obtained are summarized in Table \[tab\_sum2\] and listed here:
- We provide the first estimate of the optical turbulence extended on the whole 20 km above the Internal Antarctic Plateau.
- The Meso-NH model achieves to reconstruct the three most important parameters used to characterize the optical turbulence: the turbulent surface layer thickness, the seeing in the free atmosphere and in the surface layer for the three selected sites: Dome C, Dome A and South Pole showing results in agreement with expectations. Measurements taken at Dome C and South Pole corresponds to balloons launched during 15 nights, in both cases. The statistic is not very large but reliable for a first significant result. The selected nights correspond to the 15 nights for which measurements of the Dome C are available.
- Dome C and Dome A present a very thin surface layer size ($h_{sl,DA}$ = 37.9 $\pm$ 8.1 m and $h_{sl,DC}$ = 44.2 $\pm$ 6.6 m) while South Pole surface layer is much thicker ($h_{sl,SP}$ = 165 $\pm$ 17.4 m). If we apply the criterium (A) described by Eq.(1) integrating from the ground instead than 8 m from the ground we find similar result within a couple of meters. All these estimates are well correlated with measurements. Surface layers calculated by the model at Dome C and Dome A have a comparable thickness considering the actual sample. To better discriminate between the Dome A and Dome C surface layer thickness a richer statistic is necessary. An on-going study has started addressing this issue.
- Dome A is the site with the strongest total seeing (2.37 $\pm$ 0.27 arcsec) with respect to Dome C ($\varepsilon_{TOT,DC}$ = 1.70 $\pm$ 0.21 arcsec) and South Pole ($\varepsilon_{TOT,SP}$ = 1.82 $\pm$ 0.23 arcsec). This is explained by the stronger thermal stability near the ground with respect to the other two sites that cause large values of the optical turbulence in the thin surface layer.
- All the three sites show a very weak seeing in the free atmosphere i.e. above the correspondent mean h$_{sl}$: $\varepsilon_{FA,DA}$ = 0.23 $\pm$ 0.28 arcsec at Dome A, $\varepsilon_{FA,DC}$ = 0.30 $\pm$ 0.17 arcsec at Dome C and $\varepsilon_{FA,SP}$ = 0.36 $\pm$ 0.11 arcsec at South Pole. Dome A show the weakest seeing in the free atmosphere.
- The temporal variability of the thickness of the surface layer is more important at South Pole than above Dome A and Dome C that show a very stable trends in agreement with observations. The temporal variability of the seeing in the whole atmosphere does not show important differences above the three sites, while the variability of the seeing in the free atmosphere is almost double at Dome A than at Dome C and South Pole.
- Both, the total seeing and the seeing in the free atmosphere calculated by Meso-NH, are very well correlated with measurements at Dome C and South Pole getting the predictions done at Dome A highly reliable.
- Dealing with the criteria used to define the surface layer thickness, we proved that, at least on the sample of 15 nights investigated, the criterium defined by Eq.\[eq:bl1\] (criterium A) and the criterium using the vertical profile of the turbulent kinetic energy (TKE) taking h$_{sl}$ as the height at which the value of the TKE is less than 1$\%$ of the TKE at the lowest level near the ground (criterium B) provide very similar results.
- The mean h$_{sl}$ we estimate at Dome C (h$_{sl}$=44.2) is slightly thicker than what found by [@seg] (h$_{sl}$=27.7 m) with comparable discrepancy from measurements (h$_{sl}$ = 35.3 $\pm$ 5.1 m). The h$_{sl}$ we estimate at South Pole ($h_{sl,SP}$ = 165 $\pm$ 17.4 m) is thicker than what estimated by [@seg] ($h_{sl,SP}$ = 102) but better correlated to measurements ($h_{sl,SP}$ = 220 m) than what found by [@seg]. The h$_{sl}$ we estimate at Dome A ($h_{sl,DA}$ = 37.9 $\pm$ 8.1 m) is somehow thicker than what estimated by [@seg] ($h_{sl,DA}$ = 18 m). It is however important to note that the standard deviation of h$_{sl}$ is of the order of h$_{sl}$ itself or even larger. The statistic error $\sigma$/$\sqrt(N)$ is of the order of $\sim$ 10 m. We think therefore that at present there are no major differences in our results with respect to [@seg] with exeption of the fact that we proved that, with our model, the horizontal resolution of 1 km provides better results than a resolution of 100 km that is used by [@seg].
All these results deserve now a confirmation provided by an analysis done with a richer statistical sample. Also it would be interesting to refine this study when OT measurements above Dome A will be published. Besides, we can state that all major expectations concerning the typical features of the optical turbulence above South Pole, Dome C and Dome A have been confirmed by this study. The tendency shown by the model is obviously that in summer time, in proximity of the surface, due to the less stable regime, the turbulence thickness increases but the turbulence strength decreases. This is however, out the goals of this paper.
Acknowledgments {#acknowledgments .unnumbered}
===============
ECMWF products are extracted from the catalogue MARS, http://www.ecmwf.int, access to these data was authorized by the Meteorologic Service of the Italian Air Force. This study has been funded by the Marie Curie Excellence Grant (FOROT) - MEXT-CT-2005-023878.
Arakawa, A. & Messinger, F., 1976, GARP Tech. Rep., 17, WMO/ICSU, Geneva, Switzerland Aristidi, E., Agabi, K., Fossat, E., Azouit, M., Martin, F., Sadibekova, T., Travouillon, T., Vernin, J., Ziad, A., 2005, A&A, 444, 651 Aristidi, E., Fossat, E., Agabi, K., Mékarnia, D., Jeanneaux, F., Bondoux, E., Challita, Z., Ziad, A., Vernin, J., Trinquet, H., 2009, A&A, 499, 955 Ashley, M. et al., 2010, EAS Publications Series, 40, 79 Asselin, R., 1972, Mon. Weather. Rev., 100, 487 Bougeault, P. & Lacarrère, P., 1989, Mon. Weather. Rev., 117, 1972 Cuxart, J., Bougeault, P. and Redelsperger, J.-L., Q. J. R. Meteorol. Soc., 126, 1, 2000 Gal-Chen, T. & Sommerville, C. J., 1975, J. Comput. Phys., 17, 209 Hagelin, S., Masciadri, E., Lascaux F. and Stoesz, J., 2008, MNRAS, 387, 1499 Lafore, J.-P. et al., 1998, Annales Geophysicae, 16, 90 Lascaux, F., Masciadri, E., Stoesz, J., Hagelin, S., 2009, MNRAS, 398, 1093 Lascaux, F., Masciadri, E., Hagelin, S., 2010a, MNRAS, 403, 1714 Lascaux, F., Masciadri, E., Hagelin, S., 2010b, Ground-based and Airborne Telescopes III. Edited by Stepp, Larry M.; Gilmozzi, Roberto; Hall, Helen J. Proceedings of the SPIE, Volume 7733, pp. 77334E-77334E-8 Lawrence, J., Ashley, M., Tokovinin, A., Travouillon, T., 2004, Nature, 431, 278 Le Moigne, P., Noilhan, J., Masciadri, E., Lascaux, F., Pietroni, I., 2009, Masciadri, E. & Sarazin, M., eds, Optical Turbulence - Astronomy meets Meteorology. Imperial College Press, London, p.165 Le Moigne, P., Noilhan, J., Masciadri, E., Lascaux, F., Pietroni, I., 2010, Journal of Geophysical Research, submitted Liu, H., Jezek, K., Li, B., Zhao, Z., 2001, Digital media, National Snow and Ice Data Center, Boulder, CO, USA Marks, R.D., Vernin, J., Azouit M., Manigault J.F., Clevelin C., 1999, A&AS, 134, 161 Marks, R.D., Vernin, J., Azouit, M., Briggs, J.W., Burton, M.G., Ashley, M.C.B., Manigault, J.F., 1996, A&AS, 118, 385 Masciadri, E., Vernin, J., Bougeault, P., 1999a, A&ASS, 137, 185 Masciadri, E., Vernin, J., Bougeault, P., 1999b, A&ASS, 137, 203 Masciadri, E. & Garfias, T., 2001, A&A, 366, 708 Masciadri, E., Vernin, J., Bougeault, P., 2001, A&A, 365, 699 Masciadri, E. & Jabouille, P., 2001, A&A, 376, 727 Masciadri, E., Avila, R., Sanchez, L. J., 2004, RMxAA, 40, 3 Masciadri, E. & Egner, S., 2006, PASP, 118, 849, 1604 Noilhan J. & Planton, S., 1999, Mon. Weather. Rev., 117, 536 Stein, J., Richard, E., Lafore, J.-P., Pinty, J.-P., Asencio, N., Cosma, S., 2000, Meteorol. Atmos. Phys., 72, 203 Swain, M. & Gallée, H., 2006, PASP, 118, 1190 Trinquet, H., Agabi, K., Vernin, J., Azouit, M., Aristidi, E., Fossat, E., 2008, PASP, 120, 203
Computed temporal evolutions of 2 vertical profiles for each nights at Dome C, Dome A and South Pole
====================================================================================================
In this appendix we present all the individual figures of the 18-hours temporal evolution of the 2 for every night and at the three antarctic sites considered in this study (Figure \[fig1\_app\]: Dome A, Figure \[fig2\_app\]: Dome C and Figure \[fig3\_app\]: South Pole). The first couple of hours can be considered as spurious values because of the model adaptation to the ground.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
\[lastpage\]
[^1]: E-mail: lascaux@arcetri.astro.it; masciadri@arcetri.astro.it
[^2]: ECMWF: $http://www.ecmwf.int/$
[^3]: In the prediction of a parameterized parameter (such as the optical turbulence) there is not a 1-1 correlation with the real time. This means, with an explicative example, that is somehow meaningless to predict the turbulence at a precise time t=t$^{*}$ as we do for a parameter that we resolve explicitly such as the temperature or the wind speed. This is the reason why, in order to obtain the most representative 2 profile to be compared to measurements, we calculate the mean of the 2 in a temporal interval $\Delta$T. Such a procedure has been used in many previous papers [@m5; @m3; @m4]
|
**Dilatonic Randall-Sundrum Theory**
**and renormalization group**
[**César Gómez,**]{}[^1] [**Bert Janssen**]{}[^2] [**and**]{} [**Pedro Silva**]{}[^3]
[*Instituto de F[í]{}sica Te[ó]{}rica, C-XVI,*]{}
[*Departamento de F[í]{}sica Te[ó]{}rica, C-XI,* ]{}
[*Universidad Aut[ó]{}noma de Madrid*]{}
[*E-28006 Madrid, Spain*]{}
**ABSTRACT**
We extend Randall-Sundrum dynamics to non-conformal metrics corresponding to non-constant dilaton. We study the appareance of space-time naked singularities and the renormalization group evolution of four-dimensional Newton constant.
Introduction
============
The idea that Newtonian gravity can be localized in a three-brane world has received a lot of attention during the last year [@RS1]-[@Youm]. The original framework is a five-dimensional warped spacetime metric of the type: ds\^[2]{} = A\^[2]{}(z) d\^[2]{} - dz\^[2]{} with the warping factor $A$ depending exponentially on $z$, more precisely a $AdS_{5}$ space-time with negative cosmological constant $\Lambda$. Small gravitational fluctuations $h_{\mu\nu}$ of the metric can be written as superpositions of modes $h_{\mu\nu}=e^{ipx}\psi(z)\epsilon_{\mu\nu}$. Four-dimensional gravitons are associated with zero modes defined by the condition $p^{2}=0$. In order to get normalizable zero modes we need to cutoff the deep ultraviolet region of $AdS_{5}$. This can be done introducing a domain wall at some finite value $z$. On the domain wall metric we can now generically have gravitational zero modes that can be interpreted as bound states of the higher dimensional graviton strongly localized around the wall. This is in summary the dynamical mechanism suggested in [@RS1; @RS2] to induce four-dimensional Newtonian gravity in a brane world. If we start with $AdS_{5}$ space-time the resulting domain wall metric will have a horizon at infinity. Very likely this horizon does not have any observable effect on the physics on the brane due to the strong redshift.
From the point of view of holography [@Mald]-[@Witt], the Randall-Sundrum mechanism of inducing gravity by introducing an ultraviolet cutoff could be interpreted as the extension of the holographic map to conformal field theories coupled to gravity [@BVV; @VV].
In this letter we will address the question of extending the RS- scenario to dilatonic gravity. One reason for that is of course to make a more direct contact with string theory where the dilaton appears naturally in the definition of brane tensions. Another reason is to unravel how much of RS-dynamics depends on conformal invariance. Once we include the dilaton we have at our disposal the possibility of working with a vanishing five-dimensional cosmological constant. In this case we find bound states four-dimensional gravitons with the Newton constant fine tuned in terms of the wall tension. For non-vanishing cosmological constant we find a two-parameter family of solutions depending on the dilaton coupling and on the cosmological constant. The physics of all these cases is different from that in RS- model in the sense that in the bulk there appears a naked singularity that can be reached from the wall in finite time. This singularity can only be avoided in the conformal AdS case. This occurs as an effect of working with a non-constant dilaton.
Construction of the solutions
=============================
Our starting point is the following five-dimensional action of a dilaton $\phi$ coupled to gravity in the presence of a cosmological constant $\Lambda$: S\_[grav]{}= d\^4x dz , \[lagran\] where $a$ is a free parameter that determines the coupling of the dilaton to the cosmological constant. The domain wall solutions of this action are given by: ds\^2 &=& \^ dx\^2 - dz\^2 , e\^[-2]{}&=&\^ , \[solution\] where $z$ runs between 0 and infinity, $z_0$ is integration constant, $N(a)$ a function of the coupling $a$ and the cosmological constant: N(a)= . \[N\] Since we assume the cosmological constant to be negative, our solution only makes sense for $a$’s with values between 8 and $-8$. The above metric has clearly a naked singularity at $z=-z_0$: &&R = 2\^7 ;\
&&R\_R\^ = 2\^[13]{} . To calculate the profile of the graviton we add some small fluctuations $h_{\mu\nu}$ to the above background metric, choosing the gauge $h_\mu^\mu = \part^\mu h_{\mu\nu}=h_{z\nu}=h_{zz}=0$: ds\^2 = \^ (\_[mn]{}+h\_[mn]{}) d x\^m dx\^n - dz\^2 , \[perturbed\] Inserting the perturbed metric (\[perturbed\]) in the equations of motion of (\[lagran\]), we get, at first order in the perturbation, the following differential equation for the graviton: h\_[mn]{}=0 For the splitting of variables $h_{\mu\nu}(x,z)=\ e^{i\vec k\vec x}\ \psi(z)\
\epsilon_{\mu\nu}$, we find for the profile of the graviton zero mode (z) = \^[32/ a\^2]{}. \[zeromode\] This zero mode is not normalizable over the whole of space. If we insist on the existence of a graviton zero mode, we need to introduce a cut-off at the position $z=N(a)^{-1}$.
The cutt-off has the effect of throwing away the part of space with $z>N(a)^{-1}$, where the graviton zero mode (\[zeromode\]) becomes non-normalizable. We can replace this thrown away part by a copy of the part of space with $z<N(a)^{-1}$. At the level of the solution (\[solution\]), this is seen in the fact that we pass from the variables $z \rightarrow N(a)^{-1}- |z|$, (where $|z|$ now runs between 0 and $N(a)^{-1}-z_0$), where $N(a)z_0 \in [0, 1]$. This generates a delta function behaviour in the equations of motion, which can be compensated by introducing domain wall source terms at the bounderies: S\_[source]{} &=& \_[z=0]{} d\^4x && + \_[z=N(a)\^[-1]{}]{} d\^4x , \[source\] where $\tg_{mn}=G_{\mu\nu}\delta^\mu_m\delta^\nu_n$ is the induced metric on the domain wall, $\cL_{\rm brane}$ is the Lagrangian of a gauge theory living an the brane and $V_i$ the tensions of the branes. We thus get for the solution of the space with domain wall: ds\^2 &=& \^ dx\^2 - dz\^2 , e\^[-2]{}&=&\^ , \[|solution|\] The brane tensions $V_i$ and the dilaton coupling $b$ satisfy the matching conditions V\_0=-V\_L = , b=a . $V_0$ ($V_L$) corresponds to the tension of the so-called Planck brane (TeV-brane). What we are actually doing by introducing these source terms is making an orbifold construction $S^1/\Z_2$, where the domain walls are located at the fix points. Note that we can also take the limit in which we send the TeV-brane to the singularity, by taking the limit $z_0 \rightarrow 0$.
In these variables the Randall-Sundrum (RS) limit $a \rightarrow 0$ is singular. To get a good picture of this limit, it is instructive to go to the conformal frame via the coordinate transformation \^ = \[coordtransf\] where $|\omega|$ runs between 0 and $\omega_0= O(a)^{-1} \Bigl[1-\Bigl(N(a) z_0\Bigr)^\tfrac{a^2-16}{a^2} \Bigr]$ and $O(a) = \frac{a^2-16}{a^2} N(a)$. In this frame the solution takes the form ds\^2&=& \^ (dx\^2 - d\^2) , e\^[-2]{}&=&\^ . \[conformalsol\]
We can now make a case study for the different values of the dilaton couping $a$:
- In the conformal frame the RS limit $a\rightarrow 0$ is prefectly regular and gives us the non-dilatonic AdS${}_5$ solution of RS [@RS1; @RS2]. The graviton zero mode goes like $\psi(\omega)=\Bigl[ 1-\sqrt{\tfrac{\Lambda}{12}}\ |\omega| \Bigr]^{-3/2}$ .
- For $0<a^2<16$ we find the graviton zero mode falling off like ()= \^ , \[graviton2\] i.e. confining faster and faster as $a$ approaches the value $-4$. Note that the RS limit is the least confining case of this family.
- For $16<a^2<64$ we find that the zero mode is normalizable only in a finite interval, even in the limit $\omega_0 \rightarrow O(a)^{-1}$ [^4]. However, we can always change coordinates and make the interval $[0, \omega_0]$ infinite. For example in coordinates $1-O(a)\homega = (1-O(a)\omega)^{-1}$. Again the zero mode is confining ()= \^ , \[graviton3\] and the dilaton in these coordinates looks like e\^[-2]{}=\^ . \[dilaton3\] Note that the exponent of the graviton zero mode is bigger (smaller) than in the RS case for values of $a<\sqrt{32}$ ($a>\sqrt{32}$). However, a comparison as in the previous case is difficult since after the coordinate transformation, we are no longer in the conformal frame.
Concluding, we find that in the any of the cases discussed above there is confinement of the gravitational zero mode, in some cases even stronger than in the RS-case. However there is a big difference in the behaviour of the dilaton: for positive values of $a$ the exponent of the dilaton has the same sign as the exponent of the graviton, but for negative value of $a$ the signs are opposite. Depending on the sign of $a$ the string coupling constant $e^{\phi}$ at the space-time singularity goes to $\infty$ or zero.
Although the RS limir $a\rightarrow 0$ is well defined in the conformal frame (\[conformalsol\]), there is a singular point for the value $a=\pm4$, which needs a special analysis. Solving the equations of motion for the $a=4$ case, it becomes clear that there are only two solutions: either $\Lambda = 0$ or linear dilaton with constant warp factor i.e flat five-dimensional space-time metric. If we consider non-critical strings in five dimensions the cosmological constant term is given, in string frame, by $e^{-2\phi}\frac{(D_{cr}-D)}{3}$ with $D_{cr} = 26$ or 10. In this case the only solution is flat five-dimensional space-time and dilaton: = z . For $\Lambda = 0$ we find two solutions for two distict values of the coupling $b$, which only differ in the dilaton dependence: &b=-4:& {
[ll]{} ds\^2 =\^ dx\^2 - dz\^2 ,\
\
e\^[-2]{}= \^[-]{} ,
. \[Lambda0-1\]\
&b=4:& {
[ll]{} ds\^2 =\^ dx\^2 - dz\^2 ,\
\
e\^[-2]{}= \^ .
. \[Lambda0-2\] Here the coordinate $z$ runs between 0 and $\Bigl( (\tfrac{2}{3}\kappa V_0)^{-1} -z_0\Bigr)$ where $z_0 \in [0, (\tfrac{2}{3}\kappa V_0)^{-1} ].$ In the conformal frame, these solutions are of the form: &b=-4:& {
[ll]{} ds\^2 =\^ (dx\^2 - d\^2) ,\
\
e\^[-2]{}=\^[-2]{} ,
. \[confLambda0-1\]\
&b=4& : {
[ll]{} ds\^2 =\^ (dx\^2 - d\^2) ,\
\
e\^[-2]{}=\^[2]{} .
. \[confLambda0-2\] Again, as in the case of $16<a^2<64$ above, we can always find a coordinate system in which the graviton zero mode is confined: ()= \^[-]{} . The other singular point is at $a=-4$. This singularity however turns out to be a coordinate singularity due to the singular behaviour of the coordinate transformation (\[coordtransf\]). The solution at this point is given by ds\^2&=& e\^[-||]{} (dx\^2 - d) , e\^[-2]{}&=& e\^[-||]{} . Note that also in this case the graviton zero modes are confined.
Renormalisation group and Newton constant
=========================================
Recently a different approach to RS-dynamics based on renormalization group interpretation of holography [@Akh; @AG] has been suggested in reference [@BVV; @VV]. In this approach the Einstein gravity on the wall is replaced by the integral on the ultraviolet region of the five-dimensional effective action. The four-dimensional cosmological constant remains fixed along the renormalization group evolution and therefore can be fine tuned to zero by impossing appropiated boundary conditions in the ultraviolet region. The solutions we have been describing above correspond to particular initial conditions determined by the values of the wall tension. Notice that this value fixed by the jump equations is independent of the particular value of the UV cutoff used to locate the wall i.e. it is renormalization group invariant.
In this renormalization group scheme we can define the following beta function: \_ = A , or in terms of the “cosmological time”: = , with = the expansion rate of the four-dimensional metric. For the solutions of dilatonic gravity we have: \_ = -a , and we observe that the RS-model $a =0 $ corresponds to the conformal case $\beta_{\phi} =0$ with all other cases constant but non-vanishing beta function (positive or negative depending on the sign of $a$). The naked singularity is characterized by infinite $\gamma$ i.e by $\dot{\phi} = \infty$. For the vanishing cosmological constant case we get the beta function: \_ = , corresponding precisely to the point $a=4$ i.e to the singular line in Figure \[gravit\] and \[dilat\].
Next we will study the evolution of the Newton constant. The relevant renormalization group equations is given by: ( + ) = . This equation have a very simple physical meaning. Namely the r.h.s of the equation is simply the “time” derivative of the Newton constant $\kappa_{4}$ defined by Kaluza-Klein reduction on the bulk direction. Thus the meaning of the previous equation is simply that $\frac{\partial
\kappa_{4}}{\partial t} =0$.
This equation, once we have written $\dot{\phi}$ and $\dot{A}$ in terms of $A$ has the the solution, \_4\^[-1]{}= dz A\^2(z) + [constant]{} . For $A$ as in (\[|solution|\]), the above integral becomes \_4\^[-1]{}= \_0\^[N(a)\^[-1]{}-z\_0]{} dz \^ . In order to be able to compare with the case of zero-dilaton (Randal-Sundrum), we switch to the conformal variable $\omega$, \_4\^[-1]{}= \_0\^[\_a]{} d \^ , \[integral\] where the upper limit corresponds to the distance between the Planck brane and the place where the effective Newton constant is measured. For the case $a^2<16$, $\omega_0$ runs in a semi-infinite range. Solving the integral (\[integral\]) gives \_4\^[-1]{}= , which should be compared to the RS case: \^[-1]{}\_[4 (RS)]{}= . On the other hand, it is not straightforward how to give an adequate comparison for case $16<a^2<64$ and Randall-Sundrum. We have mentioned above that in the conformal frame the variable $\omega$ then runs over a finite range. This range can be made infinite, as done above, but again comparison to RS is hard due to the fact that we are no longer in the conformal frame.
The other physical implication is the screening of the measurements of physical quantities at the distance $z_0$ by the warp factor $A(z)$, generating a hierarchy between the Planck brane and the brane TeV-brane. In terms of the four and five-dimensional Planck-length the hierarchy is of the order of \_4 e\^[-50]{} \_[5]{} . In the conformal frame (the only frame where we can compare to the RS case), we get \_0= - (e\^[(16-a\^2)]{}-1) , which should be compared to the Randal-Sundrum case, \_0=(e\^[50]{}-1) . We see that the inclusion of the dilaton has a considerable effect on the effective Newton constant: the higher the values of dilaton coupling $a^2$, the faster the Newton constant reaches its asymptatic value. At the same time there is a screening of the constant which is bigger as the dilaton coupling grows.
Finally let us just mention a natural interpretation of the singularity from the four-dimensional physics point of view. This singularity, depending on the sign of $\beta_{\phi}$ could be interpreted either as a Landau pole or a confinement scale for the non-conformal gauge theory on the wall. It is interesting to see that this potential scale of the gauge theory is related with the four dimensional Newton scale.
[*NOTE ADDED IN PROOF:*]{} While this paper was being written we received the papers [@rest; @KSS] that partially overlap with our results.
[**Acknowledgments**]{}\
The work of C.G. and B.J. has been supported by the TMR program FMRX-CT96-0012 on [*Integrability, non-perturbative effects, and symmetry in quantum field theory*]{}. The work of P.S. was partially supported by the gouverment of Venezuela.
[99]{} L. Randall, R. Sundrum, Phys.Rev.Lett. 83 (1999) 3370, hep-ph/9905221
L. Randall, R. Sundrum, Phys.Rev.Lett. 83 (1999) 4690, hep-th/9906064
A. Brandhuber, K. Sfetsos, JHEP 9910 (1999) 013, hep-th/9908116
A. Chamblin, G. W. Gibbons, [*Supergravity on the Brane*]{}, hep-th/9909130
A. Chamblin, S. W. Hawking, H. S. Reall, [*Brane-World Black Holes*]{}, hep-th/9909205
R. Emparan, G. T. Horowitz, R. C. Myers, JHEP 0001 (2000) 007, hep-th/9911043;\
[*Exact Description of Black Holes on Branes II: Comparison with BTZ Black Holes and Black Strings*]{}, hep-th/9912135
O. DeWolfe, D. Z. Freedman, S. S. Gubser, A. Karch, [*Modeling the fifth dimension with scalars and gravity*]{}, hep-th/9909134
D. Youm, [*Solitons in Brane Worlds*]{}, hep-th/9911218
J. M. Maldacena, Int.J.Theor.Phys. 38 (1999) 1113, hep-th/9711200
S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Phys.Lett. B428 (1998) 105, hep-th/9802109
E. Witten, Adv.Theor.Math.Phys. 2 (1998) 253, hep-th/9802150
J. de Boer, E. Verlinde, H. Verlinde, [*On the Holographic Renormalization Group*]{}, hep-th/9912012
E. Verlinde, H. Verlinde, [*RG-Flow, Gravity and the Cosmological Constant*]{}, hep-th/9912018
E. T. Akhmedov, Phys.Lett. B442 (1998) 152, hep-th/9806217
E. Alvarez, C. Gomez, Nucl.Phys. B541 (1999) 441, hep-th/9807226
N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, R. Sundrum, [*A Small Cosmological Constant from a Large Extra Dimension*]{}, hep-th/0001197
S. Kachru, M. Schulz, E. Silverstein, [*Self-tuning flat domain walls in 5d gravity and string theory*]{}, hep-th/0001206
[^1]: E-mail address: [cesar.gomez@uam.es]{}
[^2]: E-mail address: [bert.janssen@uam.es]{}
[^3]: E-mail address: [psilva@delta.ft.uam.es]{}
[^4]: Note that by the definition of the range of $\omega$, $\omega_0 = O(a)^{-1}$ is the maximal value it can attain.
|
---
abstract: 'The theory of abstract kernels in non-trivial extensions for many kinds of algebraical objects, such as groups, rings and graded rings, associative algebras, Lie algebras, restricted Lie algebras, DG-algebras and DG-Lie algebras, has been widely studied since 1940’s. Gerhard Hochschild firstly treats associative algebra as an generic type in the series of kernel problems. He proves the theorem of constructing kernel by presenting many tedious relations that may lost the readers today. In this paper, we shall illustrate the formulation and recast it for Lie algebra(-oid) kernels. We also prove the independence of 3-cocycle in the case of associative algebra. Finally, we use the universal enveloping algebra of Lie algebra to reduce the difficulty of a direct construction for the derivation algebras.'
author:
- Zelong Li
bibliography:
- 'test\_mod.bib'
title: On the Associative Algebra Kernels and Obstruction
---
Introduction
============
Hochshild and Mori firstly consider the case for ordinary Lie algebra and one may consult the original ideas in [@Hoch54a], the detailed calculations [@Mori53] and some miscellaneous in [@Alek00]. Consider a split extension of Lie algebras of $\mathfrak{g}$ by $\mathfrak{h}$: $$\begin{tikzcd}
0 \ar{r} &\mathfrak{h} \ar[tail]{r}{\alpha} &\mathfrak{e} \ar[two heads]{r}{\beta} &\mathfrak{g} \ar{r} &0
\end{tikzcd}$$ where $\beta$ is an epimorphism such that $\mathfrak{h}=ker \beta$ and $\alpha$ is then a monomorphism. By splitness there is a linear map $\gamma: \mathfrak{g} \rightarrow \mathfrak{e}$ such that $\beta \gamma=id_{\mathfrak{g}}$.
This is equivalent to say that $\mathfrak{e}$ is a semidirect product of these two Lie algebras and has a one-to-one correspondence with a homomorphism: $$\begin{aligned}
\varphi':\mathfrak{g} &\rightarrow Der(\mathfrak{h}) \\
x &\mapsto ad_e(-)=[\beta^{-1}(x), -]\end{aligned}$$ where $e=\beta^{-1}(x)\in \mathfrak{e}$. Now $\mathfrak{e}=\mathfrak{h} \rtimes_{\varphi'} \mathfrak{g}$.
Passing through the adjoint map, the image of $\varphi'$ consists of elements called the **inner derivation**, so $\varphi'$ can be extended to a map $$\varphi: \mathfrak{g} \rightarrow Der(\mathfrak{h})/ad(\mathfrak{h})$$ The latter quotient is called the **outer derivation algebra**. Given any homomorphism $\varphi$ in above sense, we say $\mathfrak{g}$ and $\mathfrak{h}$ are **coupled** by $\varphi$. The pair $(h, \varphi)$ is said to be a **$\mathfrak{g}$-kernel** and **extendible** if it derived from a split extension.
Not every $\mathfrak{g}$-kernel is extendible. Given an extension we should find a proper map from $\mathfrak{g}$ to $Der(\mathfrak{h})$.
It turns out that the transversal map $\gamma$ determines a “covering map" $\sigma$ whence defines a coupling. One can summarize the details through the following diagram: $$\begin{tikzcd}
&0 \ar{r} &ad(\mathfrak{h}) \ar[tail]{r}{j} &Der(\mathfrak{h}) \ar[two heads]{r}{\sharp} &Out(\mathfrak{h}) \ar{r} &0 \\
&0 \ar{r} &\mathfrak{h} \ar[tail]{r}{\alpha} \ar{u}{ad} &\mathfrak{e} \ar[two heads]{r}{\beta} \ar{u}{ad} &\mathfrak{g} \ar{r} \ar[bend left, yshift=0.6ex]{l}{\gamma} \ar[dashed]{u}{\varphi} \ar[swap, dashed]{ul}{\exists \sigma^\gamma} &0 \\
&&&&& \mathfrak{g}\wedge \mathfrak{g} \ar[swap, dashed]{uul}{R^\varphi} \ar[bend left, dashed]{ulll}{h^\gamma}
\end{tikzcd}$$ and define $$\begin{aligned}
\sigma^\gamma_x(l)&=\alpha^{-1}\big(ad_{\beta^{-1}(x)}(\alpha(l))\big)\\
&=\alpha^{-1}\big([\beta^{-1}(x),\alpha(l)]\big)\in \mathfrak{h},\end{aligned}$$ for all $x\in \mathfrak{g}$ and $l\in \mathfrak{h}$.
There is a well-known criterion for extensibility:
A $\mathfrak{g}$-kernel $(\mathfrak{h},\varphi)$ is extendible $\Leftrightarrow $ a cohomology class in $H^3(\mathfrak{g}, Zh)$ derived from $\varphi$ vanishes
In general, a three-dimensional cohomology can arise out of the context of extension. The following pictures display the whole steps: $$\begin{tikzcd}
&&& ad(\mathfrak{h}) \ar[hook]{d} \\
0 \ar{r} &Z\mathfrak{h} \ar{r}{i} &L \ar{r}{ad} &Der(\mathfrak{h}) \ar{r}{\sharp} &Out(\mathfrak{h}) \ar{r} &0 \\
&&& \mathfrak{g} \ar[dashed]{u}{\sigma}
\ar[swap]{ur}{\varphi}
\end{tikzcd}$$
$$\begin{tikzcd}
&&& ad(\mathfrak{h}) \ar[hook]{d} \ar[dashed, bend left]{dr}{\sharp} \\
0 \ar{r} &Z\mathfrak{h} \ar{r}{i} &\mathfrak{h} \ar{r} \ar[two heads]{ur}{ad} &Der(\mathfrak{h}) \ar{r} &Out(\mathfrak{h}) \ar{r} &0 \\
&&& \mathfrak{g}\wedge \mathfrak{g} \ar{u}
\ar[dashed]{ul}{H}
\ar[bend right, crossing over, swap, near end, shift right=2.8ex]{uu}{R^\sigma}
\ar[bend right, swap, yshift=1ex]{ur}{R^\varphi}
\end{tikzcd}$$
where $R^\sigma(x_1 \wedge x_2):=[\sigma_{x_1},\sigma_{x_2}]-\sigma_{[x_1,x_2]}$ is nonzero and $\sharp \circ R^\sigma= R^\varphi=0$
$$\begin{tikzcd}
0 \ar{r} &Z\mathfrak{h} \ar{r}{i} &L \ar{r}{ad} &Der(\mathfrak{h}) \ar{r}{\sharp} &Out(\mathfrak{h}) \ar{r} &0 \\
&&& \mathfrak{g} \wedge \mathfrak{g} \wedge \mathfrak{g} \ar{ull}{\Delta^\sigma H}
\ar[swap]{u}{\Delta^\sigma R^\sigma}
\end{tikzcd}$$
where $\Delta: Alt^n(\mathfrak{g},\mathfrak{h}) \rightarrow Alt^{n+1}(\mathfrak{g},\mathfrak{h})$ happen to be a “symbolic" differential, and $$\begin{aligned}
f(\sigma, H)=f(x_1 \wedge x_2 \wedge x_3)&:=\Delta^\sigma H(x_1 \wedge x_2 \wedge x_3)\end{aligned}$$
Moreover, $\Delta^{\sigma} R^{\sigma}=0$.
Another more generic pattern of this kernel problem reduces to associative algebras. In [@Hoch46] Hochschild introduces the laborious term “bimultiplication algebra" replacing the position of Lie-wise derivation algebra, the inner and outer ones. Part **2** and **3** provides all basic definitions and derives the target cocycle. In part **4** we follow Mackenzie to present a Maurer-Cartan form in the meaning of associative algebra so that one can see the exclusive dependence of the cocycle. For completeness, we refer to **Appendix C** in checking the classical criterion for zero obstruction and extension. This follows on Mackenzie’s work in a contemporary pattern of formulation for transitive Lie algebroid in [@Mackz05]. To overcome the obstacle between the associative algebra and Lie algebra, in part **5** we shall build a fundamental bridge between them. In part **6** we state the main theorems and shortly sketch their proof. We write proofs of two structure theorems, especially the simplified one, in **Appendix D** and **E**. Finally, we develop their Lie-counterpart in **8**, under-organized in [@Hoch54a]. Note that the two consecutive **Appendix A** and **B** are real appendices in this paper, where we vainly offer the preliminary, if not being exhaustive, knowledge of classical Hochschild cohomology; see also [@CE56].
I am very indebted to Professor A. C. Mishchenko and Professor V. M. Manuilov for their constant advice to the modification of this paper.
Bimultiplication Algebra and Coupling
=====================================
A [*bimultiplication*]{} is pair of linear mappings $(u, v)$ of $K$ into itself, satisfying the following conditions: $$\begin{aligned}
u(k_1+k_2)=u(k_1)+u(k_2), &\quad v(k_1+k_2)=v(k_1)+v(k_2) \\
u(\alpha k)=\alpha u(k), &\quad v(\alpha k)=\alpha v(k) \\\end{aligned}$$ and $$\begin{aligned}
k_1 u (k_2) &= v (k_1) k_2 \\
u (k_1 k_2) &= u (k_1) k_2 \\
v (k_1 k_2) &= k_1 v (k_2),\end{aligned}$$ for any $k_1,k_2$ in $K$.
If we write $\sigma=(u, v)$, then any $\sigma$ is in $$Hom_{\mathbb{F}}(K, K) \oplus Hom_{\mathbb{F}}(K,K)^{op}$$ where $$k \mapsto \sigma k \quad \text{and} \quad
k \mapsto \sigma^* k:=k\sigma$$ We denote the family of the pairs of endomorphism of $K$ by $$Mul(K):=\big(End(K), End(K)^{op}\big).$$
The operations of addition, multiplication and scalar multiplication between the pairs of endomorphisms are defined as follows: $$\begin{aligned}
(u_1,v_1)+(u_2,v_2)&=(u_1+u_2,v_1+v_2) \\
(u_1,v_1)(u_2,v_2)&=(u_1u_2,v_2v_1) \\
\alpha(u,v)&=(\alpha u,\alpha v), \alpha \in R\end{aligned}$$
In this way, the family $Mul(K)$ forms an unital associative $\mathbb{F}$-algebra, called the [*bimultiplication algebra*]{} of $K$. Its identity element is $(1_u, 1_v)$.
For any $k_0$ in $K$, the pair $(u_{k_0}, v_{k_0})\in Mul(K)$ of endomorphisms of $K$ is called an [*inner bimultiplication produced by $k_0$*]{} if it satisfies the following conditions: $$\begin{aligned}
u_{k_0} (k) &=k_0 k \\
v_{k_0} (k) &=k k_0, \end{aligned}$$ for all $k$ in $K$. The family of inner bimultiplications produced by an element of $K$ is called the [*inner bimultiplication algebra*]{}. Denote it by $Inn(K)$.
Let us write $(u_{k_0}, v_{k_0})=(k_0^1, k_0^2)$, which treats maps as elements for the computation purpose.
$Inn(K)$ becomes a subalgebra of $Mul(K)$. Moreover, $Inn(K)\vartriangleleft Mul(K)$.
In fact, $$\begin{aligned}
(u,v)(k_0^1,k_0^2)(K)&=(u k_0^1,k_0^2 v)(K) \\
&=(u(k_0 k),v(k) k_0) \\
&=(u(k_0)k,k u(k_0)) \\
&=(u(k_0)^1,u(k_0)^2)(k)\in Inn(K)
\end{aligned}$$
Two bimultiplications $(u_1, v_1)$ and $(u_2, v_2)$ are called [*permutable*]{} if $v_2 u_1(k)=u_1 v_2(k)$ and $v_1 u_2(k)=u_2 v_1(k)$ for any $k$ in $K$.
A bimultiplication $\sigma$ is said to be **self-permutable** if $\sigma(k\sigma)=\sigma k (\sigma)$. Indeed, every inner bimultiplication of form $(k_0^1, k_0^2)$ is self-permutable. In fact, we have $k_0^1 k_0^2=k_0^2 k_0^1$. In fact,we have $k_0^1 k_0^2 (k)=k_0^1 (k k_0)=k_0 k k_0=(k_0 k) k_0=k_0^2 (k_0 k)=k_0^2 k_0^1 (k)$. The set of all self-permutatble elements needs not to be an subalgebra of $Mul(K)$, nor to be a ring. One should refer the these definitions to \[Mac58\].
The quotient algebra $Out(K):=Mul(K)/Inn(K)$ is called the [*outer bimultiplications algebra*]{} of $K$.
The [*biannihilator*]{} of $K$ is defined to be $$AnniK:=\{k\in K|k K=(0)=K k\} \vartriangleleft K,$$
The map $$\begin{aligned}
\epsilon: K &\rightarrow Mul(K) \\
k_0 &\mapsto (u_{k_0},v_{k_0})\end{aligned}$$ is an algebra homomorphism. The image subset $\epsilon(K)$ consists of elements $$\{(u_{k_0},v_{k_0})|u_{k_0}(k)=k_0 k, v_{k_0}(k) =k k_0, \forall k_0, k \in K \}$$
$Inn(K)=im\epsilon, Anni=ker\epsilon$ and $Out(K)=coker \epsilon$ such that the following sequence is exact. $$\begin{tikzcd}
&&&& im\epsilon \ar[hook]{d} \\
& 0 \ar{r} &ker\epsilon \ar{r} &K \ar{r}{\epsilon} &Mul(K) \ar{r}{\iota} &coker\epsilon \ar{r} &0 \\
\end{tikzcd}$$
Connections and Twisted Module
==============================
Let $A$ and $K$ be two associative $R$-algebras. An **A-connection** on $K$ is a linear map $\mu: A \rightarrow Mul(K)$.
A connection is said to be **flat** if it becomes a homomorphism of algebra.
A connection is said to be **regular** if the image $\mu(A)$ consist of permutable elements.
For any algebra $K$ and any $A$-connection $\mu$ on $K$, the pair $(K, \mu)$ is called a **representation** of $A$or an **$A$-module** provided the flatness of $\mu$.
In general, a connection may lose its flatness, hence there is no module structure on $K$ (somehow being“hindered").
A **coupling** of $A$ is a homomorphism of algebras $\xi:A \rightarrow Out(K)$. We also say $A$ and $K$ are **coupled** by $\xi$. In this case, the pair $(K, \xi)$ is called an **A-kernel**
Let $\natural$ be the natural projection. A regular $A$-connection $\mu$ such that the following diagram commutes $$\begin{tikzcd}
&Mul(K) \ar{r}{\natural} &Out(K) \\
&A \ar[dashed]{u}{\mu}
\ar[swap]{ur}{\xi}
\end{tikzcd}$$ is called an **(associative) bimultiplication law** that cover $\xi$.
Note that $\xi(A)$ consists of permutable elements if and only if $\mu(A)$ does, therefore the regularity of $\xi$ follows.
For each $a\in A$, we shall write the element in $\xi(A)$ by $\xi_a:=([u]_a, [v]_a)$, the pair of quotient endomorphism induced by the element in $\mu(A)$. We also write $\mu_a=(u_a, v_a)$ for indicating the potential $A$-actions on $K$: $$\begin{aligned}
&u_a: k\mapsto a\cdot_\mu k \\
&v_a: k\mapsto k\cdot_\mu a\end{aligned}$$
On the other hand, mimicking Hochscild cohomology, we may define the **twisted module** $Hom(A^{\otimes^n}, K)=\Omega^n(A, K)$ for a representation $(K, \mu)$.
A “symbolic" differential $\Delta^\mu : \Omega^n(A,K) \rightarrow \Omega^{n+1}(A,K)$ induced by the connection $\mu$ is given by $$\begin{aligned}
\Delta^\mu(f)&(a_1\otimes \cdots\otimes a_{n+1})
=u_{a_1}f(a_2\otimes \cdots \otimes a_{n+1}) \\
&+\sum_{i=1}^n (-1)^i f(a_1 \otimes \cdots \otimes a_i a_{i+1}\otimes \cdots \otimes a_{n+1})+(-1)^{n+1}v_{a_{n+1}}f(a_1\otimes \cdots \otimes a_n)\end{aligned}$$ where $f\in \Omega^n(A,K)$ and $\mu_a=(u_a, v_a)$ for any $a\in A$.
Note that $\Delta^\mu \Delta^\mu=0$ fails as one drops flatness generally.
The Emergence of 3-Cocycles
===========================
A non-flat bimultiplication law $\mu$ ensues a bilinear map $R^\mu : A\otimes A \rightarrow Mul(K)$ where $$R^\mu(a_1\otimes a_2)=\mu(a_1)\mu(a_2)-\mu(a_1a_2)$$ for any $a_1, a_2\in A$.
The difference $\mu(\cdot)\mu(\cdot)-\mu(\cdot \cdot)$ is non-zero and it lies in $Inn(K)$. Indeed, as the composition $\xi$ is a homomorphism, we have $\xi(a_1)\xi(a_2)-\xi(a_1a_2)=\natural\circ \mu(a_1) \iota \circ \mu(a_2)-\natural \circ \mu(a_1 a_2)=\natural \circ(\mu(a_1)\mu(a_2)-\mu(a_1 a_2))=\natural \circ R^\mu(a_1\otimes a_2)=0$. Therefore, $R^\mu$ induces a bilinear map $A\otimes A \rightarrow Inn(K)$ and we still denote it by $R^\mu$. It is called the **curvature** with respect to $\mu$.
For every bimultiplication law $\mu$, there are bilinear mappings $h: A\otimes A \rightarrow K$ such that $$\epsilon \circ h=R^\mu$$ In other word, each $h$ naturally lifts $R^\mu$.
Recall that $\epsilon$ is surjective and the element $R^\mu(\cdot, \cdot)$ is an inner bimultiplication produced by some $k=h(\cdot,\cdot)$ in the preimage. $$\begin{tikzcd}
&Inn(K) \subseteq Mul(K) \\
K \ar{ur}{\epsilon} &A\otimes A \ar[dashed]{l}{h} \ar[swap]{u}{R^\mu}
\end{tikzcd}$$
The map $h: A\otimes A \rightarrow K$ is called a **hindrance** of the coupling.
In other words, it hinders a (non-flat) connection that covers the coupling.
Let us concretely compute the curvature by assigning an element $k$. $$\begin{aligned}
\mu(a_2)\mu(a_3)(k)
&=(u,v)(u^\prime,v^\prime)(k)=(uu^\prime,v^\prime v)(k)=\bigl(u(u^\prime(k)), v^\prime(v(k))\bigr) \\
&=\bigl(a_2\cdot(a_3\cdot k), (k\cdot a_2)\cdot a_3 \bigr) \\
\mu(a_2a_3)(k)
&=(u^{\prime\prime},v^{\prime\prime})(k)=(u^{\prime\prime}(k), v^{\prime\prime}(k))=\bigl((a_2a_3)\cdot k,\ k\cdot(a_2 a_3)\bigr)\end{aligned}$$
So, the first coordinate in the difference of above two identities is $$a_2\cdot(a_3\cdot k)-(a_2a_3)\cdot k$$ and the second one is $$(k\cdot a_2)\cdot a_3-\ k\cdot(a_2a_3)$$
Since such an inner multiplication is produced by a hindrance, we have $$\mu(\cdot)\mu(\cdot)-\mu(\cdot\cdot)
=(uu^\prime-u^{\prime\prime},v^\prime v-v^{\prime\prime})
=\epsilon\circ h(\cdot, \cdot)
=\bigl(h(\cdot,\cdot)^1, h(\cdot,\cdot)^2\bigr)$$
More precisely, by applying a $k$, we have $$\begin{aligned}
h(a_2\otimes a_3)^1(k)&=h(a_2\otimes a_3)k \\
h(a_2\otimes a_3)^2(k)&=kh(a_2\otimes a_3)\end{aligned}$$
Comparing these coordinates, we get two important identities: $$\begin{aligned}
a_2\cdot(a_3\cdot k)-(a_2a_3)\cdot k &=h(a_2\otimes a_3)k
\tag{3.1} \\
(k\cdot a_2)\cdot a_3-k\cdot(a_2a_3) &=kh(a_2\otimes a_3)
\tag{3.2} \end{aligned}$$
We now deduce some characteristic identities involving $h$ in detail when it takes triple variables in $A$: For any $k\in K, a_r, a_s, a_t\in A$, we firstly note that $$\begin{aligned}
a_r\cdot((a_s a_t)\cdot k)-(a_r a_s a_t)\cdot k &=h(a_r\otimes a_s a_t)k
\tag{3.1$a$} \\
(a_r a_s)\cdot(a_t\cdot k)-(a_r a_s a_t)\cdot k &=h(a_ra_s\otimes a_t)k
\tag{3.1$b$} \end{aligned}$$ by viewing $a_r a_s$ as an integral symbol and then substituting it into (3.1) in two different ways.
And we have $$a_r\cdot(a_s\cdot(a_t\cdot k))-a_ra_s\cdot(a_t\cdot k)=h(a_r\otimes a_s)(a_t\cdot k)
\tag{3.1$c$}$$ by viewing $a_r\cdot k$ as an integral symbol and then substituting it into (3.1) again.
Secondly, with (3.1) and then (3.1$a$), we have $$\begin{aligned}
a_1\cdot\bigl(a_2\cdot(a_3\cdot k)\bigr) &=a_1\cdot\bigl((a_2a_3)\cdot k+h(a_2\otimes a_3)k \bigr) \\
&=a_1\cdot\bigl((a_2a_3)\cdot k \bigr)+(a_1\cdot h(a_2\otimes a_3))\cdot k \\
&=(a_1a_2a_3)\cdot k+h(a_1\otimes a_2a_3)k+(a_1\cdot h(a_2\otimes a_3))k\end{aligned}$$
On the other hand, by (2.1$c$) and then by (2.1$b$), we have $$\begin{aligned}
a_1\cdot\bigl(a_2\cdot(a_3\cdot k)\bigr) &=(a_1a_2)\cdot(a_3\cdot k)+h\{a_1\otimes a_2\}(a_3\cdot k) \\
&=(a_1a_2a_3)\cdot k+h(a_1a_2\otimes a_3)k+(h(a_1\otimes a_2)\cdot a_3)k\end{aligned}$$
We define a trilinear cochain with respect to $\mu_a=(u_a, v_a)$ and $h$ $$f(a_1\otimes a_2\otimes a_3):=u_{a_1} h(a_2\otimes a_3)-h(a_1a_2\otimes a_3)+h(a_1\otimes a_2a_3)-v_{a_3}h(a_1\otimes a_2)$$
Multiply by $k$ on the right on each side of this formula, $$f(a_1\otimes a_2\otimes a_3)\cdot k=a_1\cdot\bigl(a_2\cdot(a_3\cdot k)\bigr)-a_1\cdot\bigl(a_2\cdot(a_3\cdot k)\bigr)=0$$
Likewise, starting from (3.2), we can compute the coboundary formula by multiplying $k$ on the left and get $k\cdot f(a_1\otimes a_2\otimes a_3)=0$. These two identities imply $f$ takes its value in the biannihilator of $K$.
The coupling defines a structure of $A$-$A$-bimodule on the biannihilator of $K$, being independent from the choice of $\mu$.
Let $AnniK$ be the biannihilator of $K$. Let $n\in AnniK$, we compute $a_1(a_2\cdot n)-(a_1a_2)\cdot n=h(a_1\otimes a_2)\cdot n=0$. Similarly, we have $(n\cdot a_2)\cdot a_3-\ n\cdot(a_2a_3)=0$, either. This proves that $AnniK$ is both left and right $A$-module.
Furthermore, we need $(a_1\cdot n)\cdot a_2=a_1\cdot(n\cdot a_2)$ so that $AnniK$ becomes a bimodule. But this is just from the formula $v_2u_1(n)-u_1v_2(n)=0$, by the permutability of $\mu$.
The coupling $\xi$ induces a representation on $AnniK$ not depending one the choice of $\mu$. Indeed, let $\mu:A\rightarrow Mul(K)$ be a covering of $\xi$, then $\mu_{AnniK}:A\rightarrow Mul(K)$ defines the mapping $\mu_{AnniK}(a): n\mapsto n$ instead of $k\mapsto k$. Now if $\mu|_{AnniK}, \mu'|_{AnniK}$ are two restricted coverings of $\varphi$, then $\mu'|_{AnniK}-\mu|_{AnniK}(a)$ is an element in $Inn(K)$, as well as in $Inn(AnniK)$. Such a inner bimultiplication is produced by some elements $n\in {AnniK}$ and therefore vanishes on $AnniK$. This concludes that $AnniK$ does not rely on the choice of $\mu$.
No cochain complex follows for the twisted module $\Omega(A, K)$. However, the restricted map $\mu|_{AnniK}: A \rightarrow Mul(AnniK)$ is a homomorphism, thus it is a flat connection. We change the notation of it by $\rho^\xi$, indicating its exclusive dependence on the choice of $\xi$.
the [*central (or annihilatoral) $A$-connection*]{} on $AnniK$. Due to the flatness the pair $(AnniK^\xi, \rho^\xi)$ is the [*central (or annihilatoral) representation*]{} of $A$, and the twisted module $Hom(A^{\otimes^n}, AnniK)$ becomes the standard Hochschild cochain complex $C^n(A, AnniK)=$, together with the differential $\delta^{\rho^\xi}$ (or simply $\delta^\xi$) in the usual sense.
In other words, $\Delta^\mu$ collapse to $\delta^\xi$ on $AnniK$.
Denote $$f=\Delta^\mu(h)$$ a sort of taking the differential of $h$ (actually no applying $\delta^{xi}$ on $h$ as it takes values in the non-module $K$). Or we write $$f=f(\mu, h)$$ to indicate its stem from the law-covering $\mu$ and the hindrance $h$.
Therefore, we obtain the following theorem:
$f(\mu, h) \in C^3(A,AnniK)$
$f(\mu, h) \in Z^3(A,AnniK)$
In fact, $$\begin{aligned}
(\delta^\xi \Delta^\mu h)(a_1\otimes a_2\otimes &a_3\otimes a_4)=a_1\cdot\Delta h(a_2\otimes a_3\otimes a_4)-\Delta h(a_1a_2\otimes a_3\otimes a_4) \\
&+\Delta h(a_1\otimes a_2a_3\otimes a_4)-\Delta h(a_1\otimes a_2\otimes a_3a_4)+\Delta h(a_1\otimes a_2\otimes a_3)\cdot a_4\end{aligned}$$
Expand all $\Delta h$, $$\begin{aligned}
&=a_1\cdot \bigl(a_2\cdot h(a_3\otimes a_4)-h(a_2a_3\otimes a_4)+h(a_2\otimes a_3a_4)-h(a_2\otimes a_3)\cdot a_4 \bigr) \\
&\qquad -\bigl(a_1a_2\cdot h(a_3\otimes a_4)-h(a_1a_2a_3\otimes a_4)+h(a_1a_2\otimes a_3a_4)-h(a_1a_2\otimes a_3)\cdot a_4\bigr) \\
&\qquad +\bigl(a_1\cdot h(a_2a_3\otimes a_4)-h(a_1a_2a_3\otimes a_4)+h(a_1\otimes a_2a_3a_4)-h(a_1\otimes a_2a_3)\cdot a_4\bigr) \\
&\qquad -\bigl(a_1\cdot h(a_2\otimes a_3a_4)-h(a_1a_2\otimes a_3a_4)+h(a_1\otimes a_2a_3a_4)-h(a_1\otimes a_2)\cdot a_3a_4 \bigr) \\
&\qquad +\bigl(a_1\cdot h(a_2\otimes a_3)-h(a_1a_2\otimes a_3)+h(a_1\otimes a_2a_3)-h(a_1\otimes a_2)\cdot a_3\bigr)\cdot a_4\end{aligned}$$
Most of the terms are canceled out, so the remaining terms are the sum of the following two terms: $$\begin{aligned}
& a_1\cdot(a_2\cdot h(a_3\otimes a_4))-a_1a_2\cdot h(a_3\otimes a_4) \\
& h(a_1\otimes a_2)\cdot a_3a_4-(h(a_1\otimes a_2)\cdot a_3)\cdot a_4
\tag{$\star$}\end{aligned}$$
Since $h(\cdot, \cdot)\in K$, we can rewrite (2.1) and (2.2) as follows: $$\begin{aligned}
a_r\cdot(a_s\cdot h(\cdot,\cdot)) &-(a_ra_s)\cdot h(\cdot,\cdot)=h(a_r,a_s)h(\cdot,\cdot)
\tag{2.3} \\
(h(\cdot,\cdot)\cdot a_r)\cdot a_s &-h(\cdot,\cdot)\cdot(a_ra_s)=h(\cdot,\cdot)h(a_r,a_s)
\tag{2.4} \end{aligned}$$
Therefore, by applying these rules to $(\star)$, we have shown $(\Delta_N \Delta h)=h(a_1\otimes a_2)h(a_3\otimes a_4)-h(a_1\otimes a_2)h(a_3\otimes a_4)=0$, as desired.
Given a bimultiplication law $\mu$ that covers $\xi$, let $h$, $h'$ be two hindrances of $R^\mu$. Write $f=f(\mu, h)=\Delta^\mu h$ and $f'=f(\mu, h')=\Delta^\mu h'$ to be the corresponding cocycles. Then $h-h'=i\circ g$ for some $g\in C^2(A,ZK)$ and $f-f'=\delta^\xi g$.
1). Firstly we have $\epsilon \circ (h- h')=R^\mu-R^\mu=0$. Since $\epsilon(k)=0$ implies $k=i\{n\}$ and $i\{n\}k=0=ki\{n\}$, then there is a unique $g: A\otimes A \rightarrow N$ such that $h-h'=i\circ g$.\
2).For any $a_1, a_2, a_3\in A$ We compute $$\begin{aligned}
&i(f(a_1\otimes a_2 \otimes a_3)-f'(a_1\otimes a_2 \otimes a_3)) \\
&=i \circ (a_1\cdot (h-h')(a_2 \otimes a_3)-(h-h')(a_1a_2 \otimes a_3)+(h-h')(a_1\otimes a_2a_3)-(h-h')(a_1\otimes a_2)\cdot a_3) \\
&=i \circ (a_1\cdot (i\circ g)(a_2 \otimes a_3)-(i\circ g)(a_1a_2 \otimes a_3)+(i\circ g)(a_1\otimes a_2a_3)-(i\circ g)(a_1\otimes a_2)\cdot a_3) \\
&=i\circ \delta g(a_1\otimes a_2 \otimes a_3), \end{aligned}$$ as desired.
The Independence of the $3$-Cocycles
====================================
We have already seen that the construction of our target $3$-cocycle employs the coverings $\mu$ and hinderances $h$ and thus we write $f(\mu, h)$. Nevertheless, the choice of this cocycle does *not* rely on the choice of $\mu$ and $h$–it only depends on the given coupling $\xi$, which we shall show in this section. Once we succeed in doing that, this cohomology $\{f\}$ will be called the *obstruction* determined by $\xi$.
Firstly, we show $f$ does not rely on the choice of $\mu$.
Let $\mu$ and $\mu'$ be two bimultiplication laws that cover $\xi$. Then $$\mu'=\mu+ \epsilon \circ l$$ for some maps $l:A\rightarrow K$, and $$R^{\mu'}-R^\mu=\epsilon \circ(\Delta^\mu(l)+ l\cdot l)$$
Since $\epsilon$ is a homomorphism and $l(a_i)\in K$, then $$\begin{aligned}
&(R^{\mu'}-R^\mu)(a_1\otimes a_2) \\
&=(u_{a_1},v_{a_1})(l(a_2)^1,l(a_2)^2)+(l(a_1 )^1,l(a_1)^2)(u_{a_2},v_{a_2})+(l(a_1)^1,l(a_1)^2)(l(a_2)^1,l(a_2)^2) \\
&-(l(a_1 a_2)^1,l(a_1 a_2)^2) \\
&=(u_{a_1} l(a_2)^1,l(a_2)^2 v_{a_1}))+(l(a_1)^1 u_{a_2},v_{a_2} l(a_1)^2)+(l(a_1)^1 l(a_2)^1,l(a_2)^2 l(a_1)^2)-(l(a_1 a_2)^1,l(a_1 a_2)^2) \end{aligned}$$
The first coordinate is $$\begin{aligned}
&u_{a_1}l(a_2)^1-l(a_1 a_2)^1+l(a_1)^1 u_{a_2}+l(a_1)^1 l(a_2)^1 \\
&=a_1\cdot l(a_2)^1-l(a_1 a_2)^1+l(a_1)^1\cdot a_2+l(a_1)^1l(a_2)^1 \\
&=(\Delta^\mu (l)+l\cdot l)^1\end{aligned}$$
Likewise, we have the second coordinate $$\begin{aligned}
&u_{a_1}l(a_2)^2-l(a_1 a_2)^2+v_{a_2}l(a_1)^2+ l(a_2)^2 l(a_1)^2 \\
&=a_1\cdot l(a_2)^2-l(a_1 a_2)^2+l(a_1)^2\cdot a_2+ l(a_2)^2 l(a_1)^2 \\
&=(\Delta^\mu (l)+l\cdot l)^2\end{aligned}$$ Therefore, we have $$\begin{aligned}
&(R^{\mu'}-R^\mu)(a_1\otimes a_2) \\
&=((\Delta^\mu (l(a_1\cdot a_2))+l(a_1)l(a_2))^1,(\Delta^\mu (l(a_1\cdot a_2))+l(a_1)l(a_2))^2) \\
&=\epsilon \circ [\Delta^\mu (l)+l\cdot l]\end{aligned}$$
Let $\mu$ be a bimultiplication law that covers $\xi$ and $h$ a hindrance of $R^\mu$. Let $\mu'$ be another bimultiplication law that covers $xi$ such that $\mu'= \mu+ \epsilon \circ l$ for some maps $l: A \rightarrow K$. Then $$h' = h+ (\Delta^\mu (l)+ l\cdot l) \qquad (\star\star)$$ is a hindrance of $R^{\mu'}$. Moreover, $f(\mu, h)=f(\mu',h')$.
$$\begin{aligned}
R^{\mu'}=\epsilon \circ h' &, R^\mu=\epsilon \circ h \\
\epsilon \circ h'=\epsilon \circ h &+\epsilon \circ (\Delta^\mu (l)+l\cdot l), \end{aligned}$$
as desired.
It means that $$(h'{^1},h'{^2})(k)=(h^1,h^2)(k)+((\Delta^\mu(l)+l\cdot l)^1,(\Delta^\mu (l)+l\cdot l)^2)(k)$$ By applying $k$ for the first coordinate, for example, we have the following expression $$h'(a_1\otimes a_2)\cdot k=h(a_1\otimes a_2)\cdot k+[\Delta^\mu l(a_1\cdot a_2)+l(a_1)l(a_2)]\cdot k$$
By omitting the superscript temporarily and taking differential with respect to $\Delta^{\mu'}$, we have $$\Delta^{\mu'} h'^{1}=\Delta^{\mu'} h^1+ \Delta^{\mu'}(\Delta^\mu (l)+l\cdot l)^1$$ Then $$\Delta^{\mu'} h'^{1}-\Delta^\mu h^1=\Delta^{\mu'} h^1-\Delta^\mu h^1+ \Delta^{\mu'}(\Delta^\mu (l)+l\cdot l)^1$$ The LHS is $$\begin{aligned}
&(\Delta^{\mu'} {h'}^{1}-\Delta^\mu h^1)(a_1\otimes a_2 \otimes a_3) \\
&=u_{a_1}' h(a_2\otimes a_3)^1-h(a_1 a_2\otimes a_3)^1+h(a_1\otimes a_2 a_3 )^1+v_{a_3}' h(a_1\otimes a_2)^1 \\
&-u_{a_1}h(a_2\otimes a_3)^1+h(a_1 a_2\otimes a_3)^1-h(a_1\otimes a_2 a_3)^1-v_{a_3}h(a_1\otimes a_2)^1 \\
&=(u_{a_1}'-u_{a_1})h(a_2\otimes a_3)^1-(v_{a_3}'-v_{a_3})h(a_1\otimes a_2 )^1 \\
&=l(a_1)^1 h(a_2\otimes a_3)^1-h(a_1\otimes a_2)^1 l(a_3)^1\end{aligned}$$
And the RHS is(omit $1$) $$\begin{aligned}
& \Delta^{\mu'}(\Delta^\mu (l)+l\cdot l) \\
&=\Delta^{\mu+\epsilon \circ l} (\Delta^\mu (l)+l\cdot l) \\
&=\Delta^\mu \Delta^\mu (l)+\Delta^\mu (l\cdot l)+\Delta^{\epsilon \circ l} \Delta^\mu (l)+\Delta^{\epsilon \circ l}(l\cdot l)\end{aligned}$$
Let us compute each of the above four terms: $$\begin{aligned}
&\Delta^\mu \Delta^\mu (l)(a_1\otimes a_2\otimes a_3) \\
&=a_1\cdot \Delta l(a_2\otimes a_3)-\Delta l(a_1 a_2\otimes a_3)+\Delta l(a_1\otimes a_2 a_3)-\Delta l(a_1\otimes a_2)\cdot a_3 \\
&=a_1\cdot (a_2\cdot l(a_3))-a_1\cdot l(a_2 a_3)+a_1\cdot (l(a_2)\cdot a_3)-a_1 a_2\cdot l(a_3)+l(a_1 a_2 a_3)-l(a_1 a_2)\cdot a_3 \\
&+a_1\cdot l(a_2 a_3)+l(a_1 )\cdot a_2 a_3-(a_1\cdot l(a_2))\cdot a_3-l(a_1 a_2 a_3)+l(a_1 a_2)\cdot a_3-(l(a_1 )\cdot a_2)\cdot a_3 \\
&=a_1\cdot (a_2\cdot l(a_3))-a_1 a_2\cdot l(a_3)+l(a_1)\cdot a_2 a_3-(l(a_1)\cdot a_2)\cdot a_3\end{aligned}$$ Now we view the element $l(a_i)$ as mappings(after putting $\epsilon$ in front of it). This gives us a negative part of previous: $$h(a_1\otimes a_2)^1 l(a_3)^1-l(a_1)^1 h(a_2\otimes a_3)^1$$
For $\Delta^\mu (l\cdot l)(a_1\otimes a_2\otimes a_3)$, set $l(x)l(y)=f(x,y)$, then $$\begin{aligned}
\Delta f(x,y,z)&=xf(y,z)-f(xy,z)+f(x,yz)-f(x,y)z \\
&=x(l(y)l(z))-l(xy)l(z)+l(x)l(yz)-(l(x)l(y))z\end{aligned}$$
For $(\epsilon \circ l) \Delta^\mu (l)(a_1\otimes a_2)=l(a_1)\Delta l(a_2 \otimes a_3)-\Delta l(a_1\otimes a_2)l(a_3)$, we have $$\begin{aligned}
l(x)\Delta l(y,z)&=l(x)(yl(z))-l(x)l(yz)+l(x)l(y)z \\
-\Delta l(x,y)l(z)&=-xl(y)l(z)+l(xy)l(z)-l(x)yl(z)\end{aligned}$$
So $\Delta^\mu (l\cdot l)+(\epsilon \circ l) \Delta^\mu (l)=0$
Lastly, $$\begin{aligned}
(\epsilon \circ l)(l\cdot l)&=l(a_1)^1 (l(a_2)l(a_3 ))-l(a_3)^2 (l(a_1 )l(a_2)) \\
&=l(a_1)l(a_2)l(a_3)-l(a_1)l(a_2)l(a_3) \\
&=0 \end{aligned}$$ The second coordinate can be computed similarly, whence $f(\mu',h')-f(\mu,h)=\Delta^{\mu'}h'-\Delta^\mu h=0$, as desired.
The coupling $\xi$ of $A$ defines a cohomological class in $HH^3(A,ZK, \rho^\xi)$, elements of which does not depend on the choice of the bimultiplication law $\mu$ that covers $\xi$ and the hindrance of the law.
Such a class is called the **obstruction** derived from the coupling.
As $f(\mu, h)$ becomes the representative cocycle, independent from the two “variables", we denote it by $$f:=f^\xi$$ and the corresponding cohomological class is $\{f^\xi\}=Obs(\xi)$
It is fair to illustrate the utility of $f$ now. The hindrance $h$ can be roughly viewed as a “pre-obstruction" incarnating the difference in $R^\mu$, but it has several defects: first of all, $h$ belongs to the twisted module $C^2(A, K)$ of no cochain complex. Furthermore, $h=h(\mu)$ so it does not solely rely on the choice of the coulping $\xi$. As we have seen in this part, the obstruction cocycle $f$ nullify all the drawbacks. Indeed, this is the essence of this classical problem.
The following proposition resonates ones in [@LiMishGa14].
It can be summarized by these following digramms: $$\begin{tikzcd}
&&& Inn(K) \ar[hook]{d} \\
0 \ar{r} &AnniK \ar{r}{i} &K \ar{r}{\epsilon} &Mul(K) \ar{r}{\natural} &Out(K) \ar{r} &0 \\
&&& A\ar[dashed]{u}{\mu}
\ar[swap]{ur}{\xi}
\end{tikzcd}$$
$$\begin{tikzcd}
&&& Inn(K) \ar[hook]{d} \ar[dashed, bend left]{dr}{\natural} \\
0 \ar{r} &AnniK \ar{r}{i} &K \ar{r} \ar[two heads]{ur}{\epsilon} &Mul(K) \ar{r} &Out(K) \ar{r} &0 \\
&&& A\otimes A \ar{u}
\ar[dashed]{ul}{h}
\ar[bend right, crossing over, swap, near end, shift right=2.8ex]{uu}{R^\mu}
\ar[bend right, swap, yshift=1ex]{ur}{R^\xi}
\end{tikzcd}$$ where $\natural \circ R^\mu= R^\xi=0$
$$\begin{tikzcd}
0 \ar{r} &AnniK \ar{r}{i} &K \ar{r}{\epsilon} &Mul(K) \ar{r}{\natural} &Out(K) \ar{r} &0 \\
&&& A\otimes A\otimes A \ar{ull}{\Delta^\mu h}
\ar[swap]{u}{\Delta^\mu R^\mu}
\end{tikzcd}$$ where $\Delta^{\mu} R^{\mu}=0$.
Identifying Lie and Associative Cochains
========================================
In [@Hoch54b], Hochschild formulates a proper cochain complex for computing the cohomology of both the ordinary and restricted Lie algebra. This equivalence, identifying Chevelley-Elienberg complex and the (normalized) Cartan-Hochschild standard complex in a particular way, is recalled in Tylor Evans’ PhD thesis, [@Evan00]. We are going to generalize the definitions for computing Lie algeborid cohomology in terms of its universal enveloping algebroid.
Let $\mathfrak{g}$ be a Lie algebra over field $\mathbb{F}$ and let $\mathcal{M}$ be a $\mathfrak{g}$-module. It is well-known that there is a one-to-one correspondence between the Lie algebra representations of $\mathfrak{g}$ and the unitary representations of $U(\mathfrak{g})$. In this way, one may view $\mathcal{M}$ as a unitary $U(\mathfrak{g})$-module.
1\) Let us define the complex of “Lie type":
$$\mathcal{C}_*=\{ \mathcal{C}_n, d^{\mathcal{C}} \}$$ where $$\mathcal{C}_n :=U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!n} \mathfrak{g}$$
Clearly, each $\mathcal{C}_n $ becomes a $U(\mathfrak{g})$-module in a natural fashion.
The coboundary operator $d^{\mathcal{C}}_n: \mathcal{C}_n \rightarrow \mathcal{C}_{n-1} $ is defined by $$\begin{aligned}
d^{\mathcal{C}}_n(\mathfrak{u} \otimes x_1 \wedge \cdots \wedge x_n)&:=\sum_{i=1}^{n}(-1)^{i-1}\mathfrak{u} x_i \otimes x_1 \wedge \cdots \wedge \hat{x_i} \wedge \cdots \wedge x_n \\
& +\sum_{1 \leq s<t \leq n}(-1)^{s+t-1} \mathfrak{u} \otimes [x_s, x_t] \wedge x_1 \wedge \cdots \wedge \hat{x_s} \wedge \hat{x_t} \wedge \cdots x_n\end{aligned}$$
Consider the canonical augmentation $\epsilon: U(\mathfrak{g}) \rightarrow \mathbb{F}$ induced by the map $T(\mathfrak{g}) \mapsto \mathbb{F}$. Since the augmentation is surjective, we denote its kernel by $$U(\mathfrak{g})_+=ker \epsilon,$$ that is, all of the positive parts of tensor algebra of $\mathfrak{g}$ passing over the quotient. $U(\mathfrak{g})_+$ will then play a key component of the tensor product $\tilde{S}(\cdot)$ which is the so called [*normalized standard complex*]{}.
2)Let us now define the complex of “associative type": $$\mathcal{D}_*=\{ \mathcal{D}_n, d^{\mathcal{D}} \}$$ where $$\begin{aligned}
\mathcal{D}_n &:=U(\mathfrak{g}) \otimes {\underbrace{U(\mathfrak{g})_{+} \otimes \cdots \otimes U(\mathfrak{g})_{+}}_\text{$n$-times}} \\
&= U(\mathfrak{g}) \otimes U(\mathfrak{g})_{+}^{\otimes^{n}}\end{aligned}$$
The coboundary operator $d^{\mathcal{D}}_n: \mathcal{D}_n \rightarrow \mathcal{D}_{n-1} $ is defined by $$\begin{aligned}
d^{\mathcal{D}}_n(\mathfrak{u} \otimes x_1 \otimes \cdots \otimes x_n)&:=\mathfrak{u} x_1 \otimes \cdots \otimes x_n \\
&\sum_{i=1}^{n}(-1)^i \mathfrak{u} \otimes x_1 \otimes \cdots \otimes x_i x_{i+1} \otimes \cdots \otimes x_n\end{aligned}$$
In addition, we set $\mathcal{C}_0=\mathcal{D}_0=U(\mathfrak{g})$.
In fact, we can show that the following two augmented complexs are free resolutions(acyclic) of $U(\mathfrak{g})$-modules: $$\mathcal{C}_* \twoheadrightarrow \mathbb{F} \rightarrow 0$$ and $$\mathcal{D}_* \twoheadrightarrow \mathbb{F} \rightarrow 0$$
$$\begin{tikzcd}
\cdots \ar{r}
&U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!2}\mathfrak{g} \ar{r}{d_2^\mathcal{C}} \ar{d}{\gamma}
&U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!1}\mathfrak{g} \ar{r}{d_1^\mathcal{C}} \ar{d}{\gamma}
&U(\mathfrak{g}) \ar[two heads]{r}{\epsilon} \ar{d}{\gamma = id}
&\mathbb{F} \ar{r} \ar{d}{id_\mathbb{F}}
&0 \\
\cdots \ar{r}
&U(\mathfrak{g}) \otimes U(\mathfrak{g})_{+}^{\otimes^{2}} \ar[swap]{r}{d_2^\mathcal{D}}
&U(\mathfrak{g}) \otimes U(\mathfrak{g})_{+}^{\otimes^{1}} \ar[swap]{r}{d_1^\mathcal{D}}
&U(\mathfrak{g}) \ar[two heads, swap]{r}{\epsilon}
&\mathbb{F} \ar{r}
&0
\end{tikzcd}$$
If $\alpha$ and $\beta$ are any two of chain maps, we define the [*chain homotopy*]{} between these two complexs by assigning a family of operators $H_n: \mathcal{C}_n \rightarrow \mathcal{D}_{n+1}$ such that $d^{\mathcal{D}}_{n+1} \circ H_n + H_{n-1} \circ d^{\mathcal{C}}_n=\beta_n-\alpha_n$ for all $n$. Specifically, we require the definition: $$\begin{aligned}
H(\mathfrak{u})&=1 \otimes (\mathfrak{u}-\epsilon(\mathfrak{u})) \\
H(\mathfrak{u}\otimes x_1 \otimes \cdots \otimes x_n)&=1 \otimes (\mathfrak{u}-\epsilon(\mathfrak{u})) \otimes x_1 \otimes \cdots \otimes x_n\end{aligned}$$
Let $id=\gamma: \mathcal{C}_0 \rightarrow \mathcal{D}_0$. For $n>0$, define $\gamma:\mathcal{C}_* \rightarrow \mathcal{D}_*$ by
$$\gamma(\mathfrak{u} \otimes (x_1 \wedge \cdots \wedge x_n))=\sum_{\sigma} (sgn\sigma) \mathfrak{u}\otimes x_{\sigma(1)} \otimes \cdots \otimes x_{\sigma(n)}$$
Obviously, $\epsilon \circ id = \epsilon $. Next, $\gamma$ becomes an augmentation-preserving chain map once by justifying $\gamma \circ d_i^\mathcal{C} = d_i^\mathcal{D} \circ \gamma$. By interchanging the position of Lie and associative cochains, we can define a new map which actully serves as the inverse of $\gamma$ It follows that $\gamma$ is a chain homotopy equivalence.
Let us turn our concern back to those resolutions for while. Chopping the $-1^{th}$ terms $\mathbb{F}$ and applying the left exact contravariant functor $Hom_{U(\mathfrak{g})}(-,\mathcal{M})$ to each term of both complexs, we have $$\begin{tikzcd}
0 \ar{r}
&\mathcal{M}^\mathfrak{g} \ar{r}{\delta_0^\mathcal{C}} \ar{d}
&Hom_{U(\mathfrak{g})}\big( U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!1} \mathfrak{g}, \mathcal{M} \big) \ar{r}{\delta_1^\mathcal{C}} \ar{d}
&Hom_{U(\mathfrak{g})}\big( U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!2} \mathfrak{g}, \mathcal{M} \big) \ar{r}{\delta_2^\mathcal{C}} \ar{d}
&\cdots \\
0 \ar{r}
&^{U(\mathfrak{g})} \mathcal{M}^{\textbf{0}} \ar[swap]{r}{\delta_0^\mathcal{D}}
&Hom_{U(\mathfrak{g})} \big( U(\mathfrak{g}) \otimes U(\mathfrak{g})_{+}^{\otimes^{1}}, \mathcal{M} \big) \ar[swap]{r}{\delta_1^\mathcal{D}}
&Hom_{U(\mathfrak{g})} \big( U(\mathfrak{g}) \otimes U(\mathfrak{g})_{+}^{\otimes^{2}}, \mathcal{M} \big) \ar[swap]{r} {\delta_2^\mathcal{D}}
&\cdots
\end{tikzcd}$$ where the zeroth term $Hom_{U(\mathfrak{g})} \big(U(\mathfrak{g}), \mathcal{M} \big)$ is determined by the derived functor of $\mathfrak{g}$-**Mod** or $U(\mathfrak{g})$-$\textbf{0}$-**Bimod** mapping into $R$-**Mod** respectively.
Furthermore, we have the following “equivariant"(as in topology) isomorphisms for $n \geq 1$: $$Hom_{U(\mathfrak{g})}\big( U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!n} \mathfrak{g}, \mathcal{M} \big) \cong Hom_{\mathbb{F}}\big(\bigwedge\nolimits^{\!n} \mathfrak{g}, \mathcal{M} \big),$$ and
$$Hom_{U(\mathfrak{g})} \big( U(\mathfrak{g}) \otimes U(\mathfrak{g})_{+}^{\otimes^{n}}, \mathcal{M} \big) \cong Hom_{\mathbb{F}}\big(U(\mathfrak{g})_{+}^{\otimes^{n}}, \mathcal{M} \big).$$
Therefore, the corresponding coboundary operators $\delta_n^{\mathcal{C}}$ and $\delta_n^{\mathcal{D}}$ of vector spaces(instead of $U(\mathfrak{g})$-modules) are given in the usual sense. More precisely, $$\begin{aligned}
\delta_n^{\mathcal{C}}(f)&(x_1 \wedge \cdots \wedge x_{n+1})
:=\sum_{i=1}^{n} (-1)^{i} x_i f(x_1 \wedge \cdots \wedge \hat{x_i} \wedge \cdots \wedge x_{n+1}) \\
&+\sum_{1 \leq i<j \leq n+1} (-1)^{i+j-1} f([x_i,x_{i+1}] \wedge x_1 \wedge \cdots \wedge x_{n+1}),\end{aligned}$$ and $$\begin{aligned}
\delta_n^{\mathcal{D}}(f')&(x_1 \otimes \cdots \otimes x_{n+1})
:=x_1 f'(x_2 \otimes \cdots \otimes x_{n+1}) \\
&+\sum_{i=1}^{n}(-1)^{i}f'(x_1 \otimes \cdots \otimes x_i x_{i+1} \cdots \otimes x_{n+1}).\end{aligned}$$
Note that for the associative cochain, we made the RIGHT ACTION to be ZERO in the original definition of Hochschild differentials for associative algebra!
Thus, its Lie algebra cohomology with the differential considered above is $$\begin{aligned}
H^i_{CE}(\mathfrak{g}, \mathcal{M})&:=H^* \big(\mathcal{C}^n(\mathfrak{g}, \mathcal{M}) \big) \\
&=Ext_{U(\mathfrak{g})}^i (\mathbb{F}, \mathcal{M}),\end{aligned}$$ and the Hochschild cohomology $$\begin{aligned}
H^i_{Hoch}(U(\mathfrak{g}), \mathcal{M})&:=H^* \big(\mathcal{D}^n(U(\mathfrak{g})_+, \mathcal{M}) \big) \\
&=Ext_{U(\mathfrak{g})-\textbf{0}}^i (\mathbb{F}, \mathcal{M}).\end{aligned}$$
For every associative $n$-cochain $f \in \mathcal{D}^n(U(\mathfrak{g})_+, \mathcal{M})$, we define the Lie cochain by $$f'(x_1 \wedge \cdots \wedge x_n):=\sum_{\sigma} sgn(\sigma) f(x_{\sigma(1)} \otimes \cdots \otimes x_{\sigma(n)})$$
A direct computation shows $(\delta f)'=\delta(f')$ so that the map $f \mapsto f'$ induced a homomorphism on the cohomology groups. On the other hands, we should point out that $\gamma$ together with its inverse $\gamma^{-1}$ induce a isomorphism on the homology groups. Its dual map $\gamma^*$ is then identical to the map $f \mapsto f'$ and becomes a quasi-isomorphism as well.
Consequently, the induced map $$\begin{tikzcd}
\gamma^*: H^*_{CE}(\mathfrak{g}, \mathcal{M}) \ar{r}{\cong} &H^*_{Hoch}(U(\mathfrak{g}), \mathcal{M})
\end{tikzcd}$$ is an isomorphism for all $n$.
Main Theorems
=============
In **Appendix C** one has our first correspondence
An $A$-kernel $(K,\xi)$ is derived from an extension of algebras $\Leftrightarrow Obs(\xi)=0$ in $H^3(A, AnniK)$
Now we wish to show the second correspondence $$\begin{tikzcd}
\Big\{\big[(\xi, K)\big]_N \Big\} \ar{r}{Obs} &H^3(A, N)
\end{tikzcd}$$ is a module isomorphism where the LHS is the vector space of equivalence class of $A$-kernels with common biannihilators $N$, and the RHS is the Hochschild cohomology group. Note that if we would like to clarify the representation that induces the differential for the cohomology, then we specify it by $H^3(A, N, \rho)$.The most difficult part is to show this map is an surjection. That is to say, given any cohomology class in $H^3$, we can construct a proper $A$-kernel whose derived obstruction is identical to the class. It actually describes the structure of the kernels.
Recall that $\xi$ defines an $A$-$A$-bimodule structure on $N$ in the midway of introducing the special cohomology and it does not rely on the choice of connections or bimultiplication laws that cover $\xi$. We sometimes refer this bimodule a **nucleus** of the kernel. In terms of the set of algebra kernels with common biannihilator, we can also say they have common nucleus. So nucleus-obstruction is the only twins determined by $\xi$ exclusively, while “connection-hindrance" is otherwise. This just spells out its peculiarity.
To assign the set of $A$-kernels a linear structure, we define the addition and scalar multiplication as follows:
Define $$K_1+K_2:=(K_1 \oplus K_2)/_{\{(n,-n)|n\in N\}}$$
The factoring ideal is to cancel out those $k$ such that $x\cdot (k_1+k_2)=0$ resulting from $x=k+(-k)$ for some $x\in K_1+K_2$, and one has $Anni(K_1+K_2)=(N+N)/_{\{(n,-n)\}} \cong N$ immediately.
As $K_1+K_2$ is defined, for any two couplings $\xi_1$ and $\xi_2$, we find two (regular) bimultiplication laws $\mu_i: A \rightarrow Mul(K_i), i=1, 2$ that cover them. For any $a\in A, k_1\in K_1$ and $k_2\in K_2$, elements in the images $\mu_i(A)$ satisfy $\mu_i(a)(k_i)=(u_a, v_a)(k_i)=(a\cdot k_i, k_i \cdot a)$ for $i=1,2$.
Now $a$ acts on the direct sum of $K_i$ componentwise, after passing the quotient we have it on $[K_1\oplus K_2]$ so does on $K_1+K_2$. We denote it by $u_a([k])=a\cdot [k]$ and $v_a([k])=[k] \cdot a$ for any $[k]=[k_1\oplus k_2]\in K_1+K_2$. Thus we can define $$\mu_1+\mu_2: A \rightarrow Mul(K_1+K_2):=End(K_1+K_2) \oplus End(K_1+K_2)^{op}$$ where its image $(\mu_1+\mu_2)(A)$ consists of (permutable) elements with operations indicated as above.
Let $\xi_i: A \rightarrow Out(K_i)$ be two couplings, choose some $\mu_i$ that cover them respectively, then $\mu_1+\mu_2$ covers $\xi_1+\xi_2$. Initially, $(\xi_1+\xi_2)(A)$ consists of elements $\{([u_1+u_2]_a, [v_1+v_2]_a)\}$ derived from those elements in $\xi_1(A)$ and $\xi_2(A)$. This defines the sum of two couplings. Hence we have $(\xi_1+\xi_2, K_1+K_2)$.
Define $$_{\lambda}K=(K \oplus N)/_{\{(k, -\lambda n)\}}$$
Again, we have $Anni(_{\lambda}K)=(N\oplus N)/_{\{(n, -\lambda n)\}} \cong N$. This gives us $(_{\lambda}\xi, _{\lambda}K)$.
\
1) $Obs(\xi_1+\xi_2)=Obs(\xi_1)+Obs(\xi_2)$;\
2) $Obs(_{\lambda}\xi)=_{\lambda}Obs(\xi)$;\
3) Denote $(\cdot)^{ext}$ for an extendible kernel. If $(\xi_1, K_1)^{ext}, (\xi_2, K_2)^{ext}$, then $(\xi_1+\xi_2, K_1+K_2)^{ext}$;\
4) If $_{\lambda}(\xi, K)^{ext}$, then $(_{\lambda}\xi, _{\lambda}K )^{ext}$.
Given $\xi_i$ withe some coverings $\mu_i$, let us consider the following diagram: $$\begin{tikzcd}
& Mul(K_1) \ar{r} \ar[dashed]{dd}{\bar{\sigma}} & Out(K_1) \ar{dd} \\
A \ar{ur}{\mu_1} \ar[bend right= 5 ]{urr}{\xi_1} \ar[swap]{dr}{\mu_2} \ar[bend left= 5, swap]{drr}{\xi_2} \\
& Mul(K_2) \ar{r} & Out(K_2)
\end{tikzcd}$$ where $Anni K_1=Anni K_2=N$, that is, the kernels $(\xi_1, K_1), (\xi_2, K_2)$ with common nucleus $N$ and $\bar{\sigma}$ is induced by $\sigma: K_1 \rightarrow K_1$ with $\sigma(N)=N$.
Two $A$-kernels with common nucleus $N$ is said to be **isomorphic**, or $(\xi_1, K_1)_N \cong (\xi_2, K_2)$, if there is an isomorphism of algebras $\sigma$ fixing the biannihilator and $\bar{\sigma} \circ \mu_1=\mu_2$.
The latter statement means $\bar{\sigma} (u_1, v_1)_a(k)=(u_2, u_2)_a(k)$ is a *homomorphism* of images $\mu_i(A) \subset Mul(K_i)$ for any pairs of endomorphisms with respect to any $k$.
Two kernels with common nucleus $N$ are said to be **equivalent**, or $(\xi_1, K_1)_N \sim (\xi_2, K_2)_N$, if there are two extendible kernels $(\eta_1, S_1)^{ext}, (\eta_2, S_2)^{ext}$ with the *same* nucleus such that $$(\xi_1+\eta_1, K_1+S_1) \cong (\xi_2+\eta_2, K_2+S_2)$$
Denote the equivalence class of kernels with common nucleus by $$\big[(\xi, K)\big]_N:=(\xi, K)_N/_ \sim$$
Given any algebra $A$, an $A$-$A$-bimodule $M$ and any element in $H^3(A, M)$, we would like to find a proper kernel (and a coupling in turn) that realized it. Using the language of connection and representation from part **3**, we obtain our first structure theorem:
Given any associative algebra $A$ and let $(M, \rho)$ be an representation of $A$ where $\rho: A \rightarrow Mul(M)$ is a flat connection. Let $c$ be an element in $HH^3(A, M, \rho)$, then
1)there exists an algebra $K$ having a left $A$-module structure and such that $AnniK=M$,
2)there exists a homomorphism $\xi: A \rightarrow Out(K)$ such that the induced central representation $\rho^{\xi}: A \rightarrow Mul(AnniK)$ is equal to $\rho$, and
3\) $Obs(\xi)$ coincides with $c$.
Moreover, $(\xi, K)$ becomes the coupling of $A$.
\
According to [@Hoch46] the proof is highly constructive. We simply sketch each step here. For details, consult **Appendix D**.
- (Step 1) Define all the direct summands of $K$.
- (Step 2) Define multiplications between the components of $K$.
- (Step 3) Show the biannihilator of $L$ is trivial so that $AnniK=M$.
- (Step 4) Define the left and right $A$-actions on $K$.
- (Step 5) Show the four conditions hold whence we find a concrete connection $\bar{\mu}: A \rightarrow Mul(K)$.
- (Step 6) Set $\bar{\xi}:= \natural \circ \bar{\mu}$. Hence $\bar{\mu}|_{AnniK}$ depends on the choice of $\bar{\xi}$ and we denote it by $\bar{\mu}|_{AnniK}=\rho^{\bar{\xi}}$. The pair $(M, \rho^{\bar{\xi}})$ becomes the central representation of $A$. It induces a differential $\delta^{\bar{\xi}}$ and actually, $\delta^{\bar{\xi}}=\delta^\rho$.
- (Step 7) Write $c=\{f\}$ where the representative cocycle $f$ determines an *extension of bimodules* of $M$ by $A\otimes A\otimes A^*$ within $K$.
- (Step 8) There is a suitable cochain $\bar{h}: A \otimes A \rightarrow E$ for some $E$ with $M\subset E\subset K$ such that $\Delta|_{E} \bar{h}=f$. Moreover, $\bar{h}$ becomes the hindrance of $\xi$ (see Lemma 6.1 and 6.2). Define a proper bilinear map $$\bar{h}(a_1 \otimes a_2)=a_1 \otimes a_2 \otimes 1$$
- (Step 9) Set $F^{\bar{\xi}}:=F(\bar{\mu}, \bar{h})$. Then $Obs(\bar{\xi})=\{F^{\bar{\xi}}\}$ and is identical to $c$. Namely, $F^\xi \equiv f$.
What we really need later is an simplified version of above theorem. This refines the structure of extension of bimodules in step 7 by reducing $A\otimes A\otimes A^*$ to $A\otimes A$. We endow it with a bimodule structure through the following operations: $$\begin{aligned}
a_0 \cdot (a_1 \otimes a_2)&:=a_0 a_1 \otimes a_2 -a_0 \otimes a_1 a_2 \\
(a_1 \otimes a_2)\cdot a_0 &:=0\end{aligned}$$
The next two important lemmas are attributed to Hochschild:
Let $Q$ be any bimodule over $A$. Any element $f\in Z^3(A, Q)$ in the light of Hochschild cohomology defines a split extension of bimodules of $M$ by $A\otimes A$.
Let the underlying vector space $E=A\otimes A \oplus Q$ with a bimodule structure defined as follows $$\begin{aligned}
a \cdot (p, q)&:= (a\cdot p, f(a\otimes a_1 \otimes a_2)+a \cdot q) \\
(p,q)\cdot a&:=(p\cdot a, q\cdot a)\end{aligned}$$ Define $\pi: E \rightarrow A\otimes A$ by $\pi(p, q)=p$. We claim that $(E, \pi)$ becomes a split extension of bimodules. Indeed, $ker \pi=\{(0,q)|\pi(0,q)=0, \forall q \in Q \}$, so we can identify the sub(bi)module $(0, Q)$ with $Q$. $$\begin{tikzcd}
0 \ar{r} &Q \ar[tail]{r} &E \ar{r}{\pi} &A\otimes A \ar{r} \ar[dashed, bend left, xshift=-1mm]{l}{\gamma} &0
\end{tikzcd}$$ Set $\gamma: A \otimes A \rightarrow E$ with $$\gamma(p):=(p,0)$$ Then we have $\pi \gamma=id_{A\otimes A}$ for $\pi \circ \gamma(p)=\pi(p,0)=p$, as desired. Finally, we define $$\varphi_a(p):=a \cdot \gamma(p)- \gamma(a\cdot p)$$ Since $\pi \varphi=a\cdot p- a\cdot p=0$ then $\varphi_a(p) \in Q$, and we have $\varphi_a: A\otimes A \rightarrow Q$. Therefore the map $a \mapsto \varphi_a$ defines an element $$f^\gamma \in Hom \big(A, Hom(A\otimes A, Q) \big) \cong Hom(A\otimes A \otimes A, Q).$$ One can check that $\delta_{Hom(A\otimes A, Q)} f^\gamma=0$ with a proper bimodule structure on $Hom(A\otimes A, Q)$. Hence, $f^\gamma$ becomes a cocycle.
On the other hand, we have $\varphi_a(p)=a \cdot (p,0)- (a\cdot p, 0)=(a\cdot p, f(a\otimes a_1 \otimes a_2))-(a\cdot p,0)=(0, f(a\otimes a_1 \otimes a_2))$. We conclude that $f^\gamma \equiv f$ and the lemma is proved.
Given $f\in Z^3(A, Q)$ with the corresponding induced split extension of bimodules as above. There is an element $h_E \in C^2(A, E)$ such that $\delta_E h_E=f$.
Define $h: A\otimes A \rightarrow E$ by $$h_E(a_1 \otimes a_2):=(a_1 \otimes a_2,0)$$
Next, we compute $$\begin{aligned}
\delta_E h_E(a \otimes a_1 \otimes a_2)&=a\cdot h_E(a_1 \otimes a_2) -h_E(a a_1 \otimes a_2)+h_E(a\otimes a_1 a_2)-h(a\otimes a_1) \cdot a_2 \\
& =a \big(a_1 \otimes a_2, 0 \big)-\big(a a_1 \otimes a_2, 0 \big)+ \big(a\otimes a_1 a_2, 0 \big) \\
& =\big(a a_1 \otimes a_2-a\otimes a_1a_2, f(a\otimes a_1 \otimes a_2)\big)-\big(a a_1 \otimes a_2, 0 \big)+ \big(a\otimes a_1 a_2, 0 \big) \\
&=\big(0, f(a\otimes a_1 \otimes a_2)\big) \in (0,Q)\end{aligned}$$
As we have identified $(0,Q)$ with $Q$, then $f$ becomes a coboundary.
Since we will use the enveloping algebra of Lie algebra $\mathfrak{g}$, we appropriate the position of $A$ by denoting $U=U(\mathfrak{g})$ in next theorem:
If in particular the right $U$-action on $P_2=U \otimes U$ is trivial, then
a\) reset $h(a_1 \otimes a_2)=a_1 \otimes a_2$ such that $\epsilon \circ h = R^\mu$, and by lemma 6.2 we have $\Delta|_E h \equiv f$;
b\) in this case the structure of $K$ can be simplified.
- See **Appendix E** for its proof. Note that this is the pivotal bridge theorem where we are forwarding to the Lie algebra case.
Going to Lie
============
One of Shukla’s unproved theorem in [@Shk66] states the generalized version in terms of DG-Lie algebra. The structure theorem for any ordinary Lie algebra kernels in [@Hoch56a] is laconic and thus least readable, so we will clarify his dense writing and present a formal proof in the following paragraphs.
Let us formulate our second main theorem at first:
Given any Lie algebra $\mathfrak{g}$ and let $\mathcal{M}$ be any $\mathfrak{g}$-module. Let $\rho_{Lie}: \mathfrak{g} \rightarrow Der(\mathcal{M})$ be a The Lie algebra homomorphism (i.e. a flat $\mathfrak{g}$-connection on $\mathcal{M}$). For any element $\{f\}$ in $H^3(\mathfrak{g}, \mathcal{M}, \rho_{Lie})$,
1)there exists a Lie algebra $\mathfrak{K}$ having some module structures over $\mathfrak{g}$ and such that $Z\mathfrak{K}=\mathcal{M}$,
2)there exist a homomorphism $\Xi: \mathfrak{g} \rightarrow Out(\mathfrak{K})$ such that the induced central representation $\rho^{\Xi}: \mathfrak{g} \rightarrow Der(Z\mathfrak{K})$ coincides with $\rho_{Lie}$, and
3\) $Obs(\Xi)=f$
Moreover, $(\Xi, \mathfrak{K})$ becomes a (Lie) coupling of $\mathfrak{g}$.
We are actually beginning with the given Lie triple: $$\big(\mathfrak{g}, \mathcal{M}, f \big)$$ where $\in Z^3(\mathfrak{g}, \mathcal{M}, \rho_{Lie})$. The main technique is to transfer the Lie triple into some proper associative triples and then to go back to Lie by manipulating formulas. First of all, we take $U(\mathfrak{g})$.
The second important note is the equivalence of categories
$\textbf{Rep}(\mathfrak{g}) \Leftrightarrow U(\mathfrak{g})$-$\textbf{Mod}$
Let $M$ be a $U(\mathfrak{g})$-module corresponding to the given representation pair $(\mathcal{M}, \rho_{Lie})$.
Thirdly, we will the previous section to get $$\begin{aligned}
Hom_{U(\mathfrak{g})} \big(U(\mathfrak{g}) \otimes \bigwedge\nolimits^{\!3} \mathfrak{g}, \mathcal{M}) \big)
&\rightarrow
Hom_{U(\mathfrak{g})} \big(U(\mathfrak{g})\otimes U(\mathfrak{g})_+^{\otimes^3}, M) \big) \\
f &\mapsto f'\end{aligned}$$ such that the induced cohomology groups are isomorphic, that is $$H^3(\mathfrak{g}, \mathcal{M}, \rho_{Lie}) \cong HH^3(U(\mathfrak{g}), M, \rho)$$ where $$\begin{tikzcd}
\mathfrak{g} \ar{r}{\rho_{Lie}} \ar[swap]{d}{\epsilon} &Der(\mathcal{M}) \ar{d} \\
U(\mathfrak{g}) \ar[swap]{r}{\rho} &Mul(M)
\end{tikzcd}$$ Therefore, our associative triple is $$\big(U(\mathfrak{g}), M, f'\big)$$ where $f'\in Z^3(U(\mathfrak{g}), M, \rho)$
Due to the construction of Theorem $1$, our $K$ will be a special combination of $A=U(\mathfrak{g})$ such that $$M=ZK:=\{m|m\cdot K=K \cdot m=0\}.$$ with the listed multiplications between all possible components of $K$ and $A$-actions. The coupling $\xi$ is therefore fulfilled by this $A$-action. Write $$\begin{aligned}
\mu: U(\mathfrak{g}) &\rightarrow Mul(K) \\
a &\mapsto (u_a,v_a)\end{aligned}$$ Then $\xi=\natural \circ \mu$ for some proper linear mappings $\theta$ and the following induced central representation(only determined by the given coupling) $\rho^\xi: U(\mathfrak{g}) \rightarrow Mul(ZK)$ which coincides with $\rho$ by Theorem $1$.
Denote $\mathfrak{K}=Lie(K)$ with $[k_1, k_2]=k_1 \cdot k_2-k_1 \cdot k_2$ such that $$\mathcal{M}=Z\mathfrak{K}:=\{m|[m, K]=0 \}.$$
In the associative case, we know that the nonzero difference $R^\mu(a_1 \otimes a_2)=\mu_{a_1}\mu_{a_2}-\mu_{a_1 a_2}$ is an inner bimultiplication effected by some bilinear maps $h:U(\mathfrak{g}) \otimes U(\mathfrak{g}) \rightarrow K$. As $h$ has already built in Theorem $1$, we want to define its counterpart $H$ for Lie algebra. More precisely, beginning with the covering-hindrance pair $$(\mu, h) \quad \text{with} \quad \epsilon \circ h = R^\mu$$ we want to define the Lie-pair $$(\nabla, H) \quad \text{with} \quad ad \circ H = R^\nabla$$
For any $\mathfrak{a} \in \mathfrak{g}$, we set $$\begin{aligned}
\nabla:\mathfrak{g} & \rightarrow Der(\mathfrak{K}) \\
\mathfrak{a} & \mapsto u_{\mathfrak{a}}-v_{\mathfrak{a}} \end{aligned}$$ Recall that $\epsilon \circ h=(u_{h(a_1 \otimes a_2)}, v_{h(a_1 \otimes a_2)})$ is an inner bimultiplication produced by $h(a_1 \otimes a_2)$, so for any $k \in K$ we get $$\begin{aligned}
\epsilon \circ h (k) &= R^\mu (K) \\
(u_{h(a_1 \otimes a_2)}, v_{h(a_1 \otimes a_2)})(k)
&=(\mu_{a_1}\mu_{a_2}-\mu_{a_1 a_2})(k) \\
\Big(h(a_1\otimes a_2)k, kh(a_1\otimes a_2)\Big)
&=\Big( a_1\cdot(a_2\cdot k)-(a_1a_2)\cdot k, (k\cdot a_1)\cdot a_2-k\cdot(a_1a_2) \Big)\end{aligned}$$ $$\begin{tikzcd}
U(\mathfrak{g})\otimes U(\mathfrak{g}) \ar{r}{h}
& K \ar{d}{i} \\
\mathfrak{g} \wedge \mathfrak{g} \ar{u}{\gamma} \ar[swap, dashed]{r}{H}
&\mathfrak{K}
\end{tikzcd}$$ Then $H:=i \circ h \circ \gamma$, where $\gamma(\mathfrak{a_1} \wedge \mathfrak{a_2})=a_1 \otimes a_2 -a_2 \otimes a_1$. $$\begin{split}
\Delta^\nabla H(\mathfrak{a_1} \wedge \mathfrak{a_2} \wedge \mathfrak{a_3})
&=\Delta^\nabla \circ i \circ h \circ \gamma (\mathfrak{a_1} \wedge \mathfrak{a_2} \wedge \mathfrak{a_3}) \\
&={i^*}\Delta^\nabla \circ h \Big(\sum_{\sigma}(sgn\sigma) a_{\sigma_{(1)}} \otimes a_{\sigma_{(2)}} \otimes a_{\sigma_{(3)}} \Big) \\
&=\Delta^\mu h \Big(\mathfrak{S}\{a_1 \otimes a_2 \otimes a_3 \} \Big) \\
&=\Big(0, \mathfrak{S}\{f'(a_1 \otimes a_2 \otimes a_3) \} \Big)\\
&=\Big(0, f(\mathfrak{a_1} \wedge \mathfrak{a_2} \wedge \mathfrak{a_3}) \Big)
\end{split}$$
The curvature map for Lie algebra $$R^\nabla: \mathfrak{g} \wedge \mathfrak{g} \rightarrow ad(\mathfrak{K})$$ is given by $R^\nabla(\mathfrak{a_1} \wedge \mathfrak{a_2})=[\nabla_\mathfrak{a_1}, \nabla_\mathfrak{a_2}]-\nabla_{[\mathfrak{a_1},\mathfrak{a_2}]}$.
On the other hands, we have $R^\nabla=ad \circ H$. In details, $$\begin{split}
R^\nabla(\mathfrak{a_1} \wedge \mathfrak{a_2})
&=ad \circ i \circ h \circ \gamma(\mathfrak{a_1} \wedge \mathfrak{a_2}) \\
&=ad \circ i \circ h(a_1 \otimes a_2 -a_2 \otimes a_1) \\
\end{split}$$ For any $\mathfrak{k} \in \mathfrak{K}$, $$\begin{split}
ad_{i \circ h(a_1 \otimes a_2 -a_2 \otimes a_1)}(\mathfrak{k})
&=[i \circ h(a_1 \otimes a_2 -a_2 \otimes a_1), \mathfrak{k}] \\
&=[i \circ h(a_1 \otimes a_2), \mathfrak{k}]-[i \circ h(a_2 \otimes a_1), \mathfrak{k}] \\
&=\Big(h(a_1 \otimes a_2) k-k h(a_1 \otimes a_2) \Big)-\Big(h(a_2 \otimes a_1) k-k h(a_2 \otimes a_1) \Big) \\
\text{in addition,}
&=(u_{h(a_1 \otimes a_2)}-v_{h(a_1 \otimes a_2)})-(u_{h(a_2 \otimes a_1)}-v_{h(a_2 \otimes a_1)})(k) \\
&=(u_{h(a_1 \otimes a_2)-h(a_2 \otimes a_1)},v_{h(a_1 \otimes a_2)-h(a_2 \otimes a_1)})(k) \\
&=(u_{H(\mathfrak{a_1} \wedge \mathfrak{a_1})},v_{H(\mathfrak{a_1} \wedge \mathfrak{a_1})})(k) \\
\end{split}$$
By the definition of $\nabla$, we have $$\begin{aligned}
\nabla_\mathfrak{\mathfrak{a_1}} \nabla_\mathfrak{\mathfrak{a_2}}(\mathfrak{k})
&=(u_{\mathfrak{a_1}}-v_{\mathfrak{a_1}})(u_{\mathfrak{a_2}}-v_{\mathfrak{a_2}})(\mathfrak{k}) \\
&=u_{\mathfrak{a_1}}u_{\mathfrak{a_2}}-\mathbf{u_{\mathfrak{a_1}}v_{\mathfrak{a_2}}}-\mathbf{v_{\mathfrak{a_1}}u_{\mathfrak{a_2}}}+v_{\mathfrak{a_1}}v_{\mathfrak{a_2}} (\mathfrak{k}) \\
\nabla_\mathfrak{\mathfrak{a_2}} \nabla_\mathfrak{\mathfrak{a_1}}(\mathfrak{k})
&=(u_{\mathfrak{a_2}}-v_{\mathfrak{a_2}})(u_{\mathfrak{a_1}}-v_{\mathfrak{a_1}})(\mathfrak{k}) \\
&=u_{\mathfrak{a_2}}u_{\mathfrak{a_1}}-\mathbf{u_{\mathfrak{a_2}}v_{\mathfrak{a_1}}}-\mathbf{v_{\mathfrak{a_2}}u_{\mathfrak{a_1}}}+v_{\mathfrak{a_2}}v_{\mathfrak{a_1}} (\mathfrak{k}) \\
\nabla_{[\mathfrak{\mathfrak{a_1}},\mathfrak{\mathfrak{a_2}}]} (\mathfrak{k})
&=u_{[\mathfrak{a_1},\mathfrak{a_2}]}-v_{[\mathfrak{a_1},\mathfrak{a_2}]} (\mathfrak{k}) \\
&=u_{\mathfrak{a_1}\otimes \mathfrak{a_2}-\mathfrak{a_2} \otimes \mathfrak{a_1}}-v_{\mathfrak{a_1}\otimes \mathfrak{a_2}-\mathfrak{a_2} \otimes \mathfrak{a_1}} (\mathfrak{k}) \\
&=u_{\mathfrak{a_1} \mathfrak{a_2}}-u_{\mathfrak{a_2} \mathfrak{a_1}}-v_{\mathfrak{a_1} \mathfrak{a_2}}+v_{\mathfrak{a_2} \mathfrak{a_1}} (\mathfrak{k}) \end{aligned}$$
The last two lines hold for referring $[x \otimes y]-x\otimes y-y \otimes x$ in the canonical ideal and then abuse the tensor notation. When grouping these three nablas, the middle four bold parts are cancelled because of the permubalility condition $(a_1 \cdot k)\cdot a_2=(a_2 \cdot k)\cdot a_1$ and vice versa. Equivalently, $u_{a_1}v_{a_2}=v_{a_2}u_{a_1}$ and $u_{a_2}v_{a_1}=v_{a_1}u_{a_2}$.
Therefore, we have $$\begin{split}
R^\nabla(\mathfrak{\mathfrak{a_1}} \wedge \mathfrak{\mathfrak{a_2}})(\mathfrak{k})
&= \big( \nabla_\mathfrak{\mathfrak{a_1}} \nabla_\mathfrak{\mathfrak{a_2}}-\nabla_\mathfrak{\mathfrak{a_2}} \nabla_\mathfrak{\mathfrak{a_1}}-\nabla_{[\mathfrak{\mathfrak{a_1}},\mathfrak{\mathfrak{a_2}}]} \big)(\mathfrak{k}) \\
&=u_{\mathfrak{a_1}}u_{\mathfrak{a_2}}+v_{\mathfrak{a_1}}v_{\mathfrak{a_2}}-u_{\mathfrak{a_2}}u_{\mathfrak{a_1}}-v_{\mathfrak{a_2}}v_{\mathfrak{a_1}}-u_{\mathfrak{a_1} \mathfrak{a_2}}+u_{\mathfrak{a_2} \mathfrak{a_1}}+v_{\mathfrak{a_1} \mathfrak{a_2}}-v_{\mathfrak{a_2} \mathfrak{a_1}} (\mathfrak{k})\\
&=\Big(u_{\mathfrak{a_1}}u_{\mathfrak{a_2}}-u_{\mathfrak{a_1} \mathfrak{a_2}}-(v_{\mathfrak{a_2}}v_{\mathfrak{a_1}}+v_{\mathfrak{a_1} \mathfrak{a_2}}) \Big)-\Big(u_{\mathfrak{a_2}}u_{\mathfrak{a_1}}-u_{\mathfrak{a_2} \mathfrak{a_1}}-(v_{\mathfrak{a_1}}v_{\mathfrak{a_2}}+v_{\mathfrak{a_2} \mathfrak{a_1}}) \Big) (\mathfrak{k}) \\
&=\Big(h(\mathfrak{a_1} \otimes \mathfrak{a_2}) \mathfrak{k}-\mathfrak{k} h(\mathfrak{a_1} \otimes \mathfrak{a_2}) \Big)-\Big(h(\mathfrak{a_2} \otimes \mathfrak{a_1}) \mathfrak{k}-\mathfrak{k} h(\mathfrak{a_2} \otimes \mathfrak{a_1}) \Big)
\end{split}$$
As two results coincide, we conclude that based on our definition of $\nabla$ and $R^\nabla$, we have found a proper $H$ derived from its associative counterpart $h$ such that $R^\nabla=ad \circ H$ as desired. $$\begin{tikzcd}
\mathfrak{g} \ar[bend right, swap]{ddd}{\iota} \ar[swap]{drr}{\nabla} \ar[bend left=3]{drrr}{\Xi}
&& ad(\mathfrak{K}) \ar{d} \\
Z\mathfrak{K} \ar{r}
&\mathfrak{K} \ar{r} \ar[bend left, dashed]{ur}
&Der \big(\mathfrak{K} \big) \ar[swap]{r}{\natural'}
&Out \big(\mathfrak{K} \big) \\
ZK \ar{r} \ar{u}
&K \ar{r} \ar{u}{Lie} \ar[bend right, dashed]{dr}
&Mul(K) \ar{r}{\natural} \ar{u}
&Out(K) \ar{u} \\
U(\mathfrak{g}) \ar[bend right=3, swap]{urrr}{\xi} \ar{urr}{\theta}
&& Inn(K) \ar{u}
\end{tikzcd}$$ For some proper $\nabla$, our Lie coupling $\Xi$ can be viewed as the composition $\natural \circ \nabla$. We set $F^\Xi:=F(\nabla, H)$ then by the argument above we have $F(\nabla, H) = F(\mu, h)$, while the latter coincides with the *a priori* associative cocycle $f'$ by Theorem $1$. Again $f' \mapsto f$ is an isomorphism onto the Lie cocycle. Consequently, following with all of the equalities, we get $$F^\Xi=f$$ $$\begin{tikzcd}
F(\nabla, H) \ar{r}{\text{above}} \ar[dashed]{d} &F(\mu, h) \ar{d}{\text{Theorem 1}} \\
f & f' \ar{l}{\text{Lie-Asso id.}}
\end{tikzcd}$$ $$\begin{tikzcd}
&\mathfrak{g} \ar[bend right, swap]{ddd}{\iota} \ar[swap]{drr}{\nabla} \ar{drrr}{\Xi} \ar[swap]{dl}{\rho^{\Xi}} \\
Der(\mathcal{M})
&\mathcal{M} \ar{r} \ar[dashed]{l}
&\mathfrak{K} \ar[swap]{r}
&Der \big(\mathfrak{K} \big) \ar[swap]{r}{\natural} & Out \big(\mathfrak{K} \big) \\
Mul(M) \ar{u}
& M \ar{u} \ar{r} \ar[dashed]{l}
&K \ar[swap]{r} \ar{u}
&Mul(K) \ar{r}{\natural} \ar{u}
&Out(K) \ar{u} \\
&U(\mathfrak{g}) \ar[swap]{urrr}{\xi} \ar{urr}{\theta} \ar{ul}{\rho^{\xi}}
\end{tikzcd}$$
Appendix A: Generalities on Associative Algebras {#appendix-a-generalities-on-associative-algebras .unnumbered}
================================================
The following couple of appendices aim to give a glossary about associative algebra and Hochschild cohomology compatible with this paper. A lot of its classical knowledge occurs in [@CE56], [@Redo01] and in even those very old [@CG] and [@CT].
Let $A$ be an associative algebra over any unital commutative ring $R$. The tensor product of two $R$-algebras $A$ and $B$ is $A \otimes_R B$ with an associative multiplication defined by $(a_1 \otimes b_1)(a_2 \otimes b_2)=(a_1 a_2)\otimes (b_1 b_2)$. A $R$-algebra homomorphism is both ring homomorphism and module homomorphism over $R$.
Any $A$-$A$-bimodule $M$ is also a $R$-module in the following way: for any $r \in R$, since $(ra)\cdot m \in A$, then $(r_1 a_1) \big((r_2 a_2)\cdot m \big)=\big((r_1 a_1)(r_2 a_2) \big) \cdot m = (r_1 r_2) \big((a_1 a_2) \cdot m \big) \in A$; and similar for the right operation. Moreover, we can define a left $A \otimes_R B$-module in the following way: for any $a \in A$ and $b\in B$, let $(a \otimes b)\cdot m=a(b\cdot m)=b(a \cdot m)$.
If $M$ is a $A$-$B$-bimodule, then it can be seen as a left $A \otimes_R B^{op}$-module in the following way: $(a \otimes b^*)m=a(b^*m)= b^*(am)$, where the opposite operation is given by $a^* m := m a$. Applying the star-operation, we just get the usual bimodule condition $a(mb)=(am)b$. In this way, it is possible to identify any $A$-$A$-bimodule with the left $A \otimes_R A^{op}$-module. Write $A^e := A \otimes_R A^{op}$ to be the evenloping algebra of $A$. This is again a $R$-algebra with the same multiplication defined as above. Lemma 2.1 from \[Red00\] tells us that the category of $A$-$A$-**bimodule** is equivalent to the category of left $A^e$-**module**. Therefore, our bimodule $M$ becomes a left $A^e$-module(as well as a $R$-module).
Let us recall the classical chain complex heading to the Hochschild cohomology.
For all $n \geq 0$, let $S_n(A)=A \otimes_R A^{\otimes^n} \otimes_{R} A$ be the $(n+2)$-folds tensor product over $R$ of $R$-algebra $A$. It is an $A$-$A$-bimodule in a natural way, so it can be seen as an $A^e$-bimodule. The map $b_n: S_n(A) \rightarrow S_{n-1}(A)$ defined by $$b_n(a_1 \otimes \cdots \otimes a_n)=\sum_{i=0}^{n} (-1)^i a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n$$ is an $A$-$A$-bimodule morphism and thus an $A^e$-module morphism. Let $S_{-1}(A)=A$ and let $b_0=\epsilon: S_0(A) \rightarrow S_{-1}(A)$ such that $\epsilon(a \otimes a')=aa'$ be the augmentation(also bimodule morphism). Then $$\begin{tikzcd}
0 & S_{-1}(A) \ar[swap]{l}
& S_0(A) \ar[swap]{l}{\epsilon}
& S_1(A) \ar[swap]{l}{b_1}
& S_2(A) \ar[swap]{l}{b_2}
& \cdots \ar[swap]{l}{b_3}
\end{tikzcd}$$ $$\begin{tikzcd}
0 & A \ar[swap]{l}
& A \otimes_R A \ar[swap]{l}{\epsilon}
& A \otimes_R A \otimes_R A \ar[swap]{l}{b_1}
& A \otimes_R A^{\otimes^2} \otimes_R A \ar[swap]{l}{b_2}
& \cdots \ar[swap]{l}{b_3}
\end{tikzcd}$$ forms an acyclic complex. Indeed, consider the map $s: S_n(A) \rightarrow S_{n+1}(A)$ with $s(x)=1 \otimes x$. One can easily check that $b_n s +s b_{n-1}=id_{A^{\otimes^{n+1}}}$ and $b_0 s=id_A$. Additionally, we have $b^2=0$. If $A$ is $R$-projective, then $A^{\otimes^n}$ is also $R$-projective and $S_n(A)$ becomes $A^e$-projective. The projective resolution $(S(A), b)$ in above sense is called the ***standard complex*** or ***bar resolution*** of $A$.
Next, by chopping off the first nonzero term and applying the contravariant functor $Hom_{A^e}(-, M)=Hom_{A\otimes_{R} A^{op}}(-,M)$ to the chain resolution, we reach a cochain complex $$\begin{tikzcd}
0 \ar{r} & Hom_{A^e}\big(S_0(A), M \big) \ar{r}
& Hom_{A^e}\big(S_1(A), M \big) \ar{r}
& Hom_{A^e}\big(S_2(A), M \big) \ar{r} & \cdots
\end{tikzcd}$$ of mere left-exactness. Now consider the following form $$S_n(A)=A \otimes_R A^{\otimes^n} \otimes_R A
\cong A \otimes_R \tilde{S}_n(A) \otimes_R A
\cong A^e \otimes_R \tilde{S}_n(A),$$ where $\tilde{S_n}(A)$ is the $n$-folds tensor product of $A$ for all $n \geq 1$ and put $\tilde{S_0}(A)=R$. Then the hom functor gives $$Hom_{A^e} \big(S_n(A), M \big)
\cong Hom_{A^e} \big(A^e \otimes_R \tilde{S}_n(A), M \big)
\cong Hom_R(\tilde{S_n}(A), M).$$ Namely, $$\begin{tikzcd}
\cdots \ar{r} & Hom_{A^e}\big(S_n(A), M \big) \ar{r}{b_{n-1}^*} \ar{d}{\cong,\varphi}
& Hom_{A^e}\big(S_{n+1}(A), M \big) \ar{r} \ar{d}{\cong, \varphi}
& \cdots \\
\cdots \ar{r} & Hom_R \big(\tilde{S}_n(A), M \big) \ar[swap]{r}{\delta_n}
& Hom_R \big(\tilde{S}_{n+1}(A), M \big) \ar{r}
& \cdots \\
\end{tikzcd}$$ where $\varphi: f \mapsto \tilde{f}$ with $f(x)=\tilde{f}(1 \otimes x \otimes 1)$ and $b^*\circ f=f\circ b$. In particular, the first few entries are: $$\begin{tikzcd}
0 \rar &Hom_{A^e}(A\otimes_R A, M) \rar \dar
&Hom_{A^e}(A\otimes_R A \otimes_R A, M) \rar \dar
&\cdots \\
0 \rar &Hom_{R}(R, M) \rar
&Hom_{R}(A, M) \rar
&\cdots
\end{tikzcd}$$ One can check that the above diagram is commutative and actually define the formula of $\delta_n: Hom_R \big(\tilde{S}_n(A), M \big) \rightarrow Hom_R \big(\tilde{S}_{n+1}(A), M \big)$ for each $n$. Generally, the coboundary operator is $$\begin{aligned}
\delta_n(f)&(a_1 \otimes \cdots \otimes a_{n+1})
:=a_1 \cdot f(a_2 \otimes \cdots \otimes a_{n+1}) \\
&+\sum_{i=1}^{n}(-1)^{i}f(a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_{n+1})+(-1)^{n+1}f(a_1 \otimes \cdots \otimes a_n) \cdot a_n\end{aligned}$$
Define the ***$i^{th}$-Hochschild cohomology*** of $A$ with coefficients in an $A$-$A$-bimodule (here only consider its $R$-module structure) $M$: $$HH^n(A, M):=H^* \big( Hom_R (\tilde{S}_n(A), M) \big)$$ When using the projective resolution, we can make an alternative definition with Ext functor involved: $$HH^n(A, M):=Ext_{A^e}^i(A, M)$$ One may read \[Dowdy69\] and [@Car-E] for more detailed construction.
An elegant treatment on the interchange of these two types of cochains is the derived functor approach.
Appendix B: Derived Functor Approach {#appendix-b-derived-functor-approach .unnumbered}
====================================
Given any $R$-algebra $A$ and let $M$ be an $A$-$A$-bimodule. In the spirit of part **3**, a ***representation*** of $A$ on $M$ is a pair $(M, \rho)$ where $\rho: A \rightarrow Mul(M) $ is a $R$-algebra homomorphism. The set of representations of $A$ forms a category, $\mathbf{Rep}(A)$. For any $(M, \rho)$, we define an invariant sub(bi)module
$$M^{A^e}:= \{m \in M| {\rho_a} m-m {\rho_a}, \forall a\in A \}$$
Generally, this defines a functor:
$(-)^{A^e}: \textbf{Rep}(A)$ $\rightarrow R$-$\textbf{Mod},$
alternatively, we can express it as
$^{A}(-)^{A}: A$-$A$-$\textbf{Bimod} \rightarrow R$-$\textbf{Mod}$
such that, when writing $\rho_a m=am$ and $m \rho_a=ma$, $$^{A}M^{A}=\{ m|am-ma, \forall a\in A \},$$ for *any* bimodule $M$. **The standard complex of $A$ with coefficients in a representation $(M, \rho)$** is $Hom_R(\tilde{S}_n(A), M)$ together with a $R$-linear differential $\delta_n^{\rho}: Hom_R(\tilde{S}_n(A), M) \rightarrow Hom_R(\tilde{S}_{n+1}(A), M)$ given by $$\begin{aligned}
\delta^\rho_n(f)&(a_1\otimes \cdots\otimes a_{n+1})
=u_{a_1}f(a_2\otimes \cdots \otimes a_{n+1}) \\
&+\sum_{i=1}^n (-1)^i f(a_1 \otimes \cdots \otimes a_i a_{i+1}\otimes \cdots \otimes a_{n+1}) +(-1)^{n+1}v_{a_{n+1}}f(a_1\otimes \cdots \otimes a_n),\end{aligned}$$ for any $a \in A$ and by indicating $(\rho_a m, m\rho_a)=(u_a m, v_a m)$. Write $\delta=\delta^\rho$ for short.
We can easily verify that the functor treated above actually pushes the original Hochschild cochains forward to: $$^{A}{\big( Hom_R(\tilde{S}_n(A), M) \big)}^{A}=Hom_R(S_n(A), M)$$ for all $n \geq 0$ Therefore, we have the **$i^{th}$-Hochschild cohomology group with coeffcients in a representation** to be the right derived functor: $$HH^i(A, M, \rho):=\textbf{\textit{R}} {^{A}(-)^{A}}.$$ We shall be always highlighting the involved representations in the classical definitions of Hochschild cohomology and Lie algebra cohomology.
Appendix C: Zero Obstructions {#appendix-c-zero-obstructions .unnumbered}
=============================
Assuing we have an extension of associative algebras $(B, \beta)$ of $K$ by $A$. All essentials are pictured in the following diagram: $$\begin{tikzcd}
&0 \ar{r} &Inn(K) \ar[tail]{r}{j} &Mul(K) \ar[two heads]{r}{\natural} &Out(K) \ar{r} &0 \\
&0 \ar{r} &K \ar[tail]{r}{\alpha} \ar{u}{\epsilon} &B \ar[two heads]{r}{\beta} \ar{u} &A \ar{r} \ar[bend left, yshift=0.6ex]{l}{\gamma} \ar{u}{\xi} \ar[swap, dashed]{ul}{\mu^\gamma} &0 \\
&&&&& A\otimes A \ar[swap]{uul}{R^\xi} \ar[bend left, dashed, yshift=0.3ex]{ulll}{R^\gamma}
\end{tikzcd}$$
Maps $\alpha, \beta, \epsilon, j$ and $\natural$ are morphisms of algebras.
Define $\mu^\gamma_a: A \rightarrow Mul(K)$ such that $$\mu^\gamma_a(k)=\alpha^{-1} \big(\gamma(a)\cdot \alpha(k), \alpha(k)\cdot \gamma(a)\big)$$
Or equivalently,
$$\alpha (\mu^\gamma_a)=\big(\gamma(a)^1, \gamma(a)^2 \big)$$
For the curvature $R^{\mu^\gamma}: A\otimes A \rightarrow Inn(K)$, there exists a unique? lift $R^\gamma$ such that $$R^{\mu^\gamma}=\epsilon \circ R^\gamma$$
Since $R^{\mu^\gamma}$ takes values in $Inn(K)$ and $\epsilon$ is an epimorphism, the existence of $R^\gamma: A\otimes A \rightarrow K$ follows as before. Under the morphism of $\alpha$, we have $$\begin{aligned}
\alpha \bigl(R^{\mu^\gamma}(a_1 \otimes a_2)\bigr)&=\alpha (\mu^\gamma(a_1) \mu^\gamma(a_2)-\mu^\gamma(a_1 a_2)) \\
&=\bigl(\gamma^1(a_1)\gamma^1(a_2)-\gamma^1(a_1 a_2),\gamma^2(a_1)\gamma^2(a_2)-\gamma^2(a_1 a_2)\bigr)\end{aligned}$$ Define $R^\gamma(a_1\otimes a_2)=\gamma(a_1)\gamma(a_2)-\gamma(a_1 a_2)$. Then $\epsilon \circ R^\gamma=(R^{\gamma^1}, R^{\gamma^2})=(\gamma^1 \gamma^1-\gamma^1, \gamma^2 \gamma^2-\gamma^2)=R^{\mu^\gamma}$.
Denote $h^\gamma=R^\gamma$. This is the hindrance determined by $\gamma$ which does not clearly display in \[Hoch46\].
Another way to introduce the hindrance above is to use the notion of *produced connection* in $Mul(K)$. See more in \[Mack05\]. For $a\in A$, if $\gamma$ is a connection in $A$, then there exists a linear mapping $\mu^\gamma$ such that $\epsilon \circ \gamma =\mu^\gamma$. So we have $$\begin{aligned}
&\epsilon \circ h^\gamma(a_1\otimes a_2)=\mu^\gamma (a_1) \mu^\gamma(a_2)-\mu^\gamma(a_1 a_2)=R^{\mu^\gamma} (a_1\otimes a_2) \\
&\epsilon \circ h^\gamma(a_1\otimes a_2)=\epsilon \circ \gamma (a_1) \epsilon \circ \gamma(a_2)-\epsilon \circ \gamma (a_1 a_2)\end{aligned}$$
Hence we have $h^\gamma (a_1\circ a_2)=\gamma (a_1)\gamma (a_2)-\gamma(a_1 a_2)$ to be the hindrance to our coupling in a same way.
Coupling $\xi$ does not depend on any particular choice of the linear mapping $\gamma$. Every extension uniquely determines a bimultiplication law that covers $\xi$.
Let $\gamma'$ be another linear mapping of $A$ into $B$ such that $\beta\gamma'=id_A$. We would like to show that $\natural \circ \mu^{\gamma'}=\natural \circ \mu^\gamma$. To do this, write $\gamma'=\gamma+\alpha \circ l$ for some maps $l:A\rightarrow K$. Then passing through the surjection $\epsilon$ and $\gamma$ we have $$\begin{aligned}
\natural(\mu^{\gamma'})
&=\natural \circ (\epsilon \circ \gamma') \\
&=\natural \circ \epsilon(\gamma+\alpha \circ l) \\
&=\natural(\mu^\gamma)+ \natural(\epsilon \circ \alpha \circ l) \\
&=\natural(\mu^\gamma) \\
&=\xi\end{aligned}$$ since $\alpha$ is injective and therefore $\natural \circ \epsilon$ carries $\alpha(K)$ into $Inn(K)$ and leads to zero.
When a split extension of algebra is given, there are possibly many choices of $\gamma$ and they defines, in a one-to-one fashion, different coverings for $\xi$. By proceeding lemma, $\gamma$ thus defines a hindrance $h^\gamma$. As $Obs(\xi)$=$f(\mu^\gamma, h^\gamma)$, we conclude that $\gamma$ actually determines the obstruction class.
Coupling induced by some extensions is called **special**, due to Hochschild.
(Necessity)For every $\gamma$ derived from an extension of algebras, $f(\mu^\gamma, h^\gamma)=0$.
$$\begin{aligned}
f(\mu^\gamma, h^\gamma)&=\Delta^\alpha(\mu^\gamma) h^\gamma(a_1\otimes a_2\otimes a_3)\in B \\
&=u_{a_1}^\gamma h^\gamma(a_2\otimes a_3)-h^\gamma(a_1 a_2\otimes a_3)+h^\gamma(a_1\otimes a_2 a_3)-v_{a_3}^\gamma h^\gamma(a_1\otimes a_2) \\
&=\alpha(u_{a_1}^\gamma)\gamma(a_2)\gamma(a_3)-\alpha(u_{a_1}^\gamma)\gamma(a_2 a_3)-\gamma(a_1 a_2)\gamma(a_3) \\
&+\gamma(a_1 a_2 a_3)+\gamma(a_1)\gamma(a_2 a_3)-\gamma(a_1 a_2 a_3)-\alpha(v_{a_3}^\gamma)\gamma(a_1)\gamma(a_2)+\alpha(v_{a_3}^\gamma)\gamma(a_1 a_2) \end{aligned}$$
Since $\alpha(u_{a_1}^\gamma)(\cdot)=\gamma(a_1)\cdot_\mu (\cdot)$ and $\alpha(v_{a_3}^\gamma)=(\cdot) \cdot_\mu \gamma(a_1)$, then $$\begin{aligned}
f(\mu^\gamma,h^\gamma)&=\gamma(a_1)\gamma(a_2)\gamma(a_3)-\gamma(a_1)\gamma(a_2 a_3)-\gamma(a_1 a_2)\gamma(a_3) \\
&+\gamma(a_1)\gamma(a_2 a_3)-\gamma(a_1)\gamma(a_2)\gamma(a_3)+\gamma(a_1 a_2)\gamma(a_3) \\
&=0 \end{aligned}$$
An $A$-kernel $(K, \xi)$ is extendible if and only if $Obs(\xi)=0$.
(Sufficiency)Given $A$, $K$ and $\xi$. Let $\mu$ be a law that covers $\xi$ and there is a bilinear map $R: A\otimes A\rightarrow K$ being the lift of $\mu$ such that $f(\mu, R)=0$, then\
1) The algebras $K$ and $A$ form an extension $A'$ such that $A'=K \rtimes_{\mu} A$,\
2) For this extension we can find a linear mapping $\gamma$ making it split such that $\mu^\gamma=\mu$ and $h^\gamma=R$.
Let the underlying vector space of $A'$ be the direct sum of the underlying vector space of $A$ and $K$. The multiplication on $A'$ is defined by $$(a_1,k_1 )(a_2,k_2 )=(a_1 a_2, \ a_1\cdot k_2+k_1 \cdot a_2+k_1 k_2+h(a_1 \otimes a_2))$$ Now let us compute $$\big((a_1,k_1)(a_2,k_2)\big)(a_3,k_3)=\big(a_1 a_2, \ a_1k_2+k_1\cdot a_2+k_1 k_2+h(a_1\otimes a_2)\big)(a_3,k_3)$$ The first coordinate is $a_1 a_2 a_3$, and the second one is $$\begin{gathered}
(a_1 a_2)\cdot k_3 +\big(a_1\cdot k_2+k_1\cdot a_2+k_1 k_2+h(a_1\otimes a_2)\big)\cdot a_3 \\
+\big(a_1\cdot k_2+k_1\cdot a_2+(k_1 k_2)\cdot a_3+h(a_1\otimes a_2)\big) k_3+h(a_1\otimes a_2 \otimes a_3) \end{gathered}$$ $$\begin{gathered}
=(a_1 a_2)\cdot k_3+(a_1\cdot k_2)\cdot a_3+(k_1\cdot a_2)\cdot a_3+(k_1 k_2) \cdot a_3+h(a_1\otimes a_2)\cdot a_3 \\
+(a_1\cdot k_2)\cdot k_3+(k_1\cdot a_2)\cdot k_3+k_1 k_2 k_3+h(a_1\otimes a_2) k_3+h(a_1\otimes a_2 \otimes a_3) \end{gathered}$$ On the other hand, we compute $$(a_1,k_1)\big((a_2,k_2 )(a_3,k_3)\big)=(a_1,k_1)\big(a_2 a_3,a_2\cdot k_3+k_2\cdot a_3+k_2 k_3+h(a_2 \otimes a_3)\big)$$ Again, the first coordinate is $a_1 a_2 a_3$ and the second one is $$\begin{gathered}
a_1 \cdot \big(a_2\cdot k_3+k_2\cdot a_3+k_2 k_3+h(a_2 \otimes a_3)\big)+k_1\cdot (a_2 a_3) \\
+k_1\cdot \big(a_2\cdot k_3+k_2\cdot a_3+k_2 k_3+h(a_2 \otimes a_3)\big)+h(a_1 \otimes a_2 \otimes a_3) \end{gathered}$$ $$\begin{gathered}
=a_1\cdot (a_2\cdot k_3)+a_1\cdot (k_2\cdot a_3)+a_1\cdot (k_2 k_3 )+a_1\cdot h(a_2 \otimes a_3) \\
+k_1\cdot (a_2 a_3)+k_1\cdot (a_2\cdot k_3)+k_1\cdot (k_2\cdot a_3)+k_1 k_2 k_3+k_1 h(a_2 \otimes a_3)+h(a_1 \otimes a_2 \otimes a_3) \end{gathered}$$ By the identities $(3.1)$ and $(3.2)$, we have $$\begin{aligned}
a_1\cdot (a_2\cdot k_3)&=(a_1 a_2 )\cdot k_3+h(a_1,a_2) k_3 \\
(k_1\cdot a_2 )\cdot a_3&=k_1\cdot (a_2 a_3 )+k_1 h(a_2\otimes a_3)\end{aligned}$$ By the regularity of $\mu$, we have $$(a_1\cdot k_2)\cdot a_3=a_1 \cdot (k_2\cdot a_3)$$ By the definition of $Mul(K)$, we have $$\begin{aligned}
v(k_1 k_2)=(k_1 k_2)\cdot a_3&=k_1 (k_2\cdot a_3)=k_1 v(k_2) \\
u(k_2 k_3)=a_1\cdot (k_2 k_3)&=(a_1\cdot k_2 ) k_3=u(k_1) k_2 \\
k_1 u(k_3)=k_1\cdot (a_2\cdot k_3)&=(k_1\cdot a_2)\cdot k_3=v(k_1) k_3\end{aligned}$$ All of them, together with the term $k_1 k_2 k_3$, are identical. The remaining part is $$h(a_1 \otimes a_2)\cdot a_3+h(a_1\otimes a_2 \otimes a_3)=a_1\cdot h(a_2\otimes a_3)+h(a_1\otimes a_2\otimes a_3)$$ This implies $(\delta h)(a_1\otimes a_2\otimes a_3)=f(a_1\otimes a_2\otimes a_3)=0$ as desired. Thus, $A'$ becomes an associative algebra. Lastly, we identify the subalgebra $(0,K)$ with $K$, and define the homomorphism $\beta \big((a,k)\big)=a$. Let its inverse be $$\gamma(a)=(a,0)$$ Indeed, $\beta \gamma=id_A$ and $\gamma(a)$ produces the bimultiplication $\mu_a$. Specifically, $$\begin{aligned}
\alpha \big(\mu_a^{\gamma} (k) \big)&=\big(\gamma(a)\cdot \alpha(k), \alpha(k)\cdot \gamma(a) \big) \\
&=\big((a,0),(0,k),(0,k)(a,0) \big) \\
&=\Big(\big(0,a\cdot k+h(a\otimes 0)\big),\big(0,k\cdot a+h(0\otimes a)\big)\Big) \\
&=(a\cdot k, k\cdot a) \\
&=\mu_a \end{aligned}$$ which is identical to $\mu$. Therefore, we have shown that a coupling having trivial obstruction class determines an extension of algebra.
Appendix D: Proof of Theorem 3 {#appendix-d-proof-of-theorem-3 .unnumbered}
==============================
We start with the triple $(A, M, f)$ and build a pair $(K, \xi)$ together with some proper $\mu$ and $h$ such that the following equalities hold: $$\begin{aligned}
&AnniK=M, \quad \xi=\natural \circ \mu, \quad \epsilon \circ h=R^\mu \\
F^\xi&=F(\mu, h)=\Delta^\mu h \\
&\equiv f\end{aligned}$$ We define $$K=M \oplus L$$ where $$L:=J \oplus A\otimes A \otimes A^*$$ such that $J \triangleleft L$ and $A^*=A\oplus 1$ in the underlying vector space $$J:=C \oplus I$$ $$\begin{aligned}
&C:=\mathbb{F}e\oplus \mathbb{F}f=E \oplus F \\
&I:=E\otimes A' \oplus E\otimes A' \otimes A' \oplus E \otimes A' \otimes A' \otimes A'\end{aligned}$$
On $C$ the multiplication is defined by $$e^2=e,\quad f^2=f,\quad ef=f,\quad fe=e$$ On $I$ and $IC$ the multiplication is defined by $$II=0, \quad IC=0$$ On $CI$ the multiplication is defined by $$ev=v=fv \quad \text{for all} v\in I$$ Because of $J=I \oplus C$ we have in total $$IJ=0$$ On $(C\oplus I)(A\otimes A\otimes A^*)=J(A\otimes A\otimes A^*)$, we trivialize some of the components: $$(F \oplus E\otimes A'\otimes A' \oplus E\otimes A'\otimes A'\otimes A')(A\otimes A \otimes A^*)=0,$$ and concretize the rest ones, $(E \oplus E\otimes A')(A\otimes A \otimes A^*)$, by claiming multiplications between basis: $$\begin{aligned}
e(a_1\otimes a_2 \otimes1)&=e\otimes a_1\otimes a_2 \\
& \in E\otimes A'\otimes A', \\
e(a_1\otimes a_2\otimes a_3)&=e\otimes a_1\otimes a_2a_3-e\otimes a_1a_2\otimes a_3+e\otimes a_1\otimes a_2\otimes a_3 \\
& \in E\otimes A'\otimes A'\oplus E\otimes A'\otimes A'\otimes A', \\
(e\otimes a_1)(a_2\otimes a_3\otimes1)&=e\otimes a_1\otimes a_2\otimes a_3 \\
& \in E\otimes A'\otimes A'\otimes A', \\
(e\otimes a_1)(a_2\otimes a_3\otimes a_4)&=e\otimes a_1\otimes a_2\otimes a_3a_4-e\otimes a_1a_2\otimes a_3\otimes a_4+e\otimes a_1a_2\otimes a_3\otimes a_4 \\
& \in E\otimes A'\otimes A'\otimes A'.\end{aligned}$$
We require $MK=KM=0$ and thus $M$ is our biannihilator of $K$
We close the last unspecified product by assgining $$(A\otimes A \otimes A^*)L=0$$
It can rest assured that these would exhaust all possible multiplications between the components of $K$. Next, we shall define the $\mu$-endomorphisms for $K$ compatible with the conditions for $Mul(K)$. Notice that the most influencing $A$-actions on $K$ stand on $E=A\otimes A\otimes A^* \oplus M$.
For all elements $(p, m)\in E$ we define the two-sided actions by $$\begin{aligned}
a\cdot \big(a_1\otimes a_2\otimes a_3^{\ast}, m \big) &= \big(aa_1\otimes a_2\otimes a_3^{\ast}-a\otimes a_1a_2\otimes a_3^{\ast} +a\otimes a_1\otimes a_2 a_3^{\ast}, \\
& \qquad f(a\otimes a_1\otimes a_2) \cdot a_3^{\ast}+a\cdot m \big), \\
\big(a_1\otimes a_2\otimes a_3^{\ast}, m \big)\cdot a &=\big( a_1\otimes a_2\otimes a_3^{\ast} a, m\cdot a \big)\end{aligned}$$
Note that by definition $f(a\otimes a_1\otimes a_2) \cdot 1\in N$. On the subspace $J$ we set the left action on it by $$a\cdot J=0 \quad \text{or} \quad A\cdot J=0,$$ and all right actions on it by $$\begin{aligned}
e\cdot a &=e\otimes a\in E\otimes A', \\
f\cdot a &=0, \\
(e\otimes a_1)\cdot a &= e\otimes a_1a+e\otimes a_1\otimes a\in (E\otimes A')\oplus(E\otimes A' \otimes A'), \\
(e\otimes a_1\otimes a_2)\cdot a &=e\otimes a_1\otimes a_2a-e\otimes a_1a_2\otimes a+e\otimes a_1\otimes a_2\otimes a \\
&\in(E\otimes A' \otimes A')\oplus (E\otimes A' \otimes A' \otimes A'), \\
(e\otimes a_1\otimes a_2\otimes a_3)\cdot a &= e\otimes a_1\otimes a_2\otimes a_3a-e\otimes a_1\otimes a_2a_3\otimes a+e\otimes a_1a_2\otimes a_3\otimes a \\
&\in E\otimes A' \otimes A' \otimes A'\end{aligned}$$
Indeed, the following conditions should hold under above $K$-multiplications and $A$-actions:
$$\begin{aligned}
k_1(a\cdot k_2) &=(k_1\cdot a)k_2 \\
(a\cdot k_1)k_2 &=a\cdot(k_1k_2) \\
k_1(k_2\cdot a) &=(k_1k_2)\cdot a \\\end{aligned}$$
and permutability $$a_1\cdot (k\cdot a_2)=(a_1\cdot k)\cdot a_2,$$
It is easily to see that on the bimodule $M$ all identities hold immediately.
Take $$\bar{h}(a_1 \otimes a_2):=a_1 \otimes a_2 \otimes 1$$. Due to **Lemma 6.1** and **6.2**, the bimodule $M$ and $A\otimes A \otimes A*$ constitute an extension, from which such a $\bar{h}$ fulfills $\epsilon \circ \bar{h} =R^\mu$ and $\delta_{M\oplus A\otimes A \otimes A*}\bar{h}$ coincides with $f$ inevitably.
Recall that $$\begin{aligned}
\epsilon \circ h(a_1 \otimes a_2) (k)&=R^\mu(a_1 \otimes a_2)(k) \\
(u_{h(a_1 \otimes a_2)}, v_{h(a_1 \otimes a_2)})(k)&=\mu(a_1)\mu(a_2)-\mu(a_1 a_2)(k),\end{aligned}$$ Equivalently, $$\begin{aligned}
a_1\cdot(a_2\cdot k)-a_1 a_2\cdot k &=(a_1\otimes a_2\otimes 1)k \\
(k\cdot a_1)a_2-k\cdot (a_1a_2) &=k(a_1\otimes a_2\otimes 1)\end{aligned}$$
We dirty our hand to plug in the listed terms and get $$(a_1\otimes a_2\otimes1)k=0,$$ whence $a_1\cdot(a_2\cdot k)=a_1 a_2\cdot k$, showing that $K$ is (only) a left $A$-module(however *not* a right one, as its structure is “hindered" by $v_{h(a_1\otimes a_2)}$!).
Appendix E: Proof of Theorem 4 {#appendix-e-proof-of-theorem-4 .unnumbered}
==============================
As we have taken $U=U(\mathfrak{g})$ which plays the role of $A$ in previous theorem, we take again its underlying space $U'$ where $u' \mapsto u$ is a naive isomorphism. The tensor product of $U'$ are the product of vector space and we are able to assign some new multiplications. On the other hand, we shall find another expression of $h$ which then determines a modified structure of $K$ (especially $I\subset K$) and the table of $K$-multiplications and $A$-actions.
In order to differ from $\otimes$, we will replace $\oplus$ by $+$ for visual convenience. $$K:=M + L,$$ $$\text{where} \quad L=J + U\otimes U,$$ $$\text{where} \quad J=I+ C,$$ $$\text{where} \quad I=U' + U'\otimes U'$$ $$\text{where} \quad C=\mathbb{F}e+\mathbb{F}f$$
The list of $K$-multiplications: $$\begin{aligned}
&(U' \otimes U')L= (U \otimes U)L=0 \\
&e(u_1 \otimes u_2)=u_1' \otimes u_2', \quad \mathbb{F}e(U\otimes U)\subset U'\otimes U' \\
&f(u_1 \otimes u_2)=0, \quad \mathbb{F}f(U\otimes U)=0 \\
&u'(u_1 \otimes u_2)=(uu_1)'\otimes (u_2)'-u'\otimes (u_1 u_2)', \quad U'(U\otimes U)\subset U'\otimes U'\end{aligned}$$
The list of $A$-actions on $K$: $$\begin{aligned}
U\cdot J&=0 \\
U\cdot (U\otimes U+ M)&=\big(u u_1\otimes u_2-u\otimes u_1 u_2, f(u, u_1, u_2) \big) \\
(U'\otimes U' + U\otimes U + M)\cdot U&=0 \\
e\cdot u&=u', \qquad \mathbb{F}e\cdot U=U' \\
f\cdot u&=0, \qquad \mathbb{F}f\cdot U=0 \\
u_1'\cdot u&=(u_1 u)'+u_1'\otimes u', \qquad U'\cdot U \subset U'\oplus U'\otimes U'\end{aligned}$$ Note that $U\cdot M$ behaves what it does, and we have altogether $M\cdot U=0$ as prescribed before.
Now we are going to check all four conditions. The given lists appear to be in huge computation, but it will not dirty our hand too much to justify them. Indeed, we concentrate on the most “typical pieces" and conclude their trivialness from the left hand sides or the right hand sides, respectively.
We show that $w\cdot(k_1 k_2)=(w\cdot k_1)\cdot k_2$ for $w\in U(\mathfrak{g})$.
1)If $k_1 \in J=C+I$, then from the list, we have $$\begin{aligned}
k_1k_2\in JK&=J(M+L)\\
&=JL \\
&=(e, f, U', \mathbf{U'\otimes U'})\mathbf{L}\\
&=(e,f, \mathbf{U'})(\mathbf{J}+U'\otimes U')(J+U\otimes U) \\
&=(e,\mathbf{f})(J+\mathbf{U\otimes U}) \\
&=eJ+e(U\otimes U) \\
&=J+U'\otimes U' \\
&\subset J\end{aligned}$$ In other words, as long as $J \triangleleft L$ we have $JL\subset L$. Then in the left hand side it is $$w\cdot(k_1 k_2)\subset U\cdot J=0,$$ and in the right hand sides: $$(w\cdot k_1)\cdot k_2\subset (U\cdot J)\cdot k_2=0$$
2\) If $k_1 \in U\otimes U+M$, the from the left we have $$k_1 k_2\subset (U\otimes U+M)K=(U\otimes U)(M+L)=0$$ due to the list, and from the right $$\begin{aligned}
(w\cdot k_1)&\subset U\cdot(U\otimes U+M) \\
&\subset U'\otimes U'\end{aligned}$$ Then $$\begin{aligned}
(U'\otimes U')\cdot k_2 & \\
&\subset (U'\otimes U')K \\
&=(U'\otimes U')L=0\end{aligned}$$
In this way, we have proved the equality $w\cdot(k_1 k_2)=(w\cdot k_1)\cdot k_2$. Similarly for the other two equalities.
3\) Now we need to show that $(k\cdot w_1)w_2-k(w_1 w_2)=m(w_1 \otimes w_2)$ (this is for $\epsilon \circ h=R^\mu$!)
Let $k=\mathbf{u'_1}+u'_2\otimes u'_3 + \mathbf{e} + f + u_1 \otimes u_2 +m$. Here only the bold elements $(u'_1, e)\in U'+\mathbb{F}e$ are non-zero due to the list. That is, for any $w_1, w_2 \in U$, in the left we have $$\begin{aligned}
&(k\cdot w_1)w_2-k(w_1 w_2)= \\
&=[(u'_1+e)\cdot w_1]\cdot w_2-(u'_1+e)(w_1 w_2) \\
&=[(u_1 w_1)'+u'_1\otimes w'_1+w'_1]\cdot w_2-[(u_1 w_1 w_2)'+u_1'\otimes (w_1 w_2)'+(w_1w_2)'] \\
&=[\mathbf{U'} + U'\otimes U' + \mathbf{U'}]\cdot U-\cdots \\
&=(u_1w_1w_2)'+(u_1w_1)' \otimes w'_2+(w_1 w_2)'+w'_1 \otimes w'_2-(u_1 w_1 w_2)'-u'_1\otimes (w_1 w_2)'-(w_1 w_2)' \\
&=(u_1w_1)' \otimes w'_2+w'_1 \otimes w'_2-u'_1\otimes (w_1 w_2)',\end{aligned}$$
while in the right: $$\begin{aligned}
&m(w_1 \otimes w_2)= \\
&=(u'_1+e)(w_1 \otimes w_2) \\
&=(uw_1)'\otimes w_2-u'_1\otimes (w_1 \otimes w_2)'+w'_1\otimes w'_2\end{aligned}$$
Therefore, we get that $(k\cdot w_1)w_2-k(w_1 w_2)=m(w_1 \otimes w_2)$(meaning $K$ is *not* an right $A$-module!). Similar computation for the left one, which claims a left module on $K$, also valid in **Appendix D**)(Tips: always catch the nonzero parts and try to cancel them)
4\) We show $AnniK=M$ or $AnniL=0$
Let $l=\alpha e+\beta f+p+i+j \in \mathbb{F}e+\mathbb{F}f+U\otimes U+U'+U'\otimes U'$. By contradiction, we suppose $ZL\neq 0$, then there exists a $l'\in L$ such that $ll'=0$ or $l'l=0$. It suffices to accommodate a $l'$ deviously.
Since the arbitrariness of $l'$, at first we set $l'\in U\otimes U$ and we multiply from the left by $$(\alpha e+\beta f+p+i+j)l'\in l(U\otimes U)=0$$ Then from the list we have $$(\alpha e+i)(u_1 \otimes u_2)=0$$ The linear independence of the basis implies $$\alpha e+i=0 (*)$$
Next, let $l'=f$ and multiply from the right $$f(\alpha e+\beta f+p+i+j)=0$$ We get $$\alpha e+\beta+fv=\alpha e+\beta+v=0$$ Again,the linear independence of the basis implies $$\alpha, \beta, v=0$$
Since (\*) we have $i=0$, then immediately $v=i+j, j=0$
Finally, set $l'=e$ and multiply from the left $$(p+j)e=p\cdot e=0$$ It follows $e=0$. Therefore all constituents of $l'$ is zero and the claim is disproved, meaning the annihilator of $L$ is nihil.
|
---
abstract: 'The aim of sequential change-point detection is to issue an alarm when it is thought that certain probabilistic properties of the monitored observations have changed. This work is concerned with nonparametric, closed-end testing procedures based on differences of empirical distribution functions that are designed to be particularly sensitive to changes in the contemporary distribution of multivariate time series. The proposed detectors are adaptations of statistics used in a posteriori (offline) change-point testing and involve a weighting allowing to give more importance to recent observations. The resulting sequential change-point detection procedures are carried out by comparing the detectors to threshold functions estimated through resampling such that the probability of false alarm remains approximately constant over the monitoring period. A generic result on the asymptotic validity of such a way of estimating a threshold function is stated. As a corollary, the asymptotic validity of the studied sequential tests based on empirical distribution functions is proven when these are carried out using a dependent multiplier bootstrap for multivariate time series. Large-scale Monte Carlo experiments demonstrate the good finite-sample properties of the resulting procedures. The application of the derived sequential tests is illustrated on financial data.'
address: 'CNRS / Université de Pau et des Pays de l’Adour / E2S UPPA, Laboratoire de mathématiques et applications, IPRA, UMR 5142, B.P. 1155, 64013 Pau Cedex, France, '
author:
-
-
bibliography:
- 'biblio.bib'
title: 'Nonparametric sequential change-point detection for multivariate time series based on empirical distribution functions'
---
Introduction
============
Let $\bm X_1,\dots,\bm X_m$, $m \geq 1$, be a stretch from a $d$-dimensional stationary times series of continuous random vectors with unknown contemporary distribution function (d.f.) $F$ given by $F(\bm x) = {\mathbb{P}}(\bm X_1 \leq \bm x)$, $\bm x \in {\mathbb{R}}^d$. These available observations will be referred to as the *learning sample* as we continue. The context of this work is that of *sequential change-point detection*: new observations $\bm X_{m+1},\bm X_{m+2},\dots$ arrive sequentially and we wish to issue an alarm as soon as possible if the contemporary distribution of the most recent observations is not equal to $F$ anymore. If there is no evidence of a change in the contemporary distribution, the monitoring stops after the arrival of observation $\bm X_n$ for some $n > m$.
The theoretical framework of our investigations is that adopted in the seminal paper of [@ChuStiWhi96]. Unlike classical approaches in statistical process control (SPC) usually calibrated in terms of *average run length* (ARL) [see, e.g., @Lai01; @Mon07 for an overview] and leading in general to the rejection of the underlying null hypothesis of stationarity with probability one, the approach of [@ChuStiWhi96] guarantees that, asymptotically, stationarity, if it holds, will only be rejected with a small probability $\alpha$ to be interpreted as a type I error and thus called the *probability of false alarm*. The latter paradigm is increasingly considered in the literature; see, e.g., [@HorHusKokSte04], [@AueHorKuhSte12], [@AueHorHorHusSte12], [@Fre15], [@KirWeb18], [@DetGos19] or [@KirSto19].
Among the approaches *à la* [@ChuStiWhi96], one can distinguish between *closed-end* and *open-end* procedures. The latter can in principle continue indefinitely if no evidence against the null is observed. Our approach is of the former type in the sense that at most $n - m$ new observations will be considered before the monitoring stops.
As already mentioned, the null hypothesis of the procedure that we shall investigate is that of stationarity and can be more formally stated as follows: $$\label{eq:H0}
\begin{split}
H_0: \, &\bm X_1,\dots,\bm X_m,\bm X_{m+1},\dots, \bm X_n \text{ is a stretch from a stationary time series} \\
&\text{with contemporary d.f.\ } F.
\end{split}$$ Notice that, if one is additionally ready to assume the independence of the observations, $H_0$ simplifies to $$\label{eq:H0:ind}
H_0^{\text{ind}}: \, \bm X_1,\dots,\bm X_m,\bm X_{m+1},\dots, \bm X_n \text{ are independent random vectors with d.f.\ } F.$$ The aim of this work is to derive nonparametric sequential change-point detection procedures particularly sensitive to the alternative hypothesis $$\label{eq:H1}
\begin{split}
H_1: \, &\exists \, k^\star \in \{m,\dots,n-1\} \text{ such that } \bm X_1,\dots,\bm X_{k^\star} \text{ is a stretch from a stationary} \\
&\text{time series with contemporary d.f.\ } F \text { and } \bm X_{k^\star+1},\dots,\bm X_{n} \text{ is a stretch} \\
& \text{from a stationary time series with contemporary d.f.\ } G \neq F.
\end{split}$$ In other words, unlike some of the approaches reported for instance in [@KirWeb18], [@DetGos19] or [@KirSto19], we are not solely interested in being sensitive to a change in a given parameter of the $d$-dimensional time series such as the mean vector or the covariance matrix. We aim at deriving nonparametric monitoring procedures that, in principle, provided $m$ and $n$ are large enough, can detect all types of changes in the contemporary d.f.
In the considered context, the two main ingredients of a sequential change-point detection procedure are a sequence of positive statistics $D_m(k)$, $k \in \{m+1,\dots,n\}$, and a sequence of suitably chosen strictly positive thresholds $w_m(k)$, $k \in \{m+1,\dots,n\}$. For $k \in \{m+1,\dots,n\}$, the statistic $D_m(k)$ (called a *detector* in the literature) is used to assess a possible departure from $H_0$ in using only observations $\bm X_1, \dots, \bm X_k$ and is such that the larger $D_m(k)$, the more evidence against $H_0$. After the arrival of observation $\bm X_k$, $k \in \{m+1,\dots,n\}$, the detector $D_m(k)$ is computed and compared to the threshold $w_m(k)$. If $D_m(k) > w_m(k)$, the available evidence against stationarity is considered to be large enough and an alarm is issued resulting in the monitoring to stop. If $D_m(k) \leq w_m(k)$ and $k < n$, a new observation $\bm X_{k+1}$ is collected and the previous iteration is repeated. This monitoring process can be naturally represented by a graph illustrating the evolution of the sequence of detectors against the sequence of thresholds; see, e.g., Figures \[fig:det-thresh\] and \[fig:biv-det-thresh\]. To ensure that the monitoring can be interpreted as a testing procedure, the sequence of thresholds $w_m(k)$, $k \in \{m+1,\dots,n\}$, needs to be chosen such that, under stationarity, $$\label{eq:neg:typeI}
{\mathbb{P}}\{ D_m(m+1) \leq w_m(m+1), \dots, D_m(n) \leq w_m(n) \} \geq 1 - \alpha,$$ for some small significance level $\alpha$.
The detectors $D_m(k)$, $k \in \{m+1,\dots,n\}$, proposed in this work are defined from differences of empirical distribution functions. Specifically, as shall be explained in detail in Section \[sec:detectors\], particularly powerful monitoring procedures can be obtained by defining $D_m(k)$ as a maximized, suitably normalized difference of empirical distribution functions computed from $\bm X_1,\dots, \bm X_j$ and $\bm X_{j+1},\dots, \bm X_k$, respectively, where the maximum is taken over all $j \in \{m,\dots,k-1\}$. When sequentially investigating changes in real-valued parameters of multivariate time series, such an approach (related to what was called a Page-CUSUM procedure in [@KirWeb18] as a consequence of the work of [@Fre15]) was recently suggested in [@DetGos19] and justified through a likelihood ratio approach. The latter principle had actually already been considered in the SPC literature (to the best of our knowledge, without any asymptotic theory, however) since at least the work of [@HawQiuKan03]; see also [@HawZam05], [@RosTasAda11] and [@RosAda12]. As we shall see, our detectors can be cast into the framework of [@DetGos19] with the difference that they involve a weighting allowing to give more or less importance to recent observations. In that sense, they could also be regarded as an adaptation of the statistics considered in [@CsoSzy94b] for a posteriori (offline) change-point detection to the sequential setting considered in this work.
As far as the thresholds $w_m(k)$, $k \in \{m+1,\dots,n\}$, are considered, unlike in the recent literature in which these are calibrated through parametric functions defined up to a positive multiplicative constant such that holds [see, e.g., @KirWeb18; @DetGos19; @KirSto19], we define them through simulations or resampling with the aim that, under $H_0$ in , the probability of rejection of $H_0$ be roughly the same at every step $k \in \{m+1,\dots,n\}$ of the procedure. Except for the use of resampling, the idea is not new and seems to appear first in SPC in [@MarConWooDra95]; see also, for instance, [@HawZam05], [@Ros14], [@RosAda12] and the package `cpm` described in [@Ros15].
From the point of view of their main ingredients, the sequential change-point detection procedures *à la* [@ChuStiWhi96] studied in this work are related to the approach put forward in SPC in [@RosAda12] (without any asymptotic theory) and implemented in the package [cpm]{} [@Ros15]. However, unlike the latter, they are not restricted to univariate observations and they can deal with time series, among several other differences. To achieve this, as soon as the learning data set $\bm X_1,\dots,\bm X_m$ is multivariate ($d > 1$) or serially dependent, resampling, under the form of a *dependent multiplier bootstrap* *à la* [@BucKoj16], is used to carry out the monitoring procedures. By adapting the theoretical framework considered in [@DetGos19], the asymptotic validity of the investigated procedures is proven under strong mixing both under $H_0$ in and $H_1$ in .
This paper is organized as follows. In the second section, we propose three classes of detectors based on differences of empirical d.f.s and study their asymptotics, both under $H_0$ and $H_1$. A more general perspective is adopted in the third section: for an arbitrary detector in the considered closed-end setting, a procedure for estimating the threshold function such that the probability of false alarm remains approximately constant over the monitoring period is investigated and its asymptotic validity is proven under both $H_0$ and $H_1$ when the estimation is based on an asymptotically valid resampling scheme. The results of the third section are applied in the fourth section to the three proposed classes of detectors based on differences of empirical d.f.s. To do so, the consistency of a dependent multiplier bootstrap for the detectors is proven under strong mixing. The fifth section presents a summary of large-scale Monte-Carlo experiments demonstrating the good finite-sample properties of the resulting sequential testing procedures. An application on real financial data concludes the article.
Auxiliary results and proofs are deferred to a sequence of appendices, some of which are provided in the supplement [@KojVer20]. The studied tests will be soon available in the package [npcp]{} [@npcp] for the statistical system [@Rsystem].
The detectors and their asymptotics {#sec:detectors}
===================================
After defining three classes of detectors based on empirical distribution functions and noticing that they are margin-free under $H_0$ in , we study their asymptotics under the null and $H_1$ in .
Detectors based on empirical distribution functions {#sec:det:emp}
---------------------------------------------------
Let $F_{j:k}$ be the empirical d.f. computed from the stretch $\bm X_j,\dots,\bm X_k$ of available observations. More formally, for any integers $j,k \geq 1$ and $\bm x \in {\mathbb{R}}^d$, let $$\label{eq:Fjk}
F_{j:k}(\bm x) =
\left\{
\begin{array}{ll}
\disp \frac{1}{k-j+1}\sum_{i=j}^k {\mathbf{1}}(\bm X_i \leq \bm x), \qquad & \text{if } j \leq k, \\
0, \qquad &\text{otherwise},
\end{array}
\right.$$ where the inequalities $\bm X_i \leq \bm x$ are to be understood componentwise and ${\mathbf{1}}(\bm X_i \leq \bm x)$ is equal to 1 (resp. 0) if all (resp.some) of the $d$ underlying inequalities are true (resp. false).
After the $k$th observation has arrived, the available data take the form of the stretch $\bm X_1,\dots,\bm X_k$. If we were in the context of a posteriori change-point detection, a prototypical test statistic would be the maximally selected Kolmogorov–Smirnov-type statistic $$\label{eq:Rk}
R_k = \max_{1 \leq j \leq k-1} \frac{j (k - j)}{k^{3/2}} \sup_{\bm x \in {\mathbb{R}}^d} | F_{1:j}(\bm x) - F_{j+1:k}(\bm x) |,$$ practically considered for instance in @GomHor99 or [@HolKojQue13]. The intuition behind $R_k$ is the following: every $j \in \{1,\dots,k-1\}$ is treated as a potential break point in the sequence and the maximum in implies that $R_k$ will be large as soon as the difference between $F_{1:j}$ and $F_{j+1:k}$ is large for some $j$. The weighting $j (k - j) / k^{3/2}$ ensures that $R_k$ converges in distribution under stationarity as $k \to \infty$. As explained for instance in [@CsoSzy94b] in the case of independent observations, replacing for instance these weights simply by $\sqrt{k}$ would result in a statistic that diverges in probability to $\infty$ under stationarity. The part $j (k - j)$ in the weighting favors however the detection of potential break points in the middle of the sequence. Test statistics that are more sensitive to changes at the beginning or at the end of the sequence but still converge in distribution under stationarity can be obtained by considering weights of the form $j (k - j) / \{ k^{3/2} q(j/k) \}$ for some suitable strictly positive function $q$ on $(0,1)$; see, e.g., @CsoSzy94 [@CsoSzy94b], [@CsoHorSzy97] and [@CsoHor97].
Going back to the setting of sequential change-point detection considered in this work, a first meaningful modification of is to restrict the maximum over $j$ to $j \in \{m,\dots,k-1\}$ since a change cannot occur at the beginning of the sequence given that $\bm X_1, \dots, \bm X_m$ is the learning sample known to be a stretch from a stationary sequence. Another modification is a rescaling consisting of replacing $k^{3/2}$ by $m^{3/2}$ in the weighting. The latter is made solely for asymptotic reasons as shall become clear in Sections \[sec:asym:det:H0\] and \[sec:asym:det:H1\]. These modifications essentially lead to the first detectors considered in this work: $$\label{eq:Rmq}
R_{m,q}(k) = \max_{m \leq j \leq k-1} \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \sup_{\bm x \in {\mathbb{R}}^d} | F_{1:j}(\bm x) - F_{j+1:k}(\bm x) |, \qquad k \in \{m+1,\dots,n\}.$$ In the previous display, $q$ is a strictly positive function whose role is to potentially give more weight to recent observations. In the sequel, we consider the parametric form $$\label{eq:q}
q(s,t) = \max\{ s^\gamma (t-s)^\gamma, \delta \}, \qquad 0 \leq s \leq t,$$ where $\delta \in (0,1)$ is a small constant and $\gamma$ is a parameter in $[0,1/2]$. If $\gamma = 0$, $q$ is the constant function $1$ and $R_{m,q}(k)$ is then a straightforward adaptation of $R_k$ in to sequential change-point detection. In that case, the general form of $R_{m,q}(k)$ can also be heuristically justified through a likelihood ratio approach; see Section 2 in [@DetGos19]. When $\gamma = 1/2$, as shall be discussed in Remark \[rem:q\] (in Section \[sec:asym:det:H0\]) using asymptotic arguments, $R_{m,q}(k)$ can be regarded, under $H_0$ in , as a maximum of random variables with, approximately, the same mean and variance. This heuristically implies that all the potential break points $j \in \{m,\dots,k-1\}$ are given roughly the same weight in the computation of $R_{m,q}(k)$ unlike in the case $\gamma = 0$ in which potential break points closest to ${\lfloor k/2 \rfloor}$ are given more weight. Hence, for certain types of alternatives to $H_0$ in , choosing $\gamma \in (0,1/2]$ might accelerate the detection of the corresponding change in the contemporary distribution of the underlying time series. Examples of such alternatives will be given in Section \[sec:MC\] in which the results of numerous Monte Carlo experiments are summarized.
Two similar Cramér–von Mises-like detectors are also considered in this work. They are defined, for $k \in \{m+1,\dots,n\}$, by $$\label{eq:Smq}
\begin{split}
S_{m,q}(k) &= \max_{m \leq j \leq k-1} \int_{{\mathbb{R}}^d} \left[ \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \{ F_{1:j}(\bm x) - F_{j+1:k}(\bm x) \} \right]^2 {\mathrm{d}}F_{1:k}(\bm x) \\
&= \max_{m \leq j \leq k-1} \frac{1}{k} \sum_{i=1}^k \left[ \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \{ F_{1:j}(\bm X_i) - F_{j+1:k}(\bm X_i) \} \right]^2,
\end{split}$$ and $$\begin{aligned}
\label{eq:Tmq}
T_{m,q}(k) &= \frac{1}{m} \sum_{j=m}^{k-1} \frac{1}{k} \sum_{i=1}^k \left[ \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \{ F_{1:j}(\bm X_i) - F_{j+1:k}(\bm X_i) \} \right]^2.\end{aligned}$$
The detectors in and are related to those used in @RosAda12 [pp 104–106]. The latter are also based on differences of empirical d.f.s but deal only with independent univariate observations. The analogue of is apparently defined as a maximum of the quantities $\sup_{\bm x \in {\mathbb{R}}} | F_{1:j}(\bm x) - F_{j+1:k}(\bm x) |$, $j \in \{m,\dots,k-1\}$, previously normalized using an empirical probability integral transformation, probably based on simulations. The analogue of takes the form of a maximum of the quantities $\sum_{i=1}^k \{ F_{1:j}(\bm X_i) - F_{j+1:k}(\bm X_i) \}^2$, $j \in \{m,\dots,k-1\}$, after centering and scaling. The asymptotics of these detectors were not studied. Given that the detectors of [@RosAda12] are distribution-free (see also Section \[sec:margin:free:H0\] hereafter), that their approach assumes serially independent observations and is based on simulations, the absence of asymptotic theory is not problematic.
The detectors in , and are almost of the Page-CUSUM type considered initially in [@Fre15]; see also [@KirWeb18]. For instance, in the case of , the adaption of the latter construction to the present setting would have instead involved a maximum of the quantities $\sup_{\bm x \in {\mathbb{R}}^d} | F_{1:m}(\bm x) - F_{j+1:k}(\bm x) |$, $j \in \{m,\dots,k-1\}$. As explained in [@DetGos19], the use of such detectors may result in a loss of power in the case of a small learning sample and a rather late change point. In the Monte Carlo experiments carried out in [@DetGos19], Page-CUSUM detectors were always outperformed by their analogues of type , which is why we do not consider them in this work.
In addition to the detectors , and , we also considered in our Monte Carlo experiments the following natural competitors which are straightforward adaptions of the so-called CUSUM construction considered for instance in [@HorHusKokSte04] and [@AueHorHorHusSte12]. Their Kolomogorov–Smirnov versions and Cramér–von Mises versions are respectively given, for any $k \in \{m+1,\dots,n\}$, by $$\begin{aligned}
\label{eq:Pm}
P_m(k) =& \frac{m (k-m)}{m^{3/2}} \sup_{\bm x \in {\mathbb{R}}^d} | F_{1:m}(\bm x) - F_{m+1:k}(\bm x) |, \\
\label{eq:Qm}
Q_m(k) =& \frac{1}{k} \sum_{i=1}^k \left[ \frac{m (k-m)}{m^{3/2}} \{F_{1:m}(\bm X_i) - F_{m+1:k}(\bm X_i) \} \right]^2.
$$ The asymptotic theory for these detectors being simpler than for the detectors , and , it will not be stated in the forthcoming sections for the sake of readability.
The detectors are margin-free under the null {#sec:margin:free:H0}
--------------------------------------------
The detectors defined previously are actually margin-free under $H_0$ in , a property that shall be exploited in the forthcoming sections to carry out the corresponding sequential change-point detection procedures.
Recall that $\bm X_1,\dots,\bm X_n$ are assumed to be continuous random vectors. Saying that the detectors are margin-free under the null means that they do not depend on the $d$ univariate margins $F_1,\dots,F_d$ of $F$ (the unknown d.f. of $\bm X_1$) or, equivalently, that they can alternatively be written in terms of the unobservable random vectors $\bm U_1,\dots,\bm U_n$ defined from $\bm X_1,\dots,\bm X_n$ through marginal probability integral transformations: $$\label{eq:Ui}
\bm U_i = (F_1(X_{i1}),\dots,F_d(X_{id})).$$ Notice that we can recover the $\bm X_i$ from the $\bm U_i$ by marginal quantile transformations: $$\label{eq:Xi}
\bm X_i = (F_1^{-}(U_{i1}),\dots,F_d^{-}(U_{id})),$$ where, for any univariate d.f. $G$, $G^{-1}$ denotes its associated quantile function defined by $$\label{eq:quant:func}
G^{-1}(y) = \inf \{x \in {\mathbb{R}}: G(x) \geq y\}, \qquad y \in [0,1],$$ with the convention that the infimum of the empty set is $\infty$.
To verify that the detectors are margin-free under the null, for any integers $j,k \geq 1$ and $\bm u \in [0,1]^d$, let $$\label{eq:Cjk}
C_{j:k}(\bm u) =
\left\{
\begin{array}{ll}
\disp \frac{1}{k-j+1}\sum_{i=j}^k {\mathbf{1}}(\bm U_i \leq \bm u), \qquad &\text{if } j \leq k, \\
0, \qquad &\text{otherwise},
\end{array}
\right.$$ be the analogue of $F_{j:k}$ in based on the $\bm U_i$ in . For any $j \in \{1,\dots,d\}$, by (right) continuity of $F_j$, we have that ${\mathbf{1}}\{ F_j^-(u) \leq x \} = {\mathbf{1}}\{ u \leq F_j(x) \}$ for all $u \in [0,1]$ and $x \in {\mathbb{R}}$; see, e.g., Proposition 1 (5) in [@EmbHof13]. The latter property combined with implies that, under $H_0$ in , for any $i \in \{1,\dots,n\}$, $$\label{eq:1Xi:1Ui}
{\mathbf{1}}( \bm X_i \leq \bm x ) = {\mathbf{1}}\{ \bm U_i \leq \bm F(\bm x) \}, \qquad \bm x \in {\mathbb{R}}^d,$$ where $\bm F(\bm x) = (F_1(x_1),\dots,F_d(x_d))$. Hence, for any $k \in \{m+1,\dots,n\}$, $$\begin{aligned}
R_{m,q}(k) &= \max_{m \leq j \leq k-1} \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \sup_{\bm x \in {\mathbb{R}}^d} | C_{1:j}\{\bm F(\bm x)\} - C_{j+1:k}\{\bm F(\bm x)\} |, \\
&= \max_{m \leq j \leq k-1} \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \sup_{\bm u \in [0,1]^d} | C_{1:j}(\bm u) - C_{j+1:k}(\bm u) |.\end{aligned}$$ Similarly, $$\begin{aligned}
S_{m,q}(k) &= \max_{m \leq j \leq k-1} \frac{1}{k} \sum_{i=1}^k \left[ \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \{ C_{1:j}(\bm U_i) - C_{j+1:k}(\bm U_i) \} \right]^2, \\
T_{m,q}(k) &= \frac{1}{m} \sum_{j=m}^{k-1} \frac{1}{k} \sum_{i=1}^k \left[ \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \{ C_{1:j}(\bm U_i) - C_{j+1:k}(\bm U_i) \} \right]^2.\end{aligned}$$
In the case of univariate independent observations, the margin-free property under the null implies that the detectors are distribution-free under the null. When $d>1$, this is not true anymore as the null distribution of the detectors depends on the *copula* $C$ associated with $F$. The latter is merely the d.f. of the random vector $\bm U_1$ obtained through . Equivalently, $C$ is a $d$-dimensional d.f. with standard uniform margins further uniquely defined [see @Skl59] through the relationships $$\label{eq:F:C}
F(\bm x) = C\{F_1(x_1),\dots,F_d(x_d) \}, \qquad \bm x \in {\mathbb{R}}^d,$$ and $$\label{eq:C:F}
C(\bm u) = F\{F_1^-(u_1),\dots,F_d^-(u_d) \}, \qquad \bm u \in [0,1]^d.$$
To be able to handle both the univariate and the multivariate situations, in the rest of the paper, we adopt the convention that $C$ is the copula associated with $F$ when $d > 1$ and merely the identity function when $d=1$.
Asymptotics of the detectors under the null {#sec:asym:det:H0}
-------------------------------------------
As shall become clear in the forthcoming sections, the knowledge of the asymptotic behavior of the detectors under $H_0$ in is instrumental in showing the asymptotic validity of the corresponding sequential change-point detection procedures. To study these asymptotics, we follow [@DetGos19], among others, and set $n={\lfloor m(T+1) \rfloor}$ for some fixed real number $T > 0$. This will imply that, in the asymptotics, as the size $m$ of the learning sample goes to infinity, the maximum number of new observations considered in the monitoring increases proportionally.
Let $\Delta = \{(s,t) \in [0,T+1]^2 : s \leq t \}$ and let $$\label{eq:lambda}
\lambda_m(s,t) = ({\lfloor mt \rfloor} - {\lfloor ms \rfloor})/m, \qquad (s, t) \in \Delta.$$ Then, for any $(s, t) \in \Delta$ and $\bm u \in [0,1]^d$, let $$\label{eq:Gbm}
{\mathbb{G}}_m(s,t,\bm u) = \sqrt{m} \lambda_m(0,s) \lambda_m(s,t) \{ C_{1:{\lfloor ms \rfloor}}(\bm u) - C_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}(\bm u)\}$$ where $C_{1:{\lfloor ms \rfloor}}$ and $C_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}$ are generically defined by , and let $$\label{eq:Gbmq}
{\mathbb{G}}_{m,q}(s,t,\bm u) = \frac{{\mathbb{G}}_m(s,t,\bm u)}{q \{ \lambda_m(0,s),\lambda_m(0,t) \}},$$ where $q$ is defined in . Notice that, with the definitions adopted thus far, ${\mathbb{G}}_m(s,s,\cdot) = {\mathbb{G}}_{m,q}(s,s,\cdot) = 0$ for all $s \in [0,T+1]$.
For any $k \in \{m+1,\dots,n\}$ with $n = {\lfloor m(T+1) \rfloor}$ and any $j \in \{m,\dots,k-1\}$, there exists $(s,t) \in \Delta \cap [1,T+1]^2$ such that $k = {\lfloor mt \rfloor}$ and $j = {\lfloor ms \rfloor}$. We can thus write $R_{m,q}(k)$ as $$\label{eq:Rmq:Gmq}
\begin{split}
R_{m,q}({\lfloor mt \rfloor}) &= \max_{m \leq j \leq k-1} \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \sup_{\bm u \in [0,1]^d} | C_{1:j}(\bm u) - C_{j+1:k}(\bm u) | \\
&= \sup_{s \in [1,t]} \sup_{\bm u \in [0,1]^d} \frac{\sqrt{m} \lambda_m(0,s) \lambda_m(s,t) | C_{1:{\lfloor ms \rfloor}}(\bm u) - C_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}(\bm u) |}{q\{ \lambda_m(0,s),\lambda_m(0,t) \}} \\
&= \sup_{s \in [1,t]} \sup_{\bm u \in [0,1]^d} |{\mathbb{G}}_{m,q}(s,t,\bm u)|.
\end{split}$$ Similarly, it can be verified that $$\begin{aligned}
\label{eq:Smq:Gmq}
S_{m,q}(k) = S_{m,q}({\lfloor mt \rfloor}) &= \sup_{s \in [1,t]} \int_{[0,1]^d} \{ {\mathbb{G}}_{m,q}(s,t,\bm u) \}^2 {\mathrm{d}}C_{1:{\lfloor mt \rfloor}}(\bm u), \\
\label{eq:Tmq:Gmq}
T_{m,q}(k) = T_{m,q}({\lfloor mt \rfloor}) &= \int_1^t \int_{[0,1]^d} \{ {\mathbb{G}}_{m,q}(s,t,\bm u) \}^2 {\mathrm{d}}C_{1:{\lfloor mt \rfloor}}(\bm u) {\mathrm{d}}s.\end{aligned}$$
As we continue, we adopt the convention that $R_{m,q}(m) = S_{m,q}(m) = T_{m,q}(m) = 0$. Furthermore, given a set ${\mathcal{S}}$, the space of all bounded real-valued functions on ${\mathcal{S}}$ equipped with the uniform metric is denoted by $\ell^\infty({\mathcal{S}})$. The main purpose of this section is to study the asymptotics under the null of the elements ${\mathbb{R}}_{m,q}$, ${\mathbb{S}}_{m,q}$ and ${\mathbb{T}}_{m,q}$ of $\ell^\infty([1,T+1])$ defined respectively, for any $t \in [1,T+1]$, by $$\label{eq:det:as:func}
{\mathbb{R}}_{m,q}(t) = R_{m,q}({\lfloor mt \rfloor}), \qquad {\mathbb{S}}_{m,q}(t) = S_{m,q}({\lfloor mt \rfloor}), \qquad {\mathbb{T}}_{m,q}(t) = T_{m,q}({\lfloor mt \rfloor}).$$ Specifically, we will provide conditions under which they converge weakly in the sense of Definition 1.3.3 in [@vanWel96] under $H_0$ in . Throughout the paper, this mode of convergence will be denoted by the arrow ‘$\leadsto$’ and all convergences will be for $m \to \infty$ unless mentioned otherwise.
From the expressions given in , and , we see that, under the null, the detectors studied in this work are functionals of ${\mathbb{G}}_{m,q}$ in , and thus of ${\mathbb{G}}_m$ in . Under stationarity, the latter is in turn a functional of the *sequential empirical process* defined, for any $s \in [0,T+1]$ and $\bm u \in [0,1]^d$, by $$\label{eq:Bbm}
\begin{split}
{\mathbb{B}}_m(s,\bm u) &= \frac{1}{\sqrt{m}} \sum_{i=1}^{{\lfloor ms \rfloor}} \{ {\mathbf{1}}(\bm U_i \leq \bm u) - C(\bm u) \} = \sqrt{m} \lambda_m(0,s) \{ C_{1:{\lfloor ms \rfloor}}(\bm u) - C(\bm u)\}.
\end{split}$$ Indeed, under $H_0$ in , for any $(s, t) \in \Delta$ and $\bm u \in [0,1]^d$, $$\label{eq:GbmH0}
\begin{split}
{\mathbb{G}}_m(s,t,\bm u) &= \sqrt{m} \lambda_m(0,s) \lambda_m(s,t) \{ C_{1:{\lfloor ms \rfloor}}(\bm u) - C(\bm u) - C_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}(\bm u) + C(\bm u)\} \\
&= \lambda_m(s,t) {\mathbb{B}}_m(s, \bm u) - \lambda_m(0,s) \times \sqrt{m} \lambda_m(s,t) \{ C_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}(\bm u) - C(\bm u) \} \\
&= \{ \lambda_m(0,t) - \lambda_m(0,s) \} {\mathbb{B}}_m(s, \bm u) - \lambda_m(0,s) \{ {\mathbb{B}}_m(t, \bm u) - {\mathbb{B}}_m(s, \bm u) \} \\
&= \lambda_m(0,t) {\mathbb{B}}_m(s, \bm u) - \lambda_m(0,s) {\mathbb{B}}_m(t, \bm u).
\end{split}$$
In the forthcoming asymptotic results, we shall assume that the underlying stationary sequence $(\bm X_i)_{i \in {\mathbb{Z}}}$ (or, equivalently, the corresponding unobservable stationary sequence $(\bm U_i)_{i \in {\mathbb{Z}}}$ defined through ) is *strongly mixing*. Denote by ${\mathcal{F}}_j^k$ the $\sigma$-field generated by $(\bm X_i)_{j \leq i \leq k}$, $j, k \in {\mathbb{Z}}\cup \{-\infty,+\infty \}$, and recall that the strong mixing coefficients corresponding to the stationary sequence $(\bm X_i)_{i \in {\mathbb{Z}}}$ are then defined by $\alpha_0^{\bm X} = 1/2$, $$\alpha_r^{\bm X} = \sup_{A \in {\mathcal{F}}_{-\infty}^0,B\in {\mathcal{F}}_{r}^{+\infty}} \big| {\mathbb{P}}(A \cap B) - {\mathbb{P}}(A) {\mathbb{P}}(B) \big|, \qquad r \in {\mathbb{N}}, \, r > 0,$$ and that the sequence $(\bm X_i)_{i \in {\mathbb{Z}}}$ is said to be *strongly mixing* if $\alpha_r^{\bm X} \to 0$ as $r \to \infty$. The following result is proven in the supplement [@KojVer20].
\[prop:H0\] Assume that $H_0$ in holds and that, additionally, $\bm X_1,\dots,\bm X_n$ is a stretch from a stationary sequence $(\bm X_i)_{i \in {\mathbb{Z}}}$ of continuous $d$-dimensional random vectors whose strong mixing coefficients satisfy $\alpha_r^{\bm X} = O(r^{-a})$ for some $a > 1$ as $r \to \infty$. Then, ${\mathbb{B}}_m \leadsto {\mathbb{B}}_C$ in $\ell^\infty([0,T+1] \times [0,1]^d)$, where ${\mathbb{B}}_C$ is a tight centered Gaussian process with covariance function $${\mathrm{Cov}}\{{\mathbb{B}}_C(s,\bm u), {\mathbb{B}}_C(t,\bm v)\} = (s \wedge t) \Gamma(\bm u, \bm v), \qquad s,t \in [0,T+1], \bm u, \bm v \in [0,1]^d,$$ with $\wedge$ the minimum operator and $$\label{eq:Gamma}
\Gamma(\bm u, \bm v) = \sum_{i \in {\mathbb{Z}}} {\mathrm{Cov}}\{ {\mathbf{1}}(\bm U_0 \leq \bm u), {\mathbf{1}}(\bm U_i \leq \bm v) \}.$$ As a consequence, ${\mathbb{G}}_m \leadsto {\mathbb{G}}_C$ and ${\mathbb{G}}_{m,q} \leadsto {\mathbb{G}}_{C,q}$ in $\ell^\infty(\Delta \times [0,1]^d)$, where ${\mathbb{G}}_m$ and ${\mathbb{G}}_{m,q}$ are defined in and , respectively, and, for any $(s, t) \in \Delta$ and $\bm u \in [0,1]^d$, $$\label{eq:GbC}
{\mathbb{G}}_C(s,t, \bm u) = t {\mathbb{B}}_C(s, \bm u) - s {\mathbb{B}}_C(t, \bm u) \qquad \text{and} \qquad {\mathbb{G}}_{C,q}(s,t, \bm u) = \frac{{\mathbb{G}}_C(s,t,\bm u)}{q(s,t)}.$$ It follows that ${\mathbb{R}}_{m,q} \leadsto {\mathbb{R}}_{C,q}$, ${\mathbb{S}}_{m,q} \leadsto {\mathbb{S}}_{C,q}$ and ${\mathbb{T}}_{m,q} \leadsto {\mathbb{T}}_{C,q}$ in $\ell^\infty([1,T+1])$, where $$\label{eq:weak:limits}
\begin{split}
{\mathbb{R}}_{C,q}(t) &= \sup_{s \in [1,t]} \sup_{\bm u \in [0,1]^d} |{\mathbb{G}}_{C,q}(s,t,\bm u)|, \\
{\mathbb{S}}_{C,q}(t) &= \sup_{s \in [1,t]} \int_{[0,1]^d} \{ {\mathbb{G}}_{C,q}(s,t,\bm u) \}^2 {\mathrm{d}}C(\bm u), \\
{\mathbb{T}}_{C,q}(t) &= \int_1^t \int_{[0,1]^d} \{ {\mathbb{G}}_{C,q}(s,t,\bm u) \}^2 {\mathrm{d}}C(\bm u) {\mathrm{d}}s, \qquad t \in [1,T+1].
\end{split}$$ Furthermore, for any interval $[t_1,t_2] \subset [1,T+1]$ such that $t_2 > 1$, the distributions of $\sup_{t \in [t_1,t_2]}{\mathbb{R}}_{C,q}(t)$, $\sup_{t \in [t_1,t_2]}{\mathbb{S}}_{C,q}(t)$ and $\sup_{t \in [t_1,t_2]}{\mathbb{T}}_{C,q}(t)$ are absolutely continuous with respect to the Lebesgue measure.
Combined with a generic result on the threshold estimation procedure to be stated in Section \[sec:thresh:generic\] and additional bootstrap consistency results to be stated in Section \[sec:thresh:estim\], the last claims of Proposition \[prop:H0\] constitute a first step in proving that the derived change-point detection procedures hold their level asymptotically.
\[rem:q\] Proposition \[prop:H0\] can be used to heuristically justify the form of the weight function $q$ in appearing in the expression of the detectors , and . From , for any $(s,t) \in \Delta$ and $\bm u \in [0,1]^d$, we obtain that $$\begin{aligned}
{\mathrm{Var}}\{{\mathbb{G}}_C(s,t,\bm u) \} &= t^2 {\mathrm{Var}}\{ {\mathbb{B}}_C(s, \bm u) \} + s^2 {\mathrm{Var}}\{ {\mathbb{B}}_C(t, \bm u) \} - 2 s t {\mathrm{Cov}}\{ {\mathbb{B}}_C(s, \bm u), {\mathbb{B}}_C(t, \bm u) \} \\
&= (s t^2 + s^2 t - 2 s^2 t) \Gamma(\bm u, \bm u) = st (t-s) \Gamma(\bm u, \bm u).\end{aligned}$$ As a consequence, for any $1 \leq s < t \leq T+1$ and $\bm u \in [0,1]^d$, $${\mathrm{Var}}\{s^{-1/2} (t-s)^{-1/2} {\mathbb{G}}_C(s,t,\bm u) \} = t \Gamma(\bm u, \bm u).$$ Under the conditions of the proposition, when $\gamma = 1/2$ and $m$ is large, we could then expect that, very roughly, ${\mathrm{Var}}\{ {\mathbb{G}}_{m,q}(s, t,\bm u) \} \approx {\mathrm{Var}}\{s^{-1/2} (t-s)^{-1/2} {\mathbb{G}}_C(s,t,\bm u) \}$ does not depend on $s$ and thus regard the quantities $\sup_{\bm u \in [0,1]^d} {\mathbb{G}}_{m,q}(j/m, t,\bm u)$, $j \in \{m,\dots,{\lfloor mt \rfloor}-1\}$, appearing in the expression of $R_{m,q}({\lfloor mt \rfloor})$ in as random variables with, approximately, the same mean and variance. The latter conveys the intuition that, when $\gamma = 1/2$, all the potential break points $j \in \{m,\dots,{\lfloor mt \rfloor}-1\}$ are given roughly the same weight in the computation of $R_{m,q}({\lfloor mt \rfloor})$. This is, of course, only approximately true because of the presence of the constant $\delta$ in the expression of the weight function $q$ in . In practice, the setting $\gamma = 1/2$ might accelerate the detection of certain types of changes.
Asymptotics of the detectors under $H_1$ {#sec:asym:det:H1}
----------------------------------------
Under $H_1$ in , the detectors are not margin-free anymore. As we shall see in the forthcoming proposition, their asymptotic behavior is then a consequence of that of the process $${\mathbb{H}}_{m,q}(s,t,\bm x) = \frac{{\mathbb{H}}_m(s,t,\bm x)}{q \{ \lambda_m(0,s),\lambda_m(0,t) \}}, \qquad (s,t) \in \Delta, \bm x \in {\mathbb{R}}^d,$$ where $$\label{eq:Hbm}
{\mathbb{H}}_m(s,t,\bm x) = \sqrt{m} \lambda_m(0,s) \lambda_m(s,t) \{ F_{1:{\lfloor ms \rfloor}}(\bm x) - F_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}(\bm x)\},$$ the empirical d.f.s $F_{1:{\lfloor ms \rfloor}}$ and $F_{{\lfloor ms \rfloor}+1:{\lfloor mt \rfloor}}$ are generically defined by , and $q$ is defined in . The following result is proven in the supplement [@KojVer20]. The arrow ‘${\overset{{\mathbb{P}}}{\to}}$’ in its statement denotes convergence in probability.
\[prop:H1\] Assume that $H_1$ in holds with $k^\star = {\lfloor mc \rfloor}$ for some $c \in (1,T+1)$. Assume additionally that $\bm X_1,\dots, \bm X_{{\lfloor mc \rfloor}}$, denoted equivalently by $\bm Y_1,\dots, \bm Y_{{\lfloor mc \rfloor}}$, is a stretch from a stationary sequence $(\bm Y_i)_{i \in {\mathbb{Z}}}$ of continuous $d$-dimensional random vectors whose strong mixing coefficients satisfy $\alpha_r^{\bm Y} = O(r^{-a})$ for some $a > 1$ as $r \to \infty$, and that $\bm X_{{\lfloor mc \rfloor}+1},\dots, \bm X_{{\lfloor m(T+1) \rfloor}}$, denoted equivalently by $\bm Z_{{\lfloor mc \rfloor}+1},\dots, \bm Z_{{\lfloor m(T+1) \rfloor}}$, is a stretch from a stationary sequence $(\bm Z_i)_{i \in {\mathbb{Z}}}$ of continuous $d$-dimensional random vectors whose strong mixing coefficients satisfy $\alpha_r^{\bm Z} = O(r^{-b})$ for some $b > 1$ as $r \to \infty$. Then, $m^{-1/2} {\mathbb{H}}_m {\overset{{\mathbb{P}}}{\to}}K_c$ in $\ell^\infty(\Delta \times {\mathbb{R}}^d)$, where $$\label{eq:Kc}
K_c(s,t,\bm x) = (s \wedge c) \{(t \vee c) - (s \vee c) \} \{F(\bm x) - G(\bm x)\}, \qquad (s,t) \in \Delta, \bm x \in {\mathbb{R}}^d.$$ Consequently, $m^{-1/2} {\mathbb{H}}_{m,q} {\overset{{\mathbb{P}}}{\to}}K_{c,q}$ in $\ell^\infty(\Delta \times {\mathbb{R}}^d)$, where $K_{c,q}(s,t,\bm x) = K_c(s,t,\bm x) / q(s,t)$, $(s,t) \in \Delta$, $\bm x \in {\mathbb{R}}^d$, and $$\sup_{t \in [1,T+1]} {\mathbb{R}}_{m,q}(t) {\overset{{\mathbb{P}}}{\to}}\infty, \quad \sup_{t \in [1,T+1]} {\mathbb{S}}_{m,q}(t) {\overset{{\mathbb{P}}}{\to}}\infty \quad \text{ and } \quad \sup_{t \in [1,T+1]} {\mathbb{T}}_{m,q}(t) {\overset{{\mathbb{P}}}{\to}}\infty.$$
Combined with a generic result on the threshold estimation procedure to be stated in the forthcoming section and additional results on the asymptotic validity of adequate resampling methods to be stated in Section \[sec:thresh:estim\], the three last claims of the previous result will be instrumental in showing that the derived change-point detection procedures have asymptotic power one under $H_1$.
A generic threshold estimation procedure {#sec:thresh:generic}
========================================
In the studied context, the second ingredient of a sequential change-point detection procedure is a set of strictly positive thresholds to which detectors will be compared. In this section, we consider a generic threshold estimation procedure that can be employed with any type of detector, and provide conditions under which it is asymptotically valid. The derived results will be applied in the next section to establish the asymptotically validity of sequential change-point detection procedures based on the detectors studied in Section \[sec:detectors\].
A constant probability of false alarm at each step
--------------------------------------------------
Within the context of closed-end monitoring from time $m+1$ to time $n$, let $D_m(k)$, $k \in \{m+1,\dots,n\}$, be arbitrary detectors. As discussed in the introduction, it seems natural to choose the corresponding thresholds $w_m(k)$, $k \in \{m+1,\dots,n\}$, so that, under $H_0$ in , the probability of rejection of $H_0$ is the same at every step $k \in \{m+1,\dots,n\}$ of the procedure. More formally, this idea, possibly first appearing in [@MarConWooDra95] [see also, e.g., @HawZam05; @Ros14] consists of choosing the $w_m(k)$, $k \in \{m+1,\dots,n\}$, such that, under stationarity, for some small $\xi_m > 0$, $$\label{eq:thresh:proc}
\begin{split}
\left\{
\begin{array}{l}
{\mathbb{P}}\{ D_m(m+1) > w_m(m+1) \} = \xi_m, \\
\\
\text{and, for all } k \in \{m+2,\dots,n\} , \\
\\
{\mathbb{P}}\{ D_m(k) > w_m(k) \mid D_m(m+1) \leq w_m(m+1), \dots, D_m(k-1) \leq w_m(k-1)
\} = \xi_m.
\end{array}
\right.
\end{split}$$ We then obtain that, under $H_0$, $$\label{eq:decomp:cond}
\begin{split}
{\mathbb{P}}\{ &D_m(m+1) \leq w_m(m+1), \dots, D_m(n) \leq w_m(n) \} \\
=& {\mathbb{P}}\{ D_m(n) \leq w_m(n) \mid D_m(m+1) \leq w_m(m+1), \dots, D_m(n-1) \leq w_m(n-1) \} \\ &\times {\mathbb{P}}\{ D_m(m+1) \leq w_m(m+1), \dots \dots, D_m(n-1) \leq w_m(n-1) \} \\
=& (1-\xi_m) \times {\mathbb{P}}\{ D_m(n-1) \leq w_m(n-1) \mid D_m(m+1) \leq w_m(m+1), \dots \\
&\dots, D_m(n-2) \leq w_m(n-2) \} \\
&\times {\mathbb{P}}\{ D_m(m+1) \leq w_m(m+1), \dots, D_m(n-2) \leq w_m(n-2) \} \\
=& \dots = (1-\xi_m)^{n-m}.
\end{split}$$ Given a desired significance level $\alpha \in (0,1/2)$ for the sequential testing procedure, a simple way to ensure that holds under $H_0$ is then to choose $\xi_m$ such that $1-\alpha = (1-\xi_m)^{n-m}$, that is, $\xi_m = 1 - (1-\alpha)^{1/(n-m)}$. As one can see from , $w_m(m+1)$ is then a quantile of order $ (1-\alpha)^{1/(n-m)}$ of $D_m(m+1)$ under stationarity and, for any $k \in \{m+2,\dots,n\}$, $w_m(k)$ is a quantile of order $ (1-\alpha)^{1/(n-m)}$ of $D_m(k)$ conditionally on $D_m(m+1) \leq w_m(m+1), \dots, D_m(k-1) \leq w_m(k-1)$ under stationarity.
Before we discuss the estimation of the thresholds and its validity, let us give an alternative view of . In Sections \[sec:asym:det:H0\] and \[sec:asym:det:H1\] in which $n$ was taken equal to ${\lfloor m(T+1) \rfloor}$, we saw that the asymptotic results for the detectors are given in terms of elements of $\ell^\infty([1,T+1])$. With the convention that $D_m(m) = w_m(m) = 0$, another equivalent way of looking at sequential change-point detection procedures of the considered type is then to consider that the piecewise constant *detector function* ${\mathbb{D}}_m$ defined by ${\mathbb{D}}_m(t) = D_m({\lfloor mt \rfloor})$, $t \in [1,T+1]$, is compared to the piecewise constant *threshold function* $\tau_m$ defined by $\tau_m(t) = w_m({\lfloor mt \rfloor})$, $t \in [1,T+1]$. Let $s_k = (m+k)/m$, $k \in \{0, \dots, n-m\}$ and define the intervals $J_k = [s_k,s_{k+1})$, $k \in \{0, \dots, n-m-1\}$, and $J_{n-m} = [s_{n-m},T+1]$. Some thought reveals that is then equivalent to choosing the threshold function $\tau_m$ such that, under $H_0$ in , for any $k \in \{1, \dots, n-m\}$, $$\begin{gathered}
\label{eq:thresh:proc1}
{\mathbb{P}}\{ \exists \, t \in J_k \text{ s.t. } {\mathbb{D}}_m(t) > \tau_m(t) \mid {\mathbb{D}}_m(t) \leq \tau_m(t), \forall \, t \in J_0 \cup \dots \cup J_{k-1} \} \\ = 1 - (1-\alpha)^{1/(n-m)}.\end{gathered}$$
A formulation compatible with asymptotic validity results {#sec:thresh:asm:val:formulation}
---------------------------------------------------------
With $n = {\lfloor m(T+1) \rfloor}$, the threshold setting procedure as given in or makes no sense asymptotically since the number of (conditional) probabilities tends to infinity as $m \to \infty$. A natural solution consists of keeping the number of probabilities fixed, or, equivalently, of considering a time grid that does not depend on $m$. Let $p \geq 1$ and let $t_0 = 1 < t_1 < \dots < t_p = T+1$ be a fixed uniformly spaced time grid such that $T/p > 1/m$ (a condition that will always be satisfied for $m$ large enough). Let $\tau_m$ be a piecewise constant threshold function taking the value $g_{i,m}$ on the interval $I_i = [t_{i-1}, t_i)$, $i \in \{1,\dots,p-1\}$, and $g_{p,m}$ on the interval $I_p = [t_{p-1}, t_p]$. Mimicking , the aim is then to choose $\tau_m$ such that, under $H_0$ in , for any $i \in \{1, \dots, p\}$, $$\label{eq:thresh:proc2}
{\mathbb{P}}\{ \exists \, t \in I_i \text{ s.t. }{\mathbb{D}}_m(t) > \tau_m(t) \mid {\mathbb{D}}_m(t) \leq \tau_m(t), \forall \, t \in I_0 \cup \dots \cup I_{i-1} \} = 1 - (1-\alpha)^{1/p},$$ with the convention that $I_0 = \emptyset$. Some thought reveals that the formulation in is equivalent to choosing $\tau_m$ such that, under $H_0$ in , $$\label{eq:thresh:proc3}
\left\{
\begin{array}{l}
\disp {\mathbb{P}}\left\{ \sup_{t \in I_1} {\mathbb{D}}_m(t) > g_{1,m} \right\} = 1 - (1-\alpha)^{1/p},\\ \\
\text{and, for all } i \in \{2, \dots, p\},\\ \\
\disp {\mathbb{P}}\left\{ \sup_{t \in I_i} {\mathbb{D}}_m(t) > g_{i,m} \, \Big| \, \sup_{t \in I_1} {\mathbb{D}}_m(t) \leq g_{1,m}, \dots, \sup_{t \in I_{i-1}} {\mathbb{D}}_m(t) \leq g_{i-1,m} \right\} = 1 - (1-\alpha)^{1/p}.
\end{array}
\right.$$ In other words, $g_{1,m}$ is a quantile of order $(1-\alpha)^{1/p}$ of $\sup_{t \in I_1} {\mathbb{D}}_m(t)$ under stationarity and $g_{i,m}$, $i \in \{2,\dots,p\}$, is a quantile of order $(1-\alpha)^{1/p}$ of $\sup_{t \in I_i} {\mathbb{D}}_m(t)$ given that $\sup_{t \in I_1} {\mathbb{D}}_m(t) \leq g_{1,m}, \dots, \sup_{t \in I_{i-1}} {\mathbb{D}}_m(t) \leq g_{i-1,m}$ under stationarity. Notice that the suprema in are actually maxima since ${\mathbb{D}}_m$ is a piecewise constant function.
A further generalization of or, equivalently , would be to consider that $\tau_m$ is not necessarily piecewise constant but only defined up to a multiplicative constant on each of the intervals $I_i$, $i \in \{1,\dots,p\}$. For instance, it could have one of the parametric forms considered in @DetGos19 [Section 5], among others. For the sake of simplicity, we shall not however consider such an extension in this work.
Estimation of the threshold function {#sec:estim:thresh}
------------------------------------
As we continue, we shall focus on the threshold setting procedure as formulated in or, equivalently, , mostly because its asymptotic validity can be studied. To estimate the threshold function $\tau_m$ in , or, equivalently, the $g_{i,m}$, $i \in \{1,\dots,p\}$, in , it is thus necessary to be able to compute, at least approximately, the distribution of the $p$-dimensional random vector $$\label{eq:rvp}
\left( \sup_{t \in I_1} {\mathbb{D}}_m(t), \dots, \sup_{t \in I_p} {\mathbb{D}}_m(t) \right).$$
### Monte Carlo estimation and asymptotic validity {#sec:MC:estim}
Assume that the observations to be monitored are univariate and independent, and that ${\mathbb{D}}_m$ is distribution-free under $H_0^\text{ind}$ in . Notice that the latter implies that so is the random vector . To obtain a Monte Carlo estimate of the distribution of , it then suffices to consider a large integer $M$, generate $M$ independent samples $U_1^{[s]},\dots,U_n^{[s]}$, $s \in \{1,\dots,M\}$, of size $n$ from the standard uniform distribution and compute the corresponding realizations ${\mathbb{D}}_m^{[s]}$, $s \in \{1,\dots,M\}$, of ${\mathbb{D}}_m$. The latter can be used to obtain a Monte Carlo estimate $\tau_m^M$ of the threshold function $\tau_m$. More formally, let $$g_{i,m}^M = F_{{\mathbb{D}}_m,i}^{M,-1} \{ (1-\alpha)^{1/p} \}, \qquad i \in \{1,\dots,p\},$$ where $F_{{\mathbb{D}}_m,1}^M$ is the empirical d.f. of the sample $\sup_{t \in I_1} {\mathbb{D}}_m^{[1]}(t),\dots,\sup_{t \in I_1} {\mathbb{D}}_m^{[M]}(t)$, for any $i \in \{2,\dots,p\}$ and $x \in {\mathbb{R}}$, $$F_{{\mathbb{D}}_m,i}^M(x) = \frac{\disp \sum_{s=1}^M {\mathbf{1}}\left\{\sup_{t \in I_i} {\mathbb{D}}_m^{[s]}(t) \leq x, \sup_{t \in I_1} {\mathbb{D}}_m^{[s]}(t) \leq g_{1,m}^M, \dots, \sup_{t \in I_{i-1}} {\mathbb{D}}_m^{[s]}(t) \leq g_{i-1,m}^M \right\}}{\disp \sum_{s=1}^M {\mathbf{1}}\left\{ \sup_{t \in I_1} {\mathbb{D}}_m^{[s]}(t) \leq g_{1,m}^M, \dots, \sup_{t \in I_{i-1}} {\mathbb{D}}_m^{[s]}(t) \leq g_{i-1,m}^M \right\}},$$ and $F_{{\mathbb{D}}_m,i}^{M,-1}$, $i \in \{1,\dots,p\}$, are the associated quantile functions generically defined by . Notice that, in this particular case, the resulting estimate $\tau_m^M$ of the threshold function $\tau_m$ does not at all depend on the learning sample.
By taking a sufficiently large $M$, the Monte Carlo estimates $g_{i,m}^M$, $i \in \{1,\dots,p\}$, can be made arbitrarily close to the quantiles $g_{i,m} = F_{{\mathbb{D}}_m,i}^{-1}\{ (1-\alpha)^{1/p} \}$, $i \in \{1,\dots,p\}$, where $F_{{\mathbb{D}}_m,1}$ is the d.f. of $\sup_{t \in I_1} {\mathbb{D}}_m(t)$, and $F_{{\mathbb{D}}_m,i}$, $i \in \{2,\dots,p\}$, is the d.f. of $\sup_{t \in I_i} {\mathbb{D}}_m(t)$ given that $\sup_{t \in I_j} {\mathbb{D}}_m(t) \leq g_{j,m}$ for all $j \in \{1,\dots,i-1\}$. Interestingly enough more can be said as a consequence of the fact that Monte Carlo simulation can be regarded as a particular resampling scheme. As shall become clear in the next section, the general result stated in Theorem \[thm:thresh:boot\] hereafter can actually be used to show the asymptotically validity of the Monte Carlo based threshold estimation procedure when both $m$ and $M$ tend to infinity, under both $H_0^\text{ind}$ in and $H_1$ in . This is discussed in more detail in Remark \[rem:thresh:MC\] below.
### Bootstrap-based estimation and asymptotic validity {#sec:boot:estim}
In settings in which ${\mathbb{D}}_m$ is not distribution-free anymore, a natural alternative is to rely on a resampling scheme making use of the available learning sample $\bm X_1,\dots,\bm X_m$ known to be under $H_0$ in . Specifically, let $B$ be a large integer and suppose that we have available *bootstrap replicates* ${\mathbb{D}}_m^{[b]}$, $b \in \{1,\dots,B\}$, of ${\mathbb{D}}_m$ computed from $\bm X_1,\dots,\bm X_m$ and depending on additional sources of randomness involved in the resampling scheme. Mimicking the previous situation in which ${\mathbb{D}}_m$ was distribution-free, let $$g_{j,m}^B = F_{{\mathbb{D}}_m,j}^{B,-1} \{ (1-\alpha)^{1/p} \}, \qquad j \in \{1,\dots,p\},$$ where $F_{{\mathbb{D}}_m,1}^B$ is the empirical d.f. of the sample $\sup_{t \in I_1} {\mathbb{D}}_m^{[1]}(t),\dots,\sup_{t \in I_1} {\mathbb{D}}_m^{[B]}(t)$, and, for any $j \in \{2,\dots,p\}$ and $x \in {\mathbb{R}}$, $$F_{{\mathbb{D}}_m,j}^B(x) = \frac{\disp \sum_{b=1}^B {\mathbf{1}}\left\{ \sup_{t \in I_j} {\mathbb{D}}_m^{[b]}(t) \leq x, \sup_{t \in I_1} {\mathbb{D}}_m^{[b]}(t) \leq g_{1,m}^B, \dots, \sup_{t \in I_{j-1}} {\mathbb{D}}_m^{[b]}(t) \leq g_{j-1,m}^B \right\}}{\disp \sum_{b=1}^B {\mathbf{1}}\left\{\sup_{t \in I_1} {\mathbb{D}}_m^{[b]}(t) \leq g_{1,m}^B, \dots, \sup_{t \in I_{j-1}} {\mathbb{D}}_m^{[b]}(t) \leq g_{j-1,m}^B \right\}}.$$
As we shall see below, the main result of this section is that, essentially, as soon as the underlying resampling scheme for ${\mathbb{D}}_m$ is consistent, the above bootstrap-based version of the threshold setting procedure is asymptotically valid in the sense that, under $H_0$, ${\mathbb{P}}({\mathbb{D}}_m \leq \tau_m^B) \to 1 - \alpha$ as $m,B \to \infty$, where $\tau_m^B$ is the estimated bootstrap-based piecewise constant threshold function taking the value $g_{i,m}^B$ on the interval $I_i = [t_{i-1}, t_i)$, $i \in \{1,\dots,p-1\}$, and $g_{p,m}^B$ on the interval $I_p = [t_{p-1}, t_p]$.
\[rem:boot:val\] Following for instance @vanWel96 [Section 3.6] or @Kos08 [Section 2.2.3], a resampling scheme for ${\mathbb{D}}_m$ is typically considered consistent, if, informally, “${\mathbb{D}}_m^{[1]}$ converges weakly to the weak limit of ${\mathbb{D}}_m$ in $\ell^\infty([0,T+1])$ conditionally on $\bm X_1, \bm X_2, \dots$ in probability”. A rigorous definition of the underlying mode of convergence is more subtle than that of weak convergence. From Lemma 3.1 of [@BucKoj19], the aforementioned validity statement is actually equivalent to the joint unconditional weak convergence of ${\mathbb{D}}_m$ and two bootstrap replicates to independent copies of the same limit. Throughout the paper, all our bootstrap asymptotic validity results will take that form.
The following general result is proved in Appendix \[sec:proof:thm:thres:boot\].
\[thm:thresh:boot\] Assume that, under $H_0$ in , $$\label{eq:boot:val}
({\mathbb{D}}_m, {\mathbb{D}}_m^{[1]}, {\mathbb{D}}_m^{[2]}) \leadsto ({\mathbb{D}}_F, {\mathbb{D}}_F^{[1]}, {\mathbb{D}}_F^{[2]})$$ in $\{ \ell^\infty([1,T+1]) \}^3$, where ${\mathbb{D}}_F$ is the weak limit of ${\mathbb{D}}_m$ and ${\mathbb{D}}_F^{[1]}$ and ${\mathbb{D}}_F^{[2]}$ are independent copies of ${\mathbb{D}}_F$. Assume furthermore that the random vector $\big( \sup_{t \in I_1} {\mathbb{D}}_F(t), \dots, \sup_{t \in I_p} {\mathbb{D}}_F(t) \big)$ has a continuous d.f. Then, under $H_0$ in , as $m,B \to \infty$, $$\label{eq:thresh:val:1}
{\mathbb{P}}\left\{ \sup_{t \in I_1} {\mathbb{D}}_m(t) \leq g_{1,m}^B \right\} \to (1-\alpha)^{1/p},$$ and, for any $i \in \{2,\dots,p\}$, $$\label{eq:thresh:val:i}
{\mathbb{P}}\left\{ \sup_{t \in I_i} {\mathbb{D}}_m(t) \leq g_{i,m}^B \, \Big| \, \sup_{t \in I_1} {\mathbb{D}}_m(t) \leq g_{1,m}^B, \ldots, \sup_{t \in I_{i-1}} {\mathbb{D}}_m(t) \leq g_{i-1,m}^B \right\} \to (1-\alpha)^{1/p}.$$ As a consequence, on one hand, under $H_0$, ${\mathbb{P}}({\mathbb{D}}_m \leq \tau_m^B) \to 1 - \alpha$ as $m,B \to \infty$ and, on the other hand, when $\sup_{t \in [1,T+1]} {\mathbb{D}}_m(t) {\overset{{\mathbb{P}}}{\to}}\infty$, $${\mathbb{P}}\{ \exists \, t \in [1,T+1] \text{ s.t. } {\mathbb{D}}_m(t) > \tau_m^B(t) \} \to 1 \qquad \text{ as } m, B \to \infty.$$
\[rem:thresh:MC\] Consider the Monte Carlo estimation setting of Section \[sec:MC:estim\] in which the observations to be monitored are univariate independent and ${\mathbb{D}}_m$ is distribution-free. Then, the weak convergence ${\mathbb{D}}_m \leadsto {\mathbb{D}}_F$ in $\ell^\infty([1,T+1])$ under $H_0$ immediately implies , where ${\mathbb{D}}_m^{[1]}$ and ${\mathbb{D}}_m^{[2]}$ are (independent) Monte Carlo replicates of ${\mathbb{D}}_m$. Hence, as a consequence of Theorem \[thm:thresh:boot\], the asymptotic validity under $H_0^\text{ind}$ in of the sequential change-point detection procedure based on ${\mathbb{D}}_m$ and the Monte Carlo estimated threshold function $\tau_m^M$ defined in Section \[sec:MC:estim\] is an immediate corollary of the weak convergence under the null of ${\mathbb{D}}_m$ if $\big( \sup_{t \in I_1} {\mathbb{D}}_F(t), \dots, \sup_{t \in I_p} {\mathbb{D}}_F(t) \big)$ has a continuous d.f.
Threshold function estimation for the detectors based on empirical d.f.s {#sec:thresh:estim}
========================================================================
The aim of this section is to apply the generic results of the previous section to estimate the threshold functions for the empirical d.f.-based detector functions ${\mathbb{R}}_{m,q}$, ${\mathbb{S}}_{m,q}$ and ${\mathbb{T}}_{m,q}$ defined in . We distinguish two situations for the observations to be monitored: the independent univariate case and the possibly multivariate, time series case.
Monte Carlo estimation in the independent univariate case {#sec:thresh:ind:univ}
---------------------------------------------------------
As verified in Section \[sec:margin:free:H0\], the detector functions ${\mathbb{R}}_{m,q}$, ${\mathbb{S}}_{m,q}$ and ${\mathbb{T}}_{m,q}$ defined in are margin-free under $H_0$ in . In the univariate case, they are thus distribution-free. When dealing with independent univariate observations, one can therefore proceed exactly as explained in Section \[sec:MC:estim\] to estimate the corresponding threshold functions. Furthermore, from Proposition \[prop:H0\], Remark \[rem:thresh:MC\] and Proposition \[prop:H1\], we know that the assumptions of Theorem \[thm:thresh:boot\] are satisfied. The latter then implies that the corresponding sequential change-point detection procedures are asymptotically valid both under $H_0^\text{ind}$ in and $H_1$ in .
A dependent multiplier bootstrap in the time series case {#sec:thresh:time:series}
--------------------------------------------------------
When the monitored observations are multivariate or exhibit serial dependence, the approach considered in Section \[sec:thresh:ind:univ\] is not meaningful anymore. Having the asymptotic results of Sections \[sec:asym:det:H0\] and \[sec:boot:estim\] in mind, our aim in the considered time series context is to define suitable bootstrap replicates of ${\mathbb{B}}_m$ in such that, following Remark \[rem:boot:val\], ${\mathbb{B}}_m$ and two of its replicates jointly weakly converge to independent copies of the process ${\mathbb{B}}_C$ defined in Proposition \[prop:H0\]. Subsequently defining corresponding bootstrap replicates of the detectors functions ${\mathbb{R}}_{m,q}$, ${\mathbb{S}}_{m,q}$ and ${\mathbb{T}}_{m,q}$ defined in will lead to asymptotically valid corresponding sequential change-point detection procedures.
Following @Buh93 [Section 3.3] and [@BucKoj16], we opted for a *dependent multiplier bootstrap* in the considered time series context. In the rest of the paper, we say that a sequence of random variables $(\xi_{i,m})_{i \in {\mathbb{Z}}}$ is a [*dependent multiplier sequence*]{} if:
1. \[item:moments\] The sequence $(\xi_{i,m})_{i \in {\mathbb{Z}}}$ is stationary, independent of the available learning sample $\bm X_1,\dots,\bm X_m$ and satisfies ${\mathbb{E}}(\xi_{0,m}) = 0$, ${\mathbb{E}}(\xi_{0,m}^2) = 1$ and $\sup_{m \geq 1} {\mathbb{E}}(|\xi_{0,m}|^\nu) < \infty$ for all $\nu \geq 1$.
2. \[item:lm\] There exists a sequence $\ell_m \to \infty$ of strictly positive constants such that $\ell_m = o(m)$ and the sequence $(\xi_{i,m})_{i \in {\mathbb{Z}}}$ is $\ell_m$-dependent, i.e., $\xi_{i,m}$ is independent of $\xi_{i+h,m}$ for all $h > \ell_m$ and $i \in {\mathbb{N}}$.
3. \[item:varphi\] There exists a function $\varphi:{\mathbb{R}}\to [0,1]$, symmetric around 0, continuous at $0$, satisfying $\varphi(0)=1$ and $\varphi(x)=0$ for all $|x| > 1$ such that ${\mathbb{E}}(\xi_{0,m} \xi_{h,m}) = \varphi(h/\ell_m)$ for all $h \in {\mathbb{Z}}$.
Let $(\xi_{i,m}^{[b]})_{i \in {\mathbb{Z}}}$, $b \in {\mathbb{N}}$, be independent copies of the same dependent multiplier sequence. If we had a learning sample of size $n = {\lfloor m(T+1) \rfloor}$, following [@BucKoj16], a natural definition of a *dependent multiplier replicate* of ${\mathbb{B}}_m$ in would be $$\label{eq:check:Bbmb}
\check {\mathbb{B}}_m^{[b]}(s, \bm u) = \frac{1}{\sqrt{m}} \sum_{i=1}^{{\lfloor ms \rfloor}} \xi_{i,m}^{[b]} \{ {\mathbf{1}}(\bm U_i \leq \bm u) - C_{1:n}(\bm u) \}, \qquad s \in [0,T+1], \bm u \in [0,1]^d,b \in {\mathbb{N}},$$ where $C_{1:n}$ is generically defined by . Since threshold functions need to be estimated prior to the beginning of the monitoring and the learning sample is only of size $m$, we consider a time-rescaled version of $\check {\mathbb{B}}_m^{[b]}$ in which, roughly, $m'= {\lfloor (m/n) m \rfloor} \simeq m/(T+1)$ and $m$ play the role of $m$ and $n$, respectively. Hence, in the considered context, our definition of a dependent multiplier replicate of ${\mathbb{B}}_m$ is $$\label{eq:hat:Bbmb}
\hat {\mathbb{B}}_m^{[b]}(s, \bm u) = \frac{1}{\sqrt{m'}} \sum_{i=1}^{{\lfloor m's \rfloor}} \xi_{i,m}^{[b]} \{ {\mathbf{1}}(\bm U_i \leq \bm u) - C_{1:m}(\bm u) \}, \qquad s \in [0,T+1], \bm u \in [0,1]^d,b \in {\mathbb{N}},$$ thereby translating the fact that we can only rely on functionals computed from the learning sample to approximate the variability of the detector functions under the null.
From the two previous displays, we see that the multipliers act as random weights and that the bandwidth $\ell_m$ defined in Assumption (M\[item:lm\]) plays a role somehow similar to that of the [*block length*]{} in the block bootstrap of [@Kun89]. Note that, in our Monte Carlo experiments to be presented in Section \[sec:MC\], $\ell_m$ was estimated from the learning sample $\bm X_1,\dots,\bm X_m$ as explained in detail in Section 5.1 of [@BucKoj16] while corresponding dependent multiplier sequences were generated using the so-called *moving average approach* based on an initial standard normal random sample and Parzen’s kernel as precisely described in Section 5.2 of the same reference.
The latter construction based on a time-rescaling suggests to form a dependent multiplier replicate of ${\mathbb{G}}_m$ in as $$\label{eq:hatGbmb}
\hat {\mathbb{G}}_m^{[b]}(s,t, \bm u) = \lambda_{m'}(0,t) \hat {\mathbb{B}}_m^{[b]}(s,\bm u) - \lambda_{m'}(0,s) \hat {\mathbb{B}}_m^{[b]}(t,\bm u), \qquad (s,t) \in \Delta, \bm u \in [0,1]^d, b \in {\mathbb{N}},$$ with its weighted version being $$\hat {\mathbb{G}}_{m,q}^{[b]}(s,t, \bm u) = \frac{\hat {\mathbb{G}}_m^{[b]}(s,t, \bm u)}{q\{\lambda_{m'}(0,s),\lambda_{m'}(0,t) \}}, \qquad (s,t) \in \Delta, \bm u \in [0,1]^d, b \in {\mathbb{N}},$$ where $\lambda_{m'}$ is defined as in . Finally, for any $b \in {\mathbb{N}}$ and $t \in [1,T+1]$, let $$\label{eq:det:mult}
\begin{split}
\hat {\mathbb{R}}_{m,q}^{[b]}(t) &= \sup_{s \in [1,t]} \sup_{\bm u \in [0,1]^d} | \hat {\mathbb{G}}_{m,q}^{[b]}(s,t,\bm u)|, \\
\hat {\mathbb{S}}_{m,q}^{[b]}(t) &= \sup_{s \in [1,t]} \int_{[0,1]^d} \{ \hat {\mathbb{G}}_{m,q}^{[b]}(s,t,\bm u) \}^2 {\mathrm{d}}C_{1:{\lfloor m't \rfloor}}(\bm u), \\
\hat {\mathbb{T}}_{m,q}^{[b]}(t) &= \int_1^t \int_{[0,1]^d} \{ \hat {\mathbb{G}}_{m,q}^{[b]}(s,t,\bm u) \}^2 {\mathrm{d}}C_{1:{\lfloor m't \rfloor}}(\bm u) {\mathrm{d}}s
\end{split}$$ be dependent multiplier replicates of ${\mathbb{R}}_{m,q}$, ${\mathbb{S}}_{m,q}$ and ${\mathbb{T}}_{m,q}$, respectively, defined in , where $C_{1:{\lfloor m't \rfloor}}$ is defined generically by .
The definitions given in hide the fact that the proposed dependent multipliers replicates of the detector functions ${\mathbb{R}}_{m,q}$, ${\mathbb{S}}_{m,q}$ and ${\mathbb{T}}_{m,q}$ actually depend on the learning sample $\bm X_1,\dots,\bm X_m$. To verify that this is the case, for any $b \in {\mathbb{N}}$, $(s,t) \in \Delta$ and $\bm x \in {\mathbb{R}}^d$, let $\hat {\mathbb{F}}_m^{[b]}(s, \bm x) = \hat {\mathbb{B}}_m^{[b]}\{s, \bm F(\bm x)\}$, where $\bm F(\bm x) = (F_1(x_1),\dots,F_d(x_d))$, $$\hat {\mathbb{H}}_m^{[b]}(s,t, \bm x) = \hat {\mathbb{G}}_m^{[b]}\{s,t, \bm F(\bm x) \} = \lambda_{m'}(0,t) \hat {\mathbb{F}}_m^{[b]}(s,\bm x) - \lambda_{m'}(0,s) \hat {\mathbb{F}}_m^{[b]}(t,\bm x)$$ and $$\hat {\mathbb{H}}_{m,q}^{[b]}(s,t, \bm x) = \hat {\mathbb{G}}_{m,q}^{[b]}\{s,t, \bm F(\bm x) \} = \frac{\hat {\mathbb{H}}_m^{[b]}(s,t, \bm x)}{q\{\lambda_{m'}(0,s),\lambda_{m'}(0,t) \}}.$$ Since always holds for all $i \in \{1,\dots,m\}$, we immediately obtain that, for any $b \in {\mathbb{N}}$, $$\label{eq:hat:Fbmb}
\hat {\mathbb{F}}_m^{[b]}(s, \bm x) = \frac{1}{\sqrt{m'}} \sum_{i=1}^{{\lfloor m's \rfloor}} \xi_{i,m}^{[b]} \{ {\mathbf{1}}(\bm X_i \leq \bm x) - F_{1:m}(\bm x) \}, \qquad s \in [0,T+1], \bm x \in {\mathbb{R}}^d,$$ where $F_{1:m}$ is generically defined by , and furthermore that, for any $t \in [1,T+1]$, $$\begin{aligned}
\nonumber \hat {\mathbb{R}}_{m,q}^{[b]}(t) &= \sup_{s \in [1,t]} \sup_{\bm x \in {\mathbb{R}}^d} | \hat {\mathbb{H}}_{m,q}^{[b]}(s,t,\bm x)|, \\
\nonumber \hat {\mathbb{S}}_{m,q}^{[b]}(t) &= \sup_{s \in [1,t]} \int_{{\mathbb{R}}^d} \{ \hat {\mathbb{H}}_{m,q}^{[b]}(s,t,\bm x) \}^2 {\mathrm{d}}F_{1:{\lfloor m't \rfloor}}(\bm x), \\
\hat {\mathbb{T}}_{m,q}^{[b]}(t) &= \int_1^t \int_{{\mathbb{R}}^d} \{ \hat {\mathbb{H}}_{m,q}^{[b]}(s,t,\bm x) \}^2 {\mathrm{d}}F_{1:{\lfloor m't \rfloor}}(\bm x) {\mathrm{d}}s.\end{aligned}$$
The following result is proven in the supplement [@KojVer20].
\[prop:mult\] Assume that $H_0$ in holds and that, additionally, $\bm X_1,\dots,\bm X_n$ is a stretch from a stationary sequence $(\bm X_i)_{i \in {\mathbb{Z}}}$ of continuous $d$-dimensional random vectors whose strong mixing coefficients satisfy $\alpha_r^{\bm X} = O(r^{-a})$ for some $a > 3+3d/2$ as $r \to \infty$. If $\ell_m = O(m^{1/2-{\varepsilon}})$ for some $0 < {\varepsilon}< 1/2$, then $$({\mathbb{B}}_m, \hat {\mathbb{B}}_m^{[1]}, \hat {\mathbb{B}}_m^{[2]}) \leadsto ( {\mathbb{B}}_C, {\mathbb{B}}_C^{[1]}, {\mathbb{B}}_C^{[2]} )$$ in $\{ \ell^\infty([0,T+1] \times [0,1]^d) \}^3$, where ${\mathbb{B}}_m$ is defined in , $\hat {\mathbb{B}}_m^{[1]}$ and $\hat {\mathbb{B}}_m^{[2]}$ are defined in , ${\mathbb{B}}_C$ is the weak limit of ${\mathbb{B}}_m$ defined in Proposition \[prop:H0\], and ${\mathbb{B}}_C^{[1]}$ and ${\mathbb{B}}_C^{[2]}$ are independent copies of ${\mathbb{B}}_C$.
As a consequence, $$\begin{aligned}
&({\mathbb{R}}_{m,q},\hat {\mathbb{R}}_{m,q}^{[1]}, \hat {\mathbb{R}}_{m,q}^{[2]}) \leadsto ({\mathbb{R}}_{C,q},{\mathbb{R}}_{C,q}^{[1]}, {\mathbb{R}}_{C,q}^{[2]}), &({\mathbb{S}}_{m,q},\hat {\mathbb{S}}_{m,q}^{[1]}, \hat {\mathbb{S}}_{m,q}^{[2]}) \leadsto ({\mathbb{S}}_{C,q},{\mathbb{S}}_{C,q}^{[1]}, {\mathbb{S}}_{C,q}^{[2]}), \\
&({\mathbb{T}}_{m,q},\hat {\mathbb{T}}_{m,q}^{[1]}, \hat {\mathbb{T}}_{m,q}^{[2]}) \leadsto ({\mathbb{T}}_{C,q},{\mathbb{T}}_{C,q}^{[1]}, {\mathbb{T}}_{C,q}^{[2]}), \qquad &\text{in } \{\ell^\infty([1,T+1])\}^3,\end{aligned}$$ where $\hat {\mathbb{R}}_{m,q}^{[b]}$, $\hat {\mathbb{S}}_{m,q}^{[b]}$ and $\hat {\mathbb{T}}_{m,q}^{[b]}$, $b \in \{1,2\}$, are defined in , ${\mathbb{R}}_{C,q}$, ${\mathbb{S}}_{C,q}$ and ${\mathbb{T}}_{C,q}$ are defined in , and ${\mathbb{R}}_{C,q}^{[b]}$, ${\mathbb{S}}_{C,q}^{[b]}$ and ${\mathbb{T}}_{C,q}^{[b]}$, $b \in \{1,2\}$, are independent copies of ${\mathbb{R}}_{C,q}$, ${\mathbb{S}}_{C,q}$ and ${\mathbb{T}}_{C,q}$, respectively.
The last claims of the previous proposition along with the last claim of Proposition \[prop:H0\] and Proposition \[prop:H1\] are the assumptions of Theorem \[thm:thresh:boot\] for ${\mathbb{D}}_m \in \{{\mathbb{R}}_{m,q}, {\mathbb{S}}_{m,q}, {\mathbb{T}}_{m,q}\}$. It follows that the sequential change-point detection procedures based on these detector functions carried out as explained in Section \[sec:boot:estim\] using the above dependent multiplier replicates are asymptotically valid under $H_0$ in and $H_1$ in . Note that, in practice, since in the considered approach $m'= {\lfloor (m/n) m \rfloor}$ and $m$ play the role of $m$ and $n$, respectively, the largest possible value for $p$, the number of steps of the estimated threshold function $\tau_m^B$, is $m - m'$ and, in this case, each of the $p$ estimated thresholds covers approximately ${\lfloor n/m \rfloor}$ time steps in the monitoring.
Monte Carlo experiments {#sec:MC}
=======================
Large-scale Monte Carlo experiments were carried out to investigate the finite-sample properties of the studied sequential change-point detection procedures. The aim was in particular to try to answer the following questions:
- How well do the procedures hold their level, in particular, when the threshold functions are estimated using the dependent multiplier bootstrap of Section \[sec:thresh:time:series\]?
- What is the influence of the number of steps $p$ of the estimated threshold function (see Sections \[sec:thresh:asm:val:formulation\] and \[sec:estim:thresh\]) on the distribution of the false alarms?
- What is the effect of $p$ on the power and the *mean detection delay* (the latter is the expectation under $H_1$ of the difference between the time at which the change was detected and the time $k^\star$ at which the change really occurred)?
- What is the effect of the parameter $\gamma$ appearing in the expression of the weight function $q$ defined in on the power and mean detection delay?
- How do the detectors $R_{m,q}$, $S_{m,q}$, $T_{m,q}$, $P_m$ and $Q_m$ defined in , , , and compare in terms of power and mean detection delay?
- How do the derived procedures compare with similar, more specialized procedures in terms of power and mean detection delay?
We tried to answer these questions in detail in the univariate independent case when the estimation of the threshold functions of the sequential change-point detection procedures can be rightfully so based on the Monte Carlo approach described in Sections \[sec:MC:estim\] and \[sec:thresh:ind:univ\]. When the observations to be monitored are not univariate or independent so that resampling as described in Section \[sec:thresh:time:series\] is needed for the estimation of the threshold functions, we essentially investigated how well the procedures hold their level depending on the underlying data generating mechanism. Although many other questions could be formulated given the complexity of the problem, the following experiments should allow the reader to grasp the main finite-sample properties of the studied procedures.
Monte Carlo estimation in the independent univariate case {#monte-carlo-estimation-in-the-independent-univariate-case}
---------------------------------------------------------
As already discussed in Section \[sec:MC:estim\], the estimation of the threshold functions when monitoring independent univariate observations can be made arbitrarily precise by increasing the number $M$ of Monte Carlo samples. We used the setting $M=10^5$ in our experiments and estimated all the rejection percentages from $10^4$ samples. The change-point detection procedures were always carried out at the $\alpha = 5\%$ nominal level.
### Under the null
Unsurprisingly, all the studied tests were found to hold their level very well. Furthermore, as could have been expected given the fact that the studied detector functions have a tendency to be increasing on average, it was observed that the setting $p=1$ results in a concentration of false alarms at the end of the monitoring period, while the larger $p$, the more uniform the distribution of the false alarms over the monitoring period (these unsurprising findings are illustrated in Figure \[fig:DistFalseAlarms\_mmk\_m\_50\_n\_100\] of the supplement [@KojVer20]).
![\[fig:CompDet\_m\_50\_p\_1\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the d.f. of the standard normal and $G$ the d.f. of the $N(\delta,1)$. Right: corresponding mean detection delays. The value of $\gamma$ in is 0. The number of steps in the threshold functions is $p=1$.](figCompDet_m_50_p_1_gamma_0.pdf){width="0.7\linewidth"}
### Change in mean
To answer the aforementioned questions related to the behavior of the procedures under $H_1$ in , we first considered a simple experiment consisting of a change in the expectation of a normal distribution. Specifically, $m$, $k^\star$, $n$, $F$ and $G$ in $H_1$ were taken equal to 50, 75, 100, the d.f. of the standard normal and the d.f. of the $N(\delta,1)$, respectively. The left graph in Figure \[fig:CompDet\_m\_50\_p\_1\] displays the estimated rejection percentages for the the five detectors $R_{m,q}$, $S_{m,q}$, $T_{m,q}$, $P_m$ and $Q_m$ with $\gamma$ in set to zero and $p=1$. The right graph represents the corresponding mean detection delays which were estimated only from the samples for which neither a false alarm was obtained (which occurs when the detector function becomes larger than the threshold function before the time of change $k^\star = 75$) nor the change was undetected (which occurs when the detector function remains below the threshold function during the entire monitoring period). Because the number of steps in the threshold function was set to $p=1$, the left graph of Figure \[fig:CompDet\_m\_50\_p\_1\] is directly comparable with the top left graph given in Figure 1 of [@DetGos19]. An inspection of the latter seems to indicate that the powers of the procedures based on $R_{m,q}$, $S_{m,q}$ and $T_{m,q}$ are not substantially different from those of the mean-specialized procedures considered in Section 5.1 of [@DetGos19], even though the detectors $R_{m,q}$, $S_{m,q}$ and $T_{m,q}$ are not specifically designed to be sensitive to changes in the expectation. The graphs are not substantially different for other values of $\gamma$ and $p$. Overall, the procedures based on $R_{m,q}$, $S_{m,q}$ and $T_{m,q}$ were always observed to be more powerful and superior in terms of mean detection delay than those based on $P_m$ and $Q_m$. The latter is in full accordance with the empirical observations of [@DetGos19] for more specialized procedures. Note that the procedure based on $T_{m,q}$ seems the most powerful for the alternative under consideration.
![\[fig:GammaEffect\_mmk\_m\_50\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the d.f. of the standard normal and $G$ the d.f. of the $N(\delta,1)$ for the procedure based on $R_{m,q}$ with $p=10$. Right: corresponding mean detection delays.](figGammaEffect_mmk10_m_50.pdf){width="0.7\linewidth"}
Figure \[fig:GammaEffect\_mmk\_m\_50\] highlights the influence of the parameter $\gamma$ in on the rejection percentages and the mean detection delays of the procedure based on $R_{m,q}$ with $p=10$. The graphs are very similar for other values of $p$ or for the procedures based on $S_{m,q}$ and $T_{m,q}$. The conclusion is the same for all three procedures. For a change in expectation, while the parameter $\gamma$ does not seem to affect the power of the procedures much, it has a clear influence on the mean detection delay: the greater $\gamma$, the shorter the mean detection delay. As we shall see, this behavior is not true for all types of alternatives.
![\[fig:pEffect\_mmc\_m\_50\_gamma\_0.5\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the d.f. of the standard normal and $G$ the d.f. of the $N(\delta,1)$ for the procedure based on $S_{m,q}$ with $\gamma = 0.5$. Right: corresponding mean detection delays.](figpEffect_mmc_m_50_gamma_0.5.pdf){width="0.7\linewidth"}
Figure \[fig:pEffect\_mmc\_m\_50\_gamma\_0.5\] displays the influence of the number of steps $p$ of the threshold function on the rejection percentages and the mean detection delays of the procedure based on $S_{m,q}$ with $\gamma = 0.5$. The graphs are not qualitatively different for other values of $\gamma$ or for the procedures based on $R_{m,q}$ and $T_{m,q}$. Overall, the procedures with $p=1$ have the highest rejection percentages. The latter is due to the fact that, because $k^\star=75$, detections occur mostly at the end of the monitoring period, and, at the end of the monitoring interval, by construction, the threshold functions for $p=1$ are below the corresponding threshold functions obtained for larger values of $p$.
#### Change in variance
The setting is similar to that of the previous experiment except that, this time, it is the variance of the normal distribution that changes from 1 to $\delta^2$. The left graph in Figure \[fig:CompDet\_m\_50\_p\_50:var\] displays the estimated rejection percentages for the procedures based on $R_{m,q}$, $S_{m,q}$, $T_{m,q}$, $P_m$ and $Q_m$ with $\gamma = 0.5$ and $p=50$. The graph on the right represents the corresponding mean detection delays. Again, the procedures based on $R_{m,q}$, $S_{m,q}$ and $T_{m,q}$ appear to be substantially more powerful and superior in terms of mean detection delay than those based on $P_m$ and $Q_m$. The conclusion remains true for all values of $\gamma$ and $p$. The influence of $\gamma$ and $p$ is of the same nature as in the case of a change in mean: the greater $\gamma$, the shorter the mean detection delay and the lower $p$, the higher the power.
![\[fig:CompDet\_m\_50\_p\_50:var\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the d.f. of the standard normal and $G$ the d.f. of the $N(0,\delta^2)$. Right: corresponding mean detection delays. The value of $\gamma$ in is 0.5. The number of steps in the threshold functions is $p=50$.](figCompDet_m_50_p_50_gamma_0.5.pdf){width="0.7\linewidth"}
#### Change in distribution
As a final experiment for independent univariate observations, we considered a change in distribution that keeps the expectation and the variance constant. Specifically, $F$ and $G$ in $H_1$ were taken equal to the d.f. of the $N(1,2)$ and the d.f. of the Gamma distribution whose shape and rate parameters are both equal to 1/2, respectively. The parameter $m$ was taken to be in $\{50,100\}$, $n$ was set to $2m$ and the parameter $k^\star$ to ${\lfloor nc \rfloor}$ with $c \in \{0.51,0.56, \dots, 0.96\}$.
![\[fig:CompDet\_p\_50:dist\] Estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $F$ the d.f. of the $N(1,2)$ and $G$ the d.f. of the Gamma distribution whose shape and rate parameters are both equal to 1/2. Left: $m=50$ and $n=100$. Right: $m=100$ and $n=200$. The value of $\gamma$ in is 0 and the number of steps in the threshold functions is $p=50$.](figCompDet_p_50_gamma_0.pdf){width="0.7\linewidth"}
Figure \[fig:CompDet\_p\_50:dist\] shows the rejection percentages of $H_0^\text{ind}$ in against $k^\star$ for the procedures based on $R_{m,q}$, $S_{m,q}$, $T_{m,q}$, $P_m$ and $Q_m$ with $\gamma = 0$ and $p=50$. The graphs for other values of $\gamma$ and $p$ are very similar. As one can see, the procedures based on $P_m$ and $Q_m$ appear to be the most powerful when the change occurs at the beginning of the monitoring period. The latter could have been expected from the definition of the underlying detectors and the simulation results of [@DetGos19]. As $k^\star$ increases, the procedures based on $R_{m,q}$ and $T_{m,q}$ become more powerful.
![\[fig:GammaEffect\_mac\_m\_50:dist\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $n = 100$, $F$ the d.f. of the $N(1,2)$ and $G$ the d.f. of the Gamma distribution whose shape and rate parameters are both equal to 1/2 for the procedure based on $T_{m,q}$ with $p=10$. Right: corresponding mean detection delays.](figGammaEffect_mac10_m_50.pdf){width="0.7\linewidth"}
Figure \[fig:GammaEffect\_mac\_m\_50:dist\] displays the influence of the parameter $\gamma$ on the rejection percentages of the procedure based on $T_{m,q}$ with $p=10$. The graphs are not qualitatively different for other values of $p$ and $m$ or for the procedures based on $R_{m,q}$ and $S_{m,q}$. As one can see, unlike for a change in expectation, $\gamma$ has hardly no influence on the mean detection delay and it is the setting $\gamma = 0$ that leads to the highest rejection percentages.
![\[fig:pEffect\_m\_50\_gamma\_0:dist\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $n = 100$, $F$ the d.f. of the $N(1,2)$ and $G$ the d.f. of the Gamma distribution whose shape and rate parameters are both equal to 1/2 for the procedure based on $R_{m,q}$ with $\gamma=0$. Right: corresponding mean detection delays.](figpEffect_mmk_m_50_gamma_0.pdf){width="0.7\linewidth"}
Finally, Figure \[fig:pEffect\_m\_50\_gamma\_0:dist\] shows the influence of $p$ for the procedure based on $R_{m,q}$ with $\gamma = 0$. The graphs are not qualitatively different for other values of $\gamma$ or for the procedures based on $S_{m,q}$ and $T_{m,q}$. As in previous experiments, the setting $p=1$ leads in the highest rejection percentages. However, when the change occurs in the first third of the monitoring period, the mean detection delay for $p = 1$ is clearly substantially larger than for $p > 1$. Additional simulations show that the larger the monitoring period, the more pronounced this phenomenon.
Dependent multiplier bootstrap-based estimation in the time series case
-----------------------------------------------------------------------
The threshold function estimation approach based on the dependent multiplier bootstrap described in Section \[sec:thresh:time:series\] can in principle be used as soon as the observations to be monitored are either multivariate or serially dependent. We used the setting $B=2000$ in our experiments and estimated all the rejection percentages from 1000 samples at the $\alpha = 5\%$ nominal level.
### Under the null
One of the most important practical aspects is to assess how well the procedures hold their level when based on the dependent multiplier bootstrap. To attempt to answer this question, we conducted extensive simulations in the univariate case. For $m \in \{50,100,200,400\}$ and $T \in \{0.5,1,2,3\}$, we generated samples of size $n = {\lfloor m(T+1) \rfloor}$ from an AR(1) model with normal innovations and autoregressive parameter $\beta \in \{0,0.1,0.3,0.5\}$, and estimated the levels of the procedures based on $R_{m,q}$, $S_{m,q}$ and $T_{m,q}$ with $\gamma \in \{0,0.25,0.5\}$ and $p \in \{1,2,4\}$. The rejection percentages of $H_0$ in for $T_{m,q}$ are given in Table \[tab:H0:multARTmq\] (the missing entries in the table correspond to parameter settings for which computations took too long given our computer cluster resources). As one can see, unsurprisingly, the larger $\beta$, the more liberal the procedure based on $T_{m,q}$ tends to be. This phenomenon is particularly visible for $\beta \in \{0.3, 0.5\}$. Reassuringly however, for a given $\beta$, $T$, $p$ and $\gamma$, the estimated levels seem to get closer to the 5% nominal level as $m$ increases. Hence, unsurprisingly, the stronger the serial dependence, the larger $m$ needs to be so that the procedure can be expected to hold its nominal level. For $\beta \in \{0.3, 0.5\}$ in particular and keeping $T$, $\gamma$ and $m$ fixed, we also see that the larger $p$, the more liberal the procedure based on $T_{m,q}$ tends to be. The latter could be explained by the fact that, as $p$ increases, more thresholds need to be estimated, and that, except for the first, all the estimated thresholds are conditional empirical quantiles: the precision of the estimation of a threshold thus critically depends on the precision of the previously estimated thresholds with respect to which conditioning is performed. In other words, the fact that the empirical levels tend to become higher when $p$ increases could be explained by an error propagation effect. Finally, for $\beta$, $T$, $p$ and $m$ fixed, we also see that the larger $\gamma$, the lower the rejection percentages tend to be. Overall, for $\beta \geq 0.1$, the procedure based on $T_{m,q}$ holds its level best for $\gamma = 0.5$.
The conclusions for the procedures based on $R_{m,q}$ and $S_{m,q}$ are very similar, with the exception that the effect of $\gamma$ seems “stronger”: while the empirical levels are better for $\gamma = 0.25$ than for $\gamma = 0$, the procedures become way too conservative for $\gamma = 0.5$. The latter effect might be due to the fact the detectors $R_{m,q}$ and $S_{m,q}$ involve maxima (unlike $T_{m,q}$ which involves means) and to our too low setting of the constant $\delta$ in the definition of the weight function . The latter was arbitrarily set to $10^{-4}$ (and had *de facto* no effect in our Monte Carlo experiments given the values of $m$ that we considered).
A similar experiment was conducted by generating samples from a GARCH(1,1) model with parameters $\omega = 0.012$, $\beta = 0.919$ and $\alpha = 0.072$ to mimic SP500 daily log-returns following [@JonPooRoc07]. The empirical levels for the procedure based on $T_{m,q}$ are reported in Table \[tab:H0:multGARCHTmq\] of the supplement [@KojVer20] and appear to be closest to the 5% nominal overall when $\gamma = 0.5$.
Finally, a bivariate experiment with independent observations consisting of generating samples of size $n = 2m$ for $m \in \{50, 100, 200\}$ from a normal copula with a Kendall’s tau of $\tau \in \{-0.6, -0.3, 0, 0.3, 0.6 \}$ was carried out. The empirical levels for the procedure based on $T_{m,q}$ are reported in Table \[tab:H0:multcop\] of the supplement [@KojVer20]. The effect of $\gamma$ appears similar as in the previous experiments. For fixed $\gamma$ and $p$, it can also be observed that the procedure has a tendency of being too conservative in the case of strong negative dependence but, reassuringly, the agreement with the 5% nominal level seems to improve as $m$ increases.
![\[fig:copula\] Left: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the bivariate normal copula with a Kendall’s tau of -0.6 and $G$ the bivariate normal copula with a Kendall’s tau of $\tau$. Right: estimated rejection probabilities of $H_0^\text{ind}$ in under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the bivariate normal copula with a Kendall’s tau of 0.6 and $G$ the bivariate normal copula with a Kendall’s tau of $\tau$. The value of $\gamma$ in is 0.25. The number of steps in the threshold functions is $p=4$.](figCompDet_meth_mult.nonseq_m_50_p_4_gamma_0.25.pdf){width="0.7\linewidth"}
### Change in the copula parameter
To grasp further the finite-sample behavior of the procedures in the case of bivariate independent observations, we simulated a change in the parameter of a normal copula. The left (resp. right) graph in Figure \[fig:copula\] displays the estimated rejection probabilities of $H_0^\text{ind}$ in for the procedures based on $R_{m,q}$, $S_{m,q}$, $T_{m,q}$, $P_m$ and $Q_m$ with $\gamma = 0.25$ and $p=4$ under $H_1$ in with $m=50$, $k^\star = 75$, $n = 100$, $F$ the bivariate normal copula with a Kendall’s tau of -0.6 (resp. 0.6) and $G$ the bivariate normal copula with a Kendall’s tau of $\tau \in \{-0.6, 0.3, 0, 0.3, 0.6, 0.9\}$ (resp. $\tau \in \{-0.9, -0.6, -0.3, 0, 0.3, 0.6\}$). As one can notice, the procedure based on $T_{m,q}$ (resp. $P_m$) is always among the most (resp. least) powerful ones. Graphs for other values of $\gamma$ and $p$ are not qualitatively different. As for all previous experiments, we observed that the smaller $p$, the more powerful the procedures. For this experiment, the parameter $\gamma$ appeared to have a rather small impact on the rejection percentages of $T_{m,q}$.
Data examples
=============
![\[fig:det-thresh\] Left: closing quotes and corresponding daily log-returns of the NASDAQ composite index for the period 2019-01-02 – 2020-04-11. The solid vertical line represents the beginning of the monitoring. The dotted vertical line represent the date (2020-03-12) at which the detector function based on $T_{m,q}$ with $\gamma = 0.5$ first exceeded the two threshold functions on the right. The dashed vertical line corresponds to the estimated date of change (2020-02-20). Right: the dotted line represents the detector function based on $T_{m,q}$ with $\gamma = 0.5$. The solid (resp. dashed) line represents the threshold function with $p=4$ steps obtained using the dependent multiplier bootstrap (resp. Monte Carlo estimation).](det-thresh-mac-gamma-0.5-p-4.pdf){width="1\linewidth"}
To illustrate the use of the proposed sequential change-point detection tests, we considered two fictitious scenarios, the first (resp. second) of a univariate (resp. bivariate) nature based on closing quotes of the NASDAQ composite index (resp. Microsoft and Intel stocks) for the period 2019-01-02 – 2020-04-11. The latter were obtained using the `get.hist.quote()` function of the `tseries` package [@tseries]. In both scenarios, it was assumed that, on the last day of 2019, one wished to monitor the (univariate or bivariate) daily log-returns for a change in contemporary distribution possibly using the stretch of $m=251$ (univariate of bivariate) log-returns of 2019 as learning sample. The latter decision was confirmed after the tests of [@BucFerKoj19] implemented in the functions `stDistAutocop()` and `cpDist()` of the `npcp` package [@npcp] provided no evidence against stationarity for the two candidate learning samples. Notice that, unsurprisingly, the rank-based test of serial independence of [@GenRem04] implemented in the function `serialIndepTest()` of the `copula` package [@copula] provided weak evidence against the serial independence of the squared component time series.
The closing quotes and corresponding daily log-returns of the NASDAQ composite index are represented in the left panel of Figure \[fig:det-thresh\]. The solid vertical lines mark the beginning of the monitoring. The dotted line in the right panel represents the detector function based on $T_{m,q}$ with $\gamma = 0.5$. The latter was chosen given its overall good performance in our Monte Carlo experiments, both in terms of empirical level, power and mean detection delay. In the right panel, the solid line represents the threshold function with $p=4$ steps estimated using the dependent multiplier bootstrap with $B=10^5$, while the dashed line represents the threshold function with $p=4$ steps estimated using Monte Carlo with $M=10^5$. Note that the latter did not at all use the learning sample as it is computed under the assumption that the observations are serially independent. The relative proximity of the two threshold functions could be explained by the fact that, although present, serial dependence in the learning sample is probably very weak. The detector function exceeded the two threshold functions at the same date (2020-03-12), which is marked by the dotted vertical line in the left panel of Figure \[fig:det-thresh\] and corresponds to the 49th daily log-return of 2020. Given the definition of $T_{m,q}$ in and having that of $S_{m,q}$ in in mind, a natural estimate of a point of change for an exceedance at position $k=m+49 = 300$ is given by $$\label{eq:point:change}
\mathrm{argmax}_{m \leq j \leq k-1} \frac{1}{k} \sum_{i=1}^k \left[ \frac{j (k-j)}{m^{3/2} q(j/m,k/m)} \{ F_{1:j}(\bm X_i) - F_{j+1:k}(\bm X_i) \} \right]^2,$$ which returned 285 and corresponds to the date 2020-02-20. The latter is marked by a dashed vertical line in the left panel of Figure \[fig:det-thresh\] and corresponds to the beginning of the sharp decrease of the NASDAQ composite index as a consequence of the Covid-19 pandemic.
![\[fig:biv-det-thresh\] Monitoring of the bivariate daily log-returns of the Microsoft and Intel stocks for the period 2020-01-01 – 2020-04-11 using the 2019 bivariate log-returns as learning sample, and the procedure based on $T_{m,q}$ with $\gamma \in \{0, 0.25, 0.5\}$ and a threshold function with $p \in \{1,2,4\}$ steps estimated using the dependent multiplier bootstrap. The estimated dates of change do not depend on $p$ and are the 2020-01-24 for $\gamma = 0$, the 2020-02-14 for $\gamma = 0.25$ and the 2020-02-19 for $\gamma = 0.5$.](biv-det-thresh.pdf){width="1\linewidth"}
Figure \[fig:biv-det-thresh\] describes the monitoring of the bivariate daily log-returns of the Microsoft and Intel stocks using the procedure based on $T_{m,q}$ with $\gamma \in \{0, 0.25, 0.5\}$ and a threshold function with $p \in \{1,2,4\}$ steps estimated using the dependent multiplier bootstrap. All the dates of exceedance are between the 2020-03-12 ($\gamma=0$ and $p=2$) and the 2020-03-09 ($\gamma = 0.5$ and $p=1$). The estimated dates of change turn out not to depend on $p$ and are the 2020-01-24 for $\gamma = 0$, the 2020-02-14 for $\gamma = 0.25$ and the 2020-02-19 for $\gamma = 0.5$. This effect of $\gamma$ on was to be expected: larger values of $\gamma$ give more weight to potential break points close to $k$.
Concluding remarks
==================
In the context of closed-end sequential change-point detection, it can be argued that it is desirable that the underlying threshold function is such that the probability of false alarm remains approximately constant over the monitoring period. In this work, the asymptotic validity of the bootstrap-based estimation of such a threshold function was established for generic detectors. The latter was applied to sequential change-point tests involving detectors based on differences of empirical d.f.s that can be either simulated or resampled using a dependent multiplier bootstrap depending on whether univariate independent or multivariate serially dependent observations are monitored. The proposed detectors are adaptations of statistics used in a posteriori change-point testing and include a weight function in the spirit of [@CsoSzy94b] that can be used to give more importance to recent observations.
Extensive Monte Carlo experiments were used to investigate the finite-sample properties of the resulting sequential change-point tests. Among the proposed detectors, none led to a uniformly better testing procedure. When based on the dependent multiplier bootstrap, the procedure based on $T_{m,q}$ in was observed to have the best behavior, overall, in terms of empirical level, power and mean detection delay. In the case of univariate independent observations, when the threshold function can be estimated using Monte Carlo simulation, the number of step $p$ of the threshold function can be chosen as large as the number of monitoring steps. However, in the time series case, when the estimation of the threshold function is based on the dependent multiplier bootstrap, $p$ should not be taken too large because of an error propagation effect.
With the aim of monitoring multivariate independent observations, we were also able to establish the asymptotic validity of the proposed procedures when the threshold functions are estimated using a smooth bootstrap based on the *empirical beta copula* [@KirSegTsu19]. In future work, we plan to investigate the validity of additional bootstraps for monitoring multivariate time series. The current and future theoretical results would also need to be complemented by additional Monte Carlo simulations, involving in particular multivariate experiments. Such finite-sample investigations are however a real computational challenge given the complexity and cost of execution of such change-point detection procedures.
Auxiliary lemmas for the proof of Theorem {#sec:aux:thresh:generic}
==========================================
This section, which is, to a large extent, notationally independent of the rest of the paper, provides the proofs of two lemmas, possibly of independent interest, necessary for showing Theorem \[thm:thresh:boot\].
Let $\bm {\mathcal{X}}_n$ denote available data. No assumptions are made on $\bm {\mathcal{X}}_n$ apart from measurability. To fix ideas, one can think of $\bm {\mathcal{X}}_n$ as a sequence of $n$ multivariate serially dependent random vectors. Let $\bm S_n = \bm S_n(\bm {\mathcal{X}}_n)$ be a ${\mathbb{R}}^p$-valued statistic such that $\bm S_n = (S_{1,n},\dots,S_{p,n}) \leadsto \bm S = (S_1,\dots,S_p)$ as $n \to \infty$, where the random vector $\bm S$ is assumed to have a continuous d.f. We additionally suppose that we have available *bootstrap replicates* $\bm S_{n}^{[i]} = \bm S_{n}^{[i]}(\bm {\mathcal{X}}_n, \bm {\mathcal{W}}_n^{[i]})$ of $\bm S_n$, where the $\bm {\mathcal{W}}_n^{[i]}$, $i \in {\mathbb{N}}$, are $n$-dimensional independent and identically distributed random vectors representing the additional sources of randomness involved in the underlying bootstrap mechanism. We shall further assume that, as $n \to \infty$, $$\label{eq:jwconv}
\big(\bm S_{n},\bm S_{n}^{[1]}, \bm S_{n}^{[2]} \big) \\ \leadsto \big ( \bm S, \bm S^{[1]}, \bm S^{[2]} \big),$$ in $({\mathbb{R}}^p)^3$, where $\bm S^{[1]}$ and $\bm S^{[2]}$ are independent copies of $\bm S$. Note that, from Lemma 2.2 of [@BucKoj19], is equivalent to the usual conditional bootstrap consistency statement, that is, $$\sup_{\bm x \in {\mathbb{R}}^p} | {\mathbb{P}}(\bm S_n^{[1]} \leq \bm x \mid \bm {\mathcal{X}}_n) - {\mathbb{P}}(\bm S_n \leq \bm x)| {\overset{{\mathbb{P}}}{\to}}0, \qquad \text{as } n \to \infty.$$
Before stating and proving the two lemmas, we introduce some additional notation and list useful results. For any $q\in\{1,\dots,p\}$, $\{j_1,\dots,j_q\}\subset\{1,\dots,p\}$, $x_{j_1},\dots, x_{j_q} \in {\mathbb{R}}$ and $B \in {\mathbb{N}}$, let $$\begin{aligned}
F_{\bm S, \{j_1,\dots,j_q\}}(x_{j_1},\dots,x_{j_q}) &= {\mathbb{P}}(S_{j_1} \leq x_{j_1}, \dots, S_{j_q} \leq x_{j_q}), \\
F_{\bm S_n, \{j_1,\dots,j_q\}}(x_{j_1},\dots, x_{j_q}) &= {\mathbb{P}}(S_{j_1,n} \leq x_{j_1}, \dots, S_{j_q,n} \leq x_{j_q}), \\
F_{\bm S_n, \{j_1,\dots,j_q\}}^B(x_{j_1},\dots,x_{j_q}) &= \frac{1}{B} \sum_{i=1}^B {\mathbf{1}}(S_{j_1,n}^{[i]} \leq x_{j_1}, \dots, S_{j_q,n}^{[i]} \leq x_{j_q}).\end{aligned}$$ Since $\bm S_{n} \leadsto \bm S$ in ${\mathbb{R}}^p$ as $n \to \infty$ and $\bm S$ has a continuous d.f., we have from Lemma 2.11 of [@van98] that, for any $q\in\{1,\dots,p\}$ and $\{j_1,\dots,j_q\}\subset\{1,\dots,p\}$, as $n \to \infty$, $$\label{eq:unif}
\sup_{(x_{j_1},\dots,x_{j_q}) \in {\mathbb{R}}^q} |F_{\bm S_n, \{j_1,\dots,j_q\}}(x_{j_1},\dots,x_{j_q}) - F_{\bm S, \{j_1,\dots,j_q\}}(x_{j_1},\dots,x_{j_q})| \to 0.$$ Proceeding as in the proof of the aforementioned lemma, it can actually also be shown that, for any $q\in\{1,\dots,p\}$ and $\{j_1,\dots,j_q\}\subset\{1,\dots,p\}$, as $n \to \infty$, $$\label{eq:unif:sign}
\sup_{(x_{j_1},\dots,x_{j_q}) \in {\mathbb{R}}^q} |{\mathbb{P}}(S_{j_1,n} < x_{j_1}, \dots, S_{j_q,n} < x_{j_q}) - {\mathbb{P}}(S_{j_1} < x_{j_1}, \dots, S_{j_q} < x_{j_q})| \to 0.$$ Combining with Assertion (f) of Lemma 2.2 in [@BucKoj19] and , we further obtain that, for any $q\in\{1,\dots,p\}$ and $\{j_1,\dots,j_q\}\subset\{1,\dots,p\}$, as $n,B \to \infty$, $$\label{eq:unif:B}
\sup_{(x_{j_1},\dots,x_{j_q}) \in {\mathbb{R}}^q} |F_{\bm S_n, \{j_1,\dots,j_q\}}^B(x_{j_1},\dots,x_{j_q}) - F_{\bm S, \{j_1,\dots,j_q\}}(x_{j_1},\dots,x_{j_q})| {\overset{{\mathbb{P}}}{\to}}0.$$ Let $\xi \in (0,1)$ be arbitrary. The following notation will also be used in the lemmas. Let $$F_{\bm S, 1}(x) = F_{\bm S, \{1\}}(x) = {\mathbb{P}}(S_1 \leq x), \, x \in {\mathbb{R}}, \qquad g_1 = F_{\bm S, 1}^{-1}(1 - \xi),$$ and, recursively, for $j$ successively equal to $2,\dots,p$, $$\label{eq:Fj:gj}
F_{\bm S, j}(x) = \frac{F_{\bm S, \{1,\dots,j\}}(g_1,\dots,g_{j-1},x)}{F_{\bm S, \{1,\dots,j-1\}}(g_1,\dots,g_{j-1})}, \, x \in{\mathbb{R}}, \qquad g_j = F_{\bm S, j}^{-1}(1 - \xi).$$ Similarly, let $$F_{\bm S_n, 1}^B(x) = F_{\bm S_n, \{1\}}^B(x), \, x \in {\mathbb{R}}, \qquad g_{1,n}^B = F_{\bm S_n, 1}^{B,-1}(1 - \xi),$$ and, recursively, for $j$ successively equal to $2,\dots,p$, $$\label{eq:FjB:gjB}
F_{\bm S_n, j}^B(x) = \frac{F_{\bm S_n, \{1,\dots,j\}}^B(g_{1,n}^B,\dots,g_{j-1,n}^B,x)}{F_{\bm S_n, \{1,\dots,j-1\}}^B(g_{1,n}^B,\dots,g_{j-1,n}^B)}, \, x \in{\mathbb{R}}, \qquad g_{j,n}^B = F_{\bm S_n, j}^{B,-1}(1 - \xi).$$
The following two lemmas are instrumental for proving Theorem \[thm:thresh:boot\].
\[lem:threshold\] As $n,B \to \infty$, $$\begin{gathered}
\label{eq:rec11}
\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,1}^B(x) - F_{\bm S,1}(x)| {\overset{{\mathbb{P}}}{\to}}0, \\
\label{eq:rec12}
{\mathbb{P}}( S_{1,n} \leq g_{1,n}^B ) \to 1 - \xi, \\
\label{eq:rec13}
F_{\bm S,1}( g_{1,n}^B ) \to 1 - \xi,\end{gathered}$$ and, for any $j \in \{2,\dots,p\}$, $$\begin{gathered}
\label{eq:recj1}
\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,j}^B(x) - F_{\bm S,j}(x)| {\overset{{\mathbb{P}}}{\to}}0, \\
\label{eq:recj2}
{\mathbb{P}}( S_{j,n} \leq g_{j,n}^B \mid S_{1,n} \leq g_{1,n}^B, \dots,S_{j-1,n} \leq g_{j-1,n}^B ) \to 1 - \xi, \\
\label{eq:recj3}
F_{\bm S,j}( g_{j,n}^B ) \to 1 - \xi.\end{gathered}$$ The statements and with ‘$\leq$’ replaced by ‘$<$’ hold as well.
\[lem:thresh:inf\] Let $\bm T_n = \bm T_n(\bm {\mathcal{X}}_n) = (T_{1,n}, \dots, T_{p,n})$ be a ${\mathbb{R}}^p$-valued statistic such that $\max_{1 \leq j \leq p} T_{j,n} {\overset{{\mathbb{P}}}{\to}}\infty$ as $n \to \infty$. Then $${\mathbb{P}}(T_{1,n} \leq g_{1,n}^B,\dots,T_{p,n} \leq g_{p,n}^B) \to 0 \qquad \text{ as } n,B \to \infty.$$
The claims in and are immediate consequences of and Lemma 4.2 in [@BucKoj19], respectively. The claim follows the fact that $F_{\bm S,1}(g_1) = 1-\xi$, the triangle inequality, and .
To prove , and , we proceed by induction on $j$.
*Proof of , and for j=2:* By the triangle inequality, $|F_{\bm S_n,1}^B(g_{1,n}^B) - F_{\bm S,1}(g_1)|$ is smaller than $$|F_{\bm S_n,1}^B(g_{1,n}^B) - F_{\bm S,1}(g_{1,n}^B)| + |F_{\bm S,1}(g_{1,n}^B) - F_{\bm S_n,1}(g_{1,n}^B)|+ |F_{\bm S_n,1}(g_{1,n}^B) - F_{\bm S,1}(g_1)|.$$ The first term converges in probability to zero by as $n,B \to \infty$. The second term converges to zero by as $n,B \to \infty$. The third term converges to zero as a consequence of since $F_{\bm S,1}(g_1) = 1 - \xi$. Hence, $|F_{\bm S_n,1}^B(g_{1,n}^B) - F_{\bm S,1}(g_1)| {\overset{{\mathbb{P}}}{\to}}0$ as $n,B \to \infty$. From and , the latter implies that to prove that $\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,2}^B(x) - F_{\bm S,2}(x)| {\overset{{\mathbb{P}}}{\to}}0$ as $n,B \to \infty$, it suffices to prove that $$\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,2\}}^B(g_{1,n}^B,x) - F_{\bm S,\{1,2\}}(g_1,x)| {\overset{{\mathbb{P}}}{\to}}0$$ as $n,B \to \infty$. From the triangle inequality, and , the latter will hold if $$\label{eq:aim1}
\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,2\}}(g_{1,n}^B,x) - F_{\bm S_n,\{1,2\}}(g_1,x)| \to 0$$ as $n,B \to \infty$. For any $x \in {\mathbb{R}}$, we have $$\begin{gathered}
\label{eq:already2}
F_{\bm S_n,\{1,2\}}(g_{1,n}^B,x) - F_{\bm S_n,\{1,2\}}(g_1,x) = {\mathbb{P}}(S_{1,n} \geq g_1, S_{2,n} \leq x ) - {\mathbb{P}}(S_{1,n} = g_1, S_{2,n} \leq x ) \\ - {\mathbb{P}}( S_{1,n} \geq g_{1,n}^B, S_{2,n} \leq x ) + {\mathbb{P}}( S_{1,n} = g_{1,n}^B, S_{2,n} \leq x ).\end{gathered}$$ Hence, by the triangle inequality, $$\begin{gathered}
\label{eq:already1}
\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,2\}}(g_{1,n}^B,x) - F_{\bm S_n,\{1,2\}}(g_1,x)| \\ \leq \sup_{x \in {\mathbb{R}}} |{\mathbb{P}}( S_{1,n} \geq g_{1,n}^B, S_{2,n} \leq x ) - {\mathbb{P}}(S_{1,n} \geq g_1, S_{2,n} \leq x )| + 2 \sup_{x \in {\mathbb{R}}} {\mathbb{P}}(S_{1,n} = x).\end{gathered}$$ From the triangle inequality, and , we obtain that the last supremum on the right-hand side of the previous display converges to zero a $n \to \infty$ since $\bm S$ has a continuous d.f. Hence, to show , it remains to show that the first supremum on the right converges to zero in probability as $n,B \to \infty$. From the right-continuity of $F_{\bm S, 1}$ and $F_{\bm S_n,1}^B$, we obtain that, for any $x \in {\mathbb{R}}$, $$\begin{aligned}
|{\mathbb{P}}( S_{1,n} &\geq g_{1,n}^B, S_{2,n} \leq x ) - {\mathbb{P}}(S_{1,n} \geq g_1, S_{2,n} \leq x )| \\
&= | {\mathbb{P}}\{ S_{1,n} \geq F_{\bm S_n,1}^{B,-1}(1 -\xi), S_{2,n} \leq x \} - {\mathbb{P}}\{ S_{1,n} \geq F_{\bm S, 1}^{-1}(1 - \xi), S_{2,n} \leq x \} | \\
&= | {\mathbb{P}}\{ F_{\bm S_n,1}^B(S_{1,n}) \geq 1 -\xi, S_{2,n} \leq x \} - {\mathbb{P}}\{ F_{\bm S, 1}(S_{1,n}) \geq 1 - \xi, S_{2,n} \leq x \} |.\end{aligned}$$ Furthermore, using the fact that, for any $a, b, y \in {\mathbb{R}}$ and ${\varepsilon}> 0$, $$\label{eq:ineq:ind}
|{\mathbf{1}}(y \leq a) - {\mathbf{1}}(y \leq b)| \leq {\mathbf{1}}(|y - a| \leq {\varepsilon}) + {\mathbf{1}}(|a-b| > {\varepsilon}),$$ we obtain that, for any ${\varepsilon}> 0$, $$\label{eq:already3}
\begin{split}
| {\mathbb{P}}&\{ F_{\bm S_n,1}^B(S_{1,n}) \geq 1 -\xi, S_{2,n} \leq x \} - {\mathbb{P}}\{ F_{\bm S, 1}(S_{1,n}) \geq 1 - \xi, S_{2,n} \leq x \} | \\
&= | {\mathbb{E}}[ {\mathbf{1}}\{ F_{\bm S_n,1}^B(S_{1,n}) \geq 1 -\xi, S_{2,n} \leq x \} - {\mathbf{1}}\{ F_{\bm S, 1}(S_{1,n}) \geq 1 - \xi, S_{2,n} \leq x \} ] | \\
&\leq {\mathbb{E}}| {\mathbf{1}}\{ F_{\bm S_n,1}^B(S_{1,n}) \geq 1 -\xi, S_{2,n} \leq x \} - {\mathbf{1}}\{ F_{\bm S, 1}(S_{1,n}) \geq 1 - \xi, S_{2,n} \leq x \} | \\
&\leq {\mathbb{E}}| {\mathbf{1}}\{ F_{\bm S_n,1}^B(S_{1,n}) \geq 1 -\xi \} - {\mathbf{1}}\{ F_{\bm S, 1}(S_{1,n}) \geq 1 - \xi \} | \\
&\leq {\mathbb{P}}\{ |F_{\bm S, 1}(S_{1,n}) - 1 + \xi| \leq {\varepsilon}\} + {\mathbb{P}}\{ |F_{\bm S_n,1}^B(S_{1,n}) - F_{\bm S, 1}(S_{1,n}) | > {\varepsilon}\}.
\end{split}$$ By the continuous mapping theorem and the Portmanteau theorem, the first probability on the right converges to ${\mathbb{P}}\{ | F_{S_1}(S_1) - 1 + \xi | \leq {\varepsilon}\}$ as $n \to \infty$ and can be made arbitrarily small by decreasing ${\varepsilon}$. The second probability converges to zero as $n,B \to \infty$ by for any ${\varepsilon}> 0$. Hence, holds and so does for $j=2$.
Let us now prove for $j=2$. Since, from , ${\mathbb{P}}( S_{1,n} \leq g_{1,n}^B ) \to P(S_1 \leq g_1) = 1 - \xi$, as $n,B \to \infty$, it remains to show that $${\mathbb{P}}(S_{1,n} \leq g_{1,n}^B, S_{2,n} \leq g_{2,n}^B) \to {\mathbb{P}}(S_1 \leq g_1, S_2 \leq g_2)$$ as $n,B \to \infty$. From the triangle inequality and , if suffices to prove that, as $n,B \to \infty$, $$| F_{\bm S_n,\{1,2\}}(g_{1,n}^B, g_{2,n}^B) - F_{\bm S_n,\{1,2\}}(g_1, g_2) | \to 0.$$ The term on the left-hand side of the previous display is smaller than $$|F_{\bm S_n,\{1,2\}}(g_{1,n}^B, g_{2,n}^B) - F_{\bm S_n,\{1,2\}}(g_1, g_{2,n}^B)| + |F_{\bm S_n,\{1,2\}}(g_1, g_{2,n}^B) - F_{\bm S_n,\{1,2\}}(g_1, g_2)|.$$ The first difference between absolute values is smaller than , which was already shown to converge to zero as $n,B \to \infty$. Proceeding as in , to show that the second difference between absolute values converges to zero as $n,B \to \infty$, it suffices to prove that $$\begin{gathered}
|{\mathbb{P}}( S_{1,n} \leq g_1, S_{2,n} \geq g_{2,n}^B ) - {\mathbb{P}}(S_{1,n} \leq g_1, S_{2,n} \geq g_2 )| \\ = | {\mathbb{P}}\{ S_{1,n} \leq g_1, S_{2,n} \geq F_{\bm S_n,2}^{B,-1}(1 - \xi) \} - {\mathbb{P}}\{ S_{1,n} \leq g_1, S_{2,n} \geq F_{\bm S,2}^{-1}(1 - \xi) \} | \to 0\end{gathered}$$ as $n,B \to \infty$. Using again and proceeding as in , the probability on the left can be shown to be smaller than $${\mathbb{P}}\{ |F_{\bm S, 2}(S_{2,n}) - 1 + \xi| \leq {\varepsilon}\} + {\mathbb{P}}\{ |F_{\bm S_n,2}^B(S_{2,n}) - F_{\bm S, 2}(S_{2,n}) | > {\varepsilon}\}$$ for any ${\varepsilon}> 0$. Using the continuity of $F_{\bm S, 2}$ and the Portmanteau theorem, the first probability converges as $n \to \infty$ to a probability that can be made arbitrarily small by decreasing ${\varepsilon}$. The second probability converges to zero as $n,B \to \infty$ for any ${\varepsilon}> 0$ by for $j=2$, which was proven previously.
It remains to prove for $j=2$. From the triangle inequality, and , we have that, as $n,B \to \infty$, $$\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,2\}}(g_{1,n}^B,x) - F_{\bm S,\{1,2\}}(g_1,x)| \to 0.$$ Starting from for $j=2$, the desired result follows from the convergence in the previous display and .
*Proof of , and for all $j \in \{3,\dots,p\}$:* As mentioned previously, we proceed by induction. Let $k \in \{3,\dots,p\}$, assume that , and hold for all $j \in \{2,\dots,k-1\}$ and let us show that they also hold for $k$.
For any $x \in {\mathbb{R}}$, let $F_{\bm S_n, 1}(x) = F_{\bm S_n, \{1\}}(x)$, and, for $j$ successively equal to $2,\dots,p$, let $$F_{\bm S_n, j}(x) = \frac{F_{\bm S_n, \{1,\dots,j\}}(g_{1,n}^B,\dots,g_{j-1,n}^B,x)}{F_{\bm S_n, \{1,\dots,j-1\}}(g_{1,n}^B,\dots,g_{j-1,n}^B)}.$$ Then, from the induction hypothesis for , as $n,B \to \infty$, $$\begin{gathered}
\label{eq:decomp:F}
F_{\bm S_n,\{1,\dots,k-1\}}(g_{1,n}^B,\dots,g_{k-1}^B) = F_{\bm S_n,k-1}(g_{k-1,n}^B) F_{\bm S_n,k-2}(g_{k-2,n}^B) \cdots F_{\bm S_n,1}(g_{1,n}^B) \\ \to
F_{\bm S,k-1}(g_{k-1}) F_{\bm S,k-2}(g_{k-2}) \cdots F_{\bm S,1}(g_1) = (1 - \xi)^{k-1} = F_{\bm S,\{1,\dots,k-1\}}(g_1,\dots,g_{k-1}).\end{gathered}$$ Thus, from , as $n,B \to \infty$, $F_{\bm S_n,\{1,\dots,k-1\}}^B(g_{1,n}^B,\dots,g_{k-1}^B) {\overset{{\mathbb{P}}}{\to}}F_{\bm S,\{1,\dots,k-1\}}(g_1,\dots,g_{k-1})$. Hence, to show for $j=k$, it suffices to prove that, as $n,B \to \infty$, $$\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,\dots,k\}}^B(g_{1,n}^B,\dots,g_{k-1,n}^B,x) - F_{\bm S,\{1,\dots,k\}}(g_1,\dots,g_{k-1},x)| {\overset{{\mathbb{P}}}{\to}}0.$$ From the triangle inequality, and , the latter will hold if, as $n,B \to \infty$, $$\label{eq:aim2}
\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B,\dots,g_{k-1,n}^B,x) - F_{\bm S_n,\{1,\dots,k\}}(g_1,\dots,g_{k-1},x)| \to 0.$$ For any $x \in {\mathbb{R}}$, $|F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B,\dots,g_{k-1,n}^B,x) - F_{\bm S_n,\{1,\dots,k\}}(g_1,\dots,g_{k-1},x)|$ is smaller than the following sum of $k-1$ terms: $$\begin{aligned}
&|F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B,g_{2,n}^B,\dots,g_{k-1,n}^B,x) - F_{\bm S_n,\{1,\dots,k\}}(g_1,g_{2,n}^B,\dots,g_{k-1,n}^B,x)| \\
+& |F_{\bm S_n,\{1,\dots,k\}}(g_1,g_{2,n}^B,\dots,g_{k-1,n}^B,x) - F_{\bm S_n,\{1,\dots,k\}}(g_1,g_2,g_{3,n}^B,\dots,g_{k-1,n}^B,x)| \\
&\vdots \\
+& |F_{\bm S_n,\{1,\dots,k\}}(g_1,\dots,g_{k-2},g_{k-1,n}^B,x) - F_{\bm S_n,\{1,\dots,k\}}(g_1,\dots,g_{k-1},x)|.\end{aligned}$$ Proceeding as in , and , the $j$th term, $j \in \{1,\dots,k-1\}$, can be shown to converge to zero as $n,B \to \infty$ as a consequence of the fact that $${\mathbb{P}}\{ |F_{\bm S, j}(S_{j,n}) - 1 + \xi| \leq {\varepsilon}\} + {\mathbb{P}}\{ |F_{\bm S_n,j}^B(S_{j,n}) - F_{\bm S, j}(S_{j,n}) | > {\varepsilon}\}$$ converges to zero as $n,B \to \infty$ followed by ${\varepsilon}\downarrow 0$ using the continuity of $F_{\bm S, j}$ and the induction hypothesis. Hence, holds.
Let us now show for $j=k$. From , it suffices to prove that, as $n,B \to \infty$, $$| F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B, \dots, g_{k,n}^B ) - F_{\bm S,\{1,\dots,k\}}(g_1, \dots,g_k )| \to 0.$$ From the triangle inequality and , the latter will hold if, as $n,B \to \infty$, $$| F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B, \dots, g_{k,n}^B ) - F_{\bm S_n,\{1,\dots,k\}}(g_1, \dots,g_k )| \to 0.$$ The difference between absolute values on the left of the previous display is smaller than $$\begin{gathered}
| F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B, \dots, g_{k-1,n}^B, g_{k,n}^B ) - F_{\bm S_n,\{1,\dots,k\}}(g_1, \dots,g_{k-1},g_{k,n}^B)| \\ + | F_{\bm S_n,\{1,\dots,k\}}(g_1, \dots,g_{k-1},g_{k,n}^B) - F_{\bm S_n,\{1,\dots,k\}}(g_1, \dots, g_{k-1}, g_k )|.\end{gathered}$$ The first term converges to zero in probability as $n,B \to \infty$ as a consequence of . Proceeding as in , and , the second term can be shown to converge to zero as $n,B \to \infty$ as a consequence of the fact that $${\mathbb{P}}\{ |F_{\bm S, k}(S_{k,n}) - 1 + \xi| \leq {\varepsilon}\} + {\mathbb{P}}\{ |F_{\bm S_n,k}^B(S_{k,n}) - F_{\bm S, k}(S_{k,n}) | > {\varepsilon}\}$$ converges to zero as $n,B \to \infty$ followed by ${\varepsilon}\downarrow 0$ using the continuity of $F_{\bm S, k}$ and the already proven claim for $j=k$.
It finally remains to show for $j=k$. From the triangle inequality, and , we have that, as $n,B \to \infty$, $$\sup_{x \in {\mathbb{R}}} |F_{\bm S_n,\{1,\dots,k\}}(g_{1,n}^B,\dots,g_{k-1,n}^B,x) - F_{\bm S,\{1,\dots,k\}}(g_1,\dots,g_{k-1},x)| \to 0.$$ Starting from for $j=k$, the desired result follows by combining the latter convergence with .
The induction is thus complete. The fact that the statements and with ‘$\leq$’ replaced by ‘$<$’ hold as well is a consequence of and .
Let $M_n = \max_{1 \leq j \leq p} T_{j,n}$. Then, $$\begin{aligned}
{\mathbb{P}}(T_{1,n} \leq g_{1,n}^B,\dots,T_{p,n} \leq g_{p,n}^B) &\leq {\mathbb{P}}\left(M_n \leq \max_{1 \leq j \leq p} g_{j,n}^B \right) \\
&\leq {\mathbb{P}}(M_n \leq g_{1,n}^B ) + \dots + {\mathbb{P}}( M_n \leq g_{p,n}^B ).\end{aligned}$$ The proof is complete if we show that, for any $j \in \{1,\dots,p\}$, ${\mathbb{P}}( M_n \leq g_{j,n}^B ) \to 0$ as $n,B \to \infty$. Let $j \in \{1,\dots,p\}$ and recall the definition of the continuous d.f. $F_{\bm S,j}$ defined in . Then, $${\mathbb{P}}( M_n \leq g_{j,n}^B ) \leq {\mathbb{P}}\{ F_{\bm S,j}(M_n) \leq F_{\bm S,j} (g_{j,n}^B ) \} = {\mathbb{P}}\{ F_{\bm S,j}(M_n) - F_{\bm S,j} (g_{j,n}^B ) \leq 0\}.$$ Since $M_n {\overset{{\mathbb{P}}}{\to}}\infty$ as $n \to \infty$, we have that $F_{\bm S,j}(M_n) {\overset{{\mathbb{P}}}{\to}}1$ as $n \to \infty$. Furthermore, from Lemma \[lem:threshold\], $F_{\bm S,j}(g_{j,n}^B) \to 1 - \xi$ as $n,B \to \infty$. Hence, $F_{\bm S,j}(M_n) - F_{\bm S,j}(g_{j,n}^B) {\overset{{\mathbb{P}}}{\to}}\xi > 0$ as $n,B \to \infty$, or, equivalently, for any ${\varepsilon}> 0$, $${\mathbb{P}}\{ | F_{\bm S,j}(M_n) - F_{\bm S,j}(g_{j,n}^B) - \xi | > {\varepsilon}) \to 0$$ as $n,B \to \infty$. Hence, for any $0 < {\varepsilon}< \xi$, $${\mathbb{P}}( M_n \leq g_{j,n}^B ) \leq {\mathbb{P}}\{ F_{\bm S,j}(M_n) - F_{\bm S,j} (g_{j,n}^B ) \leq 0\} \leq {\mathbb{P}}\{ | F_{\bm S,j}(M_n) - F_{\bm S,j}(g_{j,n}^B) - \xi | > {\varepsilon}) \to 0$$ as $n,B \to \infty$, which completes the proof.
Proof of Theorem {#sec:proof:thm:thres:boot}
=================
Let $\bm S_m = \big( \sup_{t \in I_1} {\mathbb{D}}_m(t), \dots, \sup_{t \in I_p} {\mathbb{D}}_m(t) \big)$ and $$\bm S_m^{[b]} = \left( \sup_{t \in I_1} {\mathbb{D}}_m^{[b]}(t), \dots, \sup_{t \in I_p} {\mathbb{D}}_m^{[b]}(t) \right), \qquad b \in {\mathbb{N}},$$ be the corresponding bootstrap replicates of $\bm S_m$. From and the continuous mapping theorem, we immediately obtain that, under $H_0$, $(\bm S_m, \bm S_m^{[1]}, \bm S_m^{[2]}) \leadsto (\bm S, \bm S^{[1]}, \bm S^{[2]})$ in $({\mathbb{R}}^p)^3$, where $$\bm S = \left( \sup_{t \in I_1} {\mathbb{D}}_F(t), \dots, \sup_{t \in I_p} {\mathbb{D}}_F(t) \right)$$ and $\bm S^{[1]}$ and $\bm S^{[2]}$ are independent copies of $\bm S$. Since $\bm S$ is assumed to have a continuous d.f., the assumptions of Appendix \[sec:aux:thresh:generic\] are satisfied, and, therefore, Lemma \[lem:threshold\] with $n = m$ and $\xi = 1 - (1 - \alpha)^{1/p}$ implies that and hold. The fact that, under $H_0$, ${\mathbb{P}}({\mathbb{D}}_m \leq \tau_m^B) \to 1 - \alpha$ as $m,B \to \infty$ follows from a decomposition similar to the one used in .
To prove the last claim, it suffices to show that, when $\sup_{t \in [1,T+1]} {\mathbb{D}}_m(t) {\overset{{\mathbb{P}}}{\to}}\infty$, ${\mathbb{P}}( {\mathbb{D}}_m \leq \tau_m^B ) \to 0$ as $m, B \to \infty$. Let $T_{i,m} = \sup_{t \in I_i} {\mathbb{D}}_m(t)$, $i \in \{1,\dots,p\}$, which implies that $\sup_{t \in [1,T+1]} {\mathbb{D}}_m(t) = \max_{1 \leq i \leq p} T_{i,m} {\overset{{\mathbb{P}}}{\to}}\infty$. Lemma \[lem:thresh:inf\] with $n = m$ then implies that, as $m,B \to \infty$, $${\mathbb{P}}(T_{1,m} \leq g_{1,m}^B,\dots,T_{p,m} \leq g_{p,m}^B) = {\mathbb{P}}( {\mathbb{D}}_m \leq \tau_m^B ) \to 0,$$ which completes the proof.
The supplement [@KojVer20] contains the proofs of Propositions \[prop:H0\], \[prop:H1\] and \[prop:mult\] as well as additional simulation results.
|
---
author:
- |
Min Xu[^1]Yulong Wei\
[School of Mathematical Sciences, Beijing Normal University]{}\
[Laboratory of Mathematics and Complex Systems, Ministry of Education,]{}\
[Beijing, 100875, China]{}
title: '**The $h$-edge tolerable diagnosability of balanced hypercubes [^2]**'
---
[**Abstract**]{}To measure the fault diagnosis capability of a multiprocessor system with faulty links, Zhu et al. \[Theoret. Comput. Sci. 758 (2019) 1–8\] introduced the $h$-edge tolerable diagnosability. This kind of diagnosability is a generalization of the concept of traditional diagnosability. In this paper, as complement to the results in \[Theoret. Comput. Sci. 760 (2019) 1–14\], we completely determine the $h$-edge tolerable diagnosability of balanced hypercubes $BH_n$ under the PMC model and the MM$^*$ model. Thus, the traditional diagnosability of $BH_n$ is also determined.
[**Keywords**]{}Balanced hypercube; Fault diagnosis; PMC model; MM$^*$ model
0.6cm
Introduction
============
Processor failure has become an ineluctable event in a large-scale multiprocessor system. Thus, to keep the multiprocessor system performing its functions efficiently and economically, recognizing faulty processors correctly is a task of top priority. The process of recognizing faulty processors in a multiprocessor system is called [*fault diagnosis*]{}, and the [*diagnosability*]{} of a system is the maximum number of faulty processors the system can recognize. Historically, many scholars and researchers proposed different models to investigate fault diagnosis. In 1967, Preparata, Metze and Chien [@Pre] proposed the PMC model for fault diagnosis in multiprocessor systems. Under this model, all adjacent processors of a system can test one another. In 1992, by modifying the MM model [@Mae], Sengupta and Dahbura [@Sen] proposed the MM$^*$ model assuming that each processor has to test two processors if the processor is adjacent to the latter two processors. Some references related to fault diagnosis studies under the PMC model or MM$^*$ model can be seen in [@CDH; @FanL; @Gu; @Lai; @Lin17; @Pen; @WanH; @Wei; @Wei18; @Wei186; @Wei188; @XuTH; @XuTZ; @Yang; @Yang13; @YuanL; @ZhaX; @Zho; @ZhuL; @Zha].
In recent literature [@ZhuL], Zhu et al. introduced the [*$h$-edge tolerable diagnosability*]{} $t^e_h(G)$ to measure the fault diagnosis capability of a multiprocessor system $G$ with faulty links. This kind of diagnosability is a generalization of the concept of traditional diagnosability. Specifically, $t^e_h(G)$ is the minimum diagnosability of graphs $G-F_e$ which satisfy that $F_e\subseteq E(G)$ and $|F_e|\leq h$. Note that if a processor $u$ has no fault-free neighbors, it is impossible to determine whether $u$ is faulty or not in the fault diagnosis. Then $t^e_h(G)=0$ for $h\geq k$, where $G$ is a $k$-regular graph. Hence, a key issue for the $h$-edge tolerable diagnosability of a $k$-regular graph study is the case of $0\leq h\leq k$.
Let $C(G)$ be the maximum number of common neighbors of any two vertices in the graph $G$. Wei and Xu determined the $h$-edge tolerable diagnosabilities of regular graphs as follows.
\[I1\] Let $G=(V, E)$ be a $k$-regular triangle-free graph with $k\geq2$. If $C(G)\leq k-1$, then $t_h^e(G)=k-h$ under the PMC model for $0\leq h \leq k$.
\[I2\] Let $G=(V, E)$ be a connected $k$-regular triangle-free graph with $k\geq3$. If $C(G)\leq k-1$, then $$t_h^e(G)=
\left\{
\begin{array}{lll}
2 & \hbox{if $G$ is isomorphic to $G_8$ and $h=0$}; \\
\\
k-1 & \hbox{if $G$ is isomorphic to $G_{k+1, k+1}$ and $h=0$}; \\
\\
k-h & \hbox{otherwise}
\end{array}
\right.$$ under the MM$^*$ model for $0\leq h \leq k$.
In Theorem \[I2\], $G_8$ is the graph with vertex set $V(G_8)=\{x_1, x_2, \ldots, x_8\}$ and edge set $E(G_8)=\{x_ix_{i+1}\mid 1\leq i\leq7\}\cup\{x_8x_1\}\cup\{x_jx_{j+4}\mid 1\leq j\leq4\}$, and $G_{k+1, k+1}$ is the graph with vertex set $V(G_{k+1, k+1})=\{x_1, x_2, \ldots, x_{k+1}, y_1, y_2, \ldots, y_{k+1}\}$ and edge set $E(G_{k+1, k+1})=\{x_iy_j\mid 1\leq i\leq k+1, 1\leq j\leq k+1, i\neq j\}$.
In this paper, we are concerned with the fault diagnosis capability analysis of balanced hypercubes $BH_n$. The $n$-dimensional balanced hypercube $BH_n$, as one of important variants of the well-known hypercubes, was proposed by Wu and Huang [@WuH]. In recent years, $BH_n$ has received considerable attention. For example, Yang [@Yang; @Yang13] studied the conditional diagnosability of $BH_n$ under the PMC model and the MM$^*$ model. Gu et al. [@Gu] determined the $1,2$-good-neighbor diagnosability of $BH_n$ under the PMC model and the MM$^*$ model. Lin et al. [@Lin17] determined the $1,2,3$-extra conditional fault-diagnosability of $BH_n$ under the PMC model. Zhang et al. [@ZhaX] investigated the $(t, k)$-diagnosability of $BH_n$ under the PMC model. Although $BH_n$ is a $2n$-regular and triangle-free graph, $C(BH_n)=2n$. Thus, $BH_n$ does not satisfy the conditions of Theorems \[I1\] and \[I2\]. As complement to Theorems \[I1\] and \[I2\], we establish the $h$-edge tolerable diagnosability of balanced hypercubes $BH_n$ under the PMC model and the MM$^*$ model for $0\leq h \leq 2n$ and $n\geq1$.
Terminology and preliminaries {#2}
=============================
A graph $G =\big(V(G), E(G)\big)$ is used to represent a system (or a network), where each vertex of $G$ represents a processor and each edge of $G$ represents a link. The [*connectivity*]{} $\kappa(G)$ is the minimum cardinality of all vertex subsets $S\subseteq V(G)$ satisfying that $G-S$ is disconnected or trivial. The [*neighborhood*]{} $N_G(v)$ of a vertex $v$ in $G$ is the set of vertices adjacent to $v$. We refer readers to [@Bon] for terminology and notation unless stated otherwise.
In 1997, Wu and Huang proposed balanced hypercubes $BH_n$. We restate the definition of $BH_n$ as follows.
\[D0\] The $n$-dimensional balanced hypercube $BH_n=(V(BH_n), E(BH_n))$ has vertex set $V(BH_n)=\{(a_0, a_1, \ldots, a_i, \ldots, a_{n-1})\mid a_i\in\{0, 1, 2, 3\}, 0\leq i\leq n-1\}$. Each vertex $(a_0, a_1, \ldots, a_{i-1}, a_i, a_{i+1}, \ldots, a_{n-1})$ of $BH_n$ has $2n$ neighbors:
(1)
: $((a_0\pm 1)~{\rm mod~4}, a_1, \ldots, a_{i-1}, a_i, a_{i+1},\ldots, a_{n-1})$,
(2)
: $((a_0\pm 1)~{\rm mod~4}, a_1, \ldots, a_{i-1}, (a_i+(-1)^{a_0})~{\rm mod~4}, a_{i+1},\ldots, a_{n-1})$.
Figure \[DD1\] shows $BH_1$ and $BH_2$.
![Illustration of $BH_1$ and $BH_2$. []{data-label="DD1"}](DD1.pdf "fig:"){width="8cm"}\
Some basic but useful properties of $BH_n$ are presented as follows.
\[pp4\] The balanced hypercube $BH_n$ is bipartite and $\kappa(BH_n)=2n$.
\[pp8\] Let $u$ be an arbitrary vertex of $BH_n$ for $n\geq1$. Then, for an arbitrary vertex $v$ of $BH_n$, either $|N_{BH_n}(u)\cap N_{BH_n}(v)|=0$, $|N_{BH_n}(u)\cap N_{BH_n}(v)|=2$, or $|N_{BH_n}(u)\cap N_{BH_n}(v)|=2n$. Furthermore, there is exactly one vertex $w$ such that $|N_{BH_n}(u)\cap N_{BH_n}(w)|=2n$.
Now, we introduce the definition of the traditional diagnosability of a graph.
\[D3\] A graph $G=(V, E)$ of $n$ vertices is $t$-diagnosable if all faulty vertices can be detected without replacement, provided that the number of faults does not exceed $t$. The diagnosability $t(G)$ of a graph $G$ is the maximum value of $t$ such that $G$ is $t$-diagnosable.
For any two sets $A$ and $B$, we use $A-B$ to denote a set obtained by removing all elements of $B$ from $A$. The [*symmetric difference*]{} of two sets $F_1$ and $F_2$ is defined as the set $F_1\bigtriangleup F_2$ $=(F_1-F_2)\cup (F_2-F_1)$. The following lemmas give necessary and sufficient conditions for a graph to be $t$-diagnosable under the PMC model and the MM$^*$ model.
\[L01\] A graph $G =(V, E)$ is $t$-diagnosable under the PMC model if and only if for any two distinct subsets $F_1$ and $F_2$ of $V$ with $|F_1|\leq t$ and $|F_2|\leq t$, there exists a test from $V-(F_1\cup F_2)$ to $F_1\bigtriangleup F_2$ [(]{}see Figure \[L11\] [)]{}.
![ The illustration of Lemma \[L01\]. []{data-label="L11"}](L1.pdf "fig:"){width="10cm"}\
\[L02\] A graph $G =(V, E)$ is $t$-diagnosable under the MM$^*$ model if and only if for any two distinct subsets $F_1$ and $F_2$ of $V$ with $|F_1|\leq t$ and $|F_2|\leq t$, at least one of the following conditions is satisfied [(]{}see Figure \[L22\] [)]{}:
1. There are two vertices $u, w\in V-(F_1\cup F_2)$ and there is a vertex $v\in F_1\bigtriangleup F_2$ such that $uv\in E$ and $uw\in E$.
2. There are two vertices $u, v\in F_1-F_2$ and there is a vertex $w\in V-(F_1\cup F_2)$ such that $uw\in E$ and $vw\in E$.
3. There are two vertices $u, v\in F_2-F_1$ and there is a vertex $w\in V-(F_1\cup F_2)$ such that $uw\in E$ and $vw\in E$.
![ The illustration of Lemma \[L02\]. []{data-label="L22"}](L2.pdf "fig:"){width="6cm"}\
We call sets $F_1$ and $F_2$ [*distinguishable*]{} under the PMC (resp. MM$^*$) model if they satisfy the condition of Lemma \[L01\] (resp. at least one of the conditions of Lemma \[L02\]); otherwise, $F_1$ and $F_2$ are said to be [*indistinguishable*]{}.
Recently, Zhu et al. introduced the definition of the $h$-edge tolerable diagnosability of graphs as follows.
\[D1\] Given a diagnosis model and a graph $G$, $G$ is $h$-edge tolerable $t$-diagnosable under the diagnosis model if for any edge subset $F_e$ of $G$ with $|F_e|\leq h$, the graph $G-F_e$ is $t$-diagnosable under the diagnosis model. The $h$-edge tolerable diagnosability of $G$, denoted as $t_h^e(G)$, is the maximum integer $t$ such that $G$ is $h$-edge tolerable $t$-diagnosable.
Clearly, $t_0^e(G)=t(G)$ holds for any graph $G$.
Main Results {#3}
============
In this section, we investigate the $h$-edge tolerable diagnosability of a balanced hypercube $BH_n$ under the PMC model and the MM$^*$ model.
Wei and Xu gave an upper bound of the $h$-edge tolerable diagnosability of a $k$-regular graph $G$ under the PMC model and the MM$^*$ model as follows.
\[LL1\] Let $G=(V, E)$ be a $k$-regular graph with $k\geq2$. Then $t_h^e(G)\leq k-h$ under the PMC model and the MM$^*$ model for $0\leq h \leq k$.
Thus, we immediately obtain the upper bound of the $h$-edge tolerable diagnosability of a balanced hypercube $BH_n$ under the PMC model and the MM$^*$ model.
\[LLU\] Let $BH_n$ be an $n$-dimensional balanced hypercube. Then $t_h^e(BH_n)\leq 2n-h$ under the PMC model and MM$^*$ model for $0\leq h \leq 2n$.
Now, we give a lower bound of the $h$-edge tolerable diagnosability of a balanced hypercube $BH_n$ under the MM$^*$ model. In the following statements, for a vertex subset $A$ of a graph $G$, we use $N_G(A)$ to denote the set $\big(\bigcup _{v\in A}N_G(v)\big)-A$.
\[LLL\] Let $BH_n$ be an $n$-dimensional balanced hypercube with $n\geq2$. Then $t_h^e(BH_n)\geq 2n-h$ under the MM$^*$ model for $0\leq h \leq 2n$.
[[*Proof.*]{}]{}For an arbitrary edge subset $F_e\subseteq E(BH_n)$ with $|F_e|\leq h$, suppose that there exist two distinct vertex subsets $F_1, F_2\subseteq V(BH_n)$ such that $F_1$ and $F_2$ are indistinguishable in $BH_n-F_e$ under the MM$^*$ model. We will prove the lemma by showing that $|F_1|\geq 2n-h+1$ or $|F_2|\geq 2n-h+1$. If $|F_1\cap F_2|\geq 2n-h$, then $|F_1|\geq 2n-h+1$ or $|F_2|\geq 2n-h+1$. Now, we assume that $|F_1\cap F_2|\leq 2n-h-1$. Our discussion is divided into two cases as follows.
[*Case 1.*]{} For each vertex $u\in F_1\bigtriangleup F_2$, $N_{BH_n-F_e}(u)-(F_1\cup F_2)=\emptyset$.
In this case, choose a vertex $x\in F_1\bigtriangleup F_2$. Then $N_{BH_n-F_e}(x)\subseteq F_1\cup F_2$ and $|N_{BH_n-F_e}(x)|\geq 2n-h>|F_1\cap F_2|$. Thus, there exists a vertex $y\in F_1\bigtriangleup F_2$ such that $xy\in E(BH_n-F_e)$. We have $N_{BH_n-F_e}(\{x, y\})\subseteq F_1\cup F_2$. By Lemma \[pp4\], we know that $BH_n$ is a bipartite graph. Note that $|N_{BH_n-F_e}(\{x, y\})|-|F_1\cap F_2|\geq(4n-2-h)-(2n-h-1)=2n-1\geq3$ for $n\geq2$. Then, there exists a star $K_{1,3}\subseteq BH_n[F_1\bigtriangleup F_2]-F_e$. Note that $|N_{BH_n}(V(K_{1,3}))|\geq (2n-3)+(2n-1)+(2n-2)=6n-6$ (see Figure \[C1\]).
![Illustration for a lower bound of $|N_{BH_n}(V(K_{1,3}))|$.[]{data-label="C1"}](C1.pdf "fig:"){width="6cm"}\
Since $N_{BH_n-F_e}(V(K_{1,3}))\cup V(K_{1,3})\subseteq F_1\cup F_2$, $$\begin{aligned}
|F_1\cup F_2|&\geq&|N_{BH_n-F_e}(V(K_{1,3}))|+|V(K_{1,3})|\\
&\geq&(6n-6)-h+4\\
&=&(4n-2h+1)+(2n+h-3)\\
&\geq&4n-2h+1.\end{aligned}$$ The last inequality holds for $n\geq2$. Then $|F_1|\geq 2n-h+1$ or $|F_2|\geq 2n-h+1$.
[*Case 2.*]{} $N_{BH_n-F_e}(F_1\bigtriangleup F_2)-(F_1\cup F_2)\neq\emptyset$.
Without loss of generality, suppose that $u\in F_1-F_2$, $v\in V(BH_n)-(F_1\cup F_2)$ such that $uv\in E(BH_n-F_e)$. Note that $F_1$ and $F_2$ are indistinguishable in $BH_n-F_e$ under the MM$^*$ model. Then $N_{BH_n-F_e}(v)-\{u\} \subseteq F_2$ and $|N_{BH_n-F_e}(v)\cap (F_2-F_1)|\leq1$. Thus, $|F_1\cap F_2|\geq |N_{BH_n-F_e}(v)|-2\geq 2n-h-2$. If $|F_1-F_2|\geq3$ or $|F_2-F_1|\geq3$, then $|F_1|\geq 2n-h+1$ or $|F_2|\geq 2n-h+1$.
Next, we suppose that $|F_1-F_2|\leq2$ and $|F_2-F_1|\leq2$.
[*Case 2.1.*]{} For each vertex $w\in F_1\bigtriangleup F_2$, $|N_{BH_n-F_e}(w)-(F_1\cup F_2)|\leq1$.
Note that $N_{BH_n-F_e}(\{u, v\})-(F_1-F_2)\subseteq F_2$. Thus, $$\begin{aligned}
|F_2|&\geq& |N_{BH_n-F_e}(\{u, v\})-(F_1-F_2)|\\
&\geq& (2n-2)+(2n-1)-h\\
&=& (2n-h+1)+(2n-4)\\
&\geq& 2n-h+1\end{aligned}$$ for $n\geq2$. The second inequality holds for $BH_n$ is a bipartite graph.
[*Case 2.2.*]{} For some vertex $w\in F_1\bigtriangleup F_2$, $|N_{BH_n-F_e}(w)-(F_1\cup F_2)|\geq2$.
If there exists a vertex subset $\{v_1, v_2\}\subseteq N_{BH_n-F_e}(w)-(F_1\cup F_2)$ for some vertex $w\in F_1\bigtriangleup F_2$ such that $N_{BH_n}(v_1)\neq N_{BH_n}(v_2)$, then by Lemma \[pp8\], $|N_{BH_n}(v_1)\cap N_{BH_n}(v_2)|=2$. Without loss of generality, assume that $w\in F_1-F_2$. Since $F_1$ and $F_2$ are indistinguishable in $BH_n-F_e$ under the MM$^*$ model, $N_{BH_n-F_e}(v_1)-\{w\}\subseteq F_2$ and $N_{BH_n-F_e}(v_2)-\{w\}\subseteq F_2$ (see Figure \[f1\]). Thus,
![Illustration for $N_{BH_n}(v_1)\neq N_{BH_n}(v_2)$. []{data-label="f1"}](f1.pdf "fig:"){width="6cm"}\
$$\begin{aligned}
|F_2|&\geq& |N_{BH_n-F_e}(\{v_1, v_2\})-\{w\}|\\
&\geq& (2n-1)+(2n-1)-(2-1)-h\\
&=& (2n-h+1)+(2n-4)\\
&\geq& 2n-h+1.\end{aligned}$$
The last inequality holds for $n\geq2$.
Otherwise, by Lemma \[pp8\], for each vertex $w\in F_1\bigtriangleup F_2$, $|N_{BH_n-F_e}(w)-(F_1\cup F_2)|\leq2$. Without loss of generality, assume that $w\in F_1-F_2$ and let $\{v_1, v_2\}=N_{BH_n-F_e}(w)-(F_1\cup F_2)$, where $N_{BH_n}(v_1)=N_{BH_n}(v_2)$ (see Figure \[f2\]). Then $|F_1\cap F_2|\geq2n-2-\left\lfloor\dfrac{h}{2}\right\rfloor$ and $N_{BH_n-F_e}(w)-\{v_1, v_2\}\subseteq F_1\cup F_2$. Note that $(N_{BH_n-F_e}(w)-\{v_1, v_2\})\cap N_{BH_n-F_e}(v_1)=\emptyset$ and $|F_1\cup F_2|\geq |N_{BH_n-F_e}(\{w, v_1\})\cup\{w\}-\{v_2\}|\geq (2n-2)+(2n-1)+1-h=4n-h-2$.
![Illustration for $N_{BH_n}(v_1)=N_{BH_n}(v_2)$. []{data-label="f2"}](f2.pdf "fig:"){width="6cm"}\
Thus, $$\begin{aligned}
|F_1|+|F_2|&=& |F_1\cup F_2|+|F_1\cap F_2|\\
&\geq& (4n-h-2)+(2n-2-\left\lfloor\frac{h}{2}\right\rfloor)\\
&=&(4n-2h+1)+(\left\lceil\frac{h}{2}\right\rceil+2n-5).\end{aligned}$$ Note that $n\geq2$. If $n\geq3$ or $1\leq h\leq2n$, then $|F_1|+|F_2|\geq 4n-2h+1$ which means $|F_1|\geq 2n-h+1$ or $|F_2|\geq 2n-h+1$. Now, we assume that $n=2$ and $h=0$.
If $|F_1-F_2|=1$, then $$\begin{aligned}
|F_2|&\geq& |N_{BH_n-F_e}(\{w, v_1\})-\{v_2\}|\\
&\geq& (2n-2)+(2n-1)-h\\
&=& 2n-h+1\end{aligned}$$
If $|F_1-F_2|=2$ and $|F_1\cap F_2|\geq 2n-1-\left\lfloor\dfrac{h}{2}\right\rfloor$, then $|F_1|\geq2n-h+1$.
If $|F_1-F_2|=2$ and $|F_1\cap F_2|=2n-2-\left\lfloor\dfrac{h}{2}\right\rfloor=2$, then $BH_2-(F_1\cap F_2)$ is connected owing to $|F_1\cap F_2|=2<4=\kappa(BH_2)$ by Lemma \[pp4\]. By the assumption that $|N_{BH_n-F_e}(w)-(F_1\cup F_2)|\leq2$ for each vertex $w$ in $F_1\bigtriangleup F_2$, we have $|N_{BH_2}(F_1\bigtriangleup F_2)-(F_1\cup F_2)|\leq 2\times|F_1\bigtriangleup F_2|\leq8$ and so $|V(BH_2)-(F_1\cup F_2)-N_{BH_2}(F_1\bigtriangleup F_2)|= |V(BH_2)|-(|F_1-F_2|+|F_1\cap F_2|+|F_2-F_1|)-|N_{BH_2}(F_1\bigtriangleup F_2)-(F_1\cup F_2)|\geq 2^4-(2+2+2)-8\geq2$. Thus, there exists a vertex $x$ in $V(BH_2)-(F_1\cup F_2)$ with $x\notin N_{BH_2}(F_1\bigtriangleup F_2)$ connected to a vertex in $F_1\bigtriangleup F_2$ by a path of $BH_2-(F_1\cap F_2)$, which contradicts that $F_1$ and $F_2$ are indistinguishable in $BH_2$ under the MM$^*$ model by Lemma \[L02\].
Thus, $t_h^e(BH_n)\geq 2n-h$ under the MM$^*$ model for $0\leq h \leq 2n$ and $n\geq2$. ${\hfill\Box\medskip}$
Note that under the PMC model, Case 2 of Lemma \[LLL\] is non-existent and the proof of Case 1 of Lemma \[LLL\] also holds for $n\geq2$. Therefore, by Corollary \[LLU\] and Lemma \[LLL\], Theorem \[main2\] holds.
\[main2\] Let $BH_n$ be an $n$-dimensional balanced hypercube with $n\geq2$. Then $t_h^e(BH_n)=2n-h$ under the PMC model and the MM$^*$ model for $0\leq h \leq 2n$.
Now, we determine the $h$-edge tolerable diagnosability of balanced hypercube $BH_1$.
\[r\] Let $BH_1$ be a $1$-dimensional balanced hypercube. Then $$\label{}
t_h^e(BH_1)=
\left\{
\begin{array}{lll}
1, & \hbox{if $0\leq h \leq 1$}; \\
0, & \hbox{if $h=2$}
\end{array}
\right.$$ under the PMC model and $t_h^e(BH_1)=0$ under the MM$^*$ model for $0\leq h \leq 2$.
[[*Proof.*]{}]{}Note that $BH_1$ is isomorphic to a cycle with four vertices. Suppose $V(BH_1)=\{0, 1, 2, 3\}$ and $E(BH_1)=\{01, 12, 23, 30\}$ (see Figure \[DD1\]). Since $BH_1$ is $2$-regular, $t_2^e(BH_1)=0$ under both diagnosis models.
Let $F_1=\{0, 1\}$, $F_2=\{2, 3\}$ and $F_e=\emptyset$. Then $F_1$ and $F_2$ are indistinguishable in $BH_1-F_e$ under the PMC model. Thus, $t_1^e(BH_1)\leq1$ and $t_0^e(BH_1)\leq 1$ under the PMC model.
On the other hand, for an arbitrary edge subset $F_e\subseteq E(BH_1)$ with $|F_e|\leq 1$, suppose that two distinct vertex subsets $F_1, F_2\subseteq V(BH_1)$ satisfy that $|F_1|\leq1$ and $|F_2|\leq1$. Then $F_1\cap F_2=\emptyset$ and $1\leq|F_1\bigtriangleup F_2|=|F_1\cup F_2|\leq2$. Since $BH_1-F_e$ is connected for $|F_e|\leq 1$, $N_{BH_1-F_e}(F_1\bigtriangleup F_2)=N_{BH_1-F_e}(F_1\cup F_2)\neq\emptyset$. Hence, there is an edge between $F_1\bigtriangleup F_2$ and $V(BH_1-F_e)-(F_1\cup F_2)$. By Lemma \[L01\], $t_h^e(BH_1)\geq 1$ for $0\leq h\leq1$ under the PMC model.
Let $F_1=\{0\}$, $F_2=\{2\}$ and $F_e=\emptyset$. Then $F_1$ and $F_2$ are indistinguishable in $BH_1-F_e$ under the MM$^*$ model. Thus, $t_h^e(BH_1)\leq 0$ for $0\leq h\leq1$ under the MM$^*$ model. Hence, $t_1^e(BH_1)=t_0^e(BH_1)=0$ under the MM$^*$ model.
This completes the proof of Theorem \[r\]. ${\hfill\Box\medskip}$
Conclusions {#4}
===========
In this paper, we determine the $h$-edge tolerable diagnosability of balanced hypercubes $BH_n$ under the PMC model and the MM$^*$ model for $0\leq h \leq 2n$ and $n\geq1$ (see Table \[tab:2\]). In particular, the traditional diagnosability of $BH_n$ is determined. Our future research interest is to investigate the $h$-edge tolerable diagnosability of a regular graph with triangles, which will provide a more precise measure for the fault diagnosis capability of a multiprocessor system.
---------- -------------------- ------------------------ ------------------
[MM]{}$^*$[ model]{}
$0\leq h\leq 2n-1$ $h=2n$ $0\leq h\leq 2n$
$n=1$ $1$ $0$ $0$
$n\geq2$ $2n-h$
---------- -------------------- ------------------------ ------------------
: \[tab:2\] The $h$-edge tolerable diagnosability of $BH_n$
[99]{}
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan Press Ltd, New York, 1976.
N.W. Chang, W.H. Deng, S.Y. Hsieh, Conditional diagnosability of (n, k)-star networks under the comparison diagnosis model, [*IEEE Trans. Reliab.*]{}, [**64**]{} (1) (2015), 132–143.
A.T. Dahbura, G.M. Masson, An $O(n^{2.5})$ faulty identification algorithm for diagnosable systems, [*IEEE Trans. Comput.*]{}, [**33**]{} (6) (1984), 486–492.
J. Fan, X. Lin, The $t/k$-diagnosability of the BC graphs, [*IEEE Trans. Comput.*]{}, [**54**]{} (2005), 176–184.
M.M. Gu, R.X. Hao, D.X. Yang, A short note on the $1, 2$-good-neighbor diagnosability of balanced hypercubes, [*J. Interconnect. Netw.*]{}, [**16**]{} (2) (2016), pp. 1650001 (12 pages).
P.L. Lai, J.J.M. Tan, C.P. Chang, L.H. Hsu, Conditional diagnosability measures for large multiprocessor systems, [*IEEE Trans. Comput.*]{}, [**54**]{} (2) (2005), 165–175.
L. Lin, L. Xu, R. Chen, S.Y. Hsieh, D.Wang, Relating extra connectivity and extra conditional diagnosability in regular networks, [*IEEE Trans. Depend. Secure Comput.*]{} (2017), doi: 10.1109/TDSC.2017.2726541.
J. Maeng, M. Malek, A comparison connection assignment for self-diagnosis of multiprocessor systems, in: [*Proceeding of 11th International Symposium on Fault-Tolerant Computing*]{}, 1981, pp. 173–175.
S.L. Peng, C.K. Lin, J.J.M. Tan, L.H. Hsu, The $g$-good-neighbor conditional diagnosability of hypercube under the PMC model, [*Appl. Math. Comput.*]{}, [**218**]{} (21) (2012), 10406–10412.
F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosis systems, [*IEEE Trans. Electron. Comput.*]{}, [**EC-16**]{} (6) (1967), 848–854.
A. Sengupta, A. Dahbura, On self-diagnosable multiprocessor system: diagnosis by the comparison approach, [*IEEE Trans. Comput.*]{}, [**41**]{} (11) (1992), 1386–1396.
S. Wang, W. Han, The $g$-good-neighbor conditional diagnosability of $n$-dimensional hypercubes under the MM$^*$ Model, [*Inform. Process. Lett.*]{}, [**116**]{} (2016), 574–577.
Y. Wei, M. Xu, On $g$-good-neighbor conditional diagnosability of $(n, k)$-star networks, [*Theoret. Comput. Sci.*]{}, [**697**]{} (2017), 79–90.
Y. Wei, M. Xu, The $g$-good-neighbor conditional diagnosability of locally twisted cubes, [*J. Oper. Res. Soc. China*]{}, [**6**]{} (2) (2018), 333–347.
Y. Wei, M. Xu, The $1, 2$-good-neighbor conditional diagnosabilities of regular graphs, [*Appl. Math. Comput.*]{}, [**334**]{} (2018), 295–310.
Y. Wei, M. Xu, Hybrid fault diagnosis capability analysis of regular graphs, [*Theoret. Comput. Sci.*]{}, [**760**]{} (2019), 1–14.
J. Wu, K. Huang, The balanced hypercube: a cube-based system for fault-tolerant applications, [*IEEE Trans. Comput.*]{}, [**46**]{} (4) (1997), 484–490.
M. Xu, K. Thulasiraman, X.D. Hu, Conditional diagnosability of matching composition networks under the PMC model, [*IEEE Trans. Circuits Syst.*]{}, II, Express Briefs [**56**]{} (11) (2009), 875–879.
M. Xu, K. Thulasiraman, Q. Zhu, Conditional diagnosability of a class of matching composition networks under the comparison model, [*Theoret. Comput. Sci.*]{}, [**674**]{} (2017), 43–52.
M.C. Yang, Conditional diagnosability of balanced hypercubes under the PMC model, [*Inform. Sci.*]{}, [**222**]{} (2013), 754–760.
M.C. Yang, Conditional diagnosability of balanced hypercubes under the MM$^*$ model, [*J. Supercomput.*]{}, [**65**]{} (3) (2013), 1264–1278.
J. Yuan, A.X. Liu, X. Ma, X. Qin, J. Zhang, The $g$-good-neighbor conditional diagnosability of $k$-ary $n$-cubes under the PMC model and MM$^*$ model, [*IEEE Trans. Parallel Distrib. Syst.*]{}, [**26**]{} (4) (2015), 1165–1177.
S. Zhang, W. Yang, The $g$-extra conditional diagnosability and sequential $t/k$-diagnosability of hypercubes, [*Int. J. Comput. Math.*]{}, [**93**]{} (3) (2016), 482–497.
X. Zhang, L. Xu, L. Lin, Y. Huang, X. Wang, The $(t, k)$-diagnosability of balanced hypercube under the PMC model, [*Int. J. Comput. Math. Comput. Syst. Theory*]{}, [**3**]{} (4) (2018), 230–243.
S. Zhou, The conditional fault diagnosability of $(n, k)$-star graphs, [*Appl. Math. Comput.*]{}, [**218**]{} (19) (2012), 9742–9749.
Q. Zhu, L. Li, S. Liu, X. Zhang, Hybrid fault diagnosis capability analysis of hypercubes under the PMC model and MM$^*$ model, [*Theoret. Comput. Sci.*]{}, [**758**]{} (2019), 1–8.
[^1]: Corresponding author. [*E-mail address:*]{} xum@bnu.edu.cn (M. Xu).
[^2]: M. Xu’s research is supported by the National Natural Science Foundation of China (11571044, 61373021) and the Fundamental Research Funds for the Central Universities.
|
---
author:
- |
S. Sett, R. P. Sahu, S. Sinha-Ray and A.L. Yarin(ayarin@uic.edu)\
Department of Mechanical and Industrial Engineering,\
University of Illinois at Chicago,\
842 W. Taylor St., Chicago IL 60607-7022
title: '**Gravitational Drainage of Thin Films of Trisiloxane-(Poly)ethoxylate Superspreaders**'
---
![Drainage of (a) superspreader Silwet L-77, and (b) its “cousin”non-superspreader Silwet L-7607.[]{data-label="fig:combined-copy.png"}](./figures/combined-copy.png)
The aqueous solutions of trisiloxane-(poly)ethoxylate surfactants have multiple applications as agricultural adjuvants, solid modifiers, and cleaners, as well as they are widely used in pharmaceutical and cosmetic industries. The trisiloxane surfactants are commonly denoted as M(D$'$E$_n$OMe)M, where M-D$'$-M represents the trisiloxane hydrophobe, E$_n$ is the ethylene oxide, Me stands for the methyl group, and O is oxygen [@Venzmer-2011]. The superspreading ability of these surfactants when added to water drops on surfaces such as Teflon depends on the length of the (poly)ethylene oxide group. In respect to wetting hydrophobic Teflon surfaces, two drastically different “cousin” types of trisiloxane-(poly)ethoxylate surfactants with n=7.5 (which is the superspreader known under the product name Silwet L-77) and n=16 \[which is a non-superspreader, denoted as M(D E$_n$)M under the product name Silwet L-7607\] are recognized [@Venzmer-2011]. However, their effect on gravitational drainage of vertical water films with no contact with any hydrophobic surface has never been studied [@Sett-2013], and is addressed here for the first time.
Thin vertical films were formed from the 0.5 vol% aqueous solutions of Silwet L-77 and Silwet L-7607. The drainage processes of the thin films formed by the superspreader (Silwet L-77) and the non-superspreader (Silwet L-7607) were radically different, both morphologically and by the duration (Fig.\[fig:combined-copy.png\]). The drainage dynamics of the thin films were observed using the microinterferometry technique [@Sett-2013]. The thin films formed from the superspreader were stable for a much longer time in comparison with the ordinary surfactants [@Sett-2013] and the non-superspreader, albeit also much more “vigorous” (Fig.\[fig:combined-copy.png\]). The thinning of the superspreader film was practically arrested at the latter stage, whereas the “cousin” non-superspreader film continued to drain (Fig.\[fig:hvst-copy.png\]).
![Film thickness of (a) Silwet L-77 superspreader, and (b) its “cousin” non-superspreader Silwet L-7607. The data correspond to the film top.[]{data-label="fig:hvst-copy.png"}](./figures/hvst-copy.png)
The thin film formed by the superspreader (Fig.\[fig:combined-copy.png\]a;[Video: Part1]{}) revealed a “turbulent”-like motion visibly different from the marginal regeneration (Fig.\[fig:combined-copy.png\]). On the other hand, the non-superspreader film (Fig.\[fig:combined-copy.png\]b;[Video: Part2]{}) revealed the ordered drainage pattern characteristic of the ordinary surfactants [@Sett-2013]. In addition, the Dynamic Light Scattering (DLS) of the aqueous solution of the superspreader Silwet L-77 revealed the presence of the aggregates of sizes ranging from 1.6 nm to 800 nm, the latter being associated with the bilayer structures assumed for superspreaders in [@Venzmer-2011]. On the contrary, the DLS of the aqueous solution of the non-superspreader Silwet L-7607 revealed the presence of only small aggregates of the size about 3.5 nm, which are rationalized as ordinary micelles [@Venzmer-2011].\
The hydrophilic head group of the non-superspreader Silwet L-7607 is larger than that of the superspreader Silwet L-77, which thus favors the formation of spherical micelles in the former in distinction from the bilayer lamellae formed by the latter. The disjoining pressure associated with fluffy film surfaces resulting from the hanging bilayer lamellae is the physical source of the observed stabilization of the superspreader film drainage (Fig.\[fig:hvst-copy.png\]). In this case the disjoining pressure increases as p$_{disj} \sim 1/h^9$ (h is the film thickness) for 0.2-1 vol% superspreader solutions, as h diminishes in the range below 100 nm.\
Similar results were obtained for another superspreader BREAK-THRU S 278, and its non-superspreader “cousin” BREAK-THRU S 233.\
We are grateful to the United States Gypsum for support of this work.
[9]{} J. Venzmer, *Superspreading-20 years of physicochemical research*. Current Opinion in Colloid&Interface Sci. 16, 335-343 (2011).
S. Sett, S. Sinha-Ray, and A. L. Yarin, *Gravitational drainage of foam films*. Langmuir 29, 4934–4947 (2013).
|
---
abstract: |
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the non-interacting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second order calculation yields s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
\
author:
- |
Arne Neumayr\
[*Institut für Theoretische Physik C, Technische Hochschule Aachen*]{}\
[*D-52056 Aachen, Germany*]{}
- |
Walter Metzner\
[*Max-Planck-Institut für Festkörperforschung, D-70569 Stuttgart, Germany*]{}
title: |
Renormalized perturbation theory for Fermi systems:\
Fermi surface deformation and superconductivity in the two-dimensional Hubbard model
---
0[[**0**]{}]{}
Introduction
============
Unrenormalized perturbation expansions of interacting electron systems around the non-interacting part of the Hamiltonian are generally plagued by infrared divergencies. Some of the divergencies are simply due to to shifts of the Fermi surface, while others signal instabilities of the normal Fermi liquid towards qualitatively different states, such as superconducting or other ordered phases. This problem is often treated by self-consistent resummations of Feynman diagrams, where a finite or infinite subset of skeleton diagrams, with the interacting propagator $G$ on internal lines, is summed.[@BK] Symmetry breaking can be built into the structure of $G$ as an ansatz, and the size of the corresponding order parameter is determined self-consistently. This standard approach has been very useful in many cases. However, resummation schemes beyond first order (Hartree-Fock) require extensive numerics, since the full self-energy has to be determined self-consistently, and delicate low-energy structures cannot always be resolved. A more serious problem is the fact that self-energy and vertex corrections are not treated on equal footing in most feasible resummation schemes. This often leads to unphysical results.
In this work we will describe and apply an alternative procedure, which has been formulated already long ago by Nozières,[@Noz] and more recently been discussed in the mathematical literature as a way of carrying out well-defined perturbation expansions for weakly interacting Fermi systems.[@FMRT; @FKST] The basic idea is to choose an improved starting point for the perturbation expansion, by adding a suitable counterterm to the non-interacting part of the Hamiltonian, and subtracting it from the interaction part. The counterterm is quadratic in the Fermi operators and has to be determined from a self-consistency condition. In Sec. II we will describe how Fermi surface deformations and superconductivity can be treated by this method. Explicit expressions up to second order in the interaction are derived for the case of the Hubbard model in Sec. III. Results obtained from a numerical solution of the self-consistency equations in two dimensions will follow in Sec. IV. For the repulsive Hubbard model we have obtained superconducting solutions with d-wave symmetry in agreement with widespread expectations,[@Sca] and with recent renormalization group calculations which conclusively established d-wave superconductivity at weak coupling.[@RG] In addition, for Fermi levels close to the van Hove singularity, deformations which break the square lattice symmetry occur. This confirms the recently proposed possibility of symmetry-breaking Fermi surface deformations (”Pomeranchuk instabilities”).[@HM; @YK; @GKW]
Renormalized perturbation expansion
===================================
We consider a system of interacting spin-$\frac{1}{2}$ fermions with a Hamiltonian $H = H_0 + H_I$, where the non-interacting part $$H_0 = \sum_{\bk,\sg} \xi_{\bk} \, n_{\bk\sg}$$ with $\xi_{\bk} = \eps_{\bk} - \mu$ contains the kinetic energy and the chemical potential, while $H_I$ is a fermion-fermion interaction term. We are particularly interested in lattice systems, for which the dispersion relation $\eps_{\bk}$ is not isotropic. We consider only ground state properties, that is the temperature is zero throughout the whole article.
The bare propagator in a standard many-body perturbation expansion [@NO] around $H_0$ is given by $$G_0(k) = \frac{1}{i\om - \xi_{\bk}} \; ,$$ where $\om$ is the Matsubara frequency and $k = (\om,\bk)$. This progagator diverges for $\om \to 0$ and $\bk \to \bk_F$, for any Fermi momentum $\bk_F$, since $\xi_{\bk_F} = 0$. As a consequence, many Feynman diagrams diverge. A well-known singularity is the (usually) logarithmic divergency of the 1-loop particle-particle contribution to the two-particle vertex in the Cooper channel, which leads to a $(\log)^n$ divergency of the n-loop particle-particle ladder diagram. This signals a possible Cooper instability towards superconductivity. Much stronger divergencies occur in diagrams with multiple self-energy insertions on the same internal propagator line, leading to non-integrable powers of $G_0(k)$.[@FMRT; @FKST] These singularities are due to Fermi surface shifts generated by the interaction term in the Hamiltonian.
The divergency problems and the superconducting instability can be treated by splitting the Hamiltonian in a different way, namely as[@Noz] $$H = \tH_0 + \tH_I \; ,$$ where $\tH_0 = H_0 + \delta H_0$ and $\tH_I = H_I - \delta H_0$, and expanding around $\tH_0$. The [*counterterm*]{} $\delta H_0$ must be quadratic in the creation and annihilation operators to allow for a straightforward perturbation expansion based on Wick’s theorem. It is possible to chose $\delta H_0$ such that $\tH_I$ does not shift the Fermi surface corresponding to $\tH_0$ any more, and divergencies due to self-energy insertions are removed. In the superconducting state spontaneous symmetry breaking can be included already in $\delta H_0$, with an order parameter $\Delta_{\bk}$ whose value on the Fermi surface is not shifted by $\tH_I$. We will now describe this procedure in more detail.
Normal state
------------
A counterterm $\delta H_0 = \sum_{\bk,\sg} \delta\xi_{\bk} \, n_{\bk\sg}$ leads to a renormalized dispersion relation $\txi_{\bk} = \xi_{\bk} + \delta\xi_{\bk}$ in the unperturbed part of the Hamiltonian, $$\tH_0 = \sum_{\bk,\sg} \txi_{\bk} \, n_{\bk\sg} \; ,$$ and correspondingly to a new bare propagator $$\tG_0(k) = \frac{1}{i\om - \txi_{\bk}} \; .$$ The Fermi surface $\tilde{\cal F}$ associated with $\tH_0$ is given by the momenta $\tilde\bk_F$ satisfying the equation $\txi_{\bk} = 0$. The Fermi surface of the interacting system is given by the solutions of the equation $G^{-1}(0,\bk) = 0$. This surface coincides with the unperturbed one, corresponding to $\tH_0$, if the [*renormalized*]{} self-energy $\tSg = \tG_0^{-1} - G^{-1}$ vanishes on $\tilde {\cal F}$, that is if $$\tSg(0,\bk) = 0 \quad \mbox{for} \quad \bk \in {\tilde {\cal F}}
\; .$$ This imposes a self-consistency condition on the counterterms which can be solved iteratively. For isotropic systems the shift of $\xi_{\bk}$ can be chosen as a momentum independent constant, which may be interpreted as a shift of the chemical potential. For anisotropic systems, however, one generally has to adjust the whole shape of the Fermi surface. That this procedure really works at each order of the perturbation expansion has been shown rigorously for a large class of systems.[@FST]
The shift function $\delta\xi_{\bk}$ is uniquely determined by the self-consistency condition only on the (interacting) Fermi surface $\tilde {\cal F}$. For momenta away from the Fermi surface, $\delta\xi_{\bk}$ can be chosen to be any sufficiently smooth function of $\bk$ which does not lead to artificial additional zeros of $\txi_{\bk}$.
The perturbation expansion of the renormalized self-energy $\tSg$ involves two types of vertices: the usual two-particle vertex given by the interaction $H_I$ and a one-particle vertex due to the counterterm $-\delta H_0$ in $\tH_I$. In Fig. 1 we show the Feynman diagrams contributing to $\tSg$ up to second order in the interaction.
The above-mentioned divergencies of Feynman diagrams with self-energy insertions on internal propagator lines are removed in the renormalized expansion around $\tH_0$, since in products $\tG_0 \tSg \tG_0 \dots \tSg \tG_0$ only one simple pole at $k=(0,\tilde\bk_F)$ survives, all other poles being cancelled by the corresponding zeros of the self-energy $\tSg$.
Superconducting state
---------------------
To treat superconducting states we also add counterterms containing Cooper pair creation and annihilation operators, in addition to a shift of $\xi_{\bk}$. We consider only spin singlet pairing, but triplet pairing could be dealt with analogously. We thus expand around a BCS mean-field Hamiltonian $$\tH_0 = \sum_{\bk,\sg} \txi_{\bk} \, n_{\bk\sg} +
\sum_{\bk} \left[
\Delta_{\bk} \, a^{\dag}_{-\bk\down} a^{\dag}_{\bk\up} +
\Delta^*_{\bk} \, a_{\bk\up} a_{-\bk\down} \right] \; ,$$ where $\Delta_{\bk}$ is the gap-function, which has to be determined self-consistently. In terms of Nambu operators $$\Psi_{\bk} = \left( \begin{array}{l}
a_{\bk\up} \\ a^{\dag}_{-\bk\down} \end{array} \right)
\quad \mbox{and} \quad
\Psi^{\dag}_{\bk} = \left(
a^{\dag}_{\bk\up} \, , \, a_{-\bk\down} \right)$$ one can rewrite $\tH_0$ in more compact form as $$\tH_0 = \sum_{\bk} \txi_{\bk} \,
\Psi^{\dag}_{\bk} \, \sg_3 \, \Psi_{\bk} -
\sum_{\bk} \Psi^{\dag}_{\bk} \,
( \Delta'_{\bk} \, \sg_1 - \Delta''_{\bk} \, \sg_2 ) \, \Psi_{\bk}
\; ,$$ where $\sg_1,\sg_2,\sg_3$ are the Pauli matrices, and $\Delta'_{\bk}$ ($\Delta''_{\bk}$) is the real (imaginary) part of $\Delta_{\bk}$. The two expressions (7) and (9) for $\tH_0$ differ by the constant (c-number) $\sum_{\bk} \txi_{\bk}$, which must be taken into account only when absolute energies are computed. The bare Nambu matrix propagator ${\bf\tG}_0 = - \bra \Psi \Psi^{\dag} \ket_{\tilde 0}$ following from $\tH_0$ is given by $$\btG_0^{-1}(k) = \left( \begin{array}{cc}
i\om - \txi_{\bk} & \Delta_{\bk} \\
\Delta^*_{\bk} & i\om + \txi_{\bk} \end{array} \right) \; .$$ Extending the self-consistency condition for the normal state, we now require that the matrix self-energy $\btSg = \btG_0^{-1} - \bG^{-1}$ vanishes on the Fermi surface (defined by $\txi_{\bk} = 0$), that is $$\btSg(0,\bk) = 0 \quad \mbox{for} \quad \bk \in {\tilde {\cal F}}
\; .$$ Thus, for $\om=0$ and $\bk$ on the Fermi surface, neither the diagonal nor the off-diagonal elements of $\btG_0^{-1}(k)$ are shifted by the interaction term $\tH_I$. The Feynman diagrams in Fig. 1 apply also to the superconducting case, if lines are interpreted as Nambu matrix propagators.
The above renormalized perturbation theory is reminiscent of the perturbation theory for symmetry broken phases formulated by Georges and Yedidia,[@GY] where an order-parameter-dependent free energy function is constructed by adding Onsager reaction terms to the mean field contributions, and the actual order parameter is determined by minimizing this free energy.
Application to the Hubbard model
================================
In this section we derive explicit expressions for the self-energy and the counterterms for the ground state $(T=0)$ of the Hubbard model, up to second order in the renormalized perturbation expansion.
The one-band Hubbard model [@Mon] $$H = \sum_{\bi,\bj} \sum_{\sg} t_{\bi\bj} \,
c^{\dag}_{\bi\sg} c_{\bj\sg} +
U \sum_{\bj} n_{\bj\up} n_{\bj\down} -
\mu N$$ describes lattice electrons with a hopping amplitude $t_{\bi\bj}$ and a local interaction $U$. Here $c^{\dag}_{\bi\sg}$ and $c_{\bi\sg}$ are creation and annihilation operators for electrons with spin projection $\sg$ on a lattice site $\bi$, and $n_{\bj\sg} = c^{\dag}_{\bi\sg} c_{\bi\sg}$. Note that we have included the term $\mu N$ with the total particle number operator $N$ in our definition of $H$. The non-interacting part of $H$ can be written in momentum space as $H_0 = \sum_{\bk} \xi_{\bk} \, n_{\bk\sg}$ where $\xi_{\bk} = \eps_{\bk} - \mu$ and $\eps_{\bk}$ is the Fourier transform of $t_{\bi\bj}$.
Our numerical results will be given for the Hubbard model on a square lattice with a hopping amplitude $-t$ between nearest neighbors and a much smaller amplitude $-t'$ between next-nearest neighbors. The corresponding dispersion relation is $$\eps_{\bk} = -2t(\cos k_x + \cos k_y) - 4t' \cos k_x \cos k_y
\; .$$ We now derive expressions for the self-energy and the resulting self-consistency equations up to second order in $U$.
Normal state
------------
### First order
To first order in $U$ the self-energy is obtained as $$\tSg^{(1)}(k) = U \int_{k'} \tG_0(k') \, e^{i\om' 0^+}
\, - \, \delta\xi_{\bk} \; ,$$ where $\int_k$ is a short-hand notation for the frequency and momentum integral, including the usual factors of $(2\pi)^{-1}$ for each integration variable. The first term results from diagram (1a) in Fig. 1, the second from diagram (1b). Note that the tadpole diagram (1a) yields a k-independent contribution, since the Hubbard interaction is local. The self-consistency condition (6) for $\tSg^{(1)}$ yields, after carrying out the $\om'$-integration, $$\delta\xi_{\bk} =
U \int \frac{d^dk'}{(2\pi)^d} \,
\Theta(\mu - \eps_{\bk'} - \delta\xi_{\bk'}) \; ,$$ to be satisfied (at least) for $\bk \in \tilde{\cal F}$. Since the right hand side of this condition is a constant, it is natural to define $\delta\xi_{\bk}$ by this constant for all $\bk$. Using Luttinger’s theorem one can identify the above momentum integral with the particle density per spin, such that $\delta\xi_{\bk} = Un/2$, where $n$ is the total density. The self-consistency condition thus yields the $n(\mu)$ relation of the interacting system. Since the counterterm can be chosen k-independently at first order, it may be interpreted as a shift of the chemical potential.
### Second order
The diagrams (2b) and (2c) from Fig. 1 obviously cancel each other to the extent that the first order diagrams (1a) and (1b) cancel. Writing $\delta\xi_{\bk} = \delta\xi^{(1)} + \delta\xi^{(2)}_{\bk}$ with $\delta\xi^{(1)}$ given by the constant on the right hand side of Eq. (15), such that $\delta\xi^{(2)}_{\bk}$ is of order $U^2$ for all $\bk$, the sum of contributions from (2b) and (2c) is of order $U^3$ and can thus be ignored at second order. Hence, only diagram (2a) contributes to the second order self-energy. Using the Feynman rules [@NO] one obtains $$\tSg^{(2)}(k) = U^2 \int_q \tPi_0(q) \, \tG_0(k-q) \; ,$$ where $\tPi_0(q) = - \int_{k'} \tG_0(k') \, \tG_0(k'+q)$. Adding first and second order terms, one arrives at the second order self-consistency condition $$\delta\xi_{\bk} =
U \int \frac{d^dk'}{(2\pi)^d} \,
\Theta(-\txi_{\bk'}) \, + \, \tSg^{(2)}(0,\bk) \; .$$ The counterterm $\delta\xi_{\bk}$ has to be chosen such that the above equation is satisfied for all $\bk \in \tilde{\cal F}$, that is for all $\bk$ satisfying $\txi_{\bk} = 0$. Since $\tSg^{(2)}(0,\bk)$ is momentum dependent, $\delta\xi_{\bk}$ cannot be chosen constant any more. As a consequence, the Fermi surface of the interacting system will be deformed by interactions, even if the volume of the Fermi sea is kept fixed. Luttinger’s theorem can be used to determine the density from the volume of the Fermi sea as $n = 2 \int \frac{d^dk}{(2\pi)^d} \, \Theta(-\txi_{\bk})$.
Superconducting state
---------------------
For the matrix elements of the Nambu propagator we use the standard notation $$\bG(k) = \left( \begin{array}{cc}
G(k) & F(k) \\ F^*(k) & -G(-k)
\end{array} \right) \; ,$$ and the analogous expression for $\btG_0(k)$. The matrix elements of the self-energy are denoted by $$\btSg(k) = \left( \begin{array}{cc}
\tSg(k) & \tS(k) \\ \tS^*(k) & -\tSg(-k)
\end{array} \right) \; .$$
### First order
In the presence of an off-diagonal counterterm $\Delta_{\bk} \,$, the diagonal part of $\btSg$ is still given by Eq. (14) to first order, where $\tG_0(k)$ now depends on the gap function: $$\tG_0(k) = - \frac{i\om + \txi_{\bk}}
{\om^2 + \txi_{\bk}^2 + |\Delta_{\bk}|^2} \; .$$ The first order self-consistency relation (15) thus generalizes to $$\delta\xi_{\bk} = U \int \frac{d^dk'}{(2\pi)^d} \,
\frac{1}{2} \, \big( 1 - \txi_{\bk'}/E_{\bk'} \big) \; ,$$ with $E_{\bk} = \sqrt{\txi_{\bk}^2 + |\Delta_{\bk}^2|}$. Note that the above integral is the BCS formula for the average particle density per spin.
The off-diagonal matrix element of $\btSg$ is obtained from diagrams (1a) and (1b) in Fig. 1 as $$\tS^{(1)}(k) = - U \int_{k'} \tF_0(k') \, + \, \Delta_{\bk}$$ to first order in $U$, with $$\tF_0(k) = \frac{\Delta_{\bk}}
{\om^2 + \txi_{\bk}^2 + |\Delta_{\bk}|^2} \; .$$ The off-diagonal part of the self-consistency condition (11) follows as $$\Delta_{\bk} = - U \int \frac{d^dk'}{(2\pi)^d} \,
\frac{\Delta_{\bk'}}{2E_{\bk'}} \; .$$ Extended as a condition for all $\bk$ (and not just on $\tilde{\cal F}$) this is nothing but the gap equation for the Hubbard model as obtained by standard BCS theory. The self-consistency relation requires that $\Delta_{\bk}$ be constant on the Fermi surface, such that one naturally chooses a constant $\Delta_{\bk} = \Delta$ as an ansatz for all $\bk$. A non-trivial solution $\Delta \neq 0$ of this gap equation can obviously be obtained only for the attractive Hubbard model ($U < 0$).
### Second order
The diagrams (2b) and (2c) cancel each other for the same reason as in the normal state. The contribution from diagram (2a) to the diagonal part of the self-energy is still given by formula (16), with $\tG_0$ from Eq. (20) and $$\tPi_0(q) = - \int_{k'} \left[
\tG_0(k') \, \tG_0(k'+q) + \tF_0(k') \, \tF^*_0(k'+q)
\right] \; .$$ The second order contribution to the off-diagonal matrix element of $\btSg$ is $$\tS^{(2)}(k) = U^2 \int_q \tPi_0(q) \, \tF_0(k-q) \; .$$ The self-consistency relations read: $$\begin{aligned}
\delta\xi_{\bk} &=& U \int \frac{d^dk'}{(2\pi)^d} \,
\frac{1}{2} \, \big( 1 - \txi_{\bk'}/E_{\bk'} \big)
\, + \, \tSg^{(2)}(0,\bk) \; , \\[2mm]
\Delta_{\bk} &=& - U \int \frac{d^dk'}{(2\pi)^d} \,
\frac{\Delta_{\bk'}}{2E_{\bk'}}
\, - \, \tS^{(2)}(0,\bk) \; .\end{aligned}$$ In Appendix A we present more explicit expressions for $\tSg^{(2)}(0,\bk)$ and $\tS^{(2)}(0,\bk)$, obtained by carrying out the frequency integrals.
Numerical solution
------------------
The self-consistency conditions are non-linear equations for the counterterms $\delta\xi_{\bk}$ and, in the superconducting state, $\Delta_{\bk}$. The Fermi surface of the interacting system, $\tilde{\cal F}$, on which the self-consistency conditions must be satisfied, is not known a priori. The equations involve one momentum integral at first order, and two momentum integrals at second order. Such a non-linear system can only be solved iteratively. In this subsection we describe some details of our algorithm.
Since the counterterms are determined by the self-consistency conditions only on the Fermi surface, their momentum dependence away from $\tilde{\cal F}$ can be parametrized in many ways. We have chosen $\delta\xi_{\bk}$ and $\Delta_{\bk}$ as constant along the straight lines connecting the square shaped line defined by the condition $|k_x| + |k_y| = \pi$ with the points $(0,0)$ and $(\pi,\pi)$ of the Brillouin zone, respectively (see Fig. 2). For a numerical solution the remaining tangential momentum dependence is discretized by up to 256 points.
The iteration procedure starts with a tentative choice of counterterms. To be able to reach a symmetry broken solution one usually has to offer at least a small symmetry breaking counterterm in the beginning.[@fn1] In each iteration step new counterterms are determined via Eq. (17) in the normal state, and by Eqs. (27) and (28) for the superconducting state. The right hand side of these equations is evaluated using the counterterms obtained in the previous step, and $\bk$ is chosen on the Fermi surface defined by the previous $\delta\xi_{\bk}$. The momentum integrals are carried out using a Monte-Carlo routine. The iteration is continued until convergence is achieved, that is until the counterterms remain invariant within numerical accuracy from step to step. In all cases studied different choices of initial counterterms lead to the same unique solution. The symmetry breaking terms are much larger than the stochastic noise from the Monte-Carlo routine in all results shown.
The density is kept fixed by adjusting the chemical potential during the iteration procedure. To avoid a higher numerical effort we have computed the density from the Fermi surface volume in the normal state (justified by Luttinger’s theorem), and from the BCS formula for the density in superconducting solutions. The latter reduces to the Fermi surface volume in the normal state limit, such that the potential error of this approximation is very small as long as the gap is small.
Results
=======
We now discuss the most interesting results obtained within the renormalized perturbation theory described above, focussing mainly on the repulsive Hubbard model $(U>0$), for which we have found superconducting solutions with d-wave symmetry, as well as symmetry-breaking Fermi surface deformations.
Repulsive Hubbard model
-----------------------
The following results for the repulsive Hubbard model have been computed for the parameters $t' = -0.15t$ and $U = 3t$. The interaction is thus in the weak to intermediate coupling regime. For too small $U$-values it becomes very hard to resolve the small superconducting gap in the numerical solution.
We have solved the self-consistency equations for various densities ranging from $n = 0.88$ to $n = 0.90$, for which the Fermi surfaces are quite close to the saddle points of the bare dispersion relation $\eps_{\bk}$, located at $(\pi,0)$ and $(0,\pi)$. In all cases the normal state is unstable towards superconductivity. The gap function in the superconducting state obtained from the self-consistency equations has $d_{x^2-y^2}$-wave shape, with slight deviations from perfect d-wave symmetry in cases where the Fermi surface breaks the symmetry of the square lattice. This is in agreement with widespread expectations for the Hubbard model,[@Sca] and in particular with recent renormalization group arguments and calculations.[@RG] In Fig. 3 we show the gap functions obtained at the densities $n=0.88$ and $n=0.9$, respectively. We note that the size of the gap is roughly one order of magnitude smaller than the critical cutoff scale $\Lam_c$ at which Cooper pair susceptibilities diverge in 1-loop renormalization group calculations for comparable model parameters.[@RG] There are various possible reasons for this quantitative discrepancy. First, and probably most importantly, the enhancement of effective interactions due to fluctuations, especially antiferromagnetic spin fluctuations, is captured much better by a renormalization group calculation. Second, the approximate Fermi surface projection of vertices driving the renormalization group flow can lead to an overestimation of effective interactions and hence of critical energy scales. Furthermore, a renormalization group calculation within the symmetry broken phase could yield a gap that is somewhat smaller than $\Lam_c$.
While superconductivity is the only possible instability of the normal Fermi liquid state in the weak coupling limit (except for the case of perfect nesting at half-filling), at higher $U$ one should also consider the possibility of other, in particular magnetic, instabilities. This could be done within renormalized perturbation theory by allowing for counterterms introducing magnetic or charge order.
The Fermi surface is always deformed by interactions. The shifts generated by the momentum dependence of the counterterm $\delta\xi_{\bk}$ are not very large. They are more pronounced near the saddle points of $\eps_{\bk}$, where small energy shifts lead to relatively large shifts in k-space. However, the results presented in Fig. 4 show that the Fermi surface of the interacting system can nevertheless differ strikingly from the bare one. For the densities $n=0.88 - 0.889$ the Fermi surface of the interacting system obviously breaks the point group symmetry of the square lattice. For $n=0.88$ and $n=0.888$ even the topology of the Fermi surface is changed by interactions. The deformed surface has open topology in these cases, instead of being closed around the points $(0,0)$ or $(\pi,\pi)$ in the Brillouin zone. Note that the symmetry-broken Fermi surfaces shown here correspond to stable solutions of the self-consistency equations for the counterterms, while symmetric solutions are unstable.
More details about the Fermi surface shifts can be extracted from a plot of the second order counterterms, shown in Fig. 5. The actual shifts are determined by these terms plus a constant due to the first order counterterm and a shift of the chemical potential. At fixed density the interaction shifts the Fermi surface outwards at points where $\tSg(0,\tilde\bk_F)$ has an absolute minimum, and inwards at points corresponding to absolute maxima. Interactions thus reduce the curvature of the Fermi surface near the diagonals in the Brillouin zone. Fig. 5 reveals that the Fermi surface deformation is slightly asymmetric also for $n=0.9$, but the symmetry breaking is too small to be seen in Fig. 4.
If the Fermi surface breaks the square lattice symmetry, the gap function $\Delta_{\bk}$ cannot have pure d-wave symmetry any more. See, for example, the gap function at density $n=0.88$ in Fig. 3. The deviation from perfect d-wave form is however quite small, since the symmetry breaking Fermi surface deformation is small.
Interaction-induced Fermi surface deformations which break the symmetry of the square lattice have already been discussed earlier in the literature. Yamase and Kohno [@YK] have obtained symmetry-broken Fermi surfaces within a slave boson mean-field theory for the t-J model. The effective interactions obtained from 1-loop renormalization group flows for the Hubbard model also favor symmetry-breaking Pomeranchuk instabilities of the Fermi surface, if the latter is close to the van Hove points.[@HM] A systematic stability analysis of the Hubbard model using Wegner’s Hamiltonian flow equation method confirmed that symmetry breaking Fermi surface deformations are among the strongest instabilities.[@GKW] It remained an open question, however, whether such Fermi surface instabilities would be cut off by the superconducting gap. We have observed within our renormalized perturbation theory that symmetry breaking Fermi surface deformations occur indeed more easily, if the system is forced to stay in a normal state, by setting $\Delta_{\bk} = 0$. Whether a symmetry broken Fermi surface and superconductivity coexist can be seen only by performing a calculation within the symmetry-broken state. This has not yet been done using the renormalization group or flow equation methods.
From a pure symmetry-group point of view the symmetry breaking generated by the Pomeranchuk instability is equivalent to that in ”nematic” electron liquids, first discussed by Kivelson et al.[@KFE]. These authors considered doped Mott insulators, that is [*strongly*]{} interacting systems. A general theory of orientational symmetry-breaking in fully isotropic (not lattice) two- and three-dimensional Fermi liquids has been reported by Oganesyan et al.[@OKF] Superconducting nematic states, in which discrete orientational symmetry breaking develops in addition to d-wave superconductivity, have been considered recently by Vojta et al.[@VZS] Motivated by experimental properties of single-particle excitations in cuprate superconductors they performed a general classification and field-theoretic analysis of various phases with an additional order parameter on top of $d_{x^2-y^2}$-pairing.
Attractive Hubbard model
------------------------
For the attractive Hubbard model ($U<0$) the renormalized perturbation expansion yields s-wave superconductivity already at first order, which is equivalent to BCS mean-field theory.[@MRR] At this level the gap function is constant in k-space. Extending the calculation to second order, a weak momentum dependence of $\Delta_{\bk}$ is generated, as seen in Fig. 6 for the parameters $U = -2t$, $t' = -0.15t$ and $n=0.9$. More importantly, the overall size of the gap is strongly reduced by fluctuations included in the second order terms. The average gap in Fig. 6 is only one third of the corresponding mean-field gap. It has been pointed out previously that fluctuations not contained in mean-field theory reduce the size of magnetic and other order parameters even in the weak coupling limit.[@GY; @Don]
Conclusion
==========
In summary, we have formulated a renormalized perturbation theory for interacting Fermi systems, which treats Fermi surface deformations and superconductivity via additive counterterms. This method is very convenient for studying the role of fluctuations for spontaneous symmetry breaking in a controlled weak-coupling expansion. A concrete application of the expansion carried out to second order yields several non-trivial results for the two-dimensional Hubbard model. In particular, for the repulsive model we have obtained the gap function of the expected d-wave superconducting state and, for Fermi levels close to the van Hove energy, an interacting Fermi surface with broken lattice symmetry, and in some cases even open topology. The symmetry-breaking pattern of the states with symmetry-broken Fermi surfaces is equivalent to that of ”nematic” electron liquids discussed already earlier from a different point of view.[@KFE; @VZS]
The present work can be extended in several interesting directions. After fixing the counterterms one can compute the full momentum and energy dependence of the self-energy, and hence the spectral function for single-particle excitations. At second order the combined effects of symmetry breaking and quasi-particle decay are captured. Allowing for other symmetry-breaking counterterms, for example spin density waves, one can study the competition of magnetic, charge, and superconducting instabilities, as well as their possible coexistence. Finally, the formalism can be extended to finite temperature. In that case the singularities of the bare propagator are cut off by the smallest Matsubara frequency, but Fermi surface shifts and symmetry breaking can still be conveniently taken into account by counterterms.
[**Acknowledgements:**]{}\
We are grateful to A. Georges, M. Keller, D. Rohe and M. Salmhofer for valuable discussions, and to D. Rohe also for a critical reading of the manuscript.
Frequency integrals
===================
The Matsubara frequency integrals in the second order self-energy contributions can be carried out analytically by using the residue theorem. We only present the results for the superconducting case; the normal state results can be recovered by setting $\Delta_{\bk} = 0$ in the following expressions.
The frequency integrals relevant for the evaluation of $\tPi_0$ defined by Eq. (25) are $$\begin{aligned}
\int \frac{dk_0}{2\pi} \, \tG_0(k) \, \tG_0(k+q) & = &
\frac{E_{\bk} + E_{\bk+\bq}}{2 E_{\bk} E_{\bk+\bq}} \,
\frac{\txi_{\bk} \txi_{\bk+\bq} - E_{\bk} E_{\bk+\bq}}
{q_0^2 + [E_{\bk} + E_{\bk+\bq}]^2} \nonumber \\
& + &
\frac{iq_0}{2 E_{\bk} E_{\bk+\bq}} \,
\frac{\txi_{\bk} E_{\bk+\bq} - E_{\bk} \txi_{\bk+\bq}}
{q_0^2 + [E_{\bk} + E_{\bk+\bq}]^2}\end{aligned}$$ and $$\int \frac{dk_0}{2\pi} \, \tF_0(k) \, \tF^*_0(k+q) =
\frac{E_{\bk} + E_{\bk+\bq}}{2 E_{\bk} E_{\bk+\bq}} \,
\frac{\Delta_{\bk} \Delta^*_{\bk+\bq}}
{q_0^2 + [E_{\bk} + E_{\bk+\bq}]^2} \; .$$ The imaginary part of $\tPi_0$ does not contribute to $\btSg(0,\bk)$. Carrying out the $q_0$-integral in Eqs. (16) and (26) yields: $$\begin{aligned}
\tSg^{(2)}(0,\bk) & = & - U^2 \int_{\bq} \int_{\bk'}
\txi_{\bk-\bq} \, C(\bk,\bk',\bq) \\[2mm]
\tS^{(2)}(0,\bk) & = & U^2 \int_{\bq} \int_{\bk'}
\Delta_{\bk-\bq} \, C(\bk,\bk',\bq) \; ,\end{aligned}$$ where $$C(\bk,\bk',\bq) =
\frac{E_{\bk'} E_{\bk'+\bq} - \txi_{\bk'} \txi_{\bk'+\bq} -
\Delta_{\bk'} \Delta^*_{\bk'+\bq}}
{4 E_{\bk-\bq} E_{\bk'} E_{\bk'+\bq} \,
\big[ E_{\bk-\bq} + E_{\bk'} + E_{\bk'+\bq} \big]} \; .$$
[99]{}
G. Baym and L.P. Kadanoff, Phys. Rev. [**124**]{}, 287 (1961); G. Baym, Phys. Rev. [**127**]{}, 1391 (1962).
P. Nozières, [*Theory of Interacting Fermi Systems*]{} (Benjamin, New York, 1964).
J. Feldman, J. Magnen, V. Rivasseau, and E. Trubowitz, in [*The State of Matter*]{}, edited by M. Aizenmann and H. Araki, Advanced Series in Mathematical Physics Vol. 20 (World Scientific 1994).
J. Feldman, H. Knörrer, M. Salmhofer, and E. Trubowitz, J. Stat. Phys. [**94**]{}, 113 (1999).
See, for example, D.J. Scalapino, Phys. Rep. [**250**]{}, 329 (1995); D.J. Scalapino and S.R. White, Found. Phys. [**31**]{}, 27 (2001).
D. Zanchi and H.J. Schulz, Phys. Rev. B [**61**]{}, 13609 (2000); C.J. Halboth and W. Metzner, Phys. Rev. B [**61**]{}, 7364 (2000); C. Honerkamp, M. Salmhofer, N. Furukawa, and T.M. Rice, Phys. Rev. B [**63**]{}, 035109 (2001).
C.J. Halboth and W. Metzner, Phys. Rev. Lett. [**85**]{}, 5162 (2000).
H. Yamase and H. Kohno, J. Phys. Soc. Jpn. [**69**]{}, 332 (2000); [**69**]{}, 2151 (2000).
I. Grote, E. Körding, and F. Wegner, J. Low Temp. Phys. [**126**]{}, 1385 (2002); V. Hankevych, I. Grote and F. Wegner, condmat/0205213; V. Hankevych and F. Wegner, cond-mat/0207612.
See, for example, J.W. Negele and H. Orland, [*Quantum Many-Particle Systems*]{} (Addison-Wesley, Reading, MA, 1987).
J. Feldman, M. Salmhofer, and E. Trubowitz, J. Stat. Phys. [**84**]{}, 1209 (1996); Commun. Pure Appl. Math. [**51**]{}, 1133 (1998); Commun. Pure Appl. Math. [**52**]{}, 273 (1999).
A. Georges and J.S. Yedidia, Phys. Rev. B [**43**]{}, 3475 (1991).
See, for example, [*The Hubbard Model*]{}, edited by A. Montorsi (World Scientific, Singapore, 1992).
Also small numerical errors can be sufficient to start the symmetry breaking in the iteration procedure.
S.A. Kivelson, E. Fradkin, and V.J. Emery, Nature [**393**]{}, 550 (1998).
V. Oganesyan, S.A. Kivelson, and E. Fradkin, Phys. Rev. B [**64**]{}, 195109 (2001).
M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. [**85**]{}, 4940 (2000); Int. J. Mod. Phys. B [**14**]{}, 3719 (2000).
For a review of the attractive Hubbard model (and extensions), see R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. [**62**]{}, 113 (1990).
P.G.J. van Dongen, Phys. Rev. Lett. [**67**]{}, 757 (1991).
FIGURES
|
---
abstract: |
We study one-sided Markov shifts, corresponding to positively recurrent Markov chains with countable (finite or infinite) state spaces. The following classification problem is considered: when two one-sided Markov shifts are isomorphic up to a measure preserving isomorphism ? In this paper we solve the problem for the class of $\rho$-uniform (or finitely $\rho$-Bernoulli) one-sided Markov shifts considered in .
We show that every ergodic $\rho$-uniform Markov shift $T$ can be represented in a canonical form $T = T_G $ by means of a canonical (uniquely determined by $T$) stochastic graph $G$. In the canonical form, two such shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic if and only if their canonical stochastic graphs $G_1$ and $G_2$ are isomorphic.
address:
- 'Address: [*Ben Zion Rubshtein, Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel.*]{}'
- ' E-mail: [*benzion@math.bgu.ac.il*]{}'
author:
- 'Ben-Zion Rubshtein'
title: ' On a class of one-sided Markov shifts'
---
Introduction {#Intro}
============
In this paper we consider the classification problem for one-sided Markov shifts with respect to measure preserving isomorphism. Let $G$ be a finite or countable stochastic graph, i.e. a directed graph, whose edges $g \in G$ are equipped with positive weights $p(g)$. The weights $p(g)$ determine transition probabilities of a Markov chain on the discrete state space $G$. The corresponding one-sided Markov shift $T_G$ acts on the space $(X_G,m_G)$, where $ X_G = G^{{\mathbb{N}}}$ and $m_G$ is a stationary (probability) Markov measure on $X_G$. We deal only with irreducible positively recurrent Markov chains, so that such a Markov measure exists and the shift $T_G$ is an ergodic endomorphism of the Lebesgue space $(X_G,m_G)$. The problem under consideration is : When for given two stochastic graphs $G_1$ and $G_2$, does there exist an isomorphism $\; \Phi { : }X_{G_1} {\rightarrow}X_{G_2} \;$ such that $\; m_{G_2} = m_{G_1} \circ \Phi^{-1} \;$ and $\; \Phi \circ T_{G_1} \;=\; T_{G_2} \circ \Phi \;$.
It is obvious, that any (weight preserving) graph isomorphism $\phi { : }G_1 {\rightarrow}G_2$ generates such an isomorphism $\Phi = \Phi_\phi$, but nonisomorphic graphs can generate the same shift $T_G$.
Recently J. Ashley, B. Marcus and S. Tuncel [@AsMaTu] solved the classification problem for one-sided Markov shifts corresponding to [**finite**]{} Markov chains. They used an approach which is based on the following important fact: Two one-sided Markov shifts $T_{G_1}$ and $T_{G_2}$ (on finite state spaces) are isomorphic iff there exists a common extension $G$ of $G_1$ and $G_2$ by right resolving graph homomorphisms of degree $1$. The result was proved implicitly in [@BoTu], where regular isomorphisms and right closing maps for two-sided Markov shifts were studied (See also [@As], [@KiMaTr], [@Tr], [@Ki] and references cited there)
It should be noted that the classification problem for two-sided shifts is quite different from the one-sided case. Namely, any mixing two-sided Markov shift is isomorphic to the Bernoulli shift with the same entropy [@FrOr] and two-sided Bernoulli shifts are isomorphic iff they have the same entropy by the Sinai-Ornstein theorem [@Si]-[@Or].
On the other hand, let $T_\rho$ be the one-sided Bernoulli shift with a discrete state space $(I,\rho)$, where $I$ is a finite or countable set, $1 < |I| \leq \infty$, and $\rho = \{\rho_i\}_{i
\in I}$, $\sum_{i} \rho_i = 1$, $\rho_i > 0$. The endomorphism $T_\rho$ acts as the one-sided shift on the product space $(X_\rho , m_\rho) = \prod_{n=1}^{\infty} (I, \rho)$. Consider the measurable partition $T_\rho^{-1} {\varepsilon}= \{ T_\rho^{-1} x \;,\; x \in X_\rho \}$ generated by $T_\rho$ on $X_\rho$. The partition admits an independent complement ${\delta}$, which is not unique in general, but necessarily has the distribution $\rho$. This implies that one-sided Bernoulli shifts $T_{\rho_1}$ and $T_{\rho_2}$ are isomorphic iff the distributions $\rho_1$ and $\rho_2$ coincide.
This simple observation motivates the following definition. An endomorphism $T$ of a Lebesgue space $(X, m)$ is called ${\boldsymbol{\rho}}$[**-uniform**]{} (or [**finitely**]{} ${\boldsymbol{\rho}}$[**-Bernoulli**]{} according to ) if the measurable partition $T^{-1} {\varepsilon}= \{ T^{-1} x \;,\;\; x \in X \}$ admits an independent complement ${\delta}$ with $distr \; {\delta}= \rho$. We denote by ${\mathcal {UE}(\rho)}$ the class of all $\rho$-uniform endomorphisms.
Recall that the cofiltration $\xi (T)$ generated by an endomorphism $T$ is the decreasing sequence $\{\xi_n\}_{n=1}^{\infty}$ of the measurable partitions $\xi_n = T^{-n} {\varepsilon}$ of the space $X$ onto inverse images $T^{-n}x$. If two endomorphisms $T_1$ and $T_2$ are isomorphic, i.e. there exists an isomorphism $\Phi$ such that $\; \Phi \circ T_1 = T_2 \circ \Phi \;$, then $\; \Phi (T_1^{-n}x) = T_2^{-n} (\Phi x) \;$ for almost all $x \in X$, i.e. $\Phi (T_1^{-n} {\varepsilon}) = T_2^{-n} {\varepsilon}$ for all $n$. This means that the cofiltrations $\xi(T_1)$ and $\xi(T_2)$ are isomorphic.
If $T \in {\mathcal {UE}(\rho)}$, the cofiltration $\xi(T)$ is not necessarily isomorphic to the [**standard**]{} cofiltration $\xi(T_\rho)$, generated by the Bernoulli shift $T_\rho$. However, it is [**finitely isomorphic**]{} to $\xi(T_\rho)$, i.e. for every $n \in {{\mathbb{N}}}$ there exists an isomorphism $\Phi_n$ such that $\Phi_n (T^{-k} {\varepsilon}) =T_\rho^{-k} {\varepsilon}$ for all $1 \leq k
\leq n$.
The isomorphism problem for $\rho$-uniform endomorphisms is decomposed into the following two parts: When are the cofiltrations $\xi(T_1)$ and $\xi(T_2)$ isomorphic? When are $T_1$ and $T_2$ isomorphic provided that $\xi(T_1) = \xi(T_2)$?
In particular, for given $ T \in {\mathcal {UE}(\rho)}$: When is the cofiltration $\xi(T)$ standard, i.e. isomorphic to $\xi(T_\rho)$ ? When are $T_1$ and $T_2$ isomorphic provided that $\xi(T_1) = \xi(T_2)$ ?
All these problems are quite nontrivial even in the dyadic case $\rho = (\frac{1}{2},\frac{1}{2})$. Various classes of decreasing sequences of measurable partitions were considered by A.M. Vershik -, V.G. Vinokurov [@Vi], A.M. Stepin
[@St] and by author ,,-. A new remarkable progress in the theory is due to J. Feldman, D.J. Rudolph, D. Heicklen and Ch. Hoffman (See [@FeR], [@HeHo], [@HeHoR], [@Ho], [@HoR]). Note also that, as it was shown in , a $\rho$-uniform one-sided Markov shift $T_G$ is isomorphic to the Bernoulli shift $T_\rho$ iff the cofiltration $\xi(T_G)$ is isomorphic to standard cofiltration $\xi(T_\rho)$.
The purpose of this paper is to classify the $\rho$-uniform one-sided Markov shifts. We show that every ergodic $\rho$-uniform Markov shift $T$ can be represented in a [**canonical form**]{} $T = T_G $ by means of a [**canonical**]{} (uniquely determined by $T$) stochastic graph $G$. In the canonical form, two such shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic if and only if their canonical stochastic graphs $G_1$ and $G_2$ are isomorphic.
First we consider (Section \[s2\]) general $\rho$-uniform endomorphisms and use the following results from . Any ergodic $T \in {\mathcal {UE}(\rho)}$ can be represented as a skew product over $T_\rho$ on the space $X_\rho \times Y_d$, $d \in {{\mathbb{N}}}\cup \{\infty\} $, where $Y_d$ consists of $d$ atoms of equal measure $\frac{1}{d}$ for $d < \infty$ and $Y_\infty$ is a Lebesgue space with no atoms, (see Section \[ss2.2\] below). According to we introduce the [**minimal index**]{} $d(T)$ of $T \in {\mathcal {UE}(\rho)}$ as the minimal possible $d$ in the above skew product representation of $T$. The index $d(T)$ is an invariant of the endomorphism $T$ and $d(T) = 1$ iff $T$ is isomorphic to the Bernoulli shift $T_\rho$.
Other important invariants of $T \in {\mathcal {UE}(\rho)}$ (introduced also in ) are the [**partitions**]{} ${\boldsymbol{\gamma}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\beta}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ and the [**index**]{} ${\mathbf{d}}_{{\boldsymbol{\gamma}}{ : }{\boldsymbol{\beta}}}
{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$. The partition ${\gamma}(T)$ is the smallest (i.e. having the most coarse elements) measurable partition of $X$ such that almost all elements of the partition $\; {\beta}_n := {\gamma}(T) \vee
T^{-n} {\varepsilon}\;$ have homogeneous conditional measures for all $n$. The corresponding tail partition is defined by $\; {\beta}(T) =
\bigwedge_{n=1}^{\infty} {\beta}_n \geq {\gamma}(T) \;$. and the index $d_{{\gamma}{ : }{\beta}}(T) $ is the number of elements of ${\beta}(T)$ in typical elements of the partition ${\gamma}(T)$ (Proposition \[gga:gb\]).
It was proved in that $\; d(T) = d_{{\gamma}{ : }{\beta}}
(T) < \infty \;$ for any $\rho$-uniform one-sided Markov shift $T
= T_G$. This result implies, in particular, that $T_G$ is [**simple**]{} in the sence of Definition \[def SUE\]. The classification of general simple $\rho$-uniform endomorphisms is reduced to a description of equivalent $d$-extensions of the Bernoulli shift $T_\rho$ (Theorem \[simple\]).
Next we turn to $\rho$-uniform one-sided Markov shifts.
It is easy to see that a Markov shift $T_G$ is $\rho$-uniform iff the graph $G$ satisfies the following condition: For any vertex $u$ the set $G_u$ of all edges starting in $u$, equipped with the corresponding weights $\; p(g) \;,\; g \in G_u \;$, is isomorphic to $(I,\rho)$. This means that the transition probabilities of the Markov chain (starting from any fixed state) coincide with $\;
\rho(i) \;,\; i \in I \;$, up to a permutation. We call these graphs and Markov chains ${\boldsymbol{\rho}}$[**-uniform.**]{} In particular, $(I,\rho)$ itself is considered as a $\rho$-uniform graph having a single vertex. The corresponding Markov shift is the Bernoulli shift $T_\rho$.
Following [@AsMaTu] we use in the sequel graph homomorphisms of the form $\; \phi { : }G_1 {\rightarrow}G_2 \;$, which are assumed to be [**weight preserving**]{} and [**deterministic**]{}, i.e. right resolving in the terminology of [@AsMaTu], (see Definition \[hom\] for details). Thus a stochastic graph $G$ is $\rho$-uniform iff there exists a homomorphisms $\; \phi { : }G {\rightarrow}I \;$.
Two particular kinds of homomorphisms are of special interest in our explanation, they are homomorphisms of [**degree 1**]{} and [**d-extensions**]{}. A homomorphism $\; \phi { : }G_1 {\rightarrow}G_2 \;$ has degree $1$, $\; d(\phi) = 1\;$, if the corresponding factor map $\; \Phi_\phi { : }X_{G_1} {\rightarrow}X_{G_2} \;$ is an isomorphism. So that $\; \Phi_\phi \circ T_{G_1} \;=\; T_{G_2}
\circ \Phi_\phi \;$, i.e. $T_{G_1}$ and $T_{G_2}$ are isomorphic.
The d-extensions homomorphism are defined in Section \[ss3.2\] by the condition: $\; |\phi^{-1}g| = d \;,\; g \in G \;$. They can be described (up to equivalence) by the [**graph skew products**]{}, (see Example \[GSP\] and Definition \[def GSP\] in Section \[ss3.2\]).
As the first step to the construction of the canonical graph we show (Theorem \[phi bar\]) that any homomorphism $\; \phi { : }G {\rightarrow}I \;$ can be extended to a $d$-extension ${{\bar \phi}}$ by homomorphisms of degree $1$ (See Diagram \[diag phi bar\]). To this end we consider a ${\mathbf{d}}$[**-contractive**]{} semigroup ${{\mathcal S}}(\phi)$, associated with the homomorphism $\phi$, and the corresponding [**persistent**]{} sets (Section \[ss4.4\]). Thus we reduce the classification problem to the study of diagrams of the form $$\label{pipsi}
(\pi,\psi) \;{ : }\;
\xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I}$$ where ${{\bar H}}$ is a $d$-extension, $\psi$ is a degree $1$ homomorphism and the shift $T_{{\bar H}}$ is isomorphic to the shift $T_G$.
The second step is to minimize $d$ in the above Diagram \[pipsi\]. We show (Theorem \[phi bar d(T)\]) that, passing possibally to a “$n$-stringing” graph $G{^{(n)}}$, one can choose the minimal $ d = d(T)$. Note that the result is based on .
The third final step is to reduce the homomorphism $\psi$ in Diagram \[pipsi\] as much as possible. Let ${{\mathcal Ext}^d(I,\rho)}$ denotes the set of all $d$-extensions of the Bernoulli graph $(I,\rho)$ of the form (\[pipsi\]). We show that ${{\mathcal Ext}^d(I,\rho)}$ can be equipped with a natural [**partial order**]{} “$\preceq$” and [**equivalence relation**]{} “$\sim$” (Definition \[partial order\]). The minimal elements of ${{\mathcal Ext}^d(I,\rho)}$ with respect to the order are called [**irreducible**]{} (Definition \[irreduc\]). We describe these irreducible $(\pi,\psi)$-extensions by means of the persistent $d$-partitions, associated with elements of ${{\mathcal Ext}^d(I,\rho)}$ (Theorem \[reduc part\]).
Now we can formulate the main result of the paper (Theorems \[canon form\] and \[classification\]).
- [*Let $T_G$ be a $\rho$-uniform ergodic one-sided Markov shift. A stochastic graph ${{\bar H}}= {{\bar H}}(G)$ is said to be a canonical graph for the shift $T$ if there exists an irreducible $(\pi,\psi)$-extension (\[pipsi\]) from ${{\mathcal Ext}^d(I,\rho)}$ with $\; d = d(T) \;$ such that the shift $T_{{\bar H}}$ is isomorphic to $T_G$.*]{}
- [*Any $\rho$-uniform ergodic one-sided Markov shift can be represented in the canonical form $T = T_{{\bar H}}$ by a canonic graph ${{\bar H}}= {{\bar H}}(G)$.*]{}
- [*In this canonical form, two shifts $T_{{{\bar H}}_!}$ and $T_{{{\bar H}}_2}$ are isomorphic iff the canonical graphs ${{\bar H}}_1$ and ${{\bar H}}_1$ are isomorphic, and iff the corresponding irreducible $(\pi,\psi)$-extensions are equivalent.*]{}
The paper is organized as follows.
In Section \[s2\] we study general $\rho$-uniform endomorphisms (class ${\mathcal {UE}(\rho)}$) and [**simple**]{} $\rho$-uniform endomorphisms (subclass ${\mathcal {SUE}(\rho)}$). Following , we introduce the [**partitions**]{} ${\boldsymbol{\gamma}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\beta}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ and the [**index**]{} ${\mathbf{d}}_{{\boldsymbol{\gamma}}{ : }{\boldsymbol{\beta}}} {{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$. Two main conclusions of the section are Theorem \[simple\] (classification of simple $\rho$-uniform endomorphisms) and Theorem \[simp mar\], which states that every ergodic $\rho$-uniform one-sided Markov shift $T_G$ is simple and $\;
d(T_G) \;=\; d_{{\gamma}{ : }{\beta}}(T_G) \;<\; \infty \;$.
In Section \[s3\] we consider general properties of stochastic graphs and their homomorphisms. In particular, we define ${\boldsymbol{\rho}}$[**-uniform** ]{} graphs corresponding to $\rho$-uniform Markov shifts. We prove that the index $d(T_G)$ of any ergodic $\rho$-uniform Markov shift $T_G$ is finite (Theorem \[zet del\]). This follows from the finiteness of the degree $d(\phi)$ of any homomorphism $\phi { : }G {\rightarrow}I$ from any $\rho$-uniform graph $G$ onto the standard Bernoulli graph $(I,\rho)$. The degree $d(\phi)$, in turn, can be computed by means a special [**d-contractive**]{} semigroup ${{\mathcal S}}(\phi)$, induced by $\phi$ (Theorem \[d(phi)\]).
Section \[s4\] contains some essential stages of the proof of Main Theorems \[canon form\] and \[classification\]. Homomorphisms of degree $1$ and extensions of the Bernoulli graph are considered in Sections \[ss4.1\] and \[ss4.2\]. Theorem \[Equ ext\] (Section \[ss4.3\]) reduces the classification of skew product over Markov shifts $T_H$ to the classification of the corresponding graph skew product over $H$. In Sections \[ss4.4\] and \[ss4.5\], we study the set ${{\mathcal Ext}^d(I,\rho)}$ of all $(\pi,\psi)$-pairs of the form (\[pipsi\]). The main result of Section \[s4\] is Theorem \[reduc ext\], which claims the existence and uniqueness of the irreducible $(\pi,\psi)$-pair $(\pi_*,\psi_*)$, majorized by a given $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$.
In Section \[s5\] we prove Main Theorems \[canon form\] and \[classification\] and give some consequences and examples. As a consequence we prove also (Theorem \[common exten 1\]) that two shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic iff the graphs $G_1$ and $G_2$ have a common extension of degree $1$.
We do not study here the classification problem for general, not necessarily $\rho$-uniform, one-sided Markov shifts as well as the classification problem of the cofiltrations, generated by the shifts. Our approach seems to be a good tool to this end and we hope to deal with these two problems in another paper.
We do not also consider the classification problem of one-sided Markov shifts with infinite invariant measure, in particular, of null-recurrent one-sided Markov shifts. One can find a good introduction to the topic and more references in [@Aar Chapters 4 and 5].
Class of $\rho$-uniform endomorphisms {#s2}
=======================================
Lebesgue spaces and their measurable partitions {#ss2.1}
-------------------------------------------------
We use terminology and results of the Rokhlin’s theory of Lebesgue spaces and their measurable partitions (See , ). An improved and more detailed explanation can be found in [@ViRuFed]. We fix the terms “homomorphism , isomorphism, endomorphism” only for [**measure preserving**]{} maps of Lebesgue spaces.
Let $(X, {{\mathcal F}}, m)$ be a Lebesgue space with $ mX=1$. The space $X$ is called [**homogeneous** ]{} if it is non-atomic or if it consists of $d$ points of measure $\; \frac{1}{d} \;,\; d
\in {{\mathbb{N}}}\;$.
Let $ {\zeta}$ be a partition of $X$ onto mutually disjoint sets $ C
\in {\zeta}$. The element of $ {\zeta}$ containing a point $ x $ is denoted by $ C_{\zeta}(x) $. The partition $ {\zeta}$ is measurable iff there exists a measurable function $ f { : }X {\rightarrow}{{\mathbb{R}}}$ such that $$x \stackrel{{\zeta}}{\sim} y \Longleftrightarrow
C_{\zeta}(x) = C_{\zeta}(y) \Longleftrightarrow f(x)=f(y) \;,\; x,y \in X$$ Elements of $ {\zeta}$ are considered as Lebesgue spaces $\; (C, {{\mathcal F}}^C , m^C) \;,\; C \in {\zeta}\;$, with canonical system of conditional measures $\; m^C \;,\;C \in {\zeta}\;$. We shall denote also by $ m(A|C)$ the conditional measures $\; m^C(A \cap C) \;$ of a measurable set $ A \in {{\mathcal F}}$ in the element $ C$ of $ {\zeta}$.
Two measurable partitions ${\zeta}_1$ and ${\zeta}_2$ are said to be [**independent** ]{} (${\zeta}_1 \perp {\zeta}_2 $) if the corresponding ${\sigma}$-algebras $ {{\mathcal F}}({\zeta}_1) $ and $ {{\mathcal F}}({\zeta}_2)$ are independent , where ${{\mathcal F}}({\zeta})$ denotes the $m$-completion of the ${\sigma}$-algebra of all measurable ${\zeta}$-sets. We shall write also $\; {\zeta}_1 \perp {\zeta}_2 \pmod {\zeta}\;$ if the partitions ${\zeta}_1$ and $ {\zeta}_2 $ are [**conditionally independent**]{} with respect to the third measurable partition ${\zeta}$. This means that $$m(A \cap B \;|\; C_{\zeta}(x)) =
m(A|C_{\zeta}(x)) \cdot m(B \;|\; C_{\zeta}(x))$$ for all $ A \in {{\mathcal F}}({\zeta}_1), B \in {{\mathcal F}}({\zeta}_2) $ and a.a. $ x \in X $.
We denote by $ {\varepsilon}= {\varepsilon}_X $ the partition of $X$ onto separate points and by $\nu = \nu_X$ the trivial partition of $X$.
An [**independent complement**]{} of ${\delta}$ is a measurable partition $ \eta $ such that $\; {\zeta}\perp \eta \;$ and $\; {\zeta}\vee \eta =
{\varepsilon}\;$. The partition ${\zeta}$ admits an independent complement iff almost all elements $ (C,m^C) $ of ${\zeta}$ are mutually isomorphic. The collection of all independent complements of ${\zeta}$ is denoted by $IC({\zeta})$.
We shall use induced endomorphisms, which are defined as follows. Let $ A \in {{\mathcal F}}$ , $ mA > 0$ and $T$ be an endomorphism of $(X,m)$. Then the return function $$\label{ret fun}
{\varphi}_A(x) := min \{ n \geq 1 { : }T^{n}x \in A \}
\;\;,\;\; x \in A$$ is finite a.e. on $A$. The [**induced endomorphism**]{} $T_A$ on $A$ is defined now by $\; T_Ax = T^{{\varphi}_A(x)}x \;$. It is an endomorphism of $ (A,{{\mathcal F}}\cap A, m{|}_A) $ and it is ergodic if $T$ is ergodic .
$|E|$ denotes the cardinality of the set $E$
Classes ${\mathcal UE} {\boldsymbol{(}}{\boldsymbol{\rho}}{\boldsymbol{)}}$ and index ${\mathbf{d}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$. {#ss2.2}
---------------------------------------------------------------------------------------------------------------------------------------------------
Let $(I,\rho)$ be a finite or countable state space $$\rho = \{ \rho(i) \;,\; i \in I \} \;,\;\; \rho (i) > 0
\;\;,\;\; \sum_{i \in I}{\rho(i)} = 1 .$$
\[def rho Bern\] An endomorphism $T$ of a Lebesgue space $(X,m)$ is said to be ${\boldsymbol{\rho}}$[**-unform**]{} or [**finitely**]{} ${\boldsymbol{\rho}}$[**-Bernoulli**]{} endomorphism $\; (T \in {\mathcal {UE}(\rho)}\;$, if there exists a discrete measurable partition ${\delta}$ of $X$, which satisfies the following condition:
1. $\; distr \; {\delta}= \rho \;$, i.e. $\; {\delta}= \{ B(i) \}_{i \in I}$ with $\;\;m(B(i)) = \rho (i), \;\; i \in I $,
2. $\; {\delta}\in IC (T^{-1} {\varepsilon}) $ , i.e. ${\delta}\perp T^{-1} {\varepsilon}\;$ and $\; {\delta}\vee T^{-1}{\varepsilon}= {\varepsilon}$.
So $ {\mathcal {UE}(\rho)}$ denotes the class of all $\rho$-unform endomorphisms. Denote by ${\Delta}_{\rho}(T)$ the set of all partitions ${\delta}$ satisfying the condition $(i)$ and $(ii)$. Then $T \in {\mathcal {UE}(\rho)}$ means ${\Delta}_\rho(T) \neq\emptyset $.
For $T \in {\mathcal {UE}(\rho)}$ and ${\delta}\in {\Delta}_{\rho}(T)$ define $$\label{delta n}
{\delta}{^{(n)}}= T^{-n+1} {\delta}\;\;,\;\;\;
{\delta}{^{(n)}}= \{ T^{-n+1}B(i) \}_{i \in I } \;\;,\;\; n \in {{\mathbb{N}}}$$ Then $\; distr \;{\delta}_n = \rho \;$ and the partitions $ \; {\delta}_1 \;,\; {\delta}_2 \;,\;{\delta}_3 \;,\; \ldots \; $ are independent.
The partitions $$\label{delta (n)}
{\delta}{^{(n)}}= \bigvee_{k=1}^{n} {\delta}_{k} \;\;,\;\;\;
{\delta}^{(\infty)} = \bigvee_{k=1}^{\infty} {\delta}_k$$ satisfy for all $n$ the conditions $${\delta}{^{(n)}}\in IC(T^{-n} {\varepsilon}) \;\;,\;\;
{\delta}{^{(\infty)}}\perp T^{-n} {\varepsilon}\pmod { {\delta}{^{(\infty)}}\wedge T^{-n} {\varepsilon}}$$ and $${\delta}{^{(\infty)}}\vee T^{-n} {\varepsilon}= {\varepsilon}\;\;,\;\;
{\delta}{^{(\infty)}}\wedge T^{-n} {\varepsilon}\;=\; T^{-n} {\delta}{^{(\infty)}}\;.$$ In particular, let $ T=T_\rho$ be a Bernoulli endomorphism, which acts on the space $$(X_\rho, m_\rho) = \prod_{n=1}^{\infty} (I,\rho)$$ as the one-sided shift $$T_{\rho} x = \{ x_{n+1}\}_{n=1}^{\infty} \;\; , \;\;\;
x = \{ x_n \}_{n=1}^{\infty} \in X_\rho \;\;.$$ We can set $$\label{delta rho}
{\delta}_\rho = \{ B_\rho(i) \}_{i \in I} \;\;,\;\; B_\rho(i) =
\{ x = \{ x_{n}\}_{n=1}^{\infty} \in X_{\rho}
\;\; { : }\;\; x_1=i \} \;.$$ Then $\; {\delta}_\rho \in {\Delta}_\rho (T_\rho )$ and ${\delta}_{\rho}$ is an one-sided Bernoulli generator of $T_\rho$, that is $${\delta}_{\rho}^{(\infty)} = \bigvee_{n=1}^{\infty} T^{-n+1}{\delta}_{\rho}
= {\varepsilon}_{X_{\rho}} \;.$$
In general case, for $T \in {\mathcal {UE}(\rho)}$ and ${\delta}\in {\Delta}_\rho(T)$, the partition ${\delta}^{(\infty)}$ does not equal ${\varepsilon}$, but we can define the canonical factor map $$\Phi_{{\delta}} { : }X \ni x {\rightarrow}\Phi_{{\delta}} (x) = \{i_{n} (x) \}_{n=1}^{\infty} \in X_{\rho} \;,$$ where $i_{n}(x) \in I$ is uniquely defined by the inclusion $ T^{n}x \in B(i_{n}(x)) \in {\delta}$.
The homomorphism $\Phi_{{\delta}}$ satisfies $\; \Phi_{{\delta}} \circ T = T_{\rho} \circ \Phi_{{\delta}} \;$ and it determines the following representation of $T$ by a skew product over $ T_{\rho} $ (See ).
\[T decompos\] Let $T \in {\mathcal {UE}(\rho)}$ be an endomorphism of $(X,m)$ and ${\delta}\in {\Delta}_\rho(T)$. Then
1. There exists an independent complement ${\sigma}$ of the partition ${\delta}{^{(\infty)}}$.
2. The pair $\; ({\delta}{^{(\infty)}}, {\sigma}) \;$ induces decomposition of the space $(X,m)$ into the direct product $\; (X_\rho \times Y \;,\; m_{X_\rho} \times m_{Y})$ such that the factor map $\Phi_{\delta}$ coincides under the decomposition with the canonical projection $$\pi \; { : }\; X_\rho \times Y \ni (x,y) {\rightarrow}x \in X_\rho$$ and $${\delta}= \pi^{-1} {\delta}_{\rho}
\;\;, \;\; {\delta}{^{(\infty)}}= \pi^{-1} {\varepsilon}_{X_\rho}
= {\varepsilon}_{X_\rho} \times \nu_Y
\;\;,\;\; {\sigma}= \nu_{X_\rho} \times {\varepsilon}_Y$$
3. The endomorphism $T$ is identified with the following skew product over $ T_\rho $ $$\label{skew1}
{{\bar T}}(x,y) = ( T_\rho x , A(x) y)
\;\;,\;\; (x,y) \in X_{\rho} \times Y$$ where $\; \{ A(x), x \in X_\rho \} \;$ is a measurable family of automorphisms of $Y$.
4. If $T$ is ergodic, $Y$ is a homogeneous Lebesgue space.
Every homogeneous Lebesgue space $Y$ is isomorphic to $\; Y_d
\;.\;d \in {{\mathbb{N}}}\cup \{\infty\} \;$, where $\; Y_{\infty} \;$ is the Lebesgue space with a continuous measure and $\; Y_d \;$,$\;d \in
{{\mathbb{N}}}\;$, consists of $d$ points of measure $\frac{1}{d}$. Thus for any ergodic $T$ endomorphism $\; T \in {\mathcal {UE}(\rho)}\;$ and $\; {\delta}\in
{\Delta}_{\rho} (T) \;$ there exists $\; d = d(T,{\delta}) \in {{\mathbb{N}}}\cup
\{\infty\} \;$ such that $$u_{{\delta}^{(\infty)}}(x)
:= m^{C_{{\delta}^{(\infty)}} (x)}(\{x\}) = \frac{1}{d}$$ for a.a. $x \in X $.
\[def d(T)\]
1. The number $\; d(T,{\delta}) \;$ will be called the [**index**]{} of $T \in {\mathcal {UE}(\rho)}$ with respect to $\; {\delta}\in {\Delta}_{\rho}(T) \;$.
2. The [**minimal index**]{} $d(T)$ of $T$ is defined as $$\label{d(T)}
d(T) \;=\; min \; \{\; d(T,{\delta}) \;,\; {\delta}\in {\Delta}_{\rho} (T)\}$$
Note that an ergodic endomorphism $T$ is isomorphic to the Bernoulli shift $T_\rho$ iff $T \in {\mathcal {UE}(\rho)}$, and $d(T) = 1$, that is, there exists ${\delta}\in {\Delta}_\rho(T)$ such that $d(T,{\delta}) = 1$, i.e. ${\delta}{^{(\infty)}}= {\varepsilon}$.
Partitions ${\boldsymbol{\alpha}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\beta}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\gamma}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ and indices ${\mathbf{d}}_{{\boldsymbol{\alpha}}} {{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\mathbf{d}}_{{\boldsymbol{\gamma}}{ : }{\boldsymbol{\beta}}} {{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ {#ss2.3}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Let $T$ be an endomorphism of $(X,m)$ and let $ \{ \xi_n
\}_{n=1}^{\infty} $ be the decreasing sequence of measurable partitions $\; \xi_n := T^{-n}{\varepsilon}\;$, generated by $T$. The element of $\xi_n$ , containing a point $x \in X$, has the form $\; C_{\xi_n}(x) = T^{-n}(T^n x) \;$,
In order to introduce the partitions ${\gamma}(T)$ and ${\beta}(T)$, consider the measurable functions $$u_n(x) = m^{C_{\xi_n}(x)} (C_{\xi_{n-1}}(x)) \;\;\;,
\; n \in {{\mathbb{N}}}\;\;\; , \; x \in X \;,$$ where $\xi_0 := {\varepsilon}$. With these $\; u_n { : }X {\rightarrow}[0,1]
\;$ we can consider the measurable partitions $${\gamma}_n = \bigvee_{k=1}^{n}u_{k}^{-1} {\varepsilon}_{[0,1]}
\;\; ,\;\; n \in {{\mathbb{N}}}\;,$$ generated by $\;u_k\;,\; k \leq n \;$, and also $$\label{gga gbn gb}
{\gamma}= \bigvee_{n=1}^{\infty} {\gamma}_n \;\;,\;\;
{\beta}_n = {\gamma}\vee T^{-n}{\varepsilon}\;\;,\;\;
{\beta}= \bigwedge_{n=1}^{\infty} {\beta}_n$$ We shall write $\; {\gamma}_n(T) \;,\; {\gamma}(T)\;,\; {\beta}_n(T)\;,\;
{\beta}(T) \;$ to indicate $T$, if it will be necessary.
\[gga:gb\] Suppose that $T\in {\mathcal {UE}(\rho)}$ and T is ergodic. Then there exists $d \in {{\mathbb{N}}}\cup \{\infty\}$ such that $$m^{C_{{\gamma}}(x)}(C_{{\beta}}(x))=\frac{1}{d}$$ for a.a. $x\in X$.
We may define now the index $\; d_{{\gamma}{ : }{\beta}}(T) \;$ of an ergodic endomorphism $T \in {\mathcal {UE}(\rho)}$ as the number $d$ constructed in Proposition \[gga:gb\], i.e. $$d_{{\gamma}{ : }{\beta}}(T) := (m^{C_{\gamma}(x)}(C_{\beta}(x)))^{-1}$$ for a.e. $x \in X$.
We shall use the following properties of the partitions (\[gga gbn gb\])
\[gga gd d(T)\] Suppose that $\; T \in {\mathcal {UE}(\rho)}\;$, let $\; {\delta}\in {\Delta}_\rho(T) \;$ and the partitions $\; {\delta}_n \;,\; {\delta}{^{(n)}}\;,\; {\delta}{^{(\infty)}}\;$ defined by (\[delta n\]) and (\[delta (n)\]). Then
1. $\; {\gamma}{^{(n)}}\le {\delta}_n
\;\;,\;\; {\beta}_n \perp {\delta}{^{(n)}}\pmod {{\gamma}_n}
\;\;,\;\; n \in {{\mathbb{N}}}\;.$
2. ${\gamma}\le {\delta}{^{(\infty)}}\;\;,\;\; {\beta}\perp {\delta}{^{(\infty)}}\pmod {{\gamma}} \;.$
3. $\; d_{{\gamma}{ : }{\beta}}(T) \; \leq \; d(T) \;$
We shall also use the [**tail**]{} measurable partition $\; {\alpha}(T) := \bigwedge_{n=1}^{\infty} T^{-n} {\varepsilon}\;$. An endomorphism $T$ is called [**exact**]{} if ${\alpha}(T) = \nu$. The [**tail index**]{} $d_{\alpha}(T)$ (which is, in fact, the [**period** ]{} of $T$) is defined as follows: $d_{\alpha}(T)= \infty$ if $
X{/}_{{\alpha}(T)}$ is a continuous Lebesgue space and $d_{{\alpha}}(T) = d$ if $ X {/}_{{\alpha}((T)}$ consists of $d$ atoms of measure $\frac{1}{d}$. So that $d_{\alpha}(T) \in {{\mathbb{N}}}\cup \{\infty\}$.
It is easily to see, that $$\label{ga gb gga}
T^{-1} {\alpha}= {\alpha}\;\;,\;\; {\alpha}\vee {\gamma}\le {\beta}\;\;,\;\;
T^{-1} {\gamma}\le {\gamma}\;\;,\;\; T^{-1} {\beta}\le {\beta}$$ and $\; {\alpha}\perp {\delta}{^{(\infty)}}\;$ for any $ {\delta}\in {\Delta}_{\rho}(T) $.
Turning to the canonical projection $\Phi_{\delta}$ we have
\[ga(T rho)\]
1. $ {\alpha}(T_\rho) = \nu
\;\;,\;\; {\beta}(T_\rho) = {\gamma}(T_{\rho}) \;$.
2. $ {\gamma}_n(T) = \Phi_{{\delta}}^{-1}{\gamma}_n(T_\rho)
\;\;,\;\; {\gamma}(T) = \Phi_{{\delta}}^{-1}{\gamma}(T_\rho) \;$.
The stated above propositions were proved in .
Simple ${\boldsymbol{\rho}}$-uniform endomorphisms {#ss2.4}
--------------------------------------------------
We use now the partitions ${\gamma}(T)$ and $ {\beta}(T)$ to introduce an important subclass of the class ${\mathcal {UE}(\rho)}$
\[def SUE\] An endomorphism $T \in {\mathcal {UE}(\rho)}$ of a Lebesgue space $(X,m)$ is said to be a [**simple**]{} $\rho$-uniform endomorphism $(T \in {\mathcal {SUE}(\rho)})$, if there exists partition ${\delta}\in {\Delta}_\rho(T)= IC (T^{-1} {\varepsilon})$ such that $$\label{SUE1}
{\delta}^{(\infty)} \vee {\beta}(T)= {\varepsilon}$$
We denote by ${\mathcal {SUE}(\rho)}$ the class of all simple $\rho$-uniform endomorphisms.
The Bernoulli endomorphism (one-sided Bernoulli shift) $T=T_{\rho}$ belongs to ${\mathcal {SUE}(\rho)}$. In this case there exists a partition $ {\delta}= {\delta}_{\rho} \in {\Delta}_{\rho}(T)$ such that $
{\delta}{^{(\infty)}}= {\varepsilon}$ and hence $ {\beta}(T) \vee {\delta}{^{(\infty)}}= {\varepsilon}$
\[rem SUE\] It is easily to show that the condition (\[SUE1\]) holds iff there exists an independent complement ${\sigma}\in IC({\delta}^{(\infty)}) $ of ${\delta}^{(\infty)}$ that satisfies $$\label{SUE2}
{\sigma}\vee {\gamma}(T) = {\beta}(T) \;\;,\;\;
{\sigma}\in IC ({\delta}^{(\infty)}) \;\;,\;\;
{\delta}\in {\Delta}_\rho(T)= IC (T^{-1} {\varepsilon})$$
\[simple1\] Suppose $\;T \in {\mathcal {UE}(\rho)}\;$ is ergodic and $\; d(T) < \infty \;$. Then $T$ is simple iff $\; d(T) = d_{{\gamma}{ : }{\beta}}(T) \;$.
Since $\; d(T) < \infty \;$ we have, by Proposition \[gga gd d(T)\] (iii), that $\; d_{{\gamma}{ : }{\beta}}(T) \leq d(T)
< \infty \;$. Definition of the index $d_{{\gamma}{ : }{\beta}}(T)$ (Proposition \[gga:gb\]) means that $\; m^{C_{\gamma}(x)}(C_{\beta}(x)) = d^{-1} \;$ for a.a. $x \in X$ and $\; d = d_{{\gamma}{ : }{\beta}}(T) \in {{\mathbb{N}}}\;$. Almost every element of ${\gamma}(T)$ consists precisely of $d$ elements of the partition ${\beta}(T)$. On the other hand there exists ${\delta}\in {\Delta}_\rho(T)$ such that almost every element of the corresponding partition ${\delta}{^{(\infty)}}$ consists precisely of $d(T)$ points, $d \leq d(T)$. By Proposition \[gga gd d(T)\] (ii) we have $${\beta}(T) \perp {\delta}{^{(\infty)}}\pmod {{\gamma}(T)} \;\;,\;\;
{\beta}(T) \wedge {\delta}{^{(\infty)}}= {\gamma}(T) \;.$$ Whence, the condition \[SUE1\] holds iff $\; d(T) = d \;$.
Let $\; T \in {\mathcal {SUE}(\rho)}\;$. By Proposition \[T decompos\] any choice of the partition ${\sigma}$ in the equality (\[SUE2\]) determines a skew product representation (\[skew1\]) of $T={{\bar T}}$ over $T_\rho$. Herewith, all statements of Proposition \[T decompos\] hold and we have also by (\[SUE2\]) and Proposition \[ga(T rho)\]) $$\label{SUE3}
{\beta}(T_\rho)={\gamma}(T_\rho)
\;\;,\;\; {\gamma}({{\bar T}}) = {\gamma}(T_\rho) \times \nu_Y
\;\;,\;\; {\beta}({{\bar T}}) = {\gamma}(T_\rho) \times {\varepsilon}_Y$$ These arguments imply the following
\[simple\] Let $T$ be a $\rho$-uniform simple endomorphism, $\; T \in {\mathcal {SUE}(\rho)}\;$.
1. $T$ can be represented in the skew product form (\[skew1\]) $T={{\bar T}}$ over $T_\rho$ $$\label{Tbar}
{{\bar T}}(x,y) = ( T_\rho x , A(x) y)
\;\;,\;\; (x,y) \in X_{\rho} \times Y$$ with a measurable family $\; \{ A(x) \;,\; x \in X_\rho \} \;$ of automorphisms of $Y$ such that $\; {\beta}({{\bar T}}) = {\gamma}(T_\rho) \times {\varepsilon}_Y \;$.
2. Two such skew product endomorphisms ${{\bar T}}_k$, $k=1,2,$ $$\label{Tbar 1,2}
{{\bar T}}_k (x,y) = ( T_\rho x , A_k(x) y)
\;\;,\;\; (x,y) \in X_{\rho} \times Y$$ are isomorphic iff the corresponding families $\; A_1(x) \;$ and $\; A_2(x) \;$ are cohomologous, i.e. $$\label{A2W=WA1}
A_2(x) W(x) = W(T_\rho x) A_1(x) \;\;,\;\; x \in X_\rho$$ for a measurable family of $\; \{ W(x) \;,\; x \in X_\rho \} \;$ of automorphisms of $Y$.
Part (i) follows from Proposition \[T decompos\] with (\[SUE3\]).
Let ${{\bar T}}_1$ and ${{\bar T}}_2$ be two skew product endomorphisms of the form (\[Tbar 1,2\]). Denote $\; {{\tilde{W}}}(x,y):=(x,W(x)y) \;$. Then (\[A2W=WA1\]) implies $ {{\bar T}}_2 \circ S = S \circ {{\bar T}}_1 $ if we use the automorphism $S = {{\tilde{W}}}$.
Conversely, suppose there exists an automorphism $S$ such that $ {{\bar T}}_2 \circ S = S \circ {{\bar T}}_1 $. Then the partitrions $${{\bar {\gamma}}}:= {\gamma}({{\bar T}}_1) = {\gamma}({{\bar T}}_2) = {\gamma}(T_\rho) \times \nu_Y$$ and $${{\bar {\beta}}}:= {\beta}({{\bar T}}_1) = {\beta}({{\bar T}}_2) = {\gamma}(T_\rho) \times {\varepsilon}_Y$$ are invariant with respect to $S$. Moreover, ${{\bar {\gamma}}}$ is element-wise invariant with respect to $S$. Hence, $S {|}_C
({{\bar {\beta}}}{|}_C) = {{\bar {\beta}}}{|}_C $ for almost every element $C \in {{\bar {\gamma}}}$. The restriction $S{|}_C$ induces a factor automorphism $W_C$ on the factor space $ C{/}_{{{\bar {\beta}}}{|}_C} \cong Y $. We obtain a measurable family $\; W(x) := W_{C(x)} \;,\; x \in
X_\rho \;$, of automorphisms of $Y$. Since the partition $\;
{{\bar {\gamma}}}= {\gamma}(T_\rho) \times \nu_Y \;$ is ${{\bar T}}_1)$- and ${{\bar T}}_2$-invariant, the functions $ A_1(x) $ and $ A_2(x) $ (as well as $W(x)$) are constant on elements of ${\gamma}(T_\rho)$. Therefore the equality $ {{\bar T}}_2 \circ S = S \circ {{\bar T}}_1 $ implies $
{{\bar T}}_2 \circ {{\tilde{W}}} = {{\tilde{W}}} \circ {{\bar T}}_1 $ and (\[A2W=WA1\]) holds.
Consider two special cases.
[**Absolutely non-homogeneous ${\boldsymbol{\rho}}$.**]{} The distribution $\; \rho = \{\rho(i) \}_{i \in I} \;$ is called absolutely non-homogeneous if $\; \rho(i) \neq \rho(j) \;$ for all $\;i \neq j\;$.
In this case we have $\; {\gamma}_1(T) \vee T^{-1}{\varepsilon}= {\varepsilon}\;$. On the other hand $\; {\gamma}_1(T) \perp T^{-1}{\varepsilon}\;$. Thus $\; {\Delta}_\rho(T) \;$ consists of the only partition, which is $\; {\delta}= {\gamma}_1(T) \;$. Hence $${\delta}{^{(\infty)}}= {\gamma}(T) \;\;,\;\;
{\beta}_n(T) = {\gamma}_n(T) \vee T^{-n}{\varepsilon}= {\varepsilon}\;,$$ $${\beta}(T) = \bigwedge_{n=1}^{\infty} {\beta}_n(T) = {\varepsilon}\;\;,\;\;
{\beta}(T) \vee {\delta}{^{(\infty)}}= {\varepsilon}$$ Thus we have
Every $\rho$-uniform endomorphisms with absolutely non-homogeneous $\;\rho \;$ is simple.
[**Homogeneous ${\boldsymbol{\rho}}$.**]{} We have another extremal case if $\;\rho \;$ is homogeneous, i.e. if for some $\; l \in {{\mathbb{N}}}\;$,$\; I = \{1,2, \ldots ,l\;\} \;$ and $\; \rho(i) = l^{-1} \;$ for all $\; i \in I \;.$
All the functions $ \;u_n \;$, which generate the partitions $\; {\gamma}_n(T) \;$, are constant, $$u_n(x) \;=\; m^{C_{\xi_n }(x)} (C_{\xi_{n-1}}(x))
\;=\; l^{-1} \;,\; n \in {{\mathbb{N}}}\;,\; x \in X$$ We have $\; {\gamma}(T) = {\gamma}_n(T) = \nu \;,$ and $\; {\beta}_n(T) =
T^{-n}{\varepsilon}\;$, whence, $\; {\beta}(T) = \bigwedge_{n=1}^{\infty}
T^{-n}{\varepsilon}= {\alpha}(T) \;$. Therefore, for any $\;{\delta}\in {\Delta}_{\rho}(T)\;$ the equality (\[SUE1\]) is equivalent to $\; {\delta}{^{(\infty)}}\vee {\alpha}(T) = {\varepsilon}\;$. On the other hand $\; {\delta}{^{(\infty)}}\perp {\alpha}(T) \;$ for every $\;{\delta}\in {\Delta}_{\rho}(T)\;$.
Thus we have for $ T \in {\mathcal {UE}(\rho)}$ with homogeneous $\rho$
Let $ T \in {\mathcal {UE}(\rho)}$ with homogeneous $\rho$. Then
1. $T$ is simple iff there exists $\;{\delta}\in {\Delta}_{\rho}(T)\;$ such that $ {\delta}{^{(\infty)}}\in IC({\alpha}(T)) $.
2. The skew product decomposition in Theorem \[simple\] is a direct product $\; T_\rho \times S \;$ with $ S = T {/}_{{\alpha}(T)} $.
3. Two such direct products $\; T_\rho \times S_1 \;$ and $\; T_\rho \times S_2 \;$ are isomorphic iff the automorphisms $ S_1 $ and $ S_2 $ are isomorphic.
4. If, in addition, $T$ is exact, i.e. $\;{\alpha}(T) = \nu \;$, then $T$ is simple iff $T$ is isomorphic to $\; T_\rho \;$.
It is easily to construct a skew product $T$ over $T_\rho$, which is exact and has entropy $\; h(T) > h(T_\rho) = \log l \;$. Every such endomorphism is $\rho$-uniform, $\; T \in {\mathcal {UE}(\rho)}\;$, but it is not isomorphic to $T_\rho$, whence, it is not simple. See also [@FeR], [@HeHo], [@HeHoR], [@Ho], [@HoR]), for more interesting examples of such kind of endomorphisms.
It can be shown that there exist non-simple exact endomorphisms in each class $\; {\mathcal {UE}(\rho)}\;$ in the case, when $\rho$ is not absolutely non-homogeneous, i.e. $\; \rho(i) = \rho(j)\;$ for some $\;i,j \in I\;$.
The following result plays an important role in present paper.
\[simp mar\] Every ergodic $\rho$-uniform one-sided Markov shift $T_G$, corresponding to a positively recurrent Markov chain on a finite or countable state space, is simple and $$\label{d=dg:b}
d(T_G) \;=\; d_{{\gamma}{ : }{\beta}}(T_G) \;<\; \infty \;.$$
The last statement \[d=dg:b\] was proved in . It implies that $T_G$ is simple by Proposition \[simple1\].
Stochastic graphs and their homomorphisms. {#s3}
============================================
Stochastic graphs and Markov shifts {#ss3.1}
-----------------------------------
We need some terminology concerning stochastic graphs and their homomorphisms.
Consider a directed graph with countable (finite or infinite) set $G$ of edges. Denote by $G{^{(0)}}$ the set of all vertices of the graph. We also denote by $s(g)$ the starting vertex and by $t(g)$ the terminal vertex of an edge $g \in G$ $$\xymatrix{ t(g) & s(g) \ar[l]_{g} }$$ The maps $$s \; { : }\; G \ni g \; {\rightarrow}\; s(g) \in G{^{(0)}} \;\; \;,\; \;\;
t \; { : }\; G \ni g \; {\rightarrow}\; t(g) \in G{^{(0)}}$$ completely determine the structure of the graph $G$,
In the sequel we assume that both the sets $$_vG = \{ g \in G \;{ : }\; t(g) = v \} \;\;\;,\;\;\;
G_u = \{ g \in G \;{ : }\; s(g)= u \}$$ are not empty for all vertices $ \;u\;,v\;\in G{^{(0)}} $.
Denote by $\; G{^{(n)}}\;$ the set of all paths of length $n$ in $G$ , i.e. $$\label{G(n)}
G{^{(n)}}= \{ g_1 g_2 \ldots g_n \in G^n \;{ : }\;
s(g_1)=t(g_2) , \ldots , s(g_{n-1})=t(g_n) \}$$ A graph $G$ is said to be [**irreducible**]{} if for every pair of vertices $ \;u\;,v\;\in G{^{(0)}} $ there exists a finite $G$-path $ \;g_1g_2 \ldots g_n \in G{^{(n)}}\;$ such that $\; u=s(g_n) \;$ and $\; v=t(g_1) \;$.
Take into account that we use here and in the sequel the notation $\; g_1 \; g_2\; \ldots \; g_n \;$ for [**backward**]{} paths $$\xymatrix{
t(g_1) & s(g_1) = t(g_2) \ar[l]_-{g_1}
& s(g_2) = t(g_3) \ar[l]_-{g_2}
& \ar[l]_-{g_3} }
\;\;\dots\;\;
\xymatrix{ & s(g_n) \ar[l]_-{g_n} }$$ A graph $G$ is called [**stochastic**]{} if its edges $g$ are equipped with positive numbers $ p(g) $ such that $\; \sum_{g \in
G_u}\;p(g) \;=\; 1 \;$ for all $\; u \; \in G{^{(0)}} \; $. The weights $\; p(g) \;,\; g \in G \;$, determine the backward transition probabilities of the Markov chain induced by $G$.
We shall assume in the sequel that there exist stationary probabilities $ p{^{(0)}}(u) > 0 $ on $ G{^{(0)}}$ such that $$\label{PR}
\sum_{u \in G{^{(0)}}}\;p{^{(0)}}(u) \;=\; 1 \;\;\;,\;\;\;
\sum_{g \in _vG} p(g) p{^{(0)}} (s(g)) \;=\; p{^{(0)}} (v)$$ for all vertices $ \;u\;,v\;\in G{^{(0)}} $.
It is known that the stationary probabilities on $G{^{(0)}}$ exist iff the corresponding to $G$ Markov chain is positively recurrent. If, in addition, the irreducibility condition hold, the stationary probabilities $\;p{^{(0)}}(u) \;,\; u \in G{^{(0)}} \;$ on the vertices are uniquely determined by the transition probabilities $\;
p(g)\;,\;g \in G\;$ on the edges.
Thus any stochastic graph $(G,p)$ induces a Markov chain on the state space $G$ with the transition probabilities matrix $\; P
\;=\;{(P(g,h))}_{g\in G , h \in G} \;$, where $$P(g,h) \;=\;
\left\{
\begin{array}{ll}
p(h) \;, & if \;\; t(g)=s(h) \; \\
0 \;\;, & otherwise.
\end{array}
\right.$$ In the sequel we mainly deal with stochastic graphs, which induce irreducible positively recurrent Markov chains.
The one-sided Markov shift $ T_G $, induced by the stochastic graph $ G $ , is defined as follows. Let $$X_G = \{x={\{g_{n}\}}_{n=1}^{\infty} \in G^{{{\mathbb{N}}}} \;{ : }\;
s(g_1)=t(g_2)\;,\; s(g_2)=t(g_3) \;,\; \dots \; \}$$ and the Markov measure $m_G$ on $ X_G $ is given by $$p(g_{1}) \; p(g_{2}) \; \ldots \; p(g_n) \; p{^{(0)}}(s(g_n))$$ on the cylindric sets of the form $$A(g_1\;g_2\; \ldots \;g_n)
\;:=\; \{x={\{x_k\}}_{k=1}^{\infty} \in X_G
\;{ : }\; x_1=g_1 \;,\; \ldots \;,\; x_n=g_n \;\}$$ where $\; g_1\;g_2\; \ldots \;g_n) \; \in \; G{^{(n)}}\;$ is a $G$-path of length $n$ in $G$.
The one-sided shift $T_G$ acts on the probability space $\;( X_G , m_G) \; $ by $$T_G({\{x_n\}}_{n=1}^{\infty}\;) \;=\; {\{x_{n+1}\}}_{n=1}^{\infty}$$ and $T_G$ preserves the Markov measure $m_G$. The shift $T_G$ is ergodic iff the graph $G$ is irreducible. Under the irreducibility condition, the stationary probabilities $p{^{(0)}}$ on $G{^{(0)}}$ and, hence, the $T_G$-invariant Markov measure $m_G$ are uniquely determined by the stochastic graph $\; (G,p) \;$.
The coordinate functions $$Z_n \;{ : }\; X_G \ni x={\{x_k\}}_{k=1}^{\infty}\; {\rightarrow}\; x_n
\in G \;\;,\;\;n \in {{\mathbb{N}}}$$ form a stationary Markov chain on $ ( X_G , m_G) $ with the backward transition probabilities $$P(g,h) \;=\;m_G\{\; Z_n = h \;|\; Z_{n+1} = g \;\} \;=\; p(h)
\;\;,\;\; n \in {{\mathbb{N}}}$$ for all $\; (h,g) \in G^{(2)} \;$.
Consider now the partitions $${\zeta}_n = {Z_n}^{-1}{\varepsilon}_G =
T_G^{-n+1}{\zeta}_1 = \{ T_G^{-n+1}A(g) \}_{g \in G}
\;\;,\;\; n \in {{\mathbb{N}}}\;,$$ generated by $Z_n$ on $ X_G$, where $$A(g) \;=\;
\{\; x={\{x_k\}}_{k=1}^{\infty} \in X_G \;{ : }\; x_1=g \;\}$$ Setting $ {\zeta}= {\zeta}_1 $ and $ T = T_G $, we have $$\label{gen par}
\bigvee^{\infty}_{n=1}T^{-n+1}{\zeta}= {\varepsilon}$$ $$\label{Mar par}
{\zeta}\perp \bigvee^{\infty}_{n=1}T^{-n}{\zeta}\pmod { T^{-1}{\zeta}}$$
Recall that a measurable partition ${\zeta}$ of $(X,m)$ is said to be a [**one-sided Markov generator**]{} or [**one-sided Markov generating partition**]{} for an endomorphism T of $(X,m)$, if the above conditions (\[gen par\]) and (\[Mar par\]) hold.
The partition $ {\zeta}_G $ will be called the [**standard**]{} one-sided Markov generator of the one-sided Markov shift $T_G$ on $ X_G $.
\[Bern graph\] Let $(I,\rho)$ be a finite or countable alphabet and $$\rho = \{ \rho(i) \;,\; i \in I \} \;,\;\; \rho (i) > 0
\;\;,\;\; \sum_{i \in I}{\rho(i)} = 1 .$$ be a probability on $I$. We shall consider $(I,\rho)$ as a stochastic graph, which has the set of edges $i \in I$ with weights $\; \rho(i) \;$ and a single vertex, denoted by$\;"o"\;$. So $G{^{(0)}} = \{o\}$ is a singleton and $ s(i) = t(i) = o $ for all $i \in I$. We shall say that $(I,\rho)$ is the [**standard Bernoulli graph.**]{}
For instance, $\; \xymatrix{ o \ar@(ul,dl) []_{p} \ar@(dr,ur) []_{q} } \;$ if $\; |I| = 2 \;$ and $\; \rho = (p,q) \;$.
The corresponding to $(I,\rho)$ one-sided Markov shift $T_I$ coincides with the Bernoulli shift $ T_I = T_\rho $. The generating partition $ {\zeta}_I$ coincides with the standard Bernoulli generator $\; {\delta}_\rho = \{ B_\rho(i) \}_{i \in I} \;$, defined by (\[delta rho\]).
[**Induced shift**]{} $\; {\mathbf{T}}_{\mathbf{u}}\;$. For any $\; u \in G{^{(0)}} \;$, denote $$D(u) \;:=\; \{\; x={\{x_k\}}_{k=1}^{\infty} \in
X_G \;{ : }\; t(x_1)=u \;\} \;\;,\;\; u \in G{^{(0)}} \;.$$ and consider the partition $\; {\zeta}{^{(0)}} := {\{ D(u) \}}_{u \in G{^{(0)}}} \;$ on the space $X_G$. The partition ${\zeta}{^{(0)}}$ is a Markov partition with respect to shift $T_G$ , i.e. $$\label{Mar par 0}
{\zeta}{^{(0)}} \perp T_G^{-1} {\varepsilon}_{X_G} \pmod { T_G^{-1}{\zeta}{^{(0)}} } \;,$$ but it is not a one-sided generator for $T_G$, in general.
We shall use in the sequel the endomorphisms $ T_u := (T_G)_{D(u)} $, induced by the shift $T_G$ on elements $ D(u) $ of ${\zeta}{^{(0)}}$, $\; u \in G{^{(0)}} \;$. The Markov property (\[Mar par 0\]) provides that for every $u \in G{^{(0)}} $ the induced endomorphism $ T_u $ is a Bernoulli shift. More exactly, in accordance with the general definition of return functions (\[ret fun\]) we have $${\varphi}_u(x)={\varphi}_{D(u)}(x) := min \{ n \geq 1 { : }T_G^{n}x \in D(u) \}
\;\;,\;\; x \in D(u)$$ and $$T_u x \;=\; T_G^{{\varphi}_u(x)} x \;\;,\;\; x \in X_G \;.$$ Take $\; I_u = \bigcup_{n=1}^\infty I_{u,n} \;$, where $I_{u,n}$ be the set of all $ g_1g_2 \ldots g_n \in G{^{(n)}}$ such that $$\label{I u n}
t(g_1) = s(g_n) = u \;\;,\;\;
s(g_k) = t(g_{k+1}) \neq u \;,\; k = 1,2, \ldots ,n-1 \;.$$ Define also $\; \rho_u = \{\rho_u(i)\}_{i \in I_u} \;$ by $$\label{rho u}
\rho_u(i) = p(g_1) p(g_2) \ldots p(g_n) \;\;,\;\;
i = g_1g_2 \ldots g_n \in I_{u,n} \;.\; n \in {{\mathbb{N}}}\;.$$ For any $\; i = g_1g_2 \ldots g_n \in I_{u,n} \;$ we set $ B_u(i) := A(g_1g_2 \dots g_n)$ and consider the partition $\; {\zeta}_u = \{ B_u(i)\}_{i \in I_u} \;$, whose elements are enumerated by the alphabet $I_u$. The Markov property (\[Mar par 0\]) implies that the partitions $\; T_u^{-n} {\zeta}_u \;,\; n \in {{\mathbb{N}}}\;$ are independent. Thus
\[Tu\] The induced endomorphism $T_u$ is isomorphic to the Bernoulli shift $T_{\rho_u}$ and $ {\zeta}_u $ is a one-sided Bernoulli generator of$T_u$.
Graph homomorphisms and skew products {#ss3.2}
-------------------------------------
Now we want to establish the class of graph homomorphisms that we shall use.
\[hom\] Let $\; G \;$ and $\; H \;$ be two stochastic graphs.
1. A map $\; \phi { : }G {\rightarrow}H \;$ is a [**graph homomorphism**]{} if there exists a map $\; \phi{^{(0)}} : G{^{(0)}} {\rightarrow}H{^{(0)}} \;$ such that $$s(\phi (g)) = \phi{^{(0)}} (s(g)) \;\;\;,\;\;\;
t(\phi (g)) = \phi{^{(0)}} (t(g))$$ for all $\; g \in G \;$. (Note that, if the map $\; \phi{^{(0)}} \;$ exists it is unique.)
2. A graph homomorphism $\;\phi { : }G {\rightarrow}H \;$ is [**deterministic**]{} if $\; \phi{^{(0)}} (G{^{(0)}}) = H{^{(0)}} \;$ and for every $\; u \in G{^{(0)}} \;$ the restriction of $\phi $ on $ G_u$ $$\phi {|}_{G_u} \;{ : }\; G_{u} \; \rightarrow \;H_{\phi{^{(0)}} (u)}$$ is a bijection of this set onto $\; H_{\phi{^{(0)}} (u)} \;$.
3. A graph homomorphism is [**weight preserving**]{} or ${\mathbf{p}}$[**-preserving**]{} if $\; p(\phi (g)) \;=\; p(g) \;$ for all $\; g \in G \;$.
Two edges $g_1$ and $g_2$ are said to be [**congruent**]{} if $$s(g_1)=s(g_2) \;\;,\;\; t(g_1)=t(g_2) \;\;,\;\; p(g_1)=p(g_2) \;.$$ The map $ \phi{^{(0)}} $ in the above definition is uniquely determined by $ \phi $, but $ \phi{^{(0)}} $ does not determines $
\phi$ if $G$ has congruent edges.
Anyway one can use a more explicit notation $$( \phi , \phi{^{(0)}} ) \;{ : }\; (G , G{^{(0)}}) \; {\rightarrow}\; (H , H{^{(0)}} )$$ for the homomorphism $\;\phi { : }G {\rightarrow}H \;$.
We shall denote by $\; {{\mathcal Hom}}(G,H) \;$ the set of all weight preserving deterministic graph) homomorphisms $\;\phi { : }G
{\rightarrow}H \;$. In the sequel the term “homomorphism” always means just [**weight preserving deterministic graph homomorphism**]{}.
\[factor map\] Let $\; \phi { : }G {\rightarrow}H\;$ be a map.
1. If $ \phi $ is a graph homomorphism, it induces a factor map $$\Phi_\phi \;{ : }\; X_G {\rightarrow}X_H \;\;,\;\;
\Phi_\phi ( {\{x_n\}}_{n=1}^{\infty} ) \;=\;
{\{ \phi ( x_n ) \}}_{n=1}^{\infty}$$ such that $\; \Phi_\phi \circ T_G \;=\; T_H \circ {\Phi}_{\phi}\;.$
2. If, in addition, $\phi $ is weight preserving, the factor map $ \Phi_\phi $ is measure preserving, $\; ( m_H = m_G \circ {\Phi_\phi}^{-1} ) $.
3. If $\;\phi\;$ is also deterministic, the shift $T_G$ can be represented as a skew product $$\label{skew prod}
{{\bar T}}(x,y) = ( T_H x \;,\; A(x) y) \;\;,\;\; (x,y) \in X_H \times Y$$ where $\; \{ A(x) \;,\; x \in X_H \} \;$ is a measurable family of automorphisms of $Y$.
4. If $T_G$ is ergodic, $Y$ is a homogeneous Lebesgue space.
Parts (i) and (ii) follow directly from Definition \[hom\]. Part (iii) and (iv) can be proved by analogy with Proposition \[T decompos\]
Moreover
\[fin ind\] Let $\phi \in {{\mathcal Hom}}(G,H) $ and suppose that the shift $ T_G $ is ergodic. Then there exists $\;d \in {{\mathbb{N}}}\;$ such that $\; | \Phi_\phi^{-1}(x) | \;=\; d \;$ for almost all $\; x \in X_H \;$. That is, in the skew product (\[skew prod\]) the space $Y$ is finite, $\; |Y| = d \;$.
Note that Theorem \[fin ind\] claims the finiteness of $d$ even in the case, when the graph $G$ is not finite, i.e. $ |G| = \infty
$. This is a consequence of positive recurrence of the corresponding to $G$ Markov chain. The skew product decomposition (\[skew prod\]) of $T_G$ over $T_H$ is a $d$-extension.
Theorem \[fin ind\] was proved earlier in a particular case, when $H$ is a Bernoulli graph, i.e. when $ H{^{(0)}} = \{ o \} $ is a singleton (See Theorem 3.3 and Corollary 3.4 from , and also Theorem \[zet del\] below).
We omit the proof of Theorem \[fin ind\] in general case , since only the pointed out particular case is considered in this paper.
\[def d(phi)\]. The integer $d$ in Theorem \[fin ind\], i.e. the degree of the factor map $\Phi_\phi$ will be called the [**degree**]{} of the homomorphism $\phi$.
Denoting the degree by $\; d(\phi) \;$, we have $\; d(\phi) = |\Phi_\phi^{-1}(x)| \;$ for a.a. $ x \in X_H $.
The following construction plays a central role in our explanation.
\[GSP\]
Let $\;d \in {{\mathbb{N}}}\;$ and let $\; Y_d = \{ 1,2, \ldots , d \} \;$ consists of $d$ points of measure $\frac{1}{d}$. Denote by $\; {{\mathcal A}_d}= {\mathcal A}(Y_d) \;$ the full group of all permutations of $Y_d$.
Given a stochastic graph $ H $, equipped with a function $\; a { : }H \ni h {\rightarrow}a(h) \in {{\mathcal A}_d}\;$, we construct a stochastic graph $ {{\bar H}}_a $ and a homomorphism $\; \pi_H { : }{{\bar H}}_a {\rightarrow}H \;$ by $${{\bar H}}_a \;=\; H \;\times \; Y_d \;\; , \;\;
{{{\bar H}}_a}{^{(0)}} \;=\; H{^{(0)}} \; \times \; Y_d \; ,$$ with $$s({{\bar h}}) = (s(h),y) \;\;,\;\;
t({{\bar h}}) = (t(h),a(h)y) \;\;,\;\; p({{\bar h}}) = p(h)$$ for $\; {{\bar h}}= (h,y) \in {{\bar H}}_a = H \times Y_d \;$ and also $$p{^{(0)}}({{\bar u}}) = p{^{(0)}}(u) \;\;,\;\;
{{\bar u}}= (u,y) \in {{{\bar H}}_a}{^{(0)}} = H{^{(0)}} \times Y_d$$. The natural projection $$\pi_H { : }{{\bar H}}_a = H \times Y_d {\rightarrow}H \;\;,\;\;
{\pi_H}{^{(0)}} { : }{{\bar H}}_a^{(0)} = H{^{(0)}} \times Y_d {\rightarrow}H{^{(0)}}$$ is a homomorphism.
\[def GSP\] We shall say that the graph $ {{\bar H}}_a $ is a [**skew product**]{} over $H$ and the homomorphism $\;\pi_H { : }{{\bar H}}_a {\rightarrow}H \;$ is a [**graph skew product**]{} (or [**GSP**]{}) $d$-extension of $H$ .
In the above construction we have $\; | {\pi_H}^{-1} (h) | = d \;$ for all $\; h \in H \;$ and this is, in fact, a characteristic property of the graph skew product $d$-extension in the following sense
\[hom equi\] Two homomorphisms $\; \phi_k { : }G_k {\rightarrow}H \;,\; k = 1,2 \;$ are said to be [**equivalent**]{} if $\; \phi_2 = {\kappa}\circ \phi_1 \;$ for an appropriate isomorphism $\; {\kappa}{ : }G_1 {\rightarrow}G_2 \;$.
\[d-uniform\] Let $\; d \in {{\mathbb{N}}}\;$. A homomorphism $\; \phi \in {{\mathcal Hom}}(G,H) \;$ is called a [**$d$-extension**]{} if $$\label{phi-1=d}
|{\phi}^{-1} (h)| = d \;\;,\;\; h \in H \;.$$
\[d-unif=GSP\] Any $d$-extension $ \phi { : }G {\rightarrow}H $ is equivalent to a GSP $d$-extension $\; \pi_H { : }{{\bar H}}_a {\rightarrow}H \;$.
Let $\; \phi \in {{\mathcal Hom}}(G,H) \;$ is a $d$-extension. Since $\phi$ is deterministic the restrictions $\; \phi{|}_{G_u}
\;$ are bijections between $G_u$ and $ H_{\phi{^{(0)}}(u)} $ for all $
u \in H{^{(0)}} $. Hence the condition (\[phi-1=d\]) is equivalent to $$|{\phi{^{(0)}}}^{-1}(u)| = d \;,\; u \in H{^{(0)}} \;.$$ For each $u \in H{^{(0)}}$ we can choose a bijection $ w_u $ of $ {\phi{^{(0)}}}^{-1} (u) $ onto $ Y_d $. With any fixed choice of these bijections we set $$H \ni h {\rightarrow}a(h) = w_{t(h)} \circ {w_{s(h)}}^{-1} \in {{\mathcal A}_d}\;,$$ and consider the corresponding skew product graph $ {{\bar H}}_a $. The bijections $ w_u $ uniquely determine an isomorphism $\; {\kappa}{ : }G {\rightarrow}{{\bar H}}_a \;$ such that $\; \phi =\pi_H \circ {\kappa}\;$.
\[re GSP shift\] The Markov shift $T_{{{\bar H}}_a}$ corresponding to a graph skew product $ {{\bar H}}_a $ can be identified with the skew product endomorphism $ {{\bar T}}_{H,a} $, defined by $${{\bar T}}_{H,a} (x,y) = (T_H x , {a(x_1)}^{-1}y) \;\;,\;\;
x ={\{ x_n \}}_{n=1}^{\infty} \in X_H \;,\; y \in Y_d \;.$$ and thus any $d$-extension is a homomorphism of degree $d$.
Indeed, the shift $T_{{{\bar H}}_a}$ acts on the space $$X_{{{\bar H}}_a} = \{ {\{ (x_n,y_n) \}}_{n=1}^{\infty} \;{ : }\;
x = {\{ x_n \}}_{n=1}^{\infty} \in X_H \;,\;
y_n = a(x_n)y_{n+1} \in Y_d \}$$ and the map $$\Psi \;{ : }\; X_{{{\bar H}}_a} \ni {\{ x_n \}}_{n=1}^{\infty} {\rightarrow}({\{ x_n \}}_{n=1}^{\infty},y_1) \in X_H \times Y_d$$ realizes the identification, that is, $\; m_H \otimes m_{Y_d} = m_{{{\bar H}}_a} \circ \Psi^{-1} \;$ and $\; {{\bar T}}_{H,a} \circ \Psi = \Psi \circ T_{{{\bar H}}_a} \;$. Note also that $$\label{Psi gz}
\Psi \; {\zeta}_{{{\bar H}}_a} \;=\; {\zeta}_H \times {\varepsilon}_{Y_d} \;\;,\;\;
\Psi \; {\zeta}_{{{\bar H}}_a}{^{(0)}} \;=\; {\zeta}_H
{^{(0)}} \times {\varepsilon}_{Y_d} \;.$$
Consider now two skew product endomorphisms $\; {{\bar T}}_{H,a_k} \;$, corresponding to graph skew products $ {{\bar H}}_{a_k} $ with two functions $\; a_k { : }H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$.
\[cohom\]
1. Two functions $\; a_k { : }H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$, are said to be [**cohomologous**]{} with respect to $H$ if there exists a map $\; w { : }H{^{(0)}} {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{a cohom}
a_2(h) w(s(h)) = w(t(h)) a_1(h) \;,\; h \in H$$
2. Two measurable functions $\; A_k { : }X_H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$ are said to be [ **cohomologous**]{} with respect to $T_H$ if there exists a measurable map $\; W { : }X_H {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{A cohom}
A_2(x) W(x) = W(T_H x) A_1(x) \;,\; x \in X_H$$
In accordance with Definitions \[hom equi\] and \[cohom\] we can say now that the homomorphisms $\;\pi_H { : }{{\bar H}}_{a_k} {\rightarrow}H $ are equivalent iff the functions $\; a_k : H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$ are cohomologous with respect to $H$.
The equality (\[a cohom\]) is equivalent to (\[A cohom\]) if we take $$A_k(x) = {a_k(x_1)}^{-1} \;,\; k=1,2 \;\;,\;\; W(x) = w(t(x_1))$$ for $\; x = {\{ x_n \}}_{n=1}^{\infty} \in X_H \;$ and given $a_k$ and $w$. Hence if $a_1$ and $a_2$ are cohomologous with respect to $H$, then $A_1$ and $A_2$ cohomologous with respect to $T_H$.
We shall show in Section \[ss4.3\] that the inverse is also true.
\[triv exten\] Let $\chi { : }H {\rightarrow}H_1 $ be a homomorphism and $ \pi_1 : {{\bar H}}_1 {\rightarrow}H_1 $ be a $d$-extension of $H_1$ generated by a function $a_1 : H_1 {\rightarrow}{{\mathcal A}_d}$. Setting $ a(h) :=
a_1(\chi(h)) $ we obtain a $d$-extension $ \pi { : }{{\bar H}}:= {{\bar H}}_a
{\rightarrow}H $ of $H$. The map $ {{\bar \chi}}(h,y) := ({\kappa}(h),y) \;,\; (h,y)
\in {{\bar H}}$ is a homomorphism and the diagram $$\label{diag ext}
\xymatrix{
{{\bar H}}\ar[d]^{\pi} \ar[r]^{{{\bar \chi}}} & {{\bar H}}_1 \ar[d]^{\pi_1} \\
H \ar[r]^{\chi} & H_1 }$$ commutes. The homomorphism $ {{\bar \chi}}$ is called a [**trivial**]{} extension of ${{\bar \chi}}$. If, in addition, $ d(\chi) = 1 $, then $ d({{\bar \chi}}) = 1 $ and hence the corresponding endomorphisms ${{\bar T}}_{H,a}$ and ${{\bar T}}_{H_1,a_1}$ are isomorphic.
Stochastic ${\boldsymbol{\rho}}$-unform graphs {#ss3.3}
----------------------------------------------
We continue to consider $\;(I,\rho)\;$ as the standard Bernoulli stochastic graph, (Example \[Bern graph\])
\[rho uni\] A stochastic graph $ (G,p) $ is called ${\boldsymbol{\rho}}$[**-uniform**]{} if there exists a homomorphism $\; \phi \in {{\mathcal Hom}}(G,I) \;$.
For any such homomorphism $\phi$ and for every $ u \in G{^{(0)}} $ $$\phi \;{|}_{G_u} \;{ : }\;
( G_u \;,\; p \;{|}_{G_u} )\; {\rightarrow}\; (I,\rho)$$ is a weight preserving bijection. Thus the atomic probability spaces $\; ( G_u , p {|}_{G_u} ) \;$ are isomorphic to $(I,\rho)$ for every $ u \in G{^{(0)}} $.
\[p3.4\] $\; T_G \in {\mathcal {UE}(\rho)}\;$ iff $G$ is $\rho$-uniform.
Consider the partition $\; \xi_1 := T_{G}^{-1} {{\varepsilon}}_{X_G} \;$ generated by the shift $T_G$. The Markov property of the measure $m_G$ on $X_G$ implies $$m^{C_{\xi_1}(x)}(\{x\}) \;=\; m_G \{\; Z_1=x_1 \;|\; Z_2=x_2 \;\}
\;=\; p(x_1)$$ for a.a. $\; x={\{x_{n}\}}_{n=1}^{\infty} \in X_G \;$, Here $\; m^{C_{\xi_1}(x)}(\{x\}) \;$ is the conditional measure of the point $x$ in the element $C_{\xi_1}(x) = T_G^{-1}T_Gx $ of the partition $ \xi_1 $. Hence for every $ u \in G{^{(0)}} $ almost all elements $(C,m_C)$ of the partition $ \xi_1 $ are isomorphic to $\; ( G_u , p {|}_{G_u})
\;$ on the set $\; \{ x={\{x_n\}}_{n=1}^{\infty} \in X_G
\;{ : }\; s(x_1) = u \} \;$. But $\;T_G \in {\mathcal {UE}(\rho)}\;$ iff a.a. elements $(C,m_C)$ are isomorphic to $(I,\rho)$. Hence $\;T_G \in
{\mathcal {UE}(\rho)}\;$ iff $\;( G_u , p{|}_{G_u})\;$ are isomorphic to $(I,\rho)$ for every $ u \in G{^{(0)}} $.
Let $G$ be a $\rho$-uniform graph and $\; \phi \in {{\mathcal Hom}}(G,I) \;$. Consider the partition $\; \phi^{-1} {\varepsilon}_I = \{ \phi^{-1} (i) \;,\; i \in I \;\}$ of $G$. The first coordinate function $$Z_1 \;{ : }\; X_G \ni x={\{x_k\}}_{k=1}^{\infty}
\; {\rightarrow}\; x_1 \in G$$ generates the following partition $${\delta}_\phi \;=\; Z_1^{-1} ( \phi^{-1} {\varepsilon}_I )$$ of the space $ X_G $. Elements of $\; {{\delta}}_{\phi} \;$ have the form $$B(i) = Z_{1}^{-1} ( \phi^{-1} (i) ) =
\{ x={\{x_{k}\}}_{k=1}^{\infty} \in X_G \;{ : }\;
{\phi}(g) = i \} \;,\; i \in I$$ Using the standard Markov generator $${\zeta}_G = Z_{1}^{-1}{\varepsilon}_G = {\{A(g)\}}_{g \in G} \;\;,\;\;
A(g) = Z_{1}^{-1}(g)$$ of $T_G$, we have $$B(i) = {\bigcup}_{g \in {\phi}^{-1}(i)} A(g)$$ and $$m_G(B(i)) = {\sum}_{g \in {\phi}^{-1}(i)} p(g) p{^{(0)}}(s(g))
= \rho(i) {\sum}_{u \in G{^{(0)}}} p{^{(0)}}(u) = \rho(i)$$ for $\; i \in I \;$. Hence for $\; {\delta}\;=\; {\delta}_\phi \;$ we have $$\label{del phi}
{\delta}\in IC (\; T_{G}^{-1} {\varepsilon}_{X_G} \;) =
{\Delta}_\rho( T_G ) \;\;,\;\; {\delta}\leq {\zeta}_G$$ Denoting by $\; {\Delta}_\rho(T_G , {\zeta}_G) \;$ the set of all $ {\delta}$ that satisfy (\[del phi\]), we have also
\[d(T,zeta)\] $\; {\Delta}_\rho(T_G , {\zeta}_G) \;$ is precisely the set of all $ {\delta}$ of the form $\; {\delta}\;=\; {\delta}_\phi \;$.
Now we introduce a semigroup $\; {{\mathcal S}}(\phi) \;$ of maps $\; f { : }G{^{(0)}} {\rightarrow}G{^{(0)}} \;$ induced by the homomorphism $\phi$.
Let $\; i \in I \;$. Since $\; \phi \;$ is deterministic the restriction $\; \phi {|}_{G_u} \;{ : }\; G_u \; {\rightarrow}I \;$ is a bijection of $G_u$ onto $I$ for every $u \in G{^{(0)}}$. Hence for any pair $(i,u)$ there exists an unique $\; g_{i,u} \;$ such that $\; \phi (
g_{i,u} ) = i \;$ and $\; s( g_{u,i} ) = u \;$. Putting $\; f_iu =
g_{i,u} \;$, we get a map $\; f_i { : }G{^{(0)}} {\rightarrow}G{^{(0)}} \;$. Let $\; {{\mathcal S}}(\phi) \;$ be the semigroup generated by the maps $\;
\{ f_i \;,\; i \in I \} \;$.
Let $ {{\mathcal F}}{{\mathcal S}}(I) $ be the set of all finite words $\; i_1i_2
\dots i_n \;$ in the alphabet $I$. We shall consider $ {{\mathcal F}}{{\mathcal S}}(I) $ as a free semigroup with the generating set $I$ and with juxtaposition multiplication: $$i_1i_2 \dots i_m \cdot j_1j_2 \dots j_n
\;=\; i_1i_2 \dots i_mj_1j_2 \dots j_n$$ and set $$f_{i_1i_2 \dots i_n} \;=\; f_{i_1} \circ f_{i_2} \circ \; \dots \;
\circ f_{i_{n}} \;\;,\;\; i_1i_2 \dots i_n \; \in \; I^n$$ Then $\; i_1i_2 \dots i_n {\rightarrow}f_{i_1i_2 \dots i_n} \;$ is a homomorphism from the semigroup $ {{\mathcal F}}{{\mathcal S}}(I) $ onto the semigroup $${{\mathcal S}}(\phi) = \{ f_{i_1i_2 \dots i_n} \;,\;i_1i_2 \dots i_n
\in {{\mathcal F}}{{\mathcal S}}(I) \} \;,$$ generated by $\; \{ f_i \;,\; i \in I \} \;$.
Now we can describe the partitions $${\delta}_\phi = \{ B(i) \}_{i \in I} \;\;,\;\;
{\delta}_\phi{^{(n)}}\;=\; \bigvee_{k=1}^{n} T_{G}^{-k+1} {\delta}_\phi
\;,\; n \in {{\mathbb{N}}}$$ as follows.
First recall that the partition $\; {\zeta}{^{(0)}}$ consists of the atoms $D(u) = Z^{-1}(_uG) \;,\;u \in G{^{(0)}} \;$ and rename the elements $\; A(g) \;,\; g \in G \;$ of the partition ${\zeta}_G $ by $$D(i,u) := A(g_{i,u}) \;\;,\;\; u \in G{^{(0)}} \;,\; i \in I \;.$$ Then for all $i \in I$ and $u \in G{^{(0)}}$ we have $\; D(i,u) = B(i) \cap T_{G}^{-1}D(u) \;$, $$D(i,u) = \{ x={ \{ x_n \} }_{n=1}^{\infty} \in X_G
\;{ : }\; t(x_1) = u \;,\; \phi(x_1) = i \}$$ and $$\label{Bi Diu}
B(i) = \bigcup_{u \in G{^{(0)}}} D(i,u) \;\;\;,\;\;\;
D(u) = \bigcup_{v { : }f_i(v)=u} D(i,v) \;.$$ Further for any $\; g_1g_2 \dots g_n \in G{^{(n)}}\;$ there exists a unique pair $\; (i_1i_2 \dots i_n,u) \in I^n \times G{^{(0)}} \;$ such that $$\label{u i t}
u = s(g_n) \;,\; i_k = \phi (g_k) \;,\; t(g_k) = f_{i_k} (s(g_k))
\;,\; k = 1,2, \dots ,n$$ Hence any atom $\; A(g_1g_2\; \dots \,g_n) \;$ of the partition $\; {\zeta}_G{^{(n)}}= \bigvee_{k=1}^{n} T^{-k+1} {\zeta}_G \;$ can be renamed by $\;D(i_1i_2 \dots i_n,u) = A(g_1g_2\; \dots \,g_{n}) \;$, where the pair $\; (i_1i_2 \dots i_n,u) \;$ satisfies (\[u i t\]). By (\[Bi Diu\]) any atom $\; B(i_1i_2\; \dots \;i_n) \;$ of the partition $\; {\delta}_\phi{^{(n)}}= \bigvee_{k=1}^{n} T_{G}^{-k+1}
{\delta}_\phi \;$ has the form $$B(i_1i_2\; \dots \;i_n)
\;=\; \bigcup_{u \in G{^{(0)}}} D(i_1i_2 \dots i_n,u)$$ and since $$D(i_1i_2...i_n,u) \;=\; B(i_1i_2\; \dots \;i_n) \cap T^{-n} D(u)$$ we have $$m_G(B(i_1i_2\; \dots \;i_n)) \;=\;
\rho (i_1) \rho (i_2) \; \dots \; \rho (i_n),$$ $$m_G( D(i_1i_2 \dots i_n,u) ) \;=\;
\rho (i_1) \rho (i_2) \; \dots \; \rho (i_{n})p{^{(0)}}(u).$$ Any ${\zeta}_G{^{(0)}}$-set has the form $$D(E) = \{ x={ \{ x_n \} }_{n=1}^{\infty} \in X_G
\;{ : }\; t( x_1 ) \in E \}.$$ for a subset $\; E \subseteq G{^{(0)}} \;$. Then for any $ i_1i_2 \; \dots \; i_n \in I^n $ $$D(E) \cap B(i_1i_2\; \dots \;i_n) \;=\;
\bigcup_{u { : }f_{i_1i_2 \;\dots\; i_n} (u) \in E }
D(i_1i_2 \dots i_n,u).$$ Hence $$\label{D(E)}
m_G ( D(E) \; | \; B(i_1i_2 \; \dots \; i_n) ) \;=\;
p{^{(0)}} ( f^{-1}_{i_1i_2\; \dots \; i_n} (E) ).$$ Next theorem is basic for our explanation. Let $${\delta}_{\phi}^{(\infty)} \;=\; \bigvee_{n=1}^{\infty} {\delta}_\phi{^{(n)}}\;=\; \bigvee_{n=1}^{\infty} T_{G}^{-n+1} {\delta}_\phi$$
\[zet del\] Let $\; \phi \in {{\mathcal Hom}}(G,I) \;$. Then
1. $\; {\zeta}_G \; \vee \; {\delta}_{\phi}^{(\infty)}
\;=\; {\varepsilon}_{X_G} \;.$
2. $\; d(T_G) \;\leq\; d(T_G,{\delta}_{\phi}) \;=\;
d(\phi) \;<\; \infty \;$
It was proved in that if ${\zeta}$ is a one-sided Markov generator of $T$ and $$T \in {\mathcal {UE}(\rho)}\;,\; {\delta}\in {\Delta}_\rho(T) \;,\; {\delta}\; \leq \; {\zeta}\; ,$$ then $\; {\zeta}\vee {\delta}^{(\infty)} = {\varepsilon}\;$. Hence (i) follows by putting $ {\delta}= {\delta}_\phi $ and $ {\zeta}= {\zeta}_G $ .
Since the partition $ {\zeta}_G $ is finite or countable the equality (i) implies that almost all elements $(C,m_C)$ of the partition ${\delta}_{\phi}^{(\infty)}$ are atomic. Taking in to account the ergodicity of $T_G$, we see that almost all elements of ${\delta}_{\phi}^{(\infty)}$ consist of $d$ atoms of measure $\frac{1}{d}$ for an natural $d$. Herewith by Definitions \[def d(T)\] and \[def d(phi)\] we have $\; d =
d(T_G,{\delta}_{\phi}) = d(\phi)$ and, whence, (ii) follows.
We need the following sharp version of Part (i) of Theorem \[zet del\]
\[zet0 del\] $\; {\zeta}{^{(0)}}_G \; \vee \; {\delta}_{\phi}^{(\infty)} \;=\; {\varepsilon}_{X_G} \;$
Choose an increasing sequence of positive numbers $c_n > 0$ and an increasing sequence of finite subsets $E_n$ of $G{^{(0)}}$ such that $$\label{zet0 del1}
\bigcup_{n=1}^\infty E_n = G{^{(0)}} \;\;\;.\;\;\;
\sum_{n=1}^\infty(1-c_n) < \infty \;\;\;.\;\;\; p{^{(0)}}(E_n)>c_n \;.$$ Since $|E_n|<\infty$ there exist $\; i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n} \in I^{k_n} \;$ and $\; f_n := f_{i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n}} \in {{\mathcal S}}(\phi) \;$ such that $$|f_n(E_n)| \;=\; \min \{ |f(E_n)| \;{ : }\; f \in {{\mathcal S}}(\phi) \}.$$ The choice of $f_n$ provides that all restrictions $\; f{|}_{f_n(E_n)} \;,\; f \in {{\mathcal S}}(\phi) \;$ are bijections.
Consider the sets $$\begin{split}
& B_n := B(i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n}) \;\;,\;\; \\
& B'_n := B_n \bigcap T_G^{-k_n}D(E_n) \;=\;
\bigcup_{u \in E_n}D(i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n},u)
\end{split}$$ and also $$F_n \;:=\; \{ x \in X_G \;{ : }\;
T^{{\omega}_n(x)+n}x \in B^\prime _n \} \;,$$ where $${\omega}_n(x) \;:=\; \min \{k \geq 0 \;{ : }\; T^{n+k}x \in B_n \}$$ Then it is not hard to see that $$m_G(F_n) \;=\; m_G(B'_n \;|\; B_n) \;=\; p{^{(0)}}(E_n) \;>\; c_n \;.$$ Set $\; F \;:=\; \liminf_{n {\rightarrow}\infty}F_n \;$. Then we have $\; m_G(F)=1 \;$, since $\;\sum(1-c_n) < \infty \;$. By constructing, the set $F$ has the following property. Suppose $\; x={\{x_k\}}_{k=1}^{\infty} \;$ and $\;y={\{y_k\}}_{k=1}^{\infty} \;$ belong to $F$ and $$\; \Phi_{{\delta}_\phi}(x) \;=\; \Phi_{{\delta}_\phi}(y)
\;=\; {\{i_k\}}_{k=1}^{\infty}
\;\in \; X_\rho \;.$$ If $\; s(x_m) \neq s(y_m) \;$ for some $m \geq 1 $, then $$t(x_k) = f_{i_k}s(x_k) \neq t(y_k) = f_{i_k}s(y_k) \;\;,\;\;
k \;=\; 1,\,2,\;\dots \;m \;.$$ In other words, if $t(x_1) = t(y_1)$ and $\; \Phi_{{\delta}_\phi}(x) \;=\; \Phi_{{\delta}_\phi}(y) \;$ then $x=y$. Thus $\; {\zeta}{^{(0)}}_G \; \vee \; {\delta}_{\phi}^{(\infty)} \;=\;
{\varepsilon}_{X_G} \;$ on the set $F$ of measure $1$.
Semigroup $ {{\mathcal S}}({\boldsymbol{\phi}})$ and persistent ${\mathbf{d}}$-sets. {#ss3.4}
------------------------------------------------------------------------------------
Let $U$ be a finite or countable set.
\[L set\] Let $ {{\mathcal S}}$ be a semigroup of maps $\;f \;{ : }\; U \;{\rightarrow}\; U\;$ on $U$ and let $\; d \in {{\mathbb{N}}}\;$. Call the semigroup ${{\mathcal S}}$ ${\mathbf{d}}$[**-contractive**]{} if there exists a subset $\; L\; \subseteq U \;$ such that
1. $ |f(L)| \;=\; |L| \;=\; d \;$ for all $\; f \in {{\mathcal S}}\;.$
2. For every finite subset $\; E \subset U \;$ there exists $\; f \in {{\mathcal S}}\;$ with $\; f(E) \subseteq L \;$.
The sets $L$, satisfying (i) and (ii), will be called [**persistent ${\mathbf{d}}$-sets**]{} with respect to $ {{\mathcal S}}$.
Denote by $\; {{\mathcal L}}({{\mathcal S}}) \; $ the set of all such $L$. We have directly from the definition:
- For $\; L \in {{\mathcal L}}({{\mathcal S}}) \;$ and $\; f \in {{\mathcal S}}\;$ the restriction $\; f {|}_{L} \;{ : }\; L \;{\rightarrow}\; f(L) \;$ is a bijection and $\; f(L) \; \in \; {{\mathcal L}}( {{\mathcal S}}) \;$.
- The semigroup $ {{\mathcal S}}$ acts transitively on $ {{\mathcal L}}( {{\mathcal S}}) $, i.e. for every pair $\; L_1 \;,\;L_2\; \in \; {{\mathcal L}}({{\mathcal S}}) \;$ there exists $\; f \in {{\mathcal S}}\;$ such that $\; f(L_1) \;=\; L_2 \;$.
- The integer $d$ is equal to $$\label{d(G)}
d( {{\mathcal S}}) \; := \; \sup_{E \subseteq U \;{ : }\; |E| < \infty}
\;\; \min_{ f \in {{\mathcal S}}} \;\; |f(E)| \; .$$ and $\; d({{\mathcal S}}) \;=\; \min_{ f \in {{\mathcal S}}} \;\; |f(U)| \;$ if $\; |U| \;<\; \infty \;$.
Let $G$ be a $\rho$-uniform stochastic graph and $ \phi \in {{\mathcal Hom}}(G,I) $ be a homomorphism $ \phi { : }G {\rightarrow}I $.
Return to the semigroup $\; {{\mathcal S}}(\phi) \;$ which acts on $\; U = G{^{(0)}} \;$ .
\[d(phi)\] Let $T_G$ be an ergodic one-sided Markov shift corresponding to a $\rho$-uniform stochastic graph $G$ and let $ \phi \in {{\mathcal Hom}}(G,I)$. Then the semigroup $ {{\mathcal S}}(\phi)$ is $d$-contractive on $G{^{(0)}} $ and $$\label{d=d(G)}
d \;=\; d({{\mathcal S}}(\phi)) \;=\; d(T_G , {\delta}_\phi) \;=\; d(\phi)$$
To prove the theorem we shall use the partition ${\zeta}{^{(0)}}_G$ on $G{^{(0)}}$. Recall that ${\zeta}{^{(0)}}_G$ consists of all atoms of the form $\; D(u) = Z_1^{-1} (_uG) \;,\; u \in G{^{(0)}}. \;$. For any subset $E$ of $G{^{(0)}} $ we denote $$D(E) = \{ x={\{x_{n}\}}_{n=1}^{\infty} \in X_G \;{ : }\;
t( x_1 ) \in E \} = \bigcup_{u \in E} D(u) \;,$$ i.e. $ D(E) $ is a $ {\zeta}{^{(0)}}_G$-set corresponding to $E$ in the space $X_G$.
It follows from Theorem \[zet del\] Part (ii) that almost all elements $(C,m_C)$ of the partition ${\delta}_{\phi}^{(\infty)}$ are isomorphic to $Y_d$, where $\; d = d(T_G,{\delta}_{\phi}) \in {{\mathbb{N}}}\;$. Hence $$m( \{x\} \;|\; C_{{\delta}_{\phi}^{(\infty)}}(x)) \;=\; \frac{1}{d}$$ for a.e. $x \in X_G$. Then Lemma \[zet0 del\] implies that there exists a measurable family $\; \{ l(x) \;,\; x \in X \} \;$ of subsets $\; l(x) \subseteq G{^{(0)}} \;$ such that $$\label{l(x)=d}
m ( D(l(x)) \;|\; C_{{\delta}_{\phi}^{(\infty)}} (x) ) \;=\; 1
\;\;,\;\; |l(x)| \;=\; d$$ almost everywhere on $ X_G $.
For any $ L \subseteq G{^{(0)}} $ denote $$\check{L} \;:=\; \{ x \in X_G \;{ : }\; l(x) = L \} \;\;,\;\;
{{\mathcal L}}\;:=\; \{ L \subseteq G{^{(0)}} \;:\; m_G ( \check{L} ) > 0 \} \;,$$ i.e. ${{\mathcal L}}$ is the (finite or countable) set of all essential values of the function $\;x \;{\rightarrow}\; l(x) \;$. We show that $\; {{\mathcal L}}\subseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$, i.e. that every $\; L \in {{\mathcal L}}\;$ satisfies the conditions (i) and (ii) of Definition \[L set\].
Take any finite subset $E \subseteq G{^{(0)}}$ and choose $\; c > 0
\;$ such that $ \; c \;<\; \min_{u \in E}p{^{(0)}}(u) \;$. For $\; L \in {{\mathcal L}}\;$ and almost all $\; x={\{x_{n}\}}_{n=1}^{\infty} \in \check{L} \;$ we have by (\[D(E)\]) $$lim_{n {\rightarrow}\infty} m(D(L) \;|\; C_{{\delta}_\phi{^{(n)}}}(x) \}
\;=\; m_G (D(l(x)) \;|\; C_{{\delta}_\phi{^{(\infty)}}} (x)\} \;=\; 1$$ and by (\[D(E)\]) $$m(D(L) \;|\; C_{{\delta}_\phi{^{(n)}}} (x)\} \;=\;
p{^{(0)}}( f^{-1}_{x_1x_2\; \ldots \;x_n} (L)).$$ Hence we can choose $n$ and $\; (x_1x_2\; \ldots \;x_{n}) \in G{^{(n)}}\;$ such that $$m(B(x_1x_2\; \dots \;x_n) \cap \check{L} ) > 0$$ and then $$p{^{(0)}} (f^{-1}_{x_1x_2\; \dots \;x_{n}} (L)) \;>\; 1-c$$ The choice of $c$ provides $\; f^{-1}_{x_1x_2\; \ldots \;x_n}(E) \supseteq L \;$ and thus Part (ii) of Definition \[L set\] holds. Part (i) follows from the equalities $$\label{f l(x)}
f_{x_1x_2\; \dots \;x_{n}} l(T_G^nx) \;=\; l(x) \;\;,\;\; |l(x)| = d
\;\;,\;\; x \in X_G$$ by the definition of $l(x)$ . We have proved the inclusion $\; {{\mathcal L}}\subseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$ , which implies that the semigroup ${{\mathcal S}}$ is $d$-contractive with $\; d = d(T_G , {\delta}_\phi) $
\[L=L(phi)\] $\; {{\mathcal L}}= {{\mathcal L}}({{\mathcal S}}(\phi)) \;$,
It was proved above that $\; {{\mathcal L}}\subseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$. Take $ M \in {{\mathcal L}}({{\mathcal S}}(\phi)) $ and $ L \in {{\mathcal L}}$. Since also $ L \in {{\mathcal L}}({{\mathcal S}}(\phi))$, there exists $i_1i_2 \dots
i_{n} \in I^n $ such that $$\; f_{x_1x_2\;...\;x_{n}} (M) \;=\; L
\;=\; l(x) \;,\; x \in \check{L} \;$$ Then the equality (\[f l(x)\]) implies $\; M = l(T^{n}x) \;$ on a set of positive measure in $X$ and hence $\; M \in {{\mathcal L}}\;$. Thus $\; {{\mathcal L}}\supseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$.
Note that the notion of $d$-contractive semigroup was introduced in , where an analog of Theorem \[d(phi)\] was also proved. Definition \[L set\] is a generalization of what is called “point collapsing” by M. Rosenblatt , in the case $ |U| < \infty $ and $ d=1 $ . The case , when $ |U| = \infty $ and $ d=1 $, was considered in [@KuMuTo].
Graph skew product representation. {#ss3.5}
-----------------------------------
From now on let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and satisfies the positive recurrence condition.
\[phi bar\] Let $\; \phi \in {{\mathcal Hom}}(G,I) \;$ be a homomorphism of degree $\; d = d(\phi) \;$. Then there exists a commutative diagram $$\label{diag phi bar}
\xymatrix{ {{\bar H}}\ar[d]^{{{\bar \phi}}} \ar[r]^{{{\bar \psi}}} & G \ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ where the graph $ {{\bar H}}= {{\bar H}}_a $ is a graph skew product over $H$, generated by a function $\; a { : }H \ni h {\rightarrow}a(h) \in {{\mathcal A}_d}\;$, the homomorphism $ {{\bar \phi}}$ coincides with the natural projection $\pi_H$, and both the homomorphisms $\; {{\bar \psi}}\in {{\mathcal Hom}}({{\bar H}},G) \;$ and $\; \psi \in {{\mathcal Hom}}(H,I)\;$ are of degree 1. In particular, $\; ({{\bar \phi}},\psi) \in {{\mathcal Ext}^d(I,\rho)}\;$
We construct a commutative diagram $$\label{diag phi hat}
\xymatrix{
{{\hat H}}\ar[d]^{{{\hat \phi}}} \ar[r]^{{{\hat \psi}}} & G \ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ such that the homomorphism $\; {{\hat \phi}}\in {{\mathcal Hom}}({{\hat H}},H) \;$ is a $d$-extension (See Definition \[d-uniform\]) and both homomorphisms $\; {{\hat \psi}}\in {{\mathcal Hom}}({{\hat H}},G) \;$ and $\; \psi \in {{\mathcal Hom}}(H,I) \;$ are of degree 1 .
We shall use the persistent $d$-sets $ {{\mathcal L}}= {{\mathcal L}}({{\mathcal S}}(\phi)) $ of the semigroup $ {{\mathcal S}}(\phi) $, described in Theorem \[d(phi)\] (Section\[ss3.4\]). Since $ {{\mathcal L}}$ is finite or countable we can enumerate the set by an alphabet $J$, setting $ {{\mathcal L}}= \{ L_j \;,\; j \in J \} $.
Recall that the semigroup $ {{\mathcal S}}(\phi) $ is $d$-contractive with $ d = d({{\mathcal S}}(\phi)) $ (Theorem \[d(phi)\]). For any pair $\; i \in I \;,\; j \in J \;$ we have $\; |f_i(L_j)|
= |L_j| =d \;$ and the restrictions $\; f_i {|}_{L_j} \;$ is a bijection of $\; L_j \;$ onto $\;f_i(L_j) \;$, whence, $\;f_i(L_j)
\in {{\mathcal L}}\;$ for all $i$ and $j$. For any $\;i \in I \;$ denote by $f_i^J$ the map $\; J {\rightarrow}J \;$, which is defined by $\; f_i^J j
= j' \;$, where $\; f_i(L_j) = L_{j'} \;$.
To construct Diagram \[diag phi hat\] define first $\; \psi { : }H {\rightarrow}I \;$ with $ \; H := I \times J \;$,$\; H{^{(0)}} := J \;$, where $$s(i,j) = j \;\;,\;\; t(i,j) = f_i^J j \;\;,\;\; p(i,j) = \rho(i)
\;\;,\;\; \psi (i,j) = i \;.$$ Next set $${{\hat H}}{^{(0)}} \;:=\;
\{ (j,u) \in J \times G{^{(0)}} \;{ : }\; j \in J \;,\; u \in L_j \}
\;\;,\;\; {{\hat H}}\;:=\; I \times {{\hat H}}{^{(0)}}$$ with $$s(i,j,u) = (j,u) \;\;,\;\; t(i,j,u) = (f_i^J j , f_i u) \;\;,\;\;
p(i,j,u) = \rho(i) \;.$$ Finally, we define the maps ${{\hat \psi}}$ and ${{\hat \phi}}$ by $${{\hat \phi}}{ : }{{\hat H}}\ni (i,j,u) {\rightarrow}(i,j) \in H \;\;,\;\;
{{\hat \psi}}{ : }{{\hat H}}\ni (i,j,u) {\rightarrow}g_{i,u} \in G \;.$$ where $g_{i,u}$ is uniquely determined by the conditions $s(g) =
u$ and $\phi(g) = i$.
It follows directly from this constructing that $H$ and ${{\hat H}}$ are stochastic graphs, and that ${{\hat \phi}}$ , ${{\hat \psi}}$ and $\psi$ are homomorphisms, and that Diagram \[diag phi hat\] commutes. Point out only that ${{\hat \phi}}$ is a $d$-extension, since $ |L_j| = d $ for all $j$ and hence ${{\hat \phi}}$ is of degree $d$. This implies that $\hat{\psi}$ and $\psi$ are of degree 1 , since $\phi$ is of degree $d$.
It remains to apply Proposition \[d-unif=GSP\] to the homomorphism $\; {{\hat \phi}}{ : }{{\hat H}}{\rightarrow}H \;$.
Next we construct d-extensions with a minimal possible $d$. Let $G$ as above be a $\rho$-uniform stochastic graph. Recall that $G{^{(n)}}$ denotes the set of all $n$-paths in $G$, see (\[G(n)\]). We shall consider $G{^{(n)}}$ as a stochastic graph with the set of vertices $G^{(n-1)}$, where for any $\; g{^{(n)}}=
g_1\;g_2\; \ldots \;g_n \;$ $$s(g{^{(n)}}) \;=\; g_2\;g_3\; \ldots \;g_n \;\;,\;\;
t(g{^{(n)}}) \;=\; g_1\;g_2\; \ldots \;g_{n-1}$$ and $\; p(g{^{(n)}}) \;=\; p(g_1) p(g_2) \ldots p(g_n) \;$. If $G$ is $\rho$-uniform, the “$n$-stringing” graph $G{^{(n)}}$ is also $\rho$-uniform. The natural projection $$\pi{^{(n)}}{ : }G{^{(n)}}\ni g{^{(n)}}=(g_1\;g_2\;
\ldots \;g_n) {\rightarrow}g_1 \in G$$ is a homomorphism and $\; \phi \circ \pi{^{(n)}}\in {{\mathcal Hom}}(G{^{(n)}},I) \;$ for any $\; \phi \in {{\mathcal Hom}}(G,I) \;$. However, if $(I,\rho)$ has congruent edges there exist $\; \phi_1 \in {{\mathcal Hom}}(G{^{(n)}},I) \;$, which are not of the above form $\; \phi_1 = \phi \circ \pi{^{(n)}}\;$. It is an obvious fact, that $d(\pi{^{(n)}}) = 1$, i.e. $\;
\Phi_{\pi{^{(n)}}} : X_{G{^{(n)}}} {\rightarrow}X_G$ is an isomorphism. We use the index $d(T,{\delta})$ and the minimal index $d(T)$, which were defined by Definition \[def d(T)\].
\[phi bar d(T)\] Let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and satisfies the positive recurrence condition. Then there exist an integer $n \in {{\mathbb{N}}}$, a homomorphism $\; \phi
\in {{\mathcal Hom}}(G{^{(n)}},I) \;$ and a commutative diagram $$\label{diag bar H Gn}
\xymatrix{
{{\bar H}}\ar[d]^{{{\bar \phi}}} \ar[r]^{{{\bar \psi}}} & G{^{(n)}}\ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ such that
1. The graph ${{\bar H}}= {{\bar H}}_a $ is a skew product over a graph $H$, generated by a function $\; a { : }H \ni h {\rightarrow}a(h) \in {{\mathcal A}_d}\;\;,\;\;d \in {{\mathbb{N}}}\;$, and the homomorphism $ {{\bar \phi}}$ coincides with the natural projection $\pi_H$ of ${{\bar H}}$ onto $H$,
2. $\; d = d(\phi) = d(T_G) \;$,
3. The homomorphisms $\; {{\bar \psi}}\in {{\mathcal Hom}}({{\bar H}},G) \;$ and $\; \psi \in {{\mathcal Hom}}(H,I) \;$ are of degree 1 .
Let ${\zeta}= {\zeta}_G $ be the standard Markov generator of the shift $T_G$. It was proved in that there exist $ n \in {{\mathbb{N}}}$ and $ {\delta}\in {\Delta}_\rho(T_G) $ such that $${\delta}\leq {\zeta}{^{(n)}}:= \bigvee_{k=1}^{n} T^{-k+1} {\zeta}\;\;,\;\; d(T,{\delta}) = d(T) \;.$$ Take ${\zeta}{^{(n)}}$ and $G{^{(n)}}$ instead ${\zeta}$ and $G$ in Proposition \[d(T,zeta)\]. Then we obtain $\; \phi \in {{\mathcal Hom}}(G{^{(n)}},I) \;$ with $\; {\delta}= {\delta}_\phi \;$ and thus, by using Corollary \[phi bar\], we complete the proof.
Homomorphisms and finite extensions. {#s4}
====================================
Homomorphisms of degree 1 {#ss4.1}
-------------------------
Let $H$ be a $\rho$-uniform graph and consider a homomorphism $\; \psi { : }H {\rightarrow}I \;$. Suppose that $\psi$ is of degree 1. By Theorem \[d(phi)\] the semigroup ${{\mathcal S}}(\psi)$, generated by $\; f_i = f_i^\psi \;,\; i \in I \;$, is $1$-contractive and all its persistent sets are singletons. Using $\psi$ we can identify the graph $H$ with $I \times J$, where $\; J = H{^{(0)}} \;$ and for any $ h = (i,j) \in H $ $$\psi (h)=i \;,\; s(h)=j \;\;,\;\; t(h)= f_i j
\;\;,\;\; p(h)=\rho (i)$$ Since $d(\psi)=1$ the partition ${\delta}_\psi$ is a one-sided Bernoulli generator for the Markov shift $T_H$. The factor map $\; \Phi_{{\delta}_\psi} \;{ : }\; X_H {\rightarrow}X_\rho \;$ is an isomorphism, $\; \Phi_{{\delta}_\psi} \;\circ\; T_H = T_{\rho}
\;\circ\; \Phi_{{\delta}_\psi}$ and we can consider the Markov partitions $\; {\zeta}_\rho := \Phi_{{\delta}_\psi}({\zeta}_H) \;$ and $\;
{\zeta}{^{(0)}}_\rho := \Phi_{{\delta}_\psi}({\zeta}_H{^{(0)}}) \;$ for $T_\rho$ on $X_\rho$, which correspond to the Markov partitions ${\zeta}_H$ and ${\zeta}_H{^{(0)}}$ for $T_H$ on $X_H$. The partition ${\delta}_\rho = \Phi_{{\delta}_\psi}({\delta}_H) $ coincides with the standard Bernoulli generator of the Bernoulli shift $T_\rho$.
Thus we have, with the notations from Section \[ss3.3\], $${\delta}_\rho = \{ B_\rho (i) \}_{i \in I} \;\;,\;\;
{\zeta}_\rho = \{ D_\rho (i,j) \}_{ (i,j) \in I\times J} \;\;,\;\;
{\zeta}{^{(0)}}_\rho = \{ D_\rho (j) \}_{ j \in J}$$ where $$D_\rho (i,f_i(j)) = B_\rho (i) \cap T_\rho^{-1} D_\rho (j)
\;\;,\;\; (i,j) \in I\times J$$ Hence the homomorphism $\psi$ is determined by the partitions $\;
{\delta}_\rho \;,\; {\zeta}_\rho \;,\; {\zeta}{^{(0)}}_\rho \;$ uniquely up to equivalence (see Definition \[hom equi\]).
Our aim now is to construct a [**common extension of degree 1**]{} for two homomorphisms of degree 1.
\[degree 1\] Let $\; \psi_1 { : }H_1 {\rightarrow}I \;,\; \psi_2 { : }H_2 {\rightarrow}I
\;$ be two homomorphisms of $\rho$-uniform graphs $H_1$ and $H'_2$ onto the Bernoulli graph $ (I,\rho) $ and suppose that $ \psi_1 $ and $ \psi_2 $ are of degree 1. Then there exist a $\rho$-uniform graph $H$ and homomorphisms $\psi$, $\chi_1$ and $\chi_2 $ of degree 1, for which the following diagram commutes: $$\label{diag degree 1}
\xymatrix{
& H_2 \ar[rr]^{\psi_2} & & I \\
H \ar[ur]^{\chi_2} \ar[urrr]^{\psi} \ar[rr]_{\chi_1} & &
H_1 \ar[ur]_{\psi_1} & }$$
The homomorphism $\psi$ will be called a [**common extension**]{} of $\psi_1$ and $\psi_2$ of degree $1$. Denote by $ ({\zeta}_1 ,
{\zeta}{^{(0)}}_1) $ and $ ({\zeta}_2 , {\zeta}{^{(0)}}_2) $ the pairs of Markov partitions of the space $X_\rho$, which correspond to the homomorphisms $\psi_1$ and $\psi_2$. Here we omit the subscript “$\rho$” and mark the partitions and their elements by subscripts “$1$” and “$2$”, respectively.
We have to construct the desired $H$ and $\; \psi { : }H {\rightarrow}I
\;$ by means of the partitions $${\zeta}:= {\zeta}_1 \vee {\zeta}_2 \;\;,\;\;
{\zeta}{^{(0)}} := {\zeta}_1^{(0)} \vee {\zeta}_2^{(0)} \;.$$ By the identification $H_1= I \times J_1$ and $H_2= I \times J_2$, we have $${\zeta}_1 = \{ D_1 (i,j_1) \}_{ (i,j_1) \in I \times J_1} \;\;,\;\;
{\zeta}_1^{(0)} = \{ D_1 (j_1) \}_{ j_1 \in J_1} \;\;,$$ $${\zeta}_2 = \{ D_2 (i,j_2) \}_{ (i,j_2) \in I \times J_2} \;\;,\;\;
{\zeta}_2^{(0)} = \{ D_2 (j_2) \}_{ j_2 \in J_2} \;\;,$$ and then the partition $ {\zeta}{^{(0)}} $ consists of all elements $$D(j) = D_1 (j_1) \cap D_2 (j_2) \;\;,\;\; j=(j_1,j_2) \in J \;.$$ where the set $J$ is defined by $$\label{eq J}
J := \{ j=(j_1,j_2) \;{ : }\; p{^{(0)}} (j) :=
m_\rho ( D_1 (j_1) \cap D_2(j_2)) >0 \} \subset J_1 \times J_2 \;.$$ For any $i \in I$ and $ j=(j_1,j_2) \in J $ we set $ f_i j := (f_{1,i} j_1 ,f_{1,i} j_2 )$. Then $$\begin{split}
D(f_i j) :=
& D_1 (f_{1,i} j_1 ) \cap D _2(f_{2,i} j_2)
\supseteq D_1(i,j_1) \cap D_2(i,j_2) = \\
& B(i) \cap T_\rho^{-1} (D_1 ( j_1) \cap D_2( j_2)) =
B(i) \cap T_\rho^{-1} D(j)
\end{split}$$ Since $ {\delta}$ and $ T_\rho^{-1} {\varepsilon}$ are independent, this implies $$p{^{(0)}} (f_i j) = m_\rho ( D(f_i j)) \geq
m_\rho(B(i) \cap T_\rho^{-1} D(j)) = \rho(i) p{^{(0)}} (j) \;.$$ Hence $f_i j \in J$ for all $j \in J$ and $ i \in I $ .
Thus we are able to define a stochastic graph $H := I \times J $ with $H{^{(0)}} := J$ such that for any $j \in H{^{(0)}} $ and $h=(i,j) \in H $ $$s(h) := j \;,\; t(h)
:= f_i (j) \;,\; p(h) := \rho (i) \;,\; \psi (h) := i \;.$$ The construction provides that $H$ is a $\rho$-uniform graph , $p{^{(0)}}$ is a stationary probability on $H{^{(0)}} $ and $\psi { : }H
{\rightarrow}I $ is a homomorphism of index $1$. Moreover, if we set $$\chi_1(h) := (i,j_1) \;,\; \chi_2(h) := (i,j_2) \;\;,\;\;
h=(i,j_1,j_2) \in H= I \times J_1 \times J_2 \;,$$ then $\chi_1 { : }H {\rightarrow}H_1$ and $\chi_2 : H {\rightarrow}H_2$ are homomorphisms and Diagram \[diag degree 1\] commutes.
We shall use also the following sharpening of the previous theorem, which can be proved in a similar way.
\[degree 1 sharp\] Let $${\kappa}_1 { : }H_1 {\rightarrow}H_0 \;\;,\;\; {\kappa}_2 { : }H_2 {\rightarrow}H_0
\;\;,\;\; \psi_0 { : }H_0 {\rightarrow}I$$ be homomorphisms of $\rho$-uniform graphs $H_1 \;,\; H_2$ and $H_0$ and suppose they are of degree 1. Then there exist a $\rho$-uniform graph $H$ and homomorphisms $\chi$, $\chi_1$ and $\chi_2 $ of degree 1, for which the following diagram commutes $$\label{diag degree 1 sharp}
\xymatrix{
& H_2 \ar[rr]^{{\kappa}_2} & & H_0 \ar[rr]^{\psi_0} & & I \\
H \ar[ur]^{\chi_2} \ar[urrr]^{\chi} \ar[rr]_{\chi_1} & &
H_1 \ar[ur]_{{\kappa}_1} & & & }$$
Note that this theorem holds without adding of homomorphism $\psi_0$ i.e. for graphs, which are not necessary $\rho$-uniform, but we do not use the fact in this paper.
Extensions of Bernoulli graphs. {#ss4.2}
--------------------------------
Consider a very special case of the graph skew product construction ${{\bar H}}_a$ (see Example \[GSP\]), when the graph $H$ is the standard Bernoulli graph $(I,\rho)$. Let $ d \in {{\mathbb{N}}}$ and let $\; a { : }I {\rightarrow}{{\mathcal A}_d}\;$ be a function on $I$ with the values $\; a(i) , i \in I , \;$ in the group ${{\mathcal A}_d}$ of all permutations of $\; Y_d = \{1,2, \ldots ,d \} \;$. According to the general GSP-construction we have $\; {{\bar I}}_a = I
\times Y_d \;$ , $\; {{{\bar I}}_a}{^{(0)}} = Y_d \;$ and $\; \pi { : }{{\bar I}}_a {\rightarrow}I \;$, where for any $ {{\bar h}}= (i,y) \in {{\bar I}}_a $ $$s({{\bar h}}) = y \;,\; t({{\bar h}}) = a(i)y \;,\; \pi ({{\bar h}}) = i \;,\;
p({{\bar h}}) = \rho(i) \;,\, p{^{(0)}} (y) = d^{-1} \;.$$ The stochastic graph ${{\bar I}}_a$ is $\rho$-uniform and it is irreducible iff the group ${\Gamma}(a) $, generated by $\; \{ a(i) , i
\in I \} \subseteq {{\mathcal A}_d}\;$, is transitive on $Y_d$.
As it was noted in Section \[ss3.2\] (see Remark \[re GSP shift\]) the Markov shift $T_{{{\bar I}}_a} $ is isomorphic to the skew product ${{\bar T}}_{\rho,a}$ , which acts on $X_\rho \times Y_d$ by $$\label{bT rho a}
{{\bar T}}_{\rho,a} (x,y) = (T_\rho x , {a(x_1)}^{-1} y) \;\;,\;\;
x ={\{ x_n \}}_{n=1}^{\infty} \in X_\rho \;,\; y \in Y_d \;.$$
\[th rho cohom\] Let $\; \pi_k \;{ : }\; {{\bar I}}_{a_k} {\rightarrow}I \;,\; k = 1,2, \;$ be two $d$-extensions of the Bernoulli graph $(I,\rho)$, generated by functions $ a_k { : }I {\rightarrow}{{\mathcal A}_d}$, respectively. Let the functions $\; A_k { : }X_\rho {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2, \;$ are defined by $$\label{Am(x)}
A_k(x) := {a_k(x_1)}^{-1} \;\;,\;\; x
={\{x_n\}}_{n=1}^{\infty} \in X_\rho \;.$$ If there exists a measurable function $ W { : }X_\rho {\rightarrow}{{\mathcal A}_d}$ such that $$\label{eq rho cohom}
A_2(x) \cdot W(x) = W(T_\rho x) \cdot A_1(x) \;\;,\;\; x \in X_\rho$$ then $W(x)$ does not depend on $x$ , i.e. $W(x) = w_0 \in {{\mathcal A}_d}$ a.e. on $X_\rho$. Thus $A_1$ and $A_2$ are cohomologous with respect to $T_\rho$ iff $a_1$ and $a_2$ are conjugate in ${{\mathcal A}_d}$, i.e. $\; a_2(i) \cdot w_0 = w_0 \cdot a_1(i) \;,\; i \in I \;$.
Note that the last equality means the equivalence of $a_1$ and $a_2$ in the sense of Definition \[cohom\] , since $I{^{(0)}} = \{ o
\}$.
To proof the theorem we need the following simple lemma.
\[le ext Ber\] Let ${\Gamma}$ be a finite group with the identity element $e$. For any $ b { : }I {\rightarrow}{\Gamma}$ denote $$\label{be(x)}
B{^{(n)}}(x) \;:=\; b(x_1) \cdot b(x_2) \cdot \ldots \cdot b(x_n)
\;\;,\;\; x ={\{x_n\}}_{n=1}^{\infty} \in X_\rho$$ and $$\label{gwb(x)}
{\omega}_b(x) \;:=\; min \{ n \in {{\mathbb{N}}}\;{ : }\; B{^{(n)}}(x) = e \}
\;\;,\;\; x \in X_\rho \;.$$ Then the transformation $T_\rho^{{\omega}_b}$, defined by $$X_\rho \ni x {\rightarrow}T_\rho^{{\omega}_b} x := {T_\rho}^{{\omega}_b(x)}x \in
X_\rho \;,$$ is an ergodic endomorphism of $X_\rho$, which is in fact a one-sided Bernoulli shift.
Consider the ${\Gamma}$-extension of the graph $(I,\rho)$ generated by $b$.
Namely, set $\; {{\tilde{I}}}_b := \; I \times {\Gamma}\;$ and $\; {{{\tilde{I}}}_b}{^{(0)}} := {\Gamma}\;$ with $$s({\tilde{i}}) = (s(i) , {\gamma}) \;,\; t({\tilde{i}}) = (t(i) , b(i) \cdot {\gamma})
\;,\; p({\tilde{i}}) = \rho(i) \;,\; p{^{(0)}} ({\gamma}) = {|{\Gamma}|}^{-1}$$ The skew product endomorphism $ {{\tilde{T}}}_{\rho,b} $ corresponding to the stochastic graph $ {{\tilde{I}}}_b$, acts on the space $X_\rho \times {\Gamma}$ by $${{\tilde{T}}}_{\rho,b} (x , {\gamma}) = (T_\rho x \;,\; B(x) \cdot {\gamma})
\;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_\rho
\;,\; {\gamma}\in {\Gamma}\;.$$ where $\; B(x) := {b(x_1)}^{-1} \;$. The skew product $ {{\tilde{T}}}_{\rho,b} $ can be identified with the Markov shift $T_{{{\tilde{I}}}_b}$ (see Remark \[re GSP shift\] ). Under this identification the partition ${\zeta}^{(0)}_{{{\tilde{I}}}_b}$ coincides with the partition $${\zeta}{^{(0)}} = \nu_{X_\rho} \times {\varepsilon}_{{\Gamma}} =
\{{\tilde{E}}({\gamma}) \}_{{\gamma}\in {\Gamma}} \;,$$ where $${\tilde{E}}({\gamma}) :=
X_\rho \times \{{\gamma}\} \subseteq X_\rho \times {\Gamma}\;\;.\;\; {\gamma}\in {\Gamma}\;.$$ For any ${\gamma}\in {\Gamma}$ consider the endomorphism $ ({{\tilde{T}}}_{\rho,b})_{{\tilde{E}}({\gamma})} $ induced by $ {{\tilde{T}}}_{\rho,b} $ on the set $ {\tilde{E}}({\gamma}) $. Let $${\varphi}_{{\tilde{E}}({\gamma})} \;{ : }\; {\tilde{E}}({\gamma}) \ni (x,{\gamma}) {\rightarrow}{\varphi}_{{\tilde{E}}({\gamma})}(x,{\gamma}) \in {{\mathbb{N}}}$$ be the corresponding return functions (\[ret fun\]).
Since we use the left shifts on ${\Gamma}$ in the definition of the skew product $ {\tilde{T}}_b $ and they commute with the right shifts, we have $${\varphi}_{{\tilde{E}}({\gamma})}(x,{\gamma}) =
{\varphi}_{{\tilde{E}}({\gamma}\cdot {\beta})}(x,{\gamma}\cdot {\beta})
\;,\; {\gamma},{\beta}\in {\Gamma}\;,\; x \in X_\rho \;.$$ Hence with (\[gwb(x)\]) and (\[be(x)\]) we have $${\omega}^b(x) = {\varphi}_{{\tilde{E}}({\gamma})}(x,{\gamma}) \;\;,\;\;$$ and $${{{\tilde{T}}}_b}^{{\omega}^b(x)}(x,{\gamma}) = (T^{{\omega}^b(x)}x,{\gamma})
\;.\; {\gamma}\in {\Gamma}\;,\; x \in X_\rho \;,$$ Thus $T^{{\omega}^b}$ is isomorphic to the endomorphisms $(T_{{{\tilde{I}}}_b})_{D({\gamma})}$ induced by the Markov shift $T_{{{\tilde{I}}}_b}$ on elements $D({\gamma})$ of the partition $ {{\zeta}^{(0)}_{{{\tilde{I}}}_b}}$. So that $T^{{\omega}^b}$ is a Bernoulli shift by Proposition \[Tu\].
[**Proof of Theorem \[th rho cohom\]** ]{} For given two functions $a_1$ an $a_2$ put $$b { : }I \ni i {\rightarrow}b(i) :=
(a_1(i),a_2(i)) \in {\Gamma}:= {{\mathcal A}_d}\times {{\mathcal A}_d}\;.$$ and denote for $k=1,2$ $$A_k^{{\omega}_b}(x) :=
A_k(T_\rho^{{\omega}_b(x)-1}x) \cdot \ldots \cdot A_k(T_\rho x)
\cdot A_k(x) \;\;,\;\; k=1,2$$ with $A_1$ and $A_2$ defined by (\[Am(x)\]). Then by definition of $b$ and ${\omega}_b$ we have $$A_2^{{\omega}_b}(x) := A_1^{{\omega}_b}(x) = e \;\;,\;\; x \in X_\rho \;,$$ where $e$ is the identity of ${{\mathcal A}_d}$. The equality (\[eq rho cohom\]) implies $$A_2^{{\omega}_b}(x) \cdot W(x) = W(T_\rho^{{\omega}_b} x) \cdot A_1^{{\omega}_b}(x)$$ and then $\; W(T_\rho^{{\omega}_b} (x) = W(x) \;$ a.e. on $X_\rho$. By Lemma \[le ext Ber\] $\; T_\rho^{{\omega}_b} $ is ergodic and hence $W(x)$ is constant a.e. on $X_\rho$.
Equivalent extensions. {#ss4.3}
----------------------
Let $d \in {{\mathbb{N}}}$, and $H$ be an irreducible positively recurrent stochastic graph. Given a function $\; a { : }H \ni h {\rightarrow}a(h)
\in {{\mathcal A}_d}\;$ consider the graph skew product $d$-extension ${{\bar H}}_a$ of $H$ generated by the function $a$ (See Example \[GSP\]). Recall that the skew product endomorphism $ {{\bar T}}_{H,a} $, corresponding to $ {{\bar H}}_a $, acts on the space $X_H \times Y_d $ by $${{\bar T}}_{H,a} (x,y) = (T_H x , A(x) y)
\;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_H \;,\; y \in Y_d \;.$$ where $\; A(x) := {a(x_1)}^{-1} \;$. We shall use Definition \[cohom\]
\[Equ ext\] Let $\; \pi_k { : }{{\bar H}}_{a_k} {\rightarrow}H \;,\; k = 1,2 \;$, be two $d$-extensions of $H$ generated by functions $a_1$ and $a_2$, respectively, and let the functions $\; A_k { : }X_H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$ are defined by $$\label{Ak = ak -1}
A_k(x) := a_k(x_1)^{-1}
\;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_H \;.$$ Then the following two conditions are equivalent
1. $A_1$ and $A_2$ cohomologous with respect to $T_H$, i.e. there exists a measurable map $\; W { : }X_H {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{A coho}
A_2(x) \cdot W(x) = W(Tx) \cdot A_1(x) \;\;,\;\; x \in X_H \;,$$
2. $a_1$ and $a_2$ cohomologous with respect to $H$, i.e. there exists a map $\; w { : }H{^{(0)}} {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{a coho}
a_2(h) \cdot w(s(h)) = w(t(h)) \cdot a_1(h) \;,\; h \in H$$
It is obvious that (\[a coho\]) implies (\[A coho\]) with $$\label{W(x)}
W(x) = w(t(x_1)) \;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_H$$ That is (ii) implies (i).
To prove the converse, suppose that (\[A coho\]) holds with a suitable measurable function $\; W { : }X_H {\rightarrow}{{\mathcal A}_d}\;$.
We have to show that the function $W(x)$ necessarily has the form (\[W(x)\]), i.e. $W(x)$ is constant on each element $D(u) =
Z_1^{-1}(_uH)$ of the partition $\; {\zeta}_H{^{(0)}} = \{ D(u) \;,\; u
\in H{^{(0)}} \} \;$.
To this purpose we shall use induced endomorphisms, which are defined as follows.
Fix an atom $D(u)$ of the partition $ {\zeta}_H{^{(0)}} $ and consider the endomorphism $ T_u := (T_H)_{D(u)} $, induced by the shift $T_H$ on $D(u)$, see Section \[ss3.1\]. In accordance with the general definition (\[ret fun\]), the return function $${\varphi}_u(x)={\varphi}_{D(u)}(x) := min \{ n \geq 1 { : }T_H^{n}x \in D(u) \}
\;\;,\;\; x \in D(u)$$ induces $T_u$ by $\; T_ux = T_H^{{\varphi}_u(x)}x \;$. By Proposition \[Tu\] the induced endomorphism $T_u$ is isomorphic to the Bernoulli shift $T_{\rho_u}$, where $\; I_u= \bigcup_{n=1}^\infty
I_{u,n} \;$, and $\; \rho_u = \{\rho_u(i)\}_{i \in I_u} \;$ are defined by (\[I u n\]) and (\[rho u\]). That is, $I_{u,n}$ consists of all $\; h_1h_2 \ldots h_n \in H{^{(n)}}\;$ such that $$t(h_1) = s(h_n) = u \;\;,\;\; s(h_m) = t(h_{m+1}) \neq u \;,\;
m = 1,2, \ldots ,n-1$$ and $$\rho_u(i) = p(h_1) p(h_2) \ldots p(h_n) \;\;,\;\;
i = h_1h_2 \ldots h_n \in I_{u,n} \;.\; n \in {{\mathbb{N}}}\;.$$ For any $u \in H{^{(0)}}$ and $k=1,2$ set $$A_k^{{\varphi}_u}(x) := A_k(T^{{\varphi}_u(x)-1}x)
\cdot \ldots \cdot A_k(Tx) \cdot A_k(x)
\;\;.\;\; x \in D(u)$$ and $$a_k^u (i) := a_k(h_1) \cdot a_k(h_2) \cdot \ldots \cdot a_k(h_n)
\;\;,\;\; i = h_1h_2 \ldots h_n \in I_{u,n} \;.$$ Then $$A_k^{{\varphi}_u}(x) = {a_k^u(i)}^{-1} \;\;,\;\;
x \in B_u(i) \subseteq D(u) \;.\; i \in I_u \;,$$ where $$B_u(i) := \{ x ={\{ x_n \}}_{n=1}^{\infty} \in D(u) \;{ : }\;
(x_1x_2 \ldots x_n) = i \in I_{u,n} \;.$$ Then the equality (\[A coho\]) implies $$\label{Au cohom}
A_2^{{\varphi}_u}(x) \cdot W(x) =W(T_u x) \cdot A_1^{{\varphi}_u}(x)
\;\;.\;\; x \in D(u) \;,$$ i.e. $A_1^{{\varphi}_u}$ and $A_2^{{\varphi}_u}$ are cohomologous on $D(u)$ with respect to $ T_u = T_H^{{\varphi}_u} $.
Since for any fix $u \in D(u)$ the partition $\; {\zeta}_u = \{ B_u(i)
\}_{i \in I_u} \;$ is a one-sided Bernoulli generator for $T_u$, we may apply Theorem \[th rho cohom\] with the Bernoulli shift $
T_u = T_{\rho_u}$ and with the functions $a_k^u \;,\; k=1,2$. Therefore, it follows from (\[Au cohom\]) that there exists $\;
w(u) \in {{\mathcal A}_d}\;$ such that $\; W(x)=w(u) \;$ a.e. on $ D(u) $.
For every $u$ we have now an element $w(u)$ such that $ W(x) =
w(u) = w(t(x_1))$ for a.e. $ x \in D(u) $. Hence $ W(x) $ is of the form (\[W(x)\]), $\; W(T_H x) =
w(s(x_1)) \;$. Thus (\[A coho\]) implies (\[a coho\]).
As a consequence we obtain
\[Equ ext 1\] Let $\; \pi_k { : }{{\bar H}}_{a_k} {\rightarrow}H \;,\; k = 1,2 \;$, be two $d$-extensions of $H$ generated by functions $a_1$ and $a_2$, respectively. Let also $\psi { : }H {\rightarrow}I $ be an homomorphism of degree $1$. Suppose $ d = d({{\bar T}}_{H,a_1}) =
d({{\bar T}}_{H,a_2}) $. Then the endomorphisms ${{\bar T}}_{H,a_1}$ and ${{\bar T}}_{H,a_2}$ are isomorphic iff $a_1$ and $a_2$ cohomologous with respect to $H$.
Since $d(\psi) = 1$ the factor map $\Psi := \Phi_\psi
{ : }X_H {\rightarrow}X_I$ is an isomorphism. Consider two skew products over the Bernoulli shift $T_\rho$ $${{\bar T}}_k (x,y) = ( T_\rho x , B_k(x) y)
\;\;,\;\; (x,y) \in X_\rho \times Y_d
\;\;,\;\; k = 1,2$$ where $ B_k(x) := A_k(\Psi^{-1}x) $ and $A_k$ induced by $a_k$ as above (\[Ak = ak -1\]). Each of the shifts $T_{{{\bar H}}_{a_k}}$ is a simple $\rho$-uniform endomorphism by Theorem \[simp mar\]. The skew products ${{\bar T}}_{H,a_k}$ as well as the shifts $T_{{{\bar H}}_{a_k}}$, are isomorphic to ${{\bar T}}_k$. They are $\rho$-uniform endomorphisms and $d = d({{\bar T}}_1) = d({{\bar T}}_2).$ By Theorem \[simple\] ${{\bar T}}_1$ and ${{\bar T}}_2$ are isomorphic iff the functions $B_1$ and $B_2$ are cohomologous with respect to $T_\rho$. This means that the functions $A_1$ and $A_2$ are cohomologous with respect to $T_H$. Finally, by Theorem \[Equ ext\] the last condition holds iff $a_1$ and $a_2$ cohomologous with respect to $H$.
${{\mathbf{G}}{\mathbf{S}}{\mathbf{P}}}$-extensions and persistent ${\mathbf{d}}$-partitions. {#ss4.4}
----------------------------------------------------------------------------------------------
Let $H$ be a stochastic graph and $(I,\rho)$ a standard Bernoulli graph. In this section we study extensions of the form $$\label{pi psi}
(\pi,\psi) \;{ : }\; \xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I}$$ where the graph $H$ be an extension of the Bernoulli graph $(I,\rho)$ by a homomorphism $\psi$ of degree $ d(\psi) = 1 $ and $ {{\bar H}}= {{\bar H}}_a $ be a graph skew product $d$-extension of $H$ generated by a function $ a { : }H {\rightarrow}{{\mathcal A}_d}$ (See Example \[GSP\]). The diagrams of the above form (\[pi psi\]) will be referred to ${\boldsymbol{(}}{\boldsymbol{\pi}}, {\boldsymbol{\psi}}{\boldsymbol{)}}$[**-extensions**]{}. We assume that the graph $ {{\bar H}}$ is irreducible, i.e. the corresponding Markov shift $T_{{\bar H}}$ and skew product ${{\bar T}}_{H,a} $ are ergodic.
Fixing an extension (\[pi psi\]) and setting $\; J = H{^{(0)}} \;$, we identify $H$ with $ I \times J $ such that $$\psi (h) = i \;,\; s(h) = j \;,\; t(h) = f_i (j)
\;,\; p(h) = \rho(i)$$ for any $ h = (i,j) \in H = I \times J $. Here the maps $ f_i { : }J {\rightarrow}J $ are uniquely determined by $$f_ij = t(i,j) \;\;,\;\; (i,j) \in I \times J$$ and the semigroup $ {{\mathcal S}}(\psi) $, generated by $ \{ f_i \;,\; i
\in I \} $ is $1$-contractive, since $ d(\psi) = 1 \;$ (Theorem \[d(phi)\]).
The $d$-extension $ {{\bar H}}= {{\bar H}}_a $ is described now as follows: $$\label{bar H}
{{\bar H}}= I \times J \times Y_d \;\; , \;\;
{{\bar H}}{^{(0)}} = H{^{(0)}} \times Y_d = J \times Y_d \;,$$ where for any $ {{\bar h}}= (i,j,y) $ $$\label{stp bar H}
s({{\bar h}}) = (j,y) \;,\; t({{\bar h}}) = (f_ij , a(i,j)y) \;,\;
p({{\bar h}}) = \rho(i) \;,\; a(h) = a(i,j)$$ The homomorphisms $\psi$, $\pi$ and $\phi = \psi \circ \pi $ are defined by $$\pi ({{\bar h}}) = h = (i,j) \;\;,\;\; \pi{^{(0)}} (j,y) = j \;\;,\;\;
\phi ({{\bar h}}) = \psi (h) = i \;,$$ where $ d(\phi) = d(\pi) = d $ and the diagram $$\label{diag I J Yd}
\xymatrix{
I \times J \times Y_d \ar[d]^{\pi} \ar[dr]^{\phi} & \\
I \times J \ar[r]^{\psi} & I }$$ commutes. The semigroup ${{\mathcal S}}(\phi)$ can be described now as a $d$-extension ${{\bar {{\mathcal S}}}}= {{\bar {{\mathcal S}}}}(\pi,\psi)$ of the semigroup ${{\mathcal S}}(\psi)$.
Set $$\label{bfi (j,y)}
{{\bar f}}_i(j,y) := t(i,j,y) = (f_i j \;,\; a(i,j) y)
\;,\; (j,y) \in J \times Y_d \;,\; i \in I \;.$$ The maps $ {{\bar f}}_i $ act on $J \times Y_d $.
The semigroup ${{\bar {{\mathcal S}}}}$, generated by $\; {{\bar f}}_i \;,\; i \in I \;$, consists of all maps of the form: $$\{ {{\bar f}}_{i_1i_2 \ldots i_n} \;,\; i_1i_2 \ldots i_n \in I^n \;,\;
n \in {{\mathbb{N}}}\} \;,$$ where $${{\bar f}}_{i_1i_2 \ldots i_n}(j,y) = (f_{i_1i_2 \ldots i_n} j ,
a(i_1i_2 \ldots i_n,j) y)$$ and $$a(i_1i_2 \ldots i_n,j) := a(i_1,f_{i_2i_3 \ldots i_n} j )
\ldots a(i_{n-1},f_{i_n} j ) a(i_n,j) \;.$$ Note that $\; {{\mathcal S}}(\psi) \ni f {\rightarrow}{{\bar f}}\in {{\bar {{\mathcal S}}}}\;$ is an isomorphism between the semigroups.
\[S(phi) bar H\] The semigroup ${{\bar {{\mathcal S}}}}$ is $d$-contractive and its persistent $d$-sets are of the form $${{\mathcal L}}({{\bar {{\mathcal S}}}}) = \{ L_j , j \in J \} \;\;,\;\; L_j := \{ j \} \times Y_d \;.$$
By Theorem \[d(phi)\] the semigroup $ {{\bar {{\mathcal S}}}}$ is $d$-contractive and ${{\mathcal S}}(\psi)$ is $1$-contractive, since $ d(\phi) = d $ and $ d(\psi) = 1 $.
For any finite set $ F \subseteq J \times Y_d $ the set $ E :=
\pi{^{(0)}} (F) \subseteq J $ is also finite. Since the semigroup ${{\mathcal S}}(\psi)$ is $1$-contractive there exist $ i_1i_2 \ldots i_n \in
I^n $ and $j \in J$ such that $\; f_{i_1i_2 \ldots i_n}(E) = \{j\}
\;$ and hence $\; {{\bar f}}_{i_1i_2 \ldots i_n}(F) \subseteq L_j \;$. On the other hand $ d = |L_j| = |{{\bar f}}_i(L_j)| $ for all $i \in I ,
j \in J$. Thus the sets $L_j$ and only they are persistent sets for the semigroup ${{\bar {{\mathcal S}}}}$.
For every $E \subseteq J$ set $ {{\bar E}}:= {\pi{^{(0)}}}^{-1}E = E \times
Y_d $.
\[pers part\] Let the semigroup ${{\bar {{\mathcal S}}}}$ be as above.
1. A subset $ R $ of $ {{\bar H}}{^{(0)}} = J \times Y_d $ will be called [**transversal**]{} with respect to $\pi{^{(0)}}$ if $ \pi{^{(0)}} (R) = H{^{(0)}} = J $ and the restriction $ \pi{^{(0)}} {|}_R { : }R {\rightarrow}J $ is a bijection. A partition $\; r = \{ R_1,R_2, \ldots ,R_d \} \;$ will be called [**transversal**]{} with respect to $\pi{^{(0)}}$ if all the set $R_1,R_2, \ldots ,R_d$ are transversal.
2. A transversal partition $r$ will be called [ **persistent**]{} with respect to semigroup $ {{\bar {{\mathcal S}}}}$, if for every transversal partition $r_1$ and every finite subset $E \subseteq J$ there exists $ {{\bar f}}\in {{\bar {{\mathcal S}}}}$ such that $\; {{\bar f}}^{-1}r_1 \;{|}_{{\bar E}}= r \;{|}_{{\bar E}}\;.$
Denote by ${{\mathcal R}}$ the set of all transversal partitions and by ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ the set of all persistent partitions for the semigroup ${{\bar {{\mathcal S}}}}$.
If a set $R$ is transversal then for any $i \in I$ the set ${{\bar f}}^{-1}R$ is transversal. Hence for any $r = \{ R_1,R_2, \ldots
,R_d \} \in {{\mathcal R}}$ and $ {{\bar f}}\in {{\bar {{\mathcal S}}}}$ we have $${{\bar f}}^{-1} r := \{ \; {{\bar f}}^{-1}R_1 , {{\bar f}}^{-1}R_2, \ldots ,
{{\bar f}}^{-1}R_d \; \} \in {{\mathcal R}}\;.$$ Further we shall use this action $${{\mathcal R}}\ni r \; {\rightarrow}\; {{\bar f}}^{-1} r \in {{\mathcal R}}\;\;,\;\; {{\bar f}}\in {{\bar {{\mathcal S}}}}$$ of the semigroup ${{\bar {{\mathcal S}}}}$ on ${{\mathcal R}}$. The following lemma shows that $ {{\mathcal R}}({{\bar {{\mathcal S}}}})$ is an attracting set for ${{\mathcal R}}$ with respect to the action in a natural sense.
\[gL(phi)\]
1. The set ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ is not empty.
2. ${{\bar f}}^{-1} {{\mathcal R}}({{\bar {{\mathcal S}}}}) \subseteq {{\mathcal R}}({{\bar {{\mathcal S}}}})$ for all ${{\bar f}}\in {{\bar {{\mathcal S}}}}$
3. ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ is the least subset of ${{\mathcal R}}$ with the property (ii).
Consider a subset ${{\mathcal R}}_0 ({{\bar {{\mathcal S}}}})$ of ${{\mathcal R}}$ consisting of all $r \in {{\mathcal R}}$ having the following property:
1. For any finite subset $E \subseteq J $ there exists $\; f \in {{\mathcal S}}(\psi) $ such that $f(E)$ is a singleton, i.e. $|f(E)|=1$, and $ r \;{|}_{{\bar E}}= {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}\;$, where $ {{\bar E}}:= E \times Y_d $ and $ {\varepsilon}= {\varepsilon}_{J \times Y_d} $.
We show that ${{\mathcal R}}({{\bar {{\mathcal S}}}}) = {{\mathcal R}}_0({{\bar {{\mathcal S}}}}) \neq \emptyset$.
Take a sequence $\; E_n \nearrow J \;,\; |E_n| < \infty \;$. Since $d({{\mathcal S}}(\psi)) = 1 \;$ we can find a sequence $g_n \in {{\mathcal S}}(\psi)$ such that for all $n \in {{\mathbb{N}}}$ and $ f_n := g_n \cdot \ldots \cdot g_2 \cdot g_1$, the set $f_n (E_n)$ is single-point, i.e. $|f_n (E_n)|=1$. Using the decreasing sequence of partitions $${\varepsilon}\geq {{\bar f}}_1^{-1} {\varepsilon}\geq {{\bar f}}_2^{-1} {\varepsilon}\geq \; \ldots \;
\geq {{\bar f}}_n^{-1} {\varepsilon}\geq \ldots$$ set $\; r_0 := \bigwedge_{n=1}^{\infty} {{{\bar f}}_n}^{-1}{\varepsilon}\;$. Since $|f_n (E_n)|=1$ the restriction $r_0 \;{|}_{{{\bar E}}_n} $ consists of $d$ sets, whose projections on $J$ are $E_n$. Hence $
r_0 \in {{\mathcal R}}$ and $r_0 \;{|}_{{{\bar E}}_n} = {{\bar f}}_n^{-1}{\varepsilon}\;{|}_{{{\bar E}}_n} $. We see that $r_0 \in {{\mathcal R}}_0({{\bar {{\mathcal S}}}})$, i.e. ${{\mathcal R}}_0$ is not empty,
Let $r \in {{\mathcal R}}_0({{\bar {{\mathcal S}}}})$ and $r_1 \in {{\mathcal R}}$. For any finite subset $E \subseteq J$ there exists ${{\bar f}}\in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ such that $ r
\;{|}_{{\bar E}}= {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}\;$. Then $${{\bar f}}^{-1}r_1 \;{|}_{{\bar E}}\leq {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}= r \;{|}_{{\bar E}}\;,$$ Since each of the partitions consists of $d$ elements, we have also ${{\bar f}}^{-1}r_1 \;{|}_{{\bar E}}= r \;{|}_{{\bar E}}$. Hence $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$.
Conversely, let $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ and $E$ be a finite subset of $J$. There are exist ${{\bar f}}\in {{\bar {{\mathcal S}}}}$ and $r_1 \in {{\mathcal R}}$ such that $
r_1 \;{|}_{{\bar E}}= {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}\;$. On the other hand, since $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$, we can choose ${{\bar f}}_1
\in {{\bar {{\mathcal S}}}}$ for which $ {{\bar f}}_1^{-1}r_1 \;{|}_{{\bar E}}= r \;{|}_{{\bar E}}$. Hence $$r \;{|}_{{\bar E}}= {{\bar f}}_1^{-1}r_1 = {{\bar f}}_1^{-1}{{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}= ({{\bar f}}^{-1}{{\bar f}}_1)^{-1}{\varepsilon}\;{|}_{{\bar E}}.$$ We see that $r \in {{\mathcal R}}_0({{\bar {{\mathcal S}}}})$ and thus ${{\mathcal R}}({{\bar {{\mathcal S}}}}) = {{\mathcal R}}_0$ and Part $(i)$ follows.
Parts $(ii)$ and $(iii)$ follow in the same manner by the definition of ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ and by the equality ${{\mathcal R}}({{\bar {{\mathcal S}}}}) =
{{\mathcal R}}_0({{\bar {{\mathcal S}}}})$.
Irreducible ${\mathbf{d}}$-extensions. {#ss4.5}
---------------------------------------
In this section we continue to study $(\pi,\psi)$-extensions of the form (\[pi psi\]) $$(\pi,\psi) \;{ : }\; \xymatrix{
{{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I } \;,$$ where the graph $H$ is an extension of the Bernoulli graph $(I,\rho)$ by a homomorphism $\psi$ of degree $ 1 $ and $ {{\bar H}}=
{{\bar H}}_a $ be a GSP $d$-extension of $H$, generated by a function $ a
{ : }H {\rightarrow}{{\mathcal A}_d}$.
Fix $d$ and $(I,\rho)$ and consider the set $\; {{\mathcal Ext}^d(I,\rho)}\;$ of all $(\pi,\psi)$-extensions of the form (\[pi psi\]). This set is equipped with a natural partial order and with an equivalence relation as follows
\[partial order\] Let $\; (\pi,\psi) { : }\xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I } \;$ and $\; (\pi_1,\psi_1) { : }\xymatrix{{{\bar H}}_1 \ar[r]^{\pi_1} & H_1 \ar[r]^{\psi_1} & I } \;$ be two $(\pi,\psi)$-extensions from ${{\mathcal Ext}^d(I,\rho)}$. Let $a$ and $a_1$ be the functions, which generate the extensions ${{\bar H}}$ and ${{\bar H}}_1$, respectively.
1. A homomorphism $\; {{\bar {\kappa}}}{ : }{{\bar H}}{\rightarrow}{{\bar H}}_1 \;$ is said to be a [**trivializable**]{} $d$-extension of a homomorphism $\; {\kappa}{ : }H {\rightarrow}H_1 \;$, if the square part of Diagram \[diag order\] ( below ) commutes and the functions
$a_1 \circ \chi$ and $a$ are cohomologous with respect to $H$.
2. We shall say that $$(\pi_1,\psi_1) \; \preceq \; (\pi,\psi)$$ if there is a commutative diagram $$\label{diag order}
\xymatrix{
{{\bar H}}\ar[d]^{{{\bar {\kappa}}}} \ar[r]^{\pi}
& H \ar[d]^{{\kappa}} \ar[dr]^{\psi} & \\
{{\bar H}}_1 \ar[r]^{\pi_1}
& H_1 \ar[r]^{\psi_1} & I }\;,$$ where $\; {{\bar {\kappa}}}\in {{\mathcal Hom}}({{\bar H}},{{\bar H}}_1) \;$ is a trivializable $d$-extension of $\; {\kappa}\in {{\mathcal Hom}}(H,H_1) \;$.
3. We shall say that $$(\pi_1,\psi_1) \; \sim \; (\pi,\psi) \;$$ if there is commutative Diagram \[diag order\], where both $\; {\kappa}\;{ : }\; H {\rightarrow}H_1 \;$ and its $d$-extension $\; {{\bar {\kappa}}}\;{ : }\; {{\bar H}}{\rightarrow}{{\bar H}}_1 \;$ are isomorphisms.
In connection with Part (i) of the definition, note that an extension $\; {{\bar {\kappa}}}\;{ : }\; {{\bar H}}{\rightarrow}{{\bar H}}_1 \;$ is trivializable iff it is equivalent to a trivial extension of $\; {\kappa}\;:\; H
{\rightarrow}H_1 \;$ (see Remark \[triv exten\]).
It can be checked also that $\; (\pi_1,\psi_1) \preceq (\pi,\psi)
\;$ and $\; (\pi,\psi) \preceq (\pi_1,\psi_1) \;$ imply $\;
(\pi_1,\psi_1) \sim (\pi,\psi) \;$, but we do not use the fact in this paper.
Our aim now is to describe “minimal” elements of $\; (\; {{\mathcal Ext}^d(I,\rho)}\;,\; \preceq \;) \;$.
\[irreduc\] An extension $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ is called [**irreducible**]{} if $(\pi_1,\psi_1) \sim (\pi,\psi)$ as soon as $ (\pi_1,\psi_1) \in
{{\mathcal Ext}^d(I,\rho)}$ and $(\pi_1,\psi_1) \preceq (\pi,\psi)$.
\[reduc ext\] For any $\; (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}\;$ there exists a unique up to equivalence irreducible $(\pi,\psi)$-extension $\; (\pi_*,\psi_*) \in {{\mathcal Ext}^d(I,\rho)}\;$ such that $\; (\pi_*,\psi_*) \preceq (\pi,\psi) \;$.
To prove the theorem we fix a pair $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ and again use the identification (\[bar H\]). Namely, $$\label{diagr I J Yd}
(\pi, \psi) \;\;:\;\;
\xymatrix{
{{\bar H}}= I \times J \times Y_d \ar[r]^-\pi
& H = I \times J \ar[r]^-\psi & I }$$ where $H{^{(0)}} = J$ and ${{\bar H}}{^{(0)}} = H{^{(0)}} \times Y_d = J \times Y_d $ as in Section \[ss4.4\].
We construct the desired irreducible $(\pi_*,\psi_*)$-extension and a corresponding commutative diagram $$\label{diag reduc ext}
\xymatrix{
{{\bar H}}\ar[d]^{{{\bar {\kappa}}}_*} \ar[r]^{\pi}
& H \ar[d]^{{\kappa}_*} \ar[dr]^{\psi} & \\
{{\bar H}}_* \ar[r]^{\pi_*}
& H_* \ar[r]^{\psi_*} & I }$$ by means of the semigroup $ {{\bar {{\mathcal S}}}}= {{\bar {{\mathcal S}}}}(\pi,\psi) $ and its persistent partitions ${{\mathcal R}}({{\bar {{\mathcal S}}}})$.
\[reduc part\] A partition $\xi$ of $J=H{^{(0)}}$ is called [**reducing**]{} partition if the following two conditions hold
1. $f^{-1}\xi \leq \xi$ for all $f \in {{\mathcal S}}(\psi)$, i.e. $\xi$ is ${{\mathcal S}}(\psi)$-invariant
2. For any element $C \in \xi$ denote ${{\bar C}}:= \pi^{-1}C$ and let $r {|}_{{\bar C}}$ be the restriction of the partition $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ on the set ${{\bar C}}$. Then all the partitions $\; r {|}_{{\bar C}}\;,\; r \in {{\mathcal R}}({{\bar {{\mathcal S}}}}) \;$ coincide with each other.
Consider the set $\Xi$ of all reducing partitions $\xi$ on $H{^{(0)}}$.
For any $\xi \in \Xi$ we have $\; {\pi{^{(0)}}}^{-1}\xi = \xi \times \nu_{Y_d} \;$ and the partition $\; {\pi{^{(0)}}}^{-1}\xi \vee r \;$ does not depend on the choice of $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$. So that we may set $$\label{bxi}
{{\bar \xi}}:= {\pi{^{(0)}}}^{-1}\xi \vee r \;\;,\;\; \xi \in \Xi$$ and ${{\bar \Xi}}:= \{{{\bar \xi}}\;{ : }\; \xi \in \Xi \}$ on ${{\bar H}}{^{(0)}}$.
Since $\xi$ is ${{\mathcal S}}(\psi)$-invariant and ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ is ${{\bar {{\mathcal S}}}}$-invariant by Lemma \[gL(phi)\], the partition ${{\bar \xi}}$ is also ${{\bar {{\mathcal S}}}}$-invariant.
Therefore we may introduce the [**factor pair**]{} $$\label{factor pair}
\xymatrix{
{{\bar H}}{/}_{{\bar \xi}}\ar[r]^{\pi{/}_\xi}
& H{/}_\xi \ar[r]^{\psi{/}_\xi} & I }$$ Namely, we set $$H{/}_\xi := I \times J{/}_\xi \;\;,\;\;
{{\bar H}}{/}_{{\bar \xi}}:= I \times J{/}_\xi \times Y_d$$ Any element of ${{\bar \xi}}$ consists of $d$ elements of the form $\;
R_y^C \;,\; y \in Y_d \;$, where $C \in \xi$ and $\pi{^{(0)}}(R_y^C) =
C$. Hence. by possibly passing to an equivalent extension, we may assume that $R_y^C = C \times \{y\}$, i.e. ${{\bar \xi}}= \xi \times
{\varepsilon}_{Y_d}$. This means that the function $a = a(i,j)$, generating the extension ${{\bar H}}= {{\bar H}}_a$, does not depend on $j$ on the elements of $\xi$. Hence the equalities (\[stp bar H\]) and (\[bfi (j,y)\]) well define $a{/}_\xi$ and $ {{\bar H}}{/}_{{\bar \xi}}:=
(H{/}_\xi)_{a{/}_\xi}$. Thus we have shown
\[natur proj\] For any $\xi \in \Xi$ the natural projections $${\pi_{{\bar \xi}}}{^{(0)}} { : }{{\bar H}}{^{(0)}} {\rightarrow}{{\bar H}}{^{(0)}}{/}_{{\bar \xi}}\;\;,\;\;
{\pi_\xi}{^{(0)}} { : }H{^{(0)}} {\rightarrow}H{^{(0)}}{/}_\xi$$ uniquely determine $(\pi{/}_\xi,\psi{/}_\xi) \in {{\mathcal Ext}^d(I,\rho)}$ such that $ (\pi{/}_\xi,\psi{/}_\xi) \preceq (\pi,\psi) $ with the coresponding commutative diagram $$\label{diag /xi}
\xymatrix{
{{\bar H}}\ar[d]^{\pi_{{\bar \xi}}} \ar[r]^{\pi}
& H \ar[d]^{\pi_\xi} \ar[dr]^{\psi} & \\
{{\bar H}}{/}_{{\bar \xi}}\ar[r]^{\pi{/}_\xi}
& H{/}_\xi \ar[r]^{\psi{/}_\xi} & I }$$
Conversely
\[convers\] For any $\; (\pi_1,\psi_1) \; \preceq \; (\pi,\psi) \;$ there exists $\xi \in \Xi$ such that $(\pi{/}_\xi,\psi{/}_\xi) \sim (\pi_1,\psi_1)$
Take the map $ {\kappa}{^{(0)}} { : }H{^{(0)}} {\rightarrow}{H_1}{^{(0)}} $ induced by homomorphism $ {\kappa}{ : }H {\rightarrow}H_1 $ from Diagram \[diag ext\] and set $\xi := {{\kappa}{^{(0)}}}^{-1}{\varepsilon}_{{H_1}{^{(0)}}}$. Then $\xi \in \Xi$ and it is desired
[**Proof of Theorem \[reduc ext\].**]{} It is easily to see that $\Xi$ is a lattice, i.e. $ \xi_1 \vee \xi_2 \in \Xi $ and $ \xi_1
\wedge \xi_2 \in \Xi $ for all $\xi_1 , \xi_2 \;\in\;\Xi $. Herewith, $\Xi$ has the least element. Denote the least element by $\xi_* $ and let $ {{\bar \xi}}_* := \overline{(\xi_*)}$ be the corresponding partition of ${{\bar H}}{^{(0)}}$. Note that $ {{\bar \xi}}_*$ is the least element of ${{\bar \Xi}}$. Herewith $ {{\bar \xi}}_*$ is the least partition of ${{\bar H}}{^{(0)}}$ such that for all $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ and every $C \in \xi$ the restriction $r \;{|}_{{\bar C}}$ consists precisely of $d$ elements.
Putting $\xi = \xi_*$ in Diagram \[diag /xi\] (Proposition \[natur proj\]) we obtain Diagram \[diag reduc ext\] with $$H_* = H{/}_{\xi_*} \;\;,\;\; {{\bar H}}_* = {{\bar H}}{/}_{{{\bar \xi}}_*}
\;\;,\;\; \pi_* = \pi{/}_{\xi_*} \;\;,\;\; \psi_* =
\psi{/}_{\xi_*} \;.$$ and $ (\pi_*,\psi_*) \preceq (\pi,\psi) $.
Using by the above propositions and Lemma \[gL(phi)\], we see that the pair $(\pi_*,\psi_*)$ is irreducible and that it is the only (up to equivalence) irreducible pair majorized by $
(\pi,\psi) $.
\[irred ext\] The above arguments show that a pair $ (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ is irreducible iff $ (\pi_*,\psi_*) = (\pi,\psi) $, i.e. iff $ \xi_*
= {\varepsilon}_{H{^{(0)}}}$. The last equality means that the persistent partitions $ {{\mathcal R}}({{\bar {{\mathcal S}}}}) $ separate the points of $H{^{(0)}}$ in the following sense: for every pair $ u_1,u_2 \in H{^{(0)}} $ there exist $R_1 \in r_1 \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ and $R_2 \in r_2 \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ such that $${\pi{^{(0)}}}^{-1}(u_1) \cap R_1 \cap R_2 \neq \emptyset \;\;,\;\;
{\pi{^{(0)}}}^{-1}(u_2) \cap R_1 \cap R_2 = \emptyset \;.$$
Canonical form and classification. {#s5}
==================================
Main Theorems. {#ss5.1}
---------------
The following two theorems claim the existence and uniqueness of the canonical form of $\rho$-uniform one-sided Markov shifts.
\[canon form\] Let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and positively recurrent. Then there exists a $(\pi,\psi)$-extension $$\label{pi,psi}
(\pi,\psi) \;{ : }\;
\xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I }$$ such that
1. The shifts $T_G$ and $T_{{\bar H}}$ are isomorphic,
2. $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$, where $d = d(T_G)$ is the minimal index of the shift $T_G$,
3. The extension $(\pi,\psi) $ is irreducible.
Combining the results of Theorems \[reduc ext\] and \[phi bar d(T)\] we obtain from Diagrams \[diag reduc ext\] and \[diag bar H Gn\] the following commuting diagram $$\label{diag main}
\xymatrix{
{{\bar H}}\ar[dd]^{{{\bar {\kappa}}}} \ar[rd]^{\pi} \ar[rr]^{{{\bar \psi}}}
&& G{^{(n)}}\ar[dd]^{\phi} \ar[rr]^{\pi{^{(n)}}}
&& G \\
& H \ar[d]^{{\kappa}} \ar[dr]^{\psi} & && \\
{{\bar H}}_* \ar[r]^{\pi_*}
& H_* \ar[r]^{\psi_*} & I && }$$ Here, $\pi$, $\pi_*$ and $\phi$ are homomorphisms of degree $d = d(T_G)$, all other homomorphisms are of degree $1$, and the extension $(\pi_*,\psi_*) \in {{\mathcal Ext}^d(I,\rho)}$ is irreducible.
Since $G$ and $ {{\bar H}}_* $ have a common extension ${{\bar H}}$ of degree $1$, the shifts $T_G$ and $T_{{{\bar H}}_*}$ are isomorphic. Thus the the extension $(\pi_*,\psi_*) $ is desired.
\[def canon form\] We shall say that $T_{{\bar H}}$ is a [**canonical form**]{} of the shift $T_G$, if there exists an extension (\[pi,psi\]) satisfying the conditions of Theorem \[canon form\]. Herewith the graph ${{\bar H}}$ is said to be the [**canonical graph**]{} for $T_G$.
Theorem \[canon form\] states the existence of the canonical form. Turn to the uniqueness.
\[classification\] Let $G_1$ and $G_2$ be two $\rho$-uniform stochastic graphs, which are irreducible and satisfy the positive recurrence condition. Suppose the shifts $T_{G_1}$ and $T_{G_1}$ are represented in the canonical form $T_{{{\bar H}}_1}$ and $T_{{{\bar H}}_2}$, respectively, and let $$\label{pi12,psi12}
(\pi_k,\psi_k) \;{ : }\;
\xymatrix{
{{\bar H}}_k \ar[r]^{\pi_k} & H \ar[r]^{\psi_k} & I } \;\;,\;\; k=1,2$$ be corresponding canonical $(\pi,\psi) $-extensions.
Then the following conditions are equivalent
1. The shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic, $(T_{G_1} \sim T_{G_2})$.
2. The graphs ${{\bar H}}_1$ and ${{\bar H}}_2$ are isomorphic, $({{\bar H}}_1 \sim {{\bar H}}_2)$.
3. The extensions $(\pi_1,\psi_1) $ and $(\pi_1,\psi_1) $
are equivalent,$ ( (\pi_1,\psi_1) \sim (\pi_2,\psi_2) ) $.
By the definition we have $T_{G_1} \sim T_{{{\bar H}}_1}$ , $T_{G_2} \sim T_{{{\bar H}}_2}$ and $$(\pi_1,\psi_1) \sim (\pi_2,\psi_2) \;\; {\Longrightarrow}\;\; {{\bar H}}_1 \sim {{\bar H}}_2
\;\; {\Longrightarrow}\;\; T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2}$$ Thus we need to prove only $$\label{RRaro}
T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2} \;\; {\Longrightarrow}\;\; (\pi_1,\psi_1) \sim (\pi_2,\psi_2)$$ Suppose $T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2} $ and let $\; a_k { : }H_k
{\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$, be the functions generating ${{\bar H}}_k$, where $\; d = d(T_{{{\bar H}}_1}) = d(T_{{{\bar H}}_2}) \;$.
Since both of $ \psi_1 { : }H_1 {\rightarrow}I $ and $ \psi_2 : H_2
{\rightarrow}I $ are of degree $1$, we can apply Theorem \[degree 1\] and to construct a common extension $H$ of $H_1$ and $H_2$. Herewith, the corresponding Diagram \[diag degree 1\] commutes and the homomorphisms $\; \psi { : }H {\rightarrow}I \;$ , $\; \chi_1
{ : }H {\rightarrow}H_1 \;$ and $\; \chi_2 { : }H {\rightarrow}H_2 \;$ are of degree $1$.
By Remark \[triv exten\] each of homomorphisms $\; \chi_k { : }H_{b_k} {\rightarrow}H_k \;,\; k = 1,2 \;$ admits the trivial extension $\; {{\bar \chi}}_k { : }{{\bar H}}_{b_k} {\rightarrow}{{\bar H}}_k \;$ with the commuting diagram $$\label{diag ext k=1,2}
\xymatrix{
{{\bar H}}_{b_k} \ar[d]^{\pi_{b_k}} \ar[r]^{{{\bar \chi}}_k}
& {{\bar H}}_k \ar[d]^{\pi_k} \\
H \ar[r]^{\chi_k} & H_k }$$ Here ${{\bar \chi}}_k$ is of degree $1$ and $b_k := a_k \circ \chi_k $ for $k = 1,2$. Since $ d({{\bar \chi}}_1) = d({{\bar \chi}}_2) = 1 $ we have $T_{{{\bar H}}_1} \sim T_{{{\bar H}}_{b_1}} $ and $T_{{{\bar H}}_2} \sim T_{{{\bar H}}_{b_2}}
$. Therefore $T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2} $ implies that the skew products ${{\bar T}}_{H,b_1}$ and ${{\bar T}}_{H,b_2}$ are isomorphic.
Thus we have two GSP $d$-extensions $\; \pi_{b_k} { : }{{\bar H}}_{b_k} {\rightarrow}H \;,\; k = 1,2 \;,\;$ of $H$ and a homomorphism $\psi : H {\rightarrow}I $ of degree $1$. Herewith, the number $d$ is the minimal index of ${{\bar T}}_{H,b_1}$ and ${{\bar T}}_{H,b_2}$. By Theorem \[Equ ext 1\] the functions $b_1$ and $b_2$ are cohomologous with respect to $H$. Hence two constructed $(\pi,\psi)$-extensions $$(\pi_{b_k},\psi) \;{ : }\;
\xymatrix{ {{\bar H}}_{b_k} \ar[r]^{\pi}
& H \ar[r]^{\psi} & I } \;\;,\;\; k=1,2$$ are equivalent, $\; (\pi_{b_1},\psi) \sim (\pi_{b_2},\psi) \;$.
On the other hand by constructing both two diagrams $$\xymatrix{
{{\bar H}}_{b_k} \ar[d]^{{{\bar {\kappa}}}_k} \ar[r]^{\pi_{b_k}}
& H \ar[d]^{{\kappa}} \ar[dr]^{\psi} & \\
{{\bar H}}_k \ar[r]^{\pi_k}
& H_k \ar[r]^{\psi_k} & I } \;\;,\;\; k=1,2$$ commute. This means that $\; (\pi_1,\psi_1) \; \preceq \;
(\pi_{b_1},\psi) \;$ and $\; (\pi_2,\psi_2) \; \preceq \;
(\pi_{b_2},\psi) \;$.
The pairs $(\pi_1,\psi_1)$ and $(\pi_2,\psi_2)$ are irreducible and they are majorized by equivalent pairs. Hence they are equivalent.
We have shown (\[RRaro\]).
As a consequence we have also
\[common exten 1\] Under conditions of Theorem \[classification\] the shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic iff the graphs $G_1$ and $G_2$ have a common extension of degree $1$, i.e. there exists a diagram $$\label{diag com ext 1}
\xymatrix{
G_! & G \ar[l]_{\phi_1} \ar[r]^{\phi_2} & G_2 }$$ where homomorphisms $\phi_1$ and $\phi_2$ are of degree $1$.
By Theorem \[phi bar d(T)\] we have two diagram of homomorphisms $$\label{diag com ext 2}
\xymatrix{
G_k & G_k{^{(n)}}\ar[l]_{\pi{^{(n)}}}
& {{\bar H}}_k \ar[l]_{{{\bar \psi}}_k} \ar[r]^{\pi_k}
& H_k \ar[r]^{\psi_k} & I } \;\;\;\; k=1,2$$ where $d(\pi{^{(n)}}) = d({{\bar \psi}}_k) = d(\psi_k) = 1 $ and $\pi_k$ is a $d$-extension. So that $(\pi_k,\psi_k) \in {{\mathcal Ext}^d(I,\rho)}$.
By Theorem \[reduc ext\] each pair $(\pi_k,\psi_k) \;,\; k=1,2
\;$ majorizes an irreducible pair from $ {{\mathcal Ext}^d(I,\rho)}$. If the the shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic the irreducible pairs are equivalent (Theorem \[classification\]) and we may assume without loss of generality that they coincide with each other.
Thus there exists $(\pi_0,\psi_0) \in {{\mathcal Ext}^d(I,\rho)}$ with two commuting diagrams $$\label{diag reduc ext k=1,2}
\xymatrix{
{{\bar H}}_k \ar[d]^{{{\bar {\kappa}}}_k} \ar[r]^{\pi_k}
& H_k \ar[d]^{{\kappa}_k} \ar[dr]^{\psi_k} & \\
{{\bar H}}_0 \ar[r]^{\pi_0}
& H_0 \ar[r]^{\psi_0} & I } \;\;\;\; k=1,2$$ Passing possibly to equivalent extensions we may also assume that ${{\bar {\kappa}}}_1$ and ${{\bar {\kappa}}}_2$ are trivial extensions of ${\kappa}_1$ and ${\kappa}_2$.
By Theorem \[degree 1 sharp\] and Remark \[triv exten\] we find a common extension of degree 1 $$\label{diag com ext 3}
\xymatrix{ H_1 & H \ar[l]_{\chi_1} \ar[r]^{\chi_2} & H_2 }$$ of $H_1$ and $H_2$ with the trivial extensions $$\label{diag com ext 4}
\xymatrix{
{{\bar H}}_! & {{\bar H}}\ar[l]_{{{\bar \chi}}_1} \ar[r]^{{{\bar \chi}}_2} & {{\bar H}}_2 }$$ of $\chi_1$ and $\chi_2$ such that the corresponding diagram $$\label{diag com ext 5}
\xymatrix{
& {{\bar H}}_2 \ar[dd]_(.75){\pi_2} \ar[rr]^{{{\bar {\kappa}}}_2}&
& {{\bar H}}_0 \ar[dd]^{\pi_0} \ar[drrrr]^{{{\bar \psi}}_0} & &&& \\
{{\bar H}}\ar[dd]_{\pi} \ar[ur]^{{{\bar \chi}}_2} \ar[rr]^(.75){{{\bar \chi}}_1}
\ar[rrru]^{{{\bar \chi}}}
&& {{\bar H}}_1 \ar[ur]_{{{\bar {\kappa}}}_1} \ar[dd]^(.25){\pi_1} &&&&& I \\
& H_2 \ar[rr]^(.75){{\kappa}_2}&
& H_0 \ar[rrrru]^{\psi_0} &&&& \\
H \ar[ur]^{\chi_2} \ar[rr]_{\chi_1} \ar[rrru]^{\chi}
& & H_1 \ar[ur]_{{\kappa}_1} & && && \\ }$$ commutes. Therefore we have $$\label{diag com ext 6}
\xymatrix{
G_1 & G_1{^{(n)}}\ar[l]_{\pi{^{(n)}}} & {{\bar H}}_1 \ar[l]_{{{\bar \psi}}_1}
& {{\bar H}}\ar[l]_{{{\bar {\kappa}}}_1} \ar[r]^{{{\bar {\kappa}}}_2} & {{\bar H}}_2 \ar[r]^{\psi_2}
& G_2{^{(n)}}\ar[r]^{\pi{^{(n)}}} & G_2 }$$ Putting $\; G := {{\bar H}}\;$ and $\; \phi_k := {{\bar {\kappa}}}_k \circ {{\bar \psi}}_k
\circ \pi{^{(n)}}\;$ for $\; k = 1,2 \;$, we obtain the desired common extension of degree $1$ (\[diag com ext 1\]).
Consequences and examples. {#ss5.2}
--------------------------
Consider some particular cases.
[**Extensions of Bernoulli graphs.**]{} Let $(I,\rho)$ be a standard Bernoulli graph and let $ d \in {{\mathbb{N}}}$. Let $\; a { : }I {\rightarrow}{{\mathcal A}_d}\;$ be a function $\; a { : }I {\rightarrow}{{\mathcal A}_d}\;$ on $I$ with the values $\; a(i) \;,\; i \in I \;,$ in the group ${{\mathcal A}_d}$ of all permutations of $\; Y_d = \{1,2, \ldots ,d \} \;$. Consider a $d$-extension $\; {{\bar I}}_a \;$ generated by the function $a$ (See Section \[ss4.2\]). We assume that the group $ {\Gamma}(a) $, generated by $\; a(i) , i \in
I, \;$ acts transitively on $Y_d$. This provides that the shift $T_{{{\bar I}}_a} $ and the skew product ${{\bar T}}_{I,a}$ are ergodic.
We want to clarify: when is $\; {{\bar I}}_a \;$ the canonical graph for the corresponding Markov shift $T_{{{\bar I}}_a}$ (Definition \[def canon form\]). Let $\pi { : }{{\bar I}}_a {\rightarrow}I$ be the projection and $ (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$. Since every homomorphism $ \psi { : }I
{\rightarrow}I $ is an automorphism, the pair $(\pi,\psi)$ is irreducible. Therefore $\; {{\bar I}}_a \;$ is a the canonical graph iff $\; d(T_{{{\bar I}}_a}) = d \;$.
\[d = d(T bI a)\] If the function $a$ satisfies the following condition $$\label{modular a}
\rho(i) = \rho(i') \;\; {\Longrightarrow}\;\; a(i) = a(i')
\;\;,\;\; i,i' \in I$$ then $\; d(T_{{{\bar I}}_a}) = d \;$.
Suppose the condition (\[modular a\]) holds. The Markov shift $T_{{{\bar I}}_a} $ is isomorphic to the skew product ${{\bar T}}= {{\bar T}}_{\rho,a}$, which acts on $X_\rho \times Y_d$ by (\[bT rho a\]). So that we have $d(T_{{{\bar I}}_a}) = d({{\bar T}}) $ and by Theorem \[simp mar\] $\; d({{\bar T}}) \;=\; d_{{\gamma}{ : }{\beta}}({{\bar T}})
\;$.
A direct computation, using (\[modular a\]), the definition of ${\gamma}(T)$ and ${\beta}({{\bar T}})$ and Proposition \[ga(T rho)\], shows that $${\beta}({{\bar T}}) = {\gamma}(T_\rho) \times {\varepsilon}_{Y_d} \;\;\;,\;\;\;
{\gamma}({{\bar T}}) = {\gamma}(T_\rho) \times \nu_{Y_d} \;.$$ This means that any element of ${\gamma}({{\bar T}})$ consists precisely of $d$ elements of the partition ${\beta}({{\bar T}})$. By the definition of the index $ d_{{\gamma}{ : }{\beta}}({{\bar T}}) $ we have $\; d_{{\gamma}{ : }{\beta}}({{\bar T}}) = d \;$. Thus $\; d(T_{{{\bar I}}_a}) = d \;$.
Taking into account Theorem \[th rho cohom\] we have
\[cor modular a\] Let $\; \pi_k { : }{{\bar I}}_{a_k} {\rightarrow}I \;,\; k = 1,2, \;$ be two $d$-extensions of the Bernoulli graph $(I,\rho)$, generated by functions $ a_k { : }I {\rightarrow}{{\mathcal A}_d}$, respectively, and suppose both the functions $\; a_k \;,\;k=1,2 \;$ satisfy the condition \[modular a\]. Then the Markov shifts $T_{{{\bar I}}_{a_1}}$ and $T_{{{\bar I}}_{a_2}}$ are isomorphic iff $ a_1 $ and $ a_2 $ are conjugate in ${{\mathcal A}_d}$, i.e. there exists $w_0 \in {{\mathcal A}_d}$ such that $\; a_2(i) \cdot w_0 = w_0
\cdot a_1(i) \;,\; i \in I \;$.
\[rem modular a\] It can be proved that for $d$-extension ${{\bar I}}_a$, the condition \[modular a\] is equivalent to $\; d(T_{{{\bar I}}_a}) = d \;$.
[**Absolutely non-homogeneous ${\boldsymbol{\rho}}$.**]{} Consider the case , when $\rho$ is absolutely non-homogeneous (see Section \[ss2.4\]). this means that $\; \rho(i) \neq \rho(i') \;$ for all $\;i \neq i' \;$ from $I$, i.e. the Bernoulli graph $(I,\rho)$ has no congruent edges.
In this case for any $\rho$-uniform graph $G$ there exists a [**unique**]{} homomorphism $\; \phi { : }G {\rightarrow}I \;$. Therefore Theorem \[phi bar d(T)\] can be sharpened as follows
\[phi bar d(T) sharp\] Let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and satisfies the positive recurrence condition. Suppose that $\rho$ is absolutely non-homogeneous. Then there exist a unique homomorphism $\; \phi \in {{\mathcal Hom}}(G,I) \;$ and a commutative diagram $$\label{diag bar H G}
\xymatrix{ {{\bar H}}\ar[d]^{\pi} \ar[r]^{{{\bar \psi}}} & G \ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ such that
1. The pair $\; (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ is a $(\pi,\psi)$-extension.
2. $\; d = d(\phi) = d(T_G) \;$,
A natural question, which is arisen in connection with the previous theorem is:
\[gen road prob\] Does Theorem \[phi bar d(T)\] hold with $n=1$ in general case, when $\rho$ is not necessarily absolutely non-homogeneous, i.e. when $(I,\rho)$ has congruent edges ?
As we know, the problem is open even in the case, when the graph $G$ is finite (See [@AsMaTu] and references therein.)
[**Homogeneous ${\boldsymbol{\rho}}$ and Road Problem** ]{} Consider a special case, when $\rho$ is homogeneous, i.e. $\; \rho(i) =
l^{-1} \;,\; i \in I \;$ with an integer $l = |I| \in {{\mathbb{N}}}$. Theorem \[simp mar\] and arguments adduced in Section \[ss2.4\] imply
\[\] Suppose $\rho$ is homogeneous. Then every ergodic $\rho$-uniform Markov shift $\; T_G \;$ is isomorphic to a direct product $\; T_\rho \times {\sigma}_d \;$ of the Bernoulli shift $T_\rho$ and a cyclic permutation ${\sigma}_d$ of $Y_d$, where $d$ is the period of $ T_G $. If, in addition, $ T_G
$ is exact, then it is isomorphic to the Bernoulli shift $T_\rho$, herewith, there exists $n \in {{\mathbb{N}}}$ and a homomorphism $\; \phi
{ : }G{^{(n)}}{\rightarrow}I \;$ of degree $1$.
The result was proved earlier in for finite $G$ and in for general case.
If $G$ is finite and $\rho$ is homogeneous Question \[gen road prob\] is a reformulation of well-known Road Coloring Problem (See [@Fr], [@O'B], [@AdGoWe], [@Ki]). As we know, the problem is still open.
Some (p,q)-uniform graphs. {#ss5.4}
--------------------------
We construct some simple examples to illustrate the case, when the $\psi$-part in the canonical pair $(\pi,\psi)$ is not trivial.
Let $\; I = \{0,1\}\;$ and $\; \rho = (p,q) \;$, where $\; 0 < p < 1 \;$ and $\; q=1-p \;$. Given $\; n \in {{\mathbb{N}}}\;$ consider the following random walk on $\; J_n := \{ 1,2, \ldots ,n \} \;$ $$\label{FDR}
\xymatrix@C=3pc{
1 \ar@(ul,dl) []_{q} \ar@/_/ [r]_{p}
& 2 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\;\hdots\;\;\;\;
\ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& n \ar@/_/ [l]_{q} \ar@(dr,ur) []_{p} } \;,$$ which is known as a [**Finite Drunkard Ruin**]{}. We set here: $\; H:=I \times J_n \;$ , $\; H{^{(0)}} := J_n \;$ and $$s(h) = j \;,\; t(h) = f_ij \;,\; \psi (h) = i
\;\;,\;\; h = (i,j) \in H \;,$$ where the maps $\; f_i { : }J_n/ {\rightarrow}J_n \;,\; i = 0,1 \;,$ are defined by $$f_1 j = \min{(j+1,n)} \;\;,\;\; f_0 j = \max{(j-1,1)}
\;\;\;,\;\;\; j \in J_n$$ and the weights of edges $\; p(h) \;,\; h \in H \;$ are given according to (\[FDR\]) by $\; p(1,j) = p \;$, $\; p(0,j) = q
\;$.
Then the finite stochastic graph $H$ is irreducible and $\rho$-uniform, $\; \psi \in {{\mathcal Hom}}(H,I) \;$. The semigroup ${{\mathcal S}}(\psi)$, generated by $\{f_0,f_1\}$, is $1$-contractive, since $\;
(f_0)^n (J_n) = \{1\} \;$. Whence, $\; d(\psi) = 1 \;$ and the Markov shift $T_H$ is isomorphic to the Bernoulli shift $T_\psi$.
Given $p$ and $n$ we construct a ${{\mathbb{Z}}}_2$-extension ${{\bar H}}_a$ of the graph $H$, where $a : H {\rightarrow}{{\mathbb{Z}}}_2$ and ${{\mathbb{Z}}}_2 := \{0,1\}$ be the cyclic group of order $2$.
Define $\; a : H=I \times J_n \ni h=(i,j) {\rightarrow}a(h) \in {{\mathbb{Z}}}_2 \;$ by $$\label{a(h)}
a(i,j) \;=\; \left\{
\begin{array}{ll}
1 \;\;,\;\; & if \;\; (i,j) \;=\; (1,1) \; \\
0 \;\;,\;\; & if \;\; (i,j) \;\neq\; (1,1) \;.
\end{array}
\right.$$ Then the corresponding graph ${{\bar H}}_a$ has the form $$\label{FDR2}
\xymatrix@C=3pc{
**[r] 11 \ar@(ul,dl) []_{q} \ar [rd]_(.75){p}
& 21 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\hdots\;\;\;
\ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& **[l] n1 \ar@/_/ [l]_{q} \ar@(dr,ur) []_{p}
& z=1 \\
**[r] 10 \ar@(dl,ul) []^{q} \ar [ru]^(.75){p}
& 20 \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& \;\;\;\hdots\;\;\;
\ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& **[l] n0 \ar@/^/ [l]^{q} \ar@(ur,dr) []^{p}
& z=0 }$$ for $\; n > 2 \;.$ and $$\label{FDR3}
\xymatrix@C=3pc{
**[r] 11 \ar@(ul,dl) []_{q} \ar@/^3pc/ [d]^{p}
&& **[r] 11 \ar@(ul,dl) []_{q} \ar [rd]_(.75){p}
& **[l] 21 \ar@/_/ [l]_{q} \ar@(dr,ur) []_{p}
& z=1 \\
**[r] 10 \ar@(ul,dl) []_{q} \ar@/_2pc/ [u]^{p}
&& **[r] 10 \ar@(dl,ul) []^{q} \ar [ru]^(.75){p}
& **[l] 20 \ar@/^/ [l]^{q} \ar@(ur,dr) []^{p}
& z=0 }$$ for two special cases $\; n = 1,2 \;$
Suppose $\; p \neq q \;$. We claim in this case that for all $n
\in {{\mathbb{N}}}$ the graphs (\[FDR3\]) and (\[FDR2\]) are canonical. Indeed, $\; d(\pi_H) = d(T_{{{\bar H}}_a}) = 2 \;$, since $ \rho = (p,q)
$ is absolutely non-homogeneous. In order to check the irreducibility of the $2$-extension $\; (\pi_H,\psi) { : }{{\bar H}}_a
{\rightarrow}H {\rightarrow}I \;$ consider the semigroup ${{\bar {{\mathcal S}}}}= {{\bar {{\mathcal S}}}}(\pi,\psi)$ and its persistent partitions ${{\mathcal R}}({{\bar {{\mathcal S}}}})$.
The semigroup ${{\bar {{\mathcal S}}}}$ is generated by $\; \{{{\bar f}}_i , i\in I \}$, where $${{\bar f}}_i \; (j\;,\;z) \;=\; (f_i \; j \;,\; z + a(i,j) \pmod 2 ) \;\;,\;\;
(j,z) \in J_n \times {{\mathbb{Z}}}_2 \;.$$ A direct computation shows that for $\; n = 1,2 \;$ any transversal partition of $J_n \times {{\mathbb{Z}}}_2 $ is persistent in the sense of Definition \[pers part\] , and for $\; n > 2 \;$ there exists a non-persistent transversal partition. Naimly, the partition, consisting of two sets of the following “alternating” form $$\{ (1,0) , (2,1) , (3,0) , (4,1) , \ldots \}
\;\;,\;\; \{ (1,1) , (2,0) , (3,1) , (4,0) , \ldots \} \;,$$ is so. Moreover, this is the only transversal partition, which is not persistent. This implies that for every $ n \in {{\mathbb{N}}}$ the persistent partitions ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ separate points of $ J_n \times
{{\mathbb{Z}}}_2 $ in the sense of Remark \[irred ext\] and the $2$-extension $(\pi_H, \psi)$ is irreducible. Thus
- For all $n \in {{\mathbb{N}}}$ and $p \neq q$ the graphs ${{\bar H}}_a$ are canonical graphs for the corresponding shifts $T_{{{\bar H}}_a}$.
Just in the same way we can consider the following [**Infinite Drunkard Ruin**]{} $$\label{FDR4}
\xymatrix@C=3pc{
**[r] 1 \ar@(ul,dl) []_{q} \ar@/_/ [r]_{p}
& 2 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\hdots\;\;\;
\ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& n \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \ar@/_/ [l]_{q} \;\;\;\hdots }$$ where $\; H:=I \times {{\mathbb{N}}}\;$ , $\; H{^{(0)}} := {{\mathbb{N}}}\;$.
Suppose $\; p < q \;$. Then the corresponding Markov chain is positively recurrent and the Markov shift $T_H$ is isomorphic to the Bernoulli shift $T\rho$.
Again define the functions $\; a : H=I \times {{\mathbb{N}}}\ni h=(i,j) {\rightarrow}a(h) \in {{\mathbb{Z}}}_2 \;$ by (\[a(h)\]). Then ${{\mathbb{Z}}}_2$-extension ${{\bar H}}_a$ of the graph $H$ (\[FDR5\]) has the form $$\label{FDR5}
\xymatrix@C=3pc{
**[r] 11 \ar@(ul,dl) []_{q} \ar [rd]_(.75){p}
& 21 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\hdots\;\;\; \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& n1 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \ar@/_/ [l]_{q} \;\;\;\hdots \\
**[r] 10 \ar@(dl,ul) []^{q} \ar [ru]^(.75){p}
& 20 \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& \;\;\;\hdots\;\;\; \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& n0 \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& \ar@/^/ [l]^{q} \;\;\;\hdots }$$ It can be shown in this case that any transversal set is persistent. Thus
- If $p < q$ the graph ${{\bar H}}_a$ (\[FDR5\]) is the canonical graph for the shift $T_{{{\bar H}}_a}$.
Note that the shift $T_{{{\bar H}}_a}$ is a ${{\mathbb{Z}}}_2$-extension of the Bernoulli shift $T_{p,q}$, therefore, $T_{{{\bar H}}_a}$ has a $4$-element one-sided generator. On the other hand the shift is not isomorphic to Markov shifts on finite state spaces. Thus
- If $p < q$ the one-sided Markov shift $T_{{{\bar H}}_a}$ has no finite one-sided Markov generator.
[9]{}
J. Aaronson. An introduction to infinite ergodic theory. Math. Surveys and Monographs, vol. 50, 1997.
A.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts. Isr. J. Math. 27(1977), 49-63.
J. Ashley. Resolving factor maps for shifts of finite type with equal entropy. Ergod. Th. Dyn. Syst. 11(1991), 219-240.
J. Ashley, B. Marcus, S. Tuncel. The classification of one-sided Markov chains. Ergod. Th. Dyn. Syst. 17(1997), 269-295.
M. Boyle, S. Tuncel. Regular isomorphism of Markov chains is almost topological. Ergod. Th. Dyn. Syst. 10(1990), 89-100.
J. Feldman, D.J. Rudolph. Standardness of sequences of $\sigma$- fields given by certain endomorphisms. Fund. Math. 157(1998), 175-189.
J. Friedman. On the Road Coloring Problem. Proc. AMS 110(1990), 1133-1135.
N. Friedman, D. Ornstein. On the isomorphism of weak Bernoulli transformations. Adv. in Math. 5(1970), 365-394.
D. Heicklen, C. Hoffman. $(T,T^{-1})$ is not standard. Ergod. Th. Dyn. Syst. 18(1998), 875-878.
D. Heicklen, C. Hoffman, D.J. Rudolph. Entropy and dyadic equivalence of random walks on a random scenery. Adv. Math. 156(2000), 157-179.
C. Hoffman. A zero entropy $T$ such that the $(T,Id)$-endomorphism is non standard. Proc. AMS 128(2000), 183-188.
C. Hoffman, D. Rudolph. Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann, of Math. 156(2002), 79-101.
M. Keane, M. Smorodinsky. Finitary isomorphism of irreducible Markov shifts. Israel J. Math. 34(1979), 281-286.
B.P. Kitchens. Symbolic dynamic. Springer-Verlag, Berlin, 1998.
B.P. Kitchens, B. Marcus, P. Trow. Eventual factor maps and compositions of closing maps. Ergod. Th. Dyn. Syst. 11(1991), 85-113.
W. Krieger. On finitary isomorphism of Markov shifts that have finite expected coding time. Z. Wahr. 65(1983), 323-328.
J. Kubo, H. Murata, H, Totoki. On the isomorphism problem for endomorphisms of Lebesgue space. Publ. RIMS. Kyoto Univ. 9(1974), 285-296.
D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4(1970), 337-352.
O’Brien. The Road Coloring Problem. Isr. J. Math. 39(1981), 145-154.
V. Rokhlin. On the fundamental ideas of the measure theory. Mat. Sbor. 25(1949), 107-150 (Russian), Trans. AMS. 71(1952). 1-54.
V. Rokhlin. Lectures on the entropy theory of transformation with invariant measure. Usp. Mat. Nauk. 22(1967), 3-56 (Russian). Russ. Math. Surveys 22(1967), 1-52.
M. Rosenblatt. Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8(1959), 665-681.
M. Rosenblatt. Markov processes. Structure and asymptotic behavior. Springer Grundl. Math. 184, Berlin, 1971.
B.Z. Rubshtein. On decreasing sequences of measurable partitions. Sov. Math. Dok. 13(1972), 962-965.
B.Z. Rubshtein. Decreasing sequences of measurable partitions generated by endomorphisms. Ups. Math. Nauk 28(1973), 247-248.
B.Z. Rubshtein. Generating partitions of Markov endomorphism. Func. Anal. Appl. 8(1974), 84-85.
B.Z. Rubshtein. On non homogeneous finitely Bernoulli sequences of measurable partitions. Func. Anal. Appl. 10(1976), 39-44.
B.Z. Rubshtein. Lacunary isomorphism of decreasing sequences of measurable partition. Israel J. Math. 97(1997), 317-345.
B.Z. Rubshtein, On finitely Bernoulli one-sided Markov shifts and their cofiltrations, Ergod. Th. Dyn. Syst, 19(1999), 1527-1524.
Ya. G. Sinai. On weak isomorphism of measurable preserving transformations. Mat. Sb. 63(1964), 23-42, Transl. AMS 57(1966), 123-143.
A.M. Stepin. On entropy invariants of decreasing sequences of measurable partitions. Func. Anal. Appl. 5(1971). 80-84.
P. Trow. Degrees of finite to one factor maps. Israel J. Math. 71(1990), 229-238.
A.M. Vershik. A lacunary isomorphism theorem for monotone sequences of measurable partitions. Func. Anal. Appl. 2(1968). 17-21.
A.M. Vershik. Decreasing sequences of measurable partitions and their applications. Sov. Math. Doc. 11(1970), 1007-1011.
A.M. Vershik. A continuum of pairwise non-isomorphic dyadic sequences. Func. Anal. Appl. 5(1971), 16-18.
A.M. Vershik. Theory of decreasing sequences of measurable partitions. St.Peterburg Math. J. 6 (1995), 705-761.
V.G. Vinokurov. Two non-isomorphic exact endomorphisms of the Lebesgue space with isomorphic sequences of partitions. In “Random processes and related topics”. vol. 1 Tashkent 1970, 43-45 (in Russian).
V. Vinokurov. B. Rubshtein. A. Fedorov. Lebesgue spaces and their measurable partitions. Tashkent Univ. 1985, 75 pp. (in Russian).
|
---
abstract: 'I review the current status of dynamical modelling of dwarf spheroidal galaxies focusing on estimates of their dark matter content. Starting with the simplest methods using the velocity dispersion profiles I discuss the inherent issues of mass-anisotropy degeneracy and contamination by unbound stars. I then move on to methods of increasing complexity, aiming to break the degeneracy, up to recent applications of the Schwarzschild orbit superposition method. The dynamical modelling is placed in the context of possible scenarios for the formation of dwarf spheroidal galaxies, including the tidal stirring model and mergers of dwarf galaxies. The two scenarios are illustrated with examples from simulations: a comparison between the tidal evolution of dwarfs with cuspy and cored dark matter profiles and the formation of a dwarf spheroidal with prolate rotation.'
---
Introduction
============
Dwarf spheroidal (dSph) galaxies are a class of faint galaxies mostly found in the Local Group, in the vicinity of the Milky Way and Andromeda. They are gas poor, supported mostly by random motions and believed to be strongly dominated by dark matter. Their unique property is that resolved stellar populations are available for study i.e. positions and radial velocities of individual stars can be measured. Over the last two decades dSph galaxies have been the subject of intensive study, with their kinematic samples increasing from few tens to few hundred and now a few thousand of stars with measured velocities for best-studied dwarfs. This allowed us to develop more and more advanced methods of modelling their dark matter content. At the same time, a few scenarios for their dynamical evolution were proposed. In the first part of this contribution I summarize the development of the mass modelling methods, from the simplest to the most advanced, while the second part is devoted to a short description of the formation scenarios.
Mass modelling
==============
The simplest observable that can be constructed for dSph galaxies in order to obtain an estimate of their mass content is the line-of-sight velocity dispersion of the stars or its profile as a function of galactocentric radius. This profile can be modelled by solving the Jeans equation which relates the velocity dispersion to the underlying gravitational potential and the velocity anisotropy parameter $\beta$. The anisotropy parameter measures the amount of radial versus tangential motion in stellar orbits and can take the values between $\beta=1$ (radial orbits) and $\beta=-\infty$ (circular orbits) with $\beta=0$ corresponding to isotropic orbits.
Early studies have demonstrated that dSph galaxies are strongly dominated by dark matter so that their mass-to-light ratios exceed the stellar values by at least one or two orders of magnitude. It has also been shown, e.g. for the best-studied Fornax dSph galaxy ([@Lokas2002 [Ł]{}okas 2002]), that the velocity dispersion profile can be equally well reproduced by different combinations of dark matter density profiles and anisotropy parameters. In general, shallower profiles require more radial stellar orbits and steeper profiles require more tangential orbits to fit a given data set. This is a manifestation of the so-called mass-anisotropy degeneracy inherent in the modelling of any pressure-supported systems, from dSph galaxies to galaxy clusters. For a long time, the simplest and customary way to go around the problem was to assume the stellar orbits to be isotropic in order to infer the mass distribution in dSph galaxies. Such an assumption would obviously lead to biases in the estimated density profiles if the real anisotropy was different. Simulations of galaxy formation and evolution have since demonstrated that the anisotropy indeed tends to depend on radius and departs from zero.
The mass-anisotropy degeneracy can be to some extent broken by adding constraints from a higher velocity moment, the kurtosis, that could be measured from more numerous samples of stellar kinematics which became available. Modelling the kurtosis involves solving the higher order Jeans equation. Since kurtosis is mostly sensitive to the anisotropy, in combination with the modelling of dispersion, both moments can constrain both the mass distribution and anisotropy. However, this still usually requires some assumptions concerning the functional form of orbital anisotropy, for example $\beta$ to be constant with radius ([@Lokas2005 [Ł]{}okas et al. 2005], [@Lokas2009 [Ł]{}okas 2009]).
The second essential issue in the mass modelling of dSph galaxies is the question of stellar membership. In order to obtain a reliable mass estimate it is important to model a clean sample, i.e. composed only of member stars. If the sample is contaminated by tidally stripped stars or background and foreground stars from the Milky Way that do not trace the potential of the dwarf galaxy, the results will be biased. Such biased estimates will typically lead to the inference of more tangential orbits or more extended mass distributions since unbound stars tend to increase the velocity dispersion at larger galactocentric radii ([@Klimentowski2007 Klimentowski et al. 2007], [@Lokas2008 [Ł]{}okas et al. 2008]).
Recently, more advanced methods of mass modelling of dSph galaxies have been developed. In the Schwarzschild orbit superposition method the velocity moments are reconstructed from a combination of stellar orbits. No assumptions on the anisotropy are required and the anisotropy profile follows as a result of the application of the method. For spherical dwarfs the method works very well and is able to reproduce different anisotropy profiles as well as density profiles of simulated galaxies ([@Kowalczyk2017 Kowalczyk et al. 2017]). For non-spherical objects application of the method with the assumption of spherical symmetry is subject to an inherent bias depending on the line of sight; in particular, the anisotropy is underestimated, and more so for the line of sight along the shortest axis of the stellar component ([@Kowalczyk2018 Kowalczyk et al. 2018]). A recent application of the Schwarzschild method to the modelling of the Fornax dwarf used around 3300 carefully selected stars from combined catalogues and allowed to constrain both the mass and anisotropy profiles in the dwarf. The mass-to-light ratio was found to increase with radius in agreement with previous estimates and the anisotropy profile was found to be slightly decreasing with radius, but consistent with isotropy at all radii ([@Kowalczyk2019 Kowalczyk et al. 2019]).
Future improvements in the mass modelling of dSph galaxies should include: increasing the samples of member stars to reduce sampling errors, developing better membership determination methods, using multiple stellar populations, including proper motion measurements for the stars ([@Strigari2018 Strigari et al. 2018]) and extensions to non-spherical models ([@Hayashi2018 Hayashi et al. 2018]).
Formation scenarios
===================
The dynamics of dSph galaxies is best elucidated by studying their formation scenarios. Although these objects could have formed in isolation, the scenarios most studied in recent years involved interactions with other systems. The two main scenarios considered were the tidal stirring due to the proximity of the Milky Way and mergers between dwarfs. In both scenarios the progenitors of dSph galaxies were disky dwarfs. The tidal stirring model ([@Mayer2001 Mayer et al. 2001]) explains the density-morphology relation observed among the Local Group dwarfs, i.e. the fact that dSph galaxies are situated close to the Milky Way or Andromeda, tend to be stripped of gas, spheroidal and non-rotating while dwarf irregular galaxies are typically found at larger distances, contain gas, are more flattened and rotate. The merger scenario was invoked to explain peculiar structural and kinematic features found in some dwarfs.
Tidal stirring
--------------
In simulations based on the tidal stirring scenario an initially disky dwarf galaxy is placed on a typical eccentric orbit around the Milky Way-like galaxy and evolved for a few gigayears. The effects of tidal stirring involve the mass loss (in both dark matter and stars), the morphological transformation of the stellar component from a disk to a spheroid and the change of rotation into random motions of the stars. The final product of tidal stirring depends strongly on the size of the orbit: it can leave the dwarf’s disk almost unaffected for an extended orbit or transform it to a small perfect sphere for a tight orbit ([@Kazantzidis2011a Kazantzidis et al. 2011a]). The evolution also critically depends on the initial dark matter distribution in the dwarf. The dwarfs with an initial cuspy dark matter profile tend to survive for many gigayears while those with a cored profile dissolve after one or two pericenter passages. A comparison between these two cases is illustrated in Figs \[cuspcore\] and \[masses\].
![Effect of tidal stirring on two initially disky dwarf galaxies evolving around the Milky Way-like galaxy. Both dwarfs were placed on a prograde orbit with apo- to pericenter ratio equal to 100 kpc/20 kpc. The initial structural parameters of the dwarfs differed only by their initial dark matter profile. The dwarf with the initial cuspy ($r^{-1}$) density profile (left panels) forms a bar at the first pericenter passage and survives for 10 Gyr. The dwarf with a cored ($r^{-0.2}$) profile (right panels) dissolves after the second pericenter passage ([@Lokas2016 [Ł]{}okas 2016]). The stellar components of both dwarfs are shown in projection onto the orbital plane after the first (upper row) and second (lower row) pericenter.[]{data-label="cuspcore"}](lokas_fig1a.eps "fig:"){width="2.5in"} ![Effect of tidal stirring on two initially disky dwarf galaxies evolving around the Milky Way-like galaxy. Both dwarfs were placed on a prograde orbit with apo- to pericenter ratio equal to 100 kpc/20 kpc. The initial structural parameters of the dwarfs differed only by their initial dark matter profile. The dwarf with the initial cuspy ($r^{-1}$) density profile (left panels) forms a bar at the first pericenter passage and survives for 10 Gyr. The dwarf with a cored ($r^{-0.2}$) profile (right panels) dissolves after the second pericenter passage ([@Lokas2016 [Ł]{}okas 2016]). The stellar components of both dwarfs are shown in projection onto the orbital plane after the first (upper row) and second (lower row) pericenter.[]{data-label="cuspcore"}](lokas_fig1b.eps "fig:"){width="2.5in"}\
![Effect of tidal stirring on two initially disky dwarf galaxies evolving around the Milky Way-like galaxy. Both dwarfs were placed on a prograde orbit with apo- to pericenter ratio equal to 100 kpc/20 kpc. The initial structural parameters of the dwarfs differed only by their initial dark matter profile. The dwarf with the initial cuspy ($r^{-1}$) density profile (left panels) forms a bar at the first pericenter passage and survives for 10 Gyr. The dwarf with a cored ($r^{-0.2}$) profile (right panels) dissolves after the second pericenter passage ([@Lokas2016 [Ł]{}okas 2016]). The stellar components of both dwarfs are shown in projection onto the orbital plane after the first (upper row) and second (lower row) pericenter.[]{data-label="cuspcore"}](lokas_fig1c.eps "fig:"){width="2.5in"} ![Effect of tidal stirring on two initially disky dwarf galaxies evolving around the Milky Way-like galaxy. Both dwarfs were placed on a prograde orbit with apo- to pericenter ratio equal to 100 kpc/20 kpc. The initial structural parameters of the dwarfs differed only by their initial dark matter profile. The dwarf with the initial cuspy ($r^{-1}$) density profile (left panels) forms a bar at the first pericenter passage and survives for 10 Gyr. The dwarf with a cored ($r^{-0.2}$) profile (right panels) dissolves after the second pericenter passage ([@Lokas2016 [Ł]{}okas 2016]). The stellar components of both dwarfs are shown in projection onto the orbital plane after the first (upper row) and second (lower row) pericenter.[]{data-label="cuspcore"}](lokas_fig1d.eps "fig:"){width="2.5in"}
An interesting intermediate stage of the evolution from a disk to a spheroid involves the formation of a tidally induced bar in the dwarf ([@Lokas2014a [Ł]{}okas et al. 2014a], [@Gajda2017 Gajda et al. 2017]). The bar always forms at the first pericenter passage and becomes shorter in time but the spheroidal shape usually survives until the end of evolution which may explain the non-spherical appearance of dSph galaxies. If the gas is included in the progenitor dwarf the tidally induced stellar bars are weaker but the effect is strong only in gas-dominated dwarfs ([@Gajda2018 Gajda et al. 2018]). On the other hand, if the gas was stripped by ram pressure in the hot halo of the Milky Way then the bars would survive.
![Evolution of the mass contained within two disk scale-lengths $2 R_d = 0.82$ kpc for stars (upper panel) and dark matter (lower panel) for the dwarfs with initially cuspy (red upper line) and cored (blue lower line) dark matter profile. The vertical gray lines indicate pericenter passages on an orbit with apo- to pericenter of 100 kpc/20 kpc. For the cored dwarf both masses approach zero after the second pericenter passage signifying the disruption of the dwarf.[]{data-label="masses"}](lokas_fig2.eps){width="4.5in"}
Obvious candidates for tidally stirred dwarfs in the vicinity of the Milky Way include the Sagittarius and Carina dwarfs, although most of dSph galaxies in the Local Group show signs of tidal extensions. In the case of the Sagittarius dwarf it has been demonstrated that its present structural and kinematic properties can be reproduced by a model in which the initially disky progenitor has just passed the second pericenter on its orbit around the Milky Way ([@Lokas2010 [Ł]{}okas et al. 2010]). In this model the stellar component still preserves its bar-like shape (slightly inclined with respect to the orbit) and shows very little rotation. The recent compilation of data for the Carina dSph galaxy, showing its elongated shape and significant rotation, can also be reproduced by a late-stage tidally stirred disky dwarf ([@Fabrizio2016 Fabrizio et al. 2016]).
Mergers of dwarfs
-----------------
Some dwarfs show features that cannot be explained by tidal stirring, including e.g. stellar shells in Fornax. In addition, there are distant dSph galaxies in the Local Group that probably did not have enough time to strongly interact with the Milky Way or Andromeda (e.g. Cetus and Tucana). Besides, in tidally stirred dwarfs the remnant rotation is always around the short axis of the stellar component while some dwarfs show rotation around the long axis (prolate rotation), e.g. Andromeda II ([@Ho2012 Ho et al. 2012]) and Phoenix ([@Kacharov2017 Kacharov et al. 2017]). Such features point toward a different scenario for the formation of at least some dSph galaxies, such as mergers between initially disky dwarfs ([@Kazantzidis2011b Kazantzidis et al. 2011b]). This possibility is supported by constrained simulations of the Local Group, aiming to reproduce the properties of both big galaxies, the Milky Way and Andromeda, as well as their satellite population. In particular, it has been demonstrated that in such environments a significant fraction of subhaloes experienced a strong interaction with another subhalo ([@Klimentowski2010 Klimentowski et al. 2010]). Since the typical velocities of single subhaloes around the Milky Way are rather too large for such interactions to be significant, mergers usually occur between subhaloes that were accreted in pairs, whose relative velocities are low.
![Formation of a dSph galaxy similar to Andromeda II by a merger of two initially disky dwarfs ([@Lokas2014b [Ł]{}okas et al. 2014b]). The three panels show the stellar components in three stages of the merger: before, with the two galaxies approaching (upper panel), after the first encounter (middle panel) and at the end of evolution, after the merger (lower panel). The image in each panel has the size of 40 kpc $\times$ 20 kpc.[]{data-label="merger"}](lokas_fig3.eps){width="4.6in"}
The merger scenario has been proposed as a way to explain the origin of prolate rotation in Andromeda II ([@Lokas2014b [Ł]{}okas et al. 2014b]). In this scenario, the progenitors are two disky dwarfs of equal mass and similar structural parameters approaching each other on a radial orbit. Their disks are initially inclined at the right angle but both have significant components of their angular momenta aligned with the merger axis. These parts of angular momenta are preserved after the merger in the form of prolate rotation. The remnant dwarf rotates around its long axis while the rotation around the short axis is close to zero. Three different stages of such a merger are illustrated in Fig. \[merger\]. The model reproduces well the non-spherical shape of the stellar component and if the gas is included ([@Fouquet2017 Fouquet et al. 2017]) also the presence and distribution of different stellar populations, with the younger one formed during the merger. In order to lose the remaining gas, the dwarf galaxy must have additionally experienced some interaction with its host, Andromeda, whose hot gaseous halo would strip the dwarf’s gas via ram pressure. It is worth emphasizing that tidal stirring is unable to induce significant stable rotation around the longest axis. Even if it occurs, it is always much smaller than the velocity dispersion, while in Andromeda II these two quantities are comparable. We have also shown that prolate rotation can be obtained from less idealized initial conditions, i.e. from dwarfs merging on non-radial orbits and with different inclinations of the disks ([@Ebrova2015 Ebrová & [Ł]{}okas 2015]).
The scenario has been placed in the cosmological context although for more massive galaxies due to resolution limitations. Almost 60 galaxies with prolate rotation have been identified in the Illustris simulation and most of them were confirmed to be the result of a major merger ([@Ebrova2017 Ebrová & [Ł]{}okas 2017]). In most of them a very tight correlation was found between the time of merger and the time of transition to prolate rotation and prolate shape. In massive galaxies prolate rotation tends to occur only in gas-poor objects.
Although originally believed to be relatively simple, near-spherical, old-population, pressure-supported systems, dSph galaxies of the Local Group only now start to reveal their complexity. Andromeda II has been recently shown to host at least two stellar populations with different kinematics. While the older population shows prolate rotation, the younger one rotates around the minor axis ([@delPino2017b del Pino et al. 2017b]). Even larger degree of complexity has been discovered in the well-studied Fornax dwarf. The multiple stellar populations of Fornax display different distributions and kinematics with the youngest stars centrally concentrated and showing a distinct disky shape. Tracing the kinematics of different stellar populations as a function of age leads to the conclusion that also Fornax probably experienced a merger about 8 Gyr ago ([@delPino2017a del Pino et al. 2017a]).
Conclusions
===========
DSph galaxies of the Local Group remain an interesting subject of investigation. Determining the dark matter distribution in dSph galaxies remains challenging but promising pathways for improvement exist. In the era of large surveys we can expect the kinematic samples for the dwarfs to increase dramatically allowing for the modelling methods to reach the next level of refinement. These should include in the near future non-spherical orbit-based modelling and attempts to constrain the inner slope of the dark matter distribution which remained elusive so far. The measurements of proper motions for a significant number of bright stars in dSphs should put additional constraints on the models, both in terms of dark matter distribution and anisotropy of stellar orbits. These studies could also benefit from the use of multiple stellar populations that at least in principle should be able to constrain dark matter distribution at different scales.
Many dwarfs of the Local Group show signs of interactions that shaped them. The two dominant scenarios of such interactions are the tidal stirring in the vicinity of the Milky Way and Andromeda, and mergers between dwarfs. For example, tidal stirring can account for the non-spherical shape and remnant rotation in Carina and a merger can explain the prolate rotation in Andromeda II. More advanced models created in close relation to recent observational findings are needed to reproduce the properties of these quite complicated systems in more detail.
Acknowledgments {#acknowledgments .unnumbered}
===============
This contribution was supported in part by the Polish National Science Center under grant 2013/10/A/ST9/00023.
2017a, *MNRAS*, 465, 3708
2017b, *MNRAS*, 469, 4999
2015, *ApJ*, 813, 10
2017, *ApJ*, 850, 144
2016, *ApJ*, 830, 126
2017, *MNRAS*, 464, 2717
2017, *ApJ*, 842, 56
2018, *ApJ*, 868, 100
, 2018, *MNRAS*, 481, 250
2012, *ApJ*, 758, 124
2017, *MNRAS*, 466, 2006
2011, *ApJ*, 726, 98
2011, *ApJ*, 740, L24
2007, *MNRAS*, 378, 353
2010, *MNRAS*, 402, 1899
2017, *MNRAS*, 470, 3959
2018, *MNRAS*, 476, 2918
2019, *MNRAS*, 482, 5241
2002, *MNRAS*, 333, 697
2009, *MNRAS*, 394, L102
2016, *Galaxies*, 4, 74
2005, *MNRAS*, 363, 918
2008, *MNRAS*, 390, 625
2010, *ApJ*, 725, 1516
2014a, *MNRAS*, 445, 1339
2014b, *MNRAS*, 445, L6
2001, *ApJ*, 559, 754
2018, *ApJ*, 860, 56
|
---
abstract: 'We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional data $\y_i=f(\mathbf{a}_i^T\mathbf{x}_0)$ for some nonlinear (and potentially unknown) link function $f$, when the regressors $\ab_i$ are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates $\x_0$ up to a constant of proportionality $\mu_\ell$, which depends on $f$. Recently, Plan & Vershynin (2015) extended this result to the high-dimensional setting deriving sharp error bounds for the generalized Lasso. Unfortunately, both least-squares and the Lasso fail to recover $\mathbf{x}_0$ when $\mu_\ell=0$. For example, this includes all even link functions. We resolve this issue by proposing and analyzing an alternative convex recovery method. In a nutshell, our method treats such link functions as if they were linear in a lifted space of higher-dimension. Interestingly, our error analysis captures the effect of both the nonlinearity and the problem’s geometry in a few simple summary parameters.'
author:
- Christos Thrampoulidis
- Ankit Singh Rawat
bibliography:
- 'compbib.bib'
title: |
Lifting high-dimensional nonlinear models\
with Gaussian regressors
---
|
---
author:
- 'Wolf-Dieter Richter'
title: 'Skewness-kurtosis adjusted confidence estimators and significance tests'
---
Introduction {#sec:1}
============
Asymptotic normality of the distribution of the suitably centered and normalized arithmetic mean of i.i.d. random variables is one of the best studied and most often exploited facts in asymptotic statistics. It is supplemented in local asymptotic normality theory by limit theorems for the corresponding distributions under the assumption that the mean is shifted of order $n^{-1/2}$. There are many successful simulations and real applications of both types of central limit theorems, and one may ask for a more detailed explanation of those success. The present note is aimed to present such additional theoretical explanation under certain circumstances. Moreover, the note is aimed to further stimulate analogous consideration in more general situations and to stimulate checking the new results by simulation. Moreover, based upon the results presented here, it might become attractive to search for additional explanation to various known simulation results in the area of asymptotic normality which is, however, behind the scope of the present note.
Based upon a large deviation result in [@Li], skewness-kurtosis modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced here in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be exactly controlled. The orders of convergence to zero of both types of probabilities are suitably bounded below by assuming an Osipov-type condition, see [@Os], and the sample distribution is assumed to satisfy a corresponding Linnik condition.
Analogous considerations are presented for the power function when testing a hypothesis concerning the expectation both under the assumption of a true hypothesis as well as under a local alternative. Finally, applications are given for exponential families.
A concrete situation where the results of this paper apply is the case sensitive preparing of the machine settings of a machine tool. In this case, second and higher order moments of the manipulated variable do not change from one adjustment to another one and may be considered to be known over time.
It might be another aspect of stimulating further research if one asks for the derivation of limit theorems in the future being close to those in [@Li] but where higher order moments are estimated.
Let $X_1,...,X_n$ be i.i.d. random variables with the common distribution law from a shift family of distributions, $ P_\mu=P(\cdot-\mu) $, where the expectation equals $\mu,$ $ \mu \in R$, and the variance is $\sigma^2$. It is well known that $T_n=\sqrt{n}(\bar X_n -\mu)/\sigma$ is asymptotically standard normally distributed, $T_n\sim AN(0,1)$. Hence, $P_\mu(T_n>z_{1-\alpha})\rightarrow \alpha$, and under the local non-true parameter assumption, $\mu_{1,n}=\mu+\frac{\sigma}
{\sqrt{n}}(z_{1-\alpha}-z_\beta)$, i.e. if one assumes that a sample is drawn with a shift of location (or with an error in the variable), then $
P_{\mu_{1,n}}(T_n \leq z_{1-\alpha})= P_{\mu_{1,n}}(\sqrt{n}\frac{\bar X_n-\mu_{1,n}}{\sigma} \leq z_{\beta})\rightarrow \beta$ as $n\rightarrow \infty,$ where $z_q$ denotes the quantile of order $q$ of the standard Gaussian distribution.
Let $ACI^u= [\bar X_n - \frac{ \sigma}{\sqrt{n}}z_{1-\alpha}, \infty)$ denote the upper asymptotic confidence interval for $\mu$ where the true non-covering probabilities satisfy the asymptotic relation $$P_\mu(ACI^u {\; does\; not\; cover\; } \mu)\rightarrow \alpha,\; n\rightarrow \infty.$$ Because $
P_{\mu_{1,n}}(\bar X_n-\frac{\sigma}{\sqrt{n}}z_{1-\alpha}<\mu)
=
P_{\mu_{1,n}}(\sqrt{n}\frac{\bar X_n-\mu_{1,n}}{\sigma}\leq z_\beta)
$, the covering probabilities under $n^{-1/2}$-locally chosen non-true parameters satisfy $$P_{\mu_{1,n}}(ACI^u {\; covers\; } \mu) \rightarrow \beta, \; n\rightarrow \infty.$$ The aim of this note is to prove refinements of the latter two asymptotic relations where $\alpha=\alpha(n)\rightarrow 0 $ and $\beta=\beta(n)\rightarrow 0$ as $n\rightarrow \infty$, and to prove similar results for two-sided confidence intervals and for the power function when testing corresponding hypotheses.
Expectation estimation
======================
First and second kind adjusted one-sided confidence intervals {#sec:2}
-------------------------------------------------------------
According to [@Li], it is said that a random variable $X$ satisfies the Linnik condition of order $\gamma, \gamma>0, $ if $${E}_\mu
\exp\{|X-\mu|^{\frac{4\gamma}{2\gamma+1}}\}
<\infty.
\label{Lgamma}$$ Let us define the first kind (or first order) adjusted asymptotic Gaussian quantile by $$z_{1-\alpha(n)}(1)=z_{1-\alpha(n)}
+\frac{g_1}{6\sqrt{n}}
z^2_{1-\alpha(n)}$$ where $g_1=\it E (X-\it E(X))^3/\sigma^{3/2}$ is the skewness of $X$. Moreover, let the first kind (order) adjusted upper asymptotic confidence interval for $\mu$ be defined by $$ACI^u(1)=[\bar X_n -\frac{\sigma}{\sqrt{n}}
z_{1-\alpha(n)}(1), \infty)$$ and denote a first kind modified non-true local parameter choice by $$\mu_{1,n}(1)=\mu_{1,n}+\frac{\sigma g_1}
{6n} (z^2_{1-\alpha(n)}-z^2_{\beta(n)}).$$ Let us say that the probabilities $\alpha(n) $ and $\beta(n)$ satisfy an Osipov-type condition of order $\gamma$ if $$n^{\gamma}\exp\{\frac{n^{2\gamma}}{2}\}
\cdot \min\{\alpha(n),\beta(n)\}\rightarrow \infty ,\; n\rightarrow \infty.
\label{Os1}$$ This condition means that neither $\alpha(n)$ nor $\beta(n) $ tend to zero as fast as or even faster than $n^{-\gamma}\exp\{-n^{2\gamma}/2\},$ i.e. $\min\{\alpha(n), \beta(n)\}\gg n^{-\gamma}\exp\{-n^{2\gamma}/2\}
,$ and that $ \max\{z_{1-\alpha(n)},z_{1-\beta(n)}\}=
o(n^\gamma), n \rightarrow \infty.$
If two functions $f,g$ satisfy the relation $\lim\limits_{n\rightarrow\infty }f(n)/g(n)=1$ then this asymptotic equivalence will be expressed as $f(n)\sim g(n), n\rightarrow \infty.$
If $\alpha(n)\downarrow 0$, $\beta(n)\downarrow 0$ as $n\rightarrow \infty$ and conditions ($\ref{Lgamma} $) and ($\ref{Os1}$) are satisfied for $\gamma\in (\frac{1}{6},\frac{1}{4}]$ then $$P_\mu (ACI^u(1) {\; does\; not\; cover\; } \mu)\sim \alpha(n), \, n\rightarrow \infty$$ and $$P_{\mu_{1,n}(1)} (ACI^u(1) {\; covers\; } \mu)\sim \beta(n), \, n\rightarrow \infty.$$
Let us define the second kind adjusted asymptotic Gaussian quantile $$z_{1-\alpha(n)}(2)=z_{1-\alpha(n)}(1)
+\frac{3g_2-4g_1^2}{72n}
z^3_{1-\alpha(n)}$$ where $g_2=\it E (X-\it E(X))^4/\sigma^{4}-3$ is the kurtosis of $X$, the second kind adjusted upper asymptotic confidence interval for $\mu$ $$ACI^u(2)=[\bar X_n -\frac{\sigma}{\sqrt{n}}
z_{1-\alpha(n)}(2), \infty),$$ and a second kind modified non-true local parameter choice $$\mu_{1,n}(2)
=\mu_{1,n}(1)+
\frac{\sigma(3g_2-4g_1^2)}
{72n^{3/2}}(z^3_{1-\alpha(n)}-z_{\beta(n)}^3)
.$$
If $\alpha(n)\downarrow 0$, $\beta(n)\downarrow 0$ as $n\rightarrow \infty$ and conditions ($\ref{Lgamma} $) and ($\ref{Os1}$) are satisfied for $\gamma\in (\frac{1}{4},\frac{3}{10}]$ then $$P_\mu (ACI^u(2) {\; does\; not\; cover\; } \mu)\sim \alpha(n), \, n\rightarrow \infty$$ and $$P_{\mu_{1,n}(2)} (ACI^u(2) {\; covers\; } \mu)\sim \beta(n), \, n\rightarrow \infty.$$
Under the same assumptions, analogous results are true for lower asymptotic confidence intervals, i.e. for $ACI^l(s)=(-\infty, \bar X_n+\frac{\sigma}{\sqrt{n}}z^{-}_{1-\alpha}(s)), s=1,2:$ $$P_\mu(ACI^l(s)\; does\; not\; cover\; \mu)
\sim \alpha(n)$$ and $$P_{\mu^{-}_{1,n}(s)}(ACI^l(s)\; covers\; \mu)
\sim \beta(n),\, n\rightarrow \infty .$$ Here, $z^{-}_{1-\alpha}(s)$ means the quantity $z_{1-\alpha}(s)$ where $g_1$ is replaced by $-g_1, s=1,2,$ and $$\mu^{-}_{1,n}(s)=\mu-\frac{\sigma}{\sqrt{n}}
(z_{1-\alpha}-z_\beta)+\frac{\sigma g_1}{6n}
(z^2_{1-\alpha}-z^2_\beta)-
\frac{\sigma(3g_2-4g_1^2)}
{72n^{3/2}}(z^3_{1-\alpha}-
z^3_\beta)I_{\{2\}}(s).$$
In many situations where limit theorems are considered as they were in Section 1, the additional assumptions $(\ref{Lgamma})$ and $(\ref{Os1})$ may, possibly unnoticed, be fulfilled. In such situations, Theorems 1 and 2, together with the following theorem, give more insight into the asymptotic relations stated in Section 1.
Large Gaussian quantiles satisfy the asymptotic representation $$z_{1-\alpha}=\sqrt{-2\ln\alpha-\ln|\ln\alpha|-
\ln(4\pi)}\cdot
(1+O\frac{\ln|\ln\alpha|}{(\ln\alpha)^2}),
\alpha\rightarrow +0.$$
Two-sided confidence intervals
------------------------------
For $s\in\{1,2\}, \alpha>0$, put $
L(s;\alpha)=\bar X_n-\frac{\sigma}{\sqrt{n}}z_{1-\alpha}(s)$ and $ R(s;\alpha)=\bar X_n+\frac{\sigma}{\sqrt{n}}z^-_{1-\alpha}(s).$ Further, let $ \alpha_i(n)>0,\,i=1,2, \alpha_1(n)+\alpha_2(n)<1,$ and $$ACI(s;\alpha_1(n),\alpha_2(n))=[L(s;\alpha_1(n)), R(s;\alpha_2(n))].$$ If conditions $(\ref{Lgamma})$ and $(\ref{Os1})$ are fulfilled then $P_\mu((-\infty, L(s;\alpha_1(n)))\; {\it covers}\; \mu)\sim \alpha_1(n)$ and $P_\mu(( R(s;\alpha_2(n)), \infty)\; {\it covers}\; \mu)\sim \alpha_2(n)$ as $n\rightarrow \infty$.
With more detailed notation $\mu_{1,n}(s)=\mu_{1,n}(s;\alpha,\beta)$ and $\mu^-_{1,n}(s)=\mu^-_{1,n}(s;\alpha,\beta)$,
$P_{\mu_{1,n}(s;\alpha_1(n),\beta_1(n))}
((L(s;\alpha_1(n)),\infty)\; covers\; \mu)\sim \beta_1(n)$,
$P_{\mu^-_{1,n}(s;\alpha_2(n),\beta_2(n))}
((-\infty, R(s;\alpha_2(n)))\; covers\; \mu)\sim \beta_2(n), n\rightarrow \infty.$
The following corollary has thus been proved.
If $\alpha_1(n)\downarrow 0$, $\alpha_2(n)\downarrow 0$ as $n\rightarrow \infty$ and conditions ($\ref{Lgamma} $) and ($\ref{Os1}$) are satisfied for $\gamma\in (\frac{1}{6},\frac{1}{4}]$ if $s=1$ and for $\gamma\in (\frac{1}{4},\frac{3}{10}]$ if $s=2$, and with $(\alpha(n),\beta(n))=(\alpha_1(n),\alpha_2(n))$, then $$P_\mu (ACI(s;\alpha_1(n),\alpha_2(n)) {\; does\; not\; cover\; } \mu)\sim (\alpha_1(n)+\alpha_2(n)), \, n\rightarrow \infty.$$Moreover, $$\max\limits_{\nu\in\{\mu_{1,n}
(s;\alpha_1(n),\beta_1(n)),
\mu^-_{1,n}(s;\alpha_2(n),\beta_2(n)) \}}
P_{\nu} (ACI(s) {\; covers\; } \mu)\leq\max\{\beta_1(n),\beta_2(n)\}.$$
Testing
=======
Adjusted quantiles
------------------
Let us consider the problem of testing the hypothesis $H_0: \mu\leq\mu_0$ versus the alternative $H_A: \mu > \mu_0$. The first and second kind adjusted decision rules of the one-sided asymptotic Gauss test suggest to reject $H_0$ if $T_{n,0}> z_{1-\alpha(n)}(s)$ for $s=1$ or $s=2,$ respectively, where $T_{n,0}=\sqrt{n}(\bar X_n-\mu_0)/\sigma$. Because $$P_{\mu_0}(reject\; H_0)=P_{\mu_0}(ACI^u(s) \; does\; not\; cover\; \mu_0),$$ it follows from Theorems 1 and 2 that under the conditions given there the (sequence of) probabilities of an error of first kind satisfy the asymptotic relation $$P_{\mu_0}(reject\; H_0)\sim \alpha(n), n\rightarrow \infty.$$Concerning the power function of this test, because $$P_{\mu_{1,n}(s)}(do\; not\; reject\; H_0)= P_{\mu_{1,n}(s)}(ACI^u(s) \; covers\; \mu_0),$$it follows under the same assumptions that the probabilities of a second kind error in the case that the sequence of the modified local parameters is $(\mu_{1,n}(s))_{n=1,2,...}$, satisfy $$P_{\mu_{1,n}(s)}(do\; not\; reject\; H_0)\sim \beta(n), n\rightarrow \infty.$$ Similar consequences for testing $H_0: \mu>\mu_0$ or $H_0: \mu\neq\mu_0$ are omitted, here.
Adjusted statistics
-------------------
Let $T_n^{(1)}=T_n-\frac{g_1}{6\sqrt{n}}T_n^2$ and $T_n^{(2)}=T_n^{(1)}-\frac{3g_2-8g_1^2}
{72n}T_n^3$ be the first and second kind adjusted asymptotically Gaussian statistics, respectively, where $T_n=\frac{\sqrt{n}}{\sigma}(\bar X_n - \mu)$.
If the conditions ($\ref{Lgamma} $) and ($\ref{Os1}$) are satisfied for a certain $\gamma \in (\frac{s}{2s+4},\frac{s+1}{2s+6}]$ where $s\in\{1,2\}$ then $$P_{\mu_0}(T_n^{(s)}>z_{1-\alpha(n)})\sim \alpha(n), \; n\rightarrow \infty$$ and $$P_{\mu_{1,n}(s)}(T_n^{(s)}\leq z_{1-\alpha(n)})\sim \beta (n), \;
n\rightarrow \infty.$$
Clearly, the results of this theorem apply to both hypothesis testing and confidence estimation in a similar way as described in the preceding sections.
Application to exponential families
===================================
Let $\nu$ denote a $\sigma$-finite measure and assume that the distribution $P_\vartheta $ has the Radon-Nikodym density $\frac{dP_\vartheta}{d\nu}(x)=
\frac{e^{\vartheta x}}{\int
e^{\vartheta x}\nu(dx)}=e^{\vartheta x-B(\vartheta)}$, say. We assume that $X(\vartheta)\sim P_\vartheta$ and $X_1=X(\vartheta)-{ E}X(\vartheta)+\mu\sim\widetilde{P}_\mu$ where $\vartheta$ is known and $\mu$ is unknown. In the product-shift-experiment $ [R^n,{\cal B}^n,\;\{\widetilde{P}^{\times n}_\mu,\,\mu\in R\}]$, expectation estimation and testing may be done as in Sections 2 and 3, respectively, where $g_1= B^{'''}(\vartheta)/(B^{''}(\vartheta))^{3/2}$ and $g_2$ allows a similar representation.
Another problem which can be dealt with is to test the hypothesis $H_0:\vartheta\leq\vartheta_0$ versus the alternative $
H_{1n}:\vartheta\geq\vartheta_{1n}$ if one assumes that the expectation function $\vartheta\rightarrow B'(\vartheta)=E_\vartheta
X$ is strongly monotonous. For this case, we finally present just the following particular result which applies to both estimating and testing.
If conditions ($\ref{Lgamma} $) and ($\ref{Os1}$) are satisfied for $\gamma\in(\frac{1}{6},\frac{1}{4}]$ then $$P_{\vartheta_0}^{\times
n}(\sqrt{n}\frac{\overline{X}_n-B'(\vartheta_0)}{\sqrt{B''(\vartheta_0)}}>
z_{1-\alpha(n)}+\frac{B'''(\vartheta_0)}{6\sqrt{n}(B''(\vartheta_0))^{3/2}}
z^2_{1-\alpha(n)})\sim\alpha(n),n\;\rightarrow \infty.$$
Sketch of proofs
================
[*Proof of Theorems 1 and 2.*]{} If condition ($\ref{Os1}$) is satisfied then $x=z_{1-\alpha(n)}=o(n^\gamma),n\rightarrow \infty$ for $\gamma \in (\frac{1}{6},\frac{3}{10}]$, and if ($\ref{Lgamma}$) then, according to [@Li], $P_\mu(T_n>x)\sim f_{n,s}^{(X)}(x), x\rightarrow \infty$ where $
f_{n,s}^{(X)}(x)=\frac{1}{\sqrt{2\pi}x}
\exp\{-\frac{x^2}{2}+\frac{x^3}
{\sqrt{n}}\sum\limits_{k=0}^{s-1}
a_k(\frac{x}{\sqrt{n}})^k\}
$ and $s$ is an integer satisfying $\frac{s}{2(s+2)}<\gamma\leq\frac{s+1}{2(s+3)}$, i.e. $s=1$ if $\gamma\in (\frac{1}{6}, \frac{1}{4}]$ and $s=2$ if $\gamma\in (\frac{1}{4}, \frac{3}{10}]$. Here, $o(\cdot)$ stands for the small Landau symbol, and the constants $a_0=\frac{g_1}{6},\, a_1=\frac{g_2-3g_1^2}{24} $ are due to the skewness $g_1$ and kurtosis $g_2$ of $X$. Note that $\frac{g_1x^2}{6\sqrt{n}}=o(x)$ because x$=o(n^{1/2})$, thus $x+\frac{g_1x^2}{6\sqrt{n}}=o(n^\gamma)$, and $P_\mu(T_n>x+\frac{g_1x^2}{6\sqrt{n}})
\sim f_{n,1}(x+\frac{g_1x^2}{6\sqrt{n}})$. Hence, $P_\mu(T_n>x+\frac{g_1x^2}{6\sqrt{n}})\sim 1-\Phi(x).$ Similarly, $P_\mu(T_n>z_{1-\alpha(n)}(s))
\sim\alpha(n), \, s=1,2$. Further, $P_{\mu_{1,n}(s)}(T_n\leq z_{1-\alpha(n)}(s))$ $$=P_{\mu_{1,n}(s)}(\frac{\sqrt{n}}{\sigma}
(\bar X_n -\mu_{1,n}(s))< z_{1-\alpha(n)}(s)
-\frac{\sqrt{n}}{\sigma}(\mu_{1,n}(s)-\mu))
=P_0(\frac{\sqrt{n}}{\sigma}
\bar X_n< z_{\beta(n)}(s)).$$ The latter equality holds because $\{P_\mu, \mu\in (-\infty, \infty )\}$ is assumed to be a shift family. It follows that $P_{\mu_{1,n}(s)}(T_n\leq z_{1-\alpha(n)}(s))$ $$=P_0(\frac{\sqrt{n}}{\sigma}
(-\bar X_n)\geq z_{1-\beta(n)}+
\frac{-g_1}{6\sqrt{n}}z^2_{1-\beta(n)}
+I_{\{2\}}(s)
\frac{3g_2-4g_1}{72n}z^3_{1-\beta(n)}).$$
Note that $-g_1, g_2$ are skewness and kurtosis of $-X_1$. Thus, $$P_{\mu_{1,n}(s)}(T_n\leq z_{1-\alpha(n)}(s))\sim f_{n,s}^{(-X)}(z_{1-\beta(n)}(s))
\sim\beta(n), n \rightarrow \infty.$$ Because $P_\mu(ACI^u \; does \; not \; cover\; \mu)=P_{\mu}(T_n > z_{1-\alpha(n)}(s))$ and
$P_{\mu_{1,n}(s)}(ACI^u \; covers\; \mu)
=P_{\mu_{1,n}(s)}(T_n\leq z_{1-\alpha(n)}(s))$, the theorems are proved.
[*Proof of Remark 1.*]{} The first statement of the remark follows from $$P_\mu(\mu>\bar X_n+\sigma z^{-}_{1-\alpha(n)}/\sqrt{n})
=P_\mu(\sqrt{n}(-\bar X_n+\mu)/\sigma >z^{-}_{1-\alpha(n)})$$ and the second one from $$P_{\mu^{-}_{1,n}(s)}(\mu< \bar X_n+ \sigma z^{-}_{1-\alpha(n)}/\sqrt{n})
=P_0(\bar X_n>\mu-\mu^{-}_{1,n}(s)-\sigma z^{-}_{1-\alpha(n)}/\sqrt{n})$$ $$=P_0(\sqrt{n}\bar X_n/\sigma > z_{1-\beta(n)}(s)).$$
[*Proof of Theorem 3.*]{} We start from the well known relations $$\alpha=1-\Phi(z_{1-\alpha})=
(1+O(\frac{1}{z^2_{1-\alpha}}))\frac{1}{\sqrt{2\pi}z_{1-\alpha}}
e^{-\frac{z^2_{1-\alpha}}{2}},\;\alpha\rightarrow 0.$$ The solution to the approximative quantile equation $\alpha=\frac{1}{\sqrt{2\pi}x}
e^{-\frac{x^2}{2}}$ will be denoted by $x=x_{1-\alpha}$. Let us put $$xe^{\frac{x^2}{2}}=\frac{1}{\sqrt{2\pi}\alpha}=:y.
\label{QGl}$$
If $x\geq 1$ then it follows from $(\ref{QGl})$ that $ y\geq e^{\frac{x^2}{2}}$, hence $x^2\leq\ln(y^2)$. It follows again from $(\ref{QGl})$ that $ y^2\leq\ln(y^2)e^{x^2}$, thus $x^2\geq \ln(\frac{y^2}{\ln y^2}).$ After one more such step, $$\ln(\frac{y^2}{\ln y^2})\leq
x^2\leq\ln[\frac{y^2}{\ln(\frac{y^2}{\ln y^2})}].$$ The theorem now follows from $$x^2=\{\ln y^2-\ln 2-\ln\ln y\}\{1+O(\frac{\ln\ln y}{(\ln
y^2)^2})\}, y\rightarrow \infty.$$
[*Proof of Theorem 4.*]{} Recognize that if $g_{n,s}(x)=o(\frac{1}{x}), x\rightarrow \infty$ then $f^{(+/-)(X)}_{n,s}(x+g_{n,s}(x))\sim f^{(+/-)(X)}_{n,s}(x),x\rightarrow \infty.$ Let us restrict to the case $s=1$. According to [@Li], $$P_{\mu_0}(T_n^{(1)}>z_{1-\alpha(n)})\sim
P_{\mu_0}(\frac{3\sqrt{n}}{g_1}>T_n^{(1)}>z_{1-\alpha(n)}).$$ The function $f^{(1)}_n(t)=t-\frac{g_1t^2}{6\sqrt{n}}$ has a positive derivative, $f^{(1)'}_n(t)=1-
\frac{g_1t}{3\sqrt{n}}>0$, if $g_1t<3\sqrt{n}$. Denoting there the inverse function of $f_n^{(1)}$ by $f_n^{(1)^{-1}}$, it follows $f_n^{(1)^{-1}}(x)=
x+\frac{g_1x^2}{6\sqrt{n}}+O(\frac{x^3}{n})$ and $
f^{(1)}_n(f_n^{(1)^{-1}}(x)) =
x+o(\frac{1}{x}).
$ Thus,
$
P_{\mu_o}(T_n^{(1)}>z_{1-\alpha(n)})\sim
P_{\mu_o}(T_n>z_{1-\alpha(n)}(1))
\sim\alpha(n).$
Moreover, $P_{\mu_{1n}(1;\alpha(n),\beta(n))}(T_n^{(1)}\leq
z_{1-\alpha(n)})=P_{\mu_{1n}(1;\alpha(n),\beta(n))}(T_n\leq T^{(1)^{-1}}_n
(z_{1-\alpha(n)}))$\
$=P_{\mu_{1n}(1;\alpha(n),\beta(n))}(\;\sqrt{n}\frac{\overline{X}_n
-\mu_{1n}(1)}{\sigma} \leq
z_{1-\alpha(n)}(1)+\frac{z^2_{1-\alpha(n)}g_1}{6\sqrt{n}}+O(\frac{z^3_{1-\alpha(n)}}{n})-
\sqrt{n}\frac{\mu_{1n}(1)-\mu_0}{\sigma}) \sim f_{n,1}^{(-X)}(-z_{\beta(n)}(1)+ O(\frac{z^3_{1-\alpha(n)}}{n})) \sim 1-\Phi(z_{1-\beta(n)})=\beta(n).$
[99.]{} Linnik, Yu.V.: Limit theorems for sums of independent variables taking into account large deviations. I-III. Theor. Probab. Appl. 6 (1961), 131-148, 345-360; 7 (1962), 115-129; translation from Teor. Veroyatn. Primen. 6, 145-162, 377-391 (1961); 7, 121-134 (1962). Osipov, L.V.: Multidimensional limit theorems for large deviations. Theory Probab. Appl. 20, 38-56 (1975); translation from Teor. Veroyatn. Primen. 20, 40-57 (1975).
|
---
author:
- Emanuele Dotto
- Irakli Patchkoria
- Kristian Jonsson Moi
bibliography:
- 'bib.bib'
title: 'Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology'
---
Introduction {#introduction .unnumbered}
============
The various rings of Witt vectors have a prominent role in number theory and algebraic topology. They are defined as endofunctors on the category of commutative rings, and they provide a functorial way of passing from characteristic $p$ to characteristic zero. The prototypical example is the ring of $p$-typical Witt vectors of the field $\mathbb{F}_p$, which is isomorphic to the ring of $p$-adic integers $\operatorname{\mathbb{Z}}_p$. These Witt vectors functors exhibit some fundamental extra structures, such as $\lambda$-operations, $\delta$-ring structures, and Frobenius lifts, which determine several of their universal properties (see e.g. [@AT], [@Joyal]). In topology, Witt vectors appear in calculations related to topological cyclic homology [@Wittvect], cyclic $K$-theory [@Almkvist1], and in chromatic homotopy theory. Here they also exhibit extra structure, as they relate to the free Tambara functors of the cyclic groups (see [@BrunTamb]). In this paper we will provide novel additional structure on the Witt vectors, related to polynomial laws and polynomial maps.
We recall from [@Roby1] that a *multiplicative polynomial law* $f$ from a commutative ring $A$ to a commutative ring $B$ is a collection of multiplicative maps $$f_R\colon A\otimes_{\operatorname{\mathbb{Z}}}R\longrightarrow B\otimes_{\operatorname{\mathbb{Z}}}R$$ for every commutative ring $R$, which is natural with respect to ring homomorphisms in $R$. Every multiplicative polynomial law $f$ of finite degree $n$ has an underlying multiplicative map $$f_{\operatorname{\mathbb{Z}}}\colon A\longrightarrow B$$ which is $n$-polynomial, in the sense that its $(n+1)$-st cross-effect, or deviation, vanishes. The main goal of this paper is to show that various Witt vectors functors extend from the category of commutative rings and ring homomorphisms to the category of commutative rings and polynomial laws of finite degree, or to polynomial maps.
In §\[secGamma\] we introduce an axiomatic framework of “PD-functors”, to study this extended functoriality in polynomial laws. A *PD-functor* is an endofunctor of the category of commutative rings $$F\colon \operatorname{Ring}\longrightarrow \operatorname{Ring}$$ which commutes with certain limits and colimits. Examples of these functors include the Witt vectors $W_S$ for any truncation set $S\subset\mathbb{N}$, so in particular the big and $p$-typical Witt vectors, as well as their truncated versions. They also include the rational Witt vectors, the subring of the big Witt vectors of those power series with constant term one which are rational functions, which by a theorem of Almkvist [@Almkvist1; @Almkvist2] is isomorphic to the cyclic $K$-theory ring. The following is the main result of §\[secGamma\].
Any PD-functor $F\colon \operatorname{Ring}\to \operatorname{Ring}$ extends canonically to an endofunctor on the category $\operatorname{Ring^{poly}}$ of commutative rings and multiplicative polynomial laws. For any of the Witt vectors functors $W$ listed above, this is the unique extension such that for any multiplicative polynomial law $f \colon A \to B$ the diagram $$\xymatrix@C=70pt@R=15pt{W(A) \ar[d]_w \ar[r]^{W(f)} & W(B) \ar[d]^w \\ \prod A \ar[r]^{\prod f} & \prod B }$$ commutes in $\operatorname{Ring^{poly}}$, where $w$ is the ghost map of $W$ and $\prod f$ is the product polynomial law.
The theorem for a general PD-functor is proved in §\[secGammathm\]. We reduce the construction to torsion-free rings by means of a resolution argument. We then use that the $n$-homogeneous polynomial laws out of a torsion-free ring $A$ are classified by the “*universal polynomial law*” $\gamma_A=(-)^{\otimes n}\colon A\to (A^{\otimes n})^{\Sigma_n}$, which we by definition send to the map $$F(\gamma_A)\colon F(A)\xrightarrow{\gamma_{F(A)}} (F(A)^{\otimes n})^{\Sigma_n}\longrightarrow F(A^{\otimes n})^{\Sigma_n}\stackrel{\cong}{\longleftarrow}F((A^{\otimes n})^{\Sigma_n}),$$ where the last map is an isomorphism by the axioms of a PD-functor. In fact we show that our extension of $F$ to $\operatorname{Ring^{poly}}$ is the unique one that sends $\gamma_A$ to this map. When $A$ has torsion, the universal polynomial law has value in the divided powers $\Gamma_nA$, which motivates the name PD-functor. In §\[ghostlaw\] we describe the ghost components of a polynomial law for the Witt vectors functors.
By a theorem of Almkvist [@Almkvist1; @Almkvist2], our result shows that the cyclic $K$-group $\operatorname{K}^{cy}_0(A)$, defined as the group completion of the isomorphism classes of endomorphisms of finitely generated projective $A$-modules modulo the zero endomorphisms, is functorial in multiplicative polynomial laws. It is well known that $\operatorname{K}^{cy}_0$ and $\operatorname{K}_0$, as functors from additive categories, are functorial in polynomial *functors* (see [@BGMN] for a highly structured statement). It is however not clear how multiplicative polynomial laws of commutative rings relate to polynomial functors on the respective module categories. We also remark that $\operatorname{K}_0$ is not a PD-functor (Example \[K0noGamma\]), and therefore that our theorem does not provide this extra functoriality for $\operatorname{K}_0$.
In §\[secpolymap\] we turn our attention to $n$-*polynomial maps*. These are the multiplicative maps $f\colon A\to B$ which satisfy the additive condition $$(\operatorname{cr}_{n+1})f(a_1,\dots,a_{n+1}):=\sum_{\substack{U\subset \{1,\cdots, n+1\} }} (-1)^{n+1-|U|}f(\sum_{l\in U}a_l)=0.$$ As remarked above polynomial laws forget to polynomial maps, but this correspondence is neither surjective nor injective in general. However, it is bijective when the target ring is $p$-local and the degrees are at most $p-1$. By combining this observation with the theorem above we prove the following, in §\[secWpoly\]. For any integer or infinity $1\leq m\leq \infty$, let $W_m(A;p)$ denote the ring of $p$-typical $m$-truncated Witt vectors.
The functor $W_m(-;p)$ extends to the partial category of multiplicative polynomial maps of degree at most $p-1$. That is, a multiplicative $n$-polynomial map $f\colon A\to B$ induces a multiplicative $n$-polynomial map $$W_m(f)\colon W_m(A;p)\longrightarrow W_m(B;p)$$ for every $n<p$, with the property that if $f\colon A\to B$ and $g\colon B\to C$ are multiplicative and $n$ and $k$-polynomial, respectively, and $nk<p$, then $
W_m(g)\circ W_m(f)=W_m(g\circ f)$. This extension is unique with the property that in ghost coordinates $w_jW_m(f)=fw_j$ for every $0\leq j<m$, i.e., the diagram $$\xymatrix{W_m(A;p) \ar[d]^w \ar[rr]^{W_m(f)} & & W_m(B;p) \ar[d]^w \\ \prod_{j=0}^{m-1} A \ar[rr]^{\prod_{j=0}^{m-1} f} & & \prod_{j=0}^{m-1} B }$$ commutes.
Much like the universal polynomials for the sum and multiplication of $W_m(A;p)$ it does not seem to be possible to give an explicit description of the Witt components of the map $W_m(f)$, but there is an inductive procedure for finding them. The first two components of $W_m(f)$ are $$W_m(f)(a_0,a_1,\dots)=\big(f(a_0),\sum_{i=1}^{p-1}(-1)^i({p\choose i}/p)f(a_{0}^p+ia_1),\dots\big)$$ (see Example \[formula\]). When $f$ is a ring homomorphism, one can verify using standard binomial identities that the second component is equal to $f(a_1)$, so that this construction indeed extends the usual functoriality of $W_m(-;p)$ in ring homomorphisms. In §\[secWpoly\] we also discuss how the hypotheses of this corollary are necessary. Most notably, there is no further extension of this functoriality on multiplicative polynomial maps of degree $p$. For example, the map $$N\colon \operatorname{\mathbb{Z}}\longrightarrow \operatorname{\mathbb{Z}}[x]/(x^2-px)$$ that sends $a$ to $N(a)=a+\frac{a^p-a}{p}x$ is of degree $p$ and does not induce a map on $W_2(-;p)$ with the ghost components as in the corollary (see Example \[cex\]).
Our motivation for considering polynomial maps is rooted in topology. A large supply of polynomial maps is provided by the multiplicative transfers, or norms, of *Tambara functors*. A Tambara functor is a Mackey functor with a multiplication (a Green functor) and multiplicative transfers subject to certain axioms. Tambara functors naturally occur in topology, in particular in equivariant stable homotopy theory, as the components of genuine $G$-equivariant commutative ring spectra. For example the map $N$ above is the norm of the Burnside Tambara functor for the group $G=C_p$ which corresponds to the inclusion $e\to C_p$ of index $p$. This Tambara functor is the components of the initial $G$-equivariant commutative ring spectrum, namely the sphere spectrum.
For the group $G=\operatorname{\mathbb{Z}}/2$, a Tambara functor $T$ consists of two commutative rings $A$ and $B$, and maps $$\xymatrix@C=70pt{A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}\ar@(ul,dl)[]_{\tau}&B\ar[l]|-{\operatorname{res}},
}$$ subject to the axioms of [@Tambara]. In particular the involution $\tau$ and $\operatorname{res}$ are ring homomorphisms, $N$ is multiplicative $2$-polynomial, and $\operatorname{tran}$ is additive and determined by the Tambara reciprocity relation $\operatorname{tran}(a)=N(a+1)-N(a)-1$. Therefore for every odd prime $p$, we can define a diagram $$W_n(T;p):=\big(\xymatrix@C=70pt{\ar@(ul,dl)[]_{W_m(\tau)}&\hspace{-3.5cm} W_m(A;p)
&W_m(B;p) \ar[ll(.71)]|-{W_m(\operatorname{res})} \ar@{<-}@<-1ex>[ll(.71)]_-{\operatorname{tran}}\ar@{<-}@<1ex>[ll(.71)]^-{W_m(N)}\ \big),
}$$ where $W_m(\tau)$ and $W_m(\operatorname{res})$ are induced by the usual functoriality in ring homomorphisms, $W_m(N)$ is induced by the functoriality of the Corollary, and $\operatorname{tran}(x):=W_m(N)(x+1)-W_m(N)(x)-1$. The following is proved in §\[secTambara\].
Let $p$ be an odd prime and $1\leq m \leq\infty$ an integer or infinity. The diagram $W_m(T;p)$ is a $\operatorname{\mathbb{Z}}/2$-Tambara functor. It is the unique Tambara functor functorial in $T$ with underlying rings $W_m(A;p)$ and $W_m(B;p)$ such that the ghost maps define a natural morphism of Tambara functors $$\xymatrix@C=70pt@R=18pt{W_{m}(A;p)\ar[d]_w \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}&W_{m}(B;p)\ar[d]^w\ar[l]|-{\operatorname{res}}
\\
\prod_{m}A \ar@<1ex>[r]^-{\prod\operatorname{tran}}\ar@<-1ex>[r]_-{\prod N}& \prod_{m}B\ar[l]|-{\prod\operatorname{res}}\rlap{\ .}
}$$
The reader should not confuse our construction with those arising in the theory of Witt vectors for Green functors of [@GreenWitt]. In §\[secTHR\] we use this theorem to describe the components of the dihedral fixed points of the *real topological Hochschild homology* $\operatorname{THR}(E)$ of a connective commutative $\operatorname{\mathbb{Z}}/2$-equivariant ring spectrum $E$. Let $D_{p^m}$ be the dihedral group of order $2p^m$. Then $\operatorname{THR}(E)$ is a commutative $D_{p^m}$-equivariant ring spectrum, for all $m\geq 0$, defined as the (derived) dihedral Bar construction $$\operatorname{THR}(E):=B^{di}E=|[k]\longmapsto E^{\wedge k+1}|$$ with the usual cyclic structure of $\operatorname{THH}(E)$, and the involution of $E^{\wedge k+1}$ defined as the indexed smash product over the $\operatorname{\mathbb{Z}}/2$-set $\{0,1,\dots,k\}$ with the involution which reverses the order of $\{1,\dots,k\}$ (see [@THRmodels] and [@Amalie]). For every $m\geq 0$ one can define a $\operatorname{\mathbb{Z}}/2$-spectrum $$\operatorname{TRR}^{m+1}(E;p):=\operatorname{THR}(E)^{C_{p^m}},$$ which is a $\operatorname{\mathbb{Z}}/2$-equivariant refinement of the $\operatorname{TR}$-spectra of [@BHM]. In Corollary 5.2 of [@THRmodels] we identify the $\operatorname{\mathbb{Z}}/2$-Tambara functor of components $\underline{\pi}_0\operatorname{THR}(E)$, in the case $m=0$. Theorem [@Wittvect] of Hesselholt and Madsen asserts that the components of the underlying ring spectrum $$\pi_0\operatorname{TR}^{m+1}(E;p)=\pi_0\operatorname{THH}(E)^{C_{p^m}}\cong W_{m+1}(\pi_0E;p)=W_{m+1}(\pi_0\operatorname{THH}(E);p)$$ are naturally isomorphic to the ring of $p$-typical $(m+1)$-truncated Witt vectors of $\pi_0E$. Here we establish a real version of this statement for odd primes. We call a $\operatorname{\mathbb{Z}}/2$-Tambara functor *cohomological* if $N\operatorname{res}=(-)^{2}$ (for example a commutative ring with involution).
Let $E$ be a connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum, with $\underline{\pi}_0E$ cohomological. Then for every odd prime $p$ and $m\geq 0$, there is an isomorphism of $\operatorname{\mathbb{Z}}/2$-Tambara functors $$\underline{\pi}_0\operatorname{TRR}^{m+1}(E;p)=\big(
\xymatrix@C=30pt{\pi_0\operatorname{THH}(E)^{C_{p^m}}
&\pi_0\operatorname{THR}(E)^{D_{p^m}} \ar[l]|-{\operatorname{res}} \ar@{<-}@<-1ex>[l]_-{\operatorname{tran}}\ar@{<-}@<1ex>[l]^-{N}}
\big) \cong W_{m+1}(\underline{\pi}_0\operatorname{THR}(E);p)$$ which is natural in $E$. Here $W_{m+1}(-;p)$ is the Tambara functor of the previous theorem. In particular there is a natural ring isomorphism $\pi_0\operatorname{THR}(E)^{D_{p^m}}\cong W_{m+1}(\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2};p)$.
At the prime $p=2$, or if $\underline{\pi}_0E$ is not cohomological, the ring $\pi_0\operatorname{THR}(E)^{D_{p^m}}$ is in general not the Witt vectors of a ring. Moreover on the algebraic side there is no reason for the norm of $\underline{\pi}_0\operatorname{THR}(E)$ to induce a map on $2$-typical Witt vectors, since the condition $n<p$ of Corollary B is violated. For example when $E=\mathbb{S}$ is the sphere spectrum, whose components are not cohomological, $\operatorname{THR}(\mathbb{S})=\mathbb{S}$ and therefore $$\pi_0\operatorname{TRR}^{m+1}(\mathbb{S};p)^{\operatorname{\mathbb{Z}}/2}=\pi_0\mathbb{S}^{D_{p^m}}\cong \mathbb{A}(D_{p^m})$$ is the Burnside ring of the dihedral group, which is not the $p$-typical $(m+1)$-truncated Witt vectors of $\pi_0 \operatorname{THR}(\mathbb{S})^{\operatorname{\mathbb{Z}}/2}\cong \mathbb{A}(\operatorname{\mathbb{Z}}/2)$ (see Example \[different-witt\]). The dihedral fixed-points can still be described by a variant of the Witt vectors construction. For odd $p$, there are “twisted ghost maps” $\tilde{w}_j\colon \prod_{i=0}^m\pi_0\operatorname{THR}(A)^{\operatorname{\mathbb{Z}}/2}\to \pi_0\operatorname{THR}(A)^{\operatorname{\mathbb{Z}}/2}$, defined by the formula $$\tilde{w}_j(x_0,\dots,x_m):=\sum_{i=0}^j(1+\frac{(p^i-1)}{2}\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_e(1))x_i(N_{e}^{\operatorname{\mathbb{Z}}/2}\operatorname{res}_{e}^{\operatorname{\mathbb{Z}}/2}(x_i))^{\frac{p^{j-i}-1}{2}}.$$ When $\underline{\pi}_0E$, and therefore $\underline{\pi}_0\operatorname{THR}(E)$, is cohomological $\operatorname{tran}(1)=2$ and this is the usual ghost map $w_j$ of the ring $\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}$. In Theorem \[THRTamb\] we describe the dihedral fixed-points in the following terms.
Let $E$ be a connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum, and $p$ an odd prime. There is a unique ring structure $\tilde{W}_{m+1}(\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2};p)$ on the set $\prod_{i=0}^{m}\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}$ such that the maps $\tilde{w}_j$ are natural ring homomorphisms, and a natural ring isomorphism $$\pi_0\operatorname{THR}(E)^{D_{p^m}}\cong \tilde{W}_{m+1}(\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2};p)$$ for every $1\leq m\leq \infty$.
There is a similar description of $\pi_0\operatorname{THR}(E)^{D_{2^m}}$ for the prime $2$. The ring structure is again determined by a twisted version of the ghost maps $\tilde{w}_j$, which additionally take into account the action of the non-trivial Weyl group of $\operatorname{\mathbb{Z}}/2$ in $D_{2^m}$. However, as a set, $\pi_0\operatorname{THR}(E)^{D_{2^m}}$ is a quotient of the product $\prod_{i=0}^m\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}$. This quotient accounts for the fact that the transfer maps $\pi_0\operatorname{THR}(E)^{D_{2^i}}\to \pi_0\operatorname{THR}(E)^{D_{2^{i+1}}}$ are not injective for the prime $2$. This situation is analogous to the Witt vectors for non-commutative rings of [@HesselholtncW] and [@HesselholtncWcorr] where the Verschiebung is generally not injective. The description of this quotient requires a choice of free resolution of $\underline{\pi}_0E$ as a $\operatorname{\mathbb{Z}}/2$-Tambara functor, which is unsatisfying if one is interested only in the case where $E$ is a discrete ring with involution. This situation will be analyzed in a forthcoming paper with different methods.
We note that the topological applications are independent on the rest of the paper, and the readers who are only interested in the algebraic results can safely ignore §\[secTHR\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Peter Scholze for generously sharing his ideas, particularly for making us aware that in this context it is more natural to consider polynomial laws than polynomial maps. We would also like to thank Benjamin Böhme, Christopher Davis and Christian Wimmer for useful conversations.
Dotto and Patchkoria were supported by the German Research Foundation Schwerpunktprogramm 1786 and the Hausdorff Center for Mathematics at the University of Bonn. Moi was supported by the K&A Wallenberg Foundation.
The third author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Homotopy harnessing higher structures” when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1.
Notation and Conventions {#notation-and-conventions .unnumbered}
========================
All the rings of this paper will be unital and commutative, and ring homomorphisms will not necessarily be unital. We will denote by $\operatorname{Ring}$ the category of unital commutative rings and not necessarily unital ring homomorphisms (or non-unital ring homomorphisms). The category of commutative rings and unital ring homomorphisms will be denoted by $\operatorname{Ring}_{1}$. The tensor product $\otimes$ defines a symmetric monoidal structure on $\operatorname{Ring}$, which is not the coproduct. It is given by the usual tensor product of unital rings and the tensor product of non-unital maps.
PD-functors and their functoriality in polynomial laws {#secGamma}
======================================================
In this section we present an axiomatic framework that includes the various flavors of Witt vector functors. The notion of *PD-functor*, named after the French “puissances divis[é]{}es”, is an axiomatization of the properties which allow extra functoriality in polynomial laws. Since divided powers govern polynomial laws, the axioms can be thought of as sufficient conditions for compatibility with divided power structures.
Review of polynomial laws and divided powers {#recallpolylaw}
--------------------------------------------
We begin by recalling some basic results about polynomial laws, mostly from [@Roby1].
Let $A$ and $B$ be abelian groups. A *polynomial law* from $A$ to $B$ is a collection of maps of sets $f_R\colon A\otimes_{\operatorname{\mathbb{Z}}} R\to B\otimes_{\operatorname{\mathbb{Z}}}R$ for every commutative ring $R$, which is natural with respect to ring homomorphisms $R\to R'$ (i.e., a natural transformation of set valued functors). Such a collection is called *$n$-homogeneous* if $$f_R(x\cdot r)=f_R(x)\cdot r^{n}$$ for every $x\in A\otimes_{\operatorname{\mathbb{Z}}}R$, $r\in R$, and any commutative ring $R$. If $A$ and $B$ are rings, we call a polynomial law *multiplicative* if each map $f_R$ is multiplicative.
1. A homomorphism of abelian groups $f\colon A\to B$ defines a canonical $1$-homogeneous polynomial law $$f_R:=f\otimes_{\operatorname{\mathbb{Z}}} R\colon A\otimes_{\operatorname{\mathbb{Z}}} R\longrightarrow B\otimes_{\operatorname{\mathbb{Z}}}R,$$ which is multiplicative if $f$ is a ring homomorphism. This defines an embedding of the category of abelian groups and group homomorphisms into the category of abelian groups and polynomial laws. Via this embedding, we think of polynomial laws as maps $f\colon A\to B$, by keeping in mind that this is in fact a natural transformation. A similar convention will be adopted for ring homomorphisms and multiplicative polynomial laws.
2. Let $A$ be a ring, and $n$ a non-negative integer. The $n$-th power maps $$(-)^{n}\colon A\otimes_{\operatorname{\mathbb{Z}}}R\longrightarrow A\otimes_{\operatorname{\mathbb{Z}}}R$$ of the rings $A\otimes_{\operatorname{\mathbb{Z}}}R$ define a multiplicative $n$-homogeneous polynomial law.
We recall from [@Roby1] that there is a universal $n$-homogeneous law $$\gamma_n\colon A\longrightarrow \Gamma_n(A),$$ where $\Gamma_n(A)$ is the $n$-th graded piece of the divided power algebra of the abelian group $A$, and $\gamma_n$ sends an element $a\in A$ to its corresponding generator. This is universal in the following sense. Let $H_n(A,B)$ be the set of $n$-homogeneous polynomial laws from $A$ to $B$, and $M_n(A,B)$ the subset of the multiplicative laws.
For every pair $A$ and $B$ of abelian groups, there is a natural bijection $$\hom_{Ab}(\Gamma_{n}(A),B)\cong H_n(A,B)$$ which sends a group homomorphism $\phi\colon \Gamma_{n}(A)\to B$ to $\phi\circ\gamma$. If $A$ is a commutative ring so is $\Gamma_n(A)$, and restricting to ring homomorphisms gives a bijection $$\hom_{\operatorname{Ring}}(\Gamma_{n}(A),B)\cong M_n(A,B)$$ for every commutative ring $B$.
We recall from [@Roby1] that any polynomial law $f$ decomposes uniquely into a locally finite sum of homogeneous polynomial laws. If there are finitely many non-zero homogeneous pieces, we say that the polynomial law is of degree $n$, where $n$ is the largest degree of its homogeneous summands. In this case the map $f\colon A\to B$ has a Taylor decomposition $$f=f_0+f_1+\cdots+ f_n,$$ where $f_i$ is a homogeneous polynomial law of degree $i$ (in particular $f_0$ is a constant in $B$). Each $f_i$ in the Taylor decomposition of $f$ corresponds to a unique additive map $$\varphi_i \colon \Gamma_i A \longrightarrow B,$$ such that $\varphi_i \circ \gamma_i = f_i$. When $f$ is multiplicative, the maps $\varphi_i$ have the following properties (see [@Zieplies Page 71]):
- The map $\varphi_i$ is multiplicative, i.e., a not necessarily unital homomorphism of rings;
- The maps $\varphi_i$ and $\varphi_j$ are orthogonal for $i\neq j$, i.e., for any $x \in \Gamma_i A$ and $y \in \Gamma_j A$, one has $$\varphi_i(x) \cdot \varphi_j(y)=0.$$
Altogether the polynomial law $f$ admits a unique factorization $$\xymatrix@C=70pt@R=13pt{A \ar[r]^{f} \ar[d]_-{\prod_i \gamma_i} & B, \\ \prod_{i=0}^{n} \Gamma_i A \ar[ur]_{\varphi=\oplus_i \varphi_i} & }$$ and the map $\oplus_i \varphi_i$ is a morphism in $\operatorname{Ring}$. Note that if $f$ is unital, then so is $\oplus_i \varphi_i$ (but not the individual $\varphi_i$-s).
Definition and examples of PD-functors
--------------------------------------
In this section we introduce the notion of *PD-functor* which defines a class of well-behaved endofunctors on $\operatorname{Ring}_{1}$. We will show in §\[secGammathm\] that these extend in a canonical way to functors on polynomial laws, and the definition given here is taylored to this purpose.
We recall that $\operatorname{Ring}$ is the category of unital rings and not necessarily unital ring homomorphisms, and $\operatorname{Ring}_1$ is the subcategory of unital ring homomorphisms.
A functor $F \colon \operatorname{Ring}_1 \to \operatorname{Ring}_1$ is a PD functor if it preserves the following universal constructions:
1. finite products,
2. reflexive coequalizers,
3. fixed points of finite group actions.
\[exproduct\] Let $S$ be a set, and $(-)^{\times S}\colon \operatorname{Ring}\to \operatorname{Ring}$ be the functor that takes a ring $A$ to the product ring $A^{\times S}=\prod_{S}A$, and a ring homomorphism to the product map $f^{\times S}$. Then $(-)^{\times S}$ is a PD-functor. Conditions i) and iii) are satisfied, since products commute with limits. Given a reflexive coequalizer of commutative rings, it is also a reflexive coequalizer of underlying abelian groups and sets. If the set $S$ is finite, then it is clear that $(-)^{\times S}$ commutes with reflexive coequalizers, since finite products in sets do. Infinite products in sets do not commute with reflexive coequalizers in general. However, infinite products of abelian groups do commute with reflexive coequalizers and reflexive coequalizers in abelian groups are reflexive coequalizers of underlying sets.
\[tensor\] Let $R$ be a torsion-free commutative ring. The functor $(-)\otimes_{\operatorname{\mathbb{Z}}}R\colon \operatorname{Ring}_{1}\to \operatorname{Ring}_{1}$ that sends a commutative ring $A$ to $A\otimes_{\operatorname{\mathbb{Z}}}R$ is a PD-functor. The fact that $R$ has no torsion is used to show that $(-)\otimes_{\operatorname{\mathbb{Z}}}R$ commutes with invariants of finite groups.
\[Witt vectors\] For any truncation set $S$, the $S$-truncated Witt vectors functor $W_S \colon \operatorname{Ring}_{1} \to \operatorname{Ring}_{1}$ is a PD-functor. We recall that as a set $$W_S(A)=\prod_{s\in S}A,$$ and that the functoriality is given by taking the product map on underlying sets. As in Example \[exproduct\] this implies that $W_S$ satisfies Conditions i) and iii). A reflexive coequalizer of commutative rings is also a reflexive coequalizer of underlying abelian groups and sets. To show Condition ii) one can argue exactly as in Example \[exproduct\] using that $W_S$ and $()^{\times S}$ have the same underlying set.
By chosing the appropriate $S$, we see that big Witt vectors, $n$-truncated big Witt vectors, $p$-typical Witt vectors and $n$-truncated $p$-typical Witt vectors are all PD-functors. See for example [@LarsWitt; @Rabinoff] for more background on Witt vectors.
\[rationalwitt\] For a commutative ring $A$, the ring of big Witt vectors $\mathbb{W}(A)$ is isomorphic to a ring whose underlying additive group is the subgroup of the units of $A[[x]]$, the power series with constant term equal to $1$. The ring of rational Witt vectors $W_{\mathbb{Q}}(A)$ is the subring of $\mathbb{W}(A)$ corresponding to those power series which are rational functions. See [@Almkvist1; @Almkvist2; @Haze] for more details.
It is clear that $W_{\mathbb{Q}}$ preserves finite products. We verify Conditions ii) and iii) directly. For a reflexive coequalizer of unital ring maps $$\xymatrix{A \ar@<0.5ex>[r]^{f} \ar@<-0.5ex>[r]_{g} & B \ar[r]^{h} & C }$$ with a common section $s \colon B \to A$, we would like to show that $$\xymatrix{W_{\mathbb{Q}}(A) \ar@<0.5ex>[rr]^-{W_{\mathbb{Q}}(f)} \ar@<-0.5ex>[rr]_-{W_{\mathbb{Q}}(g)} & & W_{\mathbb{Q}}(B) \ar[rr]^-{W_{\mathbb{Q}}(h)} & & W_{\mathbb{Q}}(C) }$$ is a coequalizer. One only needs to check that $\operatorname{Ker}W_{\mathbb{Q}}(h) \subset \operatorname{Im}(W_{\mathbb{Q}}(f) -W_{\mathbb{Q}}(g) )$. Suppose that we have $$\frac{1+b_1x + \cdots + b_n x^n}{1+b'_1x + \cdots + b'_n x^n} \in \operatorname{Ker}W_{\mathbb{Q}}(h).$$ Then $h(b_i)=h(b_i')$ for all $i=1, \cdots, n$. This implies that there are $a_i \in A$, $i=1, \dots, n$ such that $f(a_i)=b_i$ and $g(a_i)=b'_i$ for every $i$. Consider the rational function $$\frac{1+a_1x + \cdots + a_n x^n}{1+s(b_1)x + \cdots + s(b_n) x^n} \in W_{\mathbb{Q}}(A).$$ Then the map $W_{\mathbb{Q}}(f) -W_{\mathbb{Q}}(g)$ sends this rational function to $$\begin{aligned}
&\frac{1+f(a_1)x + \cdots + f(a_n) x^n}{1+f(s(b_1))x + \cdots + f(s(b_n)) x^n} \cdot \frac{1+g(s(b_1))x + \cdots + g(s(b_n)) x^n}{1+g(a_1)x + \cdots + g(a_n) x^n}
\\&= \frac{1+b_1x + \cdots + b_n x^n}{1+b_1x + \cdots + b_n x^n} \cdot \frac{1+b_1x + \cdots + b_n x^n}{1+b'_1x + \cdots + b'_n x^n}
=\frac{1+b_1x + \cdots + b_n x^n}{1+b'_1x + \cdots + b'_n x^n}, \end{aligned}$$ where we have used that $fs=gs=1$. This shows Condition ii).
Let $G$ be a finite group and $A$ a commutative ring with $G$-action. To show Condition iii) we want to show that the canonical map $$W_{\mathbb{Q}}(A^G) \to W_{\mathbb{Q}}(A)^G$$ is an isomorphism. Since $W_{\mathbb{Q}}$ preserves injections this map is injective, so we must show surjectivity. Given a polynomial $\alpha \in A[x]$ and an element $g \in G$, we denote by $g \cdot \alpha$ the polynomial obtained by acting on the coefficients of $\alpha$ by $g$. Now let $$\frac{\alpha}{\beta} \in W_{\mathbb{Q}}(A)^G$$ be an invariant rational function. Then by definition for any element $g \in G$, we have $$g \cdot \frac{\alpha}{\beta}= \frac{g \cdot \alpha}{g \cdot \beta}= \frac{\alpha}{\beta}.$$ Hence $(g \cdot \alpha) \beta= \alpha (g \cdot \beta)$. Note that $\alpha$ and $\beta$ do not have to be $G$-invariant. However, the latter equation allows us to replace them with $G$-invariant polynomials. Indeed, we claim that $$\frac{\alpha}{\beta}= \frac{\alpha (\prod_{g \in G, g \neq 1} g \cdot \beta ) } {\prod_{g \in G} g \cdot\beta}.$$ It is clear that the denominator of the right hand fraction is $G$-invariant. We check that the numerator is invariant as well: $$\begin{aligned} h\cdot (\alpha (\prod_{g \in G, g \neq 1} g \cdot \beta ))= (h \cdot \alpha) (h h^{-1} \cdot \beta )(\prod_{g \in G, g \neq 1, h^{-1}} hg \cdot \beta )= \\ (h \cdot \alpha) \beta (\prod_{g \in G, g \neq 1, h^{-1}} hg \cdot \beta ) = \alpha (h \cdot \beta) (\prod_{g \in G, g \neq 1, h^{-1}} hg \cdot \beta )= \alpha (\prod_{g \in G, g \neq 1} g \cdot \beta ).\end{aligned}$$ The third equality uses the relation $(h \cdot \alpha) \beta= \alpha (h \cdot \beta)$. Now the quotient $$\frac{\alpha (\prod_{g \in G, g \neq 1} g \cdot \beta ) } {\prod_{g \in G} g \cdot\beta}$$ is an element of $W_{\mathbb{Q}}(A^G)$ that maps under the canonical map to $\frac{\alpha}{\beta}$ in $W_{\mathbb{Q}}(A)^G$.
\[K0noGamma\] By Almkvist’s theorem [@Almkvist1; @Almkvist2], the functor $W_{\mathbb{Q}}$ is isomorphic to the cyclic $K$-theory functor $\operatorname{K}^{cy}_0$, which is therefore a PD-functor. We remark however that the Grothendieck group $\operatorname{K}_0$ is not a PD-functor. We will see in Corollary \[surjections\] that any PD-functor preserves surjections, whereas $\operatorname{K}_0$ does not.
We end the section by remarking that any product-preserving endofunctor of $\operatorname{Ring}_1$ extends canonically to an endofunctor of $\operatorname{Ring}$. For this reason we will often implicitly consider a PD-functors as being defined on the larger category $\operatorname{Ring}$.
\[lemma:extension\]Let $F \colon \operatorname{Ring}_{1} \to \operatorname{Ring}_{1}$ be a functor that preserves finite products. Then there is a functor $E \colon \operatorname{Ring}\to \operatorname{Ring}$ and an isomorphism $\phi \colon E|_{\operatorname{Ring}_1} \to F$. The pair $(E,\phi)$ is unique up to unique isomorphism over $F$.
We first prove existence. For a possibly non-unital ring $A$ we write $A^+$ for its unitalization and $p_A \colon A^+ \to \operatorname{\mathbb{Z}}$ for the canonical projection. We set $E(A) = \ker(F(A^+) \to F(\operatorname{\mathbb{Z}}))$ and note that this defines an endofunctor of the category of non-unital rings and non-unital ring homomorphisms. If $A$ is unital there is a natural unital map $A^+ \to A$ and the induced map $A^+ \to A\times \operatorname{\mathbb{Z}}$ is an isomorphism of unital rings. Since $F$ preserves products, taking kernels of the projection to $F(\operatorname{\mathbb{Z}})$ gives an isomorphism $\phi_A \colon E(A) \to F(A)$, so we see that $E$ restricts to an endofunctor $F$ on $\operatorname{Ring}$. The maps $\phi_A$ are natural in unital ring homomorphisms so we get a natural isomorphism $\phi \colon E|_{\operatorname{Ring}_1} \to F$ as desired.
We now prove uniqueness. Let $(E,\phi)$ and $(E',\phi')$ be pairs as in the statement of the lemma and write $\psi = (\phi')^{-1} \circ \phi$. We will show that $\psi$ is natural with respect to all maps in $\operatorname{Ring}$. Any such map $f\colon A \to B$ factors as $$A \stackrel{e\cdot f}{\longrightarrow} e B \xrightarrow{\operatorname{id}\times 0} e B \times (1-e)B \cong B,$$ where $e = f(1)$ is idempotent. The left and right hand maps are unital, so it suffices to check $\psi$ is natural with respect to maps of the form $\operatorname{id}\times 0\colon R \to R \times S$. This follows since $F$, and therefore also $E$ and $E'$, preserve finite products. The uniqueness of $\psi$ is clear.
By abuse of notation we will simply write $F$ for a choice of extension of a product preserving functor $F$ on $\operatorname{Ring}_1$ to non-unital ring homomorphisms. A pair of maps $f \colon A \to C$ and $g\colon B \to C$ in $\operatorname{Ring}$ is called *orthogonal* if $f(a)\cdot g(b)=0\in C$ for all $a\in A$ and $b\in B$.
\[lemma:ext-prop\]Let $F \colon \operatorname{Ring}_1 \to \operatorname{Ring}_1$ be a functor that preserves finite products. Then the extension to the category $\operatorname{Ring}$ has the following properties:
1. \[lemma:ext-prop1\] For any pair of maps $f \colon A \to A'$ and $g \colon B \to B'$ in $\operatorname{Ring}$, the diagram $$\xymatrix{F(A) \otimes F(B) \ar[d]^c \ar[rr]^-{F(f) \otimes F(g)} && F(A') \otimes F(B') \ar[d]^{c} \\ F(A \otimes B) \ar[rr]^{F(f \otimes g)} && F(A' \otimes B') }$$ commutes, where $c$ is the canonical coproduct map for unital rings.
2. \[lemma:ext-prop2\] For any pair of orthogonal maps $f \colon A \to C$ and $g \colon B \to C$ the maps $F(f)$ and $F(g)$ are orthogonal.
3. \[lemma:ext-prop3\] For any pair of orthogonal maps $f\colon A\to C$ and $g\colon B\to C$ in $\operatorname{Ring}$ there is a commutative diagram in $\operatorname{Ring}$ $$\xymatrix@C=70pt@R=17pt{
F(A\times B)\ar[r]^-{F(f+g)}\ar[d]_-{\cong}&F(C)
\\
F(A)\times F(B).\ar[ur]_-{\ F(f)+F(g)}
}$$
Part i): Since the statement holds when both maps are unital we reduce to checking the case when $f$ is unital and $g$ is of the form $\operatorname{id}\times 0 \colon B \to B \times C$. We must show that the left hand square in the following diagram commutes $$\xymatrix{F(A) \otimes F(B) \ar[d]^c \ar[rr]^-{F(f) \otimes F(\operatorname{id}\times 0)} && F(A') \otimes F(B \times C) \ar[d]^{c} \ar[rrr]_-{\cong}^-{(\operatorname{id}\otimes F(p_1)) \times (\operatorname{id}\otimes F(p_2))} &&& (F(A')\otimes F(B)) \times (F(A')\otimes F(C)) \ar[d]^{c\times c}
\\
F(A \otimes B) \ar[rr]^-{F(f \otimes (\operatorname{id}\times 0))} && F(A' \otimes (B\times C)) \ar[rrr]_-{\cong}^-{F(\operatorname{id}\otimes p_1) \times F(\operatorname{id}\otimes p_2)} &&& F(A' \otimes B) \times F(A' \otimes C).}$$ The right hand square commutes and the right hand horizontal maps are isomorphisms, so it suffices to show that the outer rectangle commutes. The composite through the upper right hand corner is the map $(c \circ F(f)\otimes \operatorname{id}) \times c \circ (F(f) \otimes 0)$ and the other composite equals $(F(f \otimes \operatorname{id}) \circ c) \times (F(f \otimes 0) \circ c)$. The second coordinates of both maps are equal to the $0$ map and the first coordinates are equal because $\otimes$ is the coproduct on unital rings.
Part ii): Orthogonality of $f$ and $g$ is equivalent to the composite map $$A\otimes B \xrightarrow{f\otimes g} C \otimes C \xrightarrow{\mu_C} C$$ factoring through $0$, where $\mu_C$ is the multiplication of $C$. Since $F(0)=0$, Part i) implies that $$F(A)\otimes F(B) \xrightarrow{F(f)\otimes F(g)} F(C) \otimes F(C) \xrightarrow{\mu_{F(C)}} F(C)$$ factors through $0$ as well.
Part iii): The orthogonal maps $f$ and $g$ give a splitting of $C$ as $C \cong C_f \times C_g \times C'$ with $f$ and $g$ factoring through $C_f$ and $C_g$, respectively. Here $C_f=f(1)C$ and $C_g=g(1)C$. The map $f+g$ is the composite $$A \times B \xrightarrow{f^u \times g^u} C_f \times C_g \hookrightarrow C_f \times C_g \times C' \stackrel{\cong}{\longrightarrow} C,$$ where $f^u$ is the corestriction of $f$, which is unital, and similarly for $g^u$. Now consider the diagram $$\xymatrix@C=70pt@R=17pt{
F(A\times B)\ar[r]^-{F(f^u \times g^u)}\ar[d]_{\cong}& F(C_f \times C_g) \ar[d]^-{\cong} \ar[r] &F(C)
\\
F(A)\times F(B) \ar[r]^{F(f^u) \times F(g^u)} & F(C_f) \times F(C_g) \ar[ur] &
}$$ where the vertical maps are the canonical isomorphisms. This diagram commutes in $\operatorname{Ring}$ and the composite of the upper row is $F(f + g)$ while the composite from the lower left hand corner to the right hand corner is the map $F(f) + F(g)$.
Functoriality of PD-functors in multiplicative polynomial laws {#secGammathm}
--------------------------------------------------------------
Let us consider the category $\operatorname{Ring^{poly}}$ whose objects are commutative rings, whose morphisms are the multiplicative polynomial laws of finite degree, and where the composition is the composition of natural transformations. The category $\operatorname{Ring}$ is a subcategory of $\operatorname{Ring^{poly}}$ of the multiplicative polynomial laws of degree $1$ which preserve zero.
Let $A$ be a torsion-free ring and $F\colon \operatorname{Ring}\to \operatorname{Ring}$ a PD-functor. The natural transformation $\xi_A\colon \Gamma_n(A)\to (A^{\otimes n})^{\Sigma_n}$ from [@Roby1 Section III.6] is then an isomorphism, and we can consider the polynomial law $F(\gamma_A)$ defined as the composite $$\xymatrix{
F(\gamma_A)\colon F(A)\xrightarrow{(-)^{\otimes n}}(F(A)^{\otimes n})^{\Sigma_n}\longrightarrow F(A^{\otimes n})^{\Sigma_n}\stackrel{\cong}{\longleftarrow}F((A^{\otimes n})^{\Sigma_n})&F(\Gamma_n(A))\ar[l]^-{\cong}_-{F(\xi_A)}.
}$$
\[polylawfunct\] Any PD-functor $F\colon \operatorname{Ring}\to \operatorname{Ring}$ extends canonically to an endofunctor of $\operatorname{Ring^{poly}}$ which preserves the degree. This is the unique extension which sends the universal $n$-homogeneous polynomial law $\gamma_A\colon A\to \Gamma_n(A)$ to $F(\gamma_A)$ for any torsion-free $A$.
The proof of this theorem will occupy the rest of the section. The key ingredient of the proof is the construction of a natural unital ring homomorphism $$c_n \colon \Gamma_n F(A) \longrightarrow F(\Gamma_n A).$$ For $n=0$, the functor $\Gamma_0$ is constant with value $\mathbb{Z}$, and we define $c_0$ to be the unit map $$c_0\colon \mathbb{Z} \longrightarrow F(\mathbb{Z}).$$ Now suppose that $n\geq 1$. We start by defining $c_n$ for torsion-free rings, as the composite $$\xymatrix{c_n\colon \Gamma_n F(A) \ar[r]^-{\xi_{F(A)}} & (F(A)^{\otimes n})^{\Sigma_n} \ar[r]^-{c^{\Sigma_n}} & F(A^{\otimes n})^{\Sigma_n} & F((A^{\otimes n})^{\Sigma_n})\ar[l]_-{\cong} & F(\Gamma_n A).\ar[l]^-{\cong}_-{F(\xi_A)} }$$ Here $c \colon F(A)^{\otimes n} \to F(A^{\otimes n})$ is the canonical map that commutes $F$ and the coproduct, and it is $\Sigma_n$-equivariant. The third map is an isomorphism since by assumption PD-functors commute with invariants.
For general commutative rings which are not necessarily torsion-free we use resolutions by polynomial (torsion-free) rings.
\[resolution\] Let $A$ be a commutative ring. Then there is a reflexive coequalizer in $\operatorname{Ring}_{1}$ $$\xymatrix{P_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & P_0 \ar[r] & A, }$$ where $P_0$ and $P_1$ are polynomial rings over $\operatorname{\mathbb{Z}}$. Furthermore, any surjective unital ring homomorphism $Q_0 \to A$ can be completed to a reflexive coequalizer $$\xymatrix{Q_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & Q_0 \ar[r] & A. }$$ If $Q_0$ is torsion-free, then $Q_1$ can be chosen to be torsion-free too.
First we note that a reflexive coequalizer in commutative rings is a reflexive coequalizer of underlying abelian groups. The standard resolution one can use is given by $$\xymatrix{\mathbb{Z}[\mathbb{Z}[A]] \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathbb{Z}[A] \ar[r] & A, }$$ where $ \mathbb{Z}[-]$ is the polynomial ring functor. We describe the two parallel maps: The first one is given by applying $ \mathbb{Z}[-]$ to the counit morphism $\mathbb{Z}[A] \to A$. The second map is the counit for $\mathbb{Z}[A]$. Both maps are split by the ring homomorphism $\mathbb{Z}[A] \to \mathbb{Z}[\mathbb{Z}[A]]$ induced by the inclusion of the sets $A \hookrightarrow \mathbb{Z}[A]$.
Given any surjection $\sigma \colon Q_0 \to A$, we can take the pullback $$\xymatrix@C=60pt@R=17pt{Q_1 \ar[r]^{\sigma_1} \ar[d]_{\sigma_2} & Q_0 \ar[d]^{\sigma} \\ Q_0 \ar[r]^{\sigma} & A }$$ of $\sigma$ along itself. The diagonal $\Delta \colon Q_0 \to Q_1$ splits $\sigma_1$ and $\sigma_2$ and the cokernel of $\sigma_1-\sigma_2$ is $A$. Hence $$\xymatrix{Q_1 \ar@<0.5ex>[r]^{\sigma_1} \ar@<-0.5ex>[r]_{\sigma_2} & Q_0 \ar[r] & A }$$ is a reflexive coequalizer. Finally, if $Q_0$ is torsion-free then clearly $Q_1$ is as well.
\[surjections\] Any PD-functor takes surjections in $\operatorname{Ring}_1$ to surjections in $\operatorname{Ring}_1$.
Now we are ready to define the unital ring homomorphism $c_n \colon \Gamma_n F(A) \to F(\Gamma_n A)$. Choose any reflexive coequalizer $$\xymatrix{P_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & P_0 \ar[r] & A,}$$ where $P_0$ and $P_1$ are torsion-free. The functor $\Gamma_n(-)$ preserves reflexive coequalizers by [@Roby1 Section IV.10] and the functor $F$ does it by Axiom ii). Hence there exists a unique morphism $c_n \colon \Gamma_n F(A) \to F(\Gamma_n A)$ in $\operatorname{Ring}_{1}$ such that the diagram $$\xymatrix@C=60pt@R=17pt{\Gamma_n F(P_1) \ar[d]^{c_n} \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \Gamma_n F(P_0) \ar[d]^{c_n} \ar[r] & \Gamma_n F(A) \ar@{-->}[d]^{c_n} \\ F(\Gamma_n P_1) \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & F(\Gamma_n P_0) \ar[r] & F(\Gamma_n A) }$$ commutes.
\[well-defined\] The map $c_n \colon \Gamma_n F(A) \to F(\Gamma_n A)$ does not depend on the choice of resolution.
Let $\xymatrix{Q_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & Q_0 \ar[r] & A,}$ be a reflexive coequalizer with $Q_1$ and $Q_0$ torsion-free. Let us choose a reflexive coequalizer $$\xymatrix{P_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & P_0 \ar[r] & A,}$$ where $P_1$ and $P_0$ are polynomial rings. Then there is a unital ring homomorphism $P_0 \to Q_0$ such that the diagram commutes: $$\xymatrix{P_0 \ar[r] \ar[dr] & Q_0 \ar[d] \\ & A.}$$ We would like to show that the maps $c_n^P, c_n^Q \colon \Gamma_n F(A) \to F(\Gamma_n A)$ induced by the two resolutions agree. To see this, consider the following diagram: $$\xymatrix@C=40pt@R=10pt{
\Gamma_n F(P_0) \ar[dd]_-{c_n}\ar[dr] \ar@{->>}[rr] & & \Gamma_n F(A) \ar[dd]^(.3){c_n^P} \ar@{=}[dr]
\\
&\Gamma_n F(Q_0)\ar[dd]_(.3){c_n} \ar[rr] & & \Gamma_n F(A) \ar[dd]^-{c_n^Q}
\\
F(\Gamma_n P_0) \ar[rr] \ar[dr] & & F(\Gamma_n A) \ar@{=}[dr]
\\
&F(\Gamma_n Q_0) \ar[rr] & & F(\Gamma_n A)\rlap{\ .}
}$$ All squares commute except possibly the right hand face. But the map $\Gamma_n F(P_0) \to \Gamma_n F(A)$ is surjective by Corollary \[surjections\]. Therefore the right hand face commutes and we conclude that $c_n^P=c_n^Q$.
Next, we show that the maps $c_n$ are natural with respect to all maps in $\operatorname{Ring}$. This will be used later to show the extra functoriality of PD-functors with respect to multiplicative polynomial laws.
\[naturaloncom\] For any $n \geq 0$, the map $c_n \colon \Gamma_n F(A) \to F(\Gamma_n A)$ is natural with respect to morphisms in $\operatorname{Ring}$. In other words, for any not necessarily unital ring homomorphism $f \colon A \to B$, the diagram $$\xymatrix@C=60pt@R=17pt{ \Gamma_n F(A) \ar[r]^{c_n} \ar[d]_-{\Gamma_n F(f)} & F(\Gamma_n A) \ar[d]^{F(\Gamma_n f)} \\ \Gamma_n F(B) \ar[r]_{c_n} & F(\Gamma_n B)}$$ commutes.
For $n=0$ the statement is obvious, so we assume $n \geq 1$. We start by showing naturality with respect to unital homomorphisms $f \colon A \to B$. Clearly $c_n$ is natural with respect to $f$ if $A$ and $B$ are torsion-free. In general, we resolve $f$ by polynomial rings. That is, we choose polynomial rings $P$ and $Q$ and a unital ring homomorphism $f' \colon P \to Q$ such that the diagram $$\xymatrix@C=60pt@R=17pt{P \ar@{->>}[r] \ar[d]^{f'} & A \ar[d]^f \\ Q \ar@{->>}[r] & B }$$ commutes and the horizontal maps are surjective. Then consider the cube $$\xymatrix@C=40pt@R=10pt{
\Gamma_n F(P) \ar[dd]_-{c_n} \ar[dr]_-{\Gamma_n F(f')} \ar@{->>}[rr] & & \Gamma_n F(A) \ar[dd]^(.3){c_n} \ar[dr]^{\Gamma_n F(f)}
\\
&\Gamma_n F(Q)\ar[dd]_(.3){c_n} \ar[rr] & & \Gamma_n F(B) \ar[dd]^-{c_n}
\\
F(\Gamma_n P) \ar[rr] \ar[dr]_{F(\Gamma_n f')} & & F(\Gamma_n A) \ar[dr]^{F(\Gamma_n f)}
\\
&F(\Gamma_n Q) \ar[rr] & & F(\Gamma_n B)\rlap{\ .}
}$$ The left square commutes by the naturality on torsion-free rings. The top and bottom faces commute by functoriality. Finally, the back and front faces commute by the definition of $c_n$. This means that all faces except possibly the right hand one commute. Additionally, the map $\Gamma_n F(P) \to \Gamma_n F(A)$ is surjective. Hence the right square also commutes which proves the proposition for unital homomorphisms.
Now suppose that $f \colon A \to B$ is a non-unital ring homomorphism. Let $M(f) \colon M(A) \to M(B)$ be the induced non-unital map of underlying multiplicative monoids. This induces a non-unital homomorphism of monoid rings $$\mathbb{Z}(M(f)) \colon \mathbb{Z}(M(A)) \to \mathbb{Z}(M(B)).$$ The diagram $$\xymatrix@C=60pt@R=17pt{\mathbb{Z}(M(A)) \ar@{->>}[r] \ar[d]_{\mathbb{Z}(M(f))} & A \ar[d]^f \\ \mathbb{Z}(M(B)) \ar@{->>}[r] & B }$$ commutes, where the horizontal maps are the augmentation maps. We once again consider the cube $$\xymatrix@C=40pt@R=10pt{
\Gamma_n F(\mathbb{Z}(M(A))) \ar[dd]_-{c_n} \ar[dr]^-{\hspace{0.6cm} \Gamma_n F(\mathbb{Z}(M(f)))} \ar@{->>}[rr] & & \Gamma_n F(A) \ar[dd]^(.3){c_n} \ar[dr]^{\Gamma_n F(f)}
\\
&\Gamma_n F(\mathbb{Z}(M(B))) \ar[dd]_(.3){c_n} \ar[rr] & & \Gamma_n F(B) \ar[dd]^-{c_n}
\\
F(\Gamma_n \mathbb{Z}(M(A))) \ar[rr] \ar[dr]_-{\hspace{-1.0cm} F(\Gamma_n \mathbb{Z}(M(f)))} & & F(\Gamma_n A) \ar[dr]_{F(\Gamma_n f)}
\\
&F(\Gamma_n \mathbb{Z}(M(B))) \ar[rr] & & F(\Gamma_n B)
}$$ The top and bottom faces commute because of functoriality, and the back and front faces commute by the unital case. The left hand square commutes by reduction to symmetric tensors, since the rings are torsion free and then using that $F$ is a PD functor. The topmost map in the cube is surjective by Corollary \[surjections\]. Hence the right hand square also commutes.
We are now ready to define a multiplicative polynomial law $F(f)$ for any multiplicative polynomial law $f \colon A \to B$ of finite degree. We recall from §\[recallpolylaw\] that the Taylor decomposition $$f=f_0+f_1+\cdots+ f_n,$$ where $f_i$ is a homogeneous polynomial law of degree $i$, provides a factorization in $\operatorname{Ring}$ $$\xymatrix@C=70pt@R=17pt{A \ar[r]^{f} \ar[d]_-{\prod_i \gamma_i} & B, \\ \prod_{i=0}^{n} \Gamma_i A \ar[ur]_{\varphi=\oplus_i \varphi_i} & }$$ where $\varphi_i \circ \gamma_i = f_i$. We define the multiplicative polynomial law $F(f)$ to be the composite $$\xymatrix{F(A) \ar[r]^-{\prod_i \gamma_i} & \prod_{i=0}^{n} \Gamma_i F(A) \ar[r]^{\prod_i c_i} & \prod_{i=0}^{n} F(\Gamma_i A) \ar[r]^-{\cong} & F(\prod_{i=0}^{n} \Gamma_i A) \ar[r]^-{F(\varphi)} & F(B), }$$ where the third morphism is the inverse of the canonical morphism, which is an isomorphism since PD-functors commute with finite products. This definition recovers the old $F(f)$ when $f$ is a morphism in $\operatorname{Ring}$. We must show that these maps respect composition.
\[functorialityinroby\] Let $F$ be a PD-functor and $f \colon A \to B$ and $g \colon B \to C$ two multiplicative polynomial laws of finite degree. Then $F(g \circ f)= F(g) \circ F(f)$.
The proof of this theorem will require several steps. We start with the following.
\[ringsquare\] Let $F$ be a PD-functor and let $$\xymatrix@C=60pt@R=17pt{A \ar[d]_{\alpha'} \ar[r]^f & B \ar[d]^{\alpha} \\ C \ar[r]^{f'} & D }$$ a commutative square in $\operatorname{Ring^{poly}}$, where $\alpha$ and $\alpha'$ are not necessarily unital ring homomorphisms and $f$ and $f'$ are multiplicative polynomial laws of the same degree. Then the diagram $$\xymatrix@C=60pt@R=17pt{F(A) \ar[d]_{F(\alpha')} \ar[r]^{F(f)} & F(B) \ar[d]^{F(\alpha)} \\ F(C) \ar[r]^{F(f')} & F(D) }$$ commutes.
The desired square is the outer rectangle in the diagram $$\xymatrix@C=35pt{F(A) \ar[d]_{F(\alpha')} \ar[r]^-{\prod_i \gamma_i} & \prod_{i=0}^{n} \Gamma_i F(A) \ar[d]^{\prod_i \Gamma_i F(\alpha')} \ar[r]^{\prod_i c_i} & \prod_{i=0}^{n} F(\Gamma_i A) \ar[d]^{\prod_i F(\Gamma_i \alpha')} \ar[r]^{\cong} & F(\prod_{i=0}^{n} \Gamma_i A) \ar[r]^-{F(\varphi)} \ar[d]^{F(\prod_i \Gamma_i \alpha')} & F(B) \ar[d]^{F(\alpha)} \\ F(C) \ar[r]^-{\prod_i \gamma_i} & \prod_{i=0}^{n} \Gamma_i F(C) \ar[r]^{\prod_i c_i} & \prod_{i=0}^{n} F(\Gamma_i C) \ar[r]^{\cong} & F(\prod_{i=0}^{n} \Gamma_i C) \ar[r]^-{F(\varphi')} & F(D), }$$ where $\varphi$ and $\varphi'$ are induced from the Taylor decompositions of $f$ and $f'$ respectively. The second square commutes by Proposition \[naturaloncom\] and the remaining squares commute by functoriality of $F$.
The following result is an easy special case of the previous proposition, obtained by taking $\alpha'=\operatorname{id}_A$. We formulate it separately since it will be used often.
\[triangle\] Let $F$ be a PD-functor and $g=\alpha f,$ where $f$ and $g$ are multiplicative polynomial laws of finite degree and $\alpha$ is a not necessarily unital ring homomorphism. Then $F(g)=F(\alpha)F(f)$.
Suppose $f \colon A \to B$ is of degree at most $n$ and $g \colon B \to C$ is of degree at most $m$. It follows from [@Roby1], that the composite $g \circ f \colon A \to C$ has degree at most $mn$. Hence there is a commutative diagram $$\xymatrix@C=60pt@R=17pt{ A \ar[r]^f \ar[d]_{\prod_k \gamma_k} & B \ar[d]_{\prod_l \gamma_l} \ar[r]^g & C \\ \prod_{k=0}^n \Gamma_k A \ar[d]_{\prod_l \gamma_l} \ar[ur]^{\varphi} & \prod_{l=0}^m \Gamma_l B \ar[ur]_{\psi} & \\ \prod_{l=0}^m \Gamma_l (\prod_{k=0}^n \Gamma_k A) \ar[ur]^\lambda, & &}$$ where $\varphi$, $\psi$ and $\lambda$ are not necessarily unital ring homomorphisms, and $\lambda$ corresponds to the polynomial law $(\prod_l\gamma_l)\phi$. By Corollary \[triangle\], applying $F$ to the outer triangle gives a commutative diagram $$\xymatrix@C=60pt@R=17pt{F(A) \ar[d]_{F(\prod_l \gamma_l \circ \prod_k \gamma_k)} \ar[r]^{F(g \circ f)} & F(C) \\ F(\prod_{l=0}^m \Gamma_l (\prod_{k=0}^n \Gamma_k A) ) \ar[ur]_{F(\psi \circ \lambda)} \rlap{\ .}}$$ On the other hand again using Corollary \[triangle\] and Proposition \[ringsquare\], we get a commutative diagram $$\xymatrix@C=60pt@R=17pt{ F(A) \ar[r]^{F(f)} \ar[d]_{F(\prod_k \gamma_k)} & F(B) \ar[d]_{F(\prod_l \gamma_l)} \ar[r]^{F(g)} & F(C) \\ F(\prod_{k=0}^n \Gamma_k A) \ar[d]_{F(\prod_l \gamma_l)} \ar[ur]^{F(\varphi)} & F(\prod_{l=0}^m \Gamma_l B) \ar[ur]_{F(\psi)} & \\ F(\prod_{l=0}^m \Gamma_l (\prod_{k=0}^n \Gamma_k A)) \ar[ur]^{F(\lambda)}. & &}$$ Since $\lambda$ and $\psi$ are ring homomorphisms and $F$ is a functor on $\operatorname{Ring}$, we know that $F(\psi \circ \lambda)= F(\psi) \circ F(\lambda)$. Hence by combining the latter two diagrams, one sees that it suffices to show that $$F(\prod_l \gamma_l \circ \prod_k \gamma_k)= F(\prod_l \gamma_l) \circ F(\prod_k \gamma_k).$$ In other words we reduce the proof to the universal case. Using Corollary \[triangle\] and that the PD-functor $F$ commutes with finite products, we can further reduce this problem to showing that $$F(\gamma_l \circ \prod_k \gamma_k)= F(\gamma_l) \circ F(\prod_k \gamma_k)$$ for every fixed $l$ with $0 \leq l \leq m$.
Using Proposition \[ringsquare\] and Corollary \[surjections\], we can assume that $A$ is torsion-free. This allows us to work with symmetric tensors instead of divided powers. In other words, we need to show that the diagram $$\xymatrix@C=50pt{ F(A) \ar[drr]_-{\hspace{-2cm} F((-)^{\otimes l} \circ \prod_k (-)^{\otimes k} )} \ar[rr]^-{F(\prod_k (-)^{\otimes k})} & & F(\prod_{k=0}^n (A^{\otimes k})^{\Sigma_k} ) \ar[d]^-{F((-)^{\otimes l})} \\ & & F(((\prod_{k=0}^n (A^{\otimes k})^{\Sigma_k})^{\otimes l})^{\Sigma_l}) }$$ commutes, where the power map $(-)^{\otimes k} \colon A \to (A^{\otimes k})^{\Sigma_k}$ corresponds to the universal $k$-homogeneous polynomial law. Recall that there is a canonical ring isomorphism $$\xymatrix{ (\prod_{k=0}^n (A^{\otimes k})^{\Sigma_k})^{\otimes l}
\ar[rrrrr]_-{\cong}^-{\prod_{0\leq k_1, \cdots k_l \leq n} \operatorname{proj}_{k_1} \otimes \cdots \otimes \operatorname{proj}_{k_l} } & & & & & \prod_{0\leq k_1, \cdots k_l \leq n} (A^{\otimes k_1})^{\Sigma_{k_1}}
\otimes \cdots \otimes (A^{\otimes k_l})^{\Sigma_{k_l}}
. }$$ Again using Corollary \[triangle\] and the assumption that $F$ commutes with finite products and with invariants of finite groups, it is sufficient to show that the diagram $$\xymatrix@C=50pt{ F(A) \ar[drr]_-{\hspace{-2cm} F((-)^{\otimes k_1} \hat{\otimes} \dots\hat{\otimes} (-)^{\otimes k_l})} \ar[rr]^-{F(\prod_k (-)^{\otimes k})} & & F(\prod_{k=0}^n (A^{\otimes k})^{\Sigma_k} ) \ar[d]^{F(\operatorname{proj}_{k_1} \hat{\otimes} \cdots \hat{\otimes} \operatorname{proj}_{k_l})} \\ & & F((A^{\otimes k_1})^{\Sigma_{k_1}} \otimes \cdots \otimes (A^{\otimes k_l})^{\Sigma_{k_l}}) }$$ commutes for every $0\leq k_1,\dots,k_l\leq n$. Here $\hat{\otimes}$ is the tensor product of multiplicative polynomial laws $\alpha_j \colon B \to C_j$, defined as the composite $$\xymatrix{B \ar[rr]^-{\prod_j \alpha_j} & & \prod_{j=1}^l C_j \ar[r]^-\pi & C_1 \otimes \cdots \otimes C_l,}$$ where the second map $\pi$ is the canonical map from the product to the tensor product, which is a polynomial law by [@Roby1 Section I.7]. Let us now consider the diagram $$\xymatrix@R=35pt{
F(A)\ar[dr]_-{F((-)^{\otimes k_1})\hat{\otimes}\cdots\hat{\otimes}F((-)^{\otimes k_l})\hspace{1cm}}\ar[r]\ar@/^1.5pc/[rr]^-{F(\prod_k (-)^{\otimes k})}&\prod_{k=0}^nF((A^{\otimes k})^{\Sigma_k} )\ar[d]|-{\operatorname{proj}_{k_1} \hat{\otimes} \cdots \hat{\otimes} \operatorname{proj}_{k_l}} &F(\prod_{k=0}^n (A^{\otimes k})^{\Sigma_k} )\ar[d]^{F(\operatorname{proj}_{k_1} \hat{\otimes} \cdots \hat{\otimes} \operatorname{proj}_{k_l})}\ar[l]_-{\cong}\ar[dl]|-{F(\operatorname{proj}_{k_1}) \hat{\otimes} \cdots \hat{\otimes} F(\operatorname{proj}_{k_l})}
\\
&F((A^{\otimes k_1})^{\Sigma_{k_1}}) \otimes \cdots \otimes F((A^{\otimes k_l})^{\Sigma_{k_l}}) \ar[r]_c&F((A^{\otimes k_1})^{\Sigma_{k_1}} \otimes \cdots \otimes (A^{\otimes k_l})^{\Sigma_{k_l}}) \rlap{\ .}
}$$ The upper part of the diagram commutes because of Corollary \[triangle\]. The left hand and middle triangles commute by the definition of $\hat{\otimes}$. It therefore remains to show that the bottom right hand triangle commutes, and that the lower outer composite is the map $F((-)^{\otimes k_1} \hat{\otimes} \dots\hat{\otimes} (-)^{\otimes k_l})$. This follows if we show that for a torsion-free $B$ and any pair of multiplicative polynomial laws $\alpha_1\colon B\to C_1$ and $\alpha_2\colon B\to C_2$ of respective degrees $k_1$ and $k_2$, the triangle $$\xymatrix{F(B) \ar[rrr]^-{F(\alpha_1) \hat{\otimes}F(\alpha_2)} \ar[drrr]_-{ F(\alpha_1 \hat{\otimes} \alpha_2)\ \ } & & & F(C_1) \otimes F(C_2)
\ar[d]^c \\ & & & F(C_1 \otimes C_2) }$$ commutes. Let $\varphi_j\colon \Gamma_{k_j} B \to C_j$ be the ring homomorphisms corresponding to $\alpha_j$. Then $\alpha_1 \hat{\otimes} \alpha_2$ corresponds to the ring homomorphism $$\varphi_1 \hat{\otimes} \varphi_2\colon \Gamma_{k_1+k_2} B \xrightarrow{\gamma_{k_1,k_2}}\Gamma_{k_1} B \otimes \Gamma_{k_2} B\xrightarrow{\varphi_1\otimes\varphi_2} C_1\otimes C_2$$ where the first map is the ring homomorphism which corresponds to the tensor product of the universal polynomial laws $\gamma_{k_j}\colon B\to \Gamma_{k_j} B$ (The map $\gamma_{k_1,k_2} : \Gamma_{k_1+k_2} D \to \Gamma_{k_1} D \otimes \Gamma_{k_2} D$ is in fact defined for any not necessarily torsion-free commutative ring $D$.). Therefore the latter triangle will commute if we show commutativity of the outer composite of the diagram of ring homomorphisms $$\xymatrix@C=15pt{
\Gamma_{k_1+k_2} F(B)\ar[d]_{c_{k_1+k_2}} \ar[r]^-{\gamma_{k_1,k_2}}& \Gamma_{k_1} F(B)\otimes \Gamma_{k_2} F(B)\ar[rr]^-{c_{k_1}\otimes c_{k_2}}&&F(\Gamma_{k_1}B) \otimes F(\Gamma_{k_2}B)\ar[ddlll]^-{c} \ar[d]^-{F(\varphi_1)\otimes F(\varphi_2)}
\\
F(\Gamma_{k_1+k_2}B) \ar[d]_-{F(\gamma_{k_1,k_2})}&&&F(C_1)\otimes F(C_2)\ar[d]^{c}
\\
F(\Gamma_{k_1}B \otimes \Gamma_{k_2} B)\ar[rrr]_-{F(\varphi_1\otimes \varphi_2)}&&&F(C_1\otimes C_2),
}$$ where the lower right triangle commutes by Property \[lemma:ext-prop1\] of Lemma \[lemma:ext-prop\]. Thus we need to verify that the upper triangle commutes, and this requires understanding the map $\gamma_{k_1,k_2}$. One can verify by direct calculation that the diagram $$\xymatrix@C=60pt@R=17pt{
\Gamma_{k_1+k_2} X \ar[d]_-{\gamma_{k_1,k_2}}\ar[rr]^-{\Delta_\ast }&&\Gamma_{k_1+k_2}(X\times X)
\\
\Gamma_{k_1}X \otimes \Gamma_{k_2}X && \displaystyle\prod_{i+j=k_1+k_2}\Gamma_i X\otimes \Gamma_j X\ar[u]_-{\lambda}^-{\cong}\ar[ll]^{\operatorname{proj}_{k_1, k_2}}
}$$ commutes for every ring $X$, where $\Delta\colon X\to X\times X$ is the diagonal map, and the lower map projects onto the summand $(i,j)=(k_1,k_2)$. The map $\lambda$ sends an elementary tensor $a\otimes b$ in the $(i,j)$-component to $$((\operatorname{id},0)_\ast(a))\star ((0,\operatorname{id})_\ast(b)),$$ where $(\operatorname{id},0)_\ast\colon \Gamma_i X \to \Gamma_i (X\times X)$ is induced by the inclusion $(\operatorname{id},0)\colon X\to X\times X$ in the first summand, and similarly $(0,\operatorname{id})_\ast$ is induced by the inclusion in the second summand. The map $\star$ is the graded multiplication of the divided power algebra $\Gamma(X \times X)=\oplus_{n}\Gamma_n (X \times X)$ as defined in [@Roby1 Sections III.3-5]. The map $\lambda$ is an additive isomorphism by [@Roby1 Theorem III.4]. Let us verify that $\lambda$ is also multiplicative. By resolving by a free commutative ring we can assume that $X$ is torsion-free. Thus the natural transformations $\xi_X\colon \Gamma_n X \to (X^{\otimes n})^{\Sigma_{n}}$ of [@Roby1 Section III.6] are ring isomorphisms, where the target is a subring of the $n$-fold tensor product (see also [@Lak]). They moreover assemble into an isomorphism of graded algebras $$\Gamma(X) \stackrel{\cong}{\longrightarrow} \bigoplus_{n}(X^{\otimes n})^{\Sigma_{n}},$$ where the graded multiplication on the target is the shuffle product (see e.g., [@Roby1 Section III.5]) that we also denote by $\star$. We recall that on $x \in (X^{\otimes k_1})^{\Sigma_{k_1}}$ and $y \in (X^{\otimes k_2})^{\Sigma_{k_1}}$ it is given by the formula $$x \star y=\sum_{ \sigma \in S_{k_1,k_2}} \sigma (x \otimes y),$$ where $S_{k_1,k_2}$ is the set of $(k_1, k_2)$-shuffles (which are representatives of the left cosets $(\Sigma_{k_1} \times \Sigma_{k_2})\backslash\Sigma_{k_1+k_2}$), where the left action of $\sigma \in \Sigma_n$ on $X^{\otimes n}$ is defined by the formula $$\sigma (x_1 \otimes x_2 \otimes \dots \otimes x_n)=x_{\sigma^{-1}(1)} \otimes x_{\sigma^{-1}(2)} \otimes \dots \otimes x_{\sigma^{-1}(n)}$$ (there is some confusion between left and right actions in [@Roby1 Section III.5], see also [@Lak]). Thus under the isomorphism $\xi_X$ the map $\lambda$ corresponds to the map that sends the tensor product of $x \in (X^{\otimes i})^{\Sigma_i}$ and $y \in (X^{\otimes j})^{\Sigma_j}$ to $$\lambda(x\otimes y)=((\operatorname{id},0)_\ast(x))\star ((0,\operatorname{id})_\ast(x))=\sum_{ \sigma \in S_{i,j}} \sigma ((\operatorname{id},0)_\ast(x) \otimes (0,\operatorname{id})_\ast(y)),$$ for all $i+j=k_1+k_2$. The summands of this expression are (non-unital) ring homomorphisms. These homomorphisms are clearly pairwise orthogonal when $(i,j)\neq (i',j')$, and when $(i,j)= (i',j')$ they are orthogonal because the permutations in $S_{i,j}$ are $(i, j)$-shuffles. Thus $\lambda$ is a ring homomorphism.
Now we can replace $\gamma_{k_1,k_2}$ by the composite $\operatorname{proj}_{k_1, k_2} \circ \lambda^{-1} \circ \Delta_*$ into the triangle above. Using this, the definition of the maps $c_{k_1+k_2}$, $c_{k_1}$ and $c_{k_2}$, and Property \[lemma:ext-prop1\] of Lemma \[lemma:ext-prop\], one can reduce the commutativity of the triangle to the commutativity of the outer rectangle of the diagram $$\hspace{-1.3cm}
\xymatrix@C=15pt{\Gamma_{k_1} F(B) \otimes \Gamma_{k_2} F(B) \ar[d]_-{\xi_{F(B)}\otimes \xi_{F(B)}} \ar[rr]_-{(\operatorname{id},0)_\ast\otimes (0,\operatorname{id})_\ast} && \Gamma_{k_1} ((F(B)^{\times 2}) \otimes \Gamma_{k_2} ((F(B)^{\times 2}) \ar[r]^-{\star} \ar[d]_-{\xi_{F(B)^{\times 2}}\otimes \xi_{F(B)^{\times 2}}} & \Gamma_{k_1+k_2}(F(B)^{\times 2}) \ar[d]^{\xi_{F(B)^{\times 2}}}
\\
(F(B)^{\otimes k_1})^{\Sigma_{k_1}}\otimes (F(B)^{\otimes k_2})^{\Sigma_{k_2}}\ar[dd]_-{c^{\Sigma_{k_1}}\otimes c^{\Sigma_{k_2}}}\ar[rr]_-{(\operatorname{id},0)_\ast\otimes (0,\operatorname{id})_\ast}
&&((F(B)^{\times 2})^{\otimes k_1})^{\Sigma_{k_1}}\otimes ((F(B)^{\times 2})^{\otimes k_2})^{\Sigma_{k_2}}\ar[r]^-{\star}
& ((F(B)^{\times 2})^{\otimes k_1+k_2})^{\Sigma_{k_1+k_2}}
\\
&&(F(B^{\times 2})^{\otimes k_1})^{\Sigma_{k_1}}\otimes (F(B^{\times 2})^{\otimes k_2})^{\Sigma_{k_2}}\ar[r]^-{\star}\ar[u]_-{\cong}\ar[d]^-{c^{\Sigma_{k_1}}\otimes c^{\Sigma_{k_2}}}
&(F(B^{\times 2})^{\otimes k_1+k_2})^{\Sigma_{k_1+k_2}}\ar[d]^{c^{\Sigma_{k_1+k_2}}}\ar[u]_-{\cong}
\\
F(B^{\otimes k_1})^{\Sigma_{k_1}}\otimes F(B^{\otimes k_2})^{\Sigma_{k_2}}\ar[rr]_-{F(\operatorname{id},0)_\ast\otimes F(0,\operatorname{id})_\ast}
&&
F((B^{\times 2})^{\otimes k_1})^{\Sigma_{k_1}}\otimes F((B^{\times 2})^{\otimes k_2})^{\Sigma_{k_2}}\ar@{-->}[r]^-s
&F((B^{\times 2})^{\otimes k_1+k_2})^{\Sigma_{k_1+k_2}}
\\
F((B^{\otimes k_1})^{\Sigma_{k_1}})\otimes F((B^{\otimes k_2})^{\Sigma_{k_2}})\ar[u]^-{\cong}\ar[d]_-{c}&&
\\
F((B^{\otimes k_1})^{\Sigma_{k_1}}\otimes (B^{\otimes k_2})^{\Sigma_{k_2}})\ar[rrr]_-{F((\operatorname{id},0)_\ast\star (0,\operatorname{id})_\ast)}&&&F(((B^{\times 2})^{\otimes k_1+k_2})^{\Sigma_{k_1+k_2}})\ar[uu]^-{\cong}
\\
F(\Gamma_{k_1}B\otimes \Gamma_{k_2}B)\ar[u]^-{F(\xi_{B}\otimes\xi_{B})}_-{\cong}\ar[rrr]_-{F((\operatorname{id},0)_\ast\star (0,\operatorname{id})_\ast)}&&&F(\Gamma_{ k_1+k_2}(B\times B))\ar[u]_-{F(\xi_{B^{\times 2}})}^-{\cong}\rlap{\ .}
}$$
All the rectangles except the ones involving the dashed map commute by naturality, or because $\xi_X$ is a map of graded algebras. We define the dashed map by the formula $$s(z\otimes w):=\sum_{\sigma\in S_{k_1,k_2}}\sigma c(z\otimes w).$$ Here $z \in F((B^{\times 2})^{\otimes k_1})^{\Sigma_{k_1}}$ and $w \in F((B^{\times 2})^{\otimes k_2})^{\Sigma_{k_2}}$ and $c(z\otimes w)$ is the image of $z\otimes w$ under the composite $$\xymatrix{F((B^{\times 2})^{\otimes k_1})^{\Sigma_{k_1}}\otimes F((B^{\times 2})^{\otimes k_2})^{\Sigma_{k_2}} \ar[r] & F((B^{\times 2})^{\otimes k_1})\otimes F((B^{\times 2})^{\otimes k_2}) \ar[r]^-c & F((B^{\times 2})^{\otimes k_1+k_2}).}$$ It is immediate to verify that $s$ lands in the $\Sigma_{k_1+k_2}$-invariants. The rectangle above $s$ commutes since $$\begin{aligned}
c^{\Sigma_{k_1+k_2}}(x\star y)&=c^{\Sigma_{k_1+k_2}}(\sum_{\sigma\in S_{k_1,k_2}}\sigma(x\otimes y))=\sum_{\sigma\in S_{k_1,k_2}}\sigma c(x\otimes y)
\\
&=\sum_{\sigma\in S_{k_1,k_2}}\sigma c(c^{\Sigma_{k_1}}(x)\otimes c^{\Sigma_{k_2}}(y))
\\
&=s(c^{\Sigma_{k_1}}(x)\otimes c^{\Sigma_{k_2}}(y)).\end{aligned}$$ Let us denote by $\iota_{n}\colon (X^{\otimes n})^{\Sigma_{n}}\to X^{\otimes n}$ the fixed points inclusion. It is sufficient to show that the rectangle below $s$ commutes after composing with the inclusion $$F((B^{\times 2})^{\otimes k_1+k_2})^{\Sigma_{k_1+k_2}} \subset F((B^{\times 2})^{\otimes k_1+k_2}).$$ After postcomposing with this inclusion, the lower composite of this rectangle sends an element $u\otimes v \in F((B^{\otimes k_1})^{\Sigma_{k_1}})\otimes F((B^{\otimes k_2})^{\Sigma_{k_2}})$ to $$\begin{aligned}
F(\iota_{k_1+k_2})F((\operatorname{id},0)_\ast\star (0,\operatorname{id})_\ast)(c(u\otimes v))&=F(\iota_{k_1+k_2} ((\operatorname{id},0)_\ast\star (0,\operatorname{id})_\ast))(c(u\otimes v))
\\&=F(((\operatorname{id},0)_*\star (0,\operatorname{id})_*) (\iota_{k_1} \otimes \iota_{k_2}) )(c(u\otimes v))
\\
&=F(\sum_{\sigma\in S_{k_1,k_2}}\sigma ((\operatorname{id},0)_*\otimes (0,\operatorname{id})_*) (\iota_{k_1} \otimes \iota_{k_2}))(c(u\otimes v)).\end{aligned}$$ The last sum is a sum of orthogonal ring homomorphisms since the permutations of $S_{k_1,k_2}$ are shuffles, and by Property \[lemma:ext-prop3\] of Lemma \[lemma:ext-prop\] we can write this as $$\begin{aligned}
&F(\sum_{\sigma\in S_{k_1,k_2}}\sigma ((\operatorname{id},0)_*\otimes (0,\operatorname{id})_*) (\iota_{k_1} \otimes \iota_{k_2}))(c(u\otimes v))\\
&=\sum_{\sigma\in S_{k_1,k_2}}F(\sigma ((\operatorname{id},0)_*\otimes (0,\operatorname{id})_*) (\iota_{k_1} \otimes \iota_{k_2}))(c(u\otimes v))
\\&=\sum_{\sigma\in S_{k_1,k_2}}\sigma F(((\operatorname{id},0)_*\otimes (0,\operatorname{id})_*) (\iota_{k_1} \otimes \iota_{k_2}))(c(u\otimes v))
\\
&=\sum_{\sigma\in S_{k_1,k_2}}\sigma c(F((\operatorname{id},0)_*)\otimes F((0,\operatorname{id})_*)) (F(\iota_{k_1})(u) \otimes F(\iota_{k_2})(v))\end{aligned}$$ which is the value of the upper composite.
The ghost components of a polynomial law {#ghostlaw}
----------------------------------------
A consequence of Theorem \[polylawfunct\] and Example \[Witt vectors\] is that any multiplicative polynomial law $f\colon A\to B$ of finite degree induces a multiplicative polynomial law on Witt vectors $$W_S(f)\colon W_S(A)\longrightarrow W_S(B),$$ for every truncation set $S\subset\mathbb{N}$. We will describe the ghost components of $W_S(f)$, and explain how these determine the functoriality of $W_S$ in multiplicative polynomial laws.
\[wittnaturalpoly\] Let $F,G\colon \operatorname{Ring}\to \operatorname{Ring}$ be PD-functors and $\alpha \colon F \to G$ a natural transformation. Then $\alpha$ extends to a natural transformation on $\operatorname{Ring^{poly}}$. That is, for any multiplicative polynomial law $f \colon A \to B$ of finite degree, the diagram of polynomial laws $$\xymatrix@C=70pt@R=15pt{ F(A) \ar[r]^{F(f)} \ar[d]_{\alpha_A} & F(B) \ar[d]^{\alpha_B} \\ G(A) \ar[r]^{G(f)} & G(B) }$$ commutes.
Suppose that degree of $f$ is at most $n$. Recall that we have a commutative diagram of polynomial laws $$\xymatrix@C=70pt@R=15pt{A \ar[r]^{f} \ar[d]_-{\prod_i \gamma_i} & B. \\ \prod_{i=0}^{n} \Gamma_i A \ar[ur]_{\varphi=\oplus_i \varphi_i} & }$$ We need to show that the outer rectangle in the diagram $$\xymatrix@C=40pt{F(A) \ar[d]_{\alpha_A} \ar[r]^-{\prod_i \gamma_i} & \prod_{i=0}^{n} \Gamma_i F(A) \ar[d]^{\prod_i \Gamma_i (\alpha_A)} \ar[r]^{\prod_i c_i} & \prod_{i=0}^{n} F(\Gamma_i A) \ar[d]^{\prod_i \alpha_{ \Gamma_i(A)}} \ar[r]^{\cong} & F(\prod_{i=0}^{n} \Gamma_i A) \ar[r]^-{F(\varphi)} \ar[d]^{\alpha_{ \prod_i \Gamma_i(A)}} & F(B) \ar[d]^{\alpha_B} \\ G(A) \ar[r]^-{\prod_i \gamma_i} & \prod_{i=0}^{n} \Gamma_i G(A) \ar[r]^{\prod_i c_i} & \prod_{i=0}^{n} G(\Gamma_i A) \ar[r]^{\cong} & G(\prod_{i=0}^{n} \Gamma_i A) \ar[r]^-{G(\varphi)} & G(B) }$$ commutes. The first square commutes by naturality of the universal polynomial laws $\gamma_i$. The third and fourth squares commute by the naturality of $\alpha$ on $\operatorname{Ring}$. It remains to check that the second square commutes. For this it suffices to see that for any $i$, the square $$\xymatrix@C=70pt@R=15pt{ \Gamma_i F(A) \ar[d]_{\Gamma_i \alpha} \ar[r]^{c_i} & F(\Gamma_i A) \ar[d]^{\alpha \Gamma_i} \\ \Gamma_i G(A) \ar[r]^{c_i} & G(\Gamma_i A)}$$ commutes. Because $c_i$ is extended to all rings from torsion free rings it suffices to check this when $A$ is torsion-free. By the naturality of $\alpha$ this reduces to showing that for any commutative rings $C_1$ and $C_2$, the diagram $$\xymatrix@C=70pt@R=15pt{F(C_1) \otimes F(C_2) \ar[d]_{\alpha \otimes \alpha} \ar[r]^c & F(C_1 \otimes C_2) \ar[d]^{\alpha} \\ G(C_1) \otimes G(C_2) \ar[r]^c & G(C_1 \otimes C_2)}$$ commutes. This also follows from naturality of $\alpha$.
Let us recall that for any set $S$, the $S$-fold product functor $(-)^{\times S}\colon \operatorname{Ring}\to \operatorname{Ring}$ is a PD-functor, and therefore a multiplicative polynomial law $f\colon A\to B$ of finite degree induces a multiplicative polynomial law $f^{\times S}\colon A^{\times S}\to B^{\times S}$. When the set $S$ is finite this is the natural transformation $$(f^{\times S})_R\colon (A^{\times S})\otimes_{\operatorname{\mathbb{Z}}} R\cong (A\otimes_{\operatorname{\mathbb{Z}}} R)^{\times S}\xrightarrow{(f_R)^{\times S}} (B\otimes_{\operatorname{\mathbb{Z}}} R)^{\times S}\cong (B^{\times S})\otimes_{\operatorname{\mathbb{Z}}} R.$$ If $S$ is infinite there is nodirect description of this law as a natural transformation, and one needs to involve divided powers.
\[levelghost\] Let $f \colon A \to B$ be a multiplicative polynomial law of finite degree. Then for any truncation set $S$, the diagram of polynomial laws $$\xymatrix@C=70pt@R=15pt{W_S(A) \ar[d]_w \ar[r]^{W_S(f)} & W_S(B) \ar[d]^w \\ \prod_S A \ar[r]^{\prod_S f} & \prod_S B }$$ commutes, were the vertical maps are the ghost coordinates. Moreover, $W_S\colon \operatorname{Ring^{poly}}\to\operatorname{Ring^{poly}}$ is the unique extension of $W_S$ with this property.
The ghost coordinates $w\colon W_S\to (-)^{\times S}$ form a natural transformation between PD-functors. Thus by Proposition \[wittnaturalpoly\] the diagram commutes. Let $W_{S}'$ be another extension of $W_S$ to $\operatorname{Ring^{poly}}$ such that the diagram above commutes. Then we must have $wW_{S}(f)=w'W_{S}(f)$, or equivalently that the composites $$\xymatrix@C=50pt{
\prod_{i=0}^n\Gamma_iW_S(A)\ar@<.5ex>[r]^{\varphi}\ar@<-.5ex>[r]_{\varphi'}&W_S(B)\ar[r]^{w}&B^{\times S}
}$$ agree, where $\varphi$ and $\varphi'$ are the unique ring homomorphisms extending $W_S(f)$ and $W'_S(f)$, respectively. When $B$ is torsion free $w$ is injective, and this shows that $\varphi=\varphi'$, and consequently $W_S(f)=W'_S(f)$. In general, let us consider the commutative square $$\xymatrix@C=70pt@R=15pt{
\operatorname{\mathbb{Z}}[A]\ar@{->>}[d]_{\varepsilon}\ar[r]^{\prod_i\gamma_i}&\prod_{i=0}^n\Gamma_i\operatorname{\mathbb{Z}}[A]\ar[d]^{\varphi}
\\
A\ar[r]^f&B
}$$ where $\varepsilon$ is the counit from the polynomial ring, the top horizontal map is the universal polynomial law, and $\varphi$ is the unique ring homomorphism extending the polynomial law $f\varepsilon$. By applying $W_S$ and $W'_S$, respectively, to this diagram we obtain analogous commutative squares of polynomial laws, and corresponding commutative squares of ring homomorphisms $$\xymatrix@C=40pt{
\prod_{i=0}^n\Gamma_iW_S(\operatorname{\mathbb{Z}}[A])\ar@{->>}[d]^{\prod_i\Gamma_iW_S(\varepsilon)}\ar[r]^{\varphi_{W_S(\prod_i\gamma_i)}}&W_S(\prod_{i=0}^n\Gamma_i\operatorname{\mathbb{Z}}[A])\ar[d]^{W_S(\varphi)}
\\
\prod_{i=0}^n\Gamma_iW_S(A)\ar[r]_{\varphi_{W_S(f)}}& W_S(B).
}
\ \ \ \ \ \ \ \ \ \
\xymatrix@C=40pt{
\prod_{i=0}^n\Gamma_iW_S(\operatorname{\mathbb{Z}}[A])\ar@{->>}[d]^{\prod_i\Gamma_iW_S(\varepsilon)}\ar[r]^{\varphi_{W'_S(\prod_i\gamma_i)}}&W_S(\prod_{i=0}^n\Gamma_i\operatorname{\mathbb{Z}}[A])\ar[d]^{W_S(\varphi)}
\\
\prod_{i=0}^n\Gamma_iW_S(A)\ar[r]_{\varphi_{W'_S(f)}}&W_S(B).
}$$ The vertical maps of these squares agree since $W_S$ and $W_S'$ agree on ring homomorphisms by assumption. The top horizontal maps also agree by the previous argument, since $\prod_{i=0}^n\Gamma_i\operatorname{\mathbb{Z}}[A]$ is torsion-free. Since the left vertical map is surjective, the bottom horizontal maps are also equal.
The product of polynomial laws
------------------------------
We show that our construction respects the product of polynomial laws. Given two multiplicative polynomial laws $f \colon A \to B$ of degree at most $n$ and $g \colon A \to B$ of degree at most $m$, there is a multiplicative polynomial law $$(f\cdot g)_R:=f_R\cdot g_R\colon A\otimes_{\operatorname{\mathbb{Z}}}R \longrightarrow B\otimes_{\operatorname{\mathbb{Z}}} R$$ defined by the pointwise product in the ring $B\otimes_{\operatorname{\mathbb{Z}}} R$, of degree at most $n+m$.
\[productroby\] Let $F$ be a PD-functor. For any pair of finite degree multiplicative polynomial laws $f, g \colon A \to B$, we have $F(f \cdot g)=F(f) \cdot F(g)$.
This is very similar to the proof of Theorem \[functorialityinroby\]. Suppose that $f$ is of degree at most $n$ and $g $ of degree at most $m$. Then we have commutative diagrams $$\xymatrix@C=70pt@R=18pt{A \ar[r]^{f} \ar[d]_-{\prod_i \gamma_i} & B, \\ \prod_{i=0}^{n} \Gamma_i A \ar[ur]_{\varphi} & }
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\xymatrix@C=70pt@R=18pt{A \ar[r]^{g} \ar[d]_-{\prod_j \gamma_j} & B, \\ \prod_{j=0}^{m} \Gamma_j A \ar[ur]_{\psi}, & }$$ where $\varphi$ and $\psi$ are not necessarily unital ring homomorphisms. In order to prove the proposition it suffices to show that the diagram $$\xymatrix@C=28pt{& & & F(\prod_{i=0}^{n} \Gamma_i A) \otimes F(\prod_{j=0}^{m} \Gamma_i A) \ar[d]^c \ar[rr]^-{F(\varphi) \otimes F(\psi)} & & F(B) \otimes F(B) \ar@/^{4pc}/[dd]^{\mu} \ar[d]^c \\ F(A) \ar[urrr]^{\hspace{-1.5cm}F(\prod_i \gamma_i) \hat{\otimes} F(\prod_j \gamma_j)} \ar[drrrrr]_-{F(f \cdot g)} \ar[rrr]^-{\hspace{1.2cm} F((\prod_i \gamma_i) \hat{\otimes} (\prod_j \gamma_j)) } & & & F(\prod_{i=0}^{n} \Gamma_i A \otimes \prod_{j=0}^{m} \Gamma_j A) \ar[rr]^-{F(\varphi \otimes \psi)} & & F(B \otimes B) \ar[d]^{F(\mu)} \\ & & & & & F(B) }$$ commutes, where $\mu$ denotes the ring multiplications and $\hat{\otimes}$ is the tensor product of polynomial laws of the proof of Theorem \[functorialityinroby\]. Indeed, the outer composite through the upper right hand corner is $F(f) \cdot F(g)$ by functoriality of $F$. The lower triangle commutes by functoriality of $F$, and the square commutes by Lemma \[lemma:ext-prop\] \[lemma:ext-prop1\]. The upper left hand triangle commutes by the final step in the proof of Theorem \[functorialityinroby\].
On the functoriality of the Witt vectors in polynomial maps {#secpolymap}
===========================================================
In this section we show that in certain circumstances the functoriality of the Witt vectors functors in polynomial laws extends to polynomial maps. We start by reviewing some material on polynomial maps.
Review of polynomial maps
-------------------------
This is a recollection of results on polynomial maps and their relationship with polynomial laws that we will use throughout the paper. The content is classical and we do not claim originality for these results. Some of these results can be found in [@Passi1; @Passi2; @Leib; @Xan].
Let $A$ and $B$ be abelian groups and $f\colon A\to B$ a function which is not necessarily a group homomorphism, and $n\geq 0$ an integer. We recall that the *$n$-th cross-effect*, or *$n$-th deviation*, of $f$ is the function $\operatorname{cr}_n\colon A^{\times n}\to B$ defined by $$\operatorname{cr}_{n} f(a_1,\dots, a_{n}):=\sum_{\substack{U\subset \{1,\cdots, n \} }} (-1)^{n-|U|}f(\sum_{l\in U}a_l).$$
A function $f\colon A\to B$ of abelian groups is called *polynomial of degree* $\leq n$, or $n$-*polynomial*, if $\operatorname{cr}_{n+1}f=0$. It is called $n$-homogeneous if it is $n$-polynomial and $f(ka)=k^nf(a)$ for every $k\in\operatorname{\mathbb{Z}}$ and $a\in A$.
A function of rings $f\colon A\to B$ is called a *multiplicative $n$-polynomial map* if it is $n$-polynomial as a map of abelian groups, and $f(aa')=f(a)f(a')$ for any $a, a' \in A$. Similarly, it is multiplicative $n$-homogeneous if it is multiplicative $n$-polynomial and $n$-homogeneous.
The composition of an $n$-polynomial map and an $m$-polynomial map is $nm$-polynomial, by [@Leib], and similarly for $n$-homogeneous maps. Similarly the product of an $n$-polynomial map and an $m$-polynomial map is $(n+m)$-polynomial.
1. The only $0$-polynomial maps are the constant maps.
2. A multiplicative polynomial map $f\colon A\to B$ of degree $1$ is precisely a map of the form $f(a)=c + \varphi(a)$, $a \in A$, where $c$ is a constant idempotent and $\varphi$ is a not necessarily unital ring homomorphism from $A$ to $B$ which is orthogonal to $c$.
3. The exponentiation map $(-)^n\colon A\to A$ is a multiplicative $n$-homogeneous map for every commutative ring $A$. This is the case since it is the $n$-fold product of the identity map with itself, which is $1$-homogeneous.
4. The map $N\colon \operatorname{\mathbb{Z}}\to \operatorname{\mathbb{Z}}[x]/(x^2-2x)$ defined by $N(a)=a+\frac{a(a-1)}{2}x$ is multiplicative $2$-polynomial, but not homogeneous. This is the multiplicative norm of the Burnside Tambara functor of the group $\operatorname{\mathbb{Z}}/2$. It is an instance of the following more general example.
5. Let $T$ be a Tambara functor for a finite group $G$. The multiplicative transfer $$N(f)\colon T(G/H)\to T(G/K)$$ induced by a $G$-equivariant map $f\colon G/H\to G/K$ is $[K:H^g]$-polynomial, where $H^g$ is a subgroup of $K$ conjugate to $H$ with $f(eH)=gK$. This is proved in [@Tambara] [@Strickland 13.22]. For example for the groups $0=H\leq K=G=\operatorname{\mathbb{Z}}/2$, the Tambara reciprocity formula for $N=N(\operatorname{\mathbb{Z}}/2\to\ast)$ gives $$N(a+b)=N(a)+N(b)+\operatorname{tran}(a\overline{b})$$ where $\operatorname{tran}$ is the additive transfer and the bar denotes the involution of $T(\operatorname{\mathbb{Z}}/2)$. In this case one can explicitly calculate that $$\begin{aligned}
cr_3 N(a,b,c)&=N(a+b+c)-N(a+b)-N(b+c)-N(a+c)+N(a)+N(b)+N(c)=
\\ &=N(a+b)+N(c)+\operatorname{tr}((a+b)\overline{c})-N(a)-N(b)-\operatorname{tr}(a\overline{b})
-N(b)-N(c)-\operatorname{tr}(b\overline{c})\\
&-N(a)-N(c)-tr(a\overline{c})+N(a)+N(b)+N(c)
\\
&=N(a+b)-N(a)-N(b)-tr(a\overline{b})=0.\end{aligned}$$
Not all polynomial maps can be decomposed into a sum of homogeneous maps. A well-known counterexample is the degree $2$ map (see e.g., [@Gaud-Hart]) $$n \mapsto {n\choose 2} \colon \mathbb{Z} \to \mathbb{Z}.$$ However, this is possible when sufficiently many integers are invertible.
\[homdecomp\] Let $f\colon A\to B$ be $n$-polynomial, and suppose that every integer $1 \leq k \leq n$ is invertible in $B$ (e.g., if $B$ is $p$-local for some prime $p>n$). Then $f$ decomposes uniquely as $$f=\sum_{k=0}^n\varphi_k$$ where each $\varphi_k$ is $k$-polynomial and homogeneous. If moreover $f$ is multiplicative, so are the $\varphi_k$ and $$\varphi_i(x) \varphi_j(y)=0$$ for any $x, y \in A$ and $i \neq j$.
By definition the $n$-th cross-effect of an $n$-homogeneous function $h$ satisfies $$(\operatorname{cr}_n h)(x,\dots,x)=\sum_{i=0}^{n}(-1)^{n-i}{n\choose i}i^nh(x).$$ The sum $$\sum_{i=0}^{n}(-1)^{n-i}{n\choose i}i^n$$ is equal to $n!$ since the Stirling number of second kind $S(n,n)$ is equal to $1$. Hence we conclude that $$(\operatorname{cr}_n h)(x,\dots,x)=n!h(x)$$ for any $n$-homogeneous function $h$.
First let us show that the decomposition in homogeneous summands is unique. For simplicity we introduce the notation $$(\operatorname{\mathbb{cr}}_n \alpha)(x):=(\operatorname{cr}_n \alpha) (x, \cdots, x).$$ If $f=\sum_{k=0}^n\varphi_k$, then $f-\varphi_n$ is $(n-1)$-polynomial, and $$0=\operatorname{\mathbb{cr}}_{n}(f-\varphi_n)=\operatorname{\mathbb{cr}}_n f-\operatorname{\mathbb{cr}}_{n}\varphi_n=\operatorname{\mathbb{cr}}_n f-n!\varphi_n.$$ Since by assumption $n!$ is invertible, $\varphi_n$ is uniquely determined by $f$. Similarly, $f-\varphi_n-\varphi_{n-1}-\dots-\varphi_{k}$ is $(k-1)$-polynomial, and $$\begin{aligned}
0&=\operatorname{\mathbb{cr}}_{k}(f-\varphi_n-\varphi_{n-1}-\dots-\varphi_{k})=\operatorname{\mathbb{cr}}_k (f-\varphi_n-\varphi_{n-1}-\dots-\varphi_{k+1})-\operatorname{\mathbb{cr}}_{k}\varphi_k
\\
&=\operatorname{\mathbb{cr}}_k (f-\varphi_n-\varphi_{n-1}-\dots-\varphi_{k+1})-k!\varphi_k\end{aligned}$$ shows that $\varphi_k$ is inductively determined by the $\varphi_j$ for $k<j\leq n$.
The proof of uniqueness gives us an inductive procedure to define the maps $\varphi_k$. We recall that the $n$-th cross-effect $\operatorname{cr}_n f$ of an $n$-polynomial map is additive in each variable, and therefore the diagonal $\operatorname{\mathbb{cr}}_n f$ is $n$-homogeneous. We set $$\varphi_n:=\frac{\operatorname{\mathbb{cr}}_n f}{n!}.$$ Next, we see that $P_{n-1}f:=f-\varphi_n$ is $(n-1)$-polynomial, since $$\operatorname{cr}_n(f-\varphi_n)=\operatorname{cr}_nf-\operatorname{cr}_n\frac{\operatorname{\mathbb{cr}}_n f}{n!}=\operatorname{cr}_nf-\frac{n!\operatorname{cr}_n f}{n!}=0.$$ Here and below we use a classical fact about polarizations: For any symmetric multilinear function of $n$-variables $L$, the $n$-th cross effect of the diagonal $x\mapsto L(x,\cdots, x)$ is equal to $n! \cdot L$. It follows that the map defined by $$\varphi_{n-1}:=\frac{\operatorname{\mathbb{cr}}_{n-1}(f-\varphi_n)}{(n-1)!}$$ is $(n-1)$-homogeneous. Now suppose inductively that $f-\varphi_n-\dots-\varphi_{k+1}$ is $k$-polynomial. We define $$\varphi_k:=\frac{1}{k!}\operatorname{\mathbb{cr}}_k(f-\varphi_n-\dots-\varphi_{k+1}),$$ which is $k$-homogeneous, and we observe that $P_{k-1}f=(f-\varphi_n-\dots-\varphi_{k+1}-\varphi_k)$ is $(k-1)$-polynomial, since $$\begin{aligned}
\operatorname{cr}_{k}(f-\varphi_n-\dots-\varphi_{k+1}-\varphi_k)&=\operatorname{cr}_{k}(f-\varphi_n-\dots-\varphi_{k+1})-\operatorname{cr}_{k}(\varphi_k)\\&=\operatorname{cr}_{k}(f-\varphi_n-\dots-\varphi_{k+1})-\operatorname{cr}_k (\frac{1}{k!}\operatorname{\mathbb{cr}}_k(f-\varphi_n-\dots-\varphi_{k+1}))
\\&=\operatorname{cr}_{k}(f-\varphi_n-\dots-\varphi_{k+1})-k!\frac{1}{k!}\operatorname{cr}_k(f-\varphi_n-\dots-\varphi_{k+1})=0.\end{aligned}$$ This provides an inductive definition of $\varphi_k$. Finally, we obtain $
\operatorname{cr}_1(f-\sum_{i=1}^n\varphi_k)=0$. This provides a constant function $\varphi_0$ such that $$f=\sum_{i=0}^n\varphi_k.$$ This finishes the proof for the additive case. Now let us show that if $f$ is multiplicative, then so is each $\varphi_k$ and the different summands in the decomposition are orthogonal to each other. Indeed, from the equation $f(xy)=f(x)f(y)$ we see that $$\sum_{k=0}^n\varphi_k(xy)=\sum_{k,j=0}^n\varphi_k(x)\varphi_j(y)=\sum_{k=0}^n\varphi_k(x)(\sum_{j=0}^n\varphi_j(y)).$$ The function $f(xy)$ is $n$-polynomial in $x$ and $\varphi_k(xy)$ is $k$-homogeneous in $x$ for any fixed $y$. By the uniqueness of the homogeneous decomposition it follows that $$\varphi_k(xy)=\varphi_k(x)(\sum_{j=0}^n\varphi_j(y))$$ for every $k$ and every fixed $y\in A$. This is now a $k$-homogeneous polynomial map in $y$ for any fixed $x$, and again by the uniqueness of the homogeneous decomposition $$\varphi_k(xy)=\varphi_k(x)\varphi_k(y) \ \ \ \ \ \ \ \ \ \
\mbox{and}
\ \ \ \ \ \ \ \ \ \ 0=\varphi_i(x)\varphi_j(y)$$ when $i \neq j$.
Let $N^{G}_H\colon T(G/H)\to T(G/G)$ be the norm-map of a $G$-Tambara functor $T$, for some finite group $G$. Then $N^{G}_H$ is a multiplicative polynomial map whose degree is equal to the index $[G:H]$. This map does not in general extend to a polynomial law (See Example \[cex\] below), nor does it decompose into a sum of homogeneous pieces. After inverting the group-order there is an isomorphism of rings $$T(G/G)[|G|^{-1}]\stackrel{\cong}{\longrightarrow} \prod_{(K\leq G)}T(G/K)[|G|^{-1}]/J_K$$ where the product runs though the conjugacy classes of subgroups of $G$, and $J_K$ is the sum of the images of the additive transfers $T(G/L)\to T(G/K)$ where $L$ is a proper subgroup of $K$ [@Schwedeglobal Proposition 3.4.18]. The map is induced by the restrictions $\operatorname{res}^{G}_K$. The $K$-component of the composite map $$T(G/H)\stackrel{N}{\longrightarrow}T(G/G)\longrightarrow T(G/G)[|G|^{-1}]\cong \prod_{(K\leq G)}T(G/K)/J_K$$ is then homogeneous of degree $|K\backslash G/H|$. Indeed by the double coset formula this is the class modulo $J_K$ of $$\operatorname{res}_{K}^GN_{H}^G=\prod_{[g]\in K\backslash G/H}N^{K}_{H^g\cap K}c_g\operatorname{res}_{H\cap K^g}^H,$$ where the conjugation map $c_g$ and $\operatorname{res}_{H\cap K^g}^H$ are ring-homomorphisms. By reciprocity $$N^{K}_{H^g\cap K}(mx)=mN^{K}_{H^g\cap K}(x)+t(x)$$ for $m \in \operatorname{\mathbb{Z}}$, where $t(x)$ is a sum of proper transfers which vanishes in the quotient.
We remark that if $f\colon A\to B$ is an $n$-homogeneous polynomial law, the underlying map of abelian groups $$f_{\operatorname{\mathbb{Z}}} \colon A\longrightarrow B$$ obtained by evaluating the law at the ring $R=\operatorname{\mathbb{Z}}$, is an $n$-homogeneous polynomial map. This can be verified for the universal $n$-homogeneous law $\gamma_n\colon A\to \Gamma_n(A)$, where $$\begin{aligned}
\operatorname{cr}_{n+1}\gamma_n(a_1,\dots,a_n,a_{n+1})&=\operatorname{cr}_{n}(\gamma_n(a_{n+1}+(-)))(a_1,\dots ,a_n)-\operatorname{cr}_{n}\gamma_n(a_1,\dots, a_n)\\
&=\operatorname{cr}_{n}(\sum_{i+j=n}\gamma_i(a_{n+1})\gamma_{j}(-))(a_1,\dots,a_n)-\operatorname{cr}_{n}\gamma_n(a_1,\dots,a_n)\\
&=\sum_{i+j=n}\gamma_i(a_{n+1})(\operatorname{cr}_{n}\gamma_{j})(a_1,\dots,a_n)-\operatorname{cr}_{n}\gamma_n(a_1,\dots,a_n)\\
&=\gamma_0(a_{n+1})(\operatorname{cr}_{n}\gamma_{n})(a_1,\dots,a_n)-\operatorname{cr}_{n}\gamma_n(a_1,\dots,a_n)=0\end{aligned}$$ since the terms of the sum with $j<n$ vanish by induction on $n$, and $\gamma_0$ is the constant function with value $1$.
\[polymapvspolylaw\] Let $f\colon A\to B$ be an $n$-polynomial map, and $p>n$ a prime. Then the composite $f_{(p)}\colon A\to B\to B_{(p)}$ extends to a unique polynomial law of degree $n$. If $f$ is multiplicative, then so is the extension.
For convenience we will sometimes denote an element $\gamma_i(a) \in \Gamma_i(A)$ just by $a^{(i)}$.
We start by showing that any $n$-homogeneous polynomial map $\varphi \colon A\to B_{(p)}$ extends to an $n$-homogeneous law, which is multiplicative if the original map was. We will now define a unique additive extension $\overline{\varphi}\colon \Gamma_n(A) \to B_{(p)}$ of $\varphi$ as follows. The generators of the form $a^{(1)}_1\dots a^{(1)}_n$ can always be expressed as a sum of generators of the form $a^{(n)}$, by the formula $$a^{(1)}_1\dots a^{(1)}_n=(\operatorname{cr}_{n}\gamma_n)(a_1,\dots, a_n)=\sum_{U\subset n}(-1)^{n-|U|}(\sum_{i\in U}a_i)^{(n)}.$$ This formula can easily be proven by induction on $n$. Indeed it clearly holds for $n=0$ and $n=1$, and in general $$\begin{aligned}
\operatorname{cr}_{n}\gamma_n(a_1,\dots,a_{n-1},a_{n})&=\operatorname{cr}_{n-1}(\gamma_n(a_{n}+(-)))(a_1,\dots, a_{n-1})-\operatorname{cr}_{n-1}\gamma_n(a_1,\dots,a_{n-1})\\
&=\operatorname{cr}_{n-1}(\sum_{i+j=n}\gamma_i(a_{n})\gamma_{j}(-))(a_1,\dots,a_{n-1})-\operatorname{cr}_{n-1}\gamma_n(a_1,\dots,a_{n-1})\\
&=\sum_{i+j=n}\gamma_i(a_{n})(\operatorname{cr}_{n-1}\gamma_{j}(-))(a_1,\dots,a_{n-1})-\operatorname{cr}_{n-1}\gamma_n(a_1,\dots,a_{n-1})\\
&=\sum_{\substack{i+j=n\\ j<n}}\gamma_i(a_{n})(\operatorname{cr}_{n-1}\gamma_{j})(a_1,\dots,a_{n-1})\\
&=\gamma_1(a_{n})(\operatorname{cr}_{n-1}\gamma_{n-1})(a_1,\dots,a_{n-1})=\gamma_1(a_{n})a^{(1)}_1\dots a^{(1)}_{n-1}
\\
&=a^{(1)}_1\dots a^{(1)}_n,\end{aligned}$$ where the second to last equality holds if we inductively assume that the identity holds for $n-1$. Here we used that $\operatorname{cr}_{n-1}\gamma_j=0$ for $j<n-1$. Therefore we define $$\overline{\varphi}(a^{(1)}_1\dots a^{(1)}_n):=(\operatorname{cr}_{n}\varphi)(a_1,\dots, a_n).$$ Then we observe that if an additive extension $\overline{\varphi}\colon \Gamma_n(A) \to B_{(p)}$ exists, then it will factor over $\Gamma_n(A)_{(p)}$. A generic generator $a_{1}^{(n_1)}\dots a_{l}^{(n_l)}$ of $\Gamma_n (A)$ with $\sum n_i=n$ will decompose in $\Gamma_n(A)_{(p)}$ as $$a_{1}^{(n_1)}\dots a_{l}^{(n_l)}=\frac{1}{n_1!\dots n_l!}\underbrace{a_{1}^{(1)}\dots a^{(1)}_1}_{n_1}\dots \underbrace{a_{l}^{(1)}\dots a^{(1)}_l}_{n_l},$$ where the positive integers smaller or equal to $n$ are invertible since $p>n$. We therefore define $$\overline{\varphi}(a_{1}^{(n_1)}\dots a_{l}^{(n_l)}):=\frac{1}{n_1!\dots n_l!}(\operatorname{cr}_{n}\varphi)(\underbrace{a_{1},\dots, a_1}_{n_1},\dots, \underbrace{a_{l},\dots, a_l}_{n_l}).$$ This map extends $\varphi$, since $$\overline{\varphi}(a^{(n)})=\frac{1}{n!}(\operatorname{cr}_{n}\varphi)(a,\dots, a)=\varphi(a),$$ where the second equality holds because $\varphi$ is $n$-homogeneous. It remains to verify that this map respects the relations of the divided power algebra. We recall that the $n$-th cross-effect of an $n$-polynomial map is additive in each variable. It follows that $$\begin{aligned}
\overline{\varphi}((ka)^{(n)})&=\frac{1}{n!}(\operatorname{cr}_{n}\varphi)(ka,\dots, ka)=\frac{k^n}{n!}(\operatorname{cr}_{n}\varphi)(a,\dots, a)=k^n\overline{\varphi}(a^{(n)}),
\\
\overline{\varphi}((a+b)^{(n)})&=\frac{1}{n!}(\operatorname{cr}_{n}\varphi)(a+b,\dots, a+b)
=\frac{1}{n!}\sum_{i+j=n}{n\choose i} (\operatorname{cr}_{n}\varphi)(\underbrace{a,\dots, a}_i,\underbrace{b,\dots ,b}_j)
\\
&=\sum_{i+j=n}\frac{1}{i!j!}(\operatorname{cr}_{n}\varphi)(\underbrace{a,\dots, a}_{i}, \underbrace{b,\dots, b}_{j})
=\sum_{i+j=n} \overline{\varphi}(a^{(i)}b^{(j)}),
\\
\overline{\varphi}(a^{(i)}a^{(j)})&=\frac{1}{i!j!}(\operatorname{cr}_{n}\varphi)(a,\dots,a)
={i+j\choose i}
\frac{1}{(i+j)!}(\operatorname{cr}_{n}\varphi)(a,\dots,a)
=
{i+j\choose i}\overline{\varphi}(a^{(i+j)}).\end{aligned}$$ The additive map $\overline{\varphi}$ is unique since it factors through an additive map $$\widehat{\overline{\varphi}} \colon \Gamma_n(A)_{(p)} \longrightarrow B_{(p)}$$ and by the above observations the $\mathbb{Z}_{(p)}$-module $\Gamma_n(A)_{(p)}$ is generated by the image of the canonical map $$\xymatrix{A \ar[r]^-{\gamma_n} & \Gamma_n(A) \ar[r] & \Gamma_n(A)_{(p)}. }$$
Next we check that if $\varphi$ is multiplicative, then so is $\overline{\varphi}$. For this it suffices to show that $\widehat{\overline{\varphi}}$ is multiplicative. We have $$\widehat{\overline{\varphi}}(a^{(n)})=\varphi(a)$$ and the elements of the form $a^{(n)}$ generate the $\mathbb{Z}_{(p)}$-module $\Gamma_n(A)_{(p)}$. Hence $\widehat{\overline{\varphi}}$ is multiplicative on additive generators and hence in general.
Now we complete the proof for a general an $n$-polynomial map $f\colon A\to B$. By Proposition \[homdecomp\] the map $f_{(p)}$ decomposes into an orthogonal sum of homogeneous polynomial maps: $$f_{(p)}=\varphi_0+\varphi_1+\cdots+\varphi_n.$$ The previous paragraph provides an additive extension $$\oplus_i \overline{\varphi_i} \colon \prod_{i=0}^n \Gamma_i(A) \to B_{(p)}$$ of $f_{(p)}$. By Proposition \[homdecomp\] and the previous paragraph the latter extension is unique. Further if $f$ is multiplicative, then by Proposition \[homdecomp\] so are $\varphi_i$-s and the images of $\varphi_i$ and $\varphi_j$ are orthogonal if $i \neq j$. We already saw that $\overline{\varphi_i}$ is multiplicative for every $i$. Similarly, using the fact that $\Gamma_i(A)_{(p)}$ is generated by the elements of the form $a^{(i)}$, we can see the desired orthogonality property. Altogether we get that $\oplus_i \overline{\varphi_i}$ is multiplicative.
Any abelian group $A$ admits a universal polynomial map $A\to P_n(A)$ which classifies $n$-polynomial maps out of $A$. The construction can be found in [@Passi1; @Passi2]. If $A$ is a ring, then the ring multiplication on $A$, makes $P_n(A)$ also into a ring. The latter proposition implies that if $p >n$, then the $p$-localization of $P_n(A)$ is isomorphic as a ring to $\prod_{i=0}^n \Gamma_i(A)_{(p)}$. In other words, the composite $$\xymatrix{ A \ar[r]^-{\prod_i \gamma_i} & \prod_{i=0}^n \Gamma_i(A) \ar[r] & \prod_{i=0}^n \Gamma_i(A)_{(p)}}$$ is the universal multiplicative $n$-polynomial map with a $p$-local target.
Functoriality in polynomial maps {#secWpoly}
--------------------------------
Let us fix once and for all a prime number $p$. Let $1\leq m\leq\infty$ be an integer or infinity. We denote by $W_m(A)$ the ring of $p$-typical $m$-truncated Witt vectors of $A$. The case $m=\infty$ gives the full ring of $p$-typical Witt vectors for which we will use the usual notation $W(A):=W_{\infty}(A)$.
\[Wittpoly\] The functor $W_m \colon \operatorname{Ring}\to \operatorname{Ring}$ extends to the partial category of multiplicative $(p-1)$-polynomial maps. That is, a multiplicative $n$-polynomial map $f\colon A\to B$, with $n<p$, induces a multiplicative $n$-polynomial map $W_m(f)\colon W_m(A)\to W_m(B)$, with the property that if $f\colon A\to B$ and $g\colon B\to C$ are multiplicative $n$ and $k$-polynomial, respectively, and $nk<p$, then $$W_m(g)\circ W_m(f)=W_m(g\circ f).$$ This extension is unique with the property that the ghost coordinates of the map $W_m(f)$ is the product map $\prod_m f$, i.e., the square $$\xymatrix@C=70pt@R=15pt{
W_m(A)\ar[r]^-{W_m(f)}\ar[d]_{w}&W_m(B)\ar[d]^w
\\
\prod_{m} A\ar[r]_-{\prod_{m} f}&\prod_{m} B
}$$ commutes. If moreover $f,g\colon A\to B$ are multiplicative $n$ and $k$ polynomial, respectively, and $n+k<p$, then $W_m(f\cdot g)=W_m(f)\cdot W_m(g)$.
\[formula\] Using the uniqueness of the functoriality of Theorem \[Wittpoly\], one can go ahead and try to calculate the components of a polynomial map $W_m(f)$ by inductively solving the equations provided by the description in ghost components.
The first component of the image of a Witt vector $(a_0,a_1,\dots)$ in $W(A)$ by a polynomial map $W(f)\colon W(A)\to W(B)$ is $b_0=f(a_0)$. The next component $b_1$ must be the unique natural solution to the equation $$f(a_0)^{p}+pb_1=w_1(b_0,b_1)=f(w_1(a_0,a_1))=f(a^{p}_0+pa_1).$$ Since $p>n$, the map $f$ is also $p$-polynomial and from the equation $\operatorname{cr}_{p}f(a_0^p+(-))=0$ one can calculate that $$f(a^{p}_0+pa_1)=f(a_{0}^{p})+\sum_{i=1}^{p-1}(-1)^i{p\choose i}f(a_{0}^p+ia_1).$$ The binomial coefficients of this sum are all divisible by $p$, and the unique natural solution to the equation above is $$b_1=\sum_{i=1}^{p-1}(-1)^i({p\choose i}/p)f(a_{0}^p+ia_1).$$ We remark that when $f$ is a ring homomorphism, this sum is in fact equal to $f(a_1)$ so that we indeed recover the usual functoriality in ring homomorphisms.
The proof of Theorem \[Wittpoly\] will use the following well-known lemma:
\[wittpullback\] For any commutative ring $A$ and integer or infinity $1\leq m\leq \infty$, the commutative diagram $$\xymatrix@C=70pt@R=15pt{W_m(A) \ar[d]_w \ar[r] & W_m(A_{(p)}) \ar[d]^w \\ \prod_{m} A \ar[r] & \prod_{m} A_{(p)} }$$ is a pullback of rings, where the horizontal maps are induced by the canonical localization homomorphism $A \to A_{(p)}$.
The case $m=\infty$ follows from the case $m < \infty$ by passing to inverse limits, since pullbacks commute with inverse limits. For finite $m$, we prove the statement by induction. The case $m=1$ is obvious. The inductive step follows from the observation that for any commutative ring (in fact, for any abelian group), the diagram of abelian groups $$\xymatrix@C=70pt@R=15pt{A \ar[r] \ar[d]_{p^m} & A_{(p)} \ar[d]^{p^m} \\ A \ar[r] & A_{(p)} }$$ is a pullback. The latter itself follows from the fact that the induced maps on kernels and cokernels of vertical maps are isomorphisms.
We denote by $\lambda_R \colon R \to R_{(p)}$ the natural localization homomorphism. We first define $W_m(f)\colon W_m(A)\to W_m(B)$ using the pullback of Lemma \[wittpullback\]. Since pull-backs of rings are pullbacks of sets, it suffices to construct a multiplicative $n$-polynomial map $W_m(A)\to W_m(B_{(p)})$ which will make the diagram of polynomial maps $$\xymatrix@C=70pt@R=15pt{
W_m(A) \ar[d]_{(\prod_{m} f) \circ w} \ar[r] & W_m(B_{(p)}) \ar[d]^w \\ \prod_{m} B \ar[r]^-{\prod_{m} \lambda_B} & \prod_{m} B_{(p)}
}$$ commute. By Proposition \[polymapvspolylaw\] we know that the composite $A \stackrel{f}{\longrightarrow}B\stackrel{\lambda_B}{\longrightarrow} B_{(p)}$ extends uniquely to a multiplicative polynomial law. We can therefore take the map underlying the multiplicative polynomial law $W_m(\lambda_B \circ f) \colon W_m(A)\to W_m(B_{(p)})$ provided by Theorem \[polylawfunct\]. Corollary \[levelghost\] implies that the underlying multiplicative $n$-polynomial map of this polynomial law has the desired description in ghost components. We therefore obtain a map $$W_m(f)\colon W_m(A)\longrightarrow W_m(B)$$ such that $w \circ W_m(f)= (\prod_{m} f) \circ w$ and $ W_m(\lambda_B) \circ W_m(f)=W_m(\lambda_B \circ f)$. If we evaluate the polynomial law induced by the map $\lambda_B\circ f$ on the ring $\mathbb{Z}_{(p)}$, we get a multiplicative $n$-polynomial map $f_{(p)} \colon A_{(p)} \to B_{(p)}$ making the diagram $$\xymatrix@C=70pt@R=15pt{A \ar[d]_{\lambda_A} \ar[r]^f & B \ar[d]^{\lambda_B} \\ A_{(p)} \ar[r]^{f_{(p)}} & B_{(p)} }$$ commute. By Proposition \[polymapvspolylaw\], the maps $f_{(p)}$ and $\lambda_B \circ f$ are underlying maps of unique polynomial laws. Hence Theorem \[functorialityinroby\] implies that the diagram $$\xymatrix@C=70pt@R=15pt{W_m(A) \ar[d]_{W_m(\lambda_A)} \ar[r]^{W_m(f)} & W_m(B) \ar[d]^{W_m(\lambda_B)} \\ W_m(A_{(p)}) \ar[r]^{W_m(f_{(p)})} & W_m(B_{(p)}) }$$ commutes.
Next, we check the identity $W_m(g)\circ W_m(f)=W_m(g\circ f)$. Using the pullback of Lemma \[wittpullback\], it suffices to check that $W_m(g)\circ W_m(f)=W_m(g\circ f)$ holds after postcomposing with the canonical map $$W_m(\lambda_C) \colon W_m(C) \to W_m(C_{(p)}).$$ Consider the commutative diagram $$\xymatrix@C=70pt@R=15pt{A \ar[d]_{\lambda_A} \ar[r]^f & B \ar[d]^{\lambda_B} \ar[r]^g & C \ar[d]^{\lambda_C} \\ A_{(p)} \ar[r]^{f_{(p)}} & B_{(p)} \ar[r]^{g_{(p)}} & C_{(p)}.}$$ The maps $g_{(p)}$, $f_{(p)}$ and $\lambda_{(-)}$ uniquely extend to multiplicative polynomial laws by Proposition \[polymapvspolylaw\]. So do their composites, and the polynomial laws corresponding to the compositions are the compositions of the polynomial laws associated to the individual maps. Hence Theorem \[functorialityinroby\] implies that $$W_m(\lambda_C) \circ W_m(g\circ f)=W_m(\lambda_C \circ g\circ f)= W_m(g_{(p)} \circ f_{(p)} \circ \lambda_A)=W_m(g_{(p)}) \circ W_m(f_{(p)}) \circ W_m(\lambda_A).$$ Finally, using the commutative diagram $$\xymatrix@C=70pt@R=15pt{W_m(A) \ar[d]_{W_m(\lambda_A)} \ar[r]^{W_m(f)} & W_m(B) \ar[d]^{W_m(\lambda_B)} \ar[r]^{W_m(g)} & W_m(C) \ar[d]^{W_m(\lambda_C)} \\ W_m(A_{(p)}) \ar[r]^{W_m(f_{(p)})} & W_m(B_{(p)}) \ar[r]^{W_m(g_{(p)})} & W_m(C_{(p)}),}$$ we conclude that $W_m(\lambda_C) \circ W_m(g\circ f)=W_m(\lambda_C) \circ W_m(g) \circ W_m(f)$, which shows that $W_m$ is a partial functor.
The uniqueness is immediate in the torsion-free case since the ghost maps are injective. In the general case one reduces to the torsion-free case by choosing a resolution similar to the one of Proposition \[levelghost\].
Finally, arguing as above and using Proposition \[productroby\] we see that under our conditions the functor $W_m$ respects multiplications of multiplicative polynomial maps.
Now we provide examples which show that the conditions of Theorem \[Wittpoly\] are optimal.
\[cex\] The following counterexample shows that the $p-1$ bound on the degree is necessary. Let us consider the multiplicative $p$-polynomial map $$N\colon \operatorname{\mathbb{Z}}\longrightarrow \operatorname{\mathbb{Z}}[x]/(x^2-px)$$ defined by $N(a)=a+\frac{a^p-a}{p}x$. This is the norm of the Burnside Tambara functor of the cyclic group $C_p$ of order $p$. We show that the norm of the first two ghost components $(Nw_0,Nw_1)$ of $\operatorname{\mathbb{Z}}$ is not in the image of the first two ghost coordinates $(w_0,w_1)$ of the ring $\operatorname{\mathbb{Z}}[x]/(x^2-px)$. Indeed, suppose that there are elements $b_0,b_1\in \operatorname{\mathbb{Z}}[x]/(x^2-px)$ such that $(w_0,w_1)(b_0,b_1)=(Nw_0,Nw_1)(0,1)$. Then $$\begin{aligned}
(b_0,b^{p}_0+pb_1)=(Nw_0,Nw_1)(0,1)=(N(0),N(p))=(0,p+(p^{p-1}-1)x),\end{aligned}$$ and we must have that $pb_1=p+(p^{p-1}-1)x$. This equation has no solution in $\operatorname{\mathbb{Z}}[x]/(x^2-px)$ since $p^{p-1}-1$ is not divisible by $p$.
This shows that there cannot be a map on $p$-typical Witt vectors which in ghost components is the map $N$ in each coordinate. That is, for any $m \geq 2$ there cannot be any map (of sets) $f \colon W_m(\operatorname{\mathbb{Z}}) \to W_m(\operatorname{\mathbb{Z}}[x]/(x^2-px))$ making the diagram $$\xymatrix@C=60pt@R=17pt{
W_m(\operatorname{\mathbb{Z}})\ar@{-->}[rr]^-f \ar[d]_{w}&&W_m(\operatorname{\mathbb{Z}}[x]/(x^2-px))\ar[d]^w
\\
\prod_{m} \operatorname{\mathbb{Z}}\ar[rr]_-{\prod_{m} N}&&\prod_{m} \operatorname{\mathbb{Z}}[x]/(x^2-px)
}$$ commute.
Another piece of evidence on the optimality of the theorem is provided by the exponentiation maps. The map $(-)^n\colon A\to A$ induces $W((-)^n)=(-)^n\colon W(A)\to W(A)$ for $n<p$. If we try to go beyond the bound $n<p$ we see that the map $(-)^p=\operatorname{id}\colon \operatorname{\mathbb{F}}_p\to \operatorname{\mathbb{F}}_p$ should simultaneously induce the $p$-th power map and the identity on the $p$-adic integers $W(\operatorname{\mathbb{F}}_p)=\operatorname{\mathbb{Z}}_p$, contradicting the functoriality of $W$.
The functor $W$ does not extend to a functor on the subcategory of commutative rings and set maps generated by the multiplicative $(p-1)$-polynomial maps. The reason is that in general, a map can have several factorizations as compositions of $(p-1)$-polynomial maps, and the extension will depend on this choice. For example, let us take $p=5$ and the multiplicative polynomial maps $(-)^2,(-)^{3}\colon \operatorname{\mathbb{F}}_5\to \operatorname{\mathbb{F}}_5$, so that $2\cdot 3\geq 5$. These maps compose to the map $(-)^6=(-)^2\colon \operatorname{\mathbb{F}}_5\to \operatorname{\mathbb{F}}_5$, but the composite of $(-)^2$ and $(-)^{3}$ is not $(-)^2$ on $W(\operatorname{\mathbb{F}}_5) \cong \operatorname{\mathbb{Z}}_5$. This shows that the condition $nk<p$ for the composition formula in Theorem \[Wittpoly\] is necessary.
Exponentiation illustrates well the different roles played on Witt vectors by polynomial maps and polynomial laws. The key fact used in the previous examples is that $(-)^p=\operatorname{id}$ as a self polynomial map of $\mathbb{F}_p$. This equality does not however hold as polynomial laws. Indeed, the polynomial law on $\mathbb{F}_p$ defined by exponentiation by $n$ is the natural transformation $$(-)^n\colon \mathbb{F}_p\otimes R\longrightarrow \mathbb{F}_p\otimes R,$$ defined by the exponentiation of the ring $\mathbb{F}_p\otimes R$, where $R$ ranges through all commutative rings. When $R=\operatorname{\mathbb{Z}}[t]$ we clearly have that $(-)^p$ on $\mathbb{F}_p\otimes \operatorname{\mathbb{Z}}[t]=\mathbb{F}_p[t]$ is not the identity.
Similar to the case of composition, for the product of maps one cannot remove the hypothesis that $n+k<p$ and only require $n,k<p$. For example, the map $(-)^6=(-)^2\colon \operatorname{\mathbb{F}}_5\to \operatorname{\mathbb{F}}_5$ decomposes both as the product of $(-)^3$ and $(-)^3$, and of the identity with itself. However $(-)^6$ and $(-)^2$ are different on $W(\operatorname{\mathbb{F}}_5) \cong \operatorname{\mathbb{Z}}_5$.
One cannot extend the functor $W$ additively on sums of multiplicative $(p-1)$-polynomial maps. For example, the identity map $\operatorname{id}\colon \operatorname{\mathbb{F}}_p\to \operatorname{\mathbb{F}}_p$ decomposes as the sum of $(p+1)$-identities, but $(p+1)W(\operatorname{id})=(p+1) \operatorname{id}$ is different than $\operatorname{id}$ on $\operatorname{\mathbb{Z}}_p$, i.e., such an extension will not be well-defined.
Theorem \[Wittpoly\] gives a potential obstruction for decomposing an $n$-polynomial map $f\colon A\to B$ into a composition of polynomial maps of lower degree. Indeed, let $n=lk$ with $l,k\neq 1$ and choose a prime $p$ such that $l,k< p\leq lk$ (say that $l\leq k$, then a prime $p$ with $k<p<2k$ can be used, since $2k\leq lk$). By Theorem \[Wittpoly\], if $f$ is the composition of two polynomial maps of respective degree $l$ and $k$, it will induce a map on ($m$-truncated) $p$-typical Witt vectors whose ghost coordinates are $\prod f$. In particular $\prod f\circ w$ is in the image of the ghost coordinates of $B$ and one can attempt to contradict this fact as in Example \[cex\]. An example of this will be provided in §\[factor\].
Applications
============
The factorization problem for polynomial maps {#factor}
---------------------------------------------
As we pointed out above, Theorem \[Wittpoly\] can provide an obstruction for detecting if a polynomial map of degree $nk$ is the composite of some polynomial maps of degree $k$ and $n$. In this subsection we give a non-trivial example, where Theorem \[Wittpoly\] tells us that such a factorization is not possible.
Let $A_4$ be the fourth alternating group, and $A_3\leq A_4$ any copy of the third alternating group. We denote by $\mathbb{A}(A_n)$ the corresponding Burnside rings.
\[exobstruction\] The $4$-polynomial norm map $N_{A_3}^{A_4}\colon \mathbb{A}(A_3)\to \mathbb{A}(A_4)$ does not decompose as the composition of two multiplicative polynomial maps of degree $2$.
The norm $N^{G}_H$ of a Tambara functor, corresponding to a subgroup inclusion $H\leq G$, is always $[G:H]$-polynomial. Suppose that $[G:H]=nk$. If there is a subgroup $H\leq K\leq G$ of index $[G:K]=k$ we would obtain that $$N^{G}_H=N^{G}_KN^{K}_H$$ decomposes as a composite of an $n$-polynomial map and a $k$-polynomial map. Hence, in order to get an interesting example out of such norms, we need to know that $H$ is maximal in $G$ and has a non-prime index. The group $A_4$ is the smallest group which admits a maximal subgroup $A_3$ of a non-prime index. Hence the norm $N_{A_3}^{A_4}$ does not factor as the composite of two norm maps of subgroups of index $2$. Our theorem shows that it does not even decompose abstractly as the composition of two $2$-polynomial maps.
Let $W_2(-;3)$ denote the ring of $2$-truncated $3$-typical Witt vectors. Suppose that $N:=N_{A_3}^{A_4}$ is the composite $$\xymatrix{ \mathbb{A}(A_3) \ar[r]^-f & C \ar[r]^-g & \mathbb{A}(A_4) }$$ of multiplicative polynomial maps $f$ and $g$ of degree $2$, for some commutative ring $C$. Then (since $2<3$) by Theorem \[Wittpoly\], we get a commutative diagram $$\xymatrix@C=70pt@R=15pt{W_2(\mathbb{A}(A_3)) \ar[d]^{w} \ar[r]^{W_2(g) \circ W_2(f)} & W_2(\mathbb{A}(A_4)) \ar[d]^{w} \\ \mathbb{A}(A_3) \times \mathbb{A}(A_3) \ar[r]^{N \times N} & \mathbb{A}(A_4) \times \mathbb{A}(A_4). }$$ We claim that the top horizontal map making this diagram commute cannot exist. This will follow immediately if we show that $$((N \times N) \circ w) (1,1)$$ is not in the image of the ghost map of $\mathbb{A}(A_4)$. The argument is similar to the one in Example \[cex\]. If $$((N \times N) \circ w) (1,1)=(N \times N)(1,4)=(1, N(4))$$ was in the image of $w$, then $N(4)$ would be congruent to $1$ mod $3$.
Let us then show that $N(4)$ is not congruent to $1$ mod $3$. Recall that the norm $N_{H}^G$ in the Burnside Tambara functor is defined as follows. Choose orbit representatives $g_1,\dots g_n$ for the quotient $G/H$ of cardinality $n$. Then each group element $g\in G$ determines a permutation $\sigma\in \Sigma_n$ and elements $h_1,\dots,h_n\in H$, defined by the relations $gg_i=g_{\sigma(i)}h_i$ for $1\leq i\leq n$. Then $N_{H}^G$ sends an $H$-set $X$ to the set $X^{\times n}$ with the $G$ action obtained by restricting the natural $\Sigma_n\wr H$-action along the group homomorphism $$G\longrightarrow \Sigma_n\wr H \ \ \ \ \ \ \ \ \ \ \ \ g\longmapsto (\sigma,h_1,\dots,h_n).$$ In particular an integer $m\in \mathbb{A}(H)$, represented by the trivial $H$-set with $m$-elements $X$, is sent to the set $X^{\times n}$ where $G$ acts by the group homomorphism $\rho\colon G\to \Sigma_n$ that sends $g$ to $\sigma$. In the case of the groups $H=A_3\leq A_4=G$ this is the standard inclusion $A_4\to \Sigma_4$ up to an automorphism of $A_4$ (automorphisms of $A_4$ act trivially on $ \mathbb{A}(A_4)$). Indeed, the kernel of $\rho$ consists of those elements $g\in A_4$ such that $gg_i=g_{i}h_i$ for all $1\leq i\leq 4$, that is $$g\in \bigcap_{i=1}^4(g_{i}A_3g_{i}^{-1})=1.$$ Thus a trivial $A_3$-set $X$ with $m$-elements is sent by $N$ to the set $X^{\times 4}$, where $A_4$ acts by permuting the components via the standard inclusion into $\Sigma_4$. Up to conjugacy the stabilizer of an element $(x,y,z,w)\in X^{\times 4}$ only depends on how many of the components are equal. This leads to the orbits decomposition of the $A_4$-set $X^{\times 4}$, as $$N(m)=X^{\times 4}=m+(m^2-m)A_4/A_{3}+\left(\begin{array}{c}m\\ 2\end{array}\right)A_4/\operatorname{\mathbb{Z}}/2+m\left(\begin{array}{c}m-1\\ 2\end{array}\right)A_4/e+2\left(\begin{array}{c}m\\ 4\end{array}\right)A_4/e.$$ In particular $$N(4)=4+12A_4/A_{3}+6A_4/\operatorname{\mathbb{Z}}/2+14A_4/e\equiv 1+14A_4/e \ \mbox{mod}\ 3,$$ with $14$ not divisible by $3$.
The above example shows that if a product map lifts to the Witt vectors, then it must satisfy certain congruences. Indeed if $f \colon A \to B$ is a map that fits in a commutative diagram $$\xymatrix@C=60pt@R=17pt{W_2(A;p) \ar[d]^w \ar[r]^{W_2(f)} & W_2(B;p) \ar[d]^w \\ A \times A \ar[r]^{f \times f} & B \times B}$$ for some map $W_2(f)$, then for any $a_0, a_1 \in A$ we must have $$f(a_0^p+pa_1) \equiv f(a_0)^p \ \mbox{mod}\ p.$$ Similarly, an analogous argument for $W_m$ with $m > 2$ forces $f$ to satisfy higher versions of these congruences. One can in fact directly show by induction on $k$ that an $n$-polynomial map $f$ satisfies $$f(a+p^kc)=f(a)+\sum_{i_1,\dots,i_k=1}^{p-1}(-1)^{i_1+\dots+i_k}{p\choose i_1}\dots{p\choose i_k}f(a+i_1\dots i_kc)$$ for every odd prime $p>n$ and $k\geq 0$, where the sum is divisible by $p^k$. This congruence is then preserved by the composition of maps, and our proof of Proposition \[exobstruction\] shows that $N_{A_3}^{A_4}$ does not satisfy this congruence for $p=3$ and $k=1$.
The formula above is in fact sufficient to lift the product map of $f\colon A\to B$ to the Witt vectors when $A$ and $B$ are torsion-free with compatible Frobenius lifts, as this guarantees that the congruences of the Dwork Lemma are satisfied (this is for example the case for the universal $n$-polynomial map of [@Passi1; @Passi2]). Thus these congruences are closely related to the lifts of the product map on the Witt-vectors.
Witt vectors of Z/2-Tambara functors {#secTambara}
-------------------------------------
We use Theorem \[Wittpoly\] to extend the $p$-typical Witt vectors functor to the category of $\operatorname{\mathbb{Z}}/2$-Tambara functors, for odd primes $p$. We will schematically display a $\operatorname{\mathbb{Z}}/2$-Tambara functors $T$ as $$T=\xymatrix@C=70pt{\big(A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& B\ar[l]|-{\operatorname{res}}\big),}$$ where we keep in mind, but suppress from the notation, that $A$ has an involution which is part of the structure. Given a $\operatorname{\mathbb{Z}}/2$-Tambara functor $T$ as above, we let $$\prod_{n}T=\xymatrix@C=70pt{\big(\prod_{n}A \ar@<1ex>[r]^-{\prod\operatorname{tran}}\ar@<-1ex>[r]_-{\prod N}& \prod_{n}B\ar[l]|-{\prod\operatorname{res}}\big)}$$ be the $n$-fold product of $T$ in the category of $\operatorname{\mathbb{Z}}/2$-Tambara functors. The ring structures, involution, restriction, transfer and norm are all defined componentwise. The classical Witt polynomials for a prime $p$ define ghost coordinates $w\colon \prod_n A\to\prod_{n}A$ and $w\colon \prod_{n}B\to \prod_{n}B$ which are ring homomorphisms precisely when the sources are endowed with the ring structure of the $p$-typical Witt vectors.
\[WittTambara\] Let $p$ be an odd prime and $1\leq n \leq\infty$ an integer or infinity. There is a unique structure of $\operatorname{\mathbb{Z}}/2$-Tambara functor $\xymatrix@C=40pt{W_{n}(A) \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}&W_{n}(B)\ar[l]|-{\operatorname{res}}}$ such that the Witt polynomials define a natural transformation of Tambara functors $$\xymatrix@C=70pt@R=17pt{W_{n}(A)\ar[d]_w \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}&W_{n}(B)\ar[d]^w\ar[l]|-{\operatorname{res}}
\\
\prod_{n}A \ar@<1ex>[r]^-{\prod\operatorname{tran}}\ar@<-1ex>[r]_-{\prod N}& \prod_{n}B\ar[l]|-{\prod\operatorname{res}}\rlap{\ .}
}$$ We denote the resulting Tambara functor by $W_{n}(T)$.
Theorem \[WittTambara\] cannot be extended to the prime $p=2$. Indeed, Example \[cex\] shows that the norm of the $\operatorname{\mathbb{Z}}/2$-Burnside Tambara functor does not induce a map on $2$-typical Witt vectors with the above description in ghost components.
We start by defining the maps of the Tambara functor $W_n(T)$. The restriction map $\operatorname{res}\colon B\to A$ is a ring homomorphism, and therefore it induces a ring homomorphism $W_n(\operatorname{res})\colon W_n(B)\to W_n(A)$, and we define this to be the restriction of $W_n(T)$. Similarly, the involution on $A$ induces an involution on $W_n(A)$. The multiplicative transfer $N\colon A\to B$ of $T$ is multiplicative $2$-polynomial, and Theorem \[Wittpoly\] provides an induced multiplicative $2$-polynomial map for odd primes $$W_n(N)\colon W_{n}(A)\longrightarrow W_{n}(B)$$ which in ghost components is the product map. We declare this to be the norm of $W_{n}(T)$. The additive transfer $\operatorname{tran}\colon A\to B$ of a $\operatorname{\mathbb{Z}}/2$-Tambara functor is always determined by the norm $N$ by the Tambara reciprocity formula $$\operatorname{tran}(a)=N(a+1)-N(a)-1.$$ Therefore we define $W_n(\operatorname{tran}):=W_n(N)\circ (1+\operatorname{id})+W_n(N)-1$. By Theorem \[Wittpoly\] the ghost components of these maps are the product maps, and therefore it remains to show that this structure indeed defines a Tambara functor, and that it is unique.
The main tool for proving this theorem is the existence of a resolution of Tambara functors $$\xymatrix@C=70pt@R=17pt{
\operatorname{\mathbb{Z}}[A]\ar@{->>}[d]_\epsilon \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \ar[l]|-{\operatorname{res}} \mathbb{A}[A;B]\ar@{->>}[d]^\delta
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& B\ar[l]|-{\operatorname{res}}
}$$ where the vertical arrows are surjective and where $\mathbb{A}[A;B]$ is torsion free. Such a resolution is constructed in the Appendix by explicitly calculating the left adjoint of the forgetful functor from $\operatorname{\mathbb{Z}}/2$-Tambara functors to presheaves of sets on the orbit category of $\operatorname{\mathbb{Z}}/2$. The construction of $W_n(T)$ described above gives a diagram $$\xymatrix@C=70pt@R=19pt{
W_n(\operatorname{\mathbb{Z}}[A])\ar@{->>}[d]_{W_n(\epsilon)} \ar@<1ex>[r]^-{W_n(\operatorname{tran})}\ar@<-1ex>[r]_-{W_n(N)}& \ar[l]|-{W_n(\operatorname{res})} W_n(\mathbb{A}[A;B])\ar@{->>}[d]^{W_n(\delta)}
\\
W_n(A) \ar@<1ex>[r]^-{W_n(\operatorname{tran})}\ar@<-1ex>[r]_-{W_n(N)}& W_n(B)\ar[l]|-{W_n(\operatorname{res})}
}$$ where the vertical arrows are surjective ring homomorphisms, which commute with $W_n(N)$, $W_n(\operatorname{tran})$ and $W_n(\operatorname{res})$. Since the vertical maps are surjective it is sufficient to show that the relations needed for $W_n(T)$ are satisfied by the upper Tambara functor. Since $\operatorname{\mathbb{Z}}[A]$ and $\mathbb{A}[A;B]$ are torsion free their ghost coordinates are injective, where the relations hold since in ghost coordinates the maps become the product maps. The commutativity of this diagrams also shows uniqueness.
The components of the dihedral fixed-points of THR for odd primes {#secTHR}
-----------------------------------------------------------------
In this section we apply the main theorem of the previous section to topology. We describe the rings of path components of the dihedral fixed-points of real topological Hochschild homology and their multiplicative transfers in terms of the algebraic constructions of the previous section, for odd primes.
We assume that the reader is familiar with basic notions of equivariant stable homotopy theory of [@ManMay; @Schwede]. We will be mostly interested in the group $O(2)$ and its dihedral subgroups $D_{p^n}$ of order $2p^n$. We fix once and for all a group isomorphism $O(2) \cong S^{1}\rtimes \operatorname{\mathbb{Z}}/2$ by sending the generator of $\operatorname{\mathbb{Z}}/2$ to the reflection with respect to the $x$-axis. Let $E$ be an *orthogonal ring spectrum with anti-involution* (see [@THRmodels Section 2.1] for a definition), which is *flat* (Flat here refers to being underlying cofibrant in the flat model structure of [@Sto; @BrDuSt] on $\operatorname{\mathbb{Z}}/2$-spectra.). We recall from [@THRmodels] and [@Amalie] that the *real topological Hochschild homology* of $E$ is the $O(2)$-spectrum defined as the geometric realization of the dihedral nerve $$\operatorname{THR}(E):=B^{di}E=|[k]\longmapsto E^{\wedge k+1}|$$ with the usual cyclic structure of the cyclic nerve, and the involution of $E^{\wedge k+1}$ defined as the indexed smash product over the $\operatorname{\mathbb{Z}}/2$-set $\{0,1,\dots,k\}$ with the involution which reverses the order of $\{1,\dots,k\}$ and keeps $0$ fixed. We will regard $\operatorname{THR}(E)$ as a genuine $D_{p^n}$-spectrum for all primes $p$ and $n\geq 0$. For every integer $n\geq 0$, we define a $\operatorname{\mathbb{Z}}/2$-equivariant spectrum $$\operatorname{TRR}^{n+1}(E;p):=\operatorname{THR}(E)^{C_{p^n}},$$ where $(-)^{C_{p^n}}$ stands for the derived fixed points and $\operatorname{\mathbb{Z}}/2$ acts via the Weyl action of $D_{p^n}/C_{p^n}\cong \operatorname{\mathbb{Z}}/2$.
The $\operatorname{\mathbb{Z}}/2$-spectra $\operatorname{TRR}^n(E;p)$ were constructed in [@Amalie] using a version of Bökstedt’s model for $\operatorname{THR}$. This model is compared to the dihedral nerve in [@THRmodels], where an explicit zig-zag of equivalences is constructed. The same zig-zag is used in [@DMPSW] to see that the cyclotomic structures, reviewed below, agree. By combining these results we see that the two models of $\operatorname{TRR}^n(E;p)$ defined here and in [@Amalie] are equivalent.
When $E$ is a $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum, then $\operatorname{THR}(E)$ is a $D_{p^n}$-equivariant commutative orthogonal ring spectrum. In this case $\operatorname{TRR}^{n+1}(E;p)$ is a commutative $\operatorname{\mathbb{Z}}/2$-equivariant ring spectrum, and we are interested in calculating its $\operatorname{\mathbb{Z}}/2$-Tambara functor of components. In [@THRmodels Cor. 5.2] it is shown that for $n=0$ $$\underline{\pi}_0\operatorname{THR}(E)=\underline{\pi}_0\operatorname{TRR}^{1}(E;p)=\xymatrix@C=40pt{\big(\pi_0E \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \pi_0(E^{\operatorname{\mathbb{Z}}/2})\otimes_\phi\pi_0(E^{\operatorname{\mathbb{Z}}/2}) \ar[l]|-{\operatorname{res}}\big)}$$ where $\otimes_\phi$ is a quotient of the tensor product with respect to a certain Frobenius action. We recall that a $\operatorname{\mathbb{Z}}/2$-Tambara functor $T$ is called *cohomological* if $N\operatorname{res}=(-)^2$.
\[pi0TRR\] Let $E$ be a connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum, such that $\underline{\pi}_0E$ is cohomological. For an odd prime $p$ and $n \geq 0$, there is a natural isomorphism of $\operatorname{\mathbb{Z}}/2$-Tambara functors $$\underline{\pi_0}\operatorname{TRR}^{n+1}(E;p)\cong W_{n+1}(\underline{\pi}_0\operatorname{THR}(E)),$$ with the Tambara functor $W_{n+1}(-)$ of $p$-typical Witt vectors of §\[secTambara\]. In particular, the ring $\pi_0\operatorname{THR}(E)^{D_{p^n}}$ is isomorphic to $W_{n+1}( \pi_0(E^{\operatorname{\mathbb{Z}}/2})\otimes_\phi\pi_0(E^{\operatorname{\mathbb{Z}}/2}))$.
1. Since the transfer of a $\operatorname{\mathbb{Z}}/2$-Tambara functor is determined by the Tambara reciprocity formula $\operatorname{tran}(x)=N(x+1)-N(x)-1$, for cohomological Tambara functors we also have $\operatorname{tran}(1)=2$.
2. The Tambara functor associated to a commutative ring with involution $A$ is always cohomological, since $N\operatorname{res}(a)=a\overline{a}=a^2$ for all $a\in A^{\operatorname{\mathbb{Z}}/2}$.
3. The Tambara functor $$T=\xymatrix@C=40pt{\big(\operatorname{\mathbb{Z}}/2 \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \operatorname{\mathbb{Z}}/4\ar@{->>}[l]|-{\operatorname{res}}\big)}$$ is also cohomological, and it is not associated to a ring with involution. The restriction is the canonical projection, $\operatorname{tran}$ sends $0$ to $0$ and $1$ to $2$, and $N$ preserves $0$ and $1$.
4. The Burnside Tambara functor is not cohomological, since $\operatorname{tran}(1)=[\operatorname{\mathbb{Z}}/2]$ is the free transitive $\operatorname{\mathbb{Z}}/2$-set, which does not represent $2$ in the $\operatorname{\mathbb{Z}}/2$-Burnside ring.
An extension of Theorem \[pi0TRR\] to non-cohomological $\underline{\pi}_0E$ is provided at the end of the section. The proof of Theorem \[pi0TRR\] will use an inductive argument based on maps $R \colon \operatorname{TRR}^{n+1}(E;p) \to \operatorname{TRR}^{n}(E;p)$. These maps are constructed using the *real cyclotomic structure* on $\operatorname{THR}(E)$ which we now recall. We denote by $T$ the $O(2)$-spectrum $T:=\operatorname{THR}(E)$. As $E$ is flat, there is an isomorphism of $O(2)$-spectra $$\delta\colon \Phi^{C_{p}}T\stackrel{\cong}{\longrightarrow} T,$$ where $\Phi^{C_{p}}$ is the relative monoidal geometric fixed-points functor of [@ManMay Section 5.4] for a complete $O(2)$-universe (for convenience, we choose the universe $\mathcal{U}$ of [@Amalie Section 2]). Here $\Phi^{C_{p}}T$ has the residual $O(2)$-action given by the isomorphism $$O(2)/C_p \cong (S^{1}\rtimes \operatorname{\mathbb{Z}}/2)/C_p \cong (S^1/C_p) \rtimes \operatorname{\mathbb{Z}}/2 \cong S^1 \rtimes \operatorname{\mathbb{Z}}/2 \cong O(2).$$ The map $\delta$ is an $S^1$-equivariant isomorphism by work of [@sixauthors] when $E$ is cofibrant as an associative or commutative algebra, and by [@DMPSW] when $E$ is underlying flat, based on results of [@Sto; @BrDuSt]. The map $\delta$ is moreover $\operatorname{\mathbb{Z}}/2$-equivariant, and therefore an $O(2)$-equivariant isomorphism when $E$ is an orthogonal ring spectrum with anti-involution which is flat as a $\operatorname{\mathbb{Z}}/2$-spectrum. When $E$ is commutative, the map $\delta$ is an isomorphism of $\operatorname{\mathbb{Z}}/2$-equivariant commutative orthogonal ring spectra. This map $\delta$ is the real cyclotomic structure of $\operatorname{THR}(E)$. We also note that $\Phi^{C_{p}}T$ is already derived as a $\operatorname{\mathbb{Z}}/2$-spectrum when $E$ is flat. This is a real analog of [@sixauthors Theorem 4.7]. One first checks that $\Phi^{C_{p}}$ is derived on the levels of the dihedral bar construction using [@Sto §3.4.3] and the cofibrant replacement functor of [@THRmodels Appendix A.1]. After this one passes to the geometric realizations as in the proof of [@sixauthors Theorem 4.7]. By iterating this argument we see that in fact $\Phi^{C_{p}}T$ is derived as a $D_{p^{n+1}}/C_p \cong D_{p^n}$-spectrum for any $n \geq 0$.
Let $E$ be a flat orthogonal ring spectrum with anti-involution and $p$ a prime. The restriction map of $T=\operatorname{THR}(E)$ is the zig-zag of $O(2)$-equivariant maps $$R\colon T^{C_{p^n}}\stackrel{\simeq}{\longrightarrow}(T^{C_p})^{C_{p^{n-1}}}\longrightarrow (\Phi^{C_{p}}(T_f))^{C_{p^{n-1}}}\stackrel{\simeq}{\longleftarrow} (\Phi^{C_{p}}(T))^{C_{p^{n-1}}}\xrightarrow{\delta^{C_{p^{n-1}}}} T^{C_{p^{n-1}}},$$ where $(-)_f$ is a functorial fibrant replacement in the model category of $O(2)$-equivariant orthogonal spectra. We define $\operatorname{TRR}(A;p)$ to be the homotopy limit of the diagram $$\dots
\xrightarrow{\delta^{C_{p^2}}} T^{C_{p^2}}\to (\Phi^{C_{p}}(T_f))^{C_{p}}\stackrel{\simeq}{\leftarrow} (\Phi^{C_{p}}(T))^{C_{p}}\xrightarrow{\delta^{C_{p}}}
T^{C_{p}}\to \Phi^{C_{p}}(T_f)\stackrel{\simeq}{\leftarrow} \Phi^{C_{p}}T\xrightarrow{\delta} T$$ in the category of $\operatorname{\mathbb{Z}}/2$-spectra.
The third map in the zig-zag of the definition of $R$ is an equivalence since $X \to X_f$ is an acyclic cofibration, which are preserved by the relative geometric fixed points functor. It will be crucial for the proof of Theorem \[pi0TRR\] to understand the interaction between the map $\pi_0R$ and the norm of the Tambara functor $\underline{\pi_0}\operatorname{TRR}^{n+1}(E;p)$, when $E$ is commutative.
When $E$ is commutative the zig-zag defining $R$ can be arranged to take place in the category of $O(2)$-equivariant orthogonal ring spectra. This is achieved by taking $(-)_f$ to be a functorial fibrant replacement functor for the model category of $O(2)$-equivariant orthogonal ring spectra, and use that the acyclic cofibrations of ring spectra are underlying acyclic cofibrations, which are preserved by geometric fixed points. This in particular gives $\operatorname{TRR}(E;p)$ the structure of an $O(2)$-equivariant orthogonal ring spectrum, and the map $$\underline{\pi}_0 (R) \colon \underline{\pi}_0 \operatorname{TRR}^{n+1}(E;p) \to \underline{\pi}_0\operatorname{TRR}^{n}(E;p)$$ is a map of commutative $\operatorname{\mathbb{Z}}/2$-Green functors. This is however not obviously a map of $\operatorname{\mathbb{Z}}/2$-Tambara functors, since it is not clear if one can represent the $R$ map in the category of $\operatorname{\mathbb{Z}}/2$-equivariant *commutative* orthogonal ring spectra. The problem is that the cofibrations of $\operatorname{\mathbb{Z}}/2$-equivariant commutative orthogonal ring spectra are not cofibrations of underlying $\operatorname{\mathbb{Z}}/2$-spectra, and hence the third map defining $R$ will not be a weak equivalence in general. One should be able to solve this problem by working with algebras over an equivariant $E_{\infty}$-operad instead of strictly commutative $O(2)$-ring spectra. This is however outside of the main scope of this paper and we will instead explicitly show that $\underline{\pi}_0R$ is compatible with the norms, and hence is a map of $\operatorname{\mathbb{Z}}/2$-Tambara functors.
For ease of notation we write $R\colon=\pi_0(R) \colon \pi_0T^{C_{p^n}} \to \pi_0T^{C_{p^{n-1}}}$ and $R_2\colon=\pi_0^{\operatorname{\mathbb{Z}}/2}(R) \colon \pi_0 T^{D_{p^n}} \to \pi_0T^{D_{p^{n-1}}}$.
\[Rtambaramap\] Let $E$ be a connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum. Then:
1. For any $n \geq 1$ and $p$ odd, there is a commutative triangle $$\xymatrix@C=60pt@R=17pt{\pi_0 \operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2} \ar[r]^{N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} } \ar[dr]_{N_{\operatorname{\mathbb{Z}}/2}^{D_{p^{n-1}}}} & \pi_0 \operatorname{THR}(E)^{D_{p^n}} \ar[d]^{R_2} \\ & \pi_0 \operatorname{THR}(E)^{D_{p^{n-1}}} \rlap{\ .}}$$
2. For any $n \geq 1$ there is a commutative square $$\xymatrix@C=60pt@R=17pt{ \pi_0 \operatorname{THR}(E)^{C_{p^n}} \ar[d]^{R} \ar[r]^{N_{C_{p^n}}^{D_{p^n}} } & \pi_0 \operatorname{THR}(E)^{D_{p^n}} \ar[d]^{R_2} \\ \pi_0 \operatorname{THR}(E)^{C_{p^{n-1}}} \ar[r]^{N_{C_{p^{n-1}}}^{D_{p^{n-1}}} } & \pi_0 \operatorname{THR}(E)^{D_{p^{n-1}}} \rlap{\ .}}$$ Hence $\underline{\pi}_0 (R) \colon \underline{\pi}_0 \operatorname{TRR}^{n+1}(E;p) \to \underline{\pi}_0 \operatorname{TRR}^{n}(E;p)$ is a map of $\operatorname{\mathbb{Z}}/2$-Tambara functors.
For simplicity we will set $T\colon=\operatorname{THR}(E)$. We need the following construction for both parts of the lemma. For any orthogonal $G$-spectrum $X$ and a normal subgroup $N$ of $G$, there is natural transformation $$\xymatrix{ \phi\colon \pi_0^GX \cong \pi_0^GX^c \ar[r]^-{\Phi^N} & \pi_0^{G/N}\Phi^N X^c \ar[r] & \pi_0^{G/N} \Phi^N X,}$$ where $(-)^c$ is a functorial cofibrant replacement (of equivariant spectra or associative algebras). In other words this construction first takes derived geometric fixed points and then composes with the canonical map of the cofibrant replacement. The maps $R$ and $R_2$ can be described by the composites $$\xymatrix@C=20pt{R\colon \pi_0^{C_{p^n}}T \ar[r]^-\phi & \pi_0^{C_{p^{n-1}}} \Phi^{C_p}T \ar[r]^-{\delta_\ast} & \pi_0^{C_{p^{n-1}}} T }, \ \ \ \ \ \ \ \ \ \ \ \ \xymatrix@C=20pt{R_2\colon \pi_0^{D_{p^n}}T \ar[r]^-\phi & \pi_0^{D_{p^{n-1}}} \Phi^{C_p}T \ar[r]^-{\delta_\ast} & \pi_0^{D_{p^{n-1}}} T,}$$ respectively. We begin with Statement i). As a $D_{p^n}$-spectrum $\operatorname{THR}(E)$ is isomorphic to the geometric realization of a simplicial $D_{p^n}$-spectrum, defined as the Segal subdivision of the $p^n$-fold edgewise subdivision of the dihedral Bar construction. For odd $p$, its zero simplices are isomorphic to the norm $N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} (E \wedge E)$, and we let $$v_{p^n} \colon N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} (E \wedge E) \to \operatorname{THR}(E)$$ be the canonical map from the zero simplices to the geometric realization. Consider the commutative diagram $$\hspace{-2.5cm} \xymatrix{\pi_0^{\operatorname{\mathbb{Z}}/2} (E \wedge E) \ar[d]^{(v_1)_\ast} \ar[r]^-{N^{ex}} & \pi_0^{D_{p^n}} N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} (E \wedge E) \ar[d]^{N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} (v_1)_\ast} \ar@{=}[r] & \pi_0^{D_{p^n}} N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} (E \wedge E) \ar[r]^-{\phi} \ar[d]^{(v_{p^n})_\ast} & \pi_0^{D_{p^{n-1}}} \Phi^{C_p} N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} (E \wedge E) \ar[d]^{\Phi^{C_p} (v_{p^n})_\ast} & \pi_0^{D_{p^{n-1}}} N_{\operatorname{\mathbb{Z}}/2}^{D_{p^{n-1}}} (E \wedge E) \ar[l]^-{\Delta_\ast}_-{\cong} \ar[d]^{(v_{p^{n-1}})_\ast}
\\
\pi^{\operatorname{\mathbb{Z}}/2}_0 T \ar[r]^-{N^{ex}} & \pi_0^{D_{p^n}} N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}} T \ar[r]^{\epsilon_\ast} & \pi_0^{D_{p^n}} T \ar[r]^-{\phi} & \pi_0^{D_{p^{n-1}}} \Phi^{C_p} T \ar[r]^{\delta_\ast} & \pi_0^{D_{p^{n-1}}} T\rlap{\ .} }$$ Here $N^{ex}$ is the external norm and $\epsilon$ is the counit. The first and third squares commute by naturality. The second square commutes by definition of the subdivisions. The right hand square commutes by the construction of $\delta$. We also note that the diagonal $\Delta$ is an isomorphism by results of [@Sto; @BrDuSt; @sixauthors]. By definition the composite $\epsilon_\ast N^{ex}$ is the norm $N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}}$ of $\underline{\pi_0}T$. Moreover the external norm satisfies $\phi N^{ex}=\Delta_\ast N^{ex}$, and therefore $$R_2 \circ N_{\operatorname{\mathbb{Z}}/2}^{D_{p^n}}\circ(v_1)_\ast=(v_{p^{n-1}})_\ast\Delta^{-1}_\ast\phi N^{ex}=(v_{p^{n-1}})_\ast N^{ex}=N_{\operatorname{\mathbb{Z}}/2}^{D_{p^{n-1}}}\circ(v_1)_\ast,$$ where the last equality uses the first two squares for $n-1$ instead of $n$. Since $(v_1)_\ast$ is surjective this proves Part i).
For Part ii), we consider diagram $$\xymatrix@C=35pt{\pi_0^{C_{p^n}}T \ar[rr]^-{N^{ex}} \ar[d]^-\phi & & \pi_0^{D_{p^n}} N_{C_{p^n}}^{D_{p^n}} T \ar[d]^-\phi \ar[r]^{\epsilon_\ast} & \pi_0^{D_{p^n}} T \ar[d]^-\phi
\\
\pi_0^{C_{p^{n-1}}} \Phi^{C_p} T \ar[r]^-{N^{ex}} & \pi_0^{D_{p^{n-1}}} N_{C_{p^{n-1}}}^{D_{p^{n-1}}} \Phi^{C_p} T \ar[r]^-{\Delta_\ast} & \pi_0^{D_{p^{n-1}}} \Phi^{C_p} N_{C_{p^n}}^{D_{p^n}} T \ar[r]^-{(\Phi^{C_p} \epsilon)_\ast} & \pi_0^{D_{p^{n-1}}} \Phi^{C_p} T,}$$ where $\Delta \colon N_{C_{p^{n-1}}}^{D_{p^{n-1}}} \Phi^{C_p} T \to \Phi^{C_p} N_{C_{p^n}}^{D_{p^n}} T$ is the relative version of the Hill-Hopkins-Ravenel diagonal constructed in [@sixauthors]. We do not claim that it is an equivalence since $T$ is only flat rather than cofibrant. The first square commutes since it commutes after replacing $T$ cofibrantly. The second square commutes by naturality. Moreover the composite $\Phi^{C_p} (\epsilon) \circ \Delta$ is equal to the counit $\epsilon \colon N_{C_{p^{n-1}}}^{D_{p^{n-1}}} \Phi^{C_p} T \to \Phi^{C_p} T$. This can be seen by explicitly computing the adjoint of the latter composite and identifying it with the identity. Thus the lower horizontal composite is equal to the norm $$\xymatrix{N_{C_{p^{n-1}}}^{D_{p^{n-1}}} \colon \pi_0^{C_{p^{n-1}}} \Phi^{C_p} T \ar[r] & \pi_0^{D_{p^{n-1}}} \Phi^{C_p} T.}$$ Finally, the claim follows from the fact that the map $\delta \colon \Phi^{C_p} T \to T$ is a map of commutative $O(2)$-ring spectra, and therefore $\delta_\ast$ is compatible with the norms, and from the observation that $R_2$ and $R$ are the composites of $\delta_\ast$ and $\phi$.
\[Proof of Theorem \[pi0TRR\]\] We start by calculating the components of the fixed-points $$\pi_0(\operatorname{TRR}^{n+1}(E;p)^{\operatorname{\mathbb{Z}}/2})=\pi_0((\operatorname{THR}(E)^{C_{p^n}})^{\operatorname{\mathbb{Z}}/2})=\pi_0(\operatorname{THR}(E)^{D_{p^n}}),$$ using an argument analogous to [@Wittvect]. Let us denote the components of the underlying ring spectrum by $A:=\pi_0(E)$ and of the derived fixed-points by $B:=\pi_0(E^{\operatorname{\mathbb{Z}}/2})$. Let $\mathcal{R}$ be the family of subgroups of $D_{p^n}$ generated by the reflections, together with the trivial group. Let $E\mathcal{R}$ be a universal space for this family, for concreteness one could take the unit sphere $E\mathcal{R}=S(\mathbb{C}^{\infty})$ in the countably infinite direct sum of copies of $\mathbb{C}$, where $O(2)$ acts on $\mathbb{C}\cong \mathbb{R}^2$ by the standard action. Since $\mathcal{R}$ is the family of subgroups of $O(2)$ which do not contain $C_p$, using the Adams isomoprhism (see e.g., [@RV16]) and isotropy separation, we get a cofiber sequence of $D_{p^{n-1}}$-spectra $$E\mathcal{R}_+\wedge_{C_p} \operatorname{THR}(E)\longrightarrow \operatorname{THR}(E)^{C_p}\longrightarrow \Phi^{C_p}\operatorname{THR}(E).$$ (see [@Amalie] where this sequence for Bökstedt’s model is used). By postcomposing the second map with the equivalence $\delta \colon \Phi^{C_p}\operatorname{THR}(E)\stackrel{\simeq}{\to}\operatorname{THR}(E)$ and taking derived $C_{p^{n-1}}$-fixed points (and again using the Adams isomorphism), we obtain a fiber sequence of $\operatorname{\mathbb{Z}}/2$-spectra $$\xymatrix{E\mathcal{R}_+\wedge_{C_{p^{n}}} \operatorname{THR}(E)\ar[r] & \operatorname{THR}(E)^{C_{p^{n}}}\ar[r]^-{R} & \operatorname{THR}(E)^{C_{p^{n-1}}}.}$$ By the homotopy orbits spectral sequence induced by the standard filtration of $S(\mathbb{C}^{\infty})$, it is clear that $$\pi_0(E\mathcal{R}_+\wedge_{C_{p^{n}}} \operatorname{THR}(E))=\pi_0(\operatorname{THR}(E))_{C_{p^n}} \cong \pi_0(E)=A,$$ where the $C_{p^{n}}$ acts trivially on $A\cong \pi_0\operatorname{THH}(E)$ since the cyclic actions are restricted from the circle. A similar analysis on the spectral sequence converging to the homotopy groups of $(E\mathcal{R}_+\wedge_{C_{p^{n}}} \operatorname{THR}(E))^{\operatorname{\mathbb{Z}}/2}$ shows that there is an isomorphism $$\begin{aligned}
\pi_0(E\mathcal{R}_+\wedge_{C_{p^{n}}} \operatorname{THR}(E))^{\operatorname{\mathbb{Z}}/2}&\cong H^{\operatorname{\mathbb{Z}}/2}_0(E\mathcal{R}/C_{p^n};\underline{\pi}_0\operatorname{THR}(E))\cong H^{D_{p^n}}_0(E\mathcal{R};\underline{\pi}_0\operatorname{THR}(E))
\\
&\cong \operatorname*{colim}_{\mathcal{O}_R} \underline{\pi}^{(-)}_0\operatorname{THR}(E)\cong \pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}\end{aligned}$$ where the second isomorphism holds because $C_{p^n}$ acts freely on $E\mathcal{R}$. The Bredon homology group $H^{D_{p^n}}_0$ is computed as the colimit over the full subcategory $\mathcal{O}_R$ of the orbit category of $D_{p^{n}}$ generated by $\mathcal{R}$. The final isomorphism holds because when $p$ is odd there is only one conjugacy class of reflections in $D_{p^n}$. We recall that $\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}$ is a quotient of $B\otimes B$, which we denote by $B\otimes_{\phi} B$.
On homotopy groups the above fiber sequence induces a long exact sequence $$\xymatrix@C=15pt{
\dots \ar[r]&
\pi_1\operatorname{THR}(E)^{D_{p^{n-1}}}\ar[r]^-{\partial}&B\otimes_{\phi} B\ar[r]^-{V_{2}^n}&\pi_0\operatorname{THR}(E)^{D_{p^n}}\ar[r]^-{R_2}&\pi_0\operatorname{THR}(E)^{D_{p^{n-1}}}\ar[r]&0
}$$ where the map $V^{n}_2=\operatorname{tran}_{\operatorname{\mathbb{Z}}/2}^{D_{p^{n}}}$ is the transfer map from $\operatorname{\mathbb{Z}}/2$ to $D_{p^{n}}$. We claim that $V^{n}_2$ is injective, and therefore that the $\pi_0$ terms form a short exact sequence for every $n$. The restriction for the subgroup inclusion $\operatorname{\mathbb{Z}}/2\subset D_{p^n}$ defines a map $F_{2}^n\colon \pi_0\operatorname{THR}(E)^{D_{p^n}}\to B\otimes_{\phi} B$. By the double coset formula we see that $F_{2}^nV_{2}^n$ is the map $$F_{2}^nV_{2}^n=\sum_{[g]\in {_{\operatorname{\mathbb{Z}}/2}}\backslash D_{p^n}/_{\operatorname{\mathbb{Z}}/2}}\operatorname{tran}_{{}^g\operatorname{\mathbb{Z}}/2\cap \operatorname{\mathbb{Z}}/2}^{\operatorname{\mathbb{Z}}/2}c_g\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{\operatorname{\mathbb{Z}}/2\cap \operatorname{\mathbb{Z}}/2^g}=\operatorname{id}+\frac{(p^n-1)}{2}\operatorname{tran}_{1}^{\operatorname{\mathbb{Z}}/2}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{1}$$ where the conjugations are trivial since $C_{p^n}$ acts trivially on $\pi_\ast\operatorname{THH}(E)$. It follows that any element $x$ in the kernel of $V_{2}^n$ must satisfy $$x+\frac{(p^n-1)}{2}\operatorname{tran}_{1}^{\operatorname{\mathbb{Z}}/2}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{1}(x)=0.$$ Now let us consider the commutative square $$\xymatrix@C=60pt{
B\otimes_{\phi} B\ar[d]_{\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{1}}\ar[r]^-{V_{2}^n}&\pi_0\operatorname{THR}(E)^{D_{p^n}}\ar@<-1ex>[d]^{\operatorname{res}^{D_{p^n}}_{C_{p^n}}}
\\
A\ar@{>->}[r]^-{V^n}&\pi_0\operatorname{THR}(E)^{C_{p^n}}
}$$ where the bottom horizontal map $V^n=\operatorname{tran}^{C_{p^n}}_1$ is injective by [@Wittvect p. 53]. By the commutativity of this diagram if $x$ is in the kernel of $V_{2}^n$ we have that $\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{1}(x)=0$, and therefore by the formula above $$x=x+\frac{(p^n-1)}{2}\operatorname{tran}_{1}^{\operatorname{\mathbb{Z}}/2}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{1}(x)=0.$$ This shows that $V_{2}^n$ is injective.
We define maps $I^{n}_2\colon W_{n+1}(B\otimes_{\phi}B)\to \pi_0\operatorname{THR}(E)^{D_{p^n}}$ for every $n\geq 0$, by the formula $$I^{n}_2(x_0,\dots, x_n)=\sum_{i=0}^nV^{i}_2N^{p^{n-i}}_2(x_i),$$ where $N^{p^{n-i}}_2\colon =N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}\colon B\otimes_{\phi} B\cong \pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}\to \pi_0\operatorname{THR}(E)^{D_{p^{n-i}}}$ is a short notation for the norm. We claim that the following diagram commutes and that its rows are exact: $$\xymatrix@C=40pt{
0\ar[r]&W_{n}(B\otimes_{\phi} B)\ar[d]_{I^{n-1}_2}\ar[r]^-{V}&W_{n+1}(B\otimes_{\phi} B)\ar[r]^-{R^n}\ar[d]^{I_{2}^n}&B\otimes_{\phi} B\ar[d]^{I_{2}^0}_{\cong}\ar[r]&0
\\
0\ar[r]&\pi_0\operatorname{THR}(E)^{D_{p^{n-1}}}\ar[r]_{V_2}&\pi_0\operatorname{THR}(E)^{D_{p^{n}}}\ar[r]_-{R_{2}^n}&\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}\ar[r]&0\rlap{\ .}
}$$ The exactness of the top row follows from the definition of the Witt vectors. An inductive diagram chase using the latter exact sequence for $R_2$ shows that the lower row is also exact. The commutativity of the first square is clear. The second square commutes by the definition of $R_2$ and Lemma \[Rtambaramap\] (i). Further, again using the same lemma, we know that $N^{p^i}_2$ splits $R_2^i$. This implies inductively that the maps $I^{n}_2$ are all bijections, and it remains to show that $I^{n}_2$ is a ring homomorphism.
We follow the strategy of [@Wittvect] and define topological version of the ghost coordinates. That is, we consider the commutative diagram $$\xymatrix@C=70pt{
W_{n+1}(B\otimes_{\phi} B)\ar[r]^-{w}\ar[d]_-{I^{n}_2}^{\cong}&\prod_{i=0}^nB\otimes_{\phi} B
\\
\pi_0\operatorname{THR}(E)^{D_{p^{n}}}\ar[ur]_{\overline{w}}
}$$ where $\overline{w}$ is the ring homomorphism with components $\overline{w}_j=R_{2}^{n-j}\operatorname{res}^{D_{p^n}}_{D_{p^{n-j}}}=R_{2}^{n-j}F^{j}_2$. We prove that this diagram commutes in Lemma \[topghost\] below. If $B\otimes_{\phi} B$ is $p$-torsion free the ghost map $w$ is injective. Since $I^{n}_2$ is bijective $\overline{w}$ is also injective, and it is sufficient to show that $\overline{w}\circ I^{n}_2=w$ is a ring homomorphism. This is clear by the definition of the Witt vectors.
Now suppose that $B\otimes_{\phi} B$ possibly has $p$-torsion. In Lemma \[cohTambres\] we construct a $\operatorname{\mathbb{Z}}/2$-set $X$ and a map of cohomological Tambara functors $S:=\operatorname{\mathbb{Z}}[X]\twoheadrightarrow\underline{\pi}_0E$ which is pointwise surjective, where $\operatorname{\mathbb{Z}}[X]$ is regarded as a Tambara functor by the involution induced by the functoriality in $X$. Using the Eilenberg-MacLane functor $H$ from [@Ullman], we can form a homotopy pullback of commutative $\operatorname{\mathbb{Z}}/2$-ring spectra $$\xymatrix{\overline{E} \ar[d] \ar[r]^\epsilon & E \ar[d] \\ H \operatorname{\mathbb{Z}}[X] \ar@{->>}[r] & H\underline{\pi_0}E.}$$ Then $\underline{\pi}_0\overline{E}$ is isomorphic to the Tambara functor associated to $\operatorname{\mathbb{Z}}[X]$, and $\epsilon$ induces surjection of Tambara functors on $\underline{\pi}_0$. Let $\overline{B}$ denote $\pi_0 \overline{E}^{\operatorname{\mathbb{Z}}/2}$ and $\overline{A}$ denote $\pi_0 \overline{E}$ . By Lemma \[ptorfree\] below $\pi_0\operatorname{THR}( \overline{E})^{\operatorname{\mathbb{Z}}/2}=\overline{B}\otimes_\phi\overline{B}$ is torsion-free. The induced map $\overline{B}\otimes_\phi\overline{B}\to B\otimes_{\phi} B$ is also surjective, and we have a commutative diagram $$\xymatrix@C=70pt{
W_{n+1}(\overline{B}\otimes_\phi\overline{B})\ar[r]^-{I^{n}_2}\ar@{->>}[d]&\pi_0\operatorname{THR}(\overline{E})^{D_{p^n}}\ar[d]
\\
W_{n+1}(B\otimes_{\phi} B)\ar[r]_-{I^{n}_2}&\pi_0\operatorname{THR}(E)^{D_{p^n}}
}$$ where the vertical maps and the top horizontal map are ring homomorphisms, and where the left vertical map is surjective. It follows that the bottom horizontal map is also a ring homomorphism.
Let us now identify the $\operatorname{\mathbb{Z}}/2$-Tambara structure. Since the restriction map $\operatorname{res}$ and the involution $w\colon A\to A$ are ring homomorphisms, their induced maps on Witt vectors $W(\operatorname{res})$ and $W(w)$ are defined coordinatewise. We can therefore verify by direct calculation that the squares $$\xymatrix@R=20pt{
W_{n+1}(B\otimes_\phi B)\ar[d]_-{\cong}^-{I_{2}^n}\ar[r]^-{W(\operatorname{res})}&W_{n+1}(A)\ar[d]^{I_n}_{\cong}
\\
\pi_0\operatorname{THR}(E)^{D_{p^n}}\ar[r]_-{\operatorname{res}_{C_{p^n}}^{D_{p^n}}}&\pi_0\operatorname{THR}(E)^{C_{p^n}}
}
\ \ \ \ \ \ \ \ \
\xymatrix@R=20pt{
W_{n+1}(A)\ar[d]_-{\cong}^-{I^n}\ar[r]^-{W(w)}&W_{n+1}(A)\ar[d]^{I_n}_{\cong}
\\
\pi_0\operatorname{THR}(E)^{C_{p^n}}\ar[r]_-{c_{r}}&\pi_0\operatorname{THR}(E)^{C_{p^n}}
}$$ commute, where $c_{r}$ is conjugation by the preferred reflection $r=(0,\tau)\in D_{p^n}$, where $\tau$ is the generator of $\operatorname{\mathbb{Z}}/2$. The commutativity of the first square is obtained by the double coset formula for the additive and multiplicative transfers $$\begin{aligned}
\operatorname{res}_{C_{p^n}}^{D_{p^n}} I^{n}_2(a_0,\dots, a_n)&=\operatorname{res}_{C_{p^n}}^{D_{p^n}}\sum_{i=0}^n\operatorname{tran}_{D_{p^{n-i}}}^{D_{p^n}}N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)=\sum_{i=0}^n\operatorname{tran}_{C_{p^{n-i}}}^{C_{p^n}}\operatorname{res}^{D_{p^{n-i}}}_{C_{p^{n-i}}}N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)
\\&=\sum_{i=0}^n\operatorname{tran}_{C_{p^{n-i}}}^{C_{p^n}}N^{C_{p^{n-i}}}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}(a_i)=I^n(\operatorname{res}a_0,\dots, \operatorname{res}a_n),\end{aligned}$$ where we used that the double cosets $C_{p^n}\backslash D_{p^n}/D_{p^{n-i}}$ and $C_{p^{n-i}}\backslash D_{p^{n-i}}/\operatorname{\mathbb{Z}}/2$ are trivial. The second square commutes because conjugations commute with transfers and norms $$\begin{aligned}
c_r I^{n}(a_0,\dots, a_n)&=c_r\sum_{i=0}^n\operatorname{tran}_{C_{p^{n-i}}}^{C_{p^n}}N^{C_{p^{n-i}}}_{e}(a_i)=\sum_{i=0}^n\operatorname{tran}_{C_{p^{n-i}}}^{C_{p^n}}c_rN^{C_{p^{n-i}}}_{e}(a_i)
\\&=\sum_{i=0}^n\operatorname{tran}_{C_{p^{n-i}}}^{C_{p^n}}N^{C_{p^{n-i}}}_{e}(c_r a_i)=I^n(c_r a_0,\dots, c_ra_n)=I^n(w(a_0),\dots, w(a_n)),\end{aligned}$$ where we used that $C_{p^i}$ is normal in $D_{p^n}$.
Since the norm map $W(N)$ is not defined componentwise we are not able to directly show that $W(N)$ and $N_{C_{p^n}}^{D_{p^n}}$ coincide. Instead, we show that these agree in ghost components and conclude by reducing to the universal case. We observe that since the transfer is determined by the norm, this will conclude the proof. We show that the outer part of the diagram $$\xymatrix@R=5pt@C=60pt{
W_{n+1}(A)\ar[dd]_-{\cong}^-{I^n}\ar[r]^-{W(N)}&W_{n+1}(B\otimes_\phi B)\ar[dd]_{\cong}^{I_{2}^n}\ar[dr]^{w}
\\
&&\prod_{i=0}^nB\otimes_\phi B
\\
\pi_0\operatorname{THR}(E)^{C_{p^n}}\ar[r]_-{N_{C_{p^n}}^{D_{p^n}}}&\pi_0\operatorname{THR}(E)^{D_{p^n}}\ar[ur]_-{\overline{w}}
}$$ commutes. By construction (Theorem \[WittTambara\]), $w\circ W(N)=(\prod N)\circ w$, and the lower composite has components $$\begin{aligned}
\overline{w}_jN_{C_{p^n}}^{D_{p^n}}I^{n}&=R_{2}^{n-j}\operatorname{res}^{D_{p^n}}_{D_{p^{n-j}}}N_{C_{p^n}}^{D_{p^n}}I^{n}
=\operatorname{res}^{D_{p^j}}_{\operatorname{\mathbb{Z}}/2}R_{2}^{n-j}N_{C_{p^n}}^{D_{p^n}}I^{n}=\operatorname{res}^{D_{p^j}}_{\operatorname{\mathbb{Z}}/2}N_{C_{p^j}}^{D_{p^j}}R^{n-j}I^n,\end{aligned}$$ where we use Lemma \[Rtambaramap\] and that $R_2$ is induced by the fixed-points of the map $R$ of equivariant spectra. By applying the double coset formula for the norm we obtain $$\begin{aligned}
\operatorname{res}^{D_{p^j}}_{\operatorname{\mathbb{Z}}/2}N_{C_{p^j}}^{D_{p^j}}R^{n-j}I^n&=\prod_{g\in \operatorname{\mathbb{Z}}/2\backslash D_{p^j}/C_{p^j}} N_{\operatorname{\mathbb{Z}}/2 \cap C_{p^j}^g}^{\operatorname{\mathbb{Z}}/2}c_g\operatorname{res}^{C_{p^j}}_{\operatorname{\mathbb{Z}}/2^g \cap C_{p^j}}R^{n-j}I^n=N_{e}^{\operatorname{\mathbb{Z}}/2}\operatorname{res}^{C_{p^j}}_{e}R^{n-j}I^n
\\
&=N_{e}^{\operatorname{\mathbb{Z}}/2}\overline{w}_jI^n=N_{e}^{\operatorname{\mathbb{Z}}/2}w_j,\end{aligned}$$ where the last equality is from [@Wittvect Theorem 2.3]. Since the triangle in the diagram above commutes by Lemma \[topghost\] below, this proves the claim when $B\otimes_\phi B$ is $p$-torsion free. In general, the resolution $\epsilon\colon \overline{E}\to E$ above induces a diagram $$\xymatrix@C=25pt@R=10pt{
W_{n+1}(\overline{A})\ar@{->>}[dr]^{\epsilon}\ar[dd]_-{\cong}^-{I^n}\ar[rr]^-{W(N)}&&W_{n+1}(\overline{B}\otimes_\phi\overline{B})\ar[dd]_(.3){\cong}^(.3){I_{2}^n }\ar[dr]^{\epsilon}
\\
&W_{n+1}(A)\ar[rr]^(.3){W(N)}\ar[dd]_(.3){I^n}&&W_{n+1}(B\otimes_\phi B)\ar[dd]^{I_{2}^n}
\\
\pi_0\operatorname{THR}(\overline{E})^{C_{p^n}}\ar[dr]^{\epsilon}\ar[rr]_(.65){N_{C_{p^n}}^{D_{p^n}}}&&\pi_0\operatorname{THR}(\overline{E})^{D_{p^n}}\ar[dr]^{\epsilon}
\\
&\pi_0\operatorname{THR}(E)^{C_{p^n}}\ar[rr]_-{N_{C_{p^n}}^{D_{p^n}}}&&\pi_0\operatorname{THR}(E)^{D_{p^n}}\rlap{\ .}
}$$ The top and bottom faces commute since $\epsilon$ induces a morphism of Tambara functors. The side faces commute by naturality of $I^{n}$ and $I^{n}_2$. The back face commutes by the argument above, since $\overline{B}\otimes_\phi\overline{B}$ is torsion free. Since the maps $\epsilon$ are surjective the front face commutes as well.
\[ptorfree\] Let $X$ be a $\operatorname{\mathbb{Z}}/2$-set. The abelian group $\pi_0\operatorname{THR}(\mathbb{Z}[X])^{\operatorname{\mathbb{Z}}/2}$ is $p$-torsion free for every odd prime $p$.
We observe that $\mathbb{Z}[X]$ is the monoid-ring on the free commutative monoid $M(X)$ generated by the set $X$, with the involution induced functorially by the involution on $X$. It follows from [@THRmodels Proposition 5.12] that the real topological Hochschild homology spectrum of $\operatorname{\mathbb{Z}}[X]$ decomposes as $$\operatorname{THR}(\mathbb{Z}[X])\simeq \operatorname{THR}(\operatorname{\mathbb{Z}})\wedge \Sigma^{\infty}N^{di}M(X)_+$$ where $N^{di}$ is the dihedral nerve with respect to the product of spaces. In particular $$\underline{\pi}_0\operatorname{THR}(\mathbb{Z}[X])=\underline{\pi}_0\operatorname{THR}(\mathbb{Z})\Box \underline{\pi}_0(\Sigma^{\infty}N^{di}M(X)_+)\cong \underline{\operatorname{\mathbb{Z}}}\Box \underline{\pi}_0(\Sigma^{\infty}N^{di}M(X)_+),$$ where the identification of $\underline{\pi}_0\operatorname{THR}(\mathbb{Z})$ with the constant Mackey functor $\underline{\operatorname{\mathbb{Z}}}$ is in [@THRmodels Corollary 5.2]. We observe that the underlying group $\pi_0\Sigma^{\infty}N^{di}M(X)_+=\operatorname{\mathbb{Z}}[X]$ is torsion-free. Thus the result follows from the following general claim: if $L=(\xymatrix{A\ar@<.5ex>[r]^{\operatorname{tran}}&B\ar@<.5ex>[l]^{\operatorname{res}}})$ is a $\operatorname{\mathbb{Z}}/2$-Mackey functor such that $A$ is $p$-torsion-free, then the box product Mackey functor $\underline{\operatorname{\mathbb{Z}}}\Box L$ is $p$-torsion free for every odd prime $p$.
Clearly the value at the trivial group $(\underline{\operatorname{\mathbb{Z}}}\Box L)(e)=\mathbb{Z}\otimes A\cong A$ is $p$-torsion-free by assumption. The value at $\operatorname{\mathbb{Z}}/2$ is the abelian group $$(\underline{\operatorname{\mathbb{Z}}}\Box L)(\operatorname{\mathbb{Z}}/2)=(A\oplus B)/I,$$ where $I$ is the ideal generated by the elements of the form $2b-\operatorname{res}(b)$, $a-\operatorname{tran}(a)$, $\tau a-a$, for every $a\in A$ and $b\in B$, where $\tau$ is the involution of $A$. We notice that the second relation collapses the $A$-summand, and that the box product has value isomorphic to $$(\underline{\operatorname{\mathbb{Z}}}\Box L)(\operatorname{\mathbb{Z}}/2)\cong B/J$$ where $J$ is generated by the elements $2b-\operatorname{tran}\operatorname{res}(b)$. Suppose that $b\in B/J$ is $p$-torsion for some odd prime $p$. The restriction map $\operatorname{res}\colon B/J\to A$ is additive, and since $A$ is $p$-torsion-free we must have $\operatorname{res}(b)=0$. It follows that in $B/J$ $$0=2b-\operatorname{tran}\operatorname{res}(b)=2b,$$ that is that $b$ is also $2$-torsion. Since $p$ is odd $b=0$.
\[topghost\] For every odd prime $p$ and connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum $E$ with $\underline{\pi}_0E$ cohomological, we have $\overline{w}I^{n}_2=w$.
First we observe that since $R^{n-j}_2$ is induced by a map of $O(2)$-spectra it commutes with transfers, in the sense that $$R^{n-j}_2\operatorname{tran}_{D_{p^{n-i}}}^{D_{p^n}}=\operatorname{tran}_{D_{p^{j-i}}}^{D_{p^{j}}}R^{n-j}_2$$ if $i\leq j$. Since $R^{n-j}$ is induced by the canonical map to the geometric fixed-points, which kills the proper transfers, one can directly verify that $R^{n-j}_2\operatorname{tran}_{D_{p^{n-i}}}^{D_{p^n}}=0$ if $i>j$. It also commutes with restrictions, and therefore $$\begin{aligned}
\overline{w}_jI^{n}_2(a_0,\dots,a_n)&=R_{2}^{n-j}\operatorname{res}^{D_{p^n}}_{D_{p^{n-j}}}\sum_{i=0}^n\operatorname{tran}_{D_{p^{n-i}}}^{D_{p^n}}N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)
\\
&=\operatorname{res}^{D_{p^{j}}}_{\operatorname{\mathbb{Z}}/2}R_{2}^{n-j}\sum_{i=0}^n\operatorname{tran}_{D_{p^{n-i}}}^{D_{p^n}}N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)
\\
&=\operatorname{res}^{D_{p^{j}}}_{\operatorname{\mathbb{Z}}/2}\sum_{i=0}^j\operatorname{tran}_{D_{p^{j-i}}}^{D_{p^{j}}}R_{2}^{n-j}N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i).\end{aligned}$$ Moreover by Lemma \[Rtambaramap\] we have that $R_{2}^{n-j}N^{D_{p^{n-i}}}_{\operatorname{\mathbb{Z}}/2}=N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}$ for $i\leq j$. It follows that $$\begin{aligned}
\overline{w}_jI^{n}_2(a_0,\dots,a_n)=\operatorname{res}^{D_{p^{j}}}_{\operatorname{\mathbb{Z}}/2}\sum_{i=0}^j\operatorname{tran}_{D_{p^{j-i}}}^{D_{p^{j}}}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i).\end{aligned}$$ Now we apply the double coset formula for restrictions and transfers: $$\begin{aligned}
\overline{w}_jI^{n}_2(a_0,\dots,a_n)&=\sum_{i=0}^j\sum_{[g]\in (\operatorname{\mathbb{Z}}/2)\backslash D_{p^j}/D_{p^{j-i}}}\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{\operatorname{\mathbb{Z}}/2\cap (D_{p^{j-i}})^g}c_g\operatorname{res}^{D_{p^{j-i}}}_{(\operatorname{\mathbb{Z}}/2)^g\cap D_{p^{j-i}}}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i).\end{aligned}$$ The set of double cosets is isomorphic to the quotient $(\operatorname{\mathbb{Z}}/2)\backslash C_{p^i}$ of the inversion action, which has representatives $\{1,\theta,\theta^2,\dots, \theta^{(p^i-1)/2}\}$, where $\theta=\sigma^{p^{j-i}}$ for $\sigma \in C_{p^j}$ a generator. The intersection $(\operatorname{\mathbb{Z}}/2)^g\cap D_{p^{j-i}}$ is equal to $\operatorname{\mathbb{Z}}/2$ if $g=1$, and to the trivial group otherwise (since $p$ is odd). Moreover the conjugations $c_g$ are trivial for the elements of the cyclic group. Therefore we have $$\begin{aligned}
\overline{w}_jI^{n}_2(a_0,\dots,a_n)&=\sum_{i=0}^j (\operatorname{res}^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2} N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)+ (p^i-1)/2\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{D_{p^{j-i}}}_{e}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i))
\\
&=\sum_{i=0}^j (\operatorname{res}^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2} N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)+ (p^i-1)/2\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i))
\\&=\sum_{i=0}^j p^i\operatorname{res}^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i).\end{aligned}$$ where the last equality holds since $\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}(a)=\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}(1)\cdot a$, and $\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}(1)=2$ (since $\underline{\pi}_0(E)$ and hence $\underline{\pi}_0\operatorname{THR}(E)$ is cohomological). Similarly, by applying the double coset formula for the norm we have that $$\begin{aligned}
\operatorname{res}^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)&=\prod_{[g]\in (\operatorname{\mathbb{Z}}/2)\backslash D_{p^{j-i}}/(\operatorname{\mathbb{Z}}/2)}N^{\operatorname{\mathbb{Z}}/2}_{\operatorname{\mathbb{Z}}/2 \cap (\operatorname{\mathbb{Z}}/2)^g}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{(\operatorname{\mathbb{Z}}/2)^g \cap \operatorname{\mathbb{Z}}/2}(a_i)
\\&=a_i(N^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}(a_i))^{(p^{j-i}-1)/2}.\end{aligned}$$ Similarly since $\underline{\pi}_0\operatorname{THR}(E)$ is cohomological we have that $N^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}(a_i)=a_{i}^2$, and thus $$\begin{aligned}
\overline{w}_jI^{n}_2(a_0,\dots,a_n)&=\sum_{i=0}^j p^i\operatorname{res}^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}N^{D_{p^{j-i}}}_{\operatorname{\mathbb{Z}}/2}(a_i)=\sum_{i=0}^j p^ia_{i}^{p^{j-i}}=w_j(a_0,\dots,a_n).\qedhere\end{aligned}$$
Let $E$ be a connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum, such that $\underline{\pi}_0E$ is cohomological. Then the Green functor $\underline{\pi_0}\operatorname{TRR}(E;p)$ admits a structure of Tambara functor, and the isomorphisms of Theorem \[pi0TRR\] induce an isomorphism of Tambara functors $$\underline{\pi_0}\operatorname{TRR}(E;p)\cong W(\underline{\pi}_0\operatorname{THR}(E))$$ for every odd prime $p$.
By the proof of Theorem \[pi0TRR\] the connecting homomorphism $\partial\colon \pi_1\operatorname{THR}(E)^{D_{p^{n-1}}}\to \pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}$ is zero, and the $R$ maps induce surjective group homomorphisms in $\pi_1$. Thus the Mittag-Leffler condition is satisfied and there are induced ring isomorphisms $$\pi_0\operatorname{TRR}(E)^{\operatorname{\mathbb{Z}}/2}\cong \lim_n \pi_0\operatorname{THR}(E)^{D_{p^n}}\cong W(B\otimes_\phi B).$$ Combining this with the results of [@Wittvect], yields an isomorphism of Green functors $$\underline{\pi}_0 \operatorname{TRR}(E) \cong \lim_n \underline{\pi}_0 \operatorname{TRR}^n(E).$$ The right hand side of this isomorphism is canonically a Tambara functor since $\operatorname{TRR}^n(E)$ are $\operatorname{\mathbb{Z}}/2$-equivariant commutative ring spectra and the $R$ maps are compatible with the norms by Lemma \[Rtambaramap\]. Since limits of Tambara and Green functors are computed pointwise, $\underline{\pi}_0 \operatorname{TRR}(E)$ inherits a norm which defines a Tambara functor. The rest follows from Theorem \[pi0TRR\].
Let us now address the case where the flat commutative $\operatorname{\mathbb{Z}}/2$-equivariant orthogonal ring spectrum $E$ has a Tambara functor of components $\underline{\pi}_0E$ which is not necessarily cohomological. For any $\operatorname{\mathbb{Z}}/2$-Tambara functor $T=(\xymatrix@C=15pt{A \ar@<.5ex>[r]\ar@<-.5ex>[r]&B\ar[l]})$ and odd prime $p$, we define twisted ghost coordinates $\tilde{w}_j\colon \prod_{i=0}^{n}B\to B$ by the formula $$\tilde{w}_j(x_0,\dots,x_n):=\sum_{i=0}^j(1+\frac{(p^i-1)}{2}\operatorname{tran}(1))x_i(N\operatorname{res}(x_i))^{\frac{p^{j-i}-1}{2}},$$ for all $0\leq j<n+1$. When $T$ is cohomological this is the usual ghost map $w_j$ of the Witt vectors of the commutative ring $B$. If $E$ is a connective commutative $\operatorname{\mathbb{Z}}/2$-equivariant orthogonal ring spectrum, we denote by $A:=\pi_0 E$ and $B:=\pi_0 E^{\operatorname{\mathbb{Z}}/2}$.
\[THRTamb\] Let $E$ be a connective $\operatorname{\mathbb{Z}}/2$-equivariant flat commutative orthogonal ring spectrum, and $p$ an odd prime. There is a unique ring structure $\tilde{W}_{n+1}(B\otimes_\phi B)$ on the set $\prod_{i=0}^{n}B\otimes_\phi B$ such that the maps $\tilde{w}_j$ are natural ring homomorphisms, and a natural ring isomorphism $$\pi_0\operatorname{THR}(E)^{D_{p^n}}\cong \tilde{W}_{n+1}(B\otimes_\phi B)$$ for every $1\leq n\leq \infty$.
The proof of the bijectivity of $I_{2}^n\colon \prod_{i=0}^{n}B\otimes_\phi B\to \pi_0\operatorname{THR}(E)^{D_{p^n}}$ from the proof of Theorem \[pi0TRR\] does not use that $\underline{\pi}_0E$ is cohomological. Moreover the calculation of Lemma \[topghost\] shows in fact that the topological ghost coordinates correspond to the twisted algebraic ghost maps, that is $\overline{w}_jI_{2}^n=\tilde{w}_j$. The maps $\overline{w}_j=R_{2}^{n-j}F^{j}_2$ are natural ring homomorphisms. It is therefore sufficient to show that a ring structure on $\pi_0\operatorname{THR}(E)^{D_{p^n}}$ such that the maps $\overline{w}_j$ are ring homomorphisms is unique.
The product $\overline{w}$ of the maps $\overline{w}_j$ for $T:=\operatorname{THR}(E)$ fits into a commutative diagram $$\xymatrix@C=60pt@R=13pt{
\pi_0T^{D_{p^n}}\ar[r]^-{\Psi} \ar[d]_{\overline{w}}&\prod_{i=0}^{n}(\pi_0\Phi^{C_{p^i}}T\times \pi_0\Phi^{D_{p^i}}T)\ar[d]_{\cong}^{\delta}
\\
\prod_{i=0}^{n}\pi_0T^{\operatorname{\mathbb{Z}}/2}\ar[r]&\prod_{i=0}^{n}(\pi_0T\times \pi_0\Phi^{\operatorname{\mathbb{Z}}/2}T)
}$$ where $\delta$ is the product of the cyclotomic structure maps. The top horizontal map has components the composites of the restrictions from $D_{p^n}$ to $C_{p^i}$, and from $D_{p^n}$ to $D_{p^i}$, with the canonical projections to the geometric fixed-points. The bottom horizontal map is the product of the map $\pi^{\operatorname{\mathbb{Z}}/2}_0T\to \pi_0T\times \pi_0\Phi^{\operatorname{\mathbb{Z}}/2}T$ which is the restriction on the first factor and the canonical projection on the second factor. If the top horizontal map $\Psi$ is injective, then the map $\overline{w}$ is injective, and the ring structure on $\pi^{D_{p^n}}_0T$ is unique. We show that $E$ can be resolved by a commutative $\operatorname{\mathbb{Z}}/2$-ring spectrum whose map $\Psi$ is injective.
Given $E$, we resolve the Eilenberg-MacLane ring spectrum of the Tambara functor $\underline{\pi}_0E$ by the free (genuine) $\operatorname{\mathbb{Z}}/2$-$E_\infty$-algebra $\mathbb{P}(\Sigma^{\infty} Z_{+})$ of some $\operatorname{\mathbb{Z}}/2$-CW complex $Z$, and form the homotopy pullback of commutative $\operatorname{\mathbb{Z}}/2$-ring spectra $$\xymatrix@C=60pt@R=13pt{\overline{E} \ar[r] \ar[d] & E \ar[d] \\ \mathbb{P}(\Sigma^{\infty} Z_{+}) \ar[r] & H \underline{\pi}_0E. }$$ Here the vertical maps induce isomorphisms on $\underline{\pi}_0$ and the horizontal maps induce surjections. We also note that the free $\operatorname{\mathbb{Z}}/2$-$E_{\infty}$-algebra $\mathbb{P}(\Sigma^{\infty} Z_{+})$ can be modeled via a strictly commutative flat $\operatorname{\mathbb{Z}}/2$-equivariant ring spectrum and as an associative algebra it is stably equivalent to the spherical monoid-ring $\mathbb{S}\wedge M_+$ of a monoid with anti-involution $M$. The real topological Hochschild homology of the spherical monoid-ring $\operatorname{THR}(\mathbb{S}\wedge M_+)$ is a suspension spectrum by [@Amalie] and [@THRmodels 5.12], and it follows from [@Schwedeglobal 3.3.15] that the map $\Psi$ of $\mathbb{P}(\Sigma^{\infty} Z_{+})$ is injective. By the naturality of $\Psi$ the diagram $$\xymatrix@C=50pt@R=13pt{\pi_0 \operatorname{THR}(\overline{E})^{D_{p^n}} \ar[d]_-{\cong} \ar[r]^-{\Psi} & \prod_{i=0}^{n}(\pi_0\Phi^{C_{p^i}}\operatorname{THR}(\overline{E})\times \pi_0\Phi^{D_{p^i}}\operatorname{THR}(\overline{E})) \ar[d] \\ \pi_0 \operatorname{THR}(\mathbb{P}(\Sigma^{\infty} Z_{+}))^{D_{p^n}} \ar@{>->}[r]^-{\Psi} & \prod_{i=0}^{n}(\pi_0\Phi^{C_{p^i}}\operatorname{THR}(\mathbb{P}(\Sigma^{\infty} Z_{+}))\times \pi_0\Phi^{D_{p^i}}\operatorname{THR}(\mathbb{P}(\Sigma^{\infty} Z_{+}))) }$$ commutes. The left vertical map is an isomorphism since we already know that $I_2^n$ is a bijection. Hence we conclude that $\Psi$ is injective for $\overline{E}$ as well. Finally, since $\overline{E} \to E$ induces a surjection on $\underline{\pi}_0$, we get a surjective ring homomorphism $\pi_0\operatorname{THR}(\overline{E})^{\operatorname{\mathbb{Z}}/2}\cong \overline{B}\otimes_\phi \overline{B} \twoheadrightarrow B\otimes_\phi B\cong\pi_0\operatorname{THR}(E)^{\operatorname{\mathbb{Z}}/2}$, and hence a surjective ring homomorphism $$\pi_0\operatorname{THR}(\overline{E})^{D_{p^n}}\cong \prod_{i=0}^{n}\overline{B}\otimes_\phi \overline{B} \longrightarrow \prod_{i=0}^{n}B\otimes_\phi B\cong\pi_0\operatorname{THR}(E)^{D_{p^n}}.$$ Thus $\pi_0\operatorname{THR}(E)^{D_{p^n}}$ is a quotient of $\pi_0\operatorname{THR}(\overline{E})^{D_{p^n}}$ and the ring structure of the former is determined by the latter.
\[different-witt\] We demonstrate on an explicit example that the rings $\tilde{W}_{m+1}(B\otimes_\phi B)$ and $W_{m+1}(B\otimes_\phi B)$ are different in general. Consider the sphere spectrum $\mathbb{S}$ with the trivial $\operatorname{\mathbb{Z}}/2$-action. We know that $\operatorname{THR}(\mathbb{S})=\mathbb{S}$ as real cyclotomic spectra and therefore $$\pi_0\operatorname{TRR}^{m+1}(\mathbb{S};p)^{\operatorname{\mathbb{Z}}/2}=\pi_0\mathbb{S}^{D_{p^m}}\cong \mathbb{A}(D_{p^m})$$ is the Burnside ring of the dihedral group. Hence $\tilde{W}_{m+1}(B\otimes_\phi B)$ in this case is just $\mathbb{A}(D_{p^m})$. We claim that this ring is not isomorphic to $W_{m+1}(\mathbb{A}(\operatorname{\mathbb{Z}}/2))$ for $m>0$. To see this we check that the groups of units $\mathbb{A}(D_{p^m})^{*}$ and $W_{m+1}(\mathbb{A}(\operatorname{\mathbb{Z}}/2))^{*}$ are different.
The unit group of the ring $\mathbb{A}(\operatorname{\mathbb{Z}}/2)=\operatorname{\mathbb{Z}}[x]/(x^2-2x)$ is isomorphic to $\operatorname{\mathbb{Z}}/2 \times \operatorname{\mathbb{Z}}/2$, containing the elements $\pm 1$ and $\pm (x-1)$, where $x=\operatorname{tran}(1)$. The ring $\mathbb{A}(\operatorname{\mathbb{Z}}/2)$ is torsion-free. Hence the ghost map $$w \colon W_{m+1}(\mathbb{A}(\operatorname{\mathbb{Z}}/2)) \to \prod_{m+1} \mathbb{A}(\operatorname{\mathbb{Z}}/2)$$ is injective. Thus $W_{m+1}(\mathbb{A}(\operatorname{\mathbb{Z}}/2))^{*}$ injects into the units $(\prod_{m+1} \mathbb{A}(\operatorname{\mathbb{Z}}/2))^{*}$. Now using Dwork’s Lemma one can see that the image of the latter injection is isomorphic to $\operatorname{\mathbb{Z}}/2 \times \operatorname{\mathbb{Z}}/2$. Indeed, since $p$ is odd, the identity map of $\mathbb{A}(\operatorname{\mathbb{Z}}/2)$ is a Frobenius lift. This implies that a tuple $(x_0, \dots, x_m)$ is in the image of $w$ if and only if $x_{i-1} \equiv x_i \mod p^i$ for all $1 \leq i \leq m$. The latter condition implies that a tuple consisting of the elements of $\{\pm 1, \; \pm (x-1) \}$ is in the image of $w$ if and only if all the elements are equal. We conclude that $W_{m+1}(\mathbb{A}(\operatorname{\mathbb{Z}}/2))^{*} \cong \operatorname{\mathbb{Z}}/2 \times \operatorname{\mathbb{Z}}/2$. We can in fact explicitly check that all the units of $W_{m+1}(\mathbb{A}(\operatorname{\mathbb{Z}}/2))$ are given by the Teichmüler lifts of the four units of $\mathbb{A}(\operatorname{\mathbb{Z}}/2)$: $$\{\pm 1, \; \pm [x-1] \}.$$
On the other hand the unit group $\mathbb{A}(D_{p^m})^{*}$ is isomorphic to $(\operatorname{\mathbb{Z}}/2)^{m+2}$ (see e.g., [@BolPf]). Each subgroup of $C_{p^m}$ gives two units and additionally we also have $\pm 1$. The group $C_{p^m}$ has $(m+1)$ subgroups and in total we get $(\operatorname{\mathbb{Z}}/2)^{m+2}$ as the unit group. One can make these elements more explicit when $m=1$. The eight units of $\mathbb{A}(D_{p})$ are given by: $$\{\pm 1, \; \pm (1+[D_p]-[D_p/C_p]-2[D_p/\operatorname{\mathbb{Z}}/2]), \; \pm (1+[D_p]-2[D_p/\operatorname{\mathbb{Z}}/2]), \; \pm (1-[D_p/C_p]) \}.$$
The free Z/2-Tambara functor on a presheaf of sets
==================================================
Let $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{Z}}/2}$ denote the orbit category of $\operatorname{\mathbb{Z}}/2$. We compute the left adjoint of the forgetful functor that sends a Tambara functor to the underlying $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{Z}}/2}$-diagram of sets. We let $M(X)$ denote the free multiplicative abelian monoid generated by a set $X$ and $\operatorname{\mathbb{Z}}(M(X))$ its monoid ring, so that the polynomial ring $\operatorname{\mathbb{Z}}[X]=\operatorname{\mathbb{Z}}(M(X))$. More generally, we denote by $\operatorname{\mathbb{Z}}(-)$ the free abelian group functor. We define a Tambara functor $$\xymatrix@C=40pt{\mathbb{A}[X;Y]:=\big(
\operatorname{\mathbb{Z}}[X] \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \operatorname{\mathbb{Z}}(M(Y\amalg X/\operatorname{\mathbb{Z}}/2))\oplus \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2) \ar[l]|-{\operatorname{res}}
\big),}$$ where $M(X)^{free}=M(X)\setminus M(X)^{\operatorname{\mathbb{Z}}/2}$. We represent an element in the first summand as a linear combination of monomials of the form $m(x)g(y)m(\overline{x})$ where $g$ is a monomial in $Y$ and $m$ is a monomial in a set of representatives of the orbits of $X$. An element in the second summand is represented by a linear combination of monomials of the form $m(x)k(x')m(\overline{x})$ where $k$ is a monomial in the fixed-points set $X^{\operatorname{\mathbb{Z}}/2}$. An element in the third summand is represented by a linear combination of formal sums $h(x)+h(\overline{x})$ where $h$ is a monomial in $X$ which is not fixed by the involution ($h(\overline{x})+h(x)$ and $h(x)+h(\overline{x})$ are identified).
\[remA\] The motivation behind this definition is the following. First one can form the free semi-Mackey-functor on $X\leftarrow Y\colon \operatorname{res}$ by taking the free commutative monoids on $X$ and $Y$ and freely add a norm. This is the diagram $$\xymatrix{
M(X;Y)=\big(M(X) \ar@<-1ex>[r]_-{N}&M(Y\amalg X/{\operatorname{\mathbb{Z}}/2}) \ar[l]_-{\operatorname{res}}\big),
}$$ where the restriction is induced by the restriction on $Y$ and the map $X/{\operatorname{\mathbb{Z}}/2}\to X$ that sends $[x]$ to $x\overline{x}$. The norm is induced by the projection $X\to X/{\operatorname{\mathbb{Z}}/2}$. The free Tambara functor on this diagram is heuristically $\pi_0$ of the “spherical monoid ring $\mathbb{S}\wedge M(X;Y)$”. The underlying abelian group of components is then $\operatorname{\mathbb{Z}}[X]$. The fixed-points can be additively calculated using the tom Dieck splitting as $$\pi_0(\mathbb{S}\wedge (M(X;Y)^{\operatorname{\mathbb{Z}}/2}))\oplus \pi_0(\mathbb{S}\wedge M(X;Y))_{h\operatorname{\mathbb{Z}}/2}=\operatorname{\mathbb{Z}}(M(Y\amalg X/{\operatorname{\mathbb{Z}}/2}) )\oplus (\operatorname{\mathbb{Z}}[X]/{\operatorname{\mathbb{Z}}/2}),$$ where the quotient in the second summand is a quotient of abelian groups. This is therefore isomorphic to $$\operatorname{\mathbb{Z}}[X]/{\operatorname{\mathbb{Z}}/2}=\operatorname{\mathbb{Z}}(M(X)/\operatorname{\mathbb{Z}}/2)=\operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2),$$ hence the formula. In order to calculate the multiplicative structure from this formula rigorously, one needs to find an actual (or homotopy coherent) topological commutative monoid with involution $M$, such that the associated semi-Mackey functor of components is isomorphic to $M(X;Y)$. We have not investigated if this is possible. Instead of doing this, we directly define the Tambara structure on $\mathbb{A}[X;Y]$ keeping the latter heuristics in mind.
The commutative multiplication on $\operatorname{\mathbb{Z}}(M(Y\amalg X/\operatorname{\mathbb{Z}}/2))\oplus \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2)$ is defined on additive generators by the following multiplication table:
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\bullet$ $mg\overline{m}$ $mk\overline{m}$ $h+\overline{h}$
--------------------- ------------------------ -------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------
$m'g'\overline{m}'$ $mm'gg'\overline{mm'}$ $mm'\operatorname{res}(g')k\overline{mm'}$ $m'\operatorname{res}(g')\overline{m}'h+\overline{m'\operatorname{res}(g')\overline{m}'h}$
$m'k'\overline{m}'$ $2mm'kk'\overline{mm'}$ $2(m'k'h\overline{m}'+\overline{m'k'h\overline{m}'})$
$h'+\overline{h}'$ $\begin{array}{ll}
(hh'+\overline{hh'})+(h\overline{h'}+\overline{h\overline{h'}})& \mbox{if }h\overline{h'}\neq \overline{h\overline{h'}}, \ hh' \neq \overline{hh'}
\\
hh'+(h\overline{h'}+\overline{h\overline{h'}})&\mbox{if }h\overline{h'} \neq \overline{h\overline{h'}}, \ hh'= \overline{hh'}
\\
(hh'+\overline{hh'})+h\overline{h'}& \mbox{if } h\overline{h'}= \overline{h\overline{h'}}, \ hh' \neq \overline{hh'}
\end{array}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
where the elements $hh'$ and $h\overline{h'}$ in the last two cases of the bottom right case belong to the second summand. Thus, we see that $\operatorname{\mathbb{Z}}(M(Y\amalg X/\operatorname{\mathbb{Z}}/2))$ is a subring and that $ \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2)$ is an ideal. The restriction, transfer and norm maps are defined on additive generators by the formulas $$\begin{aligned}
\operatorname{res}(mg\overline{m})=m\operatorname{res}(g)\overline{m} &&\operatorname{res}(mk\overline{m})=2mk\overline{m}&&\operatorname{res}(h+\overline{h})=h+\overline{h}
\\
&&\operatorname{tran}(k)=k&&\operatorname{tran}(h)=h+\overline{h}
\\
N(m)=m1\overline{m}\end{aligned}$$ where $k$ is a fixed monomial in $X$, and $h$ is a non-fixed monomial. Here $m$ is any monomial in $X$ and its norm belongs to the first summand. The transfer and restriction are extended additively, and the norm by Tambara reciprocity: $$N(a+b)=N(a)+N(b)+\operatorname{tran}(a\overline{b}).$$ It is easy to verify that the restriction and the norm are multiplicative, and that this indeed defines a Tambara functor.
\[Tamresfree\] The functor $\mathbb{A}[-;-]$ is left adjoint to the forgetful functor that sends a Tambara functor $T=\xymatrix@C=40pt{\big(A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& B\ar[l]|-{\operatorname{res}}\big)}$ to the $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{Z}}/2}$-diagram of sets $A\leftarrow B\colon\operatorname{res}$ with the involution of $A$.
Clearly a map of Tambara functors $\mathbb{A}[X;Y]\to T$ induces maps of sets $X\to A$ and $Y\to B$, by restricting the maps of underlying rings respectively to the set of polynomial generators $X$ and along the inclusion $Y\to M(Y\amalg X/\operatorname{\mathbb{Z}}/2)$. It is easy to verify that these commute with the restriction and the involution.
Conversely, let us show that maps of sets $\alpha\colon X\to A$ and $\beta\colon Y\to B$ commuting with the restriction and the involution induce a unique map of Tambara functors $$\xymatrix@C=40pt{\mathbb{Z}[ X]\ar[d]_{\alpha_\ast} \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \operatorname{\mathbb{Z}}(M(Y\amalg X/\operatorname{\mathbb{Z}}/2))\oplus \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2) \ar@{-->}[d]^{\beta_\ast}\ar[l]|-{\operatorname{res}}
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_{N}& B,\ar[l]|-{\operatorname{res}}
}$$ whose map of underlying presheaves is again given by $\alpha$ and $\beta$. Clearly $\alpha_\ast$ is the unique map of rings which restricts to $\alpha$, and it is equivariant. Since $\alpha_\ast$ and $\beta_\ast$ must commute with transfers and norms, the map $\beta_\ast$ must satisfy the conditions $$\begin{aligned}
\beta_\ast(mg\overline{m})=\beta_\ast(m1\overline{m})\beta_\ast(g)=\beta_\ast(N(m))\beta_\ast(g)=N(\alpha_\ast(m))\beta_\ast(g)
\\
\beta_\ast(mk\overline{m})=\beta_\ast(\operatorname{tran}(mk\overline{m}))=\operatorname{tran}(\alpha_\ast(mk\overline{m}))
\\
\beta_\ast(h+\overline{h})=\beta_\ast(\operatorname{tran}(h))=\operatorname{tran}(\alpha_\ast(h))\end{aligned}$$ (here we abuse the notation and denote the monoid maps induced by $\alpha$ and $\beta$ also by $\alpha_\ast$ and $\beta_*$, respectively). Thus if $\beta_\ast$ exists it must be unique. In order to show existence we need to verify that these formulas indeed define a morphism of Tambara functors. It is immediate that $\alpha_\ast$ and $\beta_\ast$ commute with the transfers and the norms. For the restrictions, we verify that $$\begin{aligned}
\operatorname{res}\beta_\ast(mg\overline{m})&=\operatorname{res}(N(\alpha_\ast(m))\beta(g))=\operatorname{res}(N(\alpha_\ast(m)))\operatorname{res}(\beta_\ast(g))
\\
&=\alpha_\ast(m)\overline{\alpha_\ast(m)}\alpha_\ast \operatorname{res}(g)=\alpha_\ast \operatorname{res}(mg\overline{m})
\\
\operatorname{res}\beta_\ast(mk\overline{m})&=\operatorname{res}\operatorname{tran}(\alpha_\ast(mk\overline{m}))=2\alpha_\ast(mk\overline{m})=\alpha_\ast\operatorname{res}(mk\overline{m})
\\
\operatorname{res}\beta_\ast(h+\overline{h})&=\operatorname{res}\operatorname{tran}(\alpha_\ast(h))=\alpha_\ast(h)+\overline{\alpha_\ast(h)}=\alpha_\ast (\operatorname{res}(h+\overline{h})).\end{aligned}$$ It remains to verify that $\beta_\ast$ is multiplicative. We do this directly on generators: $$\begin{aligned}
\beta_\ast(mg\overline{m}\cdot m'g'\overline{m}')&=\beta_\ast(mm'gg'\overline{mm'})=N(\alpha_\ast(mm'))\beta_\ast(gg')
\\
&=N(\alpha_\ast(m))N(\alpha_\ast(m'))\beta_\ast(g)\beta_\ast(g')=\beta_\ast(mg\overline{m})\beta_\ast(m'g'\overline{m}')\end{aligned}$$ $$\begin{aligned}
\beta_\ast(mg\overline{m}\cdot m'k\overline{m}')&=\beta_\ast(mm'\operatorname{res}(g)k\overline{mm'})=\operatorname{tran}(\alpha_\ast(mm'\operatorname{res}(g)k\overline{mm'}))
\\
&=
\operatorname{tran}\big(\alpha_\ast(m'k\overline{m'})\alpha_\ast(m)\alpha_\ast(\overline{m})\alpha_\ast \operatorname{res}(g)\big)
\\&
=\operatorname{tran}\big(\alpha_\ast(m'k\overline{m'})\operatorname{res}(N(\alpha_\ast(m))\beta_\ast(g))\big)
\\
&
=N(\alpha_\ast(m))\beta(g)\operatorname{tran}(\alpha_\ast(m'k\overline{m'}))
=\beta_\ast(mg\overline{m})\beta_\ast(m'k\overline{m}')\end{aligned}$$ $$\begin{aligned}
\beta_\ast(mg\overline{m}\cdot (h+\overline{h}))&=\beta_\ast(m\operatorname{res}(g)\overline{m}h+\overline{m\operatorname{res}(g)\overline{m}h})=\operatorname{tran}(\alpha_\ast(m\operatorname{res}(g)\overline{m}h))
\\
&
=\operatorname{tran}(\alpha_\ast(h)\alpha_\ast(m)\overline{\alpha_\ast(m)}\alpha_\ast\operatorname{res}(g))
=\operatorname{tran}(\alpha_\ast(h)\operatorname{res}(N(\alpha_\ast(m))\beta_\ast(g)))
\\
&=
N(\alpha_\ast(m))\beta_\ast(g)\operatorname{tran}(\alpha_\ast(h))
=\beta_\ast(mg\overline{m})\beta_\ast(h+\overline{h})\end{aligned}$$ $$\begin{aligned}
\beta_\ast(mk\overline{m}\cdot m'k'\overline{m}')&=\beta_\ast(2mm'kk'\overline{mm'})=2\operatorname{tran}(\alpha_\ast(mm'kk'\overline{mm'}))
\\
&=2\operatorname{tran}(\alpha_\ast(mk\overline{m})\alpha_\ast(m'k'\overline{m'}))
=\operatorname{tran}(\alpha_\ast(mk\overline{m})\operatorname{res}(\operatorname{tran}(\alpha_\ast(m'k'\overline{m'}))))
\\
&=\operatorname{tran}(\alpha_\ast(mk\overline{m}))\operatorname{tran}(\alpha_\ast(m'k'\overline{m'}))=\beta_\ast(mk\overline{m})\beta_\ast(m'k'\overline{m}')\end{aligned}$$ $$\begin{aligned}
\beta_\ast(mk\overline{m}\cdot (h+\overline{h}))&=2\beta_\ast(mk\overline{m}h+\overline{mk\overline{m}h})=2\operatorname{tran}(\alpha_\ast(mk\overline{m}h))
\\
&=\operatorname{tran}(\alpha_\ast(mk\overline{m})(\alpha_\ast(h)+\overline{\alpha_\ast(h)}))=\operatorname{tran}(\alpha_\ast(mk\overline{m})\operatorname{res}\operatorname{tran}(\alpha_\ast(h)))
\\
&=
\operatorname{tran}(\alpha_\ast(mk\overline{m}))\operatorname{tran}(\alpha_\ast(h))=\beta_\ast(mk\overline{m})\beta_\ast(h+\overline{h})\end{aligned}$$ $$\begin{aligned}
\beta_\ast((h+\overline{h})\cdot(h'+\overline{h}'))&=\beta_\ast\left\{
\begin{array}{ll}
(hh'+\overline{hh'})+(h\overline{h'}+\overline{h\overline{h'}})& \mbox{if }h\overline{h'}\neq \overline{h\overline{h'}}, \ hh' \neq \overline{hh'}
\\
hh'+(h\overline{h'}+\overline{h\overline{h'}})&\mbox{if }h\overline{h'} \neq \overline{h\overline{h'}}, \ hh'= \overline{hh'}
\\
(hh'+\overline{hh'})+h\overline{h'}& \mbox{if } h\overline{h'}= \overline{h\overline{h'}}, \ hh' \neq \overline{hh'}
\end{array}
\right.
\\
&=
\operatorname{tran}(\alpha_\ast (hh'))+\operatorname{tran}(\alpha_\ast (h\overline{h'}))
=\operatorname{tran}(\alpha_\ast (h)(\alpha_\ast (h')+\alpha_\ast (\overline{h'})))
\\
&=
\operatorname{tran}(\alpha_\ast (h))\operatorname{tran}(\alpha_\ast (h'))=\beta_\ast(h+\overline{h})\beta_\ast(h'+\overline{h}')\end{aligned}$$
\[toprestamb\] There is a way of constructing the left adjoint $\mathbb{A}[-;-]$ using $\operatorname{\mathbb{Z}}/2$-equivariant stable homotopy theory. Given an $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{Z}}/2}$-diagram of sets $X\leftarrow Y\colon \operatorname{res}$ (with an involution on $X$), one can functorially construct a $\operatorname{\mathbb{Z}}/2$-CW complex $Z$ whose fixed point $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{Z}}/2}$-diagram is weakly equivalent to the latter. This uses Elmendorf’s Theorem [@Elmendorf]. Let $\mathbb{P}$ denote the free (genuine) $\operatorname{\mathbb{Z}}/2$-$E_{\infty}$-algebra functor. Then using the adjunction of [@Ullman Theorem 5.2], we can see that the components of $\mathbb{P}(\Sigma^{\infty} Z_{+})$ are the free Tambara functor, and therefore $$\underline{\pi}_0 (\mathbb{P}(\Sigma^{\infty} Z_{+}))\cong\mathbb{A}[X;Y].$$ We do not go into the details of this construction, but note that we can deduce from this abstract construction that the free Tambara functor is torsion-free. Indeed, the spectrum $\mathbb{P}(\Sigma^{\infty} Z_{+})$ is equivalent to a wedge of suspension spectra of $\operatorname{\mathbb{Z}}/2$-CW complexes. By the tom Dieck splitting, the groups of components of such spectra are in fact free-abelian.
This strategy is different than the heuristics presented in Remark \[remA\]. There we first pass to the free semi-Mackey functor and then to the free Tambara functor. Here we first go in the additive direction and create the free (honest) Mackey functor and then the associated free Tambara functor. The functors $\Sigma^{\infty}(-)_{+}$ and $\mathbb{P}$ are just topological analogs of the latter two constructions.
We can use the construction $\mathbb{A}[X;Y]$ to build resolutions for cohomological Tambara functors. Let $$T=\xymatrix@C=70pt{\big(A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& B\ar[l]|-{\operatorname{res}}\big),}$$ be a cohomological $\operatorname{\mathbb{Z}}/2$-Tambara functor, that is one for which $N\operatorname{res}=(-)^2$ (and in particular from Tambara reciprocity $\operatorname{tran}(1)=2$). Any cohomological Tambara functor can be resolved with a torsion-free commutative ring with involution.
\[cohTambres\] For any cohomological Tambara functor $T$ as above, there exists a polynomial ring with involution $S$ and a surjection of Tambara functors $$\xymatrix@C=70pt@R=13pt{S\ar@{->>}[d] \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& S^{\operatorname{\mathbb{Z}}/2} \ar@{->>}[d] \ar[l]|-{\operatorname{res}}
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_{N}& B. \ar[l]|-{\operatorname{res}}
}$$
Consider the free $\operatorname{\mathbb{Z}}/2$-set $X=\operatorname{\mathbb{Z}}/2 \times A$ which consists of two disjoint copies of the underlying set of $A$. The $\operatorname{\mathbb{Z}}/2$-action exchanges the two copies of $A$. The obvious $\operatorname{\mathbb{Z}}/2$-equivariant counit map $ \alpha \colon X \to A$ gives a map of $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{Z}}/2}$-diagram of sets $$\xymatrix@C=60pt@R=17pt{X\ar[d]_{\alpha} & \emptyset \ar[d]^{\beta} \ar[l]^-{\operatorname{res}}
\\
A & B,\ar[l]^-{\operatorname{res}}
}$$ which in turn by the adjunction of Lemma \[Tamresfree\] gives a map of Tambara functors $\mathbb{A}[X;\emptyset]\to T$: $$\xymatrix@C=40pt@R=17pt{\mathbb{Z}[ X]\ar[d]_{\alpha_\ast} \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \operatorname{\mathbb{Z}}(M(X/\operatorname{\mathbb{Z}}/2))\oplus \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2) \ar[d]^{\beta_\ast}\ar[l]|-{\operatorname{res}}
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_{N}& B. \ar[l]|-{\operatorname{res}}
}$$ The top row is not a cohomological Tambara functor. We make some identifications which will transform the top row into a cohomological Tambara functor, and which are preserved by $\beta_\ast$. For any fixed monomial $m\overline{m} \in M(X)^{\operatorname{\mathbb{Z}}/2}$, consider the difference $$2N(m)-m\overline{m}.$$ These elements generate an ideal $I$ of the upper right corner. It follows from the definitions that $\operatorname{res}(I)=0$. Since the lower row is cohomological, we know that $\operatorname{tran}(1)=2$ which implies that $\beta_*(I)=0$. Hence we get a morphism of cohomological Tambara functors $$\xymatrix@C=40pt@R=17pt{\mathbb{Z}[ X]\ar[d]_{\alpha_\ast} \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& (\operatorname{\mathbb{Z}}(M(X/\operatorname{\mathbb{Z}}/2))\oplus \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2))/I \ar[d]^{\beta_\ast}\ar[l]|-{\operatorname{res}}
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_{N}& B, \ar[l]|-{\operatorname{res}}
}$$ where we keep the notation $\beta_*$ to denote the the induced map on the quotient. An elementary calculation now shows that there is a ring isomorphism $$(\operatorname{\mathbb{Z}}(M(X/\operatorname{\mathbb{Z}}/2))\oplus \operatorname{\mathbb{Z}}(M(X)^{\operatorname{\mathbb{Z}}/2})\oplus \operatorname{\mathbb{Z}}(M(X)^{free}/\operatorname{\mathbb{Z}}/2))/I \cong \mathbb{Z}[ X]^{\operatorname{\mathbb{Z}}/2}.$$ In fact the Tambara functor associated to the commutative ring with involution $\mathbb{Z}[ X]$ is isomorphic to the top row of the latter diagram. Hence we get a map of Tambara functors $$\xymatrix@C=60pt@R=17pt{\mathbb{Z}[ X]\ar[d]_{\alpha_\ast} \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \mathbb{Z}[ X]^{\operatorname{\mathbb{Z}}/2} \ar[d]^{\beta'_\ast}\ar[l]|-{\operatorname{res}}
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_{N}& B, \ar[l]|-{\operatorname{res}}
}$$ where $\beta'_\ast(m\overline{m})=N(\alpha_*(m))$ and $\beta'_\ast(h+\overline{h})=\operatorname{tran}(\alpha_*(h))$, for any monomial $m$ and a non-fixed monomial $h$.
This almost proves the desired result, except the map $\beta'_\ast$ is not necessarily surjective. By tensoring with $\operatorname{\mathbb{Z}}[B]$, where $B$ has the trivial $\operatorname{\mathbb{Z}}/2$ action, we obtain a morphism of Tambara functors $$\xymatrix@C=40pt{\mathbb{Z}[ X] \otimes \operatorname{\mathbb{Z}}[B] \ar@{->>}[d] \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_-{N}& \mathbb{Z}[ X]^{\operatorname{\mathbb{Z}}/2} \otimes \operatorname{\mathbb{Z}}[B] \ar@{->>}[d] \ar[l]|-{\operatorname{res}}
\\
A \ar@<1ex>[r]^-{\operatorname{tran}}\ar@<-1ex>[r]_{N}& B. \ar[l]|-{\operatorname{res}}
}$$ Here we define $N(mb)=N(m)b^2$, where $b$ is a monomial in the elements of $B$. For linear combinations, the norm is extended using Tambara reciprocity. The left vertical map is induced by $\alpha_*$ and the restriction $\operatorname{res}\colon B \to A$. The right vertical map is induced by $\beta'_\ast$ and the identity on $B$. Clearly, both vertical maps are surjective. That these maps are compatible with the norm uses that $N \operatorname{res}=(-)^2$, i.e., that our Tambara functor $T$ is cohomological.
It is now easy to verify that the Tambara functor associated to the commutative ring with involution $S=\operatorname{\mathbb{Z}}[X] \otimes \operatorname{\mathbb{Z}}[B]\cong \operatorname{\mathbb{Z}}[X\amalg B]$, where $B$ has a trivial $\operatorname{\mathbb{Z}}/2$ action, is isomorphic to the top row of the latter diagram.
--------------------------------------------
Emanuele Dotto
Mathematical Institute, University of Bonn
*e-mail address:* <dotto@math.uni-bonn.de>
--------------------------------------------
\
---------------------------------
Kristian Jonsson Moi
Department of Mathematics, KTH
*e-mail address:* <krjm@kth.se>
---------------------------------
\
-----------------------------------------------
Irakli Patchkoria
Mathematical Institute, University of Bonn
*e-mail address:* <irpatchk@math.uni-bonn.de>
-----------------------------------------------
|
---
abstract: 'Mixing of binary fluids by moving stirrers is a commonplace process in many industrial applications, where even modest improvements in mixing efficiency could translate into considerable power savings or enhanced product quality. We propose a gradient-based nonlinear optimisation scheme to minimise the mix-norm of a passive scalar. The velocities of two cylindrical stirrers, moving on concentric circular paths inside a circular container, represent the control variables, and an iterative direct-adjoint algorithm is employed to arrive at enhanced mixing results. The associated stirring protocol is characterised by a complex interplay of vortical structures, generated and promoted by the stirrers’ action. Full convergence of the optimisation process requires constraints that penalise the acceleration of the moving bodies. Under these conditions, considerable mixing enhancement can be accomplished, even though an optimum cannot be guaranteed due to the non-convex nature of the optimisation problem. Various challenges and extensions of our approach are discussed.'
author:
- 'M.F. Eggl and Peter J. Schmid'
bibliography:
- 'jfm\_mixingNEW.bib'
title: Mixing enhancement in binary fluids using optimised stirring strategies
---
Introduction \[sec:intro\]
==========================
The mixing of binary fluids – the process by which a heterogeneous mixture of two miscible fluids is manipulated into a homogeneous blend of uniform composition – is at the core of a great many industrial and technological applications. The food and beverage industry, as well as the consumer product industry abound with examples where multiple fluid components are mixed into a final product. Adhesives, sealants, cosmetics, inks and paints all consist of multiple ingredients that need to be mixed into their final state during a complex industrial process. Efficiency and consistency are paramount in maintaining a quality product that is cost-effective to manufacture. Some of the strictest tolerances in mixing quality can be found in the pharmaceutical industry where medication has to be mixed into precise doses. But also chemical engineering processes, such as polymer production, rely on accurate mixing to facilitate the proper chemical reactions and to reduce undesirable by-products (for an overview of theoretical and practical aspects of mixing, see [@Handbook2003] or [@Uhl2012]).
Mixing processes can be induced actively or passively. The active strategy is commonly based on a stirrer system, where paddles or rods agitate the binary mixture, induce vortical fluid structures and ultimately blend the initial ingredients. The geometry, path and speed of the stirrers have a great influence on the effectiveness and efficiency of the mixing process and are the subject of mixing optimisation. Passive systems, on the other hand, possess no moving parts, but instead rely on a complex baffle system inside an inflow-outflow device that mixes initially separated fluid components.
In this article, we will concentrate on an active stirrer system and develop a mathematical and computational framework for the formulation and solution of a constrained optimisation problem that yields favorable stirrer protocols for enhanced mixing results in binary fluid systems. Constraints stem from speed and path restrictions on the stirrers: stirrer systems are subject to mechanical and material limitations, and paddles or rods often cannot accelerate or change directions at will or too abruptly. In addition, while a significant part of industrial mixing processes involve non-Newtonian fluids, we will, for simplicity’s sake, focus on Newtonian fluids. Furthermore, we will concentrate on inertial, laminar mixing. The inertial aspect of this parameter regime, described by a Reynolds number above the Stokes-flow regime, guarantees a rich and varied control space, taking advantage of advective, unsteady and diffusive processes, while the laminar aspect avoids divergences of the direct-adjoint optimisation scheme due to the existence of positive Lyapunov exponents linked to turbulent fluid motion. Despite these restrictions, a great many mixing processes fall into our chosen parameter regime.
Research in mixing has a long and remarkable history, covering theoretical aspects as well as technological applications. A large body of literature has been devoted to mixing in simplified fluid models, for example neglecting viscosity, surface tension, density differences or fluid inertia. The primary mechanism has been identified as streamline stretching [@Spencer1951] where the interface between two fluids is repeatedly distorted and redistributed into the bulk of the mixing volume. Among these simplifications, Stokes mixing, i.e., the mixing of highly viscous fluids where inertial effects can be neglected, has arguably received the most attention. This tendency has been further fueled by the rise of micro-mixers where multiple fluid components are injected into a micro-device and extracted affter the mixing process is completed (see, e.g., [@Orsi2013; @Galletti2015]).
More mathematical investigations studied the breakdown in scales and the statistical properties of the observed cascade of fluid filaments. Iterated maps have often been used to determine measures that describe the pertinent scale dynamics or to design optimal mixing results in these measures [@Mathew2007; @Gubanov2010; @Lin2011; @Finn2011]. Many of these findings can be found in [@Sturman2006]. Of particular interest was the rise of chaotic mixing motion from pure advection, even for laminar flows [@Aref1984; @Ottino1989; @Liu2008].
Rather recently, the inertial, but laminar mixing regime has been explored using advances in optimisation techniques. These studies build on the definition of proper mixing measures [@Mathew2005; @Thiffeault2012] and break with the focus on hydrodynamic instabilities [@Balogh2005] to increase mixing. Using wall-mounted blowing/suction control in a channel, improved mixing could be accomplished by directly targeting a mixing measure, rather than a flow instability [@Foures2014]. Further studies [@Vermach2018; @Marcotte2018] have extended this approach to higher dimensions and stratified flows.
The present article will remain in the inertial, but laminar regime and accomplish mixing of a binary fluid by embedded stirrers. These stirrers are constrained to specific paths, but can move along them in a manner that enhances mixing over a user-specified time horizon. The mixedness of the binary fluid is quantified using mix-norms [@Mathew2007; @Thiffeault2012] for a passive scalar; an objective based on this measure is then optimised by a nonlinear, gradient-based scheme, which in turn provides the associated stirring protocol.
(-2.1,-2.1) rectangle (2.1,2.1); plot ([2\*cos()]{}, [2\*sin()]{}); plot ([2\*cos()]{}, [2\*sin()]{}); (0,0) circle \[radius=1.5cm\]; (0,1.5) circle \[radius=0.2cm\]; (0,1.5) circle \[radius=0.2cm\]; (0,0) circle \[radius=0.75cm\]; (0,-0.75) circle \[radius=0.2cm\]; (0,-0.75) circle \[radius=0.2cm\]; plot ([1.6\*cos()]{}, [1.6\*sin()]{}); plot ([0.85\*cos()]{}, [0.85\*sin()]{}); (0,0) – ([0.95\*cos(110)]{},[0.95\*sin(110)]{}); ([1.25\*cos(110)]{},[1.25\*sin(110)]{}) – ([2\*cos(110)]{},[2\*sin(110)]{}); at ([1.1\*cos(110)]{},[1.1\*sin(110)]{}) [$R$]{};
(0,0) – ([0.8\*cos(135)]{},[0.8\*sin(135)]{}); ([1.1\*cos(135)]{},[1.1\*sin(135)]{}) – ([1.5\*cos(135)]{},[1.5\*sin(135)]{}); at ([0.95\*cos(135)]{},[0.95\*sin(135)]{}) [$r_1$]{};
(0,0) – ([0.225\*cos(160)]{},[0.225\*sin(160)]{}); ([0.525\*cos(160)]{},[0.525\*sin(160)]{}) – ([0.75\*cos(160)]{},[0.75\*sin(160)]{}); at ([0.375\*cos(160)]{},[0.375\*sin(160)]{}) [$r_2$]{}; at (0.8,1) [$\omega_1(t)$]{}; at (-0.85,-0.7) [$\omega_2(t)$]{}; at (0,1.5) [$1$]{}; at (0,-0.75) [$2$]{}; at (2.1,-2.1) ;
; ; plot ([+2\*cos()]{}, [-1+2\*sin()]{}); plot ([+2\*cos()]{}, [-1+2\*sin()]{}); (,-1+2) circle \[radius=0.4cm\]; (,-1+2) circle \[radius=0.4cm\]; plot ([+0.5\*cos()]{}, [-1+2+0.5\*sin()]{}); (,+1) – (,+1.5-0.1); at (+1.5,+0.8) [$\omega_1(t)$]{}; at (+0.65,+1.25) [$\Omega$]{}; at (,+1.2) [$a$]{};
Mathematical framework \[sec:math\]
===================================
Governing equations
-------------------
The focus of this study is the mixing process of a binary, miscible and Newtonian fluid by multiple circular stirrers on prescribed paths, and its optimisation by manipulating the stirring strategy within specified constraints. A two-dimensional configuration is considered. The process can be simulated by solving the fluid equations of motion, augmented by a transport equation for a passive scalar $\theta.$ We have
+ + (- \_k \_[s,k]{} ) + p - \^2 &=& 0,\
&=& 0,\
+ ( (1-) + \_k \_[s,k]{} ) - ( ) &=& 0. \[eq:GovEqu\]
with $\uu$ as the velocity vector, $p$ as the pressure field, and $\theta$ as a passive scalar (ranging from zero in one fluid to one in the other). The governing equations have been expressed in non-dimensional form using a characteristic length $L_0$ and velocity scale $u_0.$ This choice introduces the Reynolds number $\Rey$ and the P[é]{}clet number $\Pec,$ to express the kinematic viscosity and the diffusion coefficient of the mixing fluid in non-dimensional form. Furthermore, the system of equations (\[eq:GovEqu\]) contains terms that model the embedded stirrers via a Brinkmann penalisation approach [see @Angot1999]. The multiple solid stirrers, indexed by the subscript $k,$ are characterised by their velocity $\uu_{s,k},$ and are taken as circular in cross-section. The masks $\chi_k$ describing the embedded solid bodies equal one for points occupied by the $k$-th stirrer and zero outside of it. The mask $\chi$ accounts for the overall geometry, such as the domain boundaries. The constant $C_\eta$ ensures the rapid relaxation of the fluid variables towards the respective values imposed by the stirrers or the geometry. The above formalism allows the efficient treatment of objects moving through a background grid on which the motion of the surrounding fluid is describes. Details of this approach and its numerical implementation can be found in [@EgglSchmid2018]. The setup shown in (\[eq:GovEqu\]) imposes no-slip velocity boundary conditions on the stirrers and Neumann conditions for the passive scalar on the solid bodies.
Measuring mixedness
-------------------
In anticipation of our stated goal of enhancing mixing efficiency, we have to introduce a measure that quantifies the degree of mixedness of a particular flow state. This measure shall be based solely on the passive scalar field $\theta.$
In general, mixing is defined as the reduction of inhomogeneities of a given indicator field [@Handbook2003], which still leaves open a precise mathematical definition to be used in our case. Several norms of the passive scalar $\theta$ that attempt to mathematically define the measure of mixedness have been proposed and used in the past [@Mathew2005], among them the variance or the more complex negative-index and fractional-index Sobolev norms [@Thiffeault2012; @Foures2014]. As the choice of norm may influence the outcome of the optimisation, but will not affect the design of our computational optimisation platform, we will focus on the Sobolev norm of negative fractional index of the passive scalar $\theta.$ A measure of this general type downplays the role of small scales and instead directs mixing efforts towards larger fluid elements. It attains higher values for an unmixed field (with high levels of inhomogeneities) and decreases as the scalar field becomes more mixed. Mathematically, the mixedness measure is given as
$$\Vert \theta \Vert_{\rm{mix}} \equiv \frac{1}{\vert \Omega
\vert}\int_{\Omega} \Vert \nabla^{-2/3} \theta(\x,t) \Vert \text{
d}\Omega,
\label{eq:SobolevNorm}$$
with $\Omega$ denoting our computational domain, and $\vert \Omega
\vert$ representing its size (volume or area). In the above definition, we have assumed, without loss of generality, a zero mean of the passive scalar field $\theta.$ Throughout this paper we will be optimising with respect to this quantity, but we stress again that other norms can be employed without conceptual changes in the optimisation procedures. The fractional exponent of $2/3$ can be justified using arguments from optimal transport and ergodic theory. Examples of previous studies using mix-norm optimisation employed $-1/2$ [@Foures2014] or $-1$ [@Lin2011].
Mixing protocol
---------------
As a first attempt at optimising the mixing of a binary fluid, we will concentrate on a stationary circular vessel with two stirrers on circular paths or distinct radii (see figure \[fig:MixSetup\] for a sketch of this configuration). The stirrers have a circular cross-section, and their velocities along their respective paths are undetermined and subject to optimisation and constraints. The circular path is conveniently defined in polar coordinates, while we formulate the remaining equations in Cartesian coordinates, and we thus introduce the vector-valued function $\bm{l}$, which transforms between the two coordinate systems according to $(l_1(\phi),l_2(\phi))
= (-\sin(\phi),\cos(\phi))$ with $\phi$ as the angle traversed along the path of the circle. The parameterisation of the velocity of the $k$-th stirrer thus becomes $$\begin{aligned}
\bm{u}_{s,k} &=&\omega_{k}(t)\bm{r}_k(\bm{x})\bm{l}(\varphi_k(t)) + \Omega_k
{\bf{a}}_k, \label{FLUSI:US}\end{aligned}$$ where $\varphi$ is the sum of the angles travelled along the path, i.e., $$\begin{aligned}
\varphi_k(t) = \int_0^t\omega_{k}(s) \text{d}s.\end{aligned}$$ Following the notation of the governing equations, we define $\omega_k(t)$ as the rotational speed of the $k$-th solid about the centre of the vessel, ${\bf{r}}_k$ denotes the distance from the same centre, $\Omega_k$ stands for the rotation about the stirrer’s centre and ${\bf{a}}_k$ represents the stirrer’s (vectorial) radius. For simplicity, we take $\Omega_k = 0;$ the alternative choice $\Omega_k =
\omega_k$ yielded very similar results in enhancing mixing efficiency.
Constrained optimisation
------------------------
We can then state the optimisation problems as finding a time-dependent velocity protocol $\omega_k(t)$ for each of the two stirrers such that the mix-norm of the passive scalar is minimised over a prescribed time horizon. This minimum has to be achieved while satisfying the governing equations and respecting constraints and bounds on the stirrer velocities. Mathematically we have
&& { \_0\^T \_[[mix]{}]{} dt }\
&& (\[eq:GovEqu\])\
&& \_0\^T \_k \_[s,k]{} \^2 dt E\_0\
&& \_[s,[lower]{}]{} \_[s,k]{} \_[s,[upper]{}]{} k=1,2,\
&& \_[s,[lower]{}]{} \_[s,k]{} \_[s,[upper]{}]{} k=1,2. \[eq:constropt\]
The constraints on the stirrer strategy are threefold: the first constraint limits the $L_2$-norm of $\bm{u}_s$, i.e., the kinetic energy of stirrers’ motion along their paths, expended over the time horizon $T$ to a maximum value of $E_0;$ the second and third impose upper and lower bounds directly on the stirrers’ velocities and accelerations, respectively. All restrictions could conceivably stem from mechanical limitations of the mixing apparatus. In our study, we will consider the constraints successively in order to determine the influence they impose on the optimisation results.
Computational framework \[sec:comp\]
====================================
The implementation of the above optimisation problem requires the discretisation of the governing equations and the reformulation of the constrained problem (\[eq:constropt\]) in terms of an unconstrained one.
Numerical scheme for the governing equations
--------------------------------------------
Starting point for the numerical treatment of mixing enhancement is the open-source FLUSI software [@Engels2015], in which the governing equations are discretised on a Cartesian, double-periodic domain for the two-dimensional case. This formulation allows the application of Fourier-spectral techniques to represent the spatial derivatives. The outer perimeter of the mixing vessel (with radius $R$) and the two stirrers on their respective circular paths (with radius $r_1$ and $r_2$) are described by a Brinkman penalisation technique as shown in (\[eq:GovEqu\]). The original software has been augmented by the passive scalar field and embedded into a gradient-based optimisation formalism.
The Fourier-spectral discretisation allows the replacement of spatial derivatives with a multiplication by components of a wavenumber vector ${\bf{k}} = \left({\bf{k}}_1,{\bf{k}}_2\right)^T,$ with the indices ${}_{1,2}$ indicating the two coordinate directions. Mathematically, we introduce this replacement as $\partial/\partial x_j \to \AAA_j,$ with $\AAA_j = \hbox{diag}\{ i{\bf{k}}_j \}.$ The semi-discretised set of equations then reads
\_i + \_j + (\_i - \_k \_i) + \_i p - \_j \_j \_i &=& 0,\
\_j \_j &=& 0,\
+ ( ([**[1]{}**]{} - ) \_j + \_k \_j ) - \_j \_j &=& 0
where we introduced the Hadamard (element-wise) product $\circ$ (see [@Horn2012]) and assumed implicit (Einstein) summation over identical indices.
Particular care has to be exercised when evaluating the nonlinear terms, as aliasing errors can lead to inaccuracies and numerical instabilities. A low-pass Hou-Li filter [@HouLi2007] has been applied to avoid scattering of unresolved, small scales onto resolved, large scales. In addition, P3DFFT [@pekurovsky2012], a highly efficient, parallel Fourier-transform library, is used to ensure scaling on parallel computer architectures.
Finally, the representation of solid bodies on an underlying Cartesian grid calls for a transfer of geometric information onto the background mesh. This transfer is accomplished by mollified delta-functions, smoothing the otherwise discontinuous mask onto the grid and thus avoiding numerical inaccuracies and instabilities [@Kolomenskiy2009].
From constrained to unconstrained optimisation
----------------------------------------------
A common reformulation of the constrained optimisation problem (\[eq:constropt\]) as an unconstrained problem introduces Lagrange multipliers (or adjoint variables) for the dependent variables of equations (\[eq:GovEqu\]): the adjoint velocity will enforce the momentum equation, the adjoint pressure the divergence condition, and the adjoint passive scalar the transport equation for $\theta.$ The augmented Lagrangian – consisting of the cost functional and the scalar product of the adjoint variables and the governing equations – then needs to minimised. A system of equations, referred to as the KKT-system, can then be derived by setting to zero the first variation of the augmented Lagrangian with respect to direct (original) and adjoint variables. The first variation with respect to the adjoint variables recovers the original set of governing equations. The first variation with respect to the original variables produces, after considerable algebra, a set of equations governing the adjoint variables. The first variation with respect to the control variables (in our case, the velocity strategies $\omega_{1,2}(t)$) will furnish the gradients of the cost functional with respect to $\omega_{1,2}(t)$ which will be used to enhance the mixedness of our fluid system via improved stirring strategies.
Adjoint equations
-----------------
Denoting the adjoint variables (velocity, pressure, passive scalar) by $\uu^{\dag}$, $p^{\dag}$ and $\theta^{\dag},$ the governing equations for their evolution, in semi-discretized form, read
\^\_i - \^\_k - \_j\^H \[\_j \^\_i\] - \^\_i + \_j\^H \_j\^H \_i\^ &&\
-([**[1]{}**]{}-) \^ &=& 0\
\_j\^H \_j\^&=& 0\
\^ - \_j\^H \[([**[1]{}**]{}-) \_j \^\] + \_i\^H (\[([**[1]{}**]{}-) + \] \_i\^H \^) &&\
-\_j\^H \[ \_i (\_[s,i]{})\_j \^\] &=& 0 \[Adjoint:EqEnd\]
with terminal conditions $$\begin{aligned}
\bm{u}^{\dag}(\bm{x},T) = 0, \qquad
\theta^{\dag}(\bm{x},T) = \frac{2}{V_\Omega}(\mathsf{A}_i^{-2/3})^H(\mathsf{A}_i^{-2/3}\theta(\bm{x},T)).\end{aligned}$$ It is important to note that the above adjoint equations are linear in the adjoint variables, but are dependent on the direct variables $\uu_i.$ Moreover, it should become apparent that the adjoint equations have to solved backwards in time, from $t=T$ to $t=0.$
The optimality conditions, stemming from the first variation with respect to the control variables, result in the adjoint rotational velocity along the circular paths given by
$$\omega_{k}^\dag = r_i\left[l_j(\varphi(t))
+\frac{\omega_{k}}{\dot{\omega_{k}}} \frac{\partial l_j}{\partial
\varphi}\right]\chi_i^H\left(\frac{\Pi^{\dag}_j}{C_\eta}-
(\theta^{\dag}\circ[\mathsf{A}_j\theta])\right)$$
where $\Pi^{\dag}_i = \uu^{\dag}_i + \AAA^{H}_i p^{\dag}$. This value provides the gradient information in our iterative optimisation scheme and, together with a line-search routine, updates the current stirring protocol to a more effective one. The optimisation terminates when no more progress can be made, and the magnitude of the cost functional gradient drops below a prescribed threshold.
\[sec:optim\] Implementing additional constraints
-------------------------------------------------
Additional constraints that need to be enforced are incorporated into the gradient-based optimisation routine. This is accomplished by projections and thresholding. In this case, the gradient – computed from the adjoint equations and the optimality condition, without imposed constraints – is projected and properly curtailed to comply with energy constraints and velocity bounds. Details of the numerical implementation of this procedure can be found in [@EgglSchmid2018].
Summary of optimisation procedure
---------------------------------
The full optimisation scheme then proceeds along the following lines. Starting with an initial guess of the stirring protocols $\omega_{1,2}(t),$ we solve the governing equations (\[eq:GovEqu\]) forward in time over a chosen time horizon $[0,\ T].$ In a second step, the adjoint set of equations (\[Adjoint:EqEnd\]) are solved, starting with the proper terminal condition, backwards in time from $t=T$ to $t=0.$ The adjoint variables are then used to evaluate the optimality condition and retrieve the gradient of the cost functional with respect to $\omega_{1,2}.$ This gradient is then furnished to a standard optimisation routine (such as steepest descent or conjugate gradients) which, together with a line-search routine, produces a new and improved stirring protocal. In this last step, all constraints imposed on the stirring functions will be imposed by the aforementioned projections and thresholding. After this step, the next iteration is started. The optimisation terminates when no further progress can be made, within the constraints imposed on the system.
It is worth mentioning that additional complications arise from the fact that the governing equations (\[eq:GovEqu\]) are nonlinear and, as a consequence, the adjoint equations (\[Adjoint:EqEnd\]), while linear in the adjoint variables, depend on the direct variables ${\bf{u}}, \theta.$ This dependency requires the storage of direct fields during the simulation of (\[eq:GovEqu\]) and their injection into (\[Adjoint:EqEnd\]), in reverse order, during the integration of the adjoint system. For efficiency reasons, this exchange between direct and adjoint simulations is handled by checkpointing, where we trade excessive storage requirements for an increased simulation time. The library [@GriewankWalther2000] accomplishes this task in an optimal manner.
Test cases and results \[sec:results\]
======================================
We follow the setup shown in figure \[fig:MixSetup\] with two circular cylinders of radius $\Vert a_{1,2} \Vert = 1,$ moving on two concentric circular paths of radius $r_1 = 3.5$ and $r_2 = 1.5$ and embedded in a circular (stationary) vessel of radius $R = 5.$ The Reynolds number and Péclet number are chosen as $Re = Pe = 1000.$
; (0,0) – (9,0) – (9+,-0.2) – (9+2\*,0.2) – (9+3\*,0) – (12,0); (0,-0.2) – (0,0.2); (0.5,-0.2) – (0.5,0.2); (4,-0.2) – (4,0.2); (12,-0.2) – (12,0.2); at (0,2\*) ; at (0.5,2\*) ; at (4,2\*) ; at (12,2\*) ; (0,-0.75) – (0.5,-0.75); (0,-1) – (4,-1); at (4,-0.875) ; (0,-1.5) – (4,-1.5); at (4,-1.5) ; (0,-2) – (9,-2) – (9+,-2-0.2) – (9+2\*,-2+0.2) – (9+3\*,-2) – (12,-2); at (4,-2.25) ;
A further parameter in the optimisation concerns the time horizons over which (i) the control strategy is applied, and (ii) over which gradient information is gathered. The former time interval determines the window given to the stirrers to be active mixers; after this window is passed, the motion of the stirrers will stop, and only the remaining inertia of the fluid and diffusion will contribute to further mixing. The latter time interval determines the amount of information extracted from the evolution process that is used to compute an enhanced stirring protocol (applied over the former time window). The control horizon may be chosen shorter than the information (predictive) horizon: in this case, a time-compressed strategy will be employed that accounts for, and optimises over, a more expansive time window. In our case, we will juxtapose a short-term strategy with $T_{\rm{control}}=1$ and a longer-time strategy with $T_{\rm{control}}=8$ and assess the optimised strategies in either case. Both protocols, however, have access to information over a temporal interval of $T_{\rm{info}} = 8.$ Finally, the simulations have been continued over $T_{\rm{sim}}=32$ units to track the further development of the instigated mixing processes; rest inertia and diffusion will remain the only mechanisms during this stage. A summary of our choice of parameters is sketched in figure \[fig:TimeHorizons\].
Before proceeding to the various optimisation studies, it is instructive to reflect upon possible mixing mechanisms given the setup in figure \[fig:MixSetup\]. The most obvious strategy for mixing a binary fluid consists of a [**[plunging]{}**]{} motion, where the cylinders push through the initial interface, distort it and drag fluid one into regions occupied by fluid two, and vice versa (figures \[fig:Strategies\][*[a–c]{}*]{}). This type of strategy is nearly exclusively implemented in industrial mixers. Despite its omnipresence in applications, alternative strategies are often equally or more effective, foremost among them vortex shedding due to unsteady and abrupt motion of the stirrers, affably denoted as the [**[vortex cannon]{}**]{} strategy (figures \[fig:Strategies\][*[d-f]{}*]{}). In this case, the stirrer generates a sequence of startup and stopping vortices by rapid oscillations or abrupt directional changes along the circular paths. The shed vortices then act as effective autonomous mixers that, once they reach the initial or distorted interface, further deform the passive scalar field and locally (and globally) reduce the mix-norm. In this manner, a single stirrer can clone ’fluid stirrers’ (shed vortices) and thus multiply its mixing effectiveness. In a further possible strategy, vortices can be generated in the fluid that collide with each other and thus generate filaments, which are then subjected to more rapid diffusion and homogenisation (figures \[fig:Strategies\][*[g-i]{}*]{}). Of course, this [**[vortex collision]{}**]{} strategy is strongly dependent on the initial condition of the passive scalar – and for this reason, less transferable to a general, realistic mixing strategy –, nonetheless, within our computational framework, it is a viable and pervasive strategy utilised by our direct-adjoint algorithm. A far more transferable mixing strategy is the collision of vortical structures with the outer wall (figures \[fig:Strategies\][*[j-l]{}*]{}) whereby a large fluid element is broken up into smaller elements which further interact with other vortices and are subject to increased diffusion due to the breakdown in scales. Finally, the embedded physical stirrers can themselves interplay with the vortical structures they generate, acting as [**[obstructions]{}**]{} in the path of vortices (figures \[fig:Strategies\][*[m-o]{}*]{}). A collision between a stirrer and a vortex will split the vortex and yield smaller scales, hence contributing to a decrease in the mix-norm. This final strategy will continue to cause a moderate breakdown in scales, even after the control window has closed and no more stirring motion is allowed.
Given these five fundamental strategies, illustrated in figure \[fig:Strategies\] with samples from our simulations, the direct-adjoint looping algorithm will select from and combine these options into a coherent strategy, given the chosen parameters and user-specified constraints.
[cccc]{} &
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Plunging1C.png "fig:"){width="20.00000%"}]{}; (0.375,0.375) – (0.25,0.15); (-1.1,-1.1) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Plunging2C.png "fig:"){width="20.00000%"}]{}; (0.275,0.085) – (0.15,-0.135); (-1.1,-1.1) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Plunging3C.png "fig:"){width="20.00000%"}]{}; (-0.4,-0.8) – (-0.595,-0.975); (-1.1,-1.1) node ;
\
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Cannon1C.png "fig:"){width="20.00000%"}]{}; (0.28,1.475) – (0.42,1.025); (0.35,1.25) circle \[radius=0.2mm\]; (-1.1,-1.33) node ;
(0.75,1) to \[out=-65,in=90\] (0.9,0.3); (0.2,0.9) to \[out=-65,in=0\] (-0.1,0.4);
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Cannon2C.png "fig:"){width="20.00000%"}]{}; (0.375,1.21) circle \[radius=0.35mm\]; (-1.1,-1.33) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Cannon3C.png "fig:"){width="20.00000%"}]{}; (0.6,1.3) circle \[radius=0.35mm\]; (-1.1,-1.33) node ;
\
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Collision1C.png "fig:"){width="20.00000%"}]{}; (-0.4,-0.2) – (-0.15,-0.1); (-0.15,-0.1) – (0.1,0); (-1.1,-0.9) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Collision2C.png "fig:"){width="20.00000%"}]{}; (-1.1,-0.9) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Collision3C.png "fig:"){width="20.00000%"}]{}; (-1.1,-0.9) node ;
\
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/WallCollision1C.png "fig:"){width="20.00000%"}]{}; (-0.4,-0.2) – (-0.15,-0.1); (-0.15,-0.1) – (0.1,0); (-1.1,-1.9) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/WallCollision2C.png "fig:"){width="20.00000%"}]{}; (-1.1,-1.9) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/WallCollision3C.png "fig:"){width="20.00000%"}]{}; (-1.1,-1.9) node ;
\
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Obstruction1C.png "fig:"){width="20.00000%"}]{}; (0,0.55) circle \[radius=0.35mm\]; (-0.3,-0.45) to \[out=30,in=230\] (-0.1,-0.3) to \[out=50,in=265\] (0,0); (-1.1,-1.05) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Obstruction2C.png "fig:"){width="20.00000%"}]{}; (0.025,0.55) circle \[radius=0.35mm\]; (-1.1,-1.05) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:Strategies\] Various mixing strategies, from snapshots of the simulations. (a,b,c) Plunging of the stirrer through the interface, (d,e,f) casting of start-stop vortices towards the interface (vortex cannon), (g,h,i) collision of vortices, (j,k,l) collision with the vessel wall, and (m,n,o) breakup of vortical structures by stationary stirrers (obstruction).](figures/Obstruction3C.png "fig:"){width="20.00000%"}]{}; (0.035,0.525) circle \[radius=0.35mm\]; (-1.1,-1.05) node ;
Overview of test cases
-----------------------
We will consider six cases, grouped into three examples. Each example consists of a short-time strategy with a rather limited control horizon of $T_{\rm{control}}=1$ and a long-time strategy with a more generous horizon of $T_{\rm{control}}=8.$ These two $T_{\rm{control}}$-settings will impose noticeable constraints on the choice of strategies, the interplay of dynamic processes and the feasibility of the final protocol. The three examples further distinguish themselves by the number of external constraints: starting with pure energy constraints, via energy and velocity constraints, to energy, velocity and acceleration constraints. Along this course of action, algorithmic requisites and physical requirements will be encountered and discussed.
Convergence of the iterative scheme is principally governed by constraints imposed on the optimisation problem. First, the nonlinear nature of the governing equations precludes a guarantee to converge towards a global minimum; only a local minimum can be expected. More importantly, additional constraints on the stirrers, such as energy, velocity or acceleration bounds, can convert a semi-norm to a full-norm optimisation problem. Semi-norm optimisation problems [@Foures2012; @Blumenthal2017] are ’open-ended’ in the sense that the stirrer velocities increase without bounds, while the mixing measure steadily improves. In this case, the iterative optimisation scheme terminates when the adjoint variables, due to excessive direct velocities, no longer furnish useful gradient information for further improvement. In other words, the optimisation comes to an end when the signal-to-noise ratio for the gradient information drops to a value near unity, even before the cost-functional gradient attains a small value. In the full-norm case, convergence is achieved when the cost-functional gradient falls below a small, user-specified threshold, indicating that a [*[local]{}*]{} minimum has been reached. In brief, iterations are halted when either the cost-functional gradient falls below a small threshold or the signal-to-noise ratio of the adjoint gradient information reaches unity – whichever scenario comes first.
Since mixing enhancement based on complex stirring strategies is a highly dynamic process – based on a rapid sequence of abundant vortical features –, a set of static snapshots cannot do justice to the intricacies of an optimised mixing protocol. For this reason, we urge the reader to turn to the animations in the supplemental material.
Cases 1 and 2: optimisation under energy constraints
----------------------------------------------------
The first two cases follow a common procedure whereby the cost functional (mix-norm of the passive scalar) of the constrained optimisation is minimised, subject to a penalisation of the control energy that accomplishes this minimum. Since the stirrers’ kinetic energy is a measure of effort that goes into the mixing process, we add a corresponding term to the pure mix-norm cost functional. As a consequence, the energy expended by the stirrers is bounded to a user-specified value.
[ccc]{}
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint1\] Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/EnPen1/EnPen1_13.png "fig:"){height="27.00000%"}]{}; plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([0.575\*cos()]{}, [0.575\*sin()]{});
plot ([1.45\*cos()]{}, [1.45\*sin()]{}); plot ([1.2\*cos()]{}, [1.2\*sin()]{}); plot ([0.7\*cos()]{}, [0.7\*sin()]{}); plot ([0.4\*cos()]{}, [0.4\*sin()]{});
(-1.5,-1.5) node [$(a)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint1\] Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/EnPen1/EnPen1_16.png "fig:"){height="27.00000%"}]{};
(0.15,0.9) to \[out=205,in=90\] (-0.15,0.45); (0.15,0) to \[out=145,in=-90\] (-0.15,0.45);
(-1.5,-1.5) node [$(b)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint1\] Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/EnPen1/EnPen1_35.png "fig:"){height="27.00000%"}]{};
(-1.3,-1) to \[out=-45,in=180\] (-0.7,-1.2); (-0.9,-0.4) to \[out=-60,in=180\] (-0.5,-0.6);
(-1.5,-1.5) node [$(c)$]{}; (2,-1.65) node ;
\
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint1\] Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/EnPen1/EnPen1_60.png "fig:"){height="27.00000%"}]{};
(-0.7,-1.2) to \[out=0,in=-135\] (-0.1,-0.9); (-0.3,0) to \[out=35,in=-60\] (-0.1,0.9);
(-1.5,-1.5) node [$(d)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint1\] Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/EnPen1/EnPen1_190.png "fig:"){height="27.00000%"}]{};
(-0.5,-0.7) to \[out=45,in=-110\] (-1.5,0.6); (0.1,-0.9) to \[out=-5,in=45\] (0.9,-1.4); (0.1,-0.8) to \[out=-10,in=-120\] (1.4,-0.3);
(-1.5,-1.5) node [$(e)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint1\] Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/EnPen1/EnPen1_757.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(f)$]{}; (2,-1.65) node ;
[ccc]{}
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint8\] Same as figure \[fig:EnConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/EnPen8/EnPen8_46.png "fig:"){height="27.00000%"}]{}; plot ([1.4\*cos()]{}, [1.4\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.2\*cos()]{}, [1.2\*sin()]{}); plot ([0.575\*cos()]{}, [0.575\*sin()]{}); plot ([0.675\*cos()]{}, [0.675\*sin()]{}); plot ([0.475\*cos()]{}, [0.475\*sin()]{});
(-0.05,-0.2) to \[out=100,in=0\] (-0.3,0.1) to \[out=180,in=90\] (-0.5,-0.1) to \[out=270,in=170\] (-0.35,-0.2); (0.6,-0.6) to \[out=0,in=-135\] (1.1,-0.3); (-0.7,1.3) to \[out=190,in=70\] (-1.4,0.7);
(-1.5,-1.5) node [$(a)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint8\] Same as figure \[fig:EnConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/EnPen8/EnPen8_106.png "fig:"){height="27.00000%"}]{};
(0.6,-0.3) to \[out=-40,in=180\] (1.15,-0.5); (1.6,-0.3) to \[out=-110,in=0\] (1.25,-0.5); (-1.3,-0.8) to \[out=-40,in=-135\] (-0.6,-0.6); (-1.4,-0.9) to \[out=-55,in=145\] (-1,-1.3); (-0.7,0.3) to \[out=20,in=-90\] (-0.4,0.7);
(-1.5,-1.5) node [$(b)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint8\] Same as figure \[fig:EnConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/EnPen8/EnPen8_172.png "fig:"){height="27.00000%"}]{};
(1.5,0.5) to \[out=-55,in=90\] (1.65,0) to \[out=-90,in=0\] (1.4,-0.2) to \[out=180,in=-90\] (1.2,0) to \[out=90,in=-135\] (1.4,0.2);
(-1.5,-1.5) node [$(c)$]{}; (2,-1.65) node ;
\
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint8\] Same as figure \[fig:EnConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/EnPen8/EnPen8_280.png "fig:"){height="27.00000%"}]{};
(0.5,0.1) – (0.3,0.3); (-0.1,0.7) – (0.1,0.5);
(-1.5,-1.5) node [$(d)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint8\] Same as figure \[fig:EnConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/EnPen8/EnPen8_529.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(e)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:EnConstraint8\] Same as figure \[fig:EnConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/EnPen8/EnPen8_736.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(f)$]{}; (2,-1.65) node ;
Figure \[fig:EnConstraint1\] displays the results of our optimisation, visualised by iso-contours of the passive scalar $\theta$ at selected time instances. The control horizon is $T_{\rm{control}}=1.$ We observed that the optimisation does not utilise the ‘plunging’ option, as the stirrers remain nearly at their initial position. Instead, the entire energy available to the stirrers is used up in a rapid start-and-stop motion which initially causes multiple small-scale shed vortices that distort the plane interface, collide into each other and the stirrers, and merge into larger-scale vortex structures which eventually achieve good mixing. It is important to stress that for the calculation of this short-time mixing strategy, information about the full dynamics up to $T_{\rm{info}}=8$ has been incorporated into the optimisation. In other words, the consequences of the limited stirring protocol up to $T_{\rm{info}}=8$ are known to the optimisation, and adjustments to the control strategy can be made that affect the vortex dynamics beyond its active control window. The evolution of the passive scalar between $T_{\rm{info}}=8$ and $T_{\rm{sim}}=32,$ however, is neither designed nor recognized by the optimzation algorithm; it simply plays out according to the action taken during the control and optimisation windows. We include this further evolution to underline our choice of the mix-norm as the mixedness measure, whose optimisation produces the small scales that are subsequently diffused during this ‘cool-off’ window.
We conclude that the absence of any plunging option points at the suboptimality of this particular strategy in achieving an enhanced mixing process. It is thus not pursued as a viable option by the direct-adjoint optimisation technique. In the interpretation of these results, it may be tempting to conclude that a different initial placement of the cylinders – closer to the initial interface – would have resulted in strategies that included plunging. However, a simulation of the same case (not shown), with the two cylinders starting immersed in the initial interface, came to the same conclusion: while, by design, there is a small amount of plunging in this case, the vast majority of the mix-norm reduction has been accomplished by the shedding of start-and-stop vortices by a vigorous oscillatory motion of either cylinder and a subsequent collision of the generated vortices. The utilization of the stirrers’ energy to shed small “vortical stirrers” is a better strategy than the distortion of the interface by simply moving through it with the stirrers.
The later part of the stirring strategy includes vortex collisions (see figure \[fig:EnConstraint1\][*[b]{}*]{}), obstruction by the stirrers (see figures \[fig:EnConstraint1\][*[c,d]{}*]{}) and collision with the outer wall (see figure \[fig:EnConstraint1\][*[e]{}*]{}) to yield a well-mixed state at the end of the simulation horizon (figure \[fig:EnConstraint1\][*[f]{}*]{}).
Increasing the control horizon from $T_{\rm{control}}=1$ to $T_{\rm{control}}=8$ leads to similar conclusions, even though the stirring action by the cylinders is less abrupt and jarring. Still, the bulk of the mixing action is achieved by shedding start-and-stop vortices which collide with themselves, secondary vortices and the wall to produce a mixed state in the end. Again, the absence of plunging is noteworthy. This is even more remarkable, as the increased control time horizon would certainly allow the stirrers to approach and reach the interface; yet, they remain close to their initial position.
In both cases, the strategy found by the direct-adjoint optimisation technique will yield increasingly larger velocities, as long as the integrated energy is constant. Eventually, the energy expenditure becomes more and more localized in time, with the stirrers barely moving. This optimisation route is a logical consequence of our current setup. It is closely connected to the semi-norm problem (see [@Foures2012; @Blumenthal2017]): the mix-norm only contains the passive scalar $\theta$, but does not account for the other dynamic variable, the velocities, in the optimisation. The energy of the stirrers is not sufficient to arrive at realistic stirring protocols that could be implemented in an experimental or industrial setting. To ensure applicability of our stirrer strategies to real-life settings, additional constraints are required.
Cases 3 and 4: optimisation under energy and velocity constraints
-----------------------------------------------------------------
Following the findings of the previous section, in the next examples we limit the velocity of the stirrers to avoid excessive values of ${\bf{u}}_{s,i}.$ This capping of the velocity is implemented by a projection of the raw cost-functional gradient onto control strategies that satisfy the given constraints (for details of this projection technique see [@Foures2014]). The resulting limit on the stirrer velocities provides a longer time window (up to the control horizon $T_{\rm{control}}$) over which the specified energy can be expended. As a consequence, an extended and smoother movement of the stirrers is expected.
For the shorter control horizon $T_{\rm{control}}=1,$ figure \[fig:VelConstraint1\] shows the outcome of our optimisation. The top stirrer starts by an oscillatory motion, creating start-and-stop vortices. The capping of its velocity, however, keeps the shed vortices within bounds; nonetheless, the optimality of the “vortex cannon” strategy can still be exploited. Both stirrers then move closer to the (already distorted) interface. But rather than plunging through it, they abruptly stop short of it and let the overtaking stop-vortices carry out the distortion of the interface and the subsequent mixing. Again, the optimisation algorithm selects the mixing by shed vortices over the plunging of the stirrers through the interface. The remaining mixing process is characterized by vortex collision (see figures \[fig:VelConstraint1\][*[a,d]{}*]{}), collision with the wall (see figure \[fig:VelConstraint1\][*[c]{}*]{}) and stirrer obstruction (see figures \[fig:VelConstraint1\][*[c,d]{}*]{}).
Extending the horizon $T_{\rm{control}}$ over which control is applied results in a change of strategy (see figure \[fig:VelConstraint8\]). The top stirrer now plunges through the interface – but not before stopping and starting on its circular path towards it. This uneven motion creates more vortical structures in the stirrer’s wake that add to the sole plunging action of the stirrer itself. The result is a far more distorted interface (and consequently a lower mix-norm) than would be generated by a simple traversal. At the end of the motion, a back-and-forth motion is performed to generate, within the chosen energy and velocity constraints, additional shed vortices that further interact with the interface and other vortical elements. The second stirrer does not follow the strategy of the first. It engages in an oscillatory motion along its circular path and generates, as before, the resulting start-and-stop vortices that distort the interface and interact with the other vortices inside the container. Again, obstruction by the cylinders (see figures \[fig:VelConstraint8\][*[b,d]{}*]{}) and vortex and wall collisions (see figures \[fig:VelConstraint8\][*[c,d]{}*]{}) contribute to the continued mixing.
[ccc]{}
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint1\] Mixing optimisation based on energy and velocity constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/VelPen1/VelPen1_37.png "fig:"){height="27.00000%"}]{};
plot ([1.4\*cos()]{}, [1.4\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([0.575\*cos()]{}, [0.575\*sin()]{});
(0.1,-0.6) to \[out=0,in=-90\] (0.3,-0.3) to \[out=90,in=-15\] (0,0); (0.4,0.6) to \[out=0,in=90\] (0.8,0.4) to \[out=-90,in=0\] (0.5,0.2);
(-1.5,-1.5) node [$(a)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint1\] Mixing optimisation based on energy and velocity constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/VelPen1/VelPen1_85.png "fig:"){height="27.00000%"}]{};
(0.6,-0.6) to \[out=-135,in=0\] (0,-0.8); (-0.1,0.3) to \[out=160,in=-90\] (-0.5,0.9) to \[out=90,in=160\] (0.5,1.45);
(-1.5,-1.5) node [$(b)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint1\] Mixing optimisation based on energy and velocity constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/VelPen1/VelPen1_121.png "fig:"){height="27.00000%"}]{};
(-0.3,-0.5) to \[out=155,in=90\] (-1.1,-0.8) to \[out=-90,in=180\] (-0.6,-1.1); (1.2,0.9) to \[out=-45,in=90\] (1.3,0.6) to \[out=-90,in=0\] (1,0.4) to \[out=180,in=-90\] (0.7,0.7) to \[out=90,in=-135\] (0.9,1);
(-1.5,-1.5) node [$(c)$]{}; (2,-1.65) node ;
\
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint1\] Mixing optimisation based on energy and velocity constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/VelPen1/VelPen1_199.png "fig:"){height="27.00000%"}]{};
(0,-0.4) to \[out=135,in=-90\] (-0.3,0.3); (-0.6,-1) – (-0.45,-0.9); (-0.3,-1) – (-0.45,-0.9); (-0.45,-0.85) to \[out=95,in=0\] (-0.8,-0.4) to \[out=180,in=75\] (-1.05,-0.7);
(-1.5,-1.5) node [$(d)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint1\] Mixing optimisation based on energy and velocity constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/VelPen1/VelPen1_439.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(e)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint1\] Mixing optimisation based on energy and velocity constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/VelPen1/VelPen1_748.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(f)$]{}; (2,-1.65) node ;
[ccc]{}
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint8\] Same as figure \[fig:VelConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/VelPen8/VelPen8_73.png "fig:"){height="27.00000%"}]{};
plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.4\*cos()]{}, [1.4\*sin()]{});
plot ([0.475\*cos()]{}, [0.475\*sin()]{}); plot ([0.575\*cos()]{}, [0.575\*sin()]{}); plot ([0.675\*cos()]{}, [0.675\*sin()]{});
(1.7,-0.2) to \[out=95,in=0\] (1.3,0.1) to \[out=180,in=90\] (0.9,-0.3) to \[out=-90,in=135\] (1.1,-0.6);
(-1.5,-1.5) node [$(a)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint8\] Same as figure \[fig:VelConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/VelPen8/VelPen8_148.png "fig:"){height="27.00000%"}]{};
(1.5,0.7) to \[out=-65,in=90\] (1.6,0.2) to \[out=-90,in=25\] (0.4,-0.8); (1.1,-0.2) to \[out=-135,in=0\] (0.6,-0.5) to \[out=180,in=-105\] (0.3,-0.1); (-0.5,-1.5) to \[out=165,in=-45\] (-0.9,-1.3);
(-1.5,-1.5) node [$(b)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint8\] Same as figure \[fig:VelConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/VelPen8/VelPen8_358.png "fig:"){height="27.00000%"}]{};
(-0.5,-1.2) to \[out=-155,in=90\] (-0.6,-1.35) to \[out=-90,in=180\] (-0.2,-1.6); (-0.7,-1.1) to \[out=180,in=-90\] (-1.1,-0.6) to \[out=90,in=-155\] (-0.7,-0.1); (0.9,-0.3) to \[out=-35,in=110\] (1.2,-0.7);
(-1.5,-1.5) node [$(c)$]{}; (2,-1.65) node ;
\
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint8\] Same as figure \[fig:VelConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/VelPen8/VelPen8_502.png "fig:"){height="27.00000%"}]{};
(1.1,-1) to \[out=-135,in=60\] (0.9,-1.3) to \[out=-120,in=30\] (0.5,-1.6);
(-1.5,-1.5) node [$(d)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint8\] Same as figure \[fig:VelConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/VelPen8/VelPen8_601.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(e)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:VelConstraint8\] Same as figure \[fig:VelConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/VelPen8/VelPen8_691.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(f)$]{}; (2,-1.65) node ;
In both cases, a gentler stirring strategy is observed. However, the problem of converging towards a realistic mixing protocol has not been solved completely. While we explicitly avoid highly localized action of the stirrers with excessive velocities, we now tend towards favoring strategies with excessive acceleration. In other words, within our efforts to limit the total expended energy of the stirrers while capping their velocities, the optimisation algorithms tends towards strategies that are characterized by large accelerations (high velocity gradients). This should not come as a surprise as the strength of shed vortices from the stirrers’ unsteady motion is proportional to their acceleration. Our imposed constraints do account for energy and velocities, but not velocity gradients, of the stirrers. As a consequence, we can seed our binary mixture with vortical elements of nearly unlimited strength. Again, this divergence is related to the above-mentioned semi-norm problem: the velocity field of the binary mixture is not accounted for in the mix-norm, and thus the optimisation scheme can achieve high-energy [*[fluid]{}*]{} states by highly accelerating stirrers (even though the stirrers’ energy and velocities are capped). To limit the velocity of the [*[fluid]{}*]{}, we have to limit the acceleration of the stirrers. Again, additional constraints are necessary.
Cases 5 and 6: optimisation under energy, velocity and acceleration constraints
-------------------------------------------------------------------------------
For accomplishing enhanced mixing in binary fluids, the direct-adjoint optimisation technique makes heavy use of an acceleration-based strategy: shed vortices generated by the abrupt motion of the stirring cylinders are injected into both fluids, and their interactions with the interface, themselves and the container wall yield a low mix-norm. A limit on this acceleration will result in a limit on the velocities in either fluid component and thus provide the necessary restriction for a successful semi-norm optimisation. To this end, we augment our optimisation scheme by additional terms accounting for the stirrers’ acceleration. This type of penalisation is common in deblurring of images where strong gradients are detected and encouraged. In our case, additional projections are used to enforce the acceleration constraints.
For the short-term control with $T_{\rm{control}}=1$ (see figure \[fig:AccConstraint1\]), the optimal strategy now includes a plunging of the first cylinder through the interface, while the second cylinder continues in a straight manner towards the interface but stops short of it. The wake vortices of the first cylinder, as well as the (weaker) start-and-stop vortices of both cylinders, are responsible for the bulk of the mixing. As before, complex vortex collisions (see figure \[fig:AccConstraint1\][*[c]{}*]{}), stirrer obstruction (see figures \[fig:AccConstraint1\][*[c,d]{}*]{}) and wall interactions (see figure \[fig:AccConstraint1\][*[d]{}*]{}) contribute greatly to the breakdown of scales, the generation of filaments (see figure \[fig:AccConstraint1\][*[e]{}*]{}) and the eventual mixing of the binary fluid (see figure \[fig:AccConstraint1\][*[f]{}*]{}).
A longer control horizon of $T_{\rm{control}}=8$ yields a more varied stirring protocol. The first cylinder makes a farther excursion, plunging through the interface (not without stopping to generate additional shed vortices close to the interface) before stopping close to the interface and shedding two stop vortices. The second cylinder first approaches the interface, ejects a stop vortex before reversing and stopping short of the interface with another stop vortex. The generated structures interact with themselves and the wall to break down the binary fluid into a homogeneous mixture, although of less homogeneity (larger mix-norm) than for the short-term strategy. This reduced homogeneity can be attributed more to the restricted velocity range (which is due to the constraint of an equal energy budget across both time horizons) than the larger time-horizon strategy. The lower velocity maximum, combined with the limitation on acceleration, impedes the same amount of vortex shedding than for the shorter time-horizon case. For this reason, mixing cannot be as efficient.
At no point during either optimisation has the energy, velocity or acceleration of the stirrers exceeded the specified limits. As a consequence, these latter strategies are amenable to implementation in an experimental or industrial setting.
[ccc]{}
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint1\] Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/AccPen1/AccPen1_16.png "fig:"){height="27.00000%"}]{};
plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([0.575\*cos()]{}, [0.575\*sin()]{});
(0.8,0.2) to \[out=0,in=90\] (1.6,-0.2) to \[out=-90,in=50\] (1.5,-0.4); (1.3,-0.2) to \[out=-155,in=0\] (0.5,-0.5) to \[out=180,in=-55\] (0.1,-0.2);
(-1.5,-1.5) node [$(a)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint1\] Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/AccPen1/AccPen1_37.png "fig:"){height="27.00000%"}]{};
(-0.1,0.1) to \[out=135,in=-90\] (-0.3,0.5) to \[out=90,in=-155\] (0.5,1.3); (1.2,-0.2) to \[out=-110,in=0\] (0.7,-0.7) to \[out=180,in=-110\] (0.4,-0.1);
(-1.5,-1.5) node [$(b)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint1\] Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/AccPen1/AccPen1_64.png "fig:"){height="27.00000%"}]{};
(0.5,0.2) – (0.7,0.45); (0.9,0.7) – (0.7,0.45); (0.6,0.55) to \[out=135,in=-90\] (0.5,0.75) to \[out=90,in=-155\] (0.7,0.9); (0.8,0.35) to \[out=-45,in=0\] (0.5,-0.2) to \[out=180,in=-90\] (0.1,0.4) to \[out=90,in=-115\] (0.3,1); (-0.8,-0.4) to \[out=180,in=-65\] (-1.4,0);
(-1.5,-1.5) node [$(c)$]{}; (2,-1.65) node ;
\
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint1\] Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/AccPen1/AccPen1_127.png "fig:"){height="27.00000%"}]{};
(1.3,-0.5) to \[out=45,in=-90\] (1.6,0.2) to \[out=90,in=45\] (0.3,-0.1); (-1.1,-0.2) to \[out=180,in=180\] (-1.1,-0.6) to \[out=0,in=-100\] (-0.8,0.4) to \[out=80,in=-125\] (-0.5,1);
(-1.5,-1.5) node [$(d)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint1\] Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/AccPen1/AccPen1_289.png "fig:"){height="27.00000%"}]{};
(0.2,-0.3) to \[out=0,in=155\] (0.5,-0.45); (0.8,-0.3) to \[out=180,in=25\] (0.5,-0.45); (0.5,-0.5) to \[out=-90,in=5\] (-0.4,-1); (0.2,1.2) to \[out=0,in=-150\] (0.8,1.35);
(-1.5,-1.5) node [$(e)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint1\] Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is $T_{\rm{control}}=1.$ Shown are iso-contours of the passive scalar at selected instances. The optimisation algorithm includes information over a time window of $T_{\rm{info}}=8.$](figures/AccPen1/AccPen1_748.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(f)$]{}; (2,-1.65) node ;
[ccc]{}
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint8\] Same as figure \[fig:AccConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/AccPen8/AccPen8_91.png "fig:"){height="27.00000%"}]{};
plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{}); plot ([1.3\*cos()]{}, [1.3\*sin()]{});
plot ([0.575\*cos()]{}, [0.575\*sin()]{}); plot ([0.675\*cos()]{}, [0.675\*sin()]{});
(0.4,0) to \[out=35,in=-110\] (0.65,0.4); (0.8,0.1) to \[out=10,in=90\] (1.2,-0.2); (0.85,0.75) to \[out=45,in=-90\] (1,1.1); (0.85,0.75) to \[out=45,in=180\] (1.15,0.85) to \[out=0,in=95\] (1.35,0.4);
(-1.5,-1.5) node [$(a)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint8\] Same as figure \[fig:AccConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/AccPen8/AccPen8_157.png "fig:"){height="27.00000%"}]{};
(0.8,-1) to \[out=45,in=-40\] (0.55,-0.55); (1.15,-0.5) to \[out=-135,in=-50\] (0.6,-0.5); (0.4,-0.9) to \[out=-125,in=15\] (-0.2,-1.3); (-1.5,0.05) to \[out=90,in=-45\] (-1.6,0.3); (-1.1,0.05) to \[out=90,in=-60\] (-1.2,0.4);
(-1.5,-1.5) node [$(b)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint8\] Same as figure \[fig:AccConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/AccPen8/AccPen8_223.png "fig:"){height="27.00000%"}]{};
(0.5,-0.8) to \[out=160,in=-160\] (0.4,-0.1); (-0.85,0.55) to \[out=60,in=-165\] (0.4,1.3);
(-1.5,-1.5) node [$(c)$]{}; (2,-1.65) node ;
\
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint8\] Same as figure \[fig:AccConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/AccPen8/AccPen8_400.png "fig:"){height="27.00000%"}]{};
(0.6,-0.2) to \[out=-20,in=100\] (0.9,-0.8); (0.9,0.7) to \[out=-60,in=80\] (1,-0.6); (0.5,-1.2) to \[out=180,in=-75\] (-0.4,-0.5);
(-1.5,-1.5) node [$(d)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint8\] Same as figure \[fig:AccConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/AccPen8/AccPen8_628.png "fig:"){height="27.00000%"}]{}; (-1.5,-1.5) node [$(e)$]{}; (2,-1.65) node ;
&
(0,0) node\[inner sep=0\] [![\[fig:AccConstraint8\] Same as figure \[fig:AccConstraint1\], but with an extended control window of $T_{\rm{control}}=8.$](figures/AccPen8/AccPen8_727.png "fig:"){height="27.00000%"}]{};
(-1.5,-1.5) node [$(f)$]{}; (2,-1.65) node ;
Summary, conclusions and remaining challenges
=============================================
A direct-adjoint optimisation methodology has been applied to the problem of mixing of binary fluids. A circular configuration with two embedded stirrers on circular paths has been chosen, and the velocities of these two stirrers over a user-specified time interval have been determined in an attempt to enhance the homogeneity of the binary mixture. The gradient-based optimisation is effective in finding stirring protocols that yield enhanced mixing results, but convergence towards an optimum crucially relies on imposing proper constraints on the iterative algorithm. Since the mixing efficiency is based on only one dynamic variable, the passive scalar $\theta,$ but disregards the velocity fields, additional external constraints have to be imposed to properly define a feasible optimum. These constraints have to enforce limitations on the encountered fluid velocities, which are forced by accelerating the stirrers. The accelerations, in turn, inject vortical structures into both fluids by unsteady, Stokes-layer-type shedding of vortices. Thus, by restricting the stirrers’ maximum acceleration, we arrive at a properly stated optimisation problem and a convergent direct-adjoint algorithm.
Under these conditions, the optimal strategy that improves on the mixedness of the binary fluid utilises a combination of prototypical mixing techniques, consisting of plunging, unsteady vortex shedding, collisions between vortices and the wall, and obstructions by the stirrers. While unsteady vortex shedding is the key strategy for the unconstrained (or insufficiently constrained) case, a more balanced protocol ensues when excessive accelerations are increasingly penalised. Nonetheless, a rather counterintuitive optimal mixing strategy has been determined for short and longer time-horizons. As a general tendency, shorter control windows reach a lower mix-norm state, as the stirrer motion is more vigorous over a more limited horizon.
In conclusion, the above direct-adjoint approach to mixing of binary fluids – when combined with an efficient spectral simulation scheme, a Brinkman-type penalisation to accommodate moving bodies, and a systematic checkpointing technology – has proven an effective and robust tool to design stirrer strategies for the enhancement of mixing. After proper limitations on the stirrers’ acceleration have been taken into account and a proper measure of mixedness has been defined, stirring strategies exploiting the full range of fluid processes induced by fluid-structure interactions are found, that suggest realisable modifications to commonly employed stirrer-induced mixing methods for industrial applications. The accomplished increase in mixing efficiency is summarised in table \[tab:Cases\] which lists the mix-norm values for $t=1$, $t=8$ and $t=32.$ With an imposed acceleration penalisation, the short control horizon ($T_{\rm{control}}=1$) gives markedly better results than a longer one ($T_{\rm{control}}=8$), even though the difference is less pronounced after rest-inertia and diffusion set in. The number of iterations taken by the direct-adjoint algorithm is displayed as well; longer time horizons typically converge faster, owing to a less abrupt stirrer protocol. The short-horizon case with only energy penalisation (case 1, above) is included for comparison.
-------- -- ---------------------- ------------ -- -------------------------------- -------------------------------- ---------------------------------
$T_{control}$ iterations $\Vert \theta \Vert_{mix,t=1}$ $\Vert \theta \Vert_{mix,t=8}$ $\Vert \theta \Vert_{mix,t=32}$
AccPen $T_{\rm{control}}=1$ 11 0.3230 0.0745 0.0433
AccPen $T_{\rm{control}}=8$ 5 0.3885 0.2437 0.0558
EnPen $T_{\rm{control}}=1$ 12 0.3769 0.1724 0.0597
-------- -- ---------------------- ------------ -- -------------------------------- -------------------------------- ---------------------------------
: \[tab:Cases\] Summary of results for acceleration penalisation (AccPen) and energy-only penalisation (EnPen). Short ($T_{\rm{control}}=1$) and longer ($T_{\rm{control}}=8$) control horizons are displayed, together with the number of iterations taken by the direct-adjoint optimisation algorithm.
Despite a successful increase of mixing efficiency via uncommon and unexpected strategies and despite corroborating and supporting the chosen computational approach, the present study also raises a number of challenges and shortcomings. Foremost among them is the fact that gradient-based optimisation with nonlinear partial differential equations as constraints can only assure a local optimum; a globally optimal solution requires an additional, and often prohibitive, methodology, such as simulated annealing or other variations of the same concept. While a global optimum may certainly be desirable, we point out that the improvements in mixing efficiency from a local solution (as shown in this study) would already have a respectable impact on mixing results due to its omnipresence in many industrial settings. In this sense, improvements suggested by locally optimal solutions would be most welcome.
A further challenge consists of additional constraint handling, imposed by mechanical restrictions on the stirrer motion. Penalisation methods or auxiliary projections imposed on the gradient information are conceivable to address this issue.
The length of the optimisation window $T_{\rm{info}}$ places a bound on the overall algorithm. The adjoint part of the simulations computes the sensitivity of the output functional (mix norm) with respect to our control parameters (stirrer velocities along their paths). For increasingly large optimisation windows these sensitivities diverge due to the quasi-chaotic behavior of the direct problem. As a result, meaningful sensitivity to aid our optimisation will get overwhelmed by general sensitivity due to chaotic motion, and if meaningful sensitivity is lost, the optimisation algorithm will stagnate or even diverge. An advance to higher Reynolds or Péclet numbers will encounter similar issues. While techniques to overcome this predicament are currently being developed [@Blonigan], their cost-efficient application to complex systems, such as mixing, is still an open problem.
However, within the constraints of this study, with a physical problem this rich in possibilities and with a computational approach to match, there is an abundance of extensions and opportunities. Besides obvious explorations of other parameter combinations, the optimisation of the stirrers’ shape is certainly within the capabilities of the computational framework; a preliminary study in this direction can be found in [@Eggl2019] using cycloids and trochoids as cross-sectional stirrer geometries. The path of the stirrers (in our study, concentric circles) can also be optimised; a collision-avoiding constraint may pose an additional challenge in this case. Furthermore, since the wall constitutes an important component in the breakup of vortices, an optimisation of wall motion or wall corrugation may further facilitate a more rapid breakdown in scales. With a view towards industrial applications, a non-Newtonian fluid model may be implemented. Finally, injection mixing (where the unmixed fluids are introduced into a mixing device or passive baffle system, and extracted when fully mixed) could be treated within the same direct-adjoint framework. These possible extensions, some of which will be reported in future efforts, attest to the flexibility and efficacy of the computational setup; marked enhancements in mixing are expected in the above cases.
Declaration of Interests. The authors report no conflict of interest.
|
---
abstract: 'Infrared absorption cross sections near 3.3 $\mu$m have been obtained for ethane, C$_{2}$H$_{6}$. These were acquired at elevated temperatures (up to 773 K) using a Fourier transform infrared spectrometer and tube furnace with a resolution of 0.005 cm$^{-1}$. The integrated absorption was calibrated using composite infrared spectra taken from the Pacific Northwest National Laboratory (PNNL). These new measurements are the first high-resolution infrared C$_{2}$H$_{6}$ cross sections at elevated temperatures.'
address: 'Department of Chemistry, Old Dominion University, 4541 Hampton Boulevard, Norfolk, VA 23529, USA'
author:
- 'Robert J. Hargreaves'
- Eric Buzan
- Michael Dulick
- 'Peter F. Bernath'
bibliography:
- 'Hot-Ethane-BIBFILE.bib'
title: 'High-resolution absorption cross sections of C$_{2}$H$_{6}$ at elevated temperatures'
---
giant planets ,high temperatures ,exoplanets ,absorption cross sections ,infrared ,high-resolution\
*Chemical compounds:* ethane (PubChem CID: 6324)
Introduction {#sect1}
============
Ethane (C$_{2}$H$_{6}$) is the second largest component of natural gas and is primarily used in the industrial manufacture of petrochemicals. It is present as a trace gas in the Earth’s atmosphere and can be used to monitor anthropogenic (e.g., fossil fuel emission, combustion processes) and biogenic sources [@2011Natur.476..198A; @2007JQSRT.107..407K; @2011ACP....1112169T].
However, C$_{2}$H$_{6}$ is also of particular interest to astronomy. C$_{2}$H$_{6}$ is found in all four giant planets [@1974ApJ...187L..41R; @1981Sci...212..192H; @1987Icar...70....1O], Titan [@2005Natur.438..779N], comets [@1996Sci...272.1310M] and even as an ice in Kuiper Belt objects [@2007AJ....133..284B]. For Titan, observations indicate C$_{2}$H$_{6}$ is a constituent of light hydrocarbon lakes [@2008Natur.454..607B]. In the atmospheres of the giant planets and Titan, C$_{2}$H$_{6}$ is primarily formed from the photolysis of methane, CH$_{4}$ [@2007Icar..188...47N; @2008SSRv..139..191M], and subsequent recombination of methyl radicals, CH$_{3}$ [@2009JPCA..11311221W; @2009Icar..201..226K; @2014Icar..236...83K].
In Jupiter, stratospheric observations have detected hot C$_{2}$H$_{6}$ in polar auroral regions [@2009Icar..202..354K]. These hot spots occur close to similar hot CH$_{4}$ and H$^{+}_{3}$ emission [@2015Icar..257..217K] and are heated due to the channeling of particles by the strong magnetic field. The Juno mission [@2007AcAau..61..932M] is due to arrive at Jupiter in 2016 and one major objective for the Jovian Infrared Auroral Mapper, JIRAM [@2008AsBio...8..613A], is to study these auroral hot spots to determine the molecules responsible and their vertical structure.
Brown dwarfs are sub-stellar objects that do not burn hydrogen in their cores . They are warm (albeit relatively cool in comparison to stars), thereby allowing for the formation of rich molecular atmospheres. Brown dwarf atmospheric chemical models predict C$_{2}$H$_{6}$ to form deep in these objects [@2002Icar..155..393L]. Recent observations indicate that these objects can also harbour extremely bright aurorae [@2015aurora]. Similarly, exoplanets known as hot-Jupiters orbit close to their parent star and have atmospheric temperatures conducive to molecule formation. While models predict C$_{2}$H$_{6}$ may have a low thermochemical abundance in the atmosphere of these objects , disequilibrium and increased metallicity can lead to significant enhancements [@2011ApJ...738...32L; @2011ApJ...737...15M; @2013MNRAS.435.1888B]. C$_{2}$H$_{6}$ can be used as a useful temperature probe for exoplanets and brown dwarfs , but high temperature laboratory data are missing. It is therefore important to have high temperature data available for astronomical and terrestrial applications.
Due to the prevalence of C$_{2}$H$_{6}$, the infrared spectrum has been the focus of numerous studies, but complete line assignments are difficult to obtain. This is, in part, due to the $\nu_{4}$ torsional mode near 35 $\mu$m (290 cm$^{-1}$) [@1999JChPh.111.9609M; @2001JMoSp.209..228M; @2008JMoSp.250...51B; @2015JQSRT.151..123M] which produces numerous low frequency hot bands, extensive perturbations and a very dense line structure. The $\nu_{9}$ mode near 12 $\mu$m (830 cm$^{-1}$), often used in remote sensing [@2007ACP.....7.5861C; @2015Icar..250...95V], has been the focus of comprehensive analyses that have significantly improved line assignments [@2007ApJ...662..750V; @2010JQSRT.111.1234M; @2010JQSRT.111.2481M]. Line parameters and assignments have been obtained for the $\nu_{8}$ band near 6.8 $\mu$m (1470 cm$^{-1}$) [@2008JMoSp.248..134L; @2011MolPh.109.2219L; @2012PandSS...60...93D] as well as the $\nu_{5}$ and $\nu_{7}$ modes contained in the 3.3 $\mu$m (3000 cm$^{-1}$) spectral region [@2011JGRE..116.8012V; @2011JMoSp.267...71L]. Although considerable progress has been made in these recent studies, high-resolution analyses are generally incomplete and still fail to match laboratory observations precisely.
The Pacific Northwest National Laboratory (PNNL) has recorded infrared absorption cross sections for a large number of species, including C$_{2}$H$_{6}$ (see <http://nwir.pnl.gov> and Ref. [@2004ApSpe..58.1452S]), at three temperatures (278, 293 and 323 K). High-resolution (0.004 cm$^{-1}$) absorption cross sections have been provided at room temperature [@2010JQSRT.111..357H] and these measurements constitute the C$_{2}$H$_{6}$ cross sections contained in HITRAN [@2013JQSRT.130....4R]. However, the intended use of these data are for the study of the Earth’s atmosphere and will give an incorrect radiative transfer when applied to high temperature environments. High-temperature absorption cross sections of hydrocarbon species (including C$_{2}$H$_{6}$) have been recorded for combustion applications [@2014JMoSp.303....8A] [at relatively low]{} resolution ($\ge$0.16 cm$^{-1}$) [and are]{} not sufficient for high-resolution applications. The aim of this work is to provide high-resolution absorption cross sections of C$_{2}$H$_{6}$ at elevated temperatures to be used in the analysis of brown dwarfs, exoplanets and auroral hot spots of Jupiter.
Measurements {#sect2}
============
Spectra were acquired of C$_{2}$H$_{6}$ between 2200 and 5600 cm$^{-1}$ (1.8 $-$ 4.5 $\mu$m) using a Fourier transform infrared spectrometer at a resolution of 0.005 cm$^{-1}$. These spectra cover the temperatures 296 $-$ 773 K and experimental conditions are provided in Table \[tab1\].
[@lc@]{}
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
Parameter & Value*$^{a}$*\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
Temperature range & 296 $-$ 773 K\
Spectral range & 2200 $-$ 5600 cm$^{-1}$\
Resolution & 0.005 cm$^{-1}$\
Path length & 0.5 m\
Sample cell material & Quartz (SiO$_{2}$)\
External source & External globar*$^{b}$*\
Detector & Indium antimonide (InSb)\
Beam splitter & Calcium fluoride (CaF$_{2}$)\
Spectrometer windows & CaF$_{2}$\
Filter & Germanium\
Aperture & 1.5 mm\
Apodization function & Norton-Beer, weak\
Phase correction & Mertz\
Zero-fill factor & $\times$16\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
\
\
The spectrometer is combined with a tube furnace containing a sample cell made entirely from quartz, thereby allowing the cell to be contained completely within the heated portion of the furnace [@2015ApJ..inpressH]. At elevated temperatures, the C$_{2}$H$_{6}$ infrared spectrum has both emission and absorption components. The emission components [can be]{} included in the final transmittance spectra by following the same procedure outlined in Ref. [@2015ApJ..inpressH] for CH$_{4}$. This involves recording both C$_{2}$H$_{6}$ absorption ($A_{\scriptsize{\textrm{ab}}}$) and emission ($B_{\scriptsize{\textrm{em}}}$) spectra, then combining as $$\label{eqn1}
\tau = \frac{A_{\scriptsize{\textrm{ab}}} -– B_{\scriptsize{\textrm{em}}}}{A_{\scriptsize{\textrm{ref}}} –- B_{\scriptsize{\textrm{ref}}}},$$ to give the transmittance spectrum ($\tau$), where $A_{\scriptsize{\textrm{ref}}}$ and $B_{\scriptsize{\textrm{ref}}}$ are the background reference spectra of the absorption and emission, respectively. [The emission component of C$_{2}$H$_{6}$ is sufficiently strong at 673 and 773 K that an emission correction is necessary; therefore $B_{\scriptsize{\textrm{em}}}$ and $B_{\scriptsize{\textrm{ref}}}$ are required. For lower temperatures, $B_{\scriptsize{\textrm{em}}}$ and $B_{\scriptsize{\textrm{ref}}}$ equal zero and Equation \[eqn1\] reverts to the standard transmittance equation (i.e., $\tau = A_{\scriptsize{\textrm{ab}}}/A_{\scriptsize{\textrm{ref}}}$).]{}
The C$_{2}$H$_{6}$ infrared spectrum near 3000 cm$^{-1}$ (3.3 $\mu$m) contains a small number of $\nu_{7}$ $Q$-branch features that are significantly stronger than the $P$- and $R$-branches and the nearby $\nu_{5}$ mode. In order to maximize the signal from the weaker features, the C$_{2}$H$_{6}$ spectra were acquired at both “high” and “low” pressure. These high and low pressure experiments are summarised in Table \[tab2\]. The low pressure spectra were recorded to determine the absorption cross sections of these strong $Q$-branch features. The $Q$-branch cross sections were then added to the high pressure absorption cross sections in place of the high pressure $Q$-branch features, which had been intentionally saturated (see Section \[sect3\]).
[@ccccccc@]{}
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
& Sample & Sample C$_{2}$H$_{6}$ & &\
& & & Sample & Background\
Mode & Temperature & Pressure & &\
& & & Scans & Scans\
& (K) & (Torr) & &\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
& 297 & 0.276 & 400 & 550\
& 297 & 0.035 & 24 & 24\
& 473 & 0.982 & 300 & 300\
& 473 & 0.176 & 24 & 24\
& 573 & 1.476 & 300 & 300\
& 573 & 0.282 & 24 & 24\
& 673 & 2.928 & 150 & 150\
& 673 & 0.987 & 24 & 24\
& 773 & 5.026 & 150 & 150\
& 773 & 1.557 & 24 & 24\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
& 673 & 3.067 & 150 & 150\
& 673 & 1.008 & 24 & 24\
& 773 & 5.206 & 150 & 150\
& 773 & 1.534 & 24 & 24\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
Absorption cross sections {#sect3}
=========================
{width="90.00000%"}
{width="90.00000%"}
An absorption cross section, $\sigma$ (cm$^{2}$ molecule$^{-1}$), can be calculated using $$\label{crosssections}
\sigma = -\xi\frac{10^{4}k_{\textrm{B}}T}{Pl}\ln\tau,$$ where $T$ is the temperature (K), $P$ is the pressure of the absorbing gas (Pa), $l$ is the optical pathlength (m), $\tau$ is the observed transmittance spectrum, $k_{\textrm{\scriptsize{B}}}$ is the Boltzmann constant and $\xi$ is a normalization factor [@2010JQSRT.111..357H; @2015JMS..inpressH].
It has been demonstrated by numerous studies on a variety of molecular spectra that integrating an absorption cross section over an isolated band (containing primarily fundamentals) exhibits an insignificant temperature dependence [@1958JChPh..29.1042C; @1959JChPh..30.1619M; @1965JChPh..42..402B; @1976AcSpA..32.1059Y; @2010JQSRT.111.1282H; @2010JQSRT.111..357H; @2012JQSRT.113.2189H].
The PNNL infrared absorption cross sections of C$_{2}$H$_{6}$ cover the spectral range 600–6500 cm$^{-1}$ (resolution of 0.112 cm$^{-1}$) at 278, 293 and 323 K. Each PNNL cross section is a composite of approximately ten pathlength concentrations, making these data suitably accurate for calibration [@2010JQSRT.111..357H]. For the spectral region between 2500 and 3500 cm$^{-1}$ the average PNNL integrated absorption is calculated as $$\label{intint}
\begin{aligned}
\begin{split}
\int^{3500 \textrm{ cm}^{-1}}_{2500 \textrm{ cm}^{-1}} \sigma(\nu,T) d\nu = & 2.976(\pm0.011) \times 10^{-17} \\ & \textrm{ cm molecule}^{-1}.
\end{split}
\end{aligned}$$ Each individual PNNL cross section demonstrates less than 0.4% deviation from this value[^1].
The new transmittance spectra have been converted into cross sections using Equation \[crosssections\], making the original assumption that $\xi=1$. This is to allow the strong $\nu_{7}$ $Q$-branch features from the low pressure observations to be inserted in place of the same saturated (therefore distorted) $Q$-branch features in the high pressure absorption cross sections. Each replaced $Q$-branch region covered less than $\sim$0.2 cm$^{-1}$ and was chosen to be between the points where the high and low pressure cross sections [intersect]{} either side of the strong feature. These composite absorption cross sections were then integrated over the 2500 and 3500 cm$^{-1}$ spectral region. Comparisons were made to the PNNL integrated absorption cross section (Equation \[intint\]) in order to calibrate our observations. The normalization factors for each absorption cross section are provided in Table \[tab3\], alongside the calibrated pressures and calibrated integrated absorption cross sections.
[@ccccccc@]{}
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
& Normalization & C$_{2}$H$_{6}$ & Integrated absorption\
Temperature & & &\
& Factor & Effective & cross section\
(K) & & &\
& $\xi$ & Pressure (Torr) & ($\times10^{-17}$ cm molecule$^{-1}$)\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
297 & 1.028 & 0.269 & 2.978\
473 & 1.035 & 0.948 & 2.978\
573 & 1.063 & 1.388 & 2.976\
673 & 1.047 & 2.796 & 2.975\
773 & 1.033 & 4.867 & 2.976\
------------------------------------------------------------------------
\
------------------------------------------------------------------------
\
The calibrated cross sections are displayed in Figure \[fig1\] between 2600 and 3300 cm$^{-1}$ and clearly display the $\nu_{5}$ and $\nu_{7}$ [fundamental bands]{}. Figure \[fig2\] shows a 10 cm$^{-1}$ section of Figure \[fig1\] in the vicinity of three weak $\nu_{7}$ $Q$-branch features; an increase in the observed continuum at higher temperatures can be seen.
{width="75.00000%"}
The calibrated absorption cross sections described in Table \[tab3\] are available online[^2] in the standard HITRAN format [@2013JQSRT.130....4R].
Discussion {#sect4}
==========
The normalization factors are necessary to account for the difficulty in measuring the experimental parameters accurately (i.e., pathlength, pressure and temperature). The combination of the errors often leads to an underestimation of the integrated absorption cross section, which is calibrated by comparison to the PNNL data. Normalization factors are typically within 6% for measurements using similar apparatus [e.g., @2010JQSRT.111..357H]. For our measurements, the normalization factor has been used to give an effective calibrated pressure as seen in Table \[fig3\]. Based upon consideration of the experimental and photometric errors, the calibrated cross sections are expected to be accurate to within 4%. [The C$_{2}$H$_{6}$ absorption cross sections available at 194 K are also based on a calibration to the PNNL [@2010JQSRT.111..357H]]{}. These data contains C$_{2}$H$_{6}$ at 0.2208 Torr, which has been broadened by 103.86 Torr of air at a resolution of 0.015 cm$^{-1}$. Integrating the available data between 2545 and 3315 cm$^{-1}$ yields a value of 2.985$\times10^{-17}$ cm molecule$^{-1}$. This is within 0.3% of the average values contained in Table \[tab3\]. An independent quality check can be made by comparing to new C$_{2}$H$_{6}$ absorption cross sections recorded for combustion applications [@2014JMoSp.303....8A]. These data contain medium resolution (0.16-0.6 cm$^{-1}$) nitrogen-broadened cross sections of C$_{2}$H$_{6}$ between 2500 and 3400 cm$^{-1}$. At temperatures of 296, 673 and 773 K the C$_{2}$H$_{6}$ [integrated]{} absorption cross sections were calculated to be 2.81$\times10^{-17}$, 2.99$\times10^{-17}$ and 3.08$\times10^{-17}$ cm molecule$^{-1}$, respectively. While a small temperature dependence is seen when compared to our values in Table \[tab3\], the deviation is within their experimental error (5%).
Experimental spectra of C$_{2}$H$_{6}$ have not been acquired at temperatures above 773 K as the molecules begin to decompose when using a sealed cell. Evidence of CH$_{4}$ absorption was observed at 873 K; therefore reliable C$_{2}$H$_{6}$ cross sections could not be obtained.
C$_{2}$H$_{6}$ is expected to be useful as a temperature probe for exoplanets and brown dwarfs . The infrared spectrum (Figures \[fig1\] and \[fig2\]) demonstrates a continuum-like feature previously observed for CH$_{4}$ at high temperatures [@2015ApJ..inpressH]. It can be seen that as the continuum increases with temperature, the sharp $Q$-branches decrease due to a change in the population of states and they also broaden because of the increasing Doppler width (Figure \[fig3\]). However, Table \[tab3\] shows $\xi$ only exhibits a small change and the integrated intensity remains constant (within experimental error). This variation is small enough to justify the assumption that the integrated absorption cross sections are independent of temperature. For weak concentrations of C$_{2}$H$_{6}$ it may be difficult to observe a change in the continuum, particularly since the 3.3 $\mu$m region contains the prominent C-H stretch for hydrocarbons. However, the shape of the sharp $Q$-branches of the $\nu_{7}$ mode are also seen to change with increasing temperature, as shown in Figure \[fig3\]. These $Q$-branches are relatively easy to identify in congested atmospheric spectra, therefore studying the shape of these features can also be used to infer temperatures.
Conclusion {#sect5}
==========
High-resolution infrared absorption cross sections for C$_{2}$H$_{6}$ have been measured at elevated temperatures (up to 773 K) between 2500 and 3500 cm$^{-1}$. The spectra were recorded at a resolution of 0.005 cm$^{-1}$ and the integrated absorption has been calibrated to PNNL values. These data are of particular interest for simulating astronomical environments at elevated temperatures, such as brown dwarfs and exoplanet atmospheres, where C$_{2}$H$_{6}$ can be used as a temperature probe. With the imminent arrival of the Juno spacecraft into orbit around Jupiter, these data will be of particular use for observations made of aurora using the JIRAM instrument.
#### Acknowledgments
Funding was provided by the NASA Outer Planets Research Program.
References {#references .unnumbered}
==========
[^1]: PNNL units (ppm$^{-1}$m$^{-1}$ at 296 K) have been converted using the factor $k_{B}\times296\times\ln(10)\times10^{4}/0.101325 = 9.28697\times10^{-16}$ [@2012JQSRT.113.2189H]
[^2]: http://bernath.uwaterloo.ca/C2H6/
|
---
abstract: 'We establish a fundamental connection between smooth and polygonal knot energies, showing that the *Minimum Distance Energy* for polygons inscribed in a smooth knot converges to the *Möbius Energy* of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a “nice” way, and the energies must be correctly regularized. We determine an explicit error bound between the energies in terms of the number of the edges of the polygon and the *Ropelength* of the smooth curve.'
address:
- |
Department of Mathematics and Computer Science\
Duquesne University\
Pittsburgh, PA 15282
- |
Department of Mathematics\
University of Iowa\
Iowa City, IA 52242
author:
- 'Eric J. Rawdon'
- 'Jonathan K. Simon'
bibliography:
- 'JonsEricBib04-23-03.bib'
title: Polygonal approximation and energy of smooth knots
---
Polygonal Knots ,Möbius Energy ,Ropelength ,Knot Energy ,Physical Knot Theory
\#1 \#1
Introduction
============
Given a knot $K$ in 3-space, there are several ways to define an “energy function” that measures how complicated the knot is in its spatial conformation. In this paper, we establish a fundamental approximation theorem, showing that when both are appropriately normalized, the *Minimum Distance Energy* for polygons inscribed in a smooth curve converge to the *Möbius Energy* of the curve as the polygons converge to the smooth knot. We do a careful analysis, and determine an explicit error bound (Theorem \[explicit\]), from which the approximation (Theorem \[approaches\]) follows immediately.
In Section \[MainResultsAndNotation\], we state the main theorems and agree on notation for the whole paper. In Section \[lemmas\] we present a number of lemmas. These establish useful properties of curves and chords, so they may be of independent interest. In Section \[outlineproof\], we outline the proof of the error bound (Theorem \[explicit\]), especially how to divide the problem into several cases (more precisely, divide the domains into different “zones”) for which different analyses are needed. In Section \[proofsforzones\], we give the detailed analyses for the various cases, and in Section \[finalproof\], we combine the results from Section \[proofsforzones\] to obtain the overall bound.
Of course the error depends on how well the polygon approximates the smooth curve. However, there are more subtle issues to confront in controlling the error: One must reckon with the amount of curvature the knot has, and how close it is to being self-intersecting. These are captured by the [*thickness radius*]{} $r(K)$ (see later in this section for definition). Our error bound is developed in terms of the total arc-length $\ell(K)$, the number of edges of the inscribed polygon $n$, the mesh size $\delta =\frac{\ell(K)}{n}$, the thickness radius $r(K)$, and the ratio $E_L(K) =\frac{\ell(K)}{r(K)}$. Since these quantities are interrelated, there are various ways to write the bound: the one we give in Theorem \[explicit\] is stated in terms of $n$ and $E_L(K)$ to emphasize that it is invariant under change of scale.
Let $t \rightarrow x(t)$ be a unit-speed parameterization of $K$ with domain a circle C. The [*Möbius Energy*]{} or [*O’Hara Energy*]{} is $$E_0(K) = \iint_{C \times C}
\;\;\;\frac{1}{|x(t)-y(s)|^2}-
\frac{1}{|s-t|^2}\;\;ds\;dt.$$
The energy $E_{0}$ was defined and studied in [@O1; @FHW; @KS; @kusnerkim]. The subscript $0$ in $E_0$ reminds us that this version of the energy is exactly zero if $K$ is a circle.
By visualizing a smooth knot as being made of some “rope”, with a positive thickness, we obtain a fundamental measure of knot complexity. Hold the core knot $K$ fixed and thicken the rope until the moment of self-contact. Call that sup radius the [*thickness radius*]{} or [*injectivity radius*]{}, $r(K)$. Here is a more precise definition: For small enough $r$, the knot $K$ has a solid torus neighborhood consisting of pairwise disjoint disks of radius $r$ centered at the points of $K$ and orthogonal to $K$ at those centers. Gradually increase $r$ until some meridional disks touch; we call that supremum of good radii $r(K)$. The ratio $$E_L(K) = \frac{\text{arc-length of } K}{r(K)}\;,$$ called the [*Rope-Length*]{} of $K$, is a scale-invariant numerical measure of knot compaction.
The basic theorems on thickness appear in [@LSDR], although the energy $E_L$ first appeared in [@BO1]. We recall the properties of $E_L$ in Section \[lemmasaboutthickness\].
Let $P$ be a polygon with $n$ edges. The [*Minimum Distance Energy*]{} of $P$ is defined [@Si2] as follows: For each pair $X,Y$ of nonconsecutive edges of $K$, compute the minimum distance between the segments $MD(X,Y)$, define $U_{\md}(X,Y)=\frac{\text{length}(X)
\,\cdot\, \text{length}(Y) }{{[MD(X,Y)]^{2}} }$, and sum: $$U'_{md}(P) = \sum_{\text{all edges } X } \;\;\sum_{Y \neq X \text{
or adjacent}}U_{\md}(X,Y)\;\;.$$ This version of $\umd$ counts each edge-pair twice, analogous to a double integral over (most of) $K\times K$. We write $U'_{\md}$ just to distinguish from the original version [@Si2] that counted each pair once. To consider knots with varying numbers of segments, we regularize by subtracting the energy associated to a standard regular $n$-gon [@Si3] (or see [@Si5]). Note that $U'_{\md}$ is scale invariant, so we can use any regular $n$-gon and get the same number. We define $$E_{md}(P) = \upmd(P) - \upmd(\text{regular }n\text{-gon})\;\;.$$
The energy $\umd$ has been implemented in several software systems [@Wu; @HuntKED; @Sc], and studied in [@MilRaw00; @KauffHuang96; @KauffHuang98].
We shall show that for suitable polygonal approximations $P$ of a smooth curve $K$, $E_{md}(P) \approx E_{0}(K)$. While the Möbius Energy is defined for $C^{1,1}$ curves, our proof requires that the knot $K$ be $C^2$ smooth.
In order for the approximation to work, we need to be careful about what it means to say, “the polygon P is a close approximation of $K$”. First, we need to prevent extreme changes in the edge lengths of $P$ (see Figure \[unequal\]). Suppose $P$ is a polygon closely inscribed in $K$. We can slide vertex $v_3$ along $K$ towards vertex $v_4$, making edge $e_3$ arbitrarily short, making edge $e_2$ longer, and keeping the other edges of $P$ fixed. This will make the contribution of the edge-pair $(e_2,e_4)$ to $E_{md}$ arbitrarily [*large*]{}. Thus, we can make polygons $P'$ that also seem like close approximations of $K$, yet $E_{md}(P') >> E_{0}(K).$
\[unequal\]
To prevent this problem, we have to limit the variation in edge lengths of polygons inscribed in $K$; we do this by having the vertices equally spaced in arc-length along $K$. One can modify our arguments to handle other tractable approximating polygons, e.g. equal edge lengths or “equal time” subdivisions of a regularly parameterized curve.
Conversely, we can find situations where $E_{md}(P) << E_{0}(K)$. Let $P$ be the polygon (not drawn) $\langle v_1,v_2,v_3,v_4,v_1\rangle$ in Figure \[polyhigh\]. We construct the quadrilateral so that the arcs between consecutive vertices are of equal length. Keeping the vertices fixed, deform the arc $\widehat{v_1v_2}$ and the arc $\widehat{v_3v_4}$ slightly so they get arbitrarily close to intersecting (where one crosses over the other in the figure). This makes $E_{0}(K)-E_{md}(P)$ arbitrarily large. This problem is detected by the fact that $r(K)$ decreases to $0$, since normal disks of smaller and smaller radii will intersect. This is why the error bound in Theorem \[explicit\] must take into account the geometric quantity $r(K)$.
![Portions of the smooth curve can be arbitrarily close, causing $E_0$ to be very large while $E_{md}$ of the inscribed polygon remains fixed. In such a case, $E_0$ is much larger than $E_{md}$.[]{data-label="polyhigh"}](polyhigh.eps)
![We require that the order of the vertices of the inscribed polygon coincides with an orientation of the smooth knot.[]{data-label="flipflop"}](flipflop.eps)
To avoid situations as in Figure \[flipflop\], we assume the phrase “inscribed polygon” means the vertices of $P$ occur in the same order as they occur along $K$.
Finally, note that the regularizations play an essential role, making the proof more delicate than may be at first evident. See the discussion in Section \[outlineproof\].
Statement of main results and notation {#MainResultsAndNotation}
======================================
We shall use the following notation throughout the paper.
- $K$ is a $C^2$ smooth simple closed curve in $\mathbb{R}^3$.
- $\ell(K)$ is the total arc-length of $K$.
- $r(K)$ is the thickness radius of $K$.
- $E_L(K) = \frac{\ell(K)}{r(K)}$ is the Rope-Length.
- $\delta=\frac{\ell(K)}{n}$ is the mesh size of the inscribed polygon.
- $K$ is subdivided into $n$ arcs of equal length. $\delta=\frac{\ell(K)}{n}$.
- $v_1, \dots , v_n$ are the subdivision points along $K$.
- $P_n$ is the polygon formed by connecting the points $v_1, \dots , v_n, v_1$ in order.
- $\arc{x,y}$ is the length of the shorter of the two arcs of $K$ connecting $x$ and $y$.
- $|e|$ is the length of the line segment $e$.
Additional notation used in the proofs is listed at the beginning of Section \[outlineproof\].
For any smooth knot $K$, if $P_n$ are inscribed polygons as above and $n$ is large enough that $n>E_{L}(K)$, i.e. $\delta<r(K)$, then $$|E_{0}(K)-E_{md}(P_n)|\leq \Phi(n,E_{L}(K))\;,$$ where $\Phi$ is a linear combination (see final page of the paper) of six fractions of the form $\frac{E_L(K)^a}{n^b}$ for various $a>0$ and $b>0$. By combining some terms, we can take $\Phi = 550\frac{E_L(K)^{5/4}}{n^{1/4}}+ 10\frac{E_L(K)^4}{n}$. For very large $n$ ($n>E_L(K)^{11/3}$), we can use $\Phi = 560\frac{E_L(K)^{5/4}}{n^{1/4}}$. \[explicit\]
[*Remark.*]{} There are other ways to write this scale-invariant error bound, using the identity $$\frac{E_L(K)}{n} = \frac{\delta}{r(K)}\;.$$
From Theorem \[explicit\], we have immediately:
For any smooth knot $K$, if $P_n$ are inscribed polygons as above, then as $n \to \infty$, $E_{md}(P_n)\to E_{0}(K)$. \[approaches\]
The supporting lemmas and the proof of the theorem occupy the rest of the paper. The lemmas are in Section \[lemmas\]. In Section \[outlineproof\], we outline the proof and explain how the domains will be divided into zones for which different analyses are needed. We give the analysis for each zone in Section \[proofsforzones\] and put them all together in Section \[finalproof\].
There are numerous coefficients in the calculations; we constantly round up and pick the worst-case values, to keep the claims accurate and the numbers simple.
The lemmas {#lemmas}
==========
In this section, we prove the lemmas needed for the proof of the main theorem.
Lemmas about the cosine function, also chords and arcs of circles
-----------------------------------------------------------------
If $0< \phi\leq \pi$, then the following hold:
1. $1-\frac{1}{2}\phi^2 \leq \cos \phi \leq 1-\frac{1}{2}\phi^2+
\frac{1}{24}\phi^4\,,$
2. $\phi^2-\frac{1}{12}\phi^4\leq 2-2\cos \phi \leq \phi^2\,,$
3. $1-\frac{1}{12}\phi^2\leq \frac{2-2\cos \phi}{\phi^2}
\leq 1\,,$
4. $\frac{\phi^2}{2-2\cos \phi}
\leq 1+\frac{1}{2}\phi^2\,.$
\[allcosinelemmas\]
For (a), consider the Taylor series for $\cos(\phi)$. Parts (b), (c), (d) follow immediately.
1. On the unit circle $C$, for any points $x,y$, $$\frac{1}{12} <
\frac{1}{|x-y|^2} - \frac{1}{\mathrm{arc}(x,y)^2} \leq
\frac{1}{4}-\frac{1}{\pi^2}\,.$$
2. On a circle of radius $R$, $$\frac{1}{12}\;\frac{1}{R^2} <
\frac{1}{|x-y|^2} - \frac{1}{\arc{x,y}^2} \leq
\left(\frac{1}{4}-\frac{1}{\pi^2}\right)
\frac{1}{R^2}\,.$$
\[E0Integrand\]
Let $\phi$ be the angle ($\leq \pi$) between points $x$ and $y$ on the circle. Since $C$ is the unit circle, $\mathrm{arc}(x,y)=\phi$ and $|x-y|^2=2-2\cos\phi$. The function $$\frac{1}{2-2\cos\phi} -
\frac{1}{{\phi}^2}$$ is monotone, has a maximum at $\phi =
\pi$, and is bounded below by the limiting value as $\phi
\longrightarrow 0$. Part (b) is similar.
Next we want to compare the quantities $\frac{1}{|x-y|^2}$ and $\frac{1}{|X-Y|^2}$, where the points lie on circles of different sizes.
Suppose $r<R$ are radii of circles and $0< a < \pi r$. Construct any arcs of (the same) length $ a $ on the two circles and let $x,y$ and $X,Y$ be the endpoints of the two arcs. Then $$0 <
\frac{1}{|x-y|^2} - \frac{1}{|X-Y|^2} <
\left(\frac{1}{4}-\frac{1}{\pi^2}\right)\,\frac{1}{r^2}\,.$$ \[ChordsOnDifferentCircles\]
Chord length is always less than arc-length. For a fixed arc-length, as the radius gets larger, the chord length gets closer to the arc-length. Thus $|X-Y|>|x-y|$. On the other hand, applying Lemma \[E0Integrand\](b) to each circle, we have $$\frac{1}{|x-y|^2} - \frac{1}{|X-Y|^2} <
\left(\frac{1}{4}-\frac{1}{\pi^2}\right)\;
\frac{1}{r^2} - \frac{1}{12}\;\frac{1}{R^2}.$$
Lemmas about chords and arcs of general curves
----------------------------------------------
We rely a lot on Schur’s Theorem. Here is the version we need:
Let $K$ be a $C^2$ smooth curve in $\mathbb{R}^3$ whose curvature everywhere is $\leq$ some number $\kappa$. Let $C$ be a circle of curvature $\kappa$, i.e. of radius $r =
\frac{1}{\kappa}$. Let $x,y \in K, s,t \in C$ such that $\mathrm{arc}(x,y)=\mathrm{arc}(s,t) \leq \pi r$. Then the chord distances satisfy $$|x-y| \geq |s-t| \,.$$ When we write the chord length on $C$ in terms of the central angle, this becomes $$|x-y| \geq r \,\left( 2-2 \cos\left(\frac{\mathrm{arc}(s,t)}{r}
\right)
\right)^{1/2}$$ \[schursthm\]
See Schur’s Theorem in [@chern].
Let $K$ be a $C^2$ smooth curve in $\mathbb{R}^3$, with minimum radius of curvature $r$. Suppose $x:[0,\pi r] \to \mathbb{R}^3$ is a unit speed parameterization of an arc of $K$ of length $\pi r$. Then the function $|x(t)-x(0)|$ is monotone increasing. That is to say: As points move farther apart along the curve, they also move farther apart in space, so long as the arc-distance is no greater than $\pi r$. \[MonotoneArcs\]
Let $f(t) = |x(t)-x(0)|^2 = (x(t)-x(0)) \cdot
(x(t)-x(0))$. We claim $\frac{df}{dt}>0$ for $t\in (0,\pi
r)$. The derivative $\frac{df}{dt} = 2 (x(t)-x(0))\cdot x'(t)$. Thus we need to show that this dot product is positive, for all points $x(t)$ in the interior of the arc. The proof uses the same central idea as the proof of Schur’s theorem.
We have $$x(t)-x(0) = \int_0^t{x'(s)} \;ds \;,$$ so $$(x(t)-x(0)) \cdot x'(t) = \int_0^t
x'(s)\cdot x'(t) \;ds \;.$$ The dot product $x'(s)\cdot x'(t)$ is just the cosine of the angle $\leq \pi$ between the two velocity vectors. This angle is measured by the length of the geodesic arc on the unit sphere between the unit vectors $x'(s)$ and $x'(t)$. The trace of $x'(u)$, as $u$ runs from $s$ to $t$, is another path on the unit sphere between the same vectors. The length of that path gives an upper bound for the length of the geodesic path. Thus, since $\left|x''(u)\right|\leq 1/r$ (recall $r =$ minimum radius of curvature), $$\angle(x'(s),x'(t)) \leq \int_s^t{\left|x''(u)\right|} \;du
\;\leq \frac{(t-s)}{r}\;.$$
Since $0\leq s \leq t \leq \pi r$, and the cosine function is decreasing on $[0,\pi]$, we have $$\cos(\angle(x'(s),x'(t))) \geq \cos\frac{(t-s)}{r} \;.$$
Thus $$(x(t)-x(0)) \cdot (x'(t)) \geq \int_0^t
\cos\frac{(t-s)}{r} \;ds \; = r \sin(t/r)\;.$$
For $0 < t < r \pi$, $\;\sin(t/r) > 0$.
Let $K$ be a $C^2$ smooth curve in $\mathbb{R}^3$, with minimum radius of curvature $r$. Let $x:[0,\ell(K)] \to \mathbb{R}^3$ be a unit speed parameterization of $K$. Suppose $0\leq a<b<c<d\leq
\pi r$, so $x(a),x(b),x(c),x(d)$ are four points in order along $K$, contained in an arc of total length $\leq \pi r$.
Then the minimum spatial distance between line segments $\langle
x(a)x(b)\rangle$ and $\langle x(c)x(d)\rangle$ is realized at the closest endpoints. Taking into account Lemma \[MonotoneArcs\], this says, $${MD}(\;\langle{x(a)x(b)}\rangle\;,\;\langle{x(c)x(d)}\rangle\;)
=
|x(c)-x(b)|\;.$$ \[piarg\]
Without loss of generality, rescale the curve to have $r=1$. Then the four points lie in an arc of total length $\leq \pi$.
Let $A$ denote the segment $\langle{x(a)x(b)}\rangle$ and $C$ the segment $\langle{x(c)x(d)}\rangle$. We shall show that for each point $x\in A$, the point $x(c)$ is the closest point of $C$ to $x$; so $x(c)$ is the closest point of $C$ to $A$. By a symmetric argument, the point $x(b)$ is the closest point of $A$ to $C$.
Fix a point $y \in C, y \neq x(c), x(d)$. For any $x \in A$, construct the directed line segment from $x$ to $y$. We claim that the vectors satisfy $$(y-x) \cdot (x(d)-x(c)) > 0 \;.$$ If this dot product is positive, then moving $y$ along $C$ closer to $x(d)$ will increase the distance to $x$, and moving $y$ closer to $x(c)$ will decrease the distance to $x$. Thus $x(c)$ must be the closest point of $C$ to $x$.
We now show that the above dot product is positive for each $x,y$. It is convenient to think for a moment of fixing $y$ and varying $x$. Let $P_y$ be the plane through $y$ perpendicular to $C$. Rotate the entire ensemble so that the vector $x(d)-x(c)$ points “up”. Then the dot product inequality is equivalent to the assertion that the entire line segment $A$ lies [*below*]{} $P_y$. It suffices to show that each vertex $x(a),
x(b)$ lies below $P_y$.
But in fact, if $x(a)$ and $x(b)$ lie below $P_{x(c)}$, then they lie below $P_y$. We have now reduced the lemma to the following claim, an inequality that involves only the given points on $K$. The inequality is stated for parameter value $a$, and is identical for $b$. If $0
\leq a<c<d \leq \pi$, then $$(x(c)-x(a)) \cdot (x(d)-x(c)) > 0 \;.$$
The rest of the proof is similar to the proof of Lemma \[MonotoneArcs\] with some trigonometry at the end. We first express the difference vectors as integrals of derivatives, $$(x(c)-x(a)) \cdot (x(d)-x(c)) = \int_a^c\int_c^d
x'(s)\cdot\ x'(t) \;dt \;ds \;.$$
Since the cosine function is decreasing on $[0,\pi]$, we have $$\cos(\angle(x'(s),x'(t))) \geq \cos(t-s) \;.$$
As in the proof of Lemma \[MonotoneArcs\], $x'(s)\cdot x'(t) \geq \cos(t-s)$, so $$(x(c)-x(a)) \cdot (x(d)-x(c)) \geq \int_a^c\int_c^d
\cos(t-s) \;dt \;ds \;.$$ The integral evaluates to $$\cos(d-c) - \cos(d-a) - 1 + \cos(c-a)\;,$$ which is positive.
The previous two lemmas tell us that for arcs that are near each other in arc-length along a curve, the minimum spatial distance between the arcs is the same as the minimum distance between their inscribed chords. For more general pairs of arcs, the minimum distances usually will not be equal, but they still are related.
Suppose $\alpha,\beta$ are smooth arcs in $\mathbb{R}^3$, each of length $\delta$, and each having radius of curvature everywhere $\geq
r\geq \delta$. Let $e$ be the chord joining the endpoints of $\alpha$ and $f$ the corresponding chord for $\beta$.
\(a) The maximum distance between $\alpha$ and $e$ (likewise between $\beta$ and $f$) is $\leq \frac{1}{\sqrt{48}}\frac{\delta^2}{r}.$
\(b) If $\mathrm{MD}(\alpha,\beta)$ is the minimum spatial distance between $\alpha$ and $\beta$, and $\mathrm{MD}(e,f)$ is the minimum distance between the chords, then $$|\mathrm {MD}(e,f)-\mathrm {MD}(\alpha,\beta)|\leq
\frac{\sqrt{3}\delta^2}{6r} \leq \frac{\sqrt{3}}{6}\,r\;.$$
\[MaxDistArcToChord\]
For part (a), imagine the chord $e$ as a rod with “string” of length $\delta$ attached at either end, and ask, “What configuration allows the string to reach as far as possible from the rod?” The answer is when the string is pulled out to form two equal sides of an isosceles triangle, with the rod as the base. The maximum distance that any point of $\alpha$ can be from $e$ is the altitude $h$ of this isosceles triangle, so $h^2=\left(\frac{\delta}{2}\right)^2-\left(\frac{|e|}{2}\right)^2$. Since $\delta \leq r$, in particular $\delta \leq \pi r$, we can apply Schur’s theorem: By Lemma \[schursthm\] and Lemma \[allcosinelemmas\](b), $$\frac{1}{4}|e|^2
\geq\frac{1}{4}\delta^2-\frac{1}{48}\frac{\delta^4}{r^2}
\;,$$ so $$h^2\leq \frac{1}{48}\frac{\delta^4}{r^2}\,.$$
Part (b) follows from part (a), the triangle inequality, and the fact that $\delta \leq r$.
Lemmas about the thickness of a curve {#lemmasaboutthickness}
-------------------------------------
The first lemma is a characterization of the thickness radius $r(K)$ in terms of curvature and the [*critical self-distance*]{}.
Fix a point $x_0\in K$ and consider points $y$ that start at $x_0$ and gradually move along $K$. A point $y$ is a critical point for the function $|y-x_0|$ when $y=x_0$ or when $\langle xy \rangle \perp y'$. We define the [*critical self-distance of $K$*]{} (an idea attributed by J. O’Hara to N. Kuiper) to be $$\mathrm{sd} (K) = \min\left\{\; |y-x|\;\;:\;\;x\not = y
\in K \;\mathrm{ and }\; \langle xy
\rangle \perp y' \;\right\}\;.$$
The thickness of a smooth knot is bounded by the minimum radius of curvature and half the critical self-distance. In fact, $$r(K)= \min \left\{ {\mathrm{MinRad}}(K), \frac12\;
\mathrm{sd}(K)\right\}.$$ \[lsdr\]
See [@LSDR].
The next lemma is a consequence of Lemmas \[schursthm\] and \[lsdr\], and is proven in [@BRS].
Suppose $K$ is a smooth knot of thickness radius $r(K)=r$. For any $x,y \in K$ with $\arc{x,y} \geq \pi r$, we must have $|y-x| \geq 2 r$. \[distancebound\]
Let $K$ be a $C^2$ smooth closed curve in $\mathbb{R}^3$, with minimum radius of curvature $r$. Let $C$ be a circle whose total arc-length is the same as $K$, and $R$ be the radius of $C$. Then $r \leq R$ and (from Lemma \[lsdr\]) the thickness radius $r(K) \leq R$. \[fenchellemma\]
Since $r$ is the minimum radius of curvature of $K$, the maximum curvature of $K$ is $\frac{1}{r}$, so the total curvature of $K$ is at most $\frac{\ell(K)}{r}$. On the other hand, by Fenchel’s theorem [@fenchel], the total curvature of $K$ is at least $2 \pi$. Thus $2 \pi r \leq \ell(K) = 2 \pi R$.
For any $C^2$ smooth closed curve $K$, $E_L(K)\geq 2\pi$. \[ropebiggerthan2pi\]
By Lemma \[lsdr\], the curvature of $K$ is everywhere $\leq 1/r(K)$. Thus the total curvature of $K$ is $\leq \ell(K)/r(K)=E_L(K)$. But the total curvature of a closed curve is $\geq 2\pi$.
Notation and outline of proof of Theorem \[explicit\] {#outlineproof}
=====================================================
We have four objects of interest: the knot $K$, the circle $C$, the inscribed $n$-gon $P$, and the regular $n$-gon $Q$. In the following list, refer to Figure \[SetupNames\]
![The objects of study: smooth knot $K$ with arc $\alpha_i$ and vertex $v_i$, inscribed polygon $P$ with vertex $v_i$, circle $C$ with arc $\beta_i$ and vertex $b_i$ corresponding to $\alpha_i$ and $v_i$ respectively, and inscribed regular $n$-gon $Q$ with vertex $b_i$.[]{data-label="SetupNames"}](knotcirclepolygonv2.eps)
- $K$ is a $C^2$ smooth simple closed curve in $\mathbb{R}^3$.
- $C$ is a circle with total arc-length $\ell(C)=\ell(K)$.
- $r(K)$ is the thickness radius of $K$.
- $K$ is subdivided into $n$ arcs of equal length $\delta=\frac{\ell(K)}{n}$, and we are assuming $\delta<r(K)$ (so $n>E_L(K)$).
- $v_1, \dots , v_n$ are the subdivision points along $K$.
- $\alpha_i$ is the arc of $K$ with endpoints $v_i$ and $v_{i+1}$. We number the vertices modulo $n$, so $\alpha_n$ is the arc from $v_n$ to $v_1$.
- $R$ is the radius of $C$, so $R=\frac{\ell(K)}{2\pi}$.
- $t \to x(t)$ is a unit speed parameterization of $K$ from $C$.
- $b_1, \dots, b_n$ are evenly spaced points along $C$ such that $x(b_i)=v_i$.
- $\beta_i$ is the arc of $C$ corresponding to $\alpha_i$.
- $P$ is the polygon formed by connecting the points $v_i$ in order.
- $e_i$ is the edge of $P$ from $v_i$ to $v_{i+1}$, with length denoted $|e_i|$.
- $Q$ is the regular polygon inscribed in $C$, with vertices $b_1,\dots,b_n$.
- $f_i$ is the edge of $Q$ with vertices $b_i, b_{i+1}$, with length $|f_i|$.
Just to have all the important parameters specified in one place, we also define two integers, $m$ and $p$, whose role will be evident later in this section.
- $m = \lfloor \frac{\pi r(K)}{\delta} \rfloor$. For a vertex $v_i$, the vertices $v_i, v_{i+1}, \dots, v_{i+m}$ are a maximal list that lie in an arc of $K$ of length $\leq
\pi r(K)$.
- $p = \lfloor m^{\frac{3}{4}}\rfloor$. For a list of $m$ vertices as specified in the previous item, we will need to distinguish an initial bunch from the rest. It turns out that the number we need to separate off should be some fractional power of $m$ strictly greater than $1/2$, and we take $3/4$ for simplicity.
We shall analyze the energies in terms of individual pairs of arcs and/or edges.
The energies are $$\begin{aligned}
E_{0}(K) &= \int_{x\in K}\int_{y\in K}
\frac{1}{|x-y|^2} - \frac{1}{|s-t|^2}\nonumber\\ &=
\sum_{i=1}^{n} \sum_{j=1}^{n} E_{0}(\alpha_i,\alpha_j)\;,
\label{esum}\end{aligned}$$ where $$E_{0}(\alpha_i,\alpha_j)=\int_{x\in
\alpha_i}\int_{y\in \alpha_j}
\frac{1}{|x-y|^2} - \frac{1}{|s-t|^2}\;,$$ and $$\begin{aligned}
E_{md}(P) &= U'_{md}(P)-U'_{md}(Q)\nonumber\\
&= \sum_{i=1}^{n} \;\sum_{j=1}^{n}
U_{md}(e_i,e_j)-U_{md}(f_i,f_j)\;\;\;(j \neq i-1,i,i+1)\;.
\label{usum}\end{aligned}$$ Sometimes we need to treat $E_0$ as the difference between two integrals, so we also define $$E(\alpha_i,\alpha_j) = \int_{x\in
\alpha_i}\int_{y\in \alpha_j}\frac{1}{|x-y|^2}\;,$$ and likewise for $E(\beta_i,\beta_j)$ for arcs on $C$. As one might expect, our overall plan is to show that the various terms in the sum (\[esum\]) are close to the corresponding terms in (\[usum\]). However, some terms in (\[esum\]) have no corresponding term in (\[usum\]); and even when they do, there are different cases requiring different analyses. We shall, in fact, consider four kinds of pairs $(i,j)$, bound each contribution to the error, and add them to get a full error bound.
Here is a “schematic diagram” of our situation: We want to show that something of the form $\int (W - X)$ is close to something of the form $(Y-Z)$. For the edge pairs where $E_0$ has a contribution and $E_{md}$ is not defined, we show the $E_0$ contribution is small. For other edge pairs, we sometimes show that $\int (W - X)$ and $(Y-Z)$ each is small, and sometimes show that $|Y - \int W| $ and $|Z - \int
X|$ both are small. The analysis has to involve this kind of complication because the unregularized polygon energy $U'_{md}(P)$ is [**not**]{} a good approximation of the divergent integral $\iint_{K
\times K} \frac{1}{|y-x|^2}\;$, that is $|Y - \int W| $ does not get negligibly small for arc pairs (and their corresponding segment pairs) that are extremely close together along $K$. Here is a simple example to illustrate the difficulty: Consider two segments $A = [0,\epsilon]$ and $B = [2 \epsilon,3\epsilon] \subset \mathbb{R}$. Then $U_{md}(A,B)
= 1$. On the other hand, $\int_{x\in A}\int_{y \in
B}\frac{1}{|y-x|^2}\;dy\;dx =\ln \frac{4}{3}\;.$ For segments close together along the curves, we need to understand the regularizing terms rather than show the two energies are close to each other.
Following are the four types of pairs (of indices $(i,j)$, edges or arcs) that determine our four “zones” for separate analysis. The definitions are symmetric, so $(i,j)$ and $(j,i)$ are of the same type.
1. [*Adjacent Pairs*]{}: $j=i-1, i, i+1$\
For these arc pairs, we bound $\sum_{i,j}{E_0(\alpha_i, \alpha_j)}$. Since $\umd$ is only defined for non-adjacent edges, there are no corresponding edge pairs for these arc pairs.
2. [*Near Pairs*]{}: non-diagonal pairs $(i,j)$ for which the arcs $\alpha_i$ and $\alpha_j$ are contained in an arc of $K$ of length $\leq \pi r(K)$.\
Within the Near Zone, we make an additional distinction between “Very Near” and “Moderately Near”: For each vertex $v_i$, let $A$ be either of the arcs of $K$ starting at $v_i$ and having length $\ell(A) = \pi r(K)$. The vertices contained in the arc $A$ are a sequence $v_i, v_{i+1}, \hdots, v_{i+m}$ (for the other arc, we count in the other direction). The arcs contained in $A$ are $\alpha_i,
\dots, \alpha_{i+m-1}$. The vertex $v_{i+m}$ may or may not be an endpoint of $A$. We distinguish between the first $m^{3/4}$ vertices and the rest.
1. For $j=i+2, \hdots, i+p$, we call $(i,j)$ a [*very near*]{} pair.\
For such $(i,j)$, we bound $\sum_{i,j}\left(U_{md}(e_i,e_j)-U_{md}(f_i,f_j)\right)$ and $\sum_{i,j}{E_0(\alpha_i, \alpha_j)}$.
2. For $j = i+ p+ 1, \hdots, i+m-1$, we call $(i,j)$ a [*moderately near pair*]{}.\
For such $(i,j)$, we shall bound $\sum_{i,j}\left(E(\alpha_i,\alpha_j) -
U_{md}(e_i,e_j)\right)$ and $\sum_{i,j}\left(E(\beta_i, \beta_j) -
U_{md}(f_i,f_j)\right).$
3. *Far Pairs*: The pairs $(i,j)$ that are neither [*adjacent*]{} nor [*near*]{} are called [*far*]{}.\
For such pairs, we shall also bound $\sum_{i,j}{E(\alpha_i,\alpha_j) - U_{md}(e_i,e_j)}$ and $\sum_{i,j}{E(\beta_i,\beta_j) - U_{md}(f_i,f_j)},$ but we need an argument different from the moderately near pairs.
See Figure \[zones\] for an example of the zone pairings where $m=17$. We use the same terminology for corresponding pairs of arcs in $C$; that is, if $(i,j)$ are far \[resp. adjacent, very near, moderately near\] on $K$, then we call them far \[resp. adjacent, very near, moderately near\] on $C$.
![The four types of zones on which we do our analysis. Note that this is just a schematic to show the arrangement of the zones with respect to a fixed arc.[]{data-label="zones"}](zones4.eps){width="5.0in"}
In the next section, we establish the explicit error bounds in each of the different zones. In Section \[finalproof\], we collect all of the errors to determine the total error bound.
Proofs for the different zones {#proofsforzones}
==============================
Bounds for $E_0$ in Adjacent and Very Near Zones
------------------------------------------------
We establish the error bound for the combined contributions of the Adjacent and Very Near Zones to the Möbius Energy.
In the Adjacent and Very Near Zone, $$\left|\sum_{i,j}{E_0(\alpha_i,
\alpha_j)}\right|<
1.06\,\frac{E_L(K)^{5/4}}{n^{1/4}}$$ \[EoDiagAndVeryNear\]
If $x,y$ are contained in diagonal or very near arcs, then $\arc{x,y} \leq (p+1)\delta$. Thus it suffices to bound $$\left|\int_{x \in K}\int_{y \in K,\;
\mathrm{arc}(x,y)
\leq (p+1)\delta}\;\;
\frac{1}{|x-y| ^2}-\frac{1}{|s-t|
^2}\right|.$$
The calculation is independent of the choice of $x$, so we analyze $$\left|2\,\ell(K)\,\int_{y=x}^{x+(p+1)\delta}
\frac{1}{|x-y|^2}-\frac{1}{|s-t|^2}\,dy\right|\,,
\label{eq:WantToBoundVeryNear}$$ where the limits of integration are meant to indicate that we are integrating along an arc of $K$ of length $(p+1)\delta$ starting from $x$.
We are going to find upper and lower bounds for the integrand, observe that the upper bound is positive and the lower bound is negative, and conclude that the magnitude of the integrand is bounded by the difference between the upper and lower bounds. To simplify subsequent expressions, let $r$ denote $r(K)$ and $a$ denote $\arc{x,y}$.
Since $s$ and $t$ lie on a circle of radius $R$, $$\frac{1}{|s-t|
^2}=\frac{1}{R^2(2-2\cos(a/R))}\,.$$
First we get the upper bound. Since $\delta \leq r$, in particular $m \geq 2$, we have $p <
m $ and $(p+1)\delta \leq \pi r$. Thus we can apply Lemma \[schursthm\] to conclude $$|x-y|^2 \geq r^2(2-2\cos(a/r))\;.$$
So we have $$\frac{1}{|x-y|^2}-\frac{1}{|s-t|^2}
\leq \frac{1}{r^2(2-2\cos(a/r))}
-\frac{1}{R^2(2-2\cos(a/R))}\,.$$
By Lemma \[fenchellemma\], $r\leq R$. By Lemma \[schursthm\] applied to circles of different radii, or the argument in Lemma \[ChordsOnDifferentCircles\], this upper bound is nonnegative.
Now we get the lower bound. Since arc-length on any curve must be at least as large as chord length, $$|x-y|^2 \leq a^2\;.$$ Thus $$\begin{aligned}
\nonumber \frac{1}{|x-y|^2}-\frac{1}{|s-t|^2} &\geq
\frac{1}{a^2}-\frac{1}{|s-t|^2} \\ &= \nonumber
\frac{1}{a^2}-\frac{1}{R^2(2-2\cos(a/R))}\;,\end{aligned}$$ which is negative since chord length $<$ arc-length on a circle.
Taking the difference between the nonnegative upper bound and the negative lower bound, we have $$\left|\frac{1}{|x-y|^2}-
\frac{1}{|s-t|^2}\right| \leq
\frac{1}{r^2(2-2\cos(a/r))} - \frac{1}{a^2 \;.}$$
So $$(\ref{eq:WantToBoundVeryNear})
\leq 2\,\ell(K)\,\int_{0}^{(p+1)\delta}
\frac{1}{r^2(2-2\cos(a/r))}-\frac{1}{a^2}\,da\,,$$ where now we are just integrating a function of a real variable. Applying Lemma \[E0Integrand\](b), we have $$(\ref{eq:WantToBoundVeryNear}) \leq 2\,\ell(K)\,
(p+1)\,\delta\,
\left(\frac{1}{4}-\frac{1}{\pi^2}\right)\,\frac{1}{r^2}\;.$$
Since $m \geq 3$, $p \geq 2$, so $(p+1) < 1.5 p$. Combining the constants, we have $$\nonumber (\ref{eq:WantToBoundVeryNear}) < 0.45\,
\frac{\ell(K) p
\delta}{r^2}
\leq 0.45\,\frac{\ell(K) (\frac{\pi
r}{\delta})^{3/4}
\delta}{r^2}
\leq 1.06\,
\frac{E_L(K)^{5/4}}{n^{1/4}}$$ as desired.
Bound for $\emd$ in the Very Near Zone
--------------------------------------
In the Very Near Zone, $$\left|E_{md}(P)\right| = |\umd'(P)-\umd'(Q)|<
2.76\, \frac{E_L(K)^{5/4}}{n^{1/4}}\,.$$ \[polygonverynearzone\]
$$\emd(\mathrm{very\ near})=
2\,\sum_{i=1}^{n}\sum_{j=i+2}^{i+p}
\frac{| e_i|\,| e_j|}{MD(e_i,e_j)^2}-
\frac{| f_i|\,| f_j|}{MD(f_i,f_j)^2}\,.$$ We shall bound the inner sums uniformly in $i$, that is bound $$\left| \sum_{k=1}^{p-1}\;
\frac{| e_i|\,| e_{i+k+1}|}{MD(e_i,e_{i+k+1})^2}-
\frac{| f_i|\,| f_{i+k+1}|}{MD(f_i,f_{i+k+1})^2}\right|
\label{eq:EmdVeryNearEq1}$$ for arbitrary $i$, and then multiply that bound by $2n$. Here $k=j-i-1$ is the number of edges separating the two edges. As in Proposition \[EoDiagAndVeryNear\], we find a positive upper bound for each difference term, and a negative lower bound; so the difference between the upper and lower bounds is a bound for the absolute value.
On the circle $C$ of radius $R$, the edge lengths are $|f_i| = |f_j| =\sqrt{ R^2(2-2\cos(\delta/R))}\;$, and $MD(f_i,f_{i+k+1})=\sqrt{R^2(2-2\cos(k\delta/R))}$. So $$(\ref{eq:EmdVeryNearEq1}) \;=\;\left| \sum_{k=1}^{p-1}
\frac{| e_i|\,| e_{i+k+1}|}{MD(e_i,e_{i+k+1})^2}-
\frac{R^2(2-2\cos(\delta/R))}{R^2(2-2\cos(k\delta/R))}\right|\,.$$
To simplify subsequent expressions, let $r$ denote $r(K)$. If we compare $K$ locally with a circle of radius $r$, Lemma \[piarg\] and Lemma \[schursthm\] say $MD(e_i,e_{i+k+1})^2 \geq r^2(2-2\cos(k\delta/r))$. The longest an edge can be is the arc-length, so $(| e_i|\,| e_j|)
\leq \delta^2$. Thus, an upper bound for each summand is $$\mathrm{summand} \leq
\frac{\delta^2}{r^2(2-2\cos(k\delta/r))}-
\frac{R^2(2-2\cos(\delta/R))}{R^2(2-2\cos(k\delta/R))}\,.$$ We claim this upper bound is positive. First, $\delta^2 >
R^2(2-2\cos(\delta/R))$ since arc-length (now on the big circle $C$) is always $>$ chord length. Furthermore, $r^2(2-2\cos(k\delta/r)) \leq R^2(2-2\cos(k\delta/R))$ by Lemma \[ChordsOnDifferentCircles\].
We next obtain a lower bound. By Lemma \[lsdr\], $r \leq $ minimum radius of curvature of $K$. So we can apply Lemma \[schursthm\] and Lemma \[piarg\] to any points that lie in an arc of $K$ of length $\leq \pi r$. By Lemma \[schursthm\], we have $(| e_i|\,| e_{i+k+1}|) \geq r^2(2-2\cos(\delta/r))$. For the denominator, Lemma \[piarg\] gives us that $MD(e_i,e_{i+k+1}) = |v_{i+k+1}-v_{i+1}|$, the distance between points of $K$ whose arc-distance is $k\delta$. Since chord length $\leq$ arc-length, we thus have $MD(e_i,e_{i+k+1})^2\leq
(k\delta)^2$. So a lower bound for the summand is $$\mathrm{summand} \geq
\frac{r^2(2-2\cos(\delta/r))}{k^2\delta^2}-
\frac{R^2(2-2\cos(\delta/R))}{R^2(2-2\cos(k\delta/R))\textsf{}}\,.$$ Comparing numerators and denominators as we did for the upper bound, we see that this lower bound is always negative.
Thus, we can bound the absolute value of the summand by the difference between the upper and lower bounds: $$\begin{aligned}
\left|\,\mathrm{summand}\,\right| &\leq
\frac{\delta^2}{r^2(2-2\cos(k\delta/r))}-
\frac{r^2(2-2\cos(\delta/r))}{k^2\delta^2}\nonumber\\
&=\frac{1}{k^2}\left(\frac{k^2\delta^2}{r^2(2-2\cos(k\delta/r))}-
\frac{r^2(2-2\cos(\delta/r))}{\delta^2}\right)\,.
\label{eq:UmdVeryNearEq2} \end{aligned}$$
We now appeal to our lemmas on cosines and chords. To clarify how lemmas will be used, introduce angles $\theta=\delta/r$ and $\phi=k\delta/r$. Thus, the bound (\[eq:UmdVeryNearEq2\]) can be written $$(\ref{eq:UmdVeryNearEq2}) =
\frac{1}{k^2}\left(\frac{\phi^2}{2-2\cos \phi}-
\frac{2-2\cos\theta}{\theta^2}\right)\,.$$
By Lemma \[allcosinelemmas\](d), $\frac{\phi^2}{2-2\cos \phi}\leq
1+\frac{1}{2}\phi^2$. By Lemma \[allcosinelemmas\](c), $\frac{2-2\cos\theta}{\theta^2}\geq 1-\frac{1}{12}\theta^2$. Thus, $$|\mathrm{summand}|\leq
\frac{1}{k^2}\left(\frac{1}{2}\phi^2+\frac{1}{12}\theta^2\right).$$
We return to the original double sum and see that $$\begin{aligned}
2n\,\left|\sum_{k=1}^{p-1}
\frac{| e_i|\,| e_{i+k+1}|}{MD(e_i,e_{i+k+1})^2}-
\frac{| f_i|\,| f_{i+k+1}|}{MD(f_i,f_{i+k+1})^2}\right|
&\leq
2n\,\sum_{k=1}^{p-1}\frac{\frac{1}{2}\phi^2+\frac{1}{12}\theta^2}{k^2}
\nonumber\\ &=
2n\,\sum_{k=1}^{p-1}\frac{\frac{1}{2}k^2\theta^2+\frac{1}{12}\theta^2}{k^2}
\nonumber\\ &=
2n\,\frac{\delta^2}{r^2}\,\sum_{k=1}^{p-1}\left(\frac{1}{2}+
\frac{1}{12}\frac{1}{k^2}\right)\,. \nonumber
\\ &\leq 2 n\, \frac{\delta^2}{r^2} (p-1) \left(\frac{7}{12}\right)\nonumber
\\ &< \frac{7}{6}\, n\, \frac{\delta^2}{r^2} p \nonumber
\\ &< 2.76 \frac{E_L(K)^{5/4}}{n^{1/4}}
\nonumber\end{aligned}$$
[*Remark.*]{} For the Very Near Zone, we could use $p\leq$ any fractional power $m^q$. It is in the Moderately Near Zone that we need $p > 1/2$.
Bound for $|E_0(K)-\emd(P)|$ in the Moderately Near Zone
--------------------------------------------------------
In this section, we determine the error bounds in the Moderately Near Zone for $|E(K)-\umd'(K)|$ and $|E(C)-\umd'(C)|$. Recall that the Moderately Near Zone consists of pairs $(i,j)$ where $\alpha_i\,,\,\alpha_j$ \[resp. $\beta_i\,,\,\beta_j$\] are contained in an arc of $K$ \[resp. $C$\] of length $\pi\,r(K)$ but are separated by at least $p$ other arcs; that is $k=j-i-1$ runs from $p$ to $(m-2)$. The keys to the analysis in this zone are:
- The minimum distance between a given pair of arcs, or a given pair of chords, is realized at the closest endpoints along the curve.
- That vertex-to-vertex distance is bounded away from zero by Schur’s theorem.
\[ModNearZone\] In the Moderately Near Zone, $$|\teb| < 3.00\, \frac{E_L(K)^{11/4}}{n^{7/4}} + 542.84\,
\frac{E_L(K)^{3/2}}{n^{1/2}}\,.$$
As before, we use $r$ to abbreviate $r(K)$. We first analyze the error on $K$, $$\left|2 \sum_{i=1}^{n}\sum_{k=p}^{m-2}\left( \frac{| e_i|\,| e_j
|} {MD(e_i,e_j)^2}-\int_{x\in \alpha_i}\int_{y\in \alpha_j}
\frac{1}{| x-y|^2}\,dy\,dx\right)\right|\,.$$ Note: The expressions seem more clear if we use both $k$ and $j$, where $j=i+k+1$.
As in the previous case, the analysis is independent of $i$, so we work with a general $i$ and multiply that bound by $n$. To bound the above sum of differences, we introduce a third term (larger than each of the two we are studying) and use the triangle inequality.
*Claim 1*.\
$$2 n \sum_{k=p}^{m-2}\left| \frac{\delta^2} {MD(e_i,e_j)^2}-
\frac{| e_i|\,| e_j |} {MD(e_i,e_j)^2}\right| \leq
1.50\,\frac{E_L(K)^{11/4}}{n^{7/4}}\;.$$
*Claim 2*.\
$$2 n \sum_{k=p}^{m-2}\left| \frac{\delta^2}
{MD(e_i,e_j)^2}
- \int_{x\in \alpha_i}\int_{y\in
\alpha_j}
\frac{1}{| x-y|^2}\,dy\,dx \right| \leq
271.42\, \frac{E_L(K)^{3/2}}{n^{1/2}}\;.$$
[*Proof of Claim 1.*]{} Since chord length $\leq$ arc-length, $|e_i||e_j| \leq \delta^2$. So the summand without absolute value is non-negative, and any upper bound will bound the absolute value.
Since we are still within the Near Zone, Lemma \[schursthm\] and Lemma \[allcosinelemmas\](b) give $$|e_i||e_j| \geq r^2(2-2 \cos(\delta/r)) \geq
\delta^2-\frac{1}{12}\frac{\delta^4}{r^2}\;.$$
Now consider the denominator. By Lemmas \[piarg\], \[schursthm\], and \[allcosinelemmas\](b) $$\begin{aligned}
\nonumber MD(e_i,e_j)^2 = |v_j-v_{i+1}|^2 &\geq r^2(2-2
\cos(k\delta/r))
\\ \nonumber &\geq k^2 \delta^2 - \frac{1}{12}\frac{k^4
\delta^4}{r^2}\end{aligned}$$
Thus $$2 n \sum_{k=p}^{m-2}\left( \frac{\delta^2} {MD(e_i,e_j)^2}-
\frac{| e_i|\,| e_j |} {MD(e_i,e_j)^2}\right) \leq
\frac{1}{6}\, n\,\delta^2 \,
\sum_{k=p}^{m-2}\frac{1}{k^2}\;\frac{1}{(r^2-\frac{1}{12}k^2
\delta^2)}\;.$$
We next bound this denominator away from $0$. In the Near Zone, $k \delta < \pi r$, so $r^2-\frac{1}{12}k^2
\delta^2 > r^2(1-\frac{1}{12}\pi^2)$, which gives $$\frac{1}{6}\frac{1}{r^2(1-\frac{1}{12}\pi^2)} < 0.94\,
\frac{1}{r^2}\;.$$
We thus have $$\begin{aligned}
2 n \sum_{k=p}^{m-2}\left( \frac{\delta^2} {MD(e_i,e_j)^2}-
\frac{| e_i|\,| e_j |} {MD(e_i,e_j)^2}\right) &\leq
0.94 \, n\,\frac{ \delta^2}{r^2} \,
\sum_{k=p}^{m-2}\frac{1}{k^2} \nonumber
\\ &< 0.94 \, n\,\frac{
\delta^2}{r^2} \,
\sum_{k=p}^{\infty}\frac{1}{k^2} \nonumber \\
&< 0.94\, n\, \frac{\delta^2}{r^2}\,\frac{1}{p-1}\nonumber \;.\end{aligned}$$
We want to bound $\frac{1}{p-1}$ in terms of $\delta$ and $r$. Recall that $p = \lfloor m^{3/4}\rfloor$ and $m = \lfloor
\frac{\pi r}{\delta}\rfloor$. Since $\delta < r$, we have $m \geq 3$ and $p \geq 2$. So $\frac{1}{p-1} \leq \frac{3}{p+1}$ and $\frac{1}{m}\leq \frac{4}{3}\frac{1}{m+1}$. Thus, $$\frac{1}{p-1} \leq
\frac{3}{p+1}
< \frac{3}{m^{3/4}}
\leq 3\left(\frac{4}{3}\right)^{3/4}\frac{1}{(m+1)^{3/4}}
< 3.73\left(\frac{\delta}{\pi r}\right)^{3/4}
< 1.59\left(\frac{\delta}{r}\right)^{3/4}\,.$$
Then $$\begin{aligned}
2 n \sum_{k=p}^{m-2}\left( \frac{\delta^2} {MD(e_i,e_j)^2}-
\frac{| e_i|\,| e_j |} {MD(e_i,e_j)^2}\right) &<
(0.94)(1.59) \, n\,\frac{
\delta^2}{r^2}\left(\frac{\delta}{r}\right)^{3/4} \nonumber
\\ &< 1.50\, \frac{E_L(K)^{11/4}}{n^{7/4}} \;. \nonumber\end{aligned}$$
This completes the proof of Claim 1.
[*Proof of Claim 2.*]{}
We need to bound
$$\label{eq:Claim2ToBound}
2 n \sum_{k=p}^{m-2}\left|
\frac{\delta^2}{MD(e_i,e_j)^2}
- \int_{\alpha_i}\int_{\alpha_j}\;\frac{1}{|x-y|^2}\right|\;.$$
By Lemma \[piarg\], $MD(e_i,e_j) = MD(\alpha_i,\alpha_j) =
|v_j-v_{i+1}|$. Since the arcs have length $\delta$, we know that the summands without absolute value are nonnegative; so, as in Claim 1, we bound the absolute value by finding an upper bound. We are dealing with something that looks like a Riemann Sum upper estimate of a finite integral. But as $n$ increases, we are changing the domain, not just subdividing the same set and we want to control the size of the error, not just say it goes to zero as $n \rightarrow \infty$. This is where we use the choice of $p$ as a fractional power $m^q$ where $q$ is strictly larger than $1/2$.
For brevity, let $md$ denote $MD(\alpha_i,\alpha_j)=|v_j-v_{i+1}|$, where $\arc{v_{i+1},v_j}=k\delta$. Since $$|x-y| \leq md+2\delta\;,$$ we have $$\frac{\delta^2}{md^2} -
\int_{\alpha_i}\int_{\alpha_j}\;\frac{1}{|x-y|^2}
\nonumber
\leq
\frac{\delta^2}{md^2} -\frac{\delta^2}{(md+2\delta)^2}
< 4 \delta^3 \frac{1}{md^3 }\,.$$
Thus $$(\ref{eq:Claim2ToBound})
\;\;\leq\;\; 8 n \delta^3 \sum_{k=p}^{m-2} \frac{1}{md^3}\;.$$
As before, by Lemmas \[piarg\], \[schursthm\], and \[allcosinelemmas\](b), since $\arc{v_{i+1},v_j}=k\delta$, $$\begin{aligned}
\nonumber md^2 = |v_j-v_{i+1}|^2 &\geq r^2(2-2
\cos(k\delta/r))
\\ \nonumber &\geq k^2 \delta^2 - \frac{1}{12}\frac{k^4
\delta^4}{r^2}
\\ \nonumber &= k^2 \delta^2
\left(1-\frac{1}{12}\frac{k^2\delta^2}{r^2}\right)
\\ \nonumber &\geq k^2 \delta^2
\left(1-\frac{1}{12}\pi^2\right) \text{since $k\delta \leq \pi r$.}\end{aligned}$$
Thus, $\md\geq 0.42\,k\delta$, so $$\frac{1}{md^3} < 13.42\, \frac{1}{k^3\delta^3} \;.$$
With the above observation, the $\delta^3$’s cancel and we have $$\begin{aligned}
(\ref{eq:Claim2ToBound})
&< 107.36\, n \sum_{k=p}^{m-2} \frac{1}{k^3} \nonumber \\
&< 107.36\, n \sum_{k=p}^{\infty} \frac{1}{k^3} \nonumber \\
&< 107.36\, n\frac{1}{(p-1)^2}\; . \nonumber\end{aligned}$$
We showed in the proof of the prior claim that $\frac{1}{p-1}< 1.59 \left(\frac{\delta}{r}\right)^{3/4}$. Thus, $$(\ref{eq:Claim2ToBound})
\leq 107.36\,n\frac{1}{(p-1)^2}
\leq 271.42\, \frac{E_L(K)^{3/2}}{n^{1/2}}$$
Note that in the above analysis the exponent $3/4$ needs to be strictly greater than $1/2$, so that when we double it, the power of $n$ in the denominator will more than cancel the leading factor $n$.
We now need to bound the contribution from $C$, that is $|E(C)-\umd'(C)|$. The radius of $C$, $R$, is the thickness radius $r(C)$. Also, we know from Lemma \[ChordsOnDifferentCircles\] that $R\geq r$. So if arcs $\alpha_i$, $\alpha_j$ of $K$ are near, then the corresponding arcs $\beta_i$, $\beta_j$ lie within an arc of $C$ of length $\leq \pi R$. Thus, the various steps in our analysis of $K$ can be carried out on $C$. We could obtain sharper bounds for $C$, but we will settle for the same bound since they dominate anyway.
For Claim 1, we have $$|f_i|\,|f_j| = R^2(2-2\cos(\delta/R)) \geq \delta^2-
\frac{1}{12}{\delta^4}{R^4} \geq \delta^2-\frac{1}{12}\frac{\delta^4}{r^2}\,,$$ and $$MD(f_i,f_j)^2\geq k^2\delta^2 - \frac{1}{12}\frac{k^4\delta^4}{R^2}
\geq k^2\delta^2-\frac{1}{12}\frac{k^4\delta^4}{r^2}\,,$$ exactly as for $K$. Now continue the proof of Claim 1 verbatim.
For Claim 2, $$\begin{aligned}
md(f_i,f_j)^2
&=R^2(2-2\cos(k\delta/r))\\
&\geq k^2\delta^2\left(1-\frac{1}{12}\frac{k^2\delta^2}{R^2}\right)\\
&\geq k^2\delta^2\left(1-\frac{1}{12}\frac{k^2\delta^2}{r^2}\right)\,,\end{aligned}$$ and the rest follows verbatim.
Thus, our final bound for the total error in this zone is just double the values obtained in Claims 1 and 2.
Bounds for $|E_0(K)-\emd(P)|$ in the Far Zone
---------------------------------------------
As before, we use $r$ to abbreviate $r(K)$. In the Near Zones, we just needed a value for $r\leq$ minimum radius of curvature of $K$. But in the Far Zone, we need both aspects of the thickness radius.
The argument here is somewhat similar to the Moderately Near Zone, but we control the denominators in a different way. In each situation, we need to know that spatial distances between points are bounded away from zero in some way depending on their arc-length distances along $K$. For the Far Zone, we use the fact that thickness controls critical self-distance, in particular Lemma \[distancebound\], together with local analysis (Lemma \[MaxDistArcToChord\]), to relate chord-chord distances to arc-arc distances. Also, we continue to use the hypothesis $\delta \leq r$.
*Remark on notation heuristics*. In the following paragraphs and Lemma \[FarChordsMin\], think of $(\alpha,\beta)$ as $(\alpha_i,\alpha_j)$ and $(e,f)$ as $(e_i,e_j)$.
Suppose $(\alpha,\beta)$ is a pair of far arcs (on $K$ or on $C$), with $(e,f)$ the inscribed chords joining their endpoints. Then $$\mathrm{md}(\alpha,\beta) > 1.08\,r\;,$$ and $$\md(e,f)> 0.79 \, r\;.$$ \[FarChordsMin\]
We analyze $K$, and note that the same bound will work for $C$ since $r\leq R$. We establish the lower bound for arcs, then use that to bound the distance for chords. If the minimum distance between a pair of arcs is realized at points that are interior to one or both arcs, then we are dealing with singly- or doubly-critical pairs of points, so, by Lemma \[lsdr\], $\md(\alpha,\beta) \geq 2r$. Thus we just need to bound the end-point distances. Let $\alpha_0$ and $\alpha_1$ be the endpoints of the arc $\alpha$ and $\beta_0$ and $\beta_1$ the endpoints of the arc $\beta$. Choose the labels so that $\alpha_1$ and $\beta_0$ are the points which are closest with respect to arc-length. In the worst case, the arc-length from $\alpha_0$ to $\beta_1$ is $\geq \pi r$, but the arc-lengths of the arcs $\widehat{\alpha_0\beta_0}$, $\widehat{\alpha_1\beta_0}$, and $\widehat{\alpha_1\beta_1}$ are less than $\pi r$. In such a case, we have the following situation:
- $|\alpha_0-\beta_0|$\
$\pi\,r \geq {\rm
arc}(\alpha_0,\beta_0)
\geq
\pi r-\delta
\implies |\alpha_0 -
\beta_0|^2 \geq r^2(2-2\cos(\pi-1))$ by Lemma \[schursthm\], and the fact that $\delta \leq r$. So $|\alpha_0-\beta_0|> 1.75 r$.
- $|\alpha_0-\beta_1|$\
${\rm arc}(\alpha_0,\beta_1) \geq \pi r \implies
|\alpha_0 -
\beta_1| \geq 2r$ by Lemma \[distancebound\].
- $|\alpha_1-\beta_1|$\
same bound as $|\alpha_0-\beta_0|$ .
- $|\alpha_1-\beta_0|$\
${\rm arc}(\alpha_1,\beta_0) \geq \pi r-2\delta \implies
{\rm arc}(\alpha_1,\beta_0) > (\pi-2)r$, since $\delta \leq r$. Thus, by Lemma \[schursthm\],\
$|\alpha_1-\beta_0|^2 \geq r^2(2-2 \cos(\pi-2))
\implies |\alpha_1-\beta_0| > 1.08r$.
In other scenarios, the arc pair $(\alpha,\beta)$ yields three of the above four cases, but we lose the smallest. For “most” arc pairs $(\alpha,\beta)$, we have all point-to-point distances at least $2r$.
We now obtain the lower bound on chord-to-chord distances using Lemma \[MaxDistArcToChord\]: $$\md(e,f) \geq \md(\alpha,\beta) - \frac{\sqrt{3}}{6}\,r >
1.08\,r -\left(\frac{\sqrt{3}}{6}\right) r > 0.79\, r\;.$$
\[FarZoneBound\] The total error in the Far Zone is bounded by $$0.56\,\frac{E_L(K)^4}{n^2}+1.60\,\frac{E_L(K)^5}{n^2}+7.76\,\frac{E_L(K)^4}{n}\,.$$
We first analyze the error on $K$, $$2\sum_{i=1}^n\sum_{j=i+m}^{n}\left|
\frac{|e_i|\,|e_j|}{\md(e_i,e_j)^2}-
\int_{x\in \alpha_i}\int_{y\in \alpha_j}
\frac{1}{| x-y|^2}\,dy\,dx\,\right|\,.$$
We do this in three steps: Compare $\frac{|e_i|\,|e_j|}{md(e_i,e_j)^2}$ to $\frac{\delta^2}{md(e_i,e_j)^2}$, that to $\frac{\delta^2}{md(\alpha_i,\alpha_j)^2}$, and that to $\iint\frac{1}{|x-y|^2}$. After we do each step for $K$, we double that to include the contribution from $C$. Note $\frac{\delta^2}{md(e_i,e_j)^2} =
\int_{x\in\alpha_i}\int_{y\in\alpha_j}\frac{1}{md(e_i,e_j)^2}\,dy\,dx$ and similarly for $\frac{\delta^2}{md(\alpha_i,\alpha_j)^2}$.
*Claim 1:*\
$$2\sum_{i=1}^{n}\sum_{j=i+m}^n \left|
\frac{\delta^2}{md(e_i,e_j)^2}
-\frac{|e_i|\,|e_j|}{md(e_i,e_j)^2}\right|
\leq 0.28\,\frac{E_L(K)^4}{n^2}\,.$$
*Claim 2*. $$2\sum_{i=1}^{n}\sum_{j=i+m}^n \int_{x\in\alpha_i}\int_{y\in\alpha_j}
\left| \frac{1}{md(e_i,e_j)^2} -
\frac{1}{md(\alpha_i,\alpha_j)^2}\right|\,dy\,dx\,\leq
0.80\,\frac{E_L(K)^5}{m^2}.
\label{eq:mdedgearc}$$
*Claim 3*. $$2\sum_{i=1}^{n}\sum_{j=i+m}^n \int_{x\in\alpha_i}\int_{y\in\alpha_j}
\left|\frac{1}{md(\alpha_i,\alpha_j)^2} -
\frac{1}{| x-y|^2}\right|\,dy\,dx\,\leq 3.88\,\frac{E_L(K)^4}{n}\;.
\label{lastone}$$
*Proof of Claim 1*.
Since arc-length $\geq$ chord length, each summand is nonnegative without taking the absolute value, so we just need to bound the terms from above. By Lemma \[schursthm\] and Lemma \[allcosinelemmas\](b), $\delta^2-\frac{1}{12}\frac{\delta^4}{r^2}\leq
|e_i|,|e_j|$. Thus, $$\frac{\delta^2}{\md(e_i,e_j)^2}
-\frac{|e_i|\,|e_j|}{\md(e_i,e_j)^2}
\leq \frac{1}{12}\;\frac{\delta^4}{r^2 \;\md(e_i,e_j)^2}\,.$$ But Lemma \[FarChordsMin\] gives us that $\md(e_i,e_j)^2 > (0.79)^2\,r^2$, so $$\frac{\delta^2}{\md(e_i,e_j)^2}
-\frac{|e_i|\,|e_j|}{\md(e_i,e_j)^2}
< 0.14\,\frac{\delta^4}{r^4}\,.$$ Multiplying by $2n^2$ gives $$2\sum_{i=1}^{n}\sum_{j=m}^n \left(
\frac{\delta^2}{\md(e_i,e_j)^2}
-\frac{|e_i|\,|e_j|}{\md(e_i,e_j)^2}\right)
< 0.28\,\frac{n^2\delta^4}{r^4}
= 0.28\,\frac{E_L(K)^4}{n^2}$$
*Proof of Claim 2*.
The sum is bounded by $(2n^2\delta^2)\mathrm{(worst\ error\ in\ integrands)}.$ We will use Lemma \[MaxDistArcToChord\](b) to bound that. To make the algebra more evident, let $\epsilon = md(e_i,e_j)$ and $\gamma = md(\alpha_i,\alpha_j)$. The term we wish to bound is $$\left|\frac{1}{\epsilon^2}-\frac{1}{\gamma^2}\right|=
\left|\frac{\gamma^2-\epsilon^2}{\epsilon^2\gamma^2}\right|
< 1.38\,\frac{|\gamma-\epsilon|\,(\gamma+\epsilon)}{r^4}
\leq 0.40\,\frac{(\gamma+\epsilon)\delta^2}{r^5}\,,$$ since $\epsilon > 0.79 r$ and $\gamma>1.08r$ by Lemma \[FarChordsMin\], and $|\gamma-\epsilon|\leq \frac{\sqrt{3}}{6}\frac{\delta^2}{r}$ by Lemma \[MaxDistArcToChord\](b).
Now $\epsilon$, $\gamma$ are minimum distances between sets that include points of $K$, so $\epsilon,\gamma\leq \ell(K)/2$ and $\gamma+\epsilon\leq \ell(K)$. Thus, $$\left|\frac{1}{\epsilon^2}-\frac{1}{\gamma^2}\right| \leq
\frac{0.40\,\ell(K)\delta^2}{r^5}\,.$$ Multiplying by $2n^2\delta^2$, we get $$\begin{aligned}
2\sum_{i=1}^{n}\sum_{j=i+m}^n
\int_{x\in\alpha_i}\int_{y\in\alpha_j}
\left|\frac{1}{md(e_i,e_j)^2} -
\frac{1}{md(\alpha_i,\alpha_j)^2}\right|\,dy\,dx &\leq 0.80\,
\frac{n^2\delta^4\ell(K)}{r^5}\nonumber\\ &= 0.80\,
\frac{E_L(K)^5}{n^2}\nonumber\,.\end{aligned}$$
*Proof of Claim 3*.
The sum is bounded by $(2n^2\delta^2)\mathrm{(worst\ error\ in\ integrand)}.$
Let $\gamma$ denote $md(\alpha_i,\alpha_j)$. So for particular $x$, $y$ on $\alpha_i$ and $\alpha_j$, we have $|x-y|= \gamma+t$ for some $0\leq t\leq 2\delta$. The largest error is then $$\frac{1}{\gamma^2}-\frac{1}{(\gamma+t)^2}=
\frac{t(2\gamma+t)}{\gamma^2(\gamma+t)^2}
\leq \frac{t(2\gamma+t)}{\gamma^4}
<\frac{t(2\gamma+t)}{(1.08)^4r^4}
\,,$$ since $\gamma\geq 1.08r$ by Lemma \[FarChordsMin\].
Now $t\leq 2\delta$ and $\gamma\leq \ell(K)/2$. Thus, $$\begin{aligned}
\frac{t(2\gamma+t)}{1.08^4r^4}
&\leq \frac{2\delta(\ell(K)+2\delta)}{1.08^4r^4}\\
&= \frac{2}{1.08^4}\frac{\delta(n\delta+2\delta)}{r^4}\\
&= \frac{2}{1.08^4}\frac{\delta^2(n+2)}{r^4}\\
&\leq \frac{2}{1.08^4}\frac{\delta^2}{r^4}\frac{(2\pi+2) n}{2\pi}
\text{ since }n>\elk\geq 2\pi \\
&< 1.94\,\frac{\delta^2n}{r^4}\,.\end{aligned}$$
Thus, $$\begin{aligned}
2\sum_{i=1}^{n}\sum_{j=i+m}^n \int_{x\in \alpha_i}
\int_{y\in \alpha_j} \left|\frac{1}{md(\alpha_i,\alpha_j)^2} -
\frac{1}{| x-y|^2}\right|\,dy\,dx
&\leq 3.88\, n^2\delta^2\frac{\delta^2n}{r^4}\\
&= 3.88\,\frac{E_L(K)^4}{n}\end{aligned}$$
Putting it all together {#finalproof}
=======================
Here we combine the bounds from the various zones.
From Propositions \[EoDiagAndVeryNear\], \[polygonverynearzone\], \[ModNearZone\], and \[FarZoneBound\], we have $$\begin{aligned}
|E_0(K)-\emd(P)| \leq&\; 3.82\,\elk\,\left(\frac{\elk}{n}\right)^{1/4}\\
&+ 3.00\,\elk\, \left(\frac{\elk}{n}\right)^{7/4}\\
&+ 542.84\,\elk\,\left(\frac{\elk}{n}\right)^{1/2}\\
&+ 0.56\,\elk^2\,\left(\frac{\elk}{n}\right)^2\\
&+ 1.60\,\elk^3\,\left(\frac{\elk}{n}\right)^2\\
&+ 7.76\,\elk^3\left(\frac{\elk}{n}\right)\end{aligned}$$
Since $E_L(K)\geq 2\pi>1$, and $n>\elk$, we see that certain terms dominate others. So, $$|E_0(K)-\emd(P)| < 550\, \frac{\elk^{5/4}}{n^{1/4}} +
10\, \frac{\elk^4}{n}\;.$$ If $n>\elk^{11/3}$, then the total error is less than $560\, \frac{\elk^{5/4}}{n^{1/4}}$.
This completes the proof of Theorem \[explicit\].
Acknowledgments
===============
We thank J. Sullivan for asserting and experimentally confirming the correct regularization and Y.-Q. Wu for modifying [*MING*]{} to allow additional numerical confirmation. We also thank G. Buck for helpful comments.
|
---
abstract: 'We study modules over the ring $\wt{\C}$ of complex generalized numbers from a topological point of view, introducing the notions of $\wt{\C}$-linear topology and locally convex $\wt{\C}$-linear topology. In this context particular attention is given to completeness, continuity of $\wt{\C}$-linear maps and elements of duality theory for topological $\wt{\C}$-modules. As main examples we consider various Colombeau algebras of generalized functions.'
author:
- |
Claudia Garetto [^1]\
Dipartimento di Matematica\
Università di Torino,\
via Carlo Alberto 10, 10123 Torino, Italia\
`garettoc@dm.unito.it`\
date:
title: '**Topological structures in Colombeau algebras: topological $\wt{\C}$-modules and duality theory**'
---
[**[Key words:]{}**]{} modules over the ring of complex generalized numbers, algebras of generalized functions, topology, duality theory
*AMS 2000 subject classification: 46F30, 13J99, 46A20*
[10]{}
H. A. Biagioni. . Number 1421 in Lecture Notes in Math. Springer-Verlag, Berlin, 1990.
H. Biagioni and M. Oberguggenberger. Generalized solutions to [B]{}urgers’ equation. , 97(2):263–287, 1992.
H. Biagioni and M. Oberguggenberger. Generalized solutions to the [K]{}orteweg-[D]{}e [V]{}ries and the regularized long-wave equations. , 23(4):923–940, 1992.
H. A. Biagioni and J. F. Colombeau. New generalized functions and $\mathcal{C}^\infty$ functions with values in generalized complex numbers. , 33(1):169–179, 1986.
N. Bourbaki. Hermann, Paris, 1966.
J. F. Colombeau. . North Holland, Amsterdam, 1984.
J. F. Colombeau. . North-Holland Mathematics Studies 113. Elsevier Science Publishers, 1985.
J. F. Colombeau. . Number 1532 in Lecture Notes in Math. Springer-Verlag, New York, 1992.
J. F. Colombeau, A. Heibig, and M. Oberguggenberger. Generalized solutions to partial differential equations of evolution type. , 45(2):115–142, 1996.
J. F. Colombeau and M. Oberguggenberger. On a hyperbolic system with a compatible quadratic term: Generalized solutions, delta waves, and multiplication of distributions. , 15(7):905–938, 1990.
N. Dapi[ć]{}, S. Pilipovi[ć]{}, and D. Scarpal[é]{}zos. Microlocal analysis of [C]{}olombeau’s generalized functions: propagation of singularities. , 75:51–66, 1998.
E. Farkas, M. Grosser, M. Kunzinger, and R. Steinbauer. On the foundations of nonlinear generalized functions [I]{}. , 153(729), 2001.
C. Garetto. Pseudo-differential operators in algebras of generalized functions and global hypoellipticity. , 80(2):123–174, 2004.
C. Garetto. Topological structures in [C]{}olombeau algebras [II]{}: investigation into the duals of ${\Gc(\Om)}$, ${\G(\Om)}$ and ${\GS(\R^n)}$. , 2004.
C. Garetto, T. Gramchev, and M. Oberguggenberger. Pseudo-differential operators with generalized symbols and regularity theory. , 2003.
C. Garetto and G. Hörmann. Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. , 2004.
M. Grosser. On the foundations of nonlinear generalized functions [II]{}. , 153(729), 2001.
M. Grosser, G. H[ö]{}rmann, M. Kunzinger, and M. Oberguggenberger, editors. , Boca Raton, 1999. Chapman [&]{} Hall/CRC.
M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer. , volume 537 of [ *Mathematics and its Applications*]{}. Kluwer, Dordrecht, 2001.
G. H[ö]{}rmann. Integration and microlocal analysis in [C]{}olombeau algebras. , 239:332–348, 1999.
G. Hörmann. First-order hyperbolic pseudodifferential equations with generalized symbols. , 2004. ar[X]{}iv.org e-Print math.AP/0307406.
G. H[ö]{}rmann. Hölder-[Z]{}ygmund regularity in algebras of generalized functions. , 23:139–165, 2004.
G. H[ö]{}rmann and M. V. de Hoop. Geophysical modeling with [C]{}olombeau functions: [M]{}icrolocal properties and [Z]{}ygmund regularity. , 2001. ar[X]{}iv.org e-Print math.AP/0104007.
G. H[ö]{}rmann and M. V. de Hoop. Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients. , 67:173–224, 2001.
G. H[ö]{}rmann and M. Kunzinger. Microlocal analysis of basic operations in [C]{}olombeau algebras. , 261:254–270, 2001.
G. H[ö]{}rmann and M. Oberguggenberger. Elliptic regularity and solvability for partial differential equations with [C]{}olombeau coefficients. , 2004(14):1–30, 2004.
G. Hörmann, M. Oberguggenberger, and S. Pilipovic. Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients. , 2003.
J. Horváth. . Addison-Wesley, Reading, MA, 1966.
M. Kunzinger and M. Oberguggenberger. Group analysis of differential equations and generalized functions. , 31(6):1192–1213, 2000.
F. Lafon and M. Oberguggenberger. Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case. , 160:93–106, 1991.
A. H. Lightstone and A. Robinson. . North-Holland, Amsterdam.
M. Nedeljkov, S. Pilipovi[ć]{}, and D. Scarpal[é]{}zos. . Pitman Research Notes in Mathematics 385. Longman Scientific [&]{} Technical, 1998.
M. Oberguggenberger. Generalized solutions to semilinear hyperbolic systems. , 103(2):133–144, 1987.
M. Oberguggenberger. Hyperbolic systems with discontinuous coefficients: generalized solutions and a transmission problem in acoustics. , 142:452–467, 1989.
M. Oberguggenberger. The [C]{}arleman system with positive measures as initial data - generalized solutions. , 20(2-3):177–197, 1991.
M. Oberguggenberger. . Pitman Research Notes in Mathematics 259. Longman Scientific [&]{} Technical, 1992.
M. Oberguggenberger. Generalized functions in nonlinear models - a survey. , 47(8):5029–5040, 2001.
M. Oberguggenberger and M. Kunzinger. Characterization of [C]{}olombeau generalized functions by their point values. , 203:147–157, 1999.
M. Oberguggenberger, S. Pilipović, and D. Scarpalézos. Local properties of [C]{}olombeau generalized functions. , 256:88–99, 2003.
M. Oberguggenberger and T. D. Todorov. An embedding of [S]{}chwartz distributions in the algebra of asymptotic functions. , 21(3), 1998.
M. Oberguggenberger and Y. G. Wang. Regularized conservation laws and nonlinear geometric optics. , 127(1):55–66, 1999.
A. P. Robertson and W. Robertson. . Number 53 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge-New York, 1980.
A. Robinson. . North Holland, Amsterdam, 1966.
A. Robinson. Function theory on some nonarchimedean fields. , 80(6):87–109, 1973. Part II: Papers in the Foundations of Mathematics.
H. L. Royden. . Macmillan Publishing Company, New York, 1988.
D. Scarpalézos. Topologies dans les espaces de nouvelles fonctions généralisées de [C]{}olombeau. ${\widetilde{\C}}$-modules topologiques. Université Paris 7, 1992.
D. Scarpalézos. Some remarks on functoriality of [C]{}olombeau’s construction; topological and microlocal aspects and applications. , 6(1-4):295–307, 1998.
D. Scarpalézos. Colombeau’s generalized functions: topological structures; microlocal properties. a simplified point of view. , (25):89–114, 2000.
H. H. Schaefer and M. P. Wolff. . Number 3 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1999.
T. Todorov and R. S. Wolf. Hahn field representation of [R]{}obinson’s asymptotic numbers. , 2000.
[^1]: Current address: Institut für Technische Mathematik, Geometrie und Bauniformatik, Universität Innsbruck, e-mail:`claudia@mat1.uibk.ac.at`
|
---
abstract: |
In a finite abelian group $G$, define an additive matching to be a collection of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i = j = k$. In the case that $G = \mathbb{F}_2^n$, Kleinberg, building on work of Croot-Lev-Pach and Ellenberg-Gijswijt, proved a polynomial upper bound on the size of an additive matching. Fox and Lovász used this to deduce polynomial bounds on Green’s arithmetic removal lemma in $\mathbb{F}_2^n$.
If $G$ is taken to be an arbitrary finite abelian group, the questions of bounding the size of an additive matching and giving bounds for Green’s arithmetic removal lemma are much less well understood. In this note, we adapt the methods of Fox and Lovász to prove that, provided we can assume a sufficiently strong bound on the size of an additive matching in cyclic groups, a similar bound should hold in the case of removal.
author:
- 'James Aaronson[^1]'
bibliography:
- 'RemovalRefs.bib'
title: A connection between matchings and removal in abelian groups
---
Introduction {#sec:1}
============
In an abelian group $G$, define a *triangle* to be a triple of elements $x$, $y$ and $z$ with $x + y + z = 0$. Green’s arithmetic triangle removal lemma [@green2005szemeredi] states that for any ${\varepsilon}> 0$, there is a $\delta > 0$ such that the following holds. Whenever $X$, $Y$ and $Z$ are subsets of $G$ such that there are at most $\delta N^2$ triangles $x + y + z = 0$ with $x \in X$, $y \in Y$ and $z \in Z$, we can remove at most ${\varepsilon}N$ elements from $X$, $Y$ and $Z$ to remove all of the triangles. The bounds in [@green2005szemeredi] are quite weak; $1/\delta$ is given as a tower of twos of height polynomial in $1/{\varepsilon}$. The best known bounds for this problem in general are still of tower type.
In [@FLpaper], Fox and Lovász proved a much stronger bound on $\delta$ in the case of $G = \mathbb{F}_p^n$ for a fixed prime $p$; namely, that $1/\delta$ is bounded by a polynomial in $1/{\varepsilon}$. Define an *additive matching* to be a collection of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i = j = k$. These are also called tricolored sum-free sets, and can be represented by $(X,Y,Z)$, where $X = \{x_i\}$, $Y = \{y_i\}$ and $Z = \{z_i\}$. Building on the groundbreaking work on the cap set problem by Croot-Lev-Pach [@1605.01506] and subsequent work by Ellenberg-Gijswijt [@1605.09223], Kleinberg [@1605.08416] gave a polynomial upper bound for the size of an additive matching in $G$ in the case that $G = \mathbb{F}_2^n$, and Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans [@1605.06702] extended this to $\mathbb{F}_q^n$ for a fixed prime power $q$. The argument by Fox and Lovász made use of these results to prove the polynomial bounds on removal.
Polynomial bounds on removal are much stronger than could possibly hold in general groups. Indeed, using Behrend’s construction [@MR0018694] of a large subset of ${{\mathbb{Z}}/N{\mathbb{Z}}}$ with no 3-term arithmetic progressions, it is possible [@MR519318] to show that the best one could hope for is
$${\varepsilon}\ll \exp\left(-c \sqrt{\log(1/\delta)}\right).$$
The goal of this note is to adapt the arguments of Fox and Lovász to show that, in the context of cyclic groups, good bounds on additive matchings give good bounds on removal.
Assume that, in a cyclic group of order $M$, the density of an additive matching is bounded above by $f(M)$ for some function $f$. Assume that $f(M)$ can be taken to be decreasing as $M$ increases, but that $Af(A) < Bf(B)$ for $A < B$; these conditions correspond to the claim that the maximum size of an additive matching increases as the size of the group increases, but the maximum density decreases. Observe that Behrend’s example guarantees that
$$f(M) \gg \exp\left(-c \sqrt{\log(M)}\right)$$
because, if $A$ is a progression free set, then $(A, -2A, A)$ is an additive matching.
Suppose further that there exists a function $g$ such that $g(\rho)$ increases as $\rho$ decreases, $\sum_{i=1}^{\infty} \frac{1}{g(2^{-i})} \leq \frac{1}{4}$ and $g(\rho)^2 f\left(\frac{1}{g(\rho)\rho}\right)$ is decreasing as $\rho$ decreases for $\rho < \alpha$, for some absolute constant $\alpha$. $g$ plays the same role here as in [@FLpaper].
We are now ready to state Theorem \[MainTheorem\].
\[MainTheorem\] Suppose that $A$, $B$ and $C$ are subsets of ${{\mathbb{Z}}/N{\mathbb{Z}}}$ for some $N \in \mathbb{N}$ with the property that there are at most $\delta N^2$ triangles $a + b + c = 0$ with $a \in A$, $b \in B$ and $c \in C$.
Then, we can remove all of the triangles by deleting at most ${\varepsilon}N$ elements from $A$, $B$ and $C$, where ${\varepsilon}$ satisfies
$$\label{MTEQ}
{\varepsilon}\ll g(\delta) \sqrt{f\left(\frac{1}{g(\delta)\delta}\right)}.$$
We can deduce some consequences of this:
\[HappyCor\] Suppose that we have the best possible bound on the size of an additive matching, namely a Behrend-type bound. In particular, we can take $f(N)$ to be $\exp(-c\sqrt{\log{N}})$ for some constant $c$. Then $g(\rho) = k\log^2(1/\rho)$ suffices, and we deduce the bound
$${\varepsilon}\ll \exp(-c_1\sqrt{\log 1/\delta})$$
for some other constant $c_1$.
\[SadCor\] Suppose that the much more pessimistic bound $f(N) = \log^{-2 - \gamma} N$ holds, for some constant $\gamma > 0$. Then, we can take $g(\rho) = k\log^{1 + \gamma/3}(1/\rho)$, and we deduce that
$${\varepsilon}\ll \log({1/\delta})^{-O(1)}.$$
Observe that we cannot deduce anything nontrivial if the assumption on $f$ is weaker, because of the need for $\sum_{i=1}^{\infty} \frac{1}{g(2^{-i})}$ to converge.
A converse of sorts to Theorem \[MainTheorem\], namely that bounds on removal imply similar bounds on the maximal size of an additive matching, is relatively trivial. Indeed, suppose that, whenever subsets $X$, $Y$ and $Z$ of a cyclic group $G = {{\mathbb{Z}}/N{\mathbb{Z}}}$ define at most $\delta N^2$ triangles, the triangles can be removed by deleting at most ${\varepsilon}N$ elements, where ${\varepsilon}\ll f(1/\delta)$.
Then, an additive matching of size $\theta N$ defines at most $N = \frac{1}{N} N^2$ triangles, and requires removal of at least $\theta N$ elements to remove the triangles. Thus, $\theta \ll f(N)$, which can be seen to be the partial converse we wanted.
Throughout this note, we will use the notation $x \ll y$ to mean that, for some absolute constant $C$ independent of any variables, $x \leq Cy$.
The author is supported by an EPSRC grant EP/N509711/1. The author a DPhil student at Oxford University, and is grateful to his supervisor, Ben Green, for his continued support.
Theorem \[MainTheorem\] for $N$ prime
=====================================
In this section, following the approach in [@FLpaper], we we prove Theorem \[MainTheorem\], in the case that $N$ is prime. We start with a lemma, which is an analogue of Lemma 5 from [@FLpaper]:
\[BigLem\] Suppose we have three subsets of ${{\mathbb{Z}}/N{\mathbb{Z}}}$, $X$, $Y$ and $Z$, with the property that, for each $x \in X$, there are between $\delta_1 N$ and $\delta_2 N$ elements $y \in Y$ such that $z = -x -y$ is in $Z$. Suppose that the same holds with the positions of $X$, $Y$ and $Z$ permuted.
Then, we deduce that $|X|$ satisfies
$$\label{L3Conc}
|X| \ll \frac{\delta_2}{\delta_1} f(\delta_2^{-1})N,$$
and similar inequalities for $|Y|$ and $|Z|$.
Choose $a$, $b$ and $d$ uniformly and independently from ${{\mathbb{Z}}/N{\mathbb{Z}}}$ such that $d$ is nonzero. Let $L = 2l + 1$ be an odd positive integer such that $1/20 \leq L\delta_2 \leq 1/10$. We may assume that $\delta_2$ is small enough that $L > 20$ by adjusting the implicit constant in \[L3Conc\], and so such a choice of $L$ exists.
We say that a triangle $x + y + z = 0$ is *valid* if and only if $x \in I_X := a + [-l,l]d$ and $y \in I_Y := b + [-l,l]d$. Observe that this will imply that $z \in I_Z := -a-b + [-2l,2l]d$. We say that a valid triangle is *good* provided that each of $x$, $y$ and $z$ is in only one valid triangle, namely the triangle in question.
\[C1\] Given a valid triangle $x + y + z = 0$, it has a probability at least $2/5$ of being good.
We first show that the probability that $x$ is in another valid triangle is at most $1/5$. Indeed, for each $y'$ that forms a triangle with $x$, it has a probability of $\frac{L-1}{N-1} \leq \frac{L}{N}$ of lying in $I_Y$, because choosing values $a$, $b$ and $d$ such that $x + y + z = 0$ is valid is equivalent to choosing values $r, s \in [-l,l]$ so that $x = a + rd, y = b + sd$, and then choosing any value of $d$. Thus, each value of $y' \neq y$ will occur with probability $\frac{L-1}{N-1}$ since, for each choice of $t \neq s \in [-l,l]$, there is exactly one choice of $d$ such that $y = b + td$.
There are at most $\delta_2 N$ possible choices of $y'$ to consider, so the union bound guarantees that the probability that $x$ is in another valid triangle is at most $L\delta_2 \leq 1/5$.
The same argument applies to the probability that $y$ is in another valid triangle. For $z$, it turns out that the bound is even stronger, because for each $y'$ forming a valid triangle with $z$, $y'$ has a probability of at most $\frac{L}{N}$ of lying in $I_Y$. This is an upper bound for the probability that, setting $x' = -y'-z$, the triangle $x' + y' + z = 0$ is valid since $x'$ is not guaranteed to lie in $I_X$.
Thus, the probability that either $x$, $y$ or $z$ cause the triangle to be not good is at most $3/5$ by the union bound, and thus the probability that the triangle is good is at least $2/5$.
\[C2\] Given $x \in I_X$, the probability that it is in a good triangle is at least $\frac{\delta_1}{50\delta_2}$.
For each $y$ that forms a triangle with $x$, it has a probability of $\frac{L}{N}$ of lying in $I_Y$, and, conditioned on this, a probability of at least $2/5$ of forming a good triangle with $x$. In other words, for each $y$ forming a triangle with $x$, it has a probability of at least $\frac{2L}{5N} \geq \frac{1}{50\delta_2 N}$ of forming a good triangle with $x$.
By definition, $x$ can be in at most one good triangle, so these events are disjoint. There are at least $\delta_1 N$ choices of $y$ forming a triangle with $x$, and so the probability that at least one of them is good is at least $\frac{1}{50 \delta_2 N} \times \delta_1 N = \frac{\delta_1}{50\delta_2}$.
\[C3\] The expected number of $x \in X$ in good triangles is at least $\frac{|X| \delta_1}{1000 \delta_2^2 N}$.
The probability that $x$ is in $I_X$ is $\frac{L}{N} \geq \frac{1}{20 \delta_2 N}$. Conditioned on this, $x$ has a probability of at least $\frac{\delta_1}{50\delta_2}$ of being in a good triangle. Hence, each $x \in X$ has a probability of $\frac{\delta_1}{1000\delta_2^2 N}$ of lying in a good triangle. Linearity of expectation yields the result.
Thus, we can choose parameters $a$, $b$ and $d$ in such a way that there are at least $\frac{|X| \delta_1}{1000 \delta_2^2 N}$ good triangles in the corresponding sets $I_X$, $I_Y$ and $I_Z$.
Now, observe that we may map the intervals $I_X$, $I_Y$ and $I_Z$ into ${\mathbb{Z}}/M{\mathbb{Z}}$, where $M = \delta_2^{-1}$, in the obvious way. For example, $a + rd \in I_X$ for $r \in [-l,l]$ maps to $r \mod M$. It is clear that this map sends triangles to triangles; since $M > 8l$, this map also preserves the status of not being a triangle.
In other words, the at least $\frac{|X| \delta_1}{1000 \delta_2^2 N}$ good triangles we found earlier correspond to an additive matching within ${\mathbb{Z}}/M{\mathbb{Z}}$. Given our hypothesis on the size of an additive matching, we deduce that
$$\frac{|X| \delta_1}{1000 \delta_2^2 N} \leq Mf(M)$$
and so
$$\frac{|X|}{N} \leq 1000 \frac{\delta_2}{\delta_1} f(\delta_2^{-1})$$
which is exactly what we sought.
Next, we prove an analogue of Lemma 6 from [@FLpaper].
\[SmallLem\] Suppose that ${\varepsilon}, \delta > 0$ satisfy
$${\varepsilon}\gg g(\delta) \sqrt{f\left(\frac{1}{g(\delta)\delta}\right)}$$
for the functions $f$ and $g$ defined previously, and that $\delta < \alpha$ as defined immediately before Theorem \[MainTheorem\]. Suppose we have a collection of ${\varepsilon}N$ disjoint triangles $x_i + y_i + z_i = 0$, and let $X = \{x_i\}$, defining $Y$ and $Z$ analogously. Then there must be at least $\delta N^2$ triangles $x_i + y_j + z_k = 0$.
The majority of the proof is the same as that in [@FLpaper], so we will not reproduce it here; the only difference being that we do not mind if elements are in more than one out of $X,$ $Y$ and $Z$, because we are treating them separately in our proof of Lemma \[BigLem\]. Suffice it to say that we will reach a point where, for some $\delta' \leq \delta$, we have at least $\frac{\delta'}{2} N^2$ triangles in sets $X$, $Y$ and $Z$, where those sets are of size at least $\frac{{\varepsilon}}{2g(\delta')}N$. These have the property that each element is in at least $\frac{\delta'}{6{\varepsilon}} N$ and at most $\frac{g(\delta')\delta'}{{\varepsilon}} N$ triangles.
Applying Lemma \[BigLem\], we deduce that
$$\begin{aligned}
\frac{{\varepsilon}}{2g(\delta')} &\ll (6g(\delta')) f\left(\frac{{\varepsilon}}{g(\delta')\delta'}\right) \\
{\varepsilon}&\ll g(\delta')^2 {\varepsilon}^{-1} f\left(\frac{1}{g(\delta')\delta'}\right) \\
{\varepsilon}^2 &\ll g(\delta)^2 f\left(\frac{1}{g(\delta)\delta}\right)\end{aligned}$$
where in the second line we used the conditions on $f$ and in the third line we used that the right hand side decreases as $\delta'$ decreases.
We are now ready to prove Theorem \[MainTheorem\].
We follow the same strategy as in [@FLpaper]. Suppose that $A$, $B$ and $C$ are such that we cannot remove all the triangles without removing at least ${\varepsilon}N$ elements from $A$, $B$ and $C$. Any maximal set of disjoint triangles must have size at least $\frac{{\varepsilon}}{3} N$, else we could remove all of the elements of those triangles and there would be no triangles left.
Lemma \[SmallLem\] guarantees that we must have at least $\delta N^2$ triangles in total, where
$${\varepsilon}\ll g(\delta) \sqrt{f\left(\frac{1}{g(\delta)\delta}\right)}$$
as required.
Theorem \[MainTheorem\] for general $N$
=======================================
In this section, we complete the proof of Theorem \[MainTheorem\] in the case that $N$ need not be prime. First, observe that the prime case of Theorem \[MainTheorem\] implies that it holds for subsets of $[-M/2,M/2]$. Indeed, provided that the functions $f$ and $g$ exist and satisfy all of the requirements imposed upon them above, then we can deduce the following:
\[CorZed\] Suppose that $A$, $B$ and $C$ are subsets of $[-M/2,M/2]$ with the property that there are at most $\delta M^2$ triangles $a + b + c = 0$ with $a \in A$, $b \in B$ and $c \in C$.
Then, we can remove all of the triangles by deleting at most ${\varepsilon}M$ elements from $A$, $B$ and $C$, where ${\varepsilon}$ satisfies
$${\varepsilon}\ll g(\delta) \sqrt{f\left(\frac{1}{g(\delta)\delta}\right)}.$$
Suppose we have sets $A$, $B$ and $C$ which define $\delta M^2$ triangles. Select a prime $N$ such that $2M \leq N \leq 4M$, and consider the reduction modulo $N$ map $\phi$ taking $[-M/2,M/2]$ to ${{\mathbb{Z}}/N{\mathbb{Z}}}$. This preserves the status of being a triangle, as well as the status of not being a triangle.
The image of $(A,B,C)$ under $\phi$ contains at most $\delta (N/2)^2$ triangles, and thus requires removal of at most ${\varepsilon}N$ points to remove all of the triangles, where ${\varepsilon}$ satisfies
$${\varepsilon}\ll g(\delta/4) \sqrt{f\left(\frac{4}{g(\delta/4)\delta}\right)}.$$
Thus, to remove the triangles from $(A,B,C)$, the deletion of at most ${\varepsilon}N \leq 4{\varepsilon}M$ points is necessary. By adjusting the implicit constant, we deduce Corollary \[CorZed\].
We may now deduce that Theorem $\ref{MainTheorem}$ holds in arbitrary finite cyclic groups:
\[CorComp\] Theorem \[MainTheorem\] holds without the requirement that $N$ is prime.
Suppose not; then for some (composite) $N$, ${{\mathbb{Z}}/N{\mathbb{Z}}}$ contains sets $A$, $B$ and $C$ which define at most $\delta N^2$ triangles, but require deletion of at least ${\varepsilon}N$ points to remove the triangles, and where ${\varepsilon}$ does not satisfy (\[MTEQ\]) (with a slightly adjusted implicit constant).
As in the proof of Theorem \[MainTheorem\], a greedy argument guarantees the existence of $\frac{{\varepsilon}}{3} N$ disjoint triangles. Consider the map $\pi$ taking ${{\mathbb{Z}}/N{\mathbb{Z}}}$ to $[0,N-1]$ in the obvious way; a triple $(a,b,c)$ in ${{\mathbb{Z}}/N{\mathbb{Z}}}$ is a triangle if and only if its image under $\pi$ has sum either $N$ or $2N$.
For one of the two possibilities for the sum, there are at least $\frac{{\varepsilon}}{6} N$ disjoint triangles. In the first case, in which the triangles have sum $N$ in the image of $\pi$, consider $\pi(A)$, $\pi(B)$ and $\pi(C) - N$, and in the second case, consider $\pi(A)$, $\pi(B) - N$ and $\pi(C) - N$. Either way, we have at most $\frac{\delta}{2} (2N)$ triangles in $[-N,N]$, which require deletion of at least $\frac{{\varepsilon}}{12} (2N)$ points to remove, because they define at least that many disjoint triangles. Corollary \[CorZed\] gives us the result.
[^1]: Contact email:` james.aaronson@maths.ox.ac.uk`
|
---
abstract: 'We present experimental schemes that allow to study the entanglement classes of all symmetric states in multiqubit photonic systems. In addition to comparing the presented schemes in efficiency, we will highlight the relation between the entanglement properties of symmetric Dicke states and a recently proposed entanglement scheme for atoms. In analogy to the latter, we obtain a one-to-one correspondence between well-defined sets of experimental parameters and multiqubit entanglement classes inside the symmetric subspace of the photonic system.'
author:
- 'N. Kiesel'
- 'W. Wieczorek'
- 'S. Krins'
- 'T. Bastin'
- 'H. Weinfurter'
- 'E. Solano'
title: Operational multipartite entanglement classes for symmetric photonic qubit states
---
[^1]
Introduction
============
Entanglement is recognized as a fundamental resource in many quantum information tasks [@Hor09; @Guh09] like in quantum teleportation [@Ben93], quantum cryptography [@Eke91] or quantum computation [@Rau01]. In the general $N$-partite case the structure of entanglement is extremely rich and exhibits a much higher complexity than in the simplest bipartite case. There exist different kinds of entanglement and many efforts are done in trying to group them into different classes, in particular with respect to their equivalence properties under stochastic local operations and classical communication (SLOCC) [@Dur00; @Aci01; @Ver02; @Lam06; @Lam07; @Che06; @Mat09; @Bas09a; @Bas09b].
Recently, an operational approach to this classification problem has been proposed where in a *single* experimental setup a one-to-one correspondence between well-defined sets of experimental parameters and multiqubit entanglement classes of the symmetric subspace of atomic qubits is obtained [@Bas09a; @Bas09b]. When it comes to experimentally implementing different classes of entanglement, photonic qubits are widely used and so far the most flexible system [@Kie93; @Eib04; @Zei05; @Kie07; @Pan08; @Wie08; @Asp08; @Pre09; @Wie09; @vonZanthier09]. Here the observation of different types of entanglement in a *single* setup has been achieved experimentally[@Kie07; @Wie08; @Asp08; @Pre09; @Wie09; @Lan09].
Here we propose three experimental schemes that establish a one-to-one correspondence between experimental configurations and entanglement classes of photonic qubit states. Our proposed experimental schemes are based on linear optics setups making use of photons produced by single photon sources (SPSs) or spontaneous parametric down-conversion processes (SPDC). These schemes are divided into two steps. First, a photonic state ${\mbox{$\,\mid \! \psi \, \rangle$}}_I$ is obtained, where $N$ photons of well-defined polarization states occupy a single spatial mode [@Hof04; @Mit04; @Lietal]. Secondly, these photons are symmetrically distributed into $N$ separate spatial modes via polarization-independent beam splitters (BSs), i.e., essentially a multiport BS [@Rec94; @Zuk97; @Lim05]. Upon successful detection of a single photon in each of these modes the result is the observation of a symmetric state ${\mbox{$\,\mid \! \psi \, \rangle$}}_O$. Its entanglement class is fully determined by the experimental parameters of the $N$-photon source. We will compare the efficiency of the different realizations and in particular use their relation to reveal the link between the atom-based [@Bas09a; @Bas09b] and the projective measurement based scheme for symmetric Dicke states [@Kie07; @Wie09; @Pre09; @Wie09pra].
The paper is organized as follows. In Section \[secMultiport\] we establish the connection ${\mbox{$\,\mid \! \psi \, \rangle$}}_O \leftrightarrow {\mbox{$\,\mid \! \psi \, \rangle$}}_I$, which is the same for all schemes. Subsequently, in section \[secSources\] different possibilities to obtain the state ${\mbox{$\,\mid \! \psi \, \rangle$}}_I$ are presented. We study three types of photon source arrangements: overlap of SPSs via BSs, overlap of photons from entangled pairs created by SPDC and subsequent projective measurements, and projective measurements on a $2N$-photonic symmetric Dicke state.
The Multiport {#secMultiport}
=============
The multiport output setup is illustrated in Fig. \[SymSchemes\]. It fulfills the task to distribute $N$ properly polarized photons propagating in a single spatial input mode $a$ to $N$ output modes $A=a_1,\ldots,a_N$. In the following, without loss of generality, the photonic qubits are encoded in the horizontal ($|H\rangle$) and vertical ($|V\rangle$) polarization states. It is assumed that the input mode is populated with $N$ photons in the state $$\label{psiI}
{\mbox{$\,\mid \! \psi \, \rangle$}}_I = \frac{1}{\mathcal{N}\left(\alpha,\beta\right)} \prod_{i=1}^N ({\alpha_{i}\,a_H^{\dagger}+\beta_{i}\,a_V^{\dagger}}){\mbox{$\,\mid \! 0 \, \rangle$}}_a,$$ where $\alpha_i$ and $\beta_i$ are complex numbers with $|\alpha_i|^2 + |\beta_i|^2 = 1$ for $i = 1, \ldots, N$, the normalization $\mathcal{N}\left(\alpha,\beta\right)$ depends on these parameters with $\alpha=\alpha_1,\dots,\alpha_N$ and $\beta=\beta_1,\dots,\beta_N$, ${\mbox{$\,\mid \! 0 \, \rangle$}}_a$ denotes the vacuum state of the input mode $a$, and $a^{\dagger}_H$ ($a^{\dagger}_V$) is the photon creation operator for horizontally (vertically) polarized photons in that mode. Equation (\[psiI\]) can be rewritten $$\label{psiI2}
{\mbox{$\,\mid \! \psi \, \rangle$}}_I = \frac{1}{\mathcal{N}\left(\alpha,\beta\right) N!} \sum_{k = 0}^N c_k (C_N^k)^{1/2} (a^{\dagger}_V)^{k} (a^{\dagger}_H)^{N-k} {\mbox{$\,\mid \! 0 \, \rangle$}}_a,$$ with $C_N^k$ the binomial coefficient $\left(\begin{array}{c}N\\k\end{array}\right)$ and $$\label{ck}
c_k = (C_N^k)^{1/2} \sum_{1 \leqslant i_1 \neq \ldots \neq i_N \leqslant N} \beta_{i_1} \cdots \beta_{i_k} \alpha_{i_{k+1}} \cdots \alpha_{i_N}.$$ where the sum is over all $N!$ possible tuples $i_1,\dots,i_N$. Note that we choose the form of Equations (\[psiI2\]) and (\[ck\]) to resemble the ones given in Ref. [@Bas09a] for an atom-based scheme aiming at the creation of all symmetric states. Now we can also determine the normalization factor in Eq. (\[psiI\]) and Eq. (\[psiI2\]): $\mathcal{N}\left(\alpha,\beta\right)^2=(\sum_{k = 0}^{N}|c_k|^2)/N!$. The dependence on the actual coefficients $\alpha,\beta$, is due to the bosonic character of photons. The maximal value of $N!$ is obtained if all photons are equally polarized, while the minimal value of $((N/2)!)^2$ happens if orthogonal polarizations are equally populated.
The photons are distributed into the output modes via BSs. The optimal splitting ratio is achieved if the probability for a single photon to go into the different modes is equal. For the case considered in Fig. \[SymSchemes\] this implies the reflectivity $1/n$ for BS$_n$. Under the condition of collecting one photon per output of the multiport, which occurs with a probability $$p_{O}=N!/N^N,$$ each term in Eq. (\[psiI2\]) contributes equally to populate each of the $N$ output modes according to [@Lim05] $$(C_N^k)^{1/2} (a^{\dagger}_V)^{k} (a^{\dagger}_H)^{N-k} |0\rangle_a \rightarrow |D_N^{(k)}\rangle_A,$$ where $${\mbox{$\,\mid \! D_N^{(k)} \, \rangle$}}_A=(C_N^k)^{-1/2} \sum_i {\ensuremath{\mathcal{P}}}_i({\mbox{$\,\mid \! V_1,V_2,...,V_k,H_{k+1},...,H_N \, \rangle$}})$$ is the symmetric Dicke state of the $N$ output modes with $k$ vertically polarized photons and ${\ensuremath{\mathcal{P}}}_k$ denoting all possible permutations of $N$ qubits [@Dic54; @Sto03]. Consequently, the multiport transforms with probability $p_{O}$ the initial state Eq. (\[psiI\]) into the output state $$\label{psiO}
{\mbox{$\,\mid \! \psi \, \rangle$}}_O = \frac{1}{\mathcal{N}\left(\alpha,\beta\right) \sqrt{N!} } \sum_{k = 0}^N c_k |D_N^{(k)}\rangle_A.$$ Note, ${\mbox{$\,\mid \! \psi \, \rangle$}}_O$ describes a state of polarization encoded photonic qubits in different spatial modes, while ${\mbox{$\,\mid \! \psi \, \rangle$}}_I$ is a single mode multiphoton state. This scheme allows to produce any desired symmetric state in the multiport output modes: any collection of the $c_k$ coefficients in Eq. (\[psiO\]) can be obtained from initial state (\[psiI\]) with properly selected complex coefficients $\alpha_i$ and $\beta_i$. The ratios $\alpha_i/\beta_i$ must be equal to the $K$ roots of the polynomial $P(z) = \sum_{k=0}^N (-1)^k \sqrt{C_N^k} c_k z^k$, where $K$ is the polynomial degree, and the remaining $\alpha_i$ must be equal to $1$ [@Bas09a].
The entanglement SLOCC class of the generated symmetric state is then obtained from the analysis of the degeneracy configuration $\mathcal{D}$ and the diversity degree $d$ of the set of states $\{|\epsilon_1\rangle, \ldots, |\epsilon_N\rangle\}$ where $|\epsilon_i\rangle = \alpha_i |H\rangle + \beta_i |V\rangle$. The degeneracy configuration $\mathcal{D}$ is the decreasing order list of the numbers of the $|\epsilon_i\rangle$ states identical to each other (this number is $1$ for each state $|\epsilon_i\rangle$ that occurs once). The diversity degree $d$ is the dimension of this list. States differing in their degeneracy configuration are necessarily SLOCC inequivalent. This is outlined in detail in Ref. [@Bas09b], here, we will give in section \[secSPS\] an example for the three qubit case.
The Photon Sources {#secSources}
==================
In this section, different options to obtain the required state Eq. (\[psiI\]) are discussed.
Single photon sources {#secSPS}
---------------------
A direct approach is to combine photons from SPSs with BSs. This can be done with a multiport BS similar to the one used for the distribution of the photons, as shown in Fig. \[SymSchemes\](a). The input modes, denoted $e_i$, must be prepared in the states $$|\psi\rangle_{\mathrm{SPS}_{e_i}} = (\alpha_i{e_{iH}^{\dagger}}+\beta_i{e_{iV}^{\dagger}})|0\rangle_{e_i}.$$ The mode $a$ is populated by using the input multiport according to $$\begin{aligned}
\nonumber\prod_{i=1}^{N} (\alpha_i{e_{iH}^{\dagger}}+\beta_i{e_{iV}^{\dagger}})|0\rangle_{\mathrm{SPS}_{e_i}} &\stackrel{\mathrm{BSs}}{\rightarrow}& \prod_{i=1}^{N} (\alpha_i{a_{H}^{\dagger}}+\beta_i{a_{V}^{\dagger}})|0\rangle_a \\
&\equiv& {\mbox{$\,\mid \! \psi \, \rangle$}}_I.\end{aligned}$$
In this scheme, the entanglement class of the resulting final symmetric Dicke state $|\psi\rangle_O$ after passage through the output multiport is fully determined from the polarization states of the input photons in the modes $e_i$. For instance, for $N = 3$, the use of 3 identically polarized photons (corresponding to the state set $\{\epsilon_1,\epsilon_1,\epsilon_1\}$, whose degeneracy configuration is $\mathcal{D}_3$ and diversity degree $d=1$) generates a separable state $|\psi\rangle_O$, 2 identically polarized photons (corresponding to the state set $\{\epsilon_1,\epsilon_1,\epsilon_2\}$, whose degeneracy configuration is $\mathcal{D}_{2,1}$ and diversity degree $d=2$) generate a $W$ state, while photons with distinct polarization states (corresponding to the state set $\{\epsilon_1,\epsilon_2,\epsilon_3\}$, whose degeneracy configuration is $\mathcal{D}_{1,1,1}$ and diversity degree $d=3$) set the output modes of the second multiport in a GHZ class state [@Bas09b].
The latter case leading to the observation of GHZ states has been suggested in Ref. [@Hof04] in the context of states useful for super-resolving phase measurements and has been implemented experimentally with three photons [@Mit04]. In our work, we establish [*all*]{} symmetric multiphoton entanglement classes (i.e., not only the GHZ class) via the framework of operational classification of arbitrary symmetric photonic qubit states and their experimental realization, which, in the case of photonic qubits, has not been done before.
To obtain the optimal efficiency in the preparation of the desired states, we need to find a suitable BS configuration. For a *particular* state, partially polarizing BSs might be most suitable. Yet, as we aim for a flexible scheme to observe *all* symmetric states, we neglect the polarization for considering the efficiency, which is then optimized for BSs with well defined reflectivity of $1/n$ if $n$ denotes the $n$-th BS of the input multiport as shown in Fig. \[SymSchemes\](a). The total efficiency depends additionally on the amplitude of obtaining all photons in the mode $a$, which is dependent on the photon’s polarization due to interference effects and reflected in the normalization factor $\mathcal{N}\left(\alpha,\beta\right)$ in Eq. (\[psiI\]). Then, the probability $p_{I,\mathrm{SPS}}$ to obtain ${\mbox{$\,\mid \! \psi \, \rangle$}}_I $ is the product of these two contributions $$p_{I,\mathrm{SPS}}=\mathcal{N}\left(\alpha,\beta\right)^2 \prod_{n=2}^{N}\frac{(n-1)^{(n-1)}}{n^n}=\frac{\mathcal{N}\left(\alpha,\beta\right)^2 }{N^N}.$$ In an experiment involving deterministic photon creation, the rates at which the photons are supplied are given by the rates at which the single photons are prepared. As this does not scale with the photon number, this scheme is hardly comparable with the following probabilistic implementations based on SPDC sources. In contrast, for probabilistic single photon sources, we can determine a rate $R_{\mathrm{SPS}}$ for comparison with the following schemes from the rate of single photon creation $c_{\mathrm{SPS}}$: $$\label{eq:spsrate}
R_{\mathrm{SPS}}=(c_{\mathrm{SPS}})^N\cdot p_{I,\mathrm{SPS}}\cdot p_{O}=(c_{\mathrm{SPS}})^N\mathcal{N}\left(\alpha,\beta\right)^2\frac{N!}{ N^{2N}}.$$
Non-collinear SPDC and projective measurements {#sec:noncollspdc}
----------------------------------------------
The scheme exposed in the previous section requires deterministic SPSs for the $N$ input ports. With present technology this represents a limit to the achievable number of entangled photons as deterministic SPSs are not yet mature enough for multi-photon entanglement experiments [@Gra04; @Oxb05; @Lou05]. The best present alternative is given by the use of heralded SPSs realized with non-collinear SPDC (ncl) combined with conditional detection, as shown in Fig. \[SymSchemes\](b) [@Kwi95; @Roh05]. In this scheme, $N$ non-collinear SPDC sources overlap one of their modes with each other into the input mode $a$ of the multiport \[Fig. \[SymSchemes\](b)\]. Each SPDC source, numbered $1$ to $N$, is supposed to emit the antisymmetric Bell state $$\label{as}
{\mbox{$\,\mid \! \psi^- \, \rangle$}}_{\mathrm{ncl}_i} = \frac{1}{\sqrt{2}}({a_{H}^{\dagger}}{b_{iV}^{\dagger}}-{a_{V}^{\dagger}}{b_{iH}^{\dagger}}) {\mbox{$\,\mid \! 0 \, \rangle$}}_{a b_i},$$ where ${\mbox{$\,\mid \! 0 \, \rangle$}}_{a b_i}$ denotes the vacuum state in modes $a$ and $b_i$, with $b_i$ the non-overlapping output mode of the $i$-th SPDC source. In this case, the first order emissions create, before any projective measurement is performed, the $2N$-photon state $$\label{psiSPDC}
{\mbox{$\,\mid \! \psi \, \rangle$}}_{\mathrm{ncl},a b_1 \ldots b_N}=\frac{1}{\sqrt{(N+1)!}}\prod_{i=1}^{N} ({a_{H}^{\dagger}}{b_{iV}^{\dagger}}-{a_{V}^{\dagger}}{b_{iH}^{\dagger}}) {\mbox{$\,\mid \! 0 \, \rangle$}}_{a b_1 \ldots b_N},$$ where ${\mbox{$\,\mid \! 0 \, \rangle$}}_{a b_1\ldots b_N}$ denotes the vacuum state in all modes $a, b_1, \ldots, b_N$. The desired state (\[psiI2\]) is then obtained by projecting each of the output modes $b_i$ onto the polarizations orthogonal to the onces that should be combined in mode $a$, that is onto the state $${\mbox{$\,\mid \! S \, \rangle$}}_{b_1 \ldots b_N} = \prod_{i=1}^{N} (\alpha_i^* {b_{iV}^{\dagger}} - \beta_i^* {b_{iH}^{\dagger}}) {\mbox{$\,\mid \! 0 \, \rangle$}}_{b_1 \ldots b_N}.$$ Indeed, the residual state obtained in mode $a$ by this projective measurement is simply given by (denoting with $B$ the collection of modes $b_1, \ldots, b_N$) $$\begin{aligned}
\nonumber
\label{psiI_II}_B{\mbox{$\langle \, S \! \mid \,$}}\psi\rangle_{\mathrm{ncl},aB} &=& \frac{1}{\sqrt{(N+1)!}} \prod_{i=1}^{N} (\alpha_i {a_{H}^{\dagger}}+\beta_i {a_{V}^{\dagger}}) {\mbox{$\,\mid \! 0 \, \rangle$}}_a \\
&=& \frac{\mathcal{N}\left(\alpha,\beta\right) }{\sqrt{(N+1)!}} {\mbox{$\,\mid \! \psi \, \rangle$}}_{I}.\end{aligned}$$
In this scheme, the entanglement class of the final symmetric state $|\psi\rangle_O$ is fully determined from the degeneracy configuration and the diversity degree of the polarization states selected in the modes $b_i$ during the projection step. The efficiency to get the $N$-photon state ${\mbox{$\,\mid \! \psi \, \rangle$}}_{I}$ from $2N$ photons is here dependent on the probability to project onto the separable state ${\mbox{$\,\mid \! S \, \rangle$}}_B$, which is given by the normalization factor in Eq. (\[psiI\_II\]), $$\label{pproj}
p_{I,\mathrm{ncl}}=\frac{\mathcal{N}\left(\alpha,\beta\right)^2}{(N+1)!} \, .$$ For a probabilistic source with pair creation rate $c_{\mathrm{ncl}}$ \[Eq. (\[as\])\] (i.e. an $N$-pair creation rate of $(c_{\mathrm{ncl}}/2)^N(N+1)!$ \[Eq. (\[psiSPDC\])\]), the rate $R_{\mathrm{ncl}}$ to obtain the desired output state is $$\begin{aligned}
\nonumber R_{\mathrm{ncl}}&=&\left(\frac{c_{\mathrm{ncl}}}{2}\right)^N(N+1)!\cdot p_{I,\mathrm{ncl}}\cdot p_{O}\\
&=&(c_{\mathrm{ncl}})^N \mathcal{N}\left(\alpha,\beta\right)^2\frac{N!}{(2N)^N}.\end{aligned}$$ This yields for $N>2$ a higher rate than the scheme using SPSs \[Eq. (\[eq:spsrate\])\], if the rates $c_{\mathrm{ncl}}$ and $c_{\mathrm{SPS}}$ are equal.
Projective measurements on symmetric $2N$-partite Dicke states
--------------------------------------------------------------
### Analogy between non-collinear SPDC and symmetric Dicke states
In the following we will show the correspondence between the previously described scheme of section \[sec:noncollspdc\] and the property of symmetric entangled Dicke states to be projectable onto different classes of entanglement.
To this end, let us study the $2N$ photon state emergent after splitting the photons in mode $a$ in the output multiport and before projection of the photons in modes $b_i$. This corresponds to the state given in Eq. (\[psiSPDC\]) and a subsequent symmetric distribution of the photons in mode $a$ \[see Fig. \[SymSchemes\](b)\]. This state is given by $${\mbox{$\,\mid \! \psi \, \rangle$}}_{2N} = (C_{2N}^N)^{-1/2}\sum_{k=0}^{N} (-1)^k C_N^k |D_N^{(k)}\rangle_A \otimes |D_N^{(N-k)}\rangle_B.$$ Note that $C_{2N}^N=\sum_{k=0}^N{(C_N^k)^2}$, and $A$ denotes the $N$ output modes of the output multiport feeded by the mode $a$. The same expression with positive signs is obtained via a $\pi/2$-phase shift in each $b_i$ mode that transforms the states emitted from each SPDC source from the antisymmetric Bell state (\[as\]) to a symmetric Bell state $${\mbox{$\,\mid \! \psi^+ \, \rangle$}}_{\mathrm{SPDC}_i} = \frac{1}{\sqrt{2}}({a_{H}^{\dagger}}{b_{iV}^{\dagger}}+{a_{V}^{\dagger}}{b_{iH}^{\dagger}}) {\mbox{$\,\mid \! 0 \, \rangle$}}_{a b_i}.$$ The $2N$-photon state generated in that case reads $$\begin{aligned}
\nonumber{\mbox{$\,\mid \! \psi \, \rangle$}}_{2N} & = &(C_{2N}^N)^{-1/2}\sum_{k=0}^{N} C_N^k |D_N^{(k)}\rangle_A \otimes |D_N^{(N-k)}\rangle_B \\
\label{eq:dicke2nn}& \equiv& |D_{2N}^{(N)}\rangle_{A,B}.\end{aligned}$$ Thus, the resulting state is a $2N$ symmetric Dicke state with $N$ excitations [@Dic54; @Sto03]. As before, projections in the $B$ modes can be used to obtain any desired symmetric state $|\psi\rangle_O$. However, the phase shift has to be compensated for, and we need in that case to project onto $${\mbox{$\,\mid \! S \, \rangle$}}_B=\prod_{i=1}^{N} (\alpha_i^* {b_{iV}^{\dagger}} + \beta_i^* {b_{iH}^{\dagger}}) {\mbox{$\,\mid \! 0 \, \rangle$}}_{b_i},$$ in order to obtain the same state in the end. As the Dicke states are symmetric under permutation of particles it does not matter which $N$ of the $2N$ photons are projected. That means we could just as well project the photons from $A$ and observe the state in the modes $B$.
Then, the scheme is very similar to the atom scheme of Refs. [@Thi07; @Bas09a]: entangled atom-photon pairs are created, one part of each pair is mixed with all the others and, finally, symmetrically distributed to several detectors. The polarization setting at the photon detector determines the entangled state for the atoms. In our case, we consider entangled photon-photon pairs.
### Collinear SPDC for obtaining symmetric Dicke states
The symmetric Dicke state can also be obtained by a symmetric distribution of the $N$th order emission of a type II collinear down conversion (cl) [@Kie07; @Pre09; @Wie09]: $$\label{eq:spdccl}
{\mbox{$\,\mid \! \psi \, \rangle$}}_{\mathrm{cl}}=\frac{1}{N!}({a_{H}^{\dagger}}{a_{V}^{\dagger}})^N{\mbox{$\,\mid \! 0 \, \rangle$}}_a.$$ This gives rise to the scheme shown in Fig. \[SymSchemes\](c). In order to compare this approach with the previous schemes we assume a pair emission rate $c_{\mathrm{cl}}$ and obtain the $2N$-photon emission rate $(c_{\mathrm{cl}})^N (N!)^2$ \[see Eq. (\[eq:spdccl\])\]. Distribution of the $2N$ photons into separate modes occurs with a probability of $p_{I,\mathrm{cl}}=(2N)!/(2N)^{2N}$ and leads to the state of Eq. ([\[eq:dicke2nn\]]{}). Thus, the probability for a projective measurement preparing the desired state is given by Eq. ([\[pproj\]]{}). The total state preparation rate $R_{\mathrm{cl}}$ is $$\begin{aligned}
\nonumber R_{\mathrm{cl}}&=&(c_{\mathrm{cl}})^N (N!)^2\cdot\frac{(2N)!}{(2N)^{2N}}\cdot\frac{\mathcal{N}\left(\alpha,\beta\right)^2}{(N+1)!}\\
&=&(c_{\mathrm{cl}})^N\mathcal{N}\left(\alpha,\beta\right)^2\frac{N!}{(2N)^N}\frac{(2N)!}{(N+1)(2N)^N} \,\, .\end{aligned}$$ Hence, while the advantage of this scheme is its simplicity, the disadvantage is that it is by far less efficient than the other discussed implementations.
Conclusion {#secConclusion}
==========
We have presented different experimental schemes to obtain all entanglement classes of symmetric states of photonic qubits. A univocal mapping between well-defined sets of experimental parameters (i.e. the polarization of the input photons in scheme A, and the states the photons are projected on in schemes B and C) and the corresponding multiqubit entanglement classes in the symmetric subspace of the photonic system is obtained, similar to the one achieved in the atom-photon system described in Ref. [@Bas09a; @Bas09b]. This directly translates to a systematic classification of the states obtained by the well-known scheme of projective measurements on symmetric Dicke states. Comparison of the different implementations showed that for the probabilistic state-of-the-art photon sources, the scheme relying on non-collinear SPDC and subsequent projective measurements is most efficient. We are convinced that this result will initiate flexible experiments allowing the observation of photonic Dicke states belonging to well defined classes of symmetric states. Furthermore we expect that our work will stimulate the translation of the presented scheme to other physical systems. A goal for the near future is to extend this approach for devising schemes of an operational classification of non-symmetric states.
E.S. acknowledges UPV-EHU Grant No. GIU07/40 and the EuroSQIP European project. W.W. acknowledges support by QCCC of the Elite Network of Bavaria.
[99]{}
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. [**81**]{}, 865 (2009).
O. Gühne and G. Tóth, Phys. Rep. [**474**]{}, 1 (2009).
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. [**70**]{}, 1895 (1993).
A. Ekert, Phys. Rev. Lett. [**67**]{}, 661 (1991).
R. Raussendorf, H. J. Briegel, Phys. Rev. Lett. [**86**]{}, 5188 (2001).
W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A [**62**]{}, 062314 (2000).
A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. [**87**]{}, 040401 (2001).
F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A [**65**]{}, 052112 (2002).
L. Lamata, J. León, D. Salgado, and E. Solano, Phys. Rev. A [**74**]{}, 052336 (2006).
L. Lamata, J. León, D. Salgado, and E. Solano, Phys. Rev. A [**75**]{}, 022318 (2007).
L. Chen and Y.-X. Chen, Phys. Rev. A [**74**]{}, 062310 (2006).
P. Mathonet, S. Krins, M. Godefroid, L. Lamata, E. Solano, and T. Bastin, arXiv:0908.0886.
T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano and G. S. Agarwal, Phys. Rev. Lett. [**102**]{}, 053601 (2009).
T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano, Phys. Rev. Lett. [**103**]{}, 070503 (2009).
J.-W. Pan, Z.-B. Chen, M. Zukowski, H. Weinfurter, and A. Zeilinger, arXiv:0805.2853.
T. E. Kiess [*e*t al.]{}, Phys. Rev. Lett. [**71**]{}, 3893 (1993).
M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and Harald Weinfurter, Phys. Rev. Lett. [**92**]{}, 077901 (2004).
A. Zeilinger, G. Weihs, T. Jennewein, and M. Aspelmeyer, Nature [**433**]{}, 230 (2005).
W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, O. Gühne, and H. Weinfurter, Phys. Rev. Lett. [**101**]{}, 010503 (2008).
M. Aspelmeyer and J. Eisert, Nature [**455**]{}, 180 (2008).
N. Kiesel [*e*t al.]{}, Phys. Rev. Lett. [**98**]{}, 063604 (2007).
W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and Harald Weinfurter, Phys. Rev. Lett. [**103**]{}, 020504 (2009).
R. Prevedel, G. Cronenberg, M. S. Tame, M. Paternostro, P. Walther, M. S. Kim, and A. Zeilinger, Phys. Rev. Lett. [**103**]{}, 020503 (2009).
For an alternative proposal see A. Maser, R. Wiegner, U. Schilling, C. Thiel, and J. von Zanthier, arXiv:0911.5115.
B. Lanyon and N. Langford, New J. Phys. [**11**]{}, 013008 (2009).
H. F. Hofmann, Phys. Rev. A [**70**]{}, 023812 (2004).
M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature [**429**]{}, 161 (2004).
Y. Li, K. Zhang, and K. Peng, Phys. Rev. A [**77**]{}, 015802 (2008).
Y. L. Lim and A. Beige, Phys. Rev. A [**71**]{}, 062311 (2005).
M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Phys. Rev. Lett. [**73**]{}, 58 (1994).
M. Żukowski, A. Zeilinger, and M. A. Horne, Phys. Rev. A [**55**]{}, 2564 (1997).
W. Wieczorek, N. Kiesel, C. Schmid, and Harald Weinfurter, Phys. Rev. A [**79**]{}, 022311 (2009).
R. H. Dicke, Phys. Rev. [**93**]{}, 99 (1954).
J. K. Stockton, J. M. Geremia, A. C. Doherty, and H. Mabuchi, Phys. Rev. A [**67**]{}, 022112 (2003).
P. Grangier, B. Sanders, and J. Vuckovic, New J. Phys. [**6**]{} (2004).
M. Oxborrow and A.G. Sinclair, Cont. Phys. [**46**]{}, 173 (2005).
B. Lounis and M. Orrit, Rep. Prog. Phys. [**68**]{}, 1129 (2005).
P. G. Kwiat [*e*t al.]{}, Phys. Rev. Lett. [**75**]{}, 4337 (1995).
P. P. Rohde, T. C. Ralph, and M. A. Nielsen, Phys. Rev. A [**72**]{}, 052332 (2005).
C. Thiel, J. von Zanthier, T. Bastin, E. Solano and G. S. Agarwal, Phys. Rev. Lett. [**99**]{}, 193602 (2007).
[^1]: permanent address: Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria
|
---
abstract: 'We study the case of two polaritonic qubits localized in two separate cavities coupled by a fiber/additional cavity. We show that surprisingly enough, even a coherent classical pump in the intermediate cavity/fiber can lead to the creation of entanglement between the two ends in the steady state. The stationary nature of this entanglement and its survival under dissipation opens possibilities for its production under realistic laboratory conditions. To facilitate the verification of the entanglement in an experiment we also construct the relevant entanglement witness measurable by accessing only a few local variables of each polaritonic qubit.'
address:
- '$^{1}$Centre for Quantum Computation, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, UK'
- '$^{2}$Science Department, Technical University of Crete, Chania, Greece 73100'
- '$^{3}$ Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy'
- '$^{4}$Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK'
author:
- 'Dimitris G. $^{1,2}$'
- 'Stefano $^{3}$'
- 'Sougato Bose $^{4}$'
title: 'Steady state entanglement between hybrid light-matter qubits'
---
Introduction
============
Recently, there has been a growing interest in exploiting a certain class of coupled hybrid light-matter systems, namely coupled cavity polaritonic systems, for various purposes such as for realizing schemes for quantum computation [@angelakis-ekert04; @angelakis-kay07a], for communication [@angelakis-bose07b] and for simulations of quantum many-body systems [@angelakis-bose06b; @hartmann; @greentree; @cpsun; @fazio; @yamamoto; @myungshik-agarwal; @myungshik]. These cavity-atom polaritonic excitations are different from propagating polaritonic excitations in atomic gases and exciton-photon polaritons in solid state systems [@lukin-yamamoto]. This area is also distinct from those using hybrid light-matter systems in quantum computing where only the matter system (such as an atom or an electron) acts as the qubit. In the latter case the qubits are atoms and light is used exclusively as a connection bus between them [@barrett; @beige; @munro; @serafini; @MW05; @MB04; @parkins]. Promising schemes to produce steady state entanglement between atoms in distinct cavities have also been proposed [@parkins]. In these ground states of atoms have been used in order to circumvent decoherence due to spontaneous emission. In addition to auxiliary atomic levels, external driving fields as well as an unidirectional coupling between cavities are required. In polaritonic coupled cavity systems on the other hand, the localized mixed light-matter excitations, or polaritons, allow for the identification of qubits that possess the easy manipulability and measurability of atomic qubits, while also being able to naturally interact whereas separated by distances over which photons can be exchanged between them. Motivated by the rapid experimental progress in Cavity Quantum Electrodynamics and the ability to couple distinct cavities in a variety of systems [@cqed; @blockade; @toroid; @noda; @trupke], the realization of a system that could produce verifiable, steady state entanglement between two polaritonic qubits in currently realistic laboratory conditions would be extremely interesting. In that case the decoherence emerging from the photonic losses due to the mixed nature of the polaritons, in addition to that from atomic spontaneous emission, will need to be controlled. Therefore, apriori one may not expect a completely stationary entanglement of two polaritons unless the unavoidable loss of coherence due to both channels can somehow be “re-injected” into the system.
Here we show that even under strong dissipation in both the atomic and photonic parts, it is still possible to deterministically entangle two such polaritonic qubits. More precisely, we study the case of two polaritonic qubits coupled by a fiber/additional cavity and show that surprisingly enough, even a coherent [*classical pump*]{} can lead to the creation of entanglement between them in the steady state. The stationary nature of this entanglement should make easier its experimental verification. To this end we also provide a relevant operator (an “entanglement witness" [@witness]) measurable by only measuring local variables of each polariton.
The Model
=========
The Hamiltonian describing an array of $N$ identical atom-cavity systems is the sum of the free light and dopant parts and the internal photon and dopant couplings $$\begin{aligned}
H^{free}&=&\omega_{d}\sum_{k=1}^N a_k^\dagger a_k+\omega_{0}\sum_{k=1}^N
|e\rangle_{k} \langle e|, \\
H^{int}&=&g \sum_{k=1}^N(a_k^\dagger\,|g\rangle_{k}\langle e|
+a_k |e\rangle_{k}\langle g|).\end{aligned}$$ Here $a_k,a_k^{\dag}$ are the photonic field operators localized in the $k$-th system and $|e\rangle_k,|g\rangle_k$ are the excited and ground state of the dopant in the $k$-th system. Moreover, $g$ is the light-atom coupling strength and $\omega_{d}$($\omega_0$) the photonic(atomic) frequencies respectively ($\hbar=1$ throughout the paper). The $H^{free}+H^{int}$ Hamiltonian can be diagonalized in a basis of mixed photonic and atomic excitations, called [*polaritons*]{}. On resonance between atom and cavity, the polaritons are created by operators $P_{k}^{(\pm,n)\dagger}={\left | \, n\pm \right\rangle}_k{\left \langle g,0 \, \right |}$. The states ${\left | \, n\pm \right\rangle}_k=({\left | \, g,n \right\rangle}_k\pm {\left | \, e,n-1 \right\rangle}_k)/\sqrt2$ are the polaritonic states (also known as dressed states) with energies $E^{\pm}_{n}=n\omega_{d}\pm g\sqrt{n}$ and ${\left | \, n \right\rangle}_k$ denotes the $n$-photon Fock state of the $k$-th cavity.
It has been shown that in an array of these atom-cavity systems the addition of a hopping photon term $\propto \sum_j(a^{\dagger}_{j}a_{j+1}+a_{j}a^{\dagger}_{j+1})$, leads to a polaritonic Mott phase where a maximum of one excitation per site is allowed [@angelakis-bose06b]. This originates from the repulsion due to the photon blockade effect [@blockade]. In this Mott phase, the system’s Hamiltonian in the interaction picture results $$H_{I}=J \sum_{k}\left(P_{k}^{(-,1)\dagger}P_{k+1}^{(-,1)}+
P_{k}^{(-,1)}P_{k+1}^{(-,1)\dagger}\right),$$ where $J$ is the coupling due to photon hopping from cavity to cavity. Since double or more occupancy of the sites is prohibited, one can identify $P_{k}^{(-,1)\dagger}$ with $\sigma^{\dag}_k=\sigma^x_k+i\sigma^y_k$, where $\sigma^x_k$, $\sigma^y_k$ and $\sigma^z_k$ stand for the usual Pauli operators. The system’s Hamiltonian then becomes the standard $XY$ model of interacting spin qubits with spin up/down corresponding to the presence/absence of a polariton [@angelakis-bose06b].
Let us now consider a linear chain of three coupled cavities with the two extremal ones doped with a two level system as shown in Fig.1(a). Alternatively, as the central cavity in any case is undoped, one can simply replace it with an optical fiber of short length (so that the distance is greatly increased but the fiber still supports a single mode of frequency near those of the two cavities), which simplifies the setting even further, as shown in Fig.1(b). For the purposes of description, we will use the three cavity setting remembering that everything applies to the case of two cavities linked by a fiber. The fact that a classical field can drive (i.e., pump energy into) the central cavity in a three cavity setting (as also shown in Fig.1(a)) is replaced in the fiber setting by a coupler feeding light into the cavity (as also shown in Fig.1(b)).
![\[scheme\] The system under consideration. a) The cavities are coupled through direct photon hopping. b) The cavities are coupled through a fiber. The extremal cavities in each configuration are interacting with a two level system that could be an atom or a quantum dot depending the implementation technology used. c) The photon blockade allows for the ground and first dressed states of each atom-cavity system to be treated as a (polaritonic) qubit.](scheme.eps){width="60.00000%"}
Let $\sigma_{j}^{\dagger}=|1-\rangle_{j}
\langle g, 0|$ be the polaritonic spin operators for the end cavities (the index $j=1,2$ labels the two end cavities) and $a$, $a^{\dag}$ the field operators of the central empty cavity. Since the latter is not doped, there the field operators play the role of polariton operators and they couple to polariton operators of the ends cavities. Moreover, assuming that the central cavity (or fiber) is driven, the Hamiltonian describing the system dynamics will be $$\begin{aligned}
H=J\sum_{j=1}^{2}\left(\sigma_ja^{\dag}+\sigma_j^{\dag}a\right)-\Delta
a^{\dag}a +\alpha a^{\dag}+\alpha^*a,\end{aligned}$$ where $\Delta=\omega_{mid}-\omega_{pol}$ is the detuning between the central cavity mode of frequency $\omega_{mid}$ and the polaritons frequency $\omega_{pol}=\omega_0-g$. Furthermore, $\alpha$ is the product of the coupling of the driving field to the central cavity field (say $G$) and the amplitude of the driving radiation field (say $\tilde\alpha$). We also assume that $\Delta$ is much smaller than the atom-light coupling in each of the outer cavities, so that only the ground level $|\tilde{g}\rangle=|g,0\rangle$ and first excited level $|\tilde{e}\rangle=(|g,1\rangle-|e,0\rangle)/\sqrt{2}$ of the polaritons are involved (i.e., the polaritons are still good as qubits).
Suppose that the polaritons decay with the same rate $\gamma$ (this is the effective decay rate of the polariton due to both the decay of the cavity field and the atomic excited state), and the cavity radiation mode with rate $\kappa$. The quantum Langevin equations describing the dynamics will be [@qnoise] $$\begin{aligned}
\dot{\sigma}_j&=&iJ
a\sigma_j^{z}-\gamma\sigma_j+\sqrt{2\gamma}\sigma_j^{in},\qquad\quad
j=1,2\label{eqssigma}\\
\dot{a}&=&i\Delta a-iJ\left(\sigma_1+\sigma_2\right)-i\alpha-\kappa
a+\sqrt{2\kappa}a^{in},\label{eqa}\end{aligned}$$ where the superscript *in* denotes the vacuum noise operators.
If $\kappa\gg J$ the radiation mode can be adiabatically eliminated in such a way that $$a\approx\frac{J}{\Delta+i\kappa}\left(\sigma_1+\sigma_2\right)+\frac{\alpha}{\Delta+i\kappa}
+i\frac{\sqrt{2\kappa}}{\Delta+i\kappa}a^{in}. \label{light1}$$ Moreover, if the quantities $J/(2\sqrt{\kappa})$ and $\alpha/(2\sqrt{\kappa})$ are large compared to the amplitude standard deviation of the fluctuating vacuum field, the last term in Eq.(\[light1\]) can be neglected and $$a\approx\frac{J}{\Delta+i\kappa}\left(\sigma_1+\sigma_2\right)+\frac{\alpha}{\Delta+i\kappa}.
\label{aaprox}$$
Inserting Eq.(\[aaprox\]) into Eqs.(\[eqssigma\]), we get[^1] $$\begin{aligned}
\dot{\sigma}_1&=&i\frac{J^2}{\Delta+i\kappa} \sigma_2\sigma_1^{z}
+i\frac{J\alpha}{\Delta+i\kappa}\sigma_1^{z}
-\gamma\sigma_1+\sqrt{2\gamma}\sigma_1^{in},
\label{eqssigma1}\\
\dot{\sigma}_2&=&i\frac{J^2}{\Delta+i\kappa} \sigma_1\sigma_2^{z}
+i\frac{J\alpha}{\Delta+i\kappa}\sigma_2^{z}
-\gamma\sigma_2+\sqrt{2\gamma}\sigma_2^{in}, \label{eqssigma2}\end{aligned}$$ corresponding to an effective Hamiltonian for polaritons of the type $$\begin{aligned}
H_{eff}=\Re\left[\frac{J^2}{\Delta+i\kappa}\right]
\left(\sigma_1\sigma_2^{\dag}+\sigma_1^{\dag}\sigma_2\right)
+\frac{J\alpha}{\Delta+i\kappa}\left(\sigma_1^{\dag}+
\sigma_2^{\dag}\right)
+\frac{J\alpha^*}{\Delta-i\kappa}\left(\sigma_1+\sigma_2\right).\end{aligned}$$ We are using $\Re$ and $\Im$ to denote the real and imaginary part respectively.
The dynamics of the polaritons can now be described by the master equation [@qnoise] $$\begin{aligned}
\dot{\rho} = -i\left[H_{eff},\rho\right] +\sum_{j=1}^{2} L_j\rho
L_j^{\dag}-\frac{1}{2}\left\{L_j^{\dag}L_j,\rho\right\}, \label{me}\end{aligned}$$ where $L_j=\sqrt{2\gamma}\sigma_j$ are the Lindblad operators.
Steady State Entanglement
=========================
At the steady state Eq. becomes $$\begin{aligned}
0&=&-i\zeta\left[\sigma_1\sigma_2^{\dag}+\sigma_1^{\dag}\sigma_2,\rho\right]
-i\xi\left[\sigma_1^{\dag}+\sigma_2^{\dag},\rho\right]
-i\xi^*\left[\sigma_1+\sigma_2,\rho\right]\nonumber\\
&&+2\sigma_1\rho\sigma_1^{\dag}-\sigma_1^{\dag}\sigma_1\rho-\rho\sigma_1^{\dag}\sigma_1
+2\sigma_2\rho\sigma_2^{\dag}-\sigma_2^{\dag}\sigma_2\rho-\rho\sigma_2^{\dag}\sigma_2,
\label{meexp}\end{aligned}$$ where $\zeta=\Re[J^2/\gamma(\Delta+i\kappa)]$ and $\xi=\alpha
J/\gamma(\Delta+i\kappa)$.
The steady state solution of Eq.(\[meexp\]) can be found by writing the density operator and the other operators in a matrix form, in the basis $\mathbf{B}=\{|\tilde{e}\rangle_{1}|\tilde{{e}}\rangle_{2},|\tilde{g}\rangle_{1}|\tilde{{e}}\rangle_{2},
|\tilde{e}\rangle_{1}|\tilde{{g}}\rangle_{2},|\tilde{g}\rangle_{1}|\tilde{{g}}\rangle_{2}
\}$. Let us parametrize the density operator as $$\rho= \left(\begin{array}{cccc} {\cal A}&{\cal B}_{1}+i{\cal B}_{2}&
{\cal C}_{1}+i{\cal C}_{2}&{\cal D}_{1}+i{\cal D}_{2}
\\
{\cal B}_{1}-i{\cal B}_{2}&{\cal E}& {\cal F}_{1}+i{\cal
F}_{2}&{\cal G}_{1}+i{\cal G}_{2}
\\
{\cal C}_{1}-i{\cal C}_{2}&{\cal F}_{1}-i{\cal F}_{2}& {\cal
H}&{\cal I}_{1}+i{\cal I}_{2}
\\
{\cal D}_{1}-i{\cal D}_{2}&{\cal G}_{1}-i{\cal D}_{2}& {\cal
I}_{1}-i{\cal I}_{2}&1-{\cal A}-{\cal E}-{\cal H}
\end{array}\right),
\label{rhoexp}$$ where the matrix elements also respect the requirement that ${\rm
Tr}\{\rho\}=1$. The matrix representation of the other operators comes from $$\sigma_{1}= \left(\begin{array}{cccc} 0&0&0&0
\\
1&0&0&0
\\
0&0&0&0
\\
0&0&1&0
\end{array}\right)\,,
\quad \sigma_{2}= \left(\begin{array}{cccc} 0&0&0&0
\\
0&0&0&0
\\
1&0&0&0
\\
0&1&0&0
\end{array}\right)\,.
\label{sigmaexp}$$ By using matrices (\[rhoexp\]) and (\[sigmaexp\]) in the r.h.s. of Eq.(\[meexp\]), we get a single complex matrix $M$ which must be equal to zero. Then, equating to zero the entries of $M$ we get a set of equation for the entries of $\rho$. Since $M$ is Hermitian we can consider $$\begin{aligned}
M_{jj}&=&0,\quad j,k=1,2,3,4\\
{\Re}\{M_{jk}\}&=&0,\quad k>j\\
{\Im}\{M_{jk}\}&=&0,\quad k>j\end{aligned}$$ so to have a set of $16$ linear equations. They are not all independent because of the 15 unknown parameters $({\cal A}, {\cal
B}_1, {\cal B}_2,{\cal C}_1,{\cal C}_2,{\cal D}_1,{\cal D}_2,{\cal
E},{\cal F}_1,{\cal F}_2,{\cal G}_1,{\cal G}_2,{\cal H},{\cal
I}_1,{\cal I}_2)$. Explicitly the set of equations results $$\begin{aligned}
\label{rhoss}
-4{\cal A}+2\xi_2{\cal B}_1-2\xi_1{\cal B}_2+2\xi_2{\cal
C}_1-2\xi_1{\cal
C}_2&=&0, \nonumber\\
-\xi_2{\cal A}-3{\cal B}_{1}-\zeta{\cal C}_2+\xi_2{\cal
D}_1-\xi_1{\cal D}_2+\xi_2{\cal E}+\xi_2{\cal F}_1
-\xi_1{\cal F}_2&=&0, \nonumber\\
\xi_1{\cal A}-3{\cal B}_{2}+\zeta{\cal C}_1+\xi_1{\cal
D}_1+\xi_2{\cal D}_2-\xi_1{\cal E}-\xi_1{\cal F}_1
-\xi_2{\cal F}_2&=&0, \nonumber\\
-\xi_2{\cal A}-\zeta{\cal B}_2-3{\cal C}_{1}+\xi_2{\cal
D}_1-\xi_1{\cal D}_2+\xi_2{\cal F}_1
+\xi_1{\cal F}_2+\xi_2{\cal H}&=&0, \nonumber\\
\xi_1{\cal A}+\zeta{\cal B}_1-3{\cal C}_{2}+\xi_1{\cal
D}_1+\xi_2{\cal D}_2-\xi_1{\cal F}_1
+\xi_2{\cal F}_2-\xi_1{\cal H}&=&0, \nonumber\\
-\xi_1{\cal B}_{2}-\xi_1{\cal B}_{2}-\xi_2{\cal C}_{1}-\xi_1{\cal
C}_{2}-2{\cal D}_{1} +\xi_2{\cal G}_{1}+\xi_1{\cal G}_{2}+\xi_2{\cal
I}_{1}+\xi_1{\cal
I}_{2}&=&0, \nonumber\\
\xi_1{\cal B}_{1}-\xi_2{\cal B}_{2}+\xi_1{\cal C}_{1}-\xi_2{\cal
C}_{2}-2{\cal D}_{2} -\xi_1{\cal G}_{1}+\xi_2{\cal G}_{2}-\xi_1{\cal
I}_{1}+\xi_2{\cal
I}_{2}&=&0, \nonumber\\
2{\cal A}-2\xi_2{\cal B}_1+2\xi_1{\cal B}_2-2{\cal E}-2\zeta{\cal
F}_2+2\xi_2{\cal G}_1-2\xi_1{\cal G}_2
&=&0, \nonumber\\
-\xi_2{\cal B}_{1}+\xi_1{\cal B}_{2}-\xi_2{\cal C}_{1}+\xi_1{\cal
C}_{2} -2 {\cal F}_{1}+\xi_2{\cal G}_{1}-\xi_1{\cal G}_{2}
+\xi_2{\cal I}_{1}-\xi_1{\cal I}_{2}&=&0,\nonumber\\
\xi_1{\cal B}_{1}+\xi_2{\cal B}_{2}-\xi_1{\cal C}_{1}-\xi_2{\cal
C}_{2} +\zeta{\cal E} -2 {\cal F}_{2}+\xi_1{\cal G}_{1}+\xi_2{\cal
G}_{2} -\zeta{\cal H}
-\xi_1{\cal I}_{1}-\xi_2{\cal I}_{2}&=&0,\nonumber\\
-\xi_2{\cal A}+2 {\cal C}_{1}-\xi_2{\cal D}_{1}+\xi_1{\cal
D}_{2}-2\xi_2{\cal E} -\xi_2{\cal F}_{1}-\xi_1{\cal F}_{2}
-{\cal G}_{1}-\xi_2{\cal H}+\zeta{\cal I}_{2}&=&-\xi_2,\nonumber\\
\xi_1{\cal A}+2 {\cal C}_{2}-\xi_1{\cal D}_{1}-\xi_2{\cal
D}_{2}+2\xi_1{\cal E} +\xi_1{\cal F}_{1}-\xi_2{\cal F}_{2}
-{\cal G}_{2}+\xi_1{\cal H}-\zeta{\cal I}_{1}&=&\xi_1,\nonumber\\
2{\cal A}-2\xi_2{\cal C}_1+2\xi_1{\cal C}_2 +2\zeta{\cal F}_2-2{\cal
H}+2\xi_2{\cal I}_1-2\xi_1{\cal I}_2&=&0,
\nonumber\\
-\xi_2{\cal A}+2 {\cal B}_{1}-\xi_2{\cal D}_1+\xi_1{\cal
D}_2-\xi_2{\cal E} -\xi_2{\cal F}_1+\xi_1{\cal F}_2+\zeta{\cal
G}_2-2\xi_2{\cal H}-{\cal
I}_1&=&-\xi_2,\nonumber\\
\xi_1{\cal A}+2 {\cal B}_{2}-\xi_1{\cal D}_1-\xi_2{\cal
D}_2+\xi_1{\cal E} +\xi_1{\cal F}_1+\xi_2{\cal F}_2-\zeta{\cal
G}_1+2\xi_1{\cal H}-{\cal
I}_2&=&\xi_1,\nonumber\\
2{\cal E}-2\xi_2{\cal G}_1+2\xi_1{\cal G}_2+2{\cal H}-2\xi_2{\cal
I}_1+2\xi_1{\cal I}_2&=&0, \label{seteqs}\end{aligned}$$ where $\xi_1=\Re\{\xi\}$ and $\xi_2=\Im\{\xi\}$.
Solving analytically the above set of equations we obtain for $\xi_2=0$ $$\begin{aligned}
\mathcal{A}&=&\frac{\xi_1^4}{d}\,,\quad \mathcal{B}_1=0\,,\quad
\mathcal{B}_2=-\frac{\xi_1^3}{d}\,,\quad \mathcal{C}_1=0\,,\quad
\mathcal{C}_2=-\frac{\xi_1^3}{d}\,,\quad
\mathcal{D}_1=-\frac{\xi_1^2}{d}\,,\quad
\mathcal{D}_2=\zeta\frac{\xi_1^2}{d}\,,\quad
\mathcal{E}=\frac{\xi_1^2+\xi_1^4}{d}\,,\nonumber\\
\mathcal{F}_1&=&\frac{\xi_1^2}{d}\,,\quad \mathcal{F}_2=0\,,\quad
\mathcal{G}_1=-\zeta\frac{\xi_1}{d}\,,\quad
\mathcal{G}_2=-\frac{\xi_1+\xi_1^3}{d}\,,\quad
\mathcal{H}=\frac{\xi_1^2+\xi_1^4}{d}\,,\quad
\mathcal{I}_1=-\zeta\frac{\xi_1}{d}\,,\quad
\mathcal{I}_2=-\frac{\xi_1+\xi_1^3}{d}\,,\end{aligned}$$ where $$d=\zeta^2+(1+2\xi_1^2)^2.$$ Notice that for $\xi_1=0$ we have formally analogous solutions that lead to the same physical result, hence they are not reported.
Now that we know the stationary density matrix, we can use the concurrence as measure of the degree of entanglement [@Woot] $$C(\rho)=\max\left\{0,\lambda_1-\lambda_2-\lambda_3-\lambda_4\right\},$$ where $\lambda_i$’s are, in decreasing order, the nonnegative square roots of the moduli of the eigenvalues of $\rho\tilde\rho$ with $$\tilde\rho=\left(\sigma_{1}^{y}\sigma_{2}^{y}\right)\rho^*\left(\sigma_{1}^{y}\sigma_{2}^{y}\right),$$ and $\rho^*$ denotes the complex conjugate of $\rho$. With respect to the basis $\mathbf{B}$ it results $$\tilde\rho= \left(\begin{array}{cccc}
1-{\cal A}-{\cal E}-{\cal H}&-{\cal I}_{1}-i{\cal I}_{2}&
-{\cal G}_{1}-i{\cal G}_{2}&{\cal D}_{1}+i{\cal D}_{2}\\
-{\cal I}_{1}+i{\cal I}_{2}&{\cal H}& {\cal F}_{1}+i{\cal F}_{2}&-{\cal C}_{1}-i{\cal C}_{2}\\
-{\cal G}_{1}+i{\cal G}_{2}&{\cal F}_{1}-i{\cal F}_{2}&{\cal E}&-{\cal B}_{1}-i{\cal B}_{2}\\
{\cal D}_{1}-i{\cal D}_{2}&-{\cal C}_{1}+i{\cal C}_{2}&
-{\cal B}_{1}+i{\cal B}_{2}& {\cal A}
\end{array}\right),
\label{rhotil}$$
In Fig.\[concurrence\] we show the concurrence as a function of $\zeta$ and $\xi_{1}$ (the cases $\xi_1=0$ and $\xi_2=0$ give the same numerical results for the concurrence). Notice that by increasing $\zeta$, the concurrence increases quite slowly, and a maximum amount of entanglement is approximately $0.3$ for $\zeta=10$ and $\xi_{1}=2.135$. This is similar to the amount of stationary entanglement achievable with an effective interaction of the kind $\sigma_1^z\sigma_2^z$ when combined with an intricate feedback and cascading [@MW05].
![\[concurrence\] Concurrence $C$ versus $\zeta$ and $\xi_1$ (or equivalently $\xi_2$). ](concurrence.eps){width="60.00000%"}
One could try to employ entanglement witnesses to detect this entanglement [@witness]. A witness can be constructed from the density matrix corresponding to the maximum value of the concurrence. This would be a traceclass operator $W$ in the Hilbert space of the two polaritonic qubits such that ${\rm Tr}[W\rho]\ge 0$ for all separable states while ${\rm Tr}[W\rho]< 0$ for the considered entangled state. The form of such a witness in the Pauli decomposition results $$W=\sum_{j,k=id,x,y,z}
c_{j,k}\,\sigma_1^{j}\otimes \sigma_2^{k},$$ with $\sigma^{id}=I$. In Fig.\[witness\] we show the coefficients $c_{j,k}$ for the entanglement witness coming from the density matrix corresponding to the maximum value of the concurrence in Fig.\[concurrence\]. As we can see, the elements with the most significant weights (greater than 0.05) for measuring the witness, correspond to total of five measurements: two separate measurements of $\sigma^{z}$ in each polariton, and two joint measurements $\sigma_1^{z}\otimes\sigma_2^{z}$ and $\sigma_1^{x}\otimes\sigma_2^{y}$.
![\[witness\] Elements $c_{j,k}$ of the entanglement witness $W$ detecting the density matrix which maximizes the concurrence in this system.](witness.eps){width="60.00000%"}
The values of $\zeta$ and $\xi_{1}$ used in Fig.\[concurrence\] to get maximal entanglement would correspond to $\Delta=10J$, $\kappa=10J$, $G=\gamma=0.01J$ and the pumping coherent field was also taken to have roughly a hundred photons. $J$ is tunable and depends on the coupling of the photonic modes between neighboring cavities. Assuming this to be of the order of $10^{10}Hz$, this would correspond to a cavity dissipation rate $\kappa\approx 10^{11}Hz$ and a polaritonic decay rate $\gamma\approx10^{8}Hz$. These correspond to 0.1 nanoseconds lifetime of the cavity field and to ten nanoseconds for the polaritonic excitations at the two ends, which are within the near future in technologies like coupled toroidal microcavities and coupled superconducting qubits [@toroid]. Coupled defect cavities in photonic crystals arrays are also fast approaching this dissipation regime and are extremely suited in fabrication of regular arrays of many coupled defect cavities interacting with quantum dots [@noda]. In all technologies, an increase in $J$, in coupling between the cavity modes, the requirements on the various lifetimes of the polaritonic and photonic field modes can be further reduced.
Conclusion
==========
To summarize, this paper presents an example of entangling two qubits in the presence of dissipation despite the fact that each qubit has a continuously decaying state. The entanglement is not transient but stationary, and thereby easy to verify in an experiment, for which there is also a relevant witness. Though the amount of entanglement is not maximal, it is still very interesting as it is for a completely open system. As opposed to the typical case of, say, many-body systems or even the case of two purely atomic qubits in a single cavity or extremely close as to be able to directly interact, here there is the added advantage that the entangled qubits are easily individually accessible (being encoded in distinct atom-cavity systems) for measurements. It is worthwhile to point out an existing scheme to have steady state entanglement between entities in distinct cavities entangles atoms [@parkins] (as opposed to polaritons) and is much more intricate.
It is very interesting and counterintuitive that only a classical laser field driving the central cavity/connecting fiber was necessary to entangle the polaritonic qubits. A scheme feasible with current or near future technology and able to verify polaritonic entanglement as the one we have suggested in this paper, would be a significant first step towards the realization of the plethora schemes to simulate many-body systems and quantum computation using coupled cavities. Moreover, the model would also deserve to deepen counterintuitive properties of entanglement against noise (see e.g. [@stoch]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work has been supported by QIP IRC (GR/S821176/01), and the European Union through the Integrated Projects SCALA (CT-015714). SB would like to thank the Engineering and Physical Sciences Research Council (EPSRC) UK for an Advanced Research Fellowship the support of the Royal Society and the Wolfson foundation. SM thanks SB for hospitality at University College London. We would like to thank Y. Yamamoto for pointing out that a fiber can replace the central cavity in the three cavity system.
, , , and , ** ****, ().
D.G. Angelakis, and A.S. Kay, New J. of Phys. [**10**]{}, 023012 (2008).
S. Bose, D.G. Angelakis, and D. Burgarth, ** ****, ().
M.J. Hartmann, F.G.S.L. Brandao, and M.B. Plenio, ** [**2**]{}, 849 (2006).
, , and , ** ****, ().
A. Greentree, C. Tahan, J.H. Cole, and L.C. Hollenberg, ** [**2**]{}, 856 (2006).
D. Rossini, and R. Fazio, ** [**99**]{}, 186401 (2007).
M.X. Xuo, Y. Li, Z. Song, and C.P. Sun, arXiv:quant-ph/0702078.
Y.C. Neil Na, S. Utsunomiya, L. Tian, and Y. Yamamoto arXiv:quant-ph/0703219.
M. Paternostro, G.S. Agarwal, and M.S. Kim, arXiv:0707.0846.
E.K. Irish, C.D. Ogden, and M.S. Kim, arXiv:0707.1497.
H.Deng, G.Weihs, C.Santori, J. Bloch, and Y.Yamamoto, ** [**298**]{}, 199 (2002); M. Fleischhauer, and M.D. Lukin, ** [**84**]{}, 5094 (2000).
S.D. Barrett, and P. Kok, ** [**71**]{}, 060310 (2005).
Y.L. Lim, A. Beige, and L.C, Kwek, ** [**95**]{}, 030505 (2005).
T.P. Spiller, K. Nemoto, S.L. Braunstein, W.J. Munro, P. van Loock, and G.J. Milburn, ** [**8**]{}, 30 (2006).
A. Serafini, S. Mancini, and S. Bose, ** [**96**]{}, 010503 (2006).
S. Mancini, and J. Wang, ** **32**, 257 (2005).
S. Mancini, and S. Bose, ** **70**, 022307 (2004).
S. Clark, A. Peng, M. Gu, and S. Parkins, ** **91**, 177901 (2003).
G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J.M. Raimond, and S. Haroche, ** [**400**]{}, 239 (1999); P. Grangier, G. Reymond, and N. Schlosser. ** [**48**]{}, 859 (2000); P.W.H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, ** [**404**]{}, 365 (2000).
, , and , ** ****, (); *et al.*, ** ****, ().
, , and , ** ****, (); , ** ****, (); T. Aoki, B. Dayan, E. Wilcut, W.P. Bowen, A.S. Parkins, T.J. Kippenberg, K.J. Vahala, and H.J. Kimble, ** [**443**]{}, 671 (2006).
and ** ****, (); , , and , ** ****, (); , ** ****, ();
M. Trupke *et. al.*, ** [**99**]{}, 063601 (2007).
B. Terhal, ** [**323**]{}, 61 (2001); M. Lewenstein, B. Krauss, J.I. Cirac, and P. Horodecki, ** [**65**]{}, 012301 (2002); O. Guehne, P. Hyllus, D. Bruss, A.K. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, ** **50**, 1079 (2003).
, *,* ().
, ** ****, ().
S. Huelga and M.B. Plenio, ** [**98**]{}, 170601 (2007).
[^1]: The term $a\sigma_j$ in Eqs.(\[eqssigma\]) is considered as to be $(a\sigma_j+\sigma_j a)/2$.
|
---
author:
- 'N.S.Narayanaswamy'
- 'G.Ramakrishna'
bibliography:
- 'references.bib'
title: 'On Minimum Average Stretch Spanning Trees in Polygonal 2-trees [^1] [^2]'
---
**Concluding Remarks.** For a polygonal 2-tree on $n$ vertices, we have designed an $O(n \log n)$-time algorithm for the problem, <span style="font-variant:small-caps;">Mast</span>, of finding a minimum average stretch spanning tree. By using this algorithm, we have obtained a minimum fundamental cycle basis $\mathcal{B}$ of a polygonal 2-tree on $n$ vertices in $O( n \log n)$ + $\operatorname{size}(\mathcal{B})$ time. We have also shown that polygonal 2-trees have a unique minimum cycle basis and it can be computed in linear time. The problem of finding a minimum routing cost spanning tree is closely related to <span style="font-variant:small-caps;">Mast</span>. A *minimum routing cost spanning tree* is a spanning tree of a graph that minimizes the sum-total distance between every two vertices in the spanning tree. The complexity of finding a minimum routing cost spanning tree in polygonal 2-trees (also in planar graphs) is open, where as it is NP-hard in weighted undirected graphs.
[^1]: Supported by the Indo-Max Planck Centre for Computer Science Programme in the area of *Algebraic and Parameterized Complexity* for the year 2012 - 2013
[^2]: The preliminary version of this work is appeared in Eighth International Workshop on Algorithms and Computation (WALCOM) 2014
|
---
abstract: 'We propose a new way of defining entropy of a system, which gives a general form which may be nonextensive as Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a conceptually novel but simple origin and is mathematically easy to define by a very simple expression, though the probability distribution resulting from optimizing it gives rather complex, which is compared numerically with the other entropies. It may, therefore, appear as the right candidate in a physical situation where the probability distribution does not suit any of the previously defined forms.'
author:
- Fariel Shafee
title: A New Nonextensive Entropy
---
Introduction
============
Entropy is a measure of disorder or randomness, and assumes its maximal value when a system can be in a number of states randomly with equal probability, and is minimally zero when the system is in a given state, with no uncertainty in its description. Apart from this common feature shared by all definitions of entropy at two ends of the scale, variations are possible in particularizing the functional form in between [@LA1]. They lead to different forms of the probability distributions for states with different energies or some other conserved attribute. Some turn up as extensive, where the entropy of a combination of systems is simply the sum of the entropies of the systems, as in the classical case of the Shannon form, while others can be defined to be not so. Renyi entropy [@RE1] is different from Shannon, yet extensive, and hence the Shannon form is not unique with respect to the property of extensivity.
Tsallis entropy [@TS1; @TS2.; @PA1] has attracted a lot of attention in recent years, not only on account of its conceptual and theoretical novelty, but also because it can be shown in specific physical cases [@BE1; @CO1; @WO1] to be the relevant form where nonextensivity is expected on account of the interaction of the combined subsystems. In the proper limiting case it reduces to the standard Shannon entropy, indicating the consistency of the concept.
In this paper, however, we shall introduce entropy from a new perspective, which too will bear semblance to the normal form in the limit. We shall first present the rationale for this new definition and compare it briefly with the forms already being used. Then we shall find the form of the probability distribution for this entropy, which we shall henceforth call s-entropy, as it will be seen to be related to the concept of rescaling of the phase space.
DEFINING THE NEW ENTROPY
========================
Let us consider a register of only one letter. Let ${p_i}$ be the set of probabilities for each of the $N$ letters $A_i$ that can occupy this position. We are here using the language of information theory, as used for example, in the Shannon Coding Theorem, though it is trivially extensible to states $i$ of a single state of an ensemble where the individual systems can be in any $N$ states with probabilities $p_i$.
Let us now consider a small deformation of the register to a new size so that it can accommodate $q = 1+ \Delta q$ letters. The probability that the whole new phase space is occupied by the letter $p_i$ is now $p_i^q$ by the corresponding AND operation and hence the probability that the new deformed cell is occupied by any of the pure letters $A_i$ is
$$\label{eq1}
N(q) = \sum_i p_i^q$$
For $q >1 $ this would give a shortfall from the original total probability of unity for $q=1$. It is obvious that the shortfall, which we denote by
$$\label{eq2}
M(q) = 1- \sum_i p_i^q$$
represents the total probability that the mixed cell has a mixture of $A_i$ and some other $A_j$ fractionally, since the total probability that the cell is occupied by one or more (fractional included) letters must be unity. Hence the mixing probability $M(q)$ is actually a measure of the disorder introduced by increasing the cell scale from unity to $1+ \Delta q$.
The introduction of fractional values of cell numbers can be taken in the same spirit as defining the fractal (Hausdorff) dimensions of curves, and in complex systems there have been studies of diffusion [@BU1] and percolation in complex systems with effectively fractional dimensions for fluids where the special geometric constraints translate into a change in the dimension of the corresponding space to an apparently nonintuitive fractional dimension. In coding theories for optimal transmission of information [@NI1], we come across Huffmann coding, where the optimum alphabet size may be formally a fraction, though for practical purposes it may be changed to the nearest higher integer. In probabilistic optimization, we may therefore consider a fractional size of the registrar, or equivalently, an integral number of cells in the registrar with fractional sized cells to accommodate a given amount of information. Probabilistic optimization in place of the deterministic parameterization of classical Shannon information theory [@NI1] becomes inevitable in quantum computing contexts, and hence our use of the fractional cell sizes may be a classical precursor of the inevitable departure from stringent Shannon-type concepts.
For an alphabet of $m$letters we define the entropy from the information content of the registrar by
$$\label{eq3}
m^{S(q) \Delta q} = m^{(M(q+\Delta q)- M(q))}$$
so that the entropy indicates an effective change in the mixing probability due to an infinitesimal change in the cell-size of the registrar.
This leads to
$$\label{eq4}
S(q) = dM(q)/dq$$
In other words
$$\label{eq5}
S(q) = - \sum_i p_i^q \log p_i$$
[**We have some material at the other place.**]{} This differential form is analogous to but different from the Tsallis form
$$\label{eq6}
S_T(q) = - \sum_i (1- p_i^q)/ (1-q)$$
where there is an apparent singularity at $q=1$ which is the Shannon limit. The difference between the Tsallis expression and ours becomes clearer if we express entropy as the expectation value of the (generalized or ordinary) logarithm.
$$\label{eq7}
S_T(q) = <Log_q p>$$
where the generalized q-logarithm is defined as
$$\label{eq8}
Log_q p_i = 1- p_i^{(q-1)}/ ( 1- q)$$
The expectation value is defined in terms of the simple probability distribution $$\label{eq9}
<O> = \sum_i p_i O_i$$
In our case we define the expectation value with respect to the deformed probability corresponding to the extended cell, while keeping the usual logarithm
$$\label{eq10}
S_s(q) = < \log p >_q$$
with
$$\label{eq11}
<O>_q = \sum_i p_i^q O_i$$
In the limit $q \rightarrow 1$ $Log_q$ approaches the normal logarithm, and hence Tsallis entropy coincides with Shannon entropy and also as $p_i^q \rightarrow p_i$ we too get the normal Shannon entropy.
The Renyi entropy is defined by
$$\label{eq12}
S_R(q) = \log ( \sum_i p_i^q)/(1-q)$$
Like Shannon entropy this one is also extensive, i.e. simply additive for two subsystems for any value of $q$. To get Shannon entropy uniquely one needs [@KI1] a slightly different formulation of the extensivity axiom
$$\label{eq13}
S_{1+2} = S_1 + \sum_i p_{1i} S_2(i),$$
where $S_2(i)$ is the entropy of subsystem $2$ given subsystem $1$ is in state $i$.
PROBABILITY DISTRIBUTION FOR THE NEW ENTROPY
============================================
The $p_i$ can be obtained in terms of the energy of the states, or possibly also other criteria in the usual way by maximizing the entropy with constraints
$$\label{eq14}
\sum_i p_i -1=0$$
and
$$\label{eq15}
\sum_i p_i E_i - U = 0$$
.
The solution of the optimization equation gives for energy $E_i$ the probability $p_i$
$$\label{eq16}
p_i = (\frac{ - q W(z)}{(a+ b E)(q-1)})^{1/(1-q)}$$
where $$\label{eq17}
z=-e^{(q-1)/q} (a+ b E) (q-1)/q$$
and W(z) is the Lambert function defined by
$$\label{eq18}
z= w e^w$$
Here $a$ and $b$ are constants coming from the Lagrange’s multipliers for the two constraints and are related to the overall normalization and to the relative scale of energy, i.e. to temperature ( $1/(kT)$) as in the Shannon case where we get the Gibbs expression for $p_i$. In the Tsallis case $p_i$ has the well-known value
$$\label{eq19}
p_i = ( a + b (q-1)E)^{1/(1-q)}$$
which is easily seen to reduce to Shannon form for $q \rightarrow
1$.
After some algebra it can be shown that this form reduces to the Shannon form for $q \rightarrow 1$.
The nonextensivity of Tsallis entropy is seen easily by expanding
$$\begin{aligned}
\label{eq2021}
S^T{}_{1+2} = - \sum_{ij} p_i p_j (1- p_i{}^q p_j{}^q)/(1-q)^2 \\
= S^T{}_1+ S^T{}_2 + (1-q) S^T{}_1 S^T{}_2\end{aligned}$$
For Renyi entropy we have the simple additive relation
$$\label{eq22}
S^R{}_{1+2}= S^R{}_1 + S^R{}_2$$
In case of the new entropy
$$\label{eq23}
S^s{}_{1+2} = S^s{}_1 + S^s{}_2 + M_2(q)S^s{}_1 + M_1(q)S^s{}_2$$
where the $M_a$ are the mixing probability of states for subsystem $a$ as defined in Eq.2.
NUMERICAL COMPARISON
====================
In Fig.1 we show the variation of the probability function for different $E$ at different $q$ values.
![\[fig1\]Comparison of the pdf for the new entropy for values of $q=1,1.1,1.2$ and $1.3$. The solid line is for $q=1$, i.e. the Gibbs exponential distribution and the lines are in the order of $q$](fig1.eps){width="8cm"}
We note that the pdf drops increasingly rapidly for higher values of $q$, and is quite different in shape and in magnitude at high energy values from the Gibbs exponential distribution. A variation of even $10\%$ from the standard value of $q=1$ can cause a quite discernible change in the pdf and should be observable in experimental contexts fairly easily . At $q=1.3$, the shape is almost linear.
In Fig.2 and Fig.3 we show the comparison of Tsallis pdf and the pdf for the new entropy for the same values of $q$, 1.1 in the former and 1.3 in the latter. We notice that for larger $q$ values the new entropy gives much stiffer probability functions departing substantially from the Tsallis pdf’s.
![\[fig2\]Comparison of pdf’s for Tsallis nonextensive entropy and the new entropy presented here, for $q=1.1$](fig2.eps){width="8cm"}
![\[fig3\]The same as Fig. 2, but for a higher $q=1.3$](fig3.eps){width="8cm"}
CONCLUSIONS
===========
We see that the new entropy presented here based on the simple concept of the amount of mixing of states freedom introduced per unit cell of phase space leads to a nonextensive form different from any of the presently studied entropies. It leads to a complicated, but still integrable form of the pdf which departs substantially from Tsallis entropy. This entropy is also nonextensive in a fashion different from Tsallis entropy, though like Tsallis it too becomes extensive trivially in the limit $q \rightarrow 1$, as expected.
It would now be interesting to find a physical situation where such an entropy arises from first principles, though like some initial phenomenological studies of Tsallis entropy it can be also used as a parametrization scheme with $q$ as a parameter to fit experimental data. The stiffness of any data may point to its preferability to Tsallis-type entropies.
P.T. Landsberg, “Entropies Galore”, [*Braz. J. Phys.*]{} [**29**]{}, 46 (1999) A. Renyi, [*Probability Theory*]{} (North-Holland, Ams terdam, 1970) C. Tsallis,[*J. Stat. Phys.*]{}, [**52**]{}, 479(1988) P. Grigolini, C. Tsallis and B.J. West,[*Chaos, Fractals and Solitons*]{},[**13**]{}, 367 (2001) A.R. Plastino and A. Plastino, [*J. Phys. A*]{} [**27**]{}, 5707 (1994) C. Beck, “Nonextensive statistical mechanics and particle spectra”, hep-ph/0004225 (2000) O. Sotolongo-Costa et al., “A nonextensive approach to DNA breaking by ionizing radiation”, cond-mat/0201289 (2002) C. Wolf, “ Equation of state for photons admitting Tsallis statistics”, [*Fizika B*]{} [**11**]{}, 1 (2002) M. Buiatti, P. Grigolini and A. Montagnini, [*Phys. Rev. Lett.*]{} [**82**]{}, 3383 (1999) M.A. Nielsen and M. Chuang, [*Quantum computation and quantum information*]{} (Cambridge U.P., NY, 2000) A.I. Kinchin, [*Mathematical Foundationsof Information Theory*]{}, (Dover Publications, New York, 1957)
|
---
author:
- 'R. H[ü]{}bener,'
- 'Y. Sekino,'
- 'and J. Eisert'
bibliography:
- 'MatrixModelsJHEPv2.bib'
title: 'Equilibration in low-dimensional quantum matrix models'
---
Introduction
============
Matrix quantum mechanics has received significant attention in recent years, mainly for its suspected connection with quantum gravity. Most prominently, Banks, Fischler, Shenker and Susskind (BFSS) [@BFSS96] have proposed that supersymmetric matrix quantum mechanics, called matrix theory, gives a formulation of M-theory. This is an exciting proposal about the fundamental degrees of freedom in nature, which may allow us to address questions that are out of reach of semi-classical gravity. Questions of particular interest are on the quantum properties of black holes, such as their microscopic constituents, thermalisation, and the process of evaporation [@HaPr07; @Susskind].
The bosonic degrees of freedom of BFSS matrix theory are $N\times N$ Hermitian matrices $\mathfrak{X}^{(i)}$, where $i=1,\ldots, d$ is the spatial index, and the dimension of space for the BFSS theory is $d=9$. There is no mass term, and the interaction is given by a quartic term. Unsurprisingly, this is an intricate theory and it has not been solved to date. To start with, every local degree of freedom directly couples to any other degree of freedom. What is more, the strong quartic interactions render naïve approximation schemes such as perturbation theory hopeless.
The primary focus of this article is on the $d=2$, $N=2$ model, as a first paradigmatic step. Even though the original proposal of BFSS concerns a large $N$ limit, the understanding of $N=2$ models is important since one can compute the scattering of two D0-branes with $N=2$. This will allow us to probe the short-distance physics at the $11$-dimensional Planck scale [@DKPS97]. When the D0-branes are far apart, a perturbative method based on the Born-Oppenheimer approximation is valid [@OkYo99; @Taylor], but as the D0-branes approach each other, they enter the strong coupling regime where open strings between them are excited. A qualitative discussion of D0-brane scattering has been given in [@DKPS97], but a quantitative understanding of the strong coupling regime is lacking. Even with $N=2$, the matrix model will be a highly non-trivial many body system. One fact which suggests this is that Monte Carlo simulations of supersymmetric matrix quantum mechanics [@Han10; @Hanada] with $N$ as small as 2 or 3 yield results consistent with the predictions of gauge/gravity correspondence [@SeYo00; @Sekino] which is supposed to be valid in the large $N$ limit.
One cannot really define scattering amplitudes in our bosonic model, since there are no weakly-interacting asymptotic states without supersymmetric cancellations. However, our model may capture the essence of the strong coupling regime where many excited states contribute to the dynamics, since those states will not be sensitive to the detailed features of the model such as supersymmetry. Also, the gauge symmetry has an important consequence: D-branes are repelled from each other at short distances because of the centrifugal force due to angular momenta in the gauge space, as will become clear in our approach.
Even though the $d=2$, $N=2$ bosonic model (see eq. below) looks simple, it is not integrable (in the classical sense), and features rich dynamics. This model has been studied before under the name of Yang-Mills quantum mechanics [@Sa83; @Sa84; @Fu87; @JS89], and is considered to be a prototypical model for quantum chaos. Despite intensive numerical simulation, we still lack a complete understanding even in the classical realm. The chaotic nature of Yang-Mills theory may provide insight into the problems such as thermalisation effects in hadronic processes [@KMOSTY10] and confinement in Yang-Mills theory [@Ole82].
In this work we take a fresh approach to the problem, by first presenting a new equivalent formulation, taking full advantage of the gauge symmetry and exhibiting an aspect of locality in the model. It turns out that in this reformulation — which is is amenable to analytical study and precise numerical analysis — questions of equilibration [@Linden_etal09; @CramerEisert08; @InteractingThermalisation] in non-equilibrium can be precisely posed. Our considerations complement other studies, such as classical analysis of the thermalisation in matrix models [@BerensteinClassical2; @BerensteinClassical], as well as the bounds on scrambling time based on the locality of the models and Lieb-Robinson bounds [@FastScrambling]. See also recent work [@Sahakian1; @Sahakian2; @Sahakian3; @Mandal; @Kabat] for different approaches to thermalisation in matrix models.
Matrix models and relations to other models
===========================================
We will study bosonic matrix models defined by the Hamiltonian of $0+1$ dimensional Yang-Mills theory $$H = {\mathrm{Tr}} \biggl[{\frac{1}{2} \sum_{i=1}^d \mathfrak{P}^{(i)} \mathfrak{P}^{(i)} -
\frac{1}{4}\sum_{i,j=1}^d [\mathfrak{X}^{(i)}, \mathfrak{X}^{(j)}]^2}\biggr],
\label{eq:Hmatmod}$$ where $\mathfrak{X}^{(i)}= \sum_a x^{(i)}_a t_a$, with the $N \times N$ traceless Hermitian generators $t_a$ of $SU(N)$, and position operators $x^{(i)}_a$. Similarly we write $\mathfrak{P}^{(i)}= \sum_a p^{(i)}_a
t^*_a$ where $p^{(i)}_a$ are the conjugate momenta. The gauge field is not dynamic and has been set to zero by a gauge transformation; its equation of motion imposes the constraint that states under consideration are required to be singlets of $SU(N)$. In this work we will primarily treat the $d=2$, $N=2$ case[^1]. An important theorem which provides a starting point for our analysis is that the Hamiltonian with the gauge constraint can be recast into a model which has a local structure of interactions.
The Hamiltonian for $N=2$ and $d=2$, with the constraint that the states are SU(2) singlets, is equivalent with $$\label{eq:easyHam}
\begin{split}
4 H =& \left(P^2 \otimes {\mathbbm{1}}+{\mathbbm{1}}\otimes P^2 \right) \otimes {\mathbbm{1}}\\
&+
\left(
X^{-2} \otimes {\mathbbm{1}}+ {\mathbbm{1}}\otimes X^{-2}
\right) \otimes \sum_{\ell} {\ell(\ell+1) |\ell\rangle\langle \ell|} \\
&+
X^2 \otimes X^2
\otimes \sum_{\ell,\ell'} A_{\ell,\ell'}{| \ell \rangle \langle \ell' |},
\end{split}$$ acting on $\mathcal{H}= L^2( \mathbb{R}^+)\otimes L^2( \mathbb{R}^+)\otimes l^2$ where $A=A^T$ is banded, with $$\label{eq:Amatrix}
\begin{split}
A_{\ell, \ell} &= \frac 1 2-\frac{1}{8 \ell (\ell+1) - 6}\\
A_{\ell, \ell+2} = A_{\ell+2, \ell} &= -\frac{(\ell+1)(\ell+2)}{\sqrt{2 \ell+1}(2 \ell + 3)\sqrt{2 \ell +5}},
\end{split}$$ all other elements being zero.
#### Proof of equivalence.
To prove the equivalence of the above models, we start from the explicit form of 2$\times$2 traceless Hermitian matrices, $\mathfrak{X}^{(i)} = \sum_{a=1}^{3} x^{(i)}_a \sigma_a/2$ with Pauli matrices $\sigma_a$. Expressing $\mathfrak{P}^{(i)}$ similarly, we have a set of pairs of canonical coordinates, $(x^{(i)}_a,p^{(i)}_a)$, whose Hamiltonian is now written as $$4 H = (\mathbf{p}^{(1)})^2 + (\mathbf{p}^{(2)})^2 +
(\mathbf{x}^{(1)} \times \mathbf{x}^{(2)})^2.
\label{eq:4H}$$ In what follows, it is convenient to turn to the position representation in radial coordinates, in which the interaction term $(\mathbf{x}^{(1)} \times \mathbf{x}^{(2)})^2$ is identified with $r_1^2 r_2^2 \sin^2 (\theta_{1,2})$ where $\theta_{1,2}\in[0,\pi)$ is the angle between $1$ and $2$ and $r_1,r_2\in (0,\infty)$. Singlet states are given by linear combinations of $$\psi^{(0)}_{\ell}(\theta_1, \phi_1; \theta_2, \phi_2) = (2 \ell +1)^{-1/2} \sum_{m=-\ell}^{\ell} (-1)^{\ell-m} Y_\ell^m(\theta_1, \phi_1) Y_\ell^{-m}(\theta_2, \phi_2),$$ with the spherical harmonics $Y_\ell^m:[0,\pi)\times [0,2\pi)\rightarrow \mathbb{C}$. Accordingly, the state vector in the position representation takes the form $$\psi(r_1, \theta_1, \phi_1; r_2, \theta_2, \phi_2) := \frac 1 {r_1 r_2} \sum_{\ell} \rho_{\ell}(r_1,r_2) \psi^{(0)}_{\ell}(\theta_1, \phi_1; \theta_2, \phi_2),$$ with radii-dependent functions $\rho_{\ell}:(0,\infty)\times(0,\infty)\rightarrow \mathbb{C}$.
The Hamiltonian in the radial position representation becomes $$-\partial_{r_1}^2 - \partial_{r_2}^2 - \frac{\Delta_{S_1}}{r_1^2} - \frac{\Delta_{S_2}}{r_2^2} + r_1^2 r_2^2 \sin^2 (\theta_{1,2}),$$ where $\Delta_S$ is the Laplace operator on the unit sphere, and $\theta_{1,2}$ is the angle on the great circle. The interaction term, which itself is invariant under $SU(2)$, can be rewritten using the identity $$\sin^2 (\theta_{1,2}) = \frac{8 \pi}{3} \left( \psi^{(0)}_{0}(\theta_1, \phi_1; \theta_2, \phi_2) - \frac 1 {5^{1/2}} \psi^{(0)}_{2}(\theta_1, \phi_1; \theta_2, \phi_2) \right).$$ Its action on the state vector can be found from the multiplication rules for spherical harmonics, $\psi^{(0)}_{2} \psi^{(0)}_{\ell} = \sum_{\ell'} c_{\ell,2,\ell'} \psi^{(0)}_{\ell'}$. The coefficient $$c_{\ell,2,\ell'} = (-1)^{\ell - \ell'} \frac{5^{1/2}}{8 \pi}\left({(2 \ell' + 1)(2 \ell+1)}\right)^{1/2} \int_{-1}^{1} \mathrm{d}x P_{\ell}(x) P_2(x) P_{\ell'}(x),$$ is closely related to Wigner’s 3J-symbol $$\int_{-1}^{+1}\mathrm{d}x P_{\ell}(x) P_2(x) P_{\ell'}(x) =
2 \left( \begin{array}{ccc}\ell & 2 & \ell' \\ 0&0&0 \end{array} \right)^2.$$ It is non-zero only when $\ell=\ell'$ or $\ell=\ell'\pm 2$. In the end, the Hamiltonian as an operator acting on $\rho$ takes the form of with $$3A_{\ell, \ell'} = \left( 2 - c_{\ell,2,\ell} \right) \delta_{\ell,\ell'} - c_{\ell,2,\ell'} \delta_{\ell, \ell'+2} - c_{\ell,2,\ell'} \delta_{\ell, \ell'-2},$$ with the explicit form of $A_{\ell, \ell'}$ given by .[$\hfill \Box$]{}
#### Discussion of the model.
Interestingly, the bandedness of $A$ introduces a notion of locality to the model not apparent in the original form eq. . There is a fast convergence $A_{\ell,\ell} \rightarrow 1/2$, $A_{\ell,\ell \pm 2} \rightarrow - 1/4$, also $\ell(\ell+1) \approx \ell^2$, for growing values of $\ell$. Hence, the part acting on $l^2$ reminds of the harmonic oscillator on a lattice [@CGM86] where $A$ is an approximation of the Laplacian. The even and odd sublattices are decoupled, as there are no non-zero matrix elements coupling $\ell$ and $\ell \pm 1$. The dynamics on $L^2( \mathbb{R}^+)\otimes L^2( \mathbb{R}^+)$ determines joint effective “mass” and “spring constants” for the two indirectly coupled systems in $l^2$, providing indirect interaction between these systems.
The model is invariant under the spatial $SO(2)$ rotation. The generator in the representation of eq. is $\mathbf{x}^{(1)}\cdot \mathbf{p}^{(2)} -\mathbf{x}^{(2)}\cdot \mathbf{p}^{(1)}$, and we have an equivalent operator $Q$ in the representation of eq. $$\begin{split}
Q =& (X \otimes P - P \otimes X)\otimes \sum_{\ell, \ell'}
(Q_p)_{\ell, \ell'}|\ell\rangle\langle \ell'|\\
&+ (X \otimes X^{-1} - X^{-1} \otimes X)\otimes \sum_{\ell, \ell'}
(Q_x)_{\ell, \ell'}|\ell\rangle\langle \ell'|,
\end{split}$$ where $(Q_p)_{\ell,\ell+1} =(Q_p)_{\ell+1,\ell} = -\ell(4\ell^2-1)^{-1/2}$ and $(Q_x)_{\ell,\ell+1}=(Q_x)^*_{\ell+1,\ell} = -i \ell^2(4\ell^2-1)^{-1/2}$, all other elements being $0$. Note that because of the invariance of the model under the action of $Q$, we have $[Q,H]=0$, and the Hilbert space is divided into eigenspaces of $Q$. The spectral values of $Q$ are $q \in 2\mathbb{Z}$, the even integers. This angular momentum will become important in the analysis of the model.
The spectrum
============
#### Numerical analysis.
The aim of our numerical analysis is the low energy regime of the model. We compute the spectrum of $H$ in the different eigenspaces of $Q$, as well as the spatial extension of the state, $\langle X^2 \otimes {\mathbbm{1}}\otimes {\mathbbm{1}}+ {\mathbbm{1}}\otimes X^2 \otimes {\mathbbm{1}}\rangle^{1/2}$. The Hilbert space of this model is infinite dimensional, so we make use of a finite-dimensional approximation, which is the only approximation we make. We stress that without the presented reformulation, already the smallest non-trivial instance of the model — $N=2$ and $d=2$ — seems inaccessible even on super-computers, and it would be also difficult to implement the gauge constraint numerically. We restrict $L^2(\mathbb{R}^+)$ to the Hilbert space spanned by the first $h_0=107$ odd eigenfunctions of the harmonic oscillator, and we restrict the dimension of $l^2$ to some value $\ell_0=156$. The eigenvalues of $Q$ are $q \in 2 \mathbb{Z}$, the even integers. We enumerate the energies within each $Q$-eigenspace with a parameter $n \in \mathbb{N}^0$, hence we have states ${|q,n\rangle}$ with $Q{|q,n\rangle}=q{|q,n\rangle}$ and $H{|q,n\rangle}=E_{q,n}{|q,n\rangle}$. The quality of the approximation is double-checked by choosing smaller dimensions initially and then keeping only the the states whose eigenvalues remain almost unchanged when enlarging the dimension.
#### Results.
The ground state energy is $E_{GS}=1.05535\ldots$
The data suggest power laws both in $q$ and $n$. Figure \[fig:qplot\] shows, for fixed $q$, an affine function $(E_{q,n} + E_0)^\alpha$ up to certain values of $n$, where we notice almost degenerate pairs. After that, linear growth continues. Low values of $|q|$, especially $q=0$, are dominated by the degeneracies and hence the linear growth is mostly hidden, but the onset of degeneracies comes later for larger values of $|q|$. One can fit the data for fixed $n$ as well, the dependency seems correct for a range of exponents $\alpha \approx 1.5 \ldots 2.3$, provided the other constants are chosen appropriately. For $\alpha = 2$ this relation is known as the linear Regge trajectory and corresponds to the behaviour of a relativistic string [@GSW], which has been observed in the $2+1$ dimensional Yang-Mills theory [@PowerLaw]. There is a scaling argument coming from semi-classical analysis which suggests the dependence $E_{q,0} \sim
|q|^{2/3}$ [@BerensteinScaling]. The best fit we obtain when leaving all constants subject to variation yields $\alpha \approx 1.62(2)$, $E_0 \approx 1.6(1)$, but we cannot draw a definite conclusion about the value of $\alpha$ from our analysis alone. Note that the growth of $(E_{q,n}+E_0)^\alpha$ with $n$ stays always sublinear for all values of $q$. We also consider the spatial extension of the states, $\langle X^2 \otimes {\mathbbm{1}}\otimes {\mathbbm{1}}+ {\mathbbm{1}}\otimes X^2
\otimes {\mathbbm{1}}\rangle^{1/2}$. The extensions within each eigenspace of $Q$ are affine functions of $E$, except that around the degenerate pairs states have a much smaller size.
![(Large plot: energies) $H{|q,n\rangle}=E_{q,n}{|q,n\rangle}$. Depicted is the value of $(E_{q,n} + E_0)^\alpha$ on the vertical axis with $n$ on the horizontal axis, using $\alpha \approx 1.62$. The black line belongs to $q=0$ and the others to increasing $|q|$, from green to red. There is affine growth of $(E_{q,n}+E_0)^\alpha$ over both $n$ and $|q|$ in certain regions. The affine growth over $n$ stops at certain points where we notice almost degenerate pairs of energies. The growth over $n$ is always sublinear. (Small plot: sizes) Depicted is the value of $\langle X^2 \otimes {\mathbbm{1}}\otimes {\mathbbm{1}}+ {\mathbbm{1}}\otimes X^2 \otimes {\mathbbm{1}}\rangle^{1/2}$ for fixed $q=3$, as an example. This value is approximately an affine function of $\langle H \rangle$, with occasional collapses to much smaller values at the positions of the almost degenerate energies.[]{data-label="fig:qplot"}](QPlot3Ga){width="0.75\columnwidth"}
In Figure \[fig:CFplot\] we show a Chew-Frautschi plot of the model. An asymptotic affine dependence between $(E_{q,n} + E_0)^2$ and $|q|$ is clearly seen in the large energy / large angular momentum regime.
![Chew-Frautschi plot and size plot (large plot: energies). Depicted is the value of $(E_{q,n} + E_0)^2$, similarly to Figure \[fig:qplot\], but fitting the data while fixing $\alpha=2$. In this plot, we show graphs belonging to constant values of $n$, while varying over values of the angular momentum $q$. An asymptotic affine dependence between $(E_{q,n} + E_0)^2$ and $|q|$ is clearly seen in the large energy / large angular momentum regime. (Small plot: sizes) Depicted is the value of $\langle X^2 \otimes {\mathbbm{1}}\otimes {\mathbbm{1}}+ {\mathbbm{1}}\otimes X^2 \otimes {\mathbbm{1}}\rangle^{1/2}$ for fixed $n=11$, as an example, over the full range of $q$. This graph follows, asymptotically for large $|q|$ and $n$, an affine dependence as well, with a low energy region where the size and energy are smaller than a linear extrapolation of the asymptotics implies. Is resembles the behaviour of an initially strongly bound system which is less tightly bound for high energies.[]{data-label="fig:CFplot"}](QPlot4G2){width="0.75\columnwidth"}
Equilibration in matrix models
==============================
Equipped with these insights, we now turn to questions of equilibration. In the quantum setting, ‘equilibration’ refers to the situation that for most times, expectation values take values as if the system was in the time-averaged state $$\omega:=
\lim_{T\to \infty}
\frac{1}{T}\int_0^T \mathrm{d}t e^{-itH} \rho_0 e^{itH},\qquad \rho_0:= |\psi(0)\rangle\langle\psi(0)|.
\label{omega}$$ Here, we consider the sectors $H_q=\sum_n E_{q,n} {| q,n \rangle \langle q,n |}$ of the Hamiltonian $H$ for a given value of $q$. Assuming the non-degeneracy of energy levels, the time-averaged state is diagonal in the energy basis, $$\omega=\sum_{n}|\langle \psi(0)|q,n\rangle|^2
|q,n\rangle\langle q,n|.$$ A measure of the extent to which expectation values stay close to those of the time average is the effective dimension $d_{\rm eff}^{-1} := {\rm Tr}\, \omega^2
=\sum_n |\langle
\psi(0)|q,n\rangle|^4$. In fact, if we divide the system into a $d_S$-dimensional subsystem $S$ and the rest (which is often referred to as the heat bath) $B$, the reduced density matrix $\rho_S(t)={\rm Tr}_B[\rho(t)]$ is close to the time average $\omega_S={\rm Tr}_B[\omega]$ if $d_{\rm eff}$ is large [@Linden_etal09], $$\mathbb{E}\|\rho_S(t)-\omega_S\|_1 \leq d_S / d_{\rm eff}^{1/2},
\label{rhoS}$$ where $\mathbb{E}$ denotes the time average, and the distance in the state space is measured with respect to the trace-norm, $\|M\|_1={\rm Tr}\sqrt{M^\dagger M}$.
We can immediately get an estimate for the effective dimension of a random state in a micro-canonical energy window, slightly improving on ref. [@Linden_etal09]. Denote with $d_\Delta$ the number of spectral values of $H_q$ contained in an interval $[E,E + \Delta]$ for a given fixed value of $q$, and let $I$ denote the corresponding index set $k \in I \Leftrightarrow E_{q,k} \in
[E,E+\Delta]$ in this eigenspace of $Q$.
A Haar random state vector ${|\psi(0)\rangle}$ in the micro-canonical subspace spanned by $\{{|{q,n}\rangle}: n \in I\}$ satisfies $\mathbb{E}(d_{\rm eff}) \geq (1+ d_{\Delta})/2$. \[obs:effdim\]
Using the Weingarten function calculus [@Collins1; @Collins2] for computing moments of entries of Haar-random unitaries $U$, and making use of the fact that each entry is identically distributed, one finds $$\begin{gathered}
\mathbb{E} \left( d_{\rm eff}\right) = \mathbb{E} \left[
\frac{1}{\sum_{n \in I}|\langle q,n| U| q,n_{1}\rangle|^4} \right] \\
\geq \frac{1}{\mathbb{E}\left [
\sum_{n \in I}|\langle q,n | U| q,n_{1}\rangle|^4
\right]}
= \frac{1}{ d_\Delta \mathbb{E}\left [
|\langle q,n_{1} | U| q,n_{1} \rangle|^4
\right]} = \frac 1 {d_\Delta} \binom{d_\Delta+1}{2}= \frac{d_\Delta+1}{2}.\end{gathered}$$ where $|q, n_{1}\rangle$ is one of the state vectors in the microcanonical subspace. [$\hfill \Box$]{}
One can also study the equilibration of observables. Given the assumption of the non-degeneracy of energy gaps, the following relation [@Short1] holds for any operator $A$, $$\mathbb{E}\left[
|{\rm Tr}[\rho(t)A]- {\rm Tr}[\omega A]|^2\right]
\leq {\|A\|^2\over d_{\rm eff}},
\label{rhoA}$$ where $\mathbb{E}$ denotes the time average, and $\|A\|$ is the operator norm, defined as the largest singular value of the operator. If $d_{\rm eff}$ is large, the outcome of a measurement represented by the operator $A$ will be close to what we would get in the time-averaged state $\omega$, most of the time. In fact, the relation (\[rhoS\]) can be derived from (\[rhoA\]) [@Short1], and it has been shown that equilibration occurs in a finite time [@Short2]. However, having a large number $d_\Delta$ of energy levels of $H_q$ within an energy window $[E,E+\Delta]$ implies that there are observables that will take long to equilibrate.
Consider the initial state vector $$|\psi(0)\rangle = d_\Delta^{-1/2}\sum_{n \in I} {|q,n\rangle},$$ which implies $d_{\rm eff}=d_\Delta$, and the observable $$O = \sum_{i,j \in I}(\delta_{i,j-1} + \delta_{i,j+1}) |q,i\rangle\langle q,j|.$$ Then the deviation from the infinite time average $\omega = \sum_{i \in
I} {| q,i \rangle \langle q,i |}$, normalised by the operator norm of the observable $\|O\| \leq 2$, fulfills $$\frac{|{\mathrm{Tr}\left[ {O(\rho(t)-\omega)} \right]}|}{\|O\|}
\geq 1 - \frac{1}{2 d_\Delta} \sum_{(i,i+1) \in I \times I} (E_{q,i+1} - E_{q,i})^2 t^2.$$ \[obs:eqtimes\]
This follows immediately from the fact that the left hand side is bounded from below by $$\sum_{i,j \in I} \frac{\delta_{i,j+1} + \delta_{i,j-1}}{2 d_\Delta} e^{-i t (E_{q,i} - E_{q,j})} = \frac 1 {d_\Delta} \sum_{(i,i+1) \in I \times I} \cos t (E_{q,i+1} - E_{q,i}),$$ which in turn is bounded from below by the right hand side. [$\hfill \Box$]{}
We now apply Observations 2 and 3 to the matrix model under consideration. We assume that $H_q$ is not degenerate. This is supported by the numerical results, although step-like features in the spectrum exist, which are very close to degeneracies. This does not, however, invalidate the following argument unless a degeneracy is exact. Furthermore, there is strong evidence presented above that within each eigenspace of $Q$ we have sublinear growth of $(E_{q,n} + E_0)^{\alpha}$ with $n$, where $\alpha=1.62$ in Figure \[fig:qplot\]. Hence for $\Delta \ll E$, we have $d_\Delta \geq \alpha \Delta (E+E_0)^{\alpha-1}/c$, where $c$ is the $n$-proportionality factor in $(E_{q,n}+E_0)^{\alpha} = a + b |q| + c n$, such that, with Observation \[obs:effdim\], $$\mathbb{E}(d_{\rm eff}) \geq \frac{\alpha \Delta (E+E_0)^{\alpha - 1}}{2 c}+ \frac{1}{2}.$$ That is to say, for large initial energies, one expects a strong equilibration and expectation values often take the values close to the ones of the time average.
The sublinear growth of $(E_{q,n} + E_0)^{\alpha}$ allows to find an upper limit for the gaps in the spectrum. We find that for energies within one eigenspace of $Q$ and above $E$ we have $(E_{q,i+1} - E_{q,i}) \leq c / \alpha (E+E_0)^{\alpha-1}$, so we can estimate with Observation \[obs:eqtimes\] that $$\frac{|{\mathrm{Tr}\left[ {O(\rho(t)-\omega)} \right]}|}{\|O\|} \geq 1 - \frac{c^2 (d_\Delta-1)}{2 \alpha^2 d_\Delta} \frac {t^2}{(E+E_0)^{2(\alpha - 1)}}.$$ So for this specific initial condition, the system will be close to the infinite time average in expectation, but we can make the equilibration time scale as large as we want by shifting the energy up. These statements can only be concluded from the microscopic Hamiltonian once the spectrum as above has been identified, as facilitated by our new normal form.
Conclusions
===========
#### Paths towards generalisations.
To establish a relation between matrix models and black hole physics, studies of $d \ge 3$ will be important, since gravity in lower dimensions is special. The case of higher $N$ and $d$ is under study. Hamiltonian with general $N$ and $d$ can be mapped to a model which involves $d$ particles in a $N^2-1$ dimensional space of the adjoint representation of $SU(N)$. The analysis will be more complicated than in $N=2$, $d=2$, since one needs more than one quantum number to specify a singlet wave function. However, the local structure of interactions persists, which has played an essential role in the analytic and numerical treatment in this paper, since the interaction terms in the Hamiltonian can change the $SU(N)$ quantum numbers only by a certain amount.
#### Summary.
We have constructed a model that is equivalent to the $N=2$, $d=2$ bosonic matrix model. We studied its spectrum as well as its equilibration properties. We found an affine dependence of $(E_{q,n} + E_0)^\alpha$, with plausible $\alpha \approx 1.5 \ldots 2.3$, on $|q|$ and $n$ in certain regions. The dependence on $|q|$ for the lowest $n$ (the leading Regge trajectory) is well-known to be the behaviour of a string if $\alpha=2$, and so is the spatial extension, which we found to be proportional to the energy (except for certain states). It is remarkable that this kind of non-perturbative behaviour, which is usually found with extensive Monte Carlo simulations in lattice gauge theory, is found by straightforward diagonalisation in our approach.
Although the $1/N$ expansion of any gauge theory can be represented as strings [@tHooft], it is far from obvious what kind of string theory our model with $N=2$ should correspond to. In fact, the power law dependence on $n$ without any degeneracy (up to some $n$, for fixed $q$) resembles a system of one or two oscillators, such as ref. [@Yukawa:1950eq], rather than strings, which have an infinite number of modes. The states at larger $n$ that have small spatial extension may represent composite states.
Regarding the question of equilibration, we applied rigorous mathematical results, which, complemented by the numerical analysis regarding the spectrum, allowed us to estimate how the effective dimension grows as the energy of a micro-canonical subspace increases. Our result shows that for high energies, states will for the overwhelming proportion of times look like their long time average. We also pointed out that observables and state preparations exist where the equilibration takes arbitrarily long. In future investigations, these observables might be used to exhibit aspects of the complexity of states as it builds up over time. Linked (but not limited) to the notion of classical computational complexity, this means essentially the number of computation steps or other resources used to construct a state from a simpler one (see, e.g., ref. [@Nielsen1; @Nielsen2; @Briegel]). In fact, there is a recent proposal about a connection between the complexity of a quantum state and the interior geometry of a black hole, namely the volume of the Einstein-Rosen bridge [@SusskindComplexity1; @SusskindComplexity2]. Complementing this development, there has been a recent activity on the relationship between equilibration properties of Hamiltonians and the complexity of their eigenvectors, quantified in terms of the length of a quantum circuit that is required to prepare a given quantum state [@Complexity], closely related to notions of quantum Kolmogorov complexities [@Ge; @Kolmogorov]. We suspect operators of the type we studied may help in describing the physics in the black hole interior.
We regard our analysis to be a first step toward understanding the behaviour of BFSS matrix theory, treated as a small-dimensional, but fully quantum mechanical model. The local structure of interaction in the space of gauge-invariant states has been an essential tool in our analysis, but its physical consequence is yet to be understood. We hope this locality sheds some light on the unsolved problem of describing local spacetime physics in quantum gravity.
#### Acknowledgements.
We thank S. Shenker for discussions and J. Plefka, D. Berenstein, T. Yoneya, J. Nishimura and L. Susskind for helpful comments. This work has been supported by the EU (SIQS, Q-Essence, RAQUEL), the FQXi, the ERC (TAQ), and the JSPS. This work was supported in part by the National Science Foundation under Grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics.
#### Note added.
The recent ref. [@Short] also addresses quantum processes slowly equilibrating, even though in a different context and providing different bounds.
Basics of gauge theory and BFSS matrix theory
=============================================
Gauge theory
------------
Matrix quantum mechanics studied in this work is obtained from pure Yang-Mills theory in $d+1$ dimensions by a dimensional reduction ([i.e. ]{}by assuming that the fields depend only on time). The Lagrangian of the $d+1$ dimensional theory is given by $$L_{d+1}=-\frac 1 4 {\mathrm{Tr}\left[ {F_{\mu,\nu}F^{\mu,\nu}} \right]}$$ where the repeated indices $\mu, \nu=0,1,\ldots, d$ are summed over. The field strength is $$F_{\mu,\nu}=\partial_{\mu}A_{\nu}
-\partial_{\nu}A_{\mu}
+i [A_{\mu}, A_{\nu}].$$ Gauge field components $A_{\mu}$ are associated with the Lie algebra of the gauge group; for $SU(N)$, they are represented as traceless Hermitian $N\times N$ matrices. The gauge transformation by an $SU(N)$ element $U$ is given by $$A_{\mu}\mapsto i (\partial_{\mu} U)U^{-1} +U A_{\mu} U^{-1},
\quad F_{\mu,\nu}\mapsto U F_{\mu,\nu} U^{-1},$$ and the Lagrangian is invariant under this transformation. The Lagrangian of matrix quantum mechanics (Yang-Mills theory in $0+1$ dimensions) is obtained by setting the spatial derivatives to zero. Writing $A_{0}=A$ and $A_{i}=\mathfrak{X}^{(i)}$, the field strengths become $$F_{0i}=D_{0} \mathfrak{X}^{(i)}=\dot{\mathfrak{X}}^{(i)}+i [A, \mathfrak{X}^{(i)}],
\quad F_{ij}=i[\mathfrak{X}^{(i)}, \mathfrak{X}^{(j)}],$$ and we obtain $$L = {\mathrm{Tr}\left[ {\frac 1 2\left(\dot{\mathfrak{X}}^{(i)}
+i[A, \mathfrak{X}^{(i)}]\right)^2
+ \frac 1 4 [\mathfrak{X}^{(i)}, \mathfrak{X}^{(j)}]^2} \right]},$$ where the repeated indices $i, j=1, \ldots, d$ are summed over. The infinitesimal form of a gauge transformation for $U=e^{-i\Lambda}$ is $$\delta A=\dot{\Lambda} +i[A, \Lambda],
\quad \delta \mathfrak{X}^{(i)}=i[\mathfrak{X}^{(i)}, \Lambda].
\label{eq:gaugetr}$$ In $0+1$ dimension, the gauge field $A$ is not a dynamical field, since it has no kinetic term. We can set $A$ to zero by a gauge transformation; $A$ only plays the role of Lagrange multiplier, which imposes a constraint (equation of motion w.r.t. $A$) on the system. We will perform the canonical quantisation in the $A=0$ gauge. Consider each real number of the matrix elements as a dynamical variable. The trace part represents the center of mass. The corresponding momentum is conserved, and its dynamics is decoupled from the rest. So we will concentrate on the study of the relative motion.
For the explicit analysis, it is convenient to expand the fields in the basis $t_a$, $$\mathfrak{X}^{(i)}=\sum_{a=1}^{N^2-1}x_{a}^{(i)}t_a,
\quad A=\sum_{a=1}^{N^2-1}a_{a}t_a$$ where $t_{a}$ are traceless Hermitian matrices which satisfy $${\mathrm{Tr}\left[ {t_{a}t_{b}} \right]}=\frac 1 2 \delta_{a,b},
\quad [t_a, t_b]=i\sum_{c}f_{a,b,c}t_c,$$ and where $f_{a,b,c}$ is the structure constant of $SU(N)$. For the $SU(2)$ gauge group, studied in the main text, we have $t_a=\sigma_{a}/2$ and $f_{a,b,c}=\epsilon_{a,b,c}$. Gauge transformation for $x_{a}^{(i)}$ with $\Lambda=\sum_{a} \lambda_{a}t_{a}$ is $$\delta x^{(i)}_{a}= \sum_{b,c} f_{a,b,c}\lambda_{b}x^{(i)}_{c} .
\label{eq:xgauge}$$ The conjugate momenta for $x_{a}^{(i)}$ are $$p_{a}^{(i)}=\frac{\partial L}{\partial\dot{x}_{a}^{(i)}}
=\frac{\partial}{\partial\dot{x}_{a}^{(i)}}\sum_{a}
\frac{(\dot{x}_{a}^{(i)})^2}{4}=\frac 1 2 \dot{x}_{a}^{(i)},$$ and the Hamiltonian is $$H=\sum_{a}(p_{a}^{(i)})^{2}
+\frac 1 8 \sum_{a,b,c,d,e}
f_{a,b,c}f_{a,d,e}x^{(i)}_{b}x^{(j)}_{c}x^{(i)}_{d}x^{(j)}_{e}$$ For $SU(2)$, by using $\sum_{a=1}^{3} \epsilon_{a,b,c}\epsilon_{a,d,e} =\delta_{b,d}\delta_{c,e} -\delta_{b,e}\delta_{c,d}$, and redefining $x_{a}^{(i)}\mapsto 2^{1/3}x_{a}^{(i)}$, $p_{a}^{(i)}\mapsto 2^{-1/3}p_{a}^{(i)}$ and $H\mapsto 2^{-4/3}H$ we get eq. . The equation of motion for $A$ (obtained by varying the Lagrangian by $a_{a}$ and setting $a_{a}=0$ afterwards) is $$0= \frac{\partial L}{\partial a_{a}}
=\sum_{b,c}f_{a,b,c}p_{b}^{(i)}x_{c}^{(i)} \equiv V_a.
\label{eq:aeom}$$ Noether’s theorem tells us that the $V_a$ are the generators of $SU(N)$ transformations. We impose constraint by requiring that the physical state vectors ${|\psi\rangle}$ are annihilated by the generators $V_{a}$, $$V_{a}{|\psi\rangle}=0.$$ In other words, $|\psi\rangle$ are singlets of $SU(N)$.
BFSS matrix theory
------------------
Banks-Fischler-Shenker-Susskind (BFSS) matrix theory is given by the dimensional reduction of conventional $U(N)$ Yang-Mills theory with $d=9$ as described above, supplemented by fermionic degrees of freedom, which make the theory supersymmetric. The bosonic degrees of freedom are $N\times N$ Hermitian matrices $\mathfrak{X}^{(i)}$, $i=1,\ldots, 9$. Their supersymetric partners are fermionic $N\times N$ matrices $\Theta_\alpha=\sum_a \theta_{\alpha,a} t_a$, where $\theta_{\alpha, a}$ ($\alpha=1, \ldots, 16$) are are Majorana-Weyl spinors in $9+1$ dimensions which have $16$ real components. This matrix quantum mechanics with the maximal amount of supersymmetry has been proposed as a formulation of M-theory, the strong coupling limit of type IIA string theory in (9+1) dimensions [@BFSS96].
The requirement of $SO(9)$ symmetry and maximal supersymmetry determines the action uniquely[^2]. There is no mass term, and the interaction among $\mathfrak{X}^{(i)}$ is quartic in the form of the trace of a commutator squared. There are also cubic interactions involving two fermions and one boson. The only parameter in the theory is the Yang-Mills coupling $g_{YM}$ (which has a dimension of (length)$^{-3/2}$, being proportional to the square root of the string coupling $g_s$). It appears in the action as an overall factor, $1/g_{\rm YM}^2$. In our analysis, we have set $g_{YM}=1$, since it can be scaled away by redefinitions of time and the fields.
In string theory, the model is interpreted as the description of a collection of $N$ so-called D0-branes at distances shorter than the string scale $\ell_s$, when their velocities are small. The diagonal elements of the matrices $\mathfrak{X}^{(i)}$ denote the position of the D0-branes, and their off-diagonal elements are interpreted as the lowest modes of open strings stretched between the D0-branes. The $U(1)$ part ($\mathfrak{X}^{(i)}$ being proportional to the unit matrix) corresponds to the center of mass motion, and is decoupled from the rest. Thus one can concentrate to the case of traceless matrices with $SU(N)$ gauge symmetry.
D-branes have features that are unfamiliar from ordinary systems due to the presence of the off-diagonal elements. First, the spatial coordinates are non-commutative. Second, there is $U(N)$ gauge symmetry, which means that the configurations that are related by $U(N)$ transformation are to be identified. This is in a sense a generalisation of fermionic and bosonic statistics. The consequences of these features are not fully understood.
Some evidence exists that the propose matrix quantum mechanics is a formulation of M-theory. M-theory is defined in a $(10+1)$ dimensional spacetime, where one of the spatial directions is compactified into a circle. The radius of compactification is $g_s \ell_s$, so in the strong coupling limit, the circle decompactifies. A D0-brane is considered to be a Kaluza-Klein mode which has one unit of momentum in the compactified direction. The BFSS conjecture is that the $N \to \infty$ limit with $g_s$ fixed describes M-theory in the infinite momentum frame (light-cone frame). In the light-cone frame the motion becomes non-relativistic, which gives a justification for the use of non-relativistic matrix quantum mechanics for its description. Another indication which suggest the connection between this theory and M-theory is that matrix quantum mechanics has an interpretation as a regularisation of super-membranes, which are believed to be important degrees of freedom in M-theory and are known to exist in $(10+1)$ dimensions.
Some progress has been made which supports the interpretation of matrix theory as a theory of gravity. Many questions remain unanswered though, mainly because the model is very difficult to solve. Interactions between separated D0-branes have been studied by computing quantum effective actions around block diagonal configurations of matrices and shown to agree with the tree-level interactions of supergravity [@OkYo99; @Taylor]. In these studies, the distance $r$ between D0-branes introduces a mass scale, which allows one to compute the effective potential (or the phase shift in the Born-Oppenheimer approach) as a perturbative expansion in powers of $1/r$.
Supersymmetry plays a crucial role in the agreement of matrix and gravity calculations. Open string modes stretched between D0-branes have mass proportional to $r$, which gives rise to linearly increasing effective potentials when integrated out. Due to non-trivial cancellations of bosonic and fermionic fluctuations one can obtain a behaviour similar to massless gravitons, $1/r^{d-2}$ in $d$ spatial dimensions. (When D0-branes are at rest with respect to each other, exact cancellation occurs and there is no force between them.) In the bosonic model that we study, the effective potential grows linearly, and all the states will be bound states.
For an interpretation as M-theory it is needed that the maximally supersymmetric matrix models with any $N$ has a normalisable ground state with exactly zero energy. This has been shown for $N=2$ in ref. [@SeSt98] by computing the Witten index. There are also attempts to construct the ground state wave functions. See ref. [@Yin10] for a recent work. It is known that the theory has a continuous spectrum [@DLN89]. It has been considered to be a problem before the appearance of the BFSS conjecture since it suggests super-membranes are unstable, but now the continuous spectrum is regarded as a consequence of the theory describing a many-body system.
Black holes should appear in matrix theory in the form of localised excited states. Studies of the theory at finite temperature have been performed using Monte Carlo methods. Internal energy and the spatial extent of the bound state have been computed and shown to be consistent with black hole thermodynamics [@Nis98; @Nis2]. Monte Carlo studies have been applied also to correlation functions in the zero-temperature limit [@Han10; @Hanada], and the results agree in high precision with the prediction of the gauge/gravity correspondence. Interestingly, the results with $N$ as small as $2$ or $3$ agree with gravity results, too, which are, however, supposed to be valid mainly in the large $N$ limit.
Since we have strong evidence for the validity of matrix theory as a description of black holes, an important next step would be to describe black holes using a pure state and follow their dynamical evolution. The most obvious question about black holes concerns the microscopic origin of its entropy. Another question is the derivation of fast scrambling [@HaPr07; @Susskind] (thermalisation in a time logarithmic in the number of degrees of freedom, which is faster than in ordinary local field theory) from a fundamental theory. It is expected that matrix models realise fast scrambling because of the non-locality of interaction in the space of matrix elements, but no analysis of quantum dynamics has been made so far. (See ref. [@BerensteinClassical] for an interesting study of thermalisation in the classical limit.)
[^1]: The $d=2$ model is given by a dimensional reduction of pure Yang-Mills theory in $2+1$ dimensions (see ref. [@Leigh] for a recent study). The supersymmetric version of the $d=3$, $N=2$ model has been studied in ref. [@Wosiek] using a method similar to ours.
[^2]: If one breaks the SO(9) symmetry to $SO(3)\times SO(6)$, a maximally supersymmetric action with mass term exists, which is called the Berenstein-Maldacena-Nastase (BMN) matrix model [@BMN]. Application of our method to this kind of mass-deformed matrix models is an interesting subject for future investigations.
|
---
abstract: 'In this paper, we investigate the use of intelligent reflecting surfaces (IRSs) (i.e., smart mirrors) to relax the line-of-sight requirement of free space optical (FSO) systems. We characterize the impact of the physical parameters of the IRS, such as its size, position, and orientation, on the quality of the end-to-end FSO channel. In addition, we develop a statistical channel model for the geometric and the misalignment losses which accounts for the random movements of the IRS, transmitter, and receiver due to building sway. This model can be used for performance analysis of IRS-based FSO systems. Our analytical results shows that depending on the angle between the beam direction and the IRS plane, building sway for the IRS has either a smaller or larger impact on the quality of the end-to-end FSO channel than building sway for the transmitter and receiver. Furthermore, our simulation results validate the accuracy of the developed channel model and offer insight for system design.'
author:
- |
Marzieh Najafi$^{\dag}$ and Robert Schober$^{\dag}$\
$^{\dag}$Friedrich-Alexander University of Erlangen-Nuremberg, Germany
bibliography:
- 'My\_Citation\_01-05-2019.bib'
title: Intelligent Reflecting Surfaces for Free Space Optical Communications
---
Introduction
============
Intelligent reflecting surfaces (IRSs) have drawn considerable attention recently since they can be used to alter the radio frequency (RF) wireless channel for improved communication perfromance [@Rui_Zhang_RS; @MetaSurf_Akyldiz; @Vahid_RS; @Alex_RS]. For example, IRSs have been used to extend the coverage of wireless communication systems to blind spots [@Rui_Zhang_RS; @Alouini_RS] and to increase their security by improving the channel quality of the legitimate link and deteriorating the channel quality of the eavesdropper link [@Alex_RS]. Furthermore, IRSs are energy- and cost-efficient since they are composed of passive elements and can be installed on existing infrastructure, e.g., building walls.
Optical wireless systems, e.g., free space optical (FSO) systems, are a promising candidate to meet the high data rate requirements of the next generation of wireless systems and beyond [@FSO_Survey_Murat; @Steve_pointing_error; @Alouini_Pointing; @ICC_2018]. FSO systems offer the large bandwidth needed for applications such as wireless backhauling, while their transceivers are relatively cheap compared to their RF counterparts and easy to implement. However, the main requirement for establishing an FSO link is the existence of a line-of-sight (LOS) between the transceivers [@FSO_Survey_Murat]. To relax this restrictive requirement, in this paper, we propose to use IRSs (smart mirrors) in FSO systems. Similar to RF-based IRSs, the IRSs in FSO systems can be installed on the walls of buildings. In RF systems, IRSs have to be equipped with a large number of passive phase shifters in order to create a narrow beam and to adaptively change the direction of the reflected beam to track mobile users [@Rui_Zhang_RS; @Alouini_RS; @Vahid_RS; @Alex_RS]. In FSO systems, simple mirrors can be used to efficiently redirect the beam with negligible scattering [@Datasheet_Mirror]. Moreover, intelligent mirrors (i.e., optical IRSs) are able to control the direction of the reflected beam. This can be accomplished either by mechanically rotating the IRS or by electronically changing the wavefront using advanced optical metasurfaces [@OpticMetaSurf2; @OpticMetaSurf3]. In this paper, we consider the former case.
Employing reflecting surfaces (RSs) (mirrors) in FSO systems has been widely considered in the literature [@brandl2013optical; @Mirror_UAV_Exp; @FSO_Mirror]. Mostly, RSs are used in the transceiver architecture in order to guide the optical beam in a desired direction [@brandl2013optical]. Another example of optical RSs is the passive retro-reflector which reflects the incoming laser beam back to its source and the reflected beam is modulated to carry data, see [@Mirror_UAV_Exp] for an experimental demonstration of a retro-reflector. Furthermore, in [@FSO_Mirror], the concept of using IRSs in FSO links was presented as a cost-effective solution for backhauling of cellular systems. However, the focus of [@FSO_Mirror] was on network planning and the impact of IRSs on the FSO channel model was not studied.
In this paper, we characterize the FSO channel between a transmitter (Tx), an IRS, and a receiver (Rx) as a function of the area, position, and orientation of the IRS. In particular, we derive the geometric and misalignment losses (GML) of the end-to-end link, i.e., the Tx-to-IRS-to-Rx link. Moreover, since, in addition to the Tx and Rx, the IRS is also affected by random movements due to building sway, we develop a statistical channel model which accounts for the impact of building sway for all three nodes. This model can be used to analyse the perfomance of IRS-based FSO systems. Our simulation results validate the proposed channel model and offer insight for system design.
Preliminaries
=============
System Model
------------
We consider an FSO communication system, where a Tx wishes to communicate with an Rx via an FSO link. We assume that there is no LOS between Tx and Rx. Hence, communication is enabled with the help of an IRS which has a LOS to both the Tx and the Rx. In other words, we assume that the Tx has an aperture directed towards the IRS; the IRS reflects the optical beam that it receives to the Rx; and the Rx collects the optical energy with a photo detector (PD).
Channel Model
-------------
We assume an intensity modulation/direct detection (IM/DD) FSO system, where the PD responds to changes in the received optical signal power [@FSO_Survey_Murat]. Moreover, we assume that background noise is the dominant noise source at the PD and therefore the noise is independent from the signal [@FSO_Survey_Murat]. The received signal at the Rx, denoted by $y_s$, is given by
[lll]{}\[Eq:signal\] y\_s=hx\_s+n,
where $x_s\in \mathbb{R}^+$ is the transmitted optical symbol (intensity), $n\in \mathbb{R}$ is the zero-mean real-valued additive white Gaussian shot noise with variance $\sigma_n^2$ caused by ambient light at the Rx, and $h\in \mathbb{R}^+$ denotes the FSO channel gain. Moreover, we assume an average power constraint $\mathbbmss{E}\{x_s\}\leq{P}$.
The FSO channel coefficient, $h$, is affected by several phenomena and can be modeled as [@Steve_pointing_error]
[lll]{}\[Eq:channel\] h=h\_p h\_a h\_g,
where $\eta$ is the responsivity of the PD and $h_p$, $h_a$, and $h_g$ represent the atmospheric loss, atmospheric turbulence induced fading, and GML, respectively. In particular, the atmospheric loss, $h_p$, represents the power loss over a propagation path due to absorption and scattering of the light by particles in the atmosphere. The atmospheric turbulence, $h_a$, is induced by inhomogeneities in the temperature and the pressure of the atmosphere [@FSO_Survey_Murat]. The GML, $h_g$, is caused by the divergence of the optical beam along the propagation distance and the misalignment of the laser beam line[^1] and the PD center due to building sway[@FSO_Survey_Murat; @FSO_Vahid]. In this paper, our goal is to mathematically determine the impact of the IRS on the quality of the FSO channel.
Problem Statement
-----------------
The impact of IRS on the end-to-end FSO channel is reflected in $h_p$ and $h_g$ which will be disscussed in the following:
*i) Quality of reflection:* In addition to reflection, practical IRSs may also absorb or scatter some fraction of the beam power. Let $\zeta$ denote the reflection efficiency, i.e., the fraction of power reflected by the IRS. For practical IRSs, $\zeta$ usually assumes values in the range $[0.95,1]$ [@Datasheet_Mirror]. The absorption at the IRS can be regarded as a part of the atmospheric loss $h_p$.
*ii) Relative position, orientation, and size of IRS:* The relative position and orientation of the IRS with respect to (w.r.t.) the laser beam determines the distribution of the *reflected* optical power in space. The relative position and orientation of the PD w.r.t. the IRS determines the fraction of this power collected by the PD. Moreover, the size of the IRS determines which part of the PD is covered by the reflected beam. These parameters affect the *mean* of the GML $h_g$.
*iii) Building Sway:* The IRS is affected by the random movements of the building that it is installed on. This further increases the beam misalignment and affects the statistics of the GML $h_g$. In other words, the building sway of the buildings on which the Tx, Rx, and IRS are installed creates randomness in $h_g$.
Based on the above discussion, quantifying the impact of the IRS on the end-to-end FSO channel reduces to characterizing the corresponding GML $h_g$. To do so, we develop both a conditional model that accounts for the position, orientation, and size of the IRS and a statistical model that accounts for the random fluctuations of the IRS position due to building sway. As is customary for the analysis of optical systems [@VLC], we first consider a two dimensional (2D) system model. The impact of the position, orientation, and building sway on $h_g$ can also be observed in a 2D system model. We generalize our model to a 3D system model in Section V based on the insights gained from analyzing the 2D system model.
Optical Power Collected by PD
=============================
Geometry of the Considered System
---------------------------------
We first define the position and orientation of the laser source (LS), the IRS, and the PD in the considered 2D system model, which is schematically illustrated in Fig. \[Fig:Model\]. Without loss of generality, we assume that the LS is located in the origin of the coordinate system, i.e., $\mathbf{p}_{\mathrm{ls}}=(0,0)$. The center of the IRS and the PD are located at $\mathbf{c}_{\mathrm{rs}}=(x_r,y_r)$ and $\mathbf{c}_{\mathrm{pd}}=(x_p,y_p)$, respectively. The lengths of the IRS and the PD are denoted by $2a_r$ and $2a_p$, respectively. The direction of the laser beam is determined by the angle between the laser beam line and the $x$ axis denoted by $\theta_b$. The IRS and the PD have angles $\theta_{rs}$ and $\theta_p$ w.r.t. the $x$ axis, repectively. For convenience, the LS can be mirrored at the line defined by the IRS, cf. Fig. \[Fig:Model\], and the resulting virtual LS can be used in the subsequent analysis [@Phys]. The virtual laser beam has an angle of $2\theta_{rs}-\theta_b$ with the $x$ axis and the position of the virtual LS is given by
[lll]{} \_=(x\_[vs]{}, y\_[vs]{})=&((1-(2\_[rs]{}))x\_r-(2\_[rs]{})y\_r,\
&(1+(2\_[rs]{}))y\_r-(2\_[rs]{})x\_r).
Spatial Distribution of the Reflected Power Density
---------------------------------------------------
Next, we derive the power density of the reflected beam across space. We assume a Gaussian beam which dictates that the power density distribution across any line perpendicular to the direction of the wave propagation follows a Gaussian profile [@FSO_Survey_Murat; @Steve_pointing_error]. Let us consider a line that is perpendicular to the beam direction and the distance between the center of the beam footprint on the line and the LS is denoted by $d$. Then, the power density for any point on this perpendicular line with distance $r$ from the center of the beam footprint is given by [@Steve_pointing_error]
[lll]{} \[Eq:PowerOrthogonal\] I\^(r;d) = (-),
where $w(d)$ is the beam width at distance $d$ and is given by
[lll]{} \[Eq:BeamWidth\] w(d)= w\_0.
Here, $w_0$ denotes the beam waist radius, $\rho(d)=(0.55C_n^2k^2d)^{-3/5}$ is referred to as the coherence length, $k=\frac{2\pi}{\lambda}$ is the wave number, $\lambda$ denotes the optical wavelength, and $C_n^2\approx C_0\exp\left(-\frac{h}{100}\right)$ is the index of refraction structure parameter, where $C_0=1.7\times10^{-14}$ m$^{\frac{2}{3}}$ is the nominal value of the refractive index at the ground and $h$ is the operating height of the FSO transceivers [@FSO_Survey_Murat]. The following lemma provides the power density of the *beam reflected* by the IRS.
\[Lem:Truncated Gauss\] Assuming a transmitted Gaussian beam, the power density of the reflected beam on a perpendicular line w.r.t. the beam direction at point $\tilde{\mathbf{p}}=(\tilde{x},\tilde{y})$ is given by
[lll]{} \[Eq:Truncated Gauss\] I\^\_()=
(-), &\
0, &,
where $r=\|\tilde{\mathbf{p}}-\mathbf{p}_{\mathrm{c}}\|$ and $d=\|\mathbf{p}_{\mathrm{vs}}-\mathbf{p}_{\mathrm{c}}\|$ are the distances between the beam footprint center and point $\tilde{\mathbf{p}}$ and the LS, respectively. Region $\mathcal{R}$ is defined as $\mathcal{R}=\{(x,y)|s_1(x-x_{vs})+y_{vs}\leq y\leq s_2(x-x_{vs})+y_{vs}\}$, where $s_1=\frac{y_r-\sin(\theta_{rs})a_r-y_{vs}}{x_r-\cos(\theta_{rs})a_r-x_{vs}}$ and $s_2=\frac{y_r+\sin(\theta_{rs})a_r-y_{vs}}{x_r+\cos(\theta_{rs})a_r-x_{vs}}$ and $\mathbf{p}_{\mathrm{c}}$ is given by
[lll]{} \[Eq:xcyc\] \_\^ =&
(\_b-2\_[rs]{}) &1\
-(\_b-2\_[rs]{})& 1
\^[-1]{}
(\_b-2\_[rs]{})x\_[vs]{}+y\_[vs]{}\
-(\_b-2\_[rs]{})+
.\
Lemma \[Lem:Truncated Gauss\] provides several insights regarding the impact of the IRS on the reflected power distribution. In particular, the reflected beam is a *truncated* Gaussian beam which originates from the virtual LS and is confined to area $\mathcal{R}$. Moreover, the size of $\mathcal{R}$ depends on the size of the IRS as well as on its relative orientation w.r.t. the laser beam. Furthermore, for a given $d$, $I^{\mathrm{orth}}_{\mathrm{rfl}}(\tilde{\mathbf{p}})$ attains its maximum, i.e., $\frac{2}{\pi w^2(d)}$, at $r=0$, i.e., at the center of its footprint $\mathbf{p}_{\mathrm{c}}$, cf. . Note that $\mathbf{p}_{\mathrm{c}}$ depends on $\theta_{rs}$ and the value of point $\tilde{\mathbf{p}}$. Therefore, for $\tilde{\mathbf{p}}$ on the PD, for an efficient design, we should choose $\theta_{rs}$ such that $\mathbf{p}_{\mathrm{c}}$ lies in the center of the PD $\mathbf{c}_{\mathrm{pd}}=(x_p,y_p)$ and distance $d$ is the end-to-end distance between the LS and the PD, $d=d_{sr}+d_{rp}\triangleq d_{e2e}$, where $d_{sr}$ and $d_{rp}$ denote the distances between the LS to the IRS and the IRS to the PD, respectively. This leads to the optimal $\theta_{rs}^*$ which is found as the unique solution of the following equation
[lll]{} \[Eq:OptimalAngle\] (2\_[rs]{}\^\*-\_b) =.
Conditional GML Model
---------------------
In order to compute the GML, we have to integrate the reflected power density over the PD, i.e.,
[lll]{} \[Eq:Integral\] h\_g= \_ ()I\^\_() ,
where $\psi=\theta_b+\theta_p-2\theta_{rs}$ is the angle between the PD and the beam line and $\mathcal{P}$ is the set of points on the PD, i.e.,
[lll]{}=&{(x,y)|y=(\_p)(x-x\_p)+y\_p, x, y}.\
The term $\sin(\psi)\in[0,1]$ in accounts for the non-orthogonality of the PD. Let $L_c$ denote the distance between the center of the PD and the beam line, i.e., $L_c=\|\mathbf{c}_{\mathrm{pd}}-\mathbf{p}_{\mathrm{c}}\|$ for $d=d_{e2e}$. The following proposition provides a closed-form expression for the GML $h_g$. For future reference, $\mathbf{a}\leq \mathbf{b}$ indicates that all elements of $\mathbf{a}$ are smaller than the corresponding elements in $\mathbf{b}$.
\[Prop:TotalFrac\] Under the mild condition $a_p,L_c\ll d_{e2e}$, the total fraction of power that is captured by the PD is given by
[lll]{} \[Eq:Pt\] h\_g=\
()+(),&\_[12]{}=2a\_p\
|()-()|,&
where $\rho_1=\|\mathbf{p}_0-\hat{\mathbf{p}}_1\|$, $\rho_2=\|\mathbf{p}_0-\hat{\mathbf{p}}_2\|$, and $\rho_{12}=\|\mathbf{p}_0-\mathbf{p}_1\|+\|\mathbf{p}_0-\mathbf{p}_2\|$. Moreover, $\mathbf{p}_0$, $\mathbf{p}_1$, $\hat{\mathbf{p}}_1$, $\mathbf{p}_2$, and $\hat{\mathbf{p}}_2$ are given in at the top of the next page.
[ccc]{} \[Eq:r\_1\] \_0\^ =
(\_b-2\_[rs]{}) &1\
-(\_p)& 1
\^[-1]{}
(\_b-2\_[rs]{})x\_[vs]{}+y\_[vs]{}\
-(\_p)x\_p+y\_p
,\_1\^ =
x\_p+a\_p(\_p)\
y\_p+a\_p(\_p)
,\_1\^ =
-s\_2 &1\
-(\_p)& 1
\^[-1]{}\
-s\_2x\_[vs]{}+y\_[vs]{}\
-(\_p)x\_p+y\_p
,\_2\^ =
x\_p-a\_p(\_p)\
y\_p-a\_p(\_p)
,\_2\^ =
-s\_1 &1\
-(\_p)& 1
\^[-1]{}
-s\_1x\_[vs]{}+y\_[vs]{}\
-(\_p)x\_p+y\_p
,\
\_1 =
\_2, & \_1<\_2\
\_1, & \_2\_1\_1\
\_1, & \_1>\_1
,\_2 =
\_2, & \_2<\_2\
\_2, & \_2\_2\_1\
\_1, & \_2>\_1.
------------------------------------------------------------------------
Note that the conditions under which in Proposition \[Prop:TotalFrac\] holds are met in practice since 1) the physical size of the PD is much smaller than the transmission distance, i.e., $a_p\ll d_{e2e}$ holds, and 2) $L_c$ corresponds to the beam misalignment and for a properly designed system, the misalignment is much smaller than the end-to-end transmission distance, i.e., $L_c\ll d_{e2e}$ holds. The impact of the size of the IRS is reflected in the values of $\rho_{1}$ and $\rho_{2}$. In fact, if the IRS is sufficiently large such that the PD is located in region $\mathcal{R}$ defined in Lemma \[Lem:Truncated Gauss\], we obtain $\rho_1=\|\mathbf{p}_0-\mathbf{p}_1\|$, $\rho_2=\|\mathbf{p}_0-\mathbf{p}_2\|$.
\[Corol:Perp\] For the special case where $\hat{\mathbf{p}}_i=\mathbf{p}_i, i=1,2$, i.e., the IRS is sufficiently large, and the reflected beam strikes the center of the PD and its direction is perpendicular to the PD, the total fraction of power that is captured by the PD is obtained as
[lll]{} \[Eq:PtCorol\] h\_g=().
Eq. is obtained by substituting $\psi=\frac{\pi}{2}$ and $\rho_1=\rho_2=\|\mathbf{p}_0-\mathbf{p}_1\|=\|\mathbf{p}_0-\mathbf{p}_1\|=a_p$ into . This completes the proof.
For a given end-to-end distance $d_{e2e}$ and a given PD area $a_p$, the maximum fraction of power collected by the PD is given by . To attain this maximum, three conditions have to hold, namely the IRS is sufficiently large, the misalignment is zero, i.e., $\theta_{rs}=\theta_{rs}^*$, cf. , and the PD is orthogonal to the beam line, i.e., $\theta_p=\frac{\pi}{2}+2\theta_{rs}-\theta_b$.
Statistical Model - 2D System
=============================
In this section, we study the effect that building sway has on the quality of the considered FSO channel.
Building Sway Model
-------------------
We assume that the positions of the LS, IRS, and PD fluctuate because of building sway in both the $x$ and $y$ directions. In the following, we show that for the LS, IRS, and PD only the fluctuations in a certain direction have a considerably impact on the FSO channel, respectively. This observation substantially simplifies the derivation of a statistical channel model.
**LS:** The fluctuations of the position of the LS can be projected in the beam direction and the direction orthogonal to it. Let $\epsilon_s^b$ and $\epsilon_s^o$ denote the fluctuations of the LS position for the former and latter cases, respectively. Hereby, since the fluctuations of the LS in the beam direction are much smaller than the distance between the LS and the IRS, the impact of $\epsilon_s^b$ on $h_g$ can be safely neglected.
**IRS:** The fluctuations of the position of the IRS can be projected in the direction along the IRS line and the orthogonal direction denoted by $\epsilon_r^{r}$ and $\epsilon_r^{o}$, respectively. Assuming that the beam line is aligned to pass through the IRS (not necessarily its center) and that the size of the IRS is large, the impact of $\epsilon_r^{r}$ on $h_g$ is negligible. Nevertheless, $\epsilon_r^{o}$ may considerably change the position of the beam footprint center at the PD.
**PD:** Similar to the LS, let $\epsilon_p^b$ and $\epsilon_p^o$ denote the fluctuations of the position of the PD in the direction of the reflected beam and perpendicular to it, respectively. Since the distance between the IRS and the PD is much larger than the fluctuations in the reflected beam direction, we can safely neglect the impact of $\epsilon_p^b$ on $h_g$.
Let $u$ denote misalignment between the center of the beam footprint and the center of the PD. $u$ is given in the following lemma.
\[Lem:Misalignment\] The misalignment $u$ as a function of $(\epsilon_s^o,\epsilon_r^o,\epsilon_p^o)$ is obtained as
[lll]{} \[Eq:Misalignment\] u=(\_s\^o+2(\_b-\_[rs]{})\_r\^o+\_p\^o).
In , the term $\epsilon_s^o+2\cos(\theta_b-\theta_{rs})\epsilon_r^o+\epsilon_p^o$ captures the misalignment on a plane perpendicular to the direction of the reflected beam and the term $\frac{1}{\sin(\psi)}$ accounts for the non-orthogonality of the PD. Moreover, the fluctuations of the LS and PD are projected onto the perpendicular misalignment without any change, whereas the projection of the fluctuations of the IRS onto the perpendicular misalignment depends on angle $\theta_b-\theta_{rs}$ as given in . This completes the proof.
Note that $(\epsilon_s^o,\epsilon_r^o,\epsilon_p^o)$ are random variables (RVs). A widely-accepted model for building sway assumes independent zero-mean Gaussian fluctuations [@Steve_pointing_error; @ICC_2018], i.e., $\epsilon_s^o\sim\mathcal{N}(0,\sigma^2_s)$, $\epsilon_r^o\sim\mathcal{N}(0,\sigma^2_r)$, and $\epsilon_p^o\sim\mathcal{N}(0,\sigma^2_p)$, where $\sigma_i^2$ denotes the variance of $\epsilon_i^o,\,\,i\in\{s,r,p\}$. Therefore, the misalignment also follows a zero-mean Gaussian distribution, i.e., $u\sim\mathcal{N}\big(0,\sigma^2\big)$ with variance $\sigma^2=\frac{1}{\sin^2(\psi)}(\sigma_s^2+4\cos^2(\theta_b-\theta_{rs})\sigma_r^2+\sigma_p^2)$.
PDF of Power Collected by the PD
--------------------------------
In order to derive the statistical channel model for the GML $h_g$, first the power collected by the PD has to be derived as a function of $u$. To do so, we can use the exact expressions in and replace $(\rho_1,\rho_2)$ with $(|u-a_p|,u+a_p)$, assuming that the IRS is sufficiently large such that the PD is located in region $\mathcal{R}$ defined in Lemma \[Lem:Truncated Gauss\]. However, the resulting expressions are rather complicated and do not provide useful insights. Thus, to get some insights, we approximate $h_g$ as a function of $u$ as follows
[lll]{} \[Eq:A0Approx\] h\_gA\_0 (),
where $A_0=\frac{\sqrt{2}}{\sqrt{\pi}w(d_{e2e})}\mathrm{erf}\left(\nu\right)$, $t=\frac{\sqrt{\pi}\mathrm{erf}\left(\nu\right)}{2\nu\exp(-\nu^2)\sin^2(\psi)}$, and $\nu=\frac{\sqrt{2}\sin(\psi)a_p}{w(d_{e2e})}$. The derivation of is provided in . We verify the accuracy of in Section VI. Using this approximation, the PDF of $h_g$ is given in the following proposition.
\[Prop:2\] Based on and and assuming Gaussian fluctuations, $h_g$ follows a distribution with the following PDF
[lll]{} \[Eq:PDFs\] f\_[h\_g]{}(h\_g)=\^[-]{}&()\^[-1]{},\
& 0h\_gA\_0.
where $\varpi=\frac{tw^2(d_{e2e})}{4\sigma^2}$.
Eq. can be obtained by exploiting the relation between the PDF of $u$ and $h_g$ in and the fact that $u$ follows a zero-mean Gaussian distribution.
Proposition \[Prop:2\] reveals the impact of system parameters such as $d_{e2e}$ and $\sigma^2$ on the PDF of the GML.
Extension to 3D System Model
============================
For the 2D system model, we needed two position variables and one angular variable to characterize the positions and orientations of the LS, IRS, and PD, respectively, i.e., in total 9 parameters. In contrast, for a 3D system model, we require three position variables and two angular variables to characterize the positions and orientations of the nodes, i.e., in total 15 parameters. This severely complicates the analysis of the 3D system. To cope with this issue, we exploit the insights gained from analyzing the 2D system and characterize the 3D system only w.r.t. those parameters that affect the GML $h_g$. From Sections II-IV, we offer the following observations:
- Lemma \[Lem:Truncated Gauss\] reveals that in 2D systems, the impact of the IRS can be modeled via a virtual LS where the reflected beam follows a truncated Gaussian profile. The position of the virtual LS depends on the relative position and orientation of the IRS w.r.t. the beam line. Nevertheless, the distance between the virtual LS and the PD is the sum of the distances between the actual LS to the IRS and the IRS to the PD, i.e., $d_{e2e}$. Moreover, the truncation can be ignored if the IRS is sufficiently large such that the PD is completely inside region $\mathcal{R}$ defined in Lemma \[Lem:Truncated Gauss\].
- The conditional model in reveals that the overall impact of the position and orientation parameters of the IRS and the PD on the GML $h_g$ manifests itself in three variables, namely misalignment $u$, end-to-end distance $d_{e2e}$, and angle $\psi$. Due to building sway, the misalignment $u$ is an RV; however, by a proper system design, i.e., by choosing $\theta_{rs}=\theta^*_{rs}$ according to , one can make the average misalignment $u$ vanish, i.e., $\mathbbmss{E}\{u\}=0$.
In the following, we exploit the two above observations for analyzing a 3D system. Let $\psi_p$ denote the angle between the reflected beam and the PD *plane*. Assuming a circular PD of radius $a_p$, the following approximate expression was recently obtained in [@ICC_2018] for the GML of a 3D system
[lll]{} \[Eq:hg\_3D\] h\_g(u) A\_0 (-),
where $t=\sqrt{t_1t_2}$, $t_1=\frac{\sqrt{\pi}\mathrm{erf}(\nu_1)}{2\nu_1\exp(-\nu_1^2)}$ , $t_2=\frac{\sqrt{\pi}\mathrm{erf}(\nu_2)}{2\nu_2\exp(-\nu_2^2)\sin^2(\psi_p)}$, $\nu_{1}=\frac{a_p}{w(d_{e2e})}\sqrt{\frac{\pi}{2}}$, and $\nu_{2}=\nu_{1}|\sin(\psi_p)|$. Moreover, $\mathbf{u}$ denotes the vector of misalignment on the PD plane, and $A_0$ denotes the maximum fraction of optical power captured by the PD at $\|\mathbf{u}\|=0$ and is given by $A_0=\mathrm{erf}(\nu_{1})\mathrm{erf}(\nu_{2})$. Note that the exact expression for $h_g$ can be obtained in a similar manner as that obtained in Proposition \[Prop:TotalFrac\] for 2D systems but is much more involved. In the following, we derive a statistical model based on incorporating the impact of the IRS.
Similar to the statistical analysis for 2D systems given in Section IV, we assume Gaussian fluctuations due to building sway for the LS, IRS, and PD as described in the following.
**LS:** In general, the fluctuations of the position of a point in a 3D system can be modeled by three variables in three orthogonal directions. For the LS, fluctuations along the direction of the beam have negligible impact on $h_g$; hence, we need only two variables in two orthogonal directions on the plane perpendicular to the beam direction, denoted by $\epsilon_s^{o_1},\epsilon_s^{o_2}\sim\mathcal{N}(0,\sigma_s^2)$.
**IRS:** Since we assume a sufficiently large IRS, the fluctuations of the IRS along its plane can be neglected. Therefore, we need to consider only the fluctuations orthogonal to the IRS plane, denoted by $\epsilon_r^{o}\sim\mathcal{N}(0,\sigma_r^2)$.
**PD:** Similar to the LS, the fluctuations along the reflected beam direction can be neglected. Hence, we need two variables in two orthogonal directions to describe the fluctuations in the plane perpendicular to the reflected beam, denoted by $\epsilon_p^{o_1},\epsilon_p^{o_2}\sim\mathcal{N}(0,\sigma_p^2)$.
It is interesting to note that the GML is affected by the IRS only via variable $\epsilon_r^{o}$. This implies that variations of $\epsilon_r^{o}$ lead to variations of $\mathbf{u}$ along only one dimension. Without loss of generality and to simplify our notation, we choose the basis for variables $(\epsilon_s^{o_1},\epsilon_s^{o_2})$ and $(\epsilon_p^{o_1},\epsilon_p^{o_2})$ such that the variations of $\mathbf{u}$ due to $\epsilon_s^{o}$ and $\epsilon_p^{o}$ are in the same direction as those due to $\epsilon_r^{o}$. Based on this convention, the following lemma presents the misalignment vector $\mathbf{u}$.
\[Lem:Misalignment3D\] The misalignment vector $\mathbf{u}$ as a function of $(\epsilon_s^{o_1},\epsilon_s^{o_2})$, $\epsilon_r^{o}$, and $(\epsilon_p^{o_1},\epsilon_p^{o_2})$ is obtained as
[lll]{} \[Eq:Misalignment3D\] =(\_s\^[o\_1]{}+2(\_[r]{})\_r\^o+\_p\^[o\_1]{}, \_s\^[o\_2]{}+\_p\^[o\_2]{}),
where $\psi_{r}$ is the angle between the laser beam and the IRS plane.
The proof is similar to that given for Lemma \[Lem:Misalignment\] for 2D systems. The convention for the definition of the bases for $(\epsilon_s^{o_1},\epsilon_s^{o_2})$ and $(\epsilon_p^{o_1},\epsilon_p^{o_2})$ facilitates the derivation of $\mathbf{u}$ since $\epsilon_r^o$ affects only one of the dimensions of $\mathbf{u}$.
Assuming $\mathbf{u}=(u_1,u_2)$, $u_1$ and $u_2$ follow Gaussian distributions with zero mean and variances $\sigma_{u_1}^2=\frac{1}{\sin^2(\psi_p)}(\sigma_s^2+4\cos^2(\psi_r)\sigma_r^2+\sigma_p^2)$ and $\sigma_{u_2}^2=\frac{1}{\sin^2(\psi_p)}(\sigma_s^2+\sigma_p^2)$, respectively. Therefore, $\|\mathbf{u}\|$ follows a Hoyt distribution which is given by [@Alouini_Pointing]
[lll]{} \[Eq:PDFu3D\] f\_(u)=u(-u\^2)I\_0(u\^2),\
where $q=\frac{\sigma_{u_2}}{\sigma_{u_1}}$, $\Omega=\sigma_{u_1}^2+\sigma_{u_2}^2$, and $I_0(\cdot)$ is the zero-order modified Bessel function of the first kind. For the special case where $\sigma_r^2=0$, $\|\mathbf{u}\|$ is Rayleigh distributed, similar to the pointing error caused by building sway for point-to-point FSO systems without IRS [@Alouini_Pointing; @Steve_pointing_error]. Exploiting and , the PDF of $h_g$ can be obtained as
[lll]{} \[Eq:PDF\_h3D\] f\_[h\_g]{}(h\_g) = & ()\^[-1]{}\
&I\_0(-()), 0< h\_g A\_0,
where $\varpi = \frac{(1+q^2)tw^2(d_{e2e})}{4q\Omega}$ is a constant and $\ln(\cdot)$ denotes the natural logarithm.
Simulation Results
==================
Unless stated otherwise, the default values of the parameter values used for 2D simulation are $\theta_b=\frac{\pi}{4}$, $\theta_{rs}=\frac{\pi}{10}$, $\theta_p=\frac{\pi}{3}$, $a_p=10$ cm, $a_r=50$ cm, $(x_r,y_r)=(400,400)$ m, and $(x_p,y_p)=(700,350)$ m. For 3D simulation, we use parameter values that are in-line with those for 2D, i.e., $\psi_r=\frac{\pi}{4}-\frac{\pi}{10}$, $\psi_p=\frac{\pi}{3}$, $d_{sr}=400\sqrt{2}$ m, $d_{rp}=50\sqrt{37}$ m, $a_p=10$ cm, and $a_r=50$ cm. Moreover, the simulation results reported in Fig. \[Fig:PDF\] were obtained based on Monte Carlo simulation and $10^6$ realizations of RVs $\epsilon_i^{j},i\in\{s,r,p\},j\in\{o,o_1,o_2\}$.
First, in Fig. \[Fig:CondGML\], we study the impact of the size of the IRS on the conditional GML in . In this figure, we show $h_g$ vs. misalignment $u$ for $a_r=50,100$ cm. As expected, we observe from Fig. \[Fig:CondGML\] that by increasing the misalignment magnitude ($|u|$), the channel gain $h_p$ decreases. Beam truncation occurs if the misalignment exceeds a cetrain critical value, i.e., when part of the PD is outside region $\mathcal{R}$, cf. . In Fig. \[Fig:CondGML\], we use dot-dashed (dashed) lines to denote this critical misalignment for $a_r=50$ cm ($a_r=100$ cm). Fig. \[Fig:CondGML\] shows that the proposed approximation in is accurate when beam truncation does not occur. However, since the approximation neglects beam truncation, it overestimates $h_g$ when beam truncation does occur. Moreover, we observe that, for $a_p=100$ cm, the impact of beam truncation manifests itself at larger values of $|u|$ compared to $a_p=50$ cm. Furthermore, Fig. \[Fig:CondGML\] shows that for a reasonable size of the IRS, i.e., $a_r>50$ cm, the proposed approximation is accurate even for large misalignment magnitudes, e.g. $|u|>35$ cm. Finally, we note that the PD receives no optical power, i.e., $h_g=0$, when none of the points on the PD surface belongs to $\mathcal{R}$, cf. Lemma \[Lem:Truncated Gauss\].
\[Fig:CondGML\]
Next, we study the accuracy of the proposed statistical models for 2D and 3D systems in and , respectively. For the simulation results, we plot the histogram of $h_g$ given by and for 2D and 3D systems, respectively. Fig. \[Fig:PDF\] shows the PDF of $h_g$ for three fluctuation scenarios, namely Scenario 1: $(\sigma_s,\sigma_p,\sigma_r)=(5,5,5)$ cm where the building sways for the LS, IRS, and PD are similar; Scenario 2: $(\sigma_s,\sigma_p,\sigma_r)=(5,5,10)$ cm where the building sway for PD is larger than that for the LS and IRS[^2]; Scenario 3: $(\sigma_s,\sigma_p,\sigma_r)=(5,10,5)$ cm where the building sway for the IRS is larger than that for the LS and PD. Fig. \[Fig:PDF\] shows an excellent agreement between the proposed analytical statistical models and the simulation results. This is due to the fact that the impact of beam truncation is negligible as it occurs with small probability for the adopted system parameters. Moreover, we can observe from Fig. \[Fig:PDF\] that the building sway for the IRS has a larger impact than that for the PD (and LS). This is due to the factor $2\cos(\psi_{r})=1.782$ in and which enhances the variance of the corresponding building sway.
\[Fig:PDF\]
Conclusions
===========
In this paper, we proposed IRS-based FSO systems in order to relax the LOS requirement of conventional FSO systems. We developed corresponding conditional and statistical channel models which characterize the impact of the physical parameters of the IRS, such as its size, position, and orientation, on the quality of the end-to-end FSO channel. These channel models can be used for performance analysis of IRS-based FSO systems. Simulation results confirmed the validity of the developed channel models for typical IRS sizes (i.e., $a_r>50$ cm) where beam truncation is negligible. Furthermore, our results showed that depending on the angle between the beam direction and the IRS plane, building sway for the IRS could have a larger impact on the quality of the end-to-end FSO channel than building sway for the Tx and Rx for angles smaller than $\pi/3$.
[^1]: The beam line is the line that connets the laser source with the center of the beam footprint.
[^2]: Scenario 2 yields the same results as scenario $(\sigma_s,\sigma_p,\sigma_r)=(10,5,5)$ cm due to the symmetry of the problem, see and .
|
---
abstract: 'The distribution of the number of points on abelian covers of ${\mathbb{P}}^1({\mathbb{F}}_q)$ ranging over an irreducible moduli space has been answered in a recent work by the author [@M1],[@M2]. The authors of [@BDFK+] determined the distribution over the whole moduli space for curves with ${\mbox{Gal}}(K(C)/K)$ a prime cyclic. In this paper, we prove a result towards determining the distribution over the whole moduli space of curves with ${\mbox{Gal}}(K(C)/K)$ any abelian group. We successfully determine the distribution in the case ${\mbox{Gal}}(K(C)/K)$ is a power of a prime cyclic.'
author:
- Patrick Meisner
bibliography:
- 'FullSpace.bib'
title: Number of Points on the Full Moduli Space of Curves over Finite Fields
---
Introduction {#intro}
============
Let ${\mathcal{H}}$ be a family of smooth, projective curves over ${\mathbb{F}}_q$, the finite field with $q$ elements. We are interested in determining the probability that a curve, chosen randomly from our family, has a given number of points. Classical results due to Katz and Sarnak [@KS] tell us what happens if we fix the genus of the curve, $g$ and let $q\to\infty$. Progress has been made in the other situation when $q$ is fixed and $g\to\infty$.
Let $K={\mathbb{F}}_q(X)$ and $K(C)$ be the field of functions of $C$, a curve over ${\mathbb{F}}_q$. Then $K(C)$ will be a finite field extension of $K$. Moreover, if we fix a copy of ${\mathbb{P}}^1({\mathbb{F}}_q)$, then every such finite extension corresponds to a smooth, projective curve (Corollary 6.6 and Theorem 6.9 from Chapter I of [@hart]). If $K(C)$ is a Galois extension of $K$, denote ${\mbox{Gal}}(C):={\mbox{Gal}}(K(C)/K)$ and $g(C)$ to be the genus of $C$. Define $${\mathcal{H}}_{G,g} = \{C: {\mbox{Gal}}(C)=G, g(C)=g\}.$$
We want to determine the probability, that a random curve in this family has a given number of points. That is, for every $N\in {\mathbb{Z}}_{\geq 0}$, we want to determine $${\mbox{Prob}}(C\in{\mathcal{H}}_{G,g} : \#C({\mathbb{P}}^1({\mathbb{F}}_q)) = N) := \frac{|\{C\in {\mathcal{H}}_{G,g} : \#C({\mathbb{P}}^1({\mathbb{F}}_q)) = N\}|}{|{\mathcal{H}}_{G,g}|}.$$
Therefore, the first thing we need to do is determine $|{\mathcal{H}}_{G,g}|$. Wright [@wright] was the first to answer such a question. He proved that if $G$ is abelian and $q\equiv 1 {\ (\text{mod}\ \exp(G))}$, (where $\exp(G)$ is the smallest integer such that $ng=e$ for all $g\in G$) then as $g\to\infty$ $$\begin{aligned}
\label{wright}
\sum_{j=0}^{N-1}q^{-\frac{j}{N}}|{\mathcal{H}}_{G,g+j}| \sim C(K,G)g^{\phi_G(Q)-1}q^{\frac{g}{N}}\end{aligned}$$ where $C(K,G)$ is a non-zero constant, $N = |G|-\frac{|G|}{Q}$ where $Q$ is the smallest prime divisor of $|G|$ and $\phi_G(Q)$ is the number of elements of $G$ of order $Q$. (Note: Wright’s result does not require $q\equiv 1 {\ (\text{mod}\ \exp(G))}$, but we will always assume that here and it makes the formula slightly nicer).
From now on the function $\phi_G(s)$ will be the number of elements of $G$ of order $s$.
Bucur, David, Feigon, Kaplan, Lalin, Ozman and Wood [@BDFK+] shows that if $Q$ is a prime then as $g\to\infty$ $$|{\mathcal{H}}_{{\mathbb{Z}}/Q{\mathbb{Z}},g}| = \begin{cases} c_{Q-2}q^{\frac{2g+2Q-2}{Q-1}} P\left(\frac{2g+2Q-2}{Q-1}\right) + O\left(q^{(\frac{1}{2}+\epsilon) \frac{2g+2Q-2}{Q-1} }\right) & g\equiv 0 \mod{Q-1} \\ 0 & \mbox{otherwise} \end{cases}$$ where $c_{Q-2}$ is a constant that they make explicit and $P$ is a monic polynomial of degree $Q-2$.
Our first result is extending this to any abelian group. But first, we must define a quasi-polynomial.
A quasi-polynomial is a function that can be written as $$p(x) = c_n(x)x^n + c_{n-1}(x)x^{n-1} + \dots + c_0(x)$$ where $c_i(x)$ is a periodic function with integer period. We call the $c_i$ the coefficients of the quasi-polynomial. Moreover, if $c_n(x)$ is not identically the zero function then we say $p$ has degree $n$ and call it the leading coefficient.
Let ${\mathcal{R}}=[0,\dots,r_1-1]\times\dots\times[0,\dots,r_n-1]\setminus\{(0,\dots,0)\}$ be a set of inter-valued vectors. For any $\vec{\alpha}\in{\mathcal{R}}$ let $$e(\vec{\alpha}) = \underset{j=1,\dots,n}{{\mbox{lcm}}}\left(\frac{r_j}{(r_j,\alpha_j} \right)$$
\[mainthm1\]
Let $G$ be any abelian group and $q\equiv 1 \mod{\exp(G)}$. If there exists some $(d(\vec{\alpha}))_{\vec{\alpha}\in{\mathcal{R}}}$ such that $$2g+2|G|-2 = \sum_{\vec{\alpha}\in{\mathcal{R}}} \left(|G|-\frac{|G|}{e(\vec{\alpha})}\right) d(\vec{\alpha}) + |G|-\frac{|G|}{e(\vec{d})}$$ where $\vec{d}=(d_1,\dots,d_n)$ and $$d_j = \sum_{\vec{\alpha}\in{\mathcal{R}}}\alpha_jd(\vec{\alpha})$$ then $$|{\mathcal{H}}_{G,g}| = \sum_{j=1}^{\eta}P_j(2g)q^{\frac{2g+2|G|-2}{|G|-\frac{|G|}{s_j}}} + O\left(q^{\frac{(1+\epsilon)g}{|G|-\frac{|G|}{s_1}}}\right)$$ where $1=s_0<s_1<\dots<s_{\eta}=\exp(G)$ are the divisors of $\exp(G)$, $P_1$ is a quasi-polynomial of degree $\phi_G(s_1)-1$ and $P_j$ is a quasi-polynomial of degree at most $\phi_G(s_j)-1$. If no such solution exists then $|{\mathcal{H}}_{G,g}|=0$.
If we restrict to the case $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$, then we can say more about the polynomials.
\[mainthm1cor\]
If $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$ for $Q$ a prime, $q\equiv1 \mod{Q}$, and $2g+2Q^n-2\equiv 0{\ (\text{mod}\ Q^n-Q^{n-1})}$, then $$|{\mathcal{H}}_{G,g}| = P\left(\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}\right)q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}} + O\left(q^{\frac{(1+\epsilon)g}{Q^n-Q^{n-1}}}\right)$$ where $P$ is a polynomial of degree $Q^n-2$ with leading coefficient $$\frac{1}{(Q^n-2)!}\frac{q+Q^n-1}{q}\frac{L_{Q^n-2}}{\zeta_q(2)^{Q^n-1}}$$ where $L_{Q^n-2}$ is a constant defined in and $\zeta_q(s)$ is the zeta function for ${\mathbb{F}}_q[X]$ (). If $2g+2Q^n-2\not\equiv 0 {\ (\text{mod}\ Q^n-Q^{n-1})}$, then $|{\mathcal{H}}_{G,g}|=0$
If $q\equiv 1 {\ (\text{mod}\ \exp(G))}$ and $G$ is abelian, then we can we can write $${\mathcal{H}}_{G,g} = \bigcup {\mathcal{H}}^{\vec{d}(\vec{\alpha})}$$ where ${\mathcal{H}}^{\vec{d}(\vec{\alpha})}$ is an irreducible moduli space of ${\mathcal{H}}_{G,g}$ (see Section 2 of [@M1] for a full treatment of this).
For specific classes of groups, several authors determined that as $d(\vec{\alpha})\to\infty$ for all $\vec{\alpha}$, then $$\begin{aligned}
\label{irresult}
{\mbox{Prob}}(C\in{\mathcal{H}}^{\vec{d}(\vec{\alpha})} : \#C({\mathbb{P}}^1({\mathbb{F}}_q)) = N) \sim {\mbox{Prob}}\left(\sum_{i=1}^{q+1}X_i = N\right)\end{aligned}$$ where the $X_i$ are i.i.d. random variables that can be made completely explicit.
Kurlberg and Rudnick [@KR] were the first to do this for hyper-elliptic curves ($G={\mathbb{Z}}/2{\mathbb{Z}}$). Bucur, David, Feigon and Lalin [@BDFL1],[@BDFL2] then extended this to prime cyclic curves ($G={\mathbb{Z}}/Q{\mathbb{Z}}$, $Q$ a prime). Lorenzo, Meleleo and Milione [@LMM] then proved this for $n$-quadratic curves ($G=({\mathbb{Z}}/2{\mathbb{Z}})^n$). The author [@M1],[@M2] completes this for any abelian group.
The fact that these results need all the $d(\vec{\alpha})\to\infty$ is why we can not deduce the results for the whole space from the results for the subspaces. However, in the case $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$, we can deduce the main term of Corollary \[mainthm1cor\] from these results. However, the error term we get from doing this is just $(1+o(1))$. Likewise we can do the same for Corollary \[mainthm2cor\] and Theorem \[mainthm3\].
If $G={\mathbb{Z}}/r_1{\mathbb{Z}}\times\dots\times{\mathbb{Z}}/r_n{\mathbb{Z}}$, then since we assume $q\equiv1 \mod{\exp(G)}$, by Kummer Theory, we can find $F_j\in{\mathbb{F}}_q[X]$, $r_j^{th}$-power free such that $$K(C) = K\left(\sqrt[r_1]{F_1(X)}, \dots, \sqrt[r_n]{F_n(X)}\right).$$
Fix an ordering $x_1,\dots,x_{q+1}$ of ${\mathbb{P}}^1({\mathbb{F}}_q)$ such that $x_{q+1}$ is the point at infinity. This ordering will be fixed for the rest of this paper. Then, the number of points on the curve depend on the values of $$\chi_{r_j}(F_j(x_i)), j=1,\dots,n, i=1,\dots,q+1$$ where $\chi_{r_j}:{\mathbb{F}}_q\to\mu_{r_j}$ is a multiplicative character of order $r_j$. Moreover, the value of $F_j(x_{q+1})$ depends on the leading coefficient and degree of $F_j(X)$. (Again, see [@M2] for more on this.) Therefore, we will define $$\vec{k}=(k_1,\dots,k_n)\in{\mathbb{Z}}^n$$ $$E = \begin{pmatrix} \epsilon_{1,1} & \dots & \epsilon_{1,n} \\ \vdots & \ddots & \vdots \\ \epsilon_{\ell,1} & \dots & \epsilon_{\ell,n} \end{pmatrix}$$ such that $0\leq \ell \leq q$, $\epsilon_{i,j}\in \mu_{r_j}$. Define $$\begin{aligned}
{\mathcal{H}}_{G,g}(\vec{k},E) = \{C\in{\mathcal{H}}_{G,g} : & \deg(F_j)\equiv k_j {\ (\text{mod}\ r_j)}, \chi_{r_j}(F_j(x_i))=\epsilon_{i,j},\\
& i=1,\dots,\ell, j=1,\dots,n\}.\end{aligned}$$
Then we get
\[mainthm2\] Let $G$ be any abelian group and $q\equiv 1 {\ (\text{mod}\ \exp(G))}$. If there exists some $(d(\vec{\alpha}))_{\vec{\alpha}\in{\mathcal{R}}}$ such that $$2g+2|G|-2 = \sum_{\vec{\alpha}\in{\mathcal{R}}} \left(|G|-\frac{|G|}{e(\vec{\alpha})}\right) d(\vec{\alpha}) + |G|-\frac{|G|}{e(\vec{k})}$$ where $$d_j = \sum_{\vec{\alpha}\in{\mathcal{R}}}\alpha_jd(\vec{\alpha}) \equiv k_j \mod{r_j}$$ then $$|{\mathcal{H}}_{G,g}(\vec{k},E)| = \sum_{j=1}^{\eta} P_{j;\vec{k},E}(2g)q^{\frac{2g+2|G|-2}{|G|-\frac{|G|}{s_j}}} + O\left(q^{\frac{(1+\epsilon)g}{|G|-\frac{|G|}{s_1}}} \right)$$ where $1=s_0<s_1<\dots<s_{\eta}=r_n$ are the divisors of $r_n$ and $P_{j;\vec{k},E}$ is a quasi-polynomial of degree at most $\phi_G(s_j)-1$. If there no such solution exists then $|{\mathcal{H}}_{G,g}(\vec{k},E)|=0$.
Again, if $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$, then we can say more
\[mainthm2cor\]
If $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$, for $Q$ a prime, $q\equiv 1 \mod{Q}$ and $2g+2Q^n-2\equiv 0 {\ (\text{mod}\ Q^n-Q^{n-1})}$ then $$|{\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g}(\vec{k},E)| = P_{\vec{k},E}\left(\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}\right) q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}} + O\left(q^{\frac{(1+\epsilon)g}{Q^n-Q^{n-1}}} \right)$$ where $P_{\vec{k},E}$ is a polynomial of degree $Q^n-2$ with leading coefficient $$\frac{(q-1)^n}{Q^n(Q^n-2)!}\frac{L_{Q^n-2}}{\zeta_q(2)^{Q^n-1}}\left(\frac{q}{Q^n(q+Q^n-1)}\right)^{\ell}.$$ If $2g+2Q^n-2\not\equiv 0 {\ (\text{mod}\ Q^n-Q^{n-1})}$ then $|{\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g}(\vec{k},E)|=0$.
Now, using Corollaries \[mainthm1cor\] and \[mainthm2cor\] we can prove a result on the distribution of the number of points for the whole space.
\[mainthm3\] Let $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$ and fix $q$ such that $q\equiv 1 {\ (\text{mod}\ Q)}$. If $2g+2Q^n-2\equiv 0{\ (\text{mod}\ Q^n-Q^{n-1})}$ then as $g\to\infty$, $$\frac{|\{C\in{\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g} : \#C({\mathbb{P}}^1({\mathbb{F}}_q)) = M\}|}{|{\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g}|} = {\mbox{Prob}}\left(\sum_{i=1}^{q+1} X_i= M\right)\left(1+O\left(\frac{1}{g} \right)\right)$$ where the $X_i$ are $i.i.d.$ random variables taking value $0$, $Q^n$ or $Q^{n-1}$ such that $$X_i = \begin{cases} Q^{n-1} & \mbox{with probability } \frac{Q^n-1} {Q^{n-1}(q+Q^n -1)} \\ Q^n & \mbox{with probability } \frac{q}{Q^n(q+Q^n-1)} \\ 0 & \mbox{with probability } \frac{(Q^n-1)(q+Q^n-Q)}{Q^n(q+Q^n-1)} \end{cases}.$$
The proof of Theorem \[mainthm3\] follows directly from Corollaries \[mainthm1cor\] and \[mainthm2cor\] and the work done in [@M1] and [@M2]. Therefore, if we were able to determine the leading coefficients of $P_1$ in Theorem \[mainthm1\] and $P_{1,\vec{k},E}$ in Theorem \[mainthm2\], then an analogous result as Theorem \[mainthm3\] would follow from the work done in [@M1] and [@M2].
The random variables appearing in Theorem \[mainthm3\] are the same that appear in in the case $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$.
Bucur, David, Feigon, Kaplan, Lalin, Ozman and Wood [@BDFK+] prove analogous results to Corollary \[mainthm2cor\], and Theorem \[mainthm3\] for $G={\mathbb{Z}}/Q{\mathbb{Z}}$.
Notation and Setup
==================
From now on, we will assume that $G={\mathbb{Z}}/r_1{\mathbb{Z}}\times\dots\times{\mathbb{Z}}/r_n{\mathbb{Z}}$ such that $r_j|r_{j+1}$ and $q\equiv1 {\ (\text{mod}\ r_n)}$. Under these assumptions we can apply Kummer theory (Chap.14 Proposition 37 of [@DF]) to find $F_j\in{\mathbb{F}}_q[X]$, $r_j^{th}$-power free for $j=1,\dots,n$ such that $$K(C) = K\left(\sqrt[r_1]{F_1(X)},\dots,\sqrt[r_n]{F_n(X)}\right).$$
Let $${\mathcal{H}}^*_{G,g} = \{C\in{\mathcal{H}}_{G,g} : F_j \mbox{ is monic}\}$$ $${\mathcal{H}}^*_{G,g}(\vec{k},E) = \{C \in {\mathcal{H}}_{G,g}(\vec{k},E): F_j \mbox{ is monic}\}.$$ We call curves in ${\mathcal{H}}^*_{G,g}$ *monic*.
Now for each $F_j$, it’s leading coefficient can be chosen from any of the equivalence classes of ${\mathbb{F}}_q^*/({\mathbb{F}}_q^*)^{r_j}$ to give a different extension (and thus curve). Therefore we see that $|{\mathcal{H}}_{G,g}| = |G||{\mathcal{H}}^*_{G,g}|$ and $|{\mathcal{H}}_{G,g}(\vec{k},E)| = |G||{\mathcal{H}}_{G,g}(\vec{k},E)|$. Therefore, we will work with ${\mathcal{H}}^*_{G,g}$ and ${\mathcal{H}}^*_{G,g}(\vec{k},E)$ from now on.
Now, define $${\mathcal{R}}= [0,\dots,r_1-1]\times\dots\times[0,\dots,r_n-1]\setminus\{(0,\dots,0)\}$$ to be set of integer valued vectors denoted as $\vec{\alpha}=(\alpha_1,\dots,\alpha_n)$ such that not all of them are $0$. Denote $${\mathcal{R}}' = {\mathcal{R}}\cup\{(0,\dots,0)\}.$$ Then for every $\vec{\alpha}\in{\mathcal{R}}$ let $$f_{\vec{\alpha}} = \prod_{\substack{P \\ v_P(F_j)=\alpha_j}} P$$ where the product is over prime polynomials of ${\mathbb{F}}_q[X]$. Then we can write $$F_j = \prod_{\vec{\alpha}\in{\mathcal{R}}}f_{\vec{\alpha}}^{\alpha_j}$$ where we use the convention that $f^0$ is identically the constant polynomial $1$. Moreover, all the $f_{\vec{\alpha}}$ are squarefree and pairwise coprime.
In [@M2], the author uses the Riemann-Hurwitz formula (Theorem 7.16 of [@rose]) to show that the genus of $C$ satisfies the relation $$\begin{aligned}
\label{genform}
2g+2|G|-2 = \sum_{\vec{\alpha}} \left(|G|-\frac{|G|}{e(\vec{\alpha})}\right) \deg(f_{\vec{\alpha}}) + |G|-\frac{|G|}{e(\vec{d})}\end{aligned}$$ where $\vec{d}=(d_1,\dots,d_n)=(\deg(F_1),\dots,\deg(F_n))$ and for any vector $\vec{v}=(v_1,\dots,v_n)$, $$\begin{aligned}
\label{ramind}
e(\vec{v}) := \underset{j=1,\dots,n}{{\mbox{lcm}}}\left(\frac{r_j}{(r_j,v_j)}\right).\end{aligned}$$
Notice, that the genus only depends on the degree of the $f_{\vec{\alpha}}$ and the congruence class of the $d_j$ modulo $r_j$. Therefore, define $$\begin{aligned}
{\mathcal{F}}_d = \{f \in {\mathbb{F}}_q[X]: f \mbox{ is monic, squarfree and } \deg(f)=d\}\end{aligned}$$ $$\begin{aligned}
\label{Fset1}
{\mathcal{F}}_{\vec{d}(\vec{\alpha})} = \{(f_{\vec{\alpha}})\in\prod_{\vec{\alpha}\in{\mathcal{R}}} {\mathcal{F}}_{d(\vec{\alpha})} : (f_{\vec{\alpha}},f_{\vec{\beta}})=1 \mbox{ for all } \vec{\alpha}\not=\vec{\beta}\in{\mathcal{R}}\}\end{aligned}$$ where $\vec{d}(\vec{\alpha})=(d(\vec{\alpha}))_{\vec{\alpha}\in{\mathcal{R}}}$ is a vector of non-negative integers indexed by the vectors of ${\mathcal{R}}$.
Now for any $\vec{k}=(k_1,\dots,k_n)\in{\mathcal{R}}'$ consider the congruence conditions $$\begin{aligned}
\label{equiv1}
\sum_{\vec{\alpha}\in{\mathcal{R}}} \alpha_jd(\vec{\alpha}) \equiv k_j \mod{r_j}, j=1,\dots,n.\end{aligned}$$ Further, let $E$ be an $\ell\times n$ matrix such that $$\begin{aligned}
\label{matrix}
E = \begin{pmatrix} \epsilon_{1,1} & \dots & \epsilon_{1,n} \\ \vdots & \ddots & \vdots \\ \epsilon_{\ell,1} & \dots & \epsilon_{\ell,n} \end{pmatrix}\end{aligned}$$ with $\epsilon_{i,j}\in\mu_{r_j}$. Now, define $$\begin{aligned}
\label{Fset2}
{\mathcal{F}}_{\vec{d}(\vec{\alpha}); \vec{k},E} = \begin{cases} \{(f_{\vec{\alpha}})\in{\mathcal{F}}_{\vec{d}(\vec{\alpha})} : \chi_{r_j}(F_j(x_i))=\epsilon_{i,j}, i=1,\dots,\ell,j=1,\dots,n\} & \eqref{equiv1} \mbox{ is satisfied} \\ \emptyset & \mbox{otherwise} \end{cases}\end{aligned}$$
Finally, define $$\begin{aligned}
\label{Fset3}
{\mathcal{F}}_{D;\vec{k},E} = \bigcup_{\vec{d}(\vec{\alpha})} {\mathcal{F}}_{\vec{d}(\vec{\alpha});\vec{k},E}\end{aligned}$$ where the union is over all $\vec{d}(\vec{\alpha})$ that satisfies $$\begin{aligned}
\label{genform2}
D = \sum_{\vec{\alpha}\in{\mathcal{R}}} c(\vec{\alpha})d(\vec{\alpha})\end{aligned}$$ where $c(\vec{\alpha})$ are some fixed constants.
If we set $c(\vec{\alpha}) = |G|-\frac{|G|}{e(\vec{\alpha})}$ and $D = 2g+2|G|-2-c(\vec{k})$, then we see that becomes . Outside of Section \[curvesec\], we will work with arbitrary $c(\vec{\alpha})$ with the idea that eventually we will set them equal to what we need.
Therefore, it seems as if what we need to do is determine the size of ${\mathcal{F}}_{D;\vec{k},E}$ for all $D,\vec{k},E$. Unfortunately, this will actually count too many curves! However, there is still a way to determine the size of ${\mathcal{H}}^*_{G,g}$ and ${\mathcal{H}}^*_{G,g}(\vec{k},E)$ if we know $|{\mathcal{F}}_{D;\vec{k},E}|$.
Too Many Curves {#curvesec}
===============
Ideally, we would like to say that every monic curve, $C$, such that ${\mbox{Gal}}(C)=G$, $g(C)=g$, comes from an element ${\mathcal{F}}_{\vec{d}(\vec{\alpha})}$ such that $\vec{d}(\vec{\alpha})$ satisfies . Unfortunately, this is not true.
For example, if we consider the set ${\mathcal{F}}_{(0,d_2,0)}$ such that $2g+6=2d_2$ and $2d_2\equiv 0{\ (\text{mod}\ 4)}$. Then $(0,d_2,0)$ satisfies for $G={\mathbb{Z}}/4{\mathbb{Z}}$ and we would hope that this would correspond to a curve with ${\mbox{Gal}}(C)={\mathbb{Z}}/4{\mathbb{Z}}$ and $g(C)=g$. However, an element of ${\mathcal{F}}_{(0,d_2,0)}$ would look like $(1,f_2,1)$ where $f_2$ is a square-free polynomial of degree $d_2$. This would correspond to a curve with affine model $Y^4 = f_2^2$, which clearly has $K(C)=K(\sqrt{f_2})$ and so ${\mbox{Gal}}(C)={\mathbb{Z}}/2{\mathbb{Z}}$. Moreover, $$g(C) = \frac{d_2-2}{2} = \frac{g+3-2}{2} = \frac{g-1}{2}+1.$$
It is easy to see how this argument can be extended to any group $G$ that does not have prime order. Indeed, what we will show in this section is that the elements of ${\mathcal{F}}_{\vec{d}(\vec{\alpha})}$ correspond to monic curves whose Galois group is a *subgroup* of $G$.
\[curverem\]
When we talk about all the subgroups of $G$, we mean all the different possible subsets of $G$ that are subgroups of $G$. That is, two subgroups $H,H'\subset G$ are said to be the same subgroup if and only if they are equal as subsets. For example, if $G={\mathbb{Z}}/Q{\mathbb{Z}}\times{\mathbb{Z}}/Q{\mathbb{Z}}$, then the subgroups $$\{(a,0): 0\leq a \leq Q-1\}$$ $$\{(0,a): 0\leq a \leq Q-1\}$$ $$\{(a,a): 0\leq a \leq Q-1\}$$ are all different even though they are all isomorphic to ${\mathbb{Z}}/Q{\mathbb{Z}}$.
For simplicity, in this section, we will fix $c(\vec{\alpha})=|G|-\frac{|G|} {e(\vec{\alpha})}$.
\[curveprop1\] Let $$M(G,g) = \{C, \mbox{ monic } : {\mbox{Gal}}(C)=H\subset G, g(C)=\frac{g-1}{|G|/|H|}+1\}.$$ Then there is a natural bijection from elements of $$\bigcup_{\vec{d}(\vec{\alpha})} {\mathcal{F}}_{\vec{d}(\vec{\alpha})}$$ to $M(G,g)$ where the union is over all $\vec{d}(\vec{\alpha})$ that satisfies .
Let $(f_{\vec{\alpha}}) \in {\mathcal{F}}_{\vec{d}(\vec{\alpha})}$ for some fixed $\vec{d}(\vec{\alpha})$ that satisfies . Define $${\mathcal{R}}^* = \{\vec{\alpha}\in{\mathcal{R}}: d(\vec{\alpha})\not=0\}.$$ For every $\vec{\alpha}\in{\mathcal{R}}$, we can identify it as an element in $G$ in the natural way. Let $H\subset G$ be the subgroup that is generated by ${\mathcal{R}}^*$ under this identification. (From now on, in this proof, we will identify elements of $H$ and $G$ with elements of ${\mathcal{R}}$). We will show that ${\mbox{Gal}}(C)=H$.
There exists some $s_j|r_j$ (where, potentially, some of the $s_j=1$) such that $$H\cong {\mathbb{Z}}/s_1{\mathbb{Z}}\times \dots \times {\mathbb{Z}}/s_n{\mathbb{Z}}.$$
Let $\vec{\alpha}_j \in H$ be a generating set of $H$ such that the order of $\alpha_j$ is $s_j$. Therefore, if $\vec{\alpha}\in{\mathcal{R}}^*$, we can find $\alpha_j^*$ such that $0\leq \alpha_j^*\leq s_j-1$ and $$\vec{\alpha} = \sum_{j=1}^n \alpha_j^*\vec{\alpha}_j.$$
If we let $\vec{\alpha}_j=(\alpha_{j,1},\dots,\alpha_{j,n})$ then for all $\vec{\alpha}\in{\mathcal{R}}^*$, $$\alpha_k = \sum_{j=1}^n \alpha_j^*\alpha_{j,k}.$$
Now, a basis element of $K(C) = K\left(\sqrt[r_1]{F_1(X)}, \dots, \sqrt[r_n]{F_n(X)} \right)$ will be $$\prod_{k=1}^n F_k(X)^{\frac{m_k}{r_k}} = \prod_{\vec{\alpha}\in{\mathcal{R}}^*} f_{\vec{\alpha}}^{\sum_{k=1}^n \frac{\alpha_km_k}{r_k}} = \prod_{\vec{\alpha}\in{\mathcal{R}}^*} f_{\vec{\alpha}}^{\sum_{k=1}^n \frac{m_k}{r_k} \sum_{j=1}^n \alpha_j^*\alpha_{j,k} }$$ for some values of $m_k,k=1,\dots,n$. (Note, we can restrict the product down to the $\vec{\alpha}\in{\mathcal{R}}^*$ for if $\vec{\alpha}\not\in{\mathcal{R}}^*$, then $\deg(f_{\vec{\alpha}})=0$ and hence $f_{\vec{\alpha}}=1$.) Therefore, we can define an action by $h=(h_1,\dots,h_n)\in H$ on the basis elements by $$h\left(\prod_{k=1}^n F_k(X)^{\frac{m_k}{r_k}}\right) = \prod_{\vec{\alpha}\in{\mathcal{R}}^*} f_{\vec{\alpha}}^{\sum_{k=1}^n \frac{m_k}{r_k} \sum_{j=1}^n h_j\alpha_j^*\alpha_{j,k} }.$$ Therefore if $h\not=(0,\dots,0)$, there will be a $\vec{\alpha}\in{\mathcal{R}}^*$ such that $$\sum_{k=1}^n \frac{m_k}{r_k} \sum_{j=1}^n h_j\alpha_j^*\alpha_{j,k}\not\in{\mathbb{Z}}.$$ Hence, every non-trivial element of $H$ gives a non-trivial automorphism of $K(C)$ and $H\subset {\mbox{Gal}}(K(C)/K) = {\mbox{Gal}}(C)$.
Define $$F_j^* = \prod_{\vec{\alpha}\in{\mathcal{R}}^*}f_{\vec{\alpha}}^{\alpha_j^*}, j=1,\dots,n.$$
Since $\vec{\alpha}_j$ has order $s_j$ we get $s_j(\vec{\alpha}_j) = (0,\dots,0)$. Therefore, $s_j\alpha_{j,k} \equiv 0 {\ (\text{mod}\ r_k)}$ and we can find $\alpha'_{j,k}$ such that $$\alpha_{j,k} = \frac{r_k}{(s_j,r_k)}\alpha_{j,k}'.$$ Therefore, $$\sqrt[r_k]{F_k(X)} = \prod_{\vec{\alpha}\in{\mathcal{R}}^*}f_{\vec{\alpha}}^{\alpha_k/r_k} = \prod_{\vec{\alpha}\in{\mathcal{R}}^*}f_{\vec{\alpha}}^{\frac{1}{r_k}\sum_{j=1}^n \alpha_j^*\alpha_{j,k}} = \prod_{j=1}^n \left( \prod_{\vec{\alpha}\in{\mathcal{R}}^*}f_{\vec{\alpha}}^{\alpha_j^*} \right)^{\alpha_{j,k}/r_k}$$ $$= \prod_{j=1}^n \left( \prod_{\vec{\alpha}\in{\mathcal{R}}^*}f_{\vec{\alpha}}^{\alpha_j^*} \right)^{\alpha_{j,k}'/(s_j,r_k)} = \prod_{j=1}^n \left(\sqrt[s_j]{F^*_j(X)}\right)^{\alpha_{j,k}'s_j/(s_j,r_k)}.$$ Hence, $$K\left(\sqrt[r_1]{F_1(X)},\dots,\sqrt[r_n]{F_n(X)}\right) \subset K\left(\sqrt[s_1]{F^*_1(X)},\dots,\sqrt[s_n]{F^*_n(X)} \right).$$
Clearly ${\mbox{Gal}}\left(K\left(\sqrt[s_1]{F^*_1(X)},\dots,\sqrt[s_n]{F^*_n(X)} \right)/K\right) \subset H$. Thus ${\mbox{Gal}}(C)\subset H$ and therefore ${\mbox{Gal}}(C)=H$.
It remains to show that $g(C)=\frac{g-1}{|G|/|H|}+1$.
Recall, $e(\vec{\alpha}) = {\mbox{lcm}}\left(\frac{r_j}{(r_j,\alpha_j)}\right)$. Then $e(\vec{\alpha})$ will be the order of $\vec{\alpha}$ as viewed as an element in $G$. Therefore, if $\vec{\alpha}\in{\mathcal{R}}^*$, then $$e(\vec{\alpha}) = {\mbox{lcm}}\left(\frac{r_j}{(r_j,\alpha_j)}\right) = {\mbox{lcm}}\left(\frac{s_i}{(s_i,\alpha^*_i)}\right):=e^*(\vec{\alpha})$$ since $e^*(\vec{\alpha})$ would the order of $\vec{\alpha}^*$ as viewed as an element in $H$ (which would be the same as $\vec{\alpha}$ in $G$). Therefore, $$c(\vec{\alpha}) = |G|-\frac{|G|}{e(\vec{\alpha})} = \frac{|G|}{|H|}\left(|H|-\frac{|H|} {e^*(\vec{\alpha})}\right):= \frac{|G|}{|H|} c^*(\vec{\alpha}).$$ Likewise, if we define $d_j^* = \deg(F_j^*) =\sum_{\vec{\alpha}\in{\mathcal{R}}^*} \alpha^*_jd(\vec{\alpha})$, then $e^*(\vec{d}^*)=e(\vec{d})$ and $\frac{|G|}{|H|}c^*(\vec{d}^*)=c(\vec{d})$.
Since $\vec{d}(\vec{\alpha})$ satisfies , we have $$2g+2|G|-2 =\sum_{\vec{\alpha}\in{\mathcal{R}}}c(\vec{\alpha})d(\vec{\alpha}) +c(\vec{d}) = \sum_{\vec{\alpha}\in{\mathcal{R}}^*}c(\vec{\alpha})d(\vec{\alpha}) +c(\vec{d})$$ $$= \frac{|G|}{|H|}\left(\sum_{\vec{\alpha}\in{\mathcal{R}}^*}c^*(\vec{\alpha})d(\vec{\alpha}) +c^*(\vec{d}^*)\right).$$ That is, $$\left(\sum_{\vec{\alpha}\in{\mathcal{R}}^*}c^*(\vec{\alpha})d(\vec{\alpha}) +c(\vec{d}^*)\right) = 2\left(\frac{g-1}{|G|/|H|}+1\right)+2|H|-2$$ Therefore, $(f_{\vec{\alpha}})$ corresponds to a monic curve $C$ with ${\mbox{Gal}}(C) = H$ and, by the Riemann-Hurwitz formula, $g(C)$ is $\frac{g-1}{|G|/|H|}+1$.
Therefore, for any monic curve with ${\mbox{Gal}}(C)=H\subset G$ and $g(C)=\frac{g-1}{|G|/|H|}+1$, we can find $(f_{\vec{\alpha}})\in{\mathcal{F}}_{\vec{d}(\vec{\alpha})}$ such that $K(C) = K(\sqrt[r_1]{F_1(X)},\dots,\sqrt[r_n]{F_n(X)})$ where $$F_j(X) = \prod_{\vec{\alpha}\in{\mathcal{R}}} f_{\vec{\alpha}}(X)^{\alpha_j}.$$
\[curvecor1\]
For any $\vec{k}\in{\mathcal{R}}'$ and $E$ as in , let $$\begin{aligned}
M_{\vec{k},E}(G,g) = \{C, \mbox{ monic } : & {\mbox{Gal}}(C)=H\subset G, g(C)=\frac{g-1}{|G|/|H|}+1, \deg{F_j}\equiv k_j {\ (\text{mod}\ r_j)} \\
& \chi_{r_j}(F_j(x_i)) = \epsilon_{i,j}, i=1\dots, \ell, j=1,\dots,n\}.\end{aligned}$$ Then there is a natural bijection from elements of ${\mathcal{F}}_{D;\vec{k},E}$ to $M_{G,g}(\vec{k},E)$ where $D=2g+2|G|-2-c(\vec{k})$.
Follows immediately from Proposition \[curveprop1\] and the definition of ${\mathcal{F}}_{D;\vec{k},E}$.
Inclusion-Exclusion of Abelian Groups {#inex}
=====================================
Therefore, if we can determine the size of ${\mathcal{F}}_{D;\vec{k},E}$ corresponding to curves with any abelian Galois group and any genus, then we can hope to do an inclusion-exclusion type argument for abelian groups. Luckily, this was first done by Delsarte [@dels].
Let $\mathcal{G}$ be the set of all abelian groups. Define a function $$\mu:\mathcal{G} \to {\mathbb{Z}}$$ by $$\mu\left({\mathbb{Z}}/p^{a_1}{\mathbb{Z}}\times\dots\times{\mathbb{Z}}/p^{a_n}{\mathbb{Z}}\right) = \begin{cases} (-1)^n p^{\frac{n(n-1)}{2}} & a_1=\dots=a_n=1 \\ 0 & \mbox{otherwise} \end{cases}.$$ To finish the definition if $G=G_1\times G_2$ such that $(|G_1|,|G_2|)=1$, then $\mu(G)=\mu(G_1)\mu(G_2)$. Then we have the property that $$\begin{aligned}
\label{mobinv}
\sum_{H\subset G} \mu(H) = \begin{cases} 1 & G=\{e\} \\ 0 & \mbox{otherwise} \end{cases}.\end{aligned}$$
This formula requires that we sum up over all subgroups of $G$ in the sense of Remark \[curverem\]. Hence why it is important that we define $M_{G,g}$ and $M_{G,g}(\vec{k},E)$ in the way that we do.
For an example of consider the group ${\mathbb{Z}}/Q^2{\mathbb{Z}}$, for $Q$ a prime. Then the subgroups are $\{e\}, {\mathbb{Z}}/Q{\mathbb{Z}}$ and ${\mathbb{Z}}/Q^2{\mathbb{Z}}$ and each of them appear once. Therefore, $$\begin{aligned}
\sum_{H\subset {\mathbb{Z}}/Q^2{\mathbb{Z}}} \mu(H) & = \mu(\{e\}) + \mu({\mathbb{Z}}/Q{\mathbb{Z}}) + \mu({\mathbb{Z}}/Q^2{\mathbb{Z}})\\
& = 1 +(-1)+0 =0.\end{aligned}$$ Whereas if we consider the group ${\mathbb{Z}}/Q{\mathbb{Z}}\times {\mathbb{Z}}/Q{\mathbb{Z}}$, for $Q$ a prime, then the subgroups would be $\{e\}$, ${\mathbb{Z}}/Q{\mathbb{Z}}$ and ${\mathbb{Z}}/Q{\mathbb{Z}}\times{\mathbb{Z}}/Q{\mathbb{Z}}$. Obviously $\{e\}$ and ${\mathbb{Z}}/Q{\mathbb{Z}}\times{\mathbb{Z}}/Q{\mathbb{Z}}$ appear only once however, ${\mathbb{Z}}/Q{\mathbb{Z}}$ can appear many times. It is easy to see that all the subgroups of ${\mathbb{Z}}/Q{\mathbb{Z}}$ lying in ${\mathbb{Z}}/Q{\mathbb{Z}}\times {\mathbb{Z}}/Q{\mathbb{Z}}$ will be generated by $(1,a)$, $a\in{\mathbb{Z}}/Q{\mathbb{Z}}$ or $(0,1)$. That is, there are $Q+1$ different subgroups of ${\mathbb{Z}}/Q{\mathbb{Z}}$ appearing in ${\mathbb{Z}}/Q{\mathbb{Z}}\times {\mathbb{Z}}/Q{\mathbb{Z}}$. Therefore, $$\begin{aligned}
\sum_{H\subset {\mathbb{Z}}/Q{\mathbb{Z}}\times{\mathbb{Z}}/Q{\mathbb{Z}}} \mu(H) & = \mu(\{e\}) + (Q+1)\mu({\mathbb{Z}}/Q{\mathbb{Z}}) + \mu({\mathbb{Z}}/Q{\mathbb{Z}}\times{\mathbb{Z}}/Q{\mathbb{Z}})\\
& = 1 +(Q+1)(-1)+Q =0.\end{aligned}$$
This allows us to perform M$\ddot{o}$bius inversion on $M(G,g)$.
\[inexlem\] For any abelian group, $G$, and genus, $g$, $$\begin{aligned}
|{\mathcal{H}}^*_{G,g}| = \sum_{H\subset G} \mu(G/H)|M(H, \frac{g-1}{|G|/|H|}+1)|\end{aligned}$$
Likewise, for any $\vec{k}\in{\mathcal{R}}$ and $E$ as in $$\begin{aligned}
|{\mathcal{H}}^*_{G,g}(\vec{k},E)| = \sum_{H\subset G} |M_{\vec{k},E}(H,\frac{g-1}{|G|/|H|}+1)|\end{aligned}$$
Straight from the definition we get $$|M(G,g)| = \sum_{H\subset G}\left|{\mathcal{H}}^*_{H, \frac{g-1}{|G|/|H|}+1}\right|.$$ Therefore, $$\begin{aligned}
\sum_{H\subset G} \mu(G/H) \left|M\left(H,\frac{g-1}{|G|/|H|}+1\right)\right| & = \sum_{H\subset G} \mu(G/H) \sum_{H'\subset H} \left|{\mathcal{H}}^*_{H',\frac{g-1}{|G|/|H'|}+1}\right|\\
& = \sum_{H'\subset G } \left|{\mathcal{H}}^*_{H',\frac{g-1}{|G|/|H'|}+1}\right| \sum_{H'\subset H \subset G} \mu(G/H) \\
& = \sum_{H'\subset G } \left|{\mathcal{H}}^*_{H',\frac{g-1}{|G|/|H'|}+1}\right| \sum_{H''\subset G/H'} \mu(H'') \\
& = |{\mathcal{H}}^*_{G,g}|.\end{aligned}$$
The proof of the likewise is analogous.
Generating Series {#gensersec}
=================
It remains to determine the size of ${\mathcal{F}}_{D;\vec{k},E}$ as $D\to\infty$. In order to do this we will develop a generating series for this set. But first, we need indicator functions for the relations $$d_j \equiv k_j {\ (\text{mod}\ r_j)}, j=1,\dots,n$$ $$\chi_{r_j}(F(x_i))=\epsilon_{i,j}, i=1,\dots,\ell, j=1,\dots,n.$$
That is, if we let $\xi_{r_j}= e^{\frac{2\pi i}{r_j}}$, a primitive $r_j^{th}$ root of unity, then $$\begin{aligned}
\label{indicatfact1}
\frac{1}{r_1\cdots r_n}\prod_{j=1}^n \sum_{t_j=0}^{r_j-1} \xi_{r_j}^{t_j(\sum \alpha_j\deg(f_{\vec{\alpha}}) - k_j)} = \begin{cases} 1 & \sum_{\vec{\alpha}\in{\mathcal{R}}} \alpha_j\deg(f_{\vec{\alpha}}) \equiv k_j {\ (\text{mod}\ r_j)} \\ 0 & \mbox{otherwise}\end{cases}.\end{aligned}$$ Further, if we denote $h(X)= \prod_{i=1}^{\ell} (X-x_i)$, then as long as $(F_j,h)=1$ for $j=1,\dots,n$, we get $$\begin{aligned}
\label{indicatfact2}
\left(\frac{1}{r_1\cdots r_n}\right)^{\ell}\prod_{i=1}^{\ell}\prod_{j=1}^n \sum_{\nu_{i,j}=0}^{r_j-1} (\epsilon_{i,j}^{-1}\chi_{r_j}(F_j(x_i))^{\nu_{i,j}} = \begin{cases} 1 & \chi_{r_j}(F_j(x_i))=\epsilon_{i,j}, i=1,\dots,\ell, j=1,\dots,n \\ 0 & \mbox{otherwise} \end{cases}.\end{aligned}$$
The sum in the exponent in is a sum over all $\vec{\alpha}\in{\mathcal{R}}$.
For ease of notation, for every set of polynomials $(f_{\vec{\alpha}})$, let $I_{\vec{k},E}((f_{\vec{\alpha}}))$ be the indicator function defined as $$\begin{aligned}
\label{indicator1}
I_{\vec{k},E}((f_{\vec{\alpha}})) = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell+1}\left(\prod_{j=1}^n\sum_{t_j=0}^{r_j-1} \xi_{r_j}^{t_j(\sum \alpha_j\deg(f_{\vec{\alpha}}) - k_j)}\right) \left(\prod_{i=1}^{\ell}\prod_{j=1}^n \sum_{\nu_{i,j}=0}^{r_j-1} (\epsilon_{i,j}^{-1}\chi_{r_j}(F_j(x_i))^{\nu_{i,j}}\right).\end{aligned}$$
Now, define the multi-variable complex function $$\begin{aligned}
\label{genfunc1}
\mathcal{G}_{\vec{k},E}((s_{\vec{\alpha}})) = \sum_{(f_{\vec{\alpha}})} \frac{\mu^2(h\prod_{\vec{\alpha}\in{\mathcal{R}}} f_{\vec{\alpha}}) I_{\vec{k},E}((f_{\vec{\alpha}}))}{ \prod_{\vec{\alpha}\in{\mathcal{R}}} |f_{\vec{\alpha}}|^{c(\vec{\alpha})s_{\vec{\alpha}}} }.\end{aligned}$$
The sum is over *all* $r_1\cdots r_n -1$-tuples of monic polynomials $(f_{\vec{\alpha}})_{\vec{\alpha}\in{\mathcal{R}}}$. However, the factor $\mu^2(h\prod_{\vec{\alpha}\in{\mathcal{R}}} f_{\vec{\alpha}})$ means that it is zero whenever the set of polynomials $(f_{\vec{\alpha}})$ are not square-free and pairwise coprime as well as coprime to $h$ (and thus non-zero at any of the $x_i$). Moreover, as usual, we let $|f_{\vec{\alpha}}|= q^{\deg(f_{\vec{\alpha}})}$.
Now, if we let $z_{\vec{\alpha}}=q^{-s_{\vec{\alpha}}}$ and define $F_{\vec{k},E}((z_{\vec{\alpha}})) = \mathcal{G}_{\vec{k},E}((q^{-s_{\vec{\alpha}}}))$, then $$\begin{aligned}
F_{\vec{k},E}((z_{\vec{\alpha}})) & = \sum_{(f_{\vec{\alpha}})} \mu^2(h\prod_{\vec{\alpha}\in{\mathcal{R}}} f_{\vec{\alpha}}) I_{\vec{k},E}((f_{\vec{\alpha}})) \prod_{\vec{\alpha}\in{\mathcal{R}}} z_{\vec{\alpha}}^{c(\vec{\alpha})\deg(f_{\vec{\alpha}})} \\
& = \sum_{\substack{ d(\vec{\alpha})=0 \\ \vec{\alpha}\in{\mathcal{R}}}}^{\infty} |{\mathcal{F}}_{\vec{d}(\vec{\alpha});\vec{k},E}| \prod_{\vec{\alpha}\in{\mathcal{R}}} z_{\vec{\alpha}}^{c(\vec{\alpha})d(\vec{\alpha})}.\end{aligned}$$
With some abuse of notation, if we let $F_{\vec{k},E}(z)$ be the function that sets all the $z_{\vec{\alpha}}=z$ to be the same in $F_{\vec{k},E}((z_{\vec{\alpha}}))$, then we get $$\begin{aligned}
F_{\vec{k},E}(z) &= \sum_{\substack{ d(\vec{\alpha})=0 \\ \vec{\alpha}\in{\mathcal{R}}}}^{\infty} |{\mathcal{F}}_{\vec{d}(\vec{\alpha});\vec{k},E}| z^{\sum_{\vec{\alpha}\in{\mathcal{R}}}c(\vec{\alpha})d(\vec{\alpha})} \label{genfunc2} \\
& = \sum_{D=0}^{\infty} |{\mathcal{F}}_{D;\vec{k},E}|z^D. \nonumber\end{aligned}$$
Ideally, we would like to write $F_{\vec{k},E}(z)$ as an Euler product. However, this is not possible. We can, though, write it as a sum of functions that can be written as a Euler product. But first we need some notation.Let $${\mathcal{M}}:= \left\{ \nu = \begin{pmatrix} \nu_{1,1} & \dots & \nu_{1,n} \\ \vdots & & \vdots \\ \nu_{\ell,1} & \dots & \nu_{\ell,n} \end{pmatrix} \in M_{\ell,n} : \nu_{i,j} \in {\mathbb{Z}}/r_j{\mathbb{Z}}\right\}.$$ We can define an action on ${\mathcal{R}}'$ and $E$ by ${\mathcal{M}}$ by $$\begin{aligned}
\nu\vec{\alpha} & := \begin{pmatrix} \sum_{j=1}^n \frac{r_n}{r_j}\nu_{1,j}\alpha_j \\ \vdots \\ \sum_{j=1}^n \frac{r_n}{r_j} \nu_{\ell,j} \alpha_j \end{pmatrix} \in ({\mathbb{Z}}/r_n{\mathbb{Z}})^{\ell}\\
E^{\nu} & := \prod_{i=1}^{\ell} \prod_{j=1}^n \epsilon_{i,j}^{\nu_{i,j}} \in\mu_{r_n}\end{aligned}$$ Moreover, for any $\vec{\alpha},\vec{\beta}\in{\mathcal{R}}'$ define $$\begin{aligned}
\vec{\alpha}\cdot\vec{\beta} = \sum_{j=1}^n \frac{r_n}{r_j}\alpha_j\beta_j \in {\mathbb{Z}}/r_n{\mathbb{Z}}.\end{aligned}$$
With this notation, we can rewrite as $$\frac{1}{r_1\cdots r_n} \prod_{j=1}^n \sum_{t_j=0}^{r_j-1} \xi_{r_j}^{t_j(\sum \alpha_j\deg(f_{\vec{\alpha}}) - k_j)} = \frac{1}{r_1\cdots r_n} \sum_{\vec{t}\in{\mathcal{R}}'} \prod_{j=1}^n \xi_{r_j}^{t_j(\sum \alpha_j\deg(f_{\vec{\alpha}}) - k_j)}$$ $$= \frac{1}{r_1\cdots r_n} \sum_{\vec{t}\in{\mathcal{R}}'} \xi_{r_n}^{-\vec{t}\cdot\vec{k}} \prod_{\vec{\alpha}\in{\mathcal{R}}} \xi_{r_n}^{\vec{t}\cdot\vec{\alpha}\deg(f_{\vec{\alpha}})}.$$
Recall $h(X) = \prod_{i=1}^{\ell} (X-x_i)$. For every $\nu\in{\mathcal{M}}$ and $\vec{\alpha}\in{\mathcal{R}}$, define $$\begin{aligned}
\label{character}
\chi^{\nu\vec{\alpha}}_{r_n}(F(X)) = \begin{cases}\prod_{i=1}^{\ell} \chi_{r_n}^{(\nu\vec{\alpha})_i}(F(x_i)) & (F,h)=1 \\ 0 & \mbox{otherwise}\end{cases}.\end{aligned}$$ Then, $\chi^{\nu\vec{\alpha}}_{r_n}$ if a multiplicative character on ${\mathbb{F}}_q[X]$ modulo $h(X)$. Moreover, it will be trivial if and only if $\nu\vec{\alpha}=\vec{0}$. Hence, we can rewrite as $$\left(\frac{1}{r_1\cdots r_n}\right)^{\ell} \prod_{i=1}^{\ell}\prod_{j=1}^n \sum_{\nu_{i,j}=0}^{r_j-1} (\epsilon_{i,j}^{-1}\chi_{r_j}(F_j(x_i))^{\nu_{i,j}} = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell} \sum_{\nu\in{\mathcal{M}}} \prod_{i=1}^{\ell}\prod_{j=1}^n (\epsilon_{i,j}^{-1}\chi_{r_j}(F_j(x_i))^{\nu_{i,j}}$$ $$= \left(\frac{1}{r_1\cdots r_n}\right)^{\ell} \sum_{\nu\in{\mathcal{M}}} E^{-\nu} \prod_{\vec{\alpha}\in{\mathcal{R}}} \prod_{i=1}^{\ell}\prod_{j=1}^n \chi^{\nu_{i,j}}_{r_j}(f_{\vec{\alpha}}^{\alpha_j}(x_i)) = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell} \sum_{\nu\in{\mathcal{M}}} E^{-\nu} \prod_{\vec{\alpha}\in{\mathcal{R}}} \chi^{\nu\vec{\alpha}}_{r_n}(f_{\vec{\alpha}}(X)).$$
Therefore, we can rewrite the indicator function in as $$\begin{aligned}
\label{indicator2}
I_{\vec{k},E}((f_{\vec{\alpha}})) = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell+1}\sum_{\vec{t}\in{\mathcal{R}}'} \sum_{\nu\in{\mathcal{M}}} E^{-\nu} \xi_{r_n}^{-\vec{t}\cdot\vec{k}} \prod_{\vec{\alpha}\in{\mathcal{R}}} \xi_{r_n}^{\vec{t}\cdot\vec{\alpha}\deg(f_{\vec{\alpha}})} \chi^{\nu\vec{\alpha}}_{r_n}(f_{\vec{\alpha}}(X)).\end{aligned}$$
We can rewrite $F_{\vec{k},E}(z)$ using this new notation.
$$\begin{aligned}
F_{\vec{k},E}(z) & = \sum_{(f_{\vec{\alpha}})} \mu^2(h\prod_{\vec{\alpha}\in{\mathcal{R}}} f_{\vec{\alpha}}) I_{\vec{k},E}((f_{\vec{\alpha}})) z^{\sum c(\vec{\alpha})\deg(f_{\vec{\alpha}})}\\
& = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell+1} \sum_{(f_{\vec{\alpha}})} \mu^2(h\prod_{\vec{\alpha}\in{\mathcal{R}}} f_{\vec{\alpha}})\sum_{\vec{t}\in{\mathcal{R}}'} \sum_{\nu\in{\mathcal{M}}} E^{-\nu} \xi_{r_n}^{-\vec{t}\cdot\vec{k}} \prod_{\vec{\alpha}\in{\mathcal{R}}} \left( \chi_{r_n}^{\nu\vec{\alpha}}(f_{\vec{\alpha}}) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{ c(\vec{\alpha})})^{\deg(f_{\vec{\alpha}})}\right) \\
& = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell+1} \sum_{\vec{t}\in{\mathcal{R}}'} \sum_{\nu\in{\mathcal{M}}} E^{-\nu} \xi_{r_n}^{-\vec{t}\cdot\vec{k}} A_{\vec{t},\nu}(z)\end{aligned}$$
where $$\begin{aligned}
A_{\vec{t},\nu}(z) & := \sum_{(f_{\vec{\alpha}})} \mu^2(h\prod_{\vec{\alpha}} f_{\vec{\alpha}}) \prod_{\vec{\alpha}\in{\mathcal{R}}}\left( \chi_{r_n}^{\nu\vec{\alpha}}(f_{\vec{\alpha}}) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{ c(\vec{\alpha})})^{\deg(f_{\vec{\alpha}})}\right).\end{aligned}$$
We call a function $G : {\mathbb{F}}_q[X]^n\to\mathbb{C}$ an **$n$-dimensional multiplicative** function if $$G(f_1,\dots,f_n) = \prod_P G(P^{v_P(f_1)},\dots,P^{v_P(f_n)})$$ where the product is over all prime polynomial $P$ dividing $f_1 \cdots f_n$.
Therefore, if $G$ is an $n$-dimensional multiplicative function, then $$\sum_{f_1,\dots,f_n} G(f_1,\dots,f_n) = \prod_{P} \left(1 + \sum_{(a_1,\dots,a_n)\not=(0,\dots,0)}G(P^{a_1},\dots,P^{a_n})\right).$$ where the sum is over all monic polynomials in ${\mathbb{F}}_q[X]$ and the product is over all monic prime polynomials.
Now, $$G((f_{\vec{\alpha}})) = \mu^2(h\prod_{\vec{\alpha}} f_{\vec{\alpha}}) \prod_{\vec{\alpha}\in{\mathcal{R}}}\left( \chi_{r_n}^{\nu\vec{\alpha}}(f_{\vec{\alpha}}) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{ c(\vec{\alpha})})^{\deg(f_{\vec{\alpha}})}\right)$$ is an $|R|$-dimensional multiplicative function. Moreover, if $P$ is a prime polynomial coprime to $h$, then $$G((P^{a_{\vec{\alpha}}})) = \begin{cases}\chi^{\nu\vec{\alpha_0}}_{r_n}(P)(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha_0}} z^{c(\vec{\alpha_0})})^{\deg(P)} & a_{\vec{\alpha_0}}=1 \mbox{ for some } \vec{\alpha_0}, a_{\vec{\beta}}=0 \mbox{ for all } \vec{\beta}\not=\vec{\alpha_0} \\ 0 & \mbox{otherwise} \end{cases}$$
Therefore, $$\begin{aligned}
A_{\vec{t},\nu}(z) & = \sum_{\substack{f_{\vec{\alpha}} \\ \vec{\alpha}\in{\mathcal{R}}}} \mu^2(h\prod_{\vec{\alpha}} f_{\vec{\alpha}}) \prod_{\vec{\alpha}\in{\mathcal{R}}}\left( \chi_{r_n}^{\nu\vec{\alpha}}(f_{\vec{\alpha}}) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{ c(\vec{\alpha})})^{\deg(f_{\vec{\alpha}})}\right) \\
&= \prod_{\substack{P \\ (P,h)=1}} \left(1 + \sum_{\vec{\alpha}\in{\mathcal{R}}} \chi^{\nu\vec{\alpha}}_{r_n}(P)(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} \right).\end{aligned}$$
Now, if we let $c_1<c_2<\dots<c_{\eta}$ be the unique values of the $c(\vec{\alpha})$, then we see that if $|z|< q^{-1/c_1}$, then $A_{\vec{t},\nu}(z)$ absolutely converges for all $\vec{t},\nu$ and, hence, so does $F_{\vec{k},E}(z)$. Therefore, we can express $|{\mathcal{F}}_{D;\vec{k},E}|$ as a contour integral of $F_{\vec{k},E}(z)$.
\[firstresprop\]
If $c_1 = \min(c(\vec{\alpha}))$ and $0<\delta_1<q^{-1/c_1}$ then let $C_{\delta_1} = \{z\in \mathbb{C} : |z|=\delta_1\}$, oriented counterclockwise. Then $$\begin{aligned}
\label{firstres}
\frac{1}{2\pi i}\oint_{C_{\delta_1}} \frac{F_{\vec{k},E}(z)}{z^{D+1}} dz = |{\mathcal{F}}_{D;\vec{k},E}|.\end{aligned}$$
By , we have $$F_{\vec{k},E}(z) = \sum_{D=0}^{\infty} |{\mathcal{F}}_{D;\vec{k},E}|z^D.$$ By our discussion above, $\frac{F_{\vec{k},E}(z)}{z^{D+1}}$ has only one pole at $0$ in the region contained in $C_{\delta_1}$ and it’s residue is $|{\mathcal{F}}_{D;\vec{k},E}|$.
Analytic Continuation of $A_{\vec{t},\nu}(z)$
=============================================
In this section, we will calculate an analytic continuation for $A_{\vec{t},\nu}(z)$ for all $\vec{t},\nu$. Then in the next section, we will use this analytic continuation to analyze the poles of $A_{\vec{t},\nu}(z)$.
Recall $h(X) = \prod_{i=1}^{\ell}(X-x_i)$ and define ${\mathcal{R}}_{\nu} = \{\vec{\alpha}\in{\mathcal{R}}: \nu\vec{\alpha}=\vec{0}\}$, then the character $\chi_{r_n}^{\nu\vec{\alpha}}$ (as defined in ) will be trivial if and only if $\vec{\alpha}\in{\mathcal{R}}_{\nu}$. Therefore, $$\begin{aligned}
A_{\vec{t},\nu}(z) = & \prod_{\substack{P \\ (P,h)=1}} \left(1 + \sum_{\vec{\alpha}\in{\mathcal{R}}} \chi^{\nu\vec{\alpha}}_{r_n}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} \right) \\
= & \prod_{\substack{P \\ (P,h)=1}} \left( 1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} + \sum_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}} \chi_{r_n}^{\nu\vec{\alpha}}(P)(\xi_{r_n}^{\vec{t}\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right)\\
= & \prod_{P} \left( 1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} + \sum_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}} \chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right) \times\\
& \prod_{P|h}\left(1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right)^{-1}\\
= & \prod_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}\prod_{P}\left(1 + (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right) H_{\vec{t},\nu}(z)\\
= & \prod_{\vec{\alpha}\in{\mathcal{R}}_{\nu}} \frac{Z_K(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})} {Z_K(\xi_{r_n}^{2\vec{t}\cdot\vec{\alpha}}z^{2c(\vec{\alpha})})} H_{\vec{t},\nu}(z)\end{aligned}$$ where $$Z_K(z) = \prod_P \left(1-z^{\deg(P)}\right)^{-1} = (1-qz)^{-1}$$ is the zeta-function of $K$ in the $z$-variable and $$H_{\vec{t},\nu}(z) = \prod_{P} \left(\frac{ 1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} + \sum_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}} \chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}}{\prod_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}\left(1 + (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right) }\right) \times$$ $$\prod_{P|h}\left(1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right)^{-1}.$$
Now, for all $\vec{\alpha}\in{\mathcal{R}}$, $$\frac{Z_K(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})}{Z_K(\xi_{r_n}^{2\vec{t}\cdot\vec{\alpha}} z^{2c(\vec{\alpha})})}$$ is a meromorphic function with simple poles when $z^{c(\vec{\alpha})}=(q\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}) ^{-1/c(\vec{\alpha})}$. So it remains to determine where $H_{\vec{t},\nu}(z)$ converges.
\[analyticlem\]
$H_{\vec{t},\nu}(z)$ absolutely converges for all $|z|<q^{-1/2c_1}$.
Since $$\prod_{P|h}\left(1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right)^{-1}$$ is a finite product, it will always converge and thus we need only consider the factor consisting of the infinite product.
$$\prod_{P} \left(\frac{ 1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} + \sum_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}} \chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}}{\prod_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}\left(1 + (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right) }\right)$$ $$= \prod_{\vec{\alpha}\not\in{\mathcal{R}}_\nu} \prod_P \left(1+\chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right) H^*_{\vec{t},\nu}(z).$$
Since, for all $\vec{\alpha}\not\in{\mathcal{R}}_\nu$, $\chi_{r_n}^{\nu\vec{\alpha}}$ is a non-trivial character we get that $$\prod_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}}\prod_P \left(1+\chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right)$$ is an entire function. Moreover, $$\begin{aligned}
H^*_{\vec{t},\nu}(z) & = \prod_{P} \left(\frac{ 1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} + \sum_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}} \chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}}{\prod_{\vec{\alpha}\in{\mathcal{R}}_\nu}\left(1 + (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right) \prod_{\vec{\alpha}\not\in{\mathcal{R}}_\nu} \left(1+\chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right) }\right)\\
&= \prod_{P} \left(1 - \frac{h_P(z)}{\prod_{\vec{\alpha}\in{\mathcal{R}}_\nu}\left(1 + (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right) \prod_{\vec{\alpha}\not\in{\mathcal{R}}_\nu} \left(1+\chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right)}\right)\end{aligned}$$ where $$\begin{aligned}
h_p(z) = & \prod_{\vec{\alpha}\in{\mathcal{R}}_\nu}\left(1 + (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)}\right) \prod_{\vec{\alpha}\not\in{\mathcal{R}}_\nu} \left(1+\chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right)\\
& - \left(1 + \sum_{\vec{\alpha}\in{\mathcal{R}}_{\nu}}(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}} z^{c(\vec{\alpha})})^{\deg(P)} + \sum_{\vec{\alpha}\not\in{\mathcal{R}}_{\nu}} \chi_{r_n}^{\nu\vec{\alpha}}(P) (\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})^{\deg(P)}\right)\\
= & O\left(z^{\underset{\vec{\alpha}\not=\vec{\beta}}{\min}(c(\vec{\alpha}) + c(\vec{\beta}))}\right)= O\left(z^{2c_1}\right).\end{aligned}$$
Therefore, if $|z|<q^{-1/2c_1}$, then $H^*_{\vec{t},\nu}(z)$ converges absolutely and hence so does $H_{\vec{t},\nu}(z)$.
For $0\leq a \leq r_n-1$, and $i=1,\dots,\eta$, define $$\begin{aligned}
\label{Rpoleset}
{\mathcal{R}}_{\vec{t},\nu;a,i} = \{\vec{\alpha}\in{\mathcal{R}}_{\nu} : c(\vec{\alpha})=c_i \mbox{ and }\vec{t}\cdot\vec{\alpha}\equiv a {\ (\text{mod}\ r_n)}\}\end{aligned}$$ and let $$\begin{aligned}
\label{poleorder}
m_{\vec{t},\nu;a,i} = |{\mathcal{R}}_{\vec{t},\nu;a,i}|.\end{aligned}$$
\[analyticcor\]
$A_{\vec{t},\nu}(z)$ is meromorphic on the disc $|z|<q^{-1/2c_1}$ with poles of order $m_{\vec{t},\nu;a,i}$ at $$z=\xi_{c_i}^k\left(q\xi_{r_n}^a\right)^{-1/c_i}$$ for $k=1,\dots,c_i$.
Immediate from Lemma \[analyticlem\] and the factors of $Z_K(z)$ appearing.
It is highly possible that $m_{\vec{t},\nu;a,i}=0$ for some values of $\vec{t},\nu,a,i$. In this case when we say a pole of order $0$, we mean there is no pole.
Residue Calculations {#rescalcsec}
====================
Now, we can calculate the residues of $A_{\vec{t},\nu}(z)$ at each of its poles.
\[rescalclem1\]
Let $a,i$ be such that $m_{\vec{t},\nu;a,i}\not=0$, then for any $1\leq k \leq c_i$, $${\mbox{Res}}_{z=\xi_{c_i}^k \left( q\xi_{r_n}^a \right)^{-1/c_i}}\left( \frac{A_{\vec{t},\nu}(z)}{z^{D+1}} \right) = P_{\vec{t},\nu;a,i,k}(D) q^{\frac{D}{c_i}}$$ where $P_{\vec{t},\nu;a,i,k}$ is a quasi-polynomial of degree $(m_{\vec{t},\nu;a,i}-1)$ with leading coefficient $-C_{\vec{t},\nu;a,i,k}$ such that $$C_{\vec{t},\nu;a,i,k} = \frac{1}{(m_{\vec{t},\nu;a,i}-1)!}\left(\frac{1-q^{-1}}{c_i}\right)^{m_{\vec{t},\nu;a,i}} \xi_{c_i}^{-kD}\left(\xi_{r_n}^a\right)^{\frac{D}{c_i}} H_{\vec{t},\nu;a,i}(\xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i})$$ and $H_{\vec{t},\nu;a,i}$ is defined in the proof.
$$\begin{aligned}
\frac{A_{\vec{t},\nu}(z)}{z^{D+1}} & = \frac{1}{z^{D+1}} \prod_{\vec{\alpha}\in{\mathcal{R}}_{\nu}} \frac{Z_K(\xi_{r_n}^{\vec{t}\cdot\vec{\alpha}}z^{c(\vec{\alpha})})} {Z_K(\xi_{r_n}^{2\vec{t}\cdot\vec{\alpha}}z^{2c(\vec{\alpha})})} H_{\vec{t},\nu}(z) \\
& = \frac{1}{z^{D+1}} \underset{m_{\vec{t},\nu;b,j\not=0}} {\prod_{j=1}^{\eta} \prod_{b=0}^{r_n-1}} \left(\frac{Z_K(\xi_{r_n}^{b}z^{c_j})} {Z_K(\xi_{r_n}^{2b}z^{2c_j})}\right)^ {m_{\vec{t},\nu;b,j}} H_{\vec{t},\nu}(z)\\
& = \frac{1}{z^{D+1}}\left(\frac{1-q\xi_{r_n}^{2a}z^{2c_i}}{1-q\xi_{r_n}^{a}z^{c_i}}\right)^{m_{\vec{t},\nu;a,i}} H_{\vec{t},\nu;a,i}(z)\end{aligned}$$
where $$H_{\vec{t},\nu;a,i}(z) = \underset{\substack{(b,j)\not=(a,i) \\ m_{\vec{t},\nu;b,j\not=0}}} {\prod_{j=1}^{\eta} \prod_{b=0}^{r_n-1}}\left(\frac{Z_K(\xi_{r_n}^{b}z^{c_j})} {Z_K(\xi_{r_n}^{2b}z^{2c_j})}\right)^ {m_{\vec{t},\nu;b,j}} H_{\vec{t},\nu}(z).$$
Therefore, for any $1\leq k \leq c_i$, if we let $$R_{a,i,k}(z) = \frac{z^{c_i}-(q\xi_{r_n}^a)^{-1}}{z-\xi_{c_i}^k(q\xi_{r_n}^a)^{-1/c_i}},$$ then
$$\begin{aligned}
& (m_{\vec{t},\nu;a,i}-1)!{\mbox{Res}}_{z=\xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i}} \left(\frac{A_{\vec{t},\nu}(z)}{z^{D+1}}\right)\\
= & \lim_{z\to \xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i}} \frac{d^{m_{\vec{t},\nu;a,i}-1}}{dz^{m_{\vec{t},\nu;a,i}-1}} \frac{(z-\xi_{c_i}^k(\xi_{r_n}^aq)^{-1/c_i})^{m_{\vec{t},\nu;a,i}}}{z^{D+1}}\left(\frac{1-q\xi_{r_n}^{2a}z^{2c_i}}{1-q\xi_{r_n}^{a} z^{c_i}} \right)^{m_{\vec{t},\nu;a,i}} H_{\vec{t},\nu;a,i}(z) \\
= & \lim_{z\to \xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i}} \frac{d^{m_{\vec{t},\nu;a,i}-1}}{dz^{m_{\vec{t},\nu;a,i}-1}} \frac{1}{z^{D+1}}\left(\frac{1-q\xi_{r_n}^{2a}z^{2c_i}}{-q\xi_{r_n}^aR_{a,i,k}(z)} \right)^{m_{\vec{t},\nu;a,i}} H_{\vec{t},\nu;a,i}(z) \\
= & \lim_{z\to \xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i}}\sum_{j=0}^{m_{\vec{t},\nu;a,i}-1}\binom{m_{\vec{t},\nu;a,i}-1}{j} \frac{d^j}{dz^j} \left(\frac{1}{z^{D+1}}\right) \frac{d^{m_{\vec{t},\nu;a,i}-1-j}}{dz^{m_{\vec{t},\nu;a,i}-1-j}} \left(\frac{1-q\xi_{r_n}^{2a}z^{2c_i}}{-q\xi_{r_n}^aR_{a,i,k}(z)} \right)^{m_{\vec{t},\nu;a,i}}\times \\
& H_{\vec{t},\nu;a,i}(z) \\
= & \sum_{j=0}^{m_{\vec{t},\nu;a,i}-1}\binom{m_{\vec{t},\nu;a,i}-1}{j} (-1)^j(D+1)\cdots(D+j) \xi_{c_i}^{-k(D+j+1)} (\xi_{r_n}^aq)^{(D+j+1)/c_i} \times \\
& \frac{d^{m_{\vec{t},\nu;a,i}-1-j}}{dz^{m_{\vec{t},\nu;a,i}-1-j}}\left(\frac{1-q\xi_{r_n}^{2a}z^{2c_i}} {-q\xi_{r_n}^aR_{a,i,k} (z)} \right)^{m_{\vec{t},\nu;a,i}} H_{\vec{t},\nu;a,i}(z) \bigg|_{z = \xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i}}\\
= & P_{\vec{t},\nu;a,i,k}(D)q^{\frac{D}{c_i}}\end{aligned}$$
where $P_{\vec{t},\nu;a,i,k}$ is a quasi-polynomial of degree $m_{\vec{t},\nu;a,i}-1$. Moreover, we see that the leading coefficient of $P_{\vec{t},\nu;a,i,k}$ arises when $j=m_{\vec{t},\nu;a,i}-1$. That is $$\begin{aligned}
P_{\vec{t},\nu;a,i,k}(D)q^{\frac{D}{c_i}} = & (-D)^{m_{\vec{t},\nu;a,i}-1} \xi_{c_i}^{-k(D+m_{\vec{t},\nu;a,i})} \left(\xi^a_{r_n}q\right)^{(D+m_{\vec{t},\nu;a,i})/c_i} \left(\frac{1-q^{-1}}{-c_i\xi_{c_i}^{k(c_i-1)} \left(\xi_{r_n}^aq \right)^{1/c_i} }\right)^{m_{\vec{t},\nu;a,i}} \times \\
& H_{\vec{t},\nu;a,i}(\xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i})\left(1+O\left(\frac{1}{D}\right)\right) \\
= & -\left(\frac{1-q^{-1}}{c_i}\right)^{m_{\vec{t},\nu;a,i}} \xi_{c_i}^{-kD} D^{m_{\vec{t},\nu;a,i}-1} \left(\xi_{r_n}^aq\right)^{\frac{D}{c_i}} H_{\vec{t},\nu;a,i}(\xi^k_{c_i}(\xi_{r_n}^aq)^{-1/c_i}) \times \\
& \left(1+O\left(\frac{1}{D}\right)\right).\end{aligned}$$
\[rescalccor1\]
Let $m_{\vec{t},\nu,i} = \underset{0\leq a \leq r_n-1}{\max}(m_{\vec{t},\nu;a,i})$. Let $0<\delta_1<q^{-1/c_1}$, $\delta_2 = \frac{1+\epsilon}{2c_1}$ for some $\epsilon>0$ and let $C_{\delta_1} = \{z\in\mathbb{C} : |z|=\delta_1\}$ oriented counterclockwise and $C_{\delta_2} = \{z\in\mathbb{C}: |z| = q^{-\delta_2}\}$ oriented clockwise. Then $$\frac{1}{2\pi i} \oint_{C_{\delta_1}+C_{\delta_2}} \frac{A_{\vec{t},\nu}(z)}{z^{D+1}} dz = \sum_{i=1}^{\eta}P_{\vec{t},\nu,i}(D)q^{\frac{D}{c_i}}$$ where $P_{\vec{t},\nu,i}$ is a quasi-polynomial such that $$P_{\vec{t},\nu,i}(D) = C_{\vec{t},\nu,i}D^{m_{\vec{t},\nu;i}-1} + O\left(D^{m_{\vec{t},\nu;i}-2}\right)$$ with $$C_{\vec{t},\nu,i} = \sum_{\substack{a=0 \\ m_{\vec{t},\nu;a,i} = m_{\vec{t},\nu;i}}}^{r_n-1} \sum_{k=0}^{c_i-1} C_{\vec{t},\nu;a,i,k}.$$
\[rescalccor2\]
By Cauchy’s Residue Theorem, and the fact that the larger disc, $C_{\delta_2}$, is oriented clockwise, $$\begin{aligned}
\frac{1}{2\pi i} \oint_{C_{\delta_1}+C_{\delta_2}} \frac{A_{\vec{t},\nu}(z)}{z^{D+1}} dz & = \underset{m_{\vec{t},\nu;a,i}\not=0} {\sum_{i=1}^{\eta} \sum_{a=0}^{r_n-1}} \sum_{k=0}^{c_i} -{\mbox{Res}}_{z=\xi_{c_i}^k \left( q\xi_{r_n}^a \right)^{-1/c_i}}\left( \frac{A_{\vec{t},\nu}(z)}{z^{D+1}} \right)\\
& = \underset{m_{\vec{t},\nu;a,i}\not=0} {\sum_{i=1}^{\eta} \sum_{a=0}^{r_n-1} } \sum_{k=0}^{c_i} -P_{\vec{t},\nu;a,i,k}(D) q^{\frac{D}{c_i}}\\
& = \sum_{i=1}^{\eta}P_{\vec{t},\nu,i}(D)q^{\frac{D}{c_i}}.\end{aligned}$$ The fact that $P_{\vec{t},\nu,i}(D)$ satisfies the conditions in the statement follow directly from Lemma \[rescalclem1\]
Now, we are unable to determine if $C_{\vec{t},\nu,i}$ is non-zero. Hence we can only give a bound on the degree of the $P_{\vec{t},\nu,i}$. This is why in the statement of the main theorems we say “of degree at most” instead of give the exact degree.
\[rescalcprop1\] Let $$m_i = \underset{\substack{ \vec{t}\in{\mathcal{R}}\\ \nu\in{\mathcal{M}}}}{\max}(m_{\vec{t},\nu,i}).$$ If there exists a solution to and , then for every $\epsilon>0$, $$|{\mathcal{F}}_{D;\vec{k},E}| = \sum_{i=1}^{\eta} P_i(D)q^{\frac{D}{c_i}} + O\left(q^{(\frac{1}{2}+\epsilon)\frac{D}{c_1}}\right)$$ where $P_i$ is a quasi-polynomial such that of degree at most $(m_i-1)$. Otherwise, if there does not exist a solution to and , then $|{\mathcal{F}}_{D,\vec{k},E}|=0$.
Recall that $${\mathcal{F}}_{D;\vec{k},E} = \bigcup_{\vec{d}(\vec{\alpha})} {\mathcal{F}}_{\vec{d}(\vec{\alpha});\vec{k},E}$$ where the union is over all solutions to where $D=2g+2|G|-2-c(\vec{k})$. Therefore, if there are no solutions to , we have an empty union, so ${\mathcal{F}}_{D;\vec{k},E}=\emptyset$. Further, if there are solution to but none of which that satisfy then ${\mathcal{F}}_{D;\vec{k},E}$ would be a union of empty sets and thus empty itself. Therefore, from now on, we will always assume there is a solution to and .
Let $C_{\delta_1}$ and $C_{\delta_2}$ be as defined in Corollary \[rescalccor1\]. Then $$\begin{aligned}
\frac{1}{2\pi i}\oint_{C_{\delta_1}+C_{\delta_2}} \frac{F_{\vec{k},E}(z)}{z^{D+1}} dz & = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell+1} \sum_{\vec{t}\in{\mathcal{R}}'} \sum_{\nu\in{\mathcal{M}}} \xi_{r_n}^{-\vec{t}\cdot\vec{k}}E^{-\nu} \frac{1}{2\pi i}\oint_{C_{\delta_1}+C_{\delta_2}} \frac{A_{\vec{t},\nu}(z)}{z^{D+1}} dz \\
& = \left(\frac{1}{r_1\cdots r_n}\right)^{\ell+1} \sum_{\vec{t}\in{\mathcal{R}}'} \sum_{\nu\in{\mathcal{M}}} \xi_{r_n}^{-\vec{t}\cdot\vec{k}}E^{-\nu} \sum_{i=1}^{\eta}P_{\vec{t},\nu,i}(D)q^{\frac{D}{c_i}}\\
& = \sum_{i=1}^{\eta} P_i(D)q^{\frac{D}{c_i}}\end{aligned}$$ where $P_i$ is a quasi-polynomial of degree at most $m_i$.
Now, by Proposition \[firstresprop\], we know that $$\frac{1}{2\pi i}\oint_{C_{\delta_1}}\frac{F_{\vec{k},E}(z)}{z^{D+1}}dz = -|{\mathcal{F}}_{D;\vec{k},E}|.$$ Moreover, $$\left|\frac{1}{2\pi i}\oint_{C_{\delta_2}} \frac{F_{\vec{k},E}(z)}{z^{D+1}}dz \right| = O\left( q^{(\frac{1}{2}+\epsilon) \frac{D}{c_1}} \right)$$ where the implied constant is the maximum values of $F_{\vec{k},E}(z)$ on $C_{\delta_2}$.
If we let $c(\vec{\alpha})$ be any integers, then we could have that $\frac{D}{c_i} \leq \frac{D}{2c_1}$ and thus part of the main term could be absorbed into the error term. However, if we let $c(\vec{\alpha}) = |G|-\frac{|G|}{e(\vec{\alpha})}$, then we actually have that $\frac{D}{c_i}>\frac{D}{2c_1}$ for all $i=1,\dots,\eta$. So for small enough $\epsilon$, none of our main terms can be absorbed into the error term.
Proofs of the Main Theorems {#proofsec}
===========================
All that remains is to combine Proposition \[rescalcprop1\] and Lemma \[inexlem\]. From now on, we will fix the $c(\vec{\alpha})=|G|-\frac{|G|}{e(\vec{\alpha})}$. But first, we present a little more notation in order to deal with the subgroups of $G$ as used in Lemma \[inexlem\].
As in Section \[curvesec\], there is a natural bijection from $G\setminus\{e\}$ to ${\mathcal{R}}$. For every $H\subset G$, let ${\mathcal{R}}_H$ be the image of $H$ under this natural bijection. Recall that $\eta=\eta_G$ is the number of non-trivial divisors of $\exp(G)=r_n$. Then, for all $\vec{t}\in{\mathcal{R}}'$, $\nu\in{\mathcal{M}}$, $0\leq a \leq r_n-1$ and $1\leq i \leq \eta_G$, define the analogous objects $$\begin{aligned}
{\mathcal{R}}_{H,\nu} & = \{\vec{\alpha}\in{\mathcal{R}}_H: \nu\vec{\alpha}=0\}\\
{\mathcal{R}}_{H,\vec{t},\nu;a,i} & = \{\vec{\alpha}\in{\mathcal{R}}_{H,\nu} : c(\vec{\alpha})=c_i \mbox{ and } \vec{t}\cdot\vec{\alpha}\equiv a {\ (\text{mod}\ r_n)}\} \\
m_{H,t,\nu;a,i} & = |{\mathcal{R}}_{H,\vec{t},\nu;a,i}|\\
m_{H,\vec{t},\nu,i} & = \underset{0\leq a \leq r_n-1}{\max}(m_{H,\vec{t},\nu;a,i})\\
m_{H,i} & = \underset{\substack{\vec{t}\in{\mathcal{R}}' \\ \nu\in{\mathcal{M}}}}{\max}(m_{H,\vec{t},\nu,i})\end{aligned}$$
Now, $$m_{H,i} = m_{H,0,0;0,i} = |\{\vec{\alpha}\in{\mathcal{R}}_H: c(\vec{\alpha})=c_i\}| = \phi_{H}(s_i)$$ since $c_i = |G|-\frac{|G|}{e(\vec{\alpha})}$ and $e(\vec{\alpha})$ is the order of $\vec{\alpha}$ as seen as an element in $G$. So, if $\vec{\alpha}\in{\mathcal{R}}_H$, then it can be seen as element in $H$ and will have the same order. Notice, however, that we could have $\phi_H(s)=0$ even if $\phi_G(s)\not=0$.
Lemma \[inex\] and Corollary \[curvecor1\] tell us that $${\mathcal{H}}^*_{G,g}(\vec{k},E) = \sum_{H \subset G} \sum_{\vec{d}(\vec{\alpha})} {\mathcal{F}}_{\vec{d}(\vec{\alpha});\vec{k},E}$$ where the inner sum is over all $\vec{d}(\vec{\alpha})$ that satisfy $$\begin{aligned}
\label{genform10}
&d(\vec{\alpha}) =0, \vec{\alpha}\not\in{\mathcal{R}}_H\nonumber \\
&d_j = \sum_{\vec{\alpha}\in{\mathcal{R}}}\alpha_jd(\vec{\alpha}) \equiv k_j {\ (\text{mod}\ r_j)}, j=1,\dots,n \\
&\sum_{\vec{\alpha}\in{\mathcal{R}}} c(\vec{\alpha})d(\vec{\alpha}) = 2g+2|G|-2-c(\vec{k}). \nonumber\end{aligned}$$
Therefore, if there are no solutions to and , then the above sum is empty and we have that $|{\mathcal{H}}_{G,g}(\vec{k},E)|=|{\mathcal{H}}^*_{G,g}(\vec{k},E)| = 0$. From now on, we will assume that there exists a solution to and so that the above sum is non-empty. Further, note that if $g\not\equiv 1 {\ (\text{mod}\ |G|/|H|)}$ for some $H$ then there would be no solutions to as this would correspond to a curve with a non-integer genus, which is impossible.
Moreover, if $H\cong {\mathbb{Z}}/s_1{\mathbb{Z}}\times\dots{\mathbb{Z}}/s_n{\mathbb{Z}}$ where $s_j|r_j$, then ${\mathcal{R}}_H$ can be identified with the set $$[0,\dots,s_1-1]\times\dots\times[0,\dots,s_n-1]\setminus\{(0,\dots,0)\}.$$ This allows us to apply Proposition \[rescalcprop1\] to obtain $$\begin{aligned}
|{\mathcal{H}}^*_{G,g}(\vec{k},E)|& =\sum_{H\subset G} \mu(G/H)\left( \sum_{j=1}^{\eta_H} P_{H,j;\vec{k},E}(2g)q^{\frac{2g+2|G|-2}{|G|-\frac{|G|}{s_{H,j}}}} + O\left(q^{\frac{(1+\epsilon)g}{|G|-\frac{|G|}{s_{H,1}}}}\right) \right)\end{aligned}$$ where $\eta_H$ is the number of non-trivial divisors of $\exp(H)$ and $1=s_{H,0}<s_{H,1}<\dots<s_{H,\eta_H}=\exp(H)$ are the divisor of $\exp(H)$ and $P_{H,j;\vec{k},E}$ is a quasi-polynomial of degree at most $\phi_H(s_{H,j})-1$ if $g\equiv 1 {\ (\text{mod}\ |G|/|H|)}$ and identically the $0$ polynomial otherwise. Since $\exp(H)|\exp(G)$ for all $H\subset G$ and $\phi_H(s_{H,j})\leq \phi_G(s_{G,j})$, we can write $$\begin{aligned}
|{\mathcal{H}}_{G,g}(\vec{k},E)|& =\sum_{j=1}^{\eta} P_{j;\vec{k},E}(2g)q^{\frac{2g+2|G|-2}{c_j}} + O\left(q^{\frac{(1+\epsilon)g}{c_1}} \right)\end{aligned}$$ where $c_j$ and $\eta=\eta_G$ are as above and $P_{j;\vec{k},E}$ is a quasi-polynomial of degree at most $\phi_G(s_j)-1$.
If we set $\ell=0$, then we get $E=\emptyset$ is an empty matrix and thus the condition on it vanishes in ${\mathcal{H}}_{\vec{k},\emptyset}(G,g)$. Therefore, $$\begin{aligned}
|{\mathcal{H}}(G,g)| & = \sum_{\vec{k}\in{\mathcal{R}}'} |{\mathcal{H}}_{\vec{k},\emptyset}(G,g)|\\
& = \sum_{\vec{k}\in{\mathcal{R}}'} \sum_{j=1}^{\eta} P_{j;\vec{k},\emptyset}(2g) q^{\frac{2g+2|G|-2}{c_j}} + O\left(q^{\frac{(1+\epsilon)g}{c_1}} \right)\\
& = \sum_{j=1}^{\eta}\sum_{\vec{k}\in{\mathcal{R}}'} P_{j;\vec{k},\emptyset}(2g )q^{\frac{2g+2|G|-2}{c_j}} + O\left(q^{\frac{(1+\epsilon)g}{c_1}} \right) \\
& = \sum_{j=1}^{\eta}P_{j}(2g)q^{\frac{2g+2|G|-2}{c_j}} + O\left(q^{\frac{(1+\epsilon)g}{c_1}} \right)\end{aligned}$$ where $P_j$ is a quasi-polynomial of degree at most $\phi_G(s_j)-1$. To show that $P_1$ has exact degree, suppose $$P_1(2g) = a_0(g)(2g)^{\phi_G(s_j)-1} + a_1(g)(2g)^{\phi_G(s_j-2)} + \dots$$ for some periodic function $a_i$ with integer period. Now, our results, shows that $$\sum_{j=0}^{c_1-1}q^{-\frac{j}{c_1}}|{\mathcal{H}}_{G,g+j}| \sim \sum_{j=0}^{N-1} a_0(g+j) (2g)^{\phi_G(s_1)-1}q^{\frac{2g+2|G|-2}{c_1}}.$$ Further, then tells us that $$\sum_{j=0}^{c_1-1} a_0(g+j) = C(K,G)\not=0.$$ Therefore, $a_0$ will be non-zero for at least one integer in every interval of length $c_1$. That is, $a_0$ is not identically $0$ and $P_1$ has degree exactly $\phi_G(s_1)-1$.
$G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$ {#Q^n}
==================================
In this section we will determine the leading coefficient of $P_{\vec{k},E,1}$ and $P_1$ that appear in Corollaries \[mainthm1cor\] and \[mainthm2cor\] in the case that $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$.
The reason we are able to determine the leading coefficient of $P_1$ in this case is that the genus and M$\ddot{o}$bius inversion formulas become simpler when $G=({\mathbb{Z}}/Q{\mathbb{Z}})^n$. Indeed, in this case becomes $$\begin{aligned}
\label{genformQ}
2g+2Q^n-2 = \begin{cases} (Q^n-Q^{n-1})\sum_{\vec{\alpha}\in{\mathcal{R}}}d(\vec{\alpha}) & d_j\equiv 0 {\ (\text{mod}\ Q)}, j=1,\dots,n \\ (Q^n-Q^{n-1})(\sum_{\vec{\alpha}\in{\mathcal{R}}}d(\vec{\alpha}) +1) & \mbox{otherwise} \end{cases}\end{aligned}$$
Therefore, by Theorem \[mainthm2\], we get that if $2g+2Q^n-2\equiv 0 {\ (\text{mod}\ Q^n-Q^{n-1})}$ then $${\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g}(\vec{k},E) = \begin{cases} P_{\vec{0},E}(2g)q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}} & \vec{k}=\vec{0} \\ P_{\vec{k},E}(2g)q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}-1} & \vec{k}\not=\vec{0} \end{cases} + O\left(q^{(1+\epsilon)\frac{g}{Q^n-Q^{n-1}}}\right)$$ for some quasi-polynomial $P_{\vec{k},E}$ whose degree is at most $\phi_G(Q)-1=Q^n-2$. In fact we will show the is, in fact, a polynomials and it has exact degree $Q^n-2$. For the rest of this section we will always be assuming that $2g+2Q^n-2 \equiv 0 {\ (\text{mod}\ Q^n-Q^{n-1})}$.
We see that in this case we get $c(\vec{\alpha})=c(\vec{d}) = Q^n-Q^{n-1}$ for all $\vec{\alpha}\in{\mathcal{R}}$. Therefore, since we always assumed $c(\vec{\alpha})$ was arbitrary we can apply the results therein to this case $c(\vec{\alpha})=c(\vec{d})=1$ and $D=\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}\in\mathbb{N}$ in order to find the leading coefficient of $P_{\vec{k},E}$.
If we look back at where the quasi-polynomials come from, it is because we have a factor of the form $\zeta_{c_i}^{-kD}$ appearing in the constant $C_{\vec{t},\nu;a,i,k}$ in Lemma \[rescalclem1\]. Therefore, since in the case we can assume $c_i=1$ for all $i$, these terms disappear and we in fact get that $P_{\vec{k},E}$ is a polynomial and not a quasi-polynomial.
By setting $D=\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}$ instead of just $2g+2Q^n-2$, we are now counting by *conductor* instead of discriminant (genus). This is more analogous to what Bucur, et al. did in [@BDFK+]. Because we can easily switch to counting by conductor is why it is easier to compute the constant in this case
In this setting, for all $\vec{t}\in{\mathcal{R}}'$ and $\nu\in{\mathcal{M}}$, we have that $A_{\vec{t},\nu}(z)$ will have poles of order $m_{\vec{t},\nu;a}$ when $z=(q\xi_Q^a)^{-1}$ where $$m_{\vec{t},\nu;a} = |\{\vec{\alpha}\in{\mathcal{R}}_{\nu} : \vec{t}\cdot\vec{\alpha}\equiv a {\ (\text{mod}\ Q)}\}|.$$ Now, since $\left({\mathbb{Z}}/Q{\mathbb{Z}}\right)^n$ can be viewed as a vector space over the field ${\mathbb{Z}}/Q{\mathbb{Z}}$, we get that the action of $\nu$ and $\vec{t}$ on ${\mathcal{R}}$ are vector space morphisms. Therefore, the set $$\{\vec{\alpha}\in{\mathcal{R}}_{\nu} : \vec{t}\cdot\vec{\alpha}\equiv a {\ (\text{mod}\ Q)}\}\subsetneq {\mathcal{R}}$$ unless $\nu=0$, $\vec{t}=\vec{0}$ and $a=0$. In which case we get $$m_{\vec{0},0;0} = |{\mathcal{R}}|=Q^n-1.$$
Therefore, combining the results of Section \[rescalcsec\], we get that the leading coefficient of $P_{\vec{k},E}$ is $$C_{\vec{k},E} = \frac{1}{(Q^n-2)!}\left(1-q^{-1}\right)^{Q^n-1}\prod_{P}\left(\frac{ |P|^{Q^n-1}+(Q^n-1) |P|^{Q^n-2}}{(|P|+1)^{Q^n-1}} \right) \left(\frac{q}{Q^n(q+Q^n-1)}\right)^{\ell}$$ $$= \frac{1}{(Q^n-2)!} \frac{L_{Q^n-2}}{\zeta_q(2)^{Q^n-1}} \left(\frac{q}{Q^n(q+Q^n-1)}\right)^{\ell}$$ where $$\begin{aligned}
\label{Lnum}
L_m = \prod_{j=1}^m \prod_{P}\left(1 - \frac{j}{(|P|-1)(|P|+j)}\right)\end{aligned}$$ and $$\begin{aligned}
\label{zeta}
\zeta_q(s) = \sum_{F\in{\mathbb{F}}_q[X]} \frac{1}{|F|^s} = \frac{1}{1-q^{1-s}}\end{aligned}$$ is the zeta function where $|F|=q^{\deg{F}}$.
Notice that $C_{\vec{k},E}$ does not depend on $\vec{k}$ or $E$. Therefore, if we set $\ell=0$ and sum over all $\vec{k}$, we get $$\begin{aligned}
{\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g} = & \sum_{\vec{k}\in{\mathcal{R}}'} {\mathcal{H}}_{({\mathbb{Z}}/Q{\mathbb{Z}})^n,g)}(\vec{k},\emptyset) \\
= & P_{\vec{0},\emptyset}\left(\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}\right)q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}} + \sum_{\vec{k}\in{\mathcal{R}}} P_{\vec{k},\emptyset}\left(\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}\right)q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}-1}\\
& + O\left(q^{(1+\epsilon)\frac{g}{Q^n-Q^{n-1}}}\right)\\
= & P\left(\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}\right)q^{\frac{2g+2Q^n-2}{Q^n-Q^{n-1}}} + O\left(q^{(1+\epsilon)\frac{g}{Q^n-Q^{n-1}}}\right)\end{aligned}$$ where $P$ is a polynomial of degree $Q^n-2$ with leading coefficient $$\begin{aligned}
C & = C_{0,\emptyset} + \sum_{\vec{k}\in{\mathcal{R}}}C_{\vec{k},\emptyset}q^{-1}\\
& = \frac{1}{(Q^n-2)!}\frac{q+Q^n-1}{q} \frac{L_{Q^n-2}}{\zeta_q(2)^{Q^n-1}}\end{aligned}$$ which is exactly the analogue of the constant in [@BDFK+].
Since the condition $F_j(x_{q+1})\not=0$, where $x_{q+1}$ is the point at infinity, is equivalent to saying $\deg(F_j)\equiv 0 {\ (\text{mod}\ r_j)}$ for $j=1,\dots,n$, we get that if $\epsilon_{i,j}\in\mu_{r_j}$ for $i=1,\dots,q+1$ and $j=1,\dots,n$. Then as $g\to\infty$
$$\begin{aligned}
& \frac{|\{C\in{\mathcal{H}}_{G,g} : \chi_{r_j}(F_j(x_i))=\epsilon_{i,j}, i=1,\dots,q+1, j=1,\dots,n\}|}{|{\mathcal{H}}_{G,g}|}\\
= & \frac{\frac{1}{Q^n}|{\mathcal{H}}_{G,g}(0,E)|}{|{\mathcal{H}}_{G,g}|} = \left(\frac{q}{Q^n(q+Q^n-1)}\right)^{q+1} \left(1+O\left(\frac{1}{g}\right)\right)\end{aligned}$$
where the $\frac{1}{Q^n}$ factor in the first equality comes from the fact the leading coefficients of the $F_j$ must satisfy $\chi_{r_j}(c_j)=\epsilon_{q+1,j}$.
Finally, from this result, the exact same argument will work to show that as $g\to\infty$, $$\frac{|\{C\in{\mathcal{H}}_{G,g} : \#C({\mathbb{P}}^1({\mathbb{F}}_q)) = M\}|}{|{\mathcal{H}}_{G,g}|} = {\mbox{Prob}}\left(\sum_{i=1}^{q+1} X_i= M\right)\left(1+O\left(\frac{1}{g} \right)\right)$$ where the $X_i$ are $i.i.d.$ random variables taking value $0$, $Q^n$ or $Q^{n-1}$ such that $$X_i = \begin{cases} Q^{n-1} & \mbox{with probability } \frac{Q^n-1} {Q^{n-1}(q+Q^n -1)} \\ Q^n & \mbox{with probability } \frac{q}{Q^n(q+Q^n-1)} \\ 0 & \mbox{with probability } \frac{(Q^n-1)(q+Q^n-Q)}{Q^n(q+Q^n-1)} \end{cases}.$$
|
---
abstract: 'We derive a reation between four-fermion QED Green functions of different covariant gauges which defines the gauge dependence completely. We use the derived gauge dependence to check the gauge invariance of atom-like bound state calculations. We find that the existing QED procedure does not provide gauge invariant binding energies. A way to a corrected gauge invariant procedure is pointed out.'
author:
- |
Grigorii B. Pivovarov[^1]\
Institute for Nuclear Research\
of the Russian Academy of Sciences, Moscow 117312, Russia
title: 'Gauge Dependence of Four-Fermion QED Green Function and Atom-Like Bound State Calculations'
---
3000 3000
40by
Introduction
============
The persisting discrepancy between theory and experiment for positronium width [@Westbrook] is a chalenge for QED. At the moment the hope is on taking into account corrections of relative order $\alpha^{2}$ [@Lepage; @Khriplovich]. In the circumstances the question of self-consistency of the calculations, in particular, of gauge invariance of the result is of prime concern.
The modern way to calculate parameters of two-particle atom-like bound states is to extract them from corresponding four-fermion QED Green function (see, for example, and this paper below). Thus, to check the gauge independence of the calculated bound state parameters, one should carry the gauge parameter through all the extraction procedure. (An example of this see in [@Adkins] where the gauge independence of the correction to the positronium width of relative order $\alpha$ was checked.) The extraction procedure gets more and more complicated with an increase in order of radiative corrections and direct order by order check of gauge invariance becomes impractical as a check of self-consistency of the calculations. Instead, one would like to exploit gauge invariance choosing a most convenient gauge and switching from one gauge to another in the process of the calculations. In view of these complications, it seems pertinent to make a step out of the concrete practice of bound-state calculations and to study first the gauge dependence of the four-fermion QED Green function itself without taking into account the complications of the bound-state parameter calculations.
In the present paper we derive a relation between four-fermion QED Green functions of different values of gauge-fixing parameter (we consider the covariant gauges only). The relation completely defines the evolution of the Green function in the gauge-fixing parameter. Our derivation does not use perturbation theory. Next, we use our relation to check gauge invariance of the extraction procedure of atom-like bound-state parameters. The result is negative. It turns out that the existing procedure provides gauge-dependent answers for binding energies. We find a flaw in the procedure which is responsible for the gauge-dependence of the result and point the way to its correction.
Next section contains a derivation of the evolution in the gauge-fixing parameter; section 3 comprises a brief recall of the extraction procedure and an utilization of the general evolution formula from section 2 for an analysis of gauge-dependence of the extraction; in the last, fourth, section we point out the reason for the gauge dependence and the way to the correct procedure.
Evolution in Gauge-Fixing Parameter
===================================
Let us consider the four-fermion QED Green function $$\label{Gf}
G_{\beta}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i})\equiv
i\int D\psi DA\, \exp\left(iS_{QED}(\beta)\right)
(\overline{\psi}(\overline{x}_{f}) \psi(x_{f}))
(\overline{\psi}({x}_{i}) \psi(\overline{x}_{i}))\, ,$$ where $x_{f}(\overline{x}_{f})$ is a coordinate of outgoing particle (antiparticle) and $x_{i}(\overline{x}_{i})$ is the same for ingoing pair. The definition of gauge fixing parameter $\beta$ is given by corresponding photon propagator: $$\label{gfix}
D_{\mu \nu}(\beta,x) = \int\frac{dk}{(2\pi)^{4}}
\left(-g_{\mu \nu} + \beta\frac{k_{\mu}k_{\nu}}{k^{2}}\right)
\frac{i}{k^{2}}e^{ikx}.$$
Our aim is to study the dependence of $G_{\beta}$ on $\beta$. To this end, it is useful to consider a Green function in external photon field, $G(A)$, which is a result of integration over the fermion field in the rhs of (\[Gf\]). From the one hand, it is simply connected to the Green function [@Vass]: $$\label{connection}
G_{\beta} = (e^{L_{\beta}}G(A))_{A=0}\,,\;
L_{\beta}\equiv\frac{1}{2}\frac{\delta}{\delta A_{\mu}}D_{\mu \nu}(\beta)
\frac{\delta}{\delta A_{\nu}}.$$ (In this formula each $L_{\beta}$ generates a photon propagator; the dependence on the coordinates of ingoing and outgoing particles is suppressed for brevity.) From the other hand, $G(A)$ is siply connected to a gauge invariant object $G_{inv}(A)$: $$\label{coninv}
G(A) = G_{inv}(A) \exp\left(ie\int^{x_{f}}_{\overline{x}_{f}}A_{\mu}dx^{\mu}
-ie\int^{x_{i}}_{\overline{x}_{i}}A_{\mu}dx^{\mu}
\right).$$ The gauge invariance of $G_{inv}$ means that it is independent of the longitudinal component of $A$: $$\label{gi}
\partial_{\mu}\frac{\delta}{\delta A_{\mu}}G_{inv}(A) = 0$$ and is a consequence of gauge invariance of the combination $$\overline\psi(x)\exp\left(ie\int^{x}_{y}A_{\mu}dz^{\mu}
\right)\psi(y).$$
A substitution of (\[coninv\]) into (\[connection\]) yields $$\label{hot}
G_{\beta} = \left (e^{L_{\beta}}G_{inv}(A)
\exp\left(ie\int^{x_{f}}_{\overline{x}_{f}}A_{\mu}dx^{\mu}
-ie\int^{x_{i}}_{\overline{x}_{i}}A_{\mu}dx^{\mu}\right)\right)_{A=0}.$$ Let us take a $\beta$-derivative of both sides of this equation: $$\label{almost eq}
\frac{\partial}{\partial\beta}G_{\beta} =
\left (e^{L_{\beta}}(\partial_{\beta}L_{\beta})G_{inv}(A)
\exp\left(ie\int^{x_{f}}_{\overline{x}_{f}}A_{\mu}dx^{\mu}
-ie\int^{x_{i}}_{\overline{x}_{i}}A_{\mu}dx^{\mu}\right)\right )_{A=0} .$$ To get an evolution equation, one needs to express the rhs of this equation in terms of $G_{\beta}$. It is possible because $(\partial_{\beta}L_{\beta})$ commutes with $G_{inv}(A)$ and gives a $c$-factor when acts on the consequent exponential. So, (\[almost eq\]) transforms itself into $$\label{equation}
\frac{\partial}{\partial\beta}
G_{\beta}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i}) =
F(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i})
G_{\beta}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i}),$$ where we have restored the $x$-dependence and used $F$ to denote the action of $(\partial_{\beta}L_{\beta})$ on the exponential: $$\begin{aligned}
\label{F-def}
\lefteqn{(\partial_{\beta}L_{\beta})
\exp\left(ie\int^{x_{f}}_{\overline{x}_{f}}A_{\mu}dx^{\mu}
-ie\int^{x_{i}}_{\overline{x}_{i}}A_{\mu}dx^{\mu}
\right) \equiv}\nonumber \\
& & F(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i})
\exp\left(ie\int^{x_{f}}_{\overline{x}_{f}}A_{\mu}dx^{\mu}
-ie\int^{x_{i}}_{\overline{x}_{i}}A_{\mu}dx^{\mu}
\right).\end{aligned}$$
An explanation is in order: In deriving (\[equation\]) we have used a commutativity of $(\partial_{\beta}L_{\beta})$ and $G_{inv}(A)$; it is a direct consequence of gauge invariance of $G_{inv}$ (see (\[gi\])) and the fact that $(\partial_{\beta}L_{\beta})$ contains only derivatives in longitudinal components of $A$ (see (\[connection\]) for a definition of $L_{\beta}$ and (\[gfix\]) for $\beta$-dependence of $D_{\mu \nu}$).
The solution of eq.(\[equation\]) for $\beta$-evolution is $$\label{solution}
G_{\beta}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i}) =
\exp\left((\beta-\beta_{0})
F(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i})
\right)
G_{\beta_{0}}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i}).$$
To get the final answer one needs an explicite view of $F$ from (\[solution\]). It is easily deduced from the $F$-definition (\[F-def\]) and the following representation for the longitudinal part of the photon propagator: $$\label{representation}
\partial_{\beta}D_{\mu \nu}(\beta,x) =
-\frac{1}{16\pi^{2}}\partial_{\mu}\partial_{\nu}
\ln((x^{2}-i\varepsilon)m^{2}),$$ where $m$ is an arbitrary mass scale which is fixed, for defineteness, on the fermion mass. Then, up to an additive constant, $$\label{repres}
F = \frac{\alpha}{4\pi}\left(
\ln\frac{1}{m^{4}(x_{f}-\overline{x}_{f})^{2}(x_{i}-\overline{x}_{i})^{2}}
+\ln\frac{(x_{f}-x_{i})^{2}(\overline{x}_{f}-\overline{x}_{i})^{2}}
{(x_{f}-\overline{x}_{i})^{2}(\overline{x}_{f}-x_{i})^{2}}
\right).$$
Substituting (\[repres\]) into (\[solution\]), we get our final aswer for $\beta$-evolution: $$\begin{aligned}
\label{answer}
G_{\beta}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i})&=&
\left[
\frac{Z(x_{f}-x_{i})^{2}(\overline{x}_{f}-\overline{x}_{i})^{2}}
{m^{4}(x_{f}-\overline{x}_{f})^{2}(x_{i}-\overline{x}_{i})^{2}
(x_{f}-\overline{x}_{i})^{2}(\overline{x}_{f}-x_{i})^{2}}
\right]^{\frac{\alpha}{4\pi}(\beta-\beta_{0})} \times\nonumber\\
&&G_{\beta_{0}}(x_{f},\overline{x}_{f},x_{i},\overline{x}_{i}) .\end{aligned}$$ The normalization $Z$ is infinite before the ultraviolet renormalization. After the renormalization it is scheme-dependent and calculable order by order in perturbation theory. We will not need its value in what follows.
The Bound State Parameters And The Four-Fermion QED Green Function
==================================================================
The four-fermion QED Green function contains too much information for one who just going to calculate bound-sate parameters. Ona can throw away unnessesary information by putting senter of mass space-time coordinate of ingoing pair and relative times of both ingoing and outgoing pairs to zero: $$\label{eqtimes}
G_{(et) \beta}(t,{\bf x},{\bf r'},{\bf r})\equiv
G_{\beta}\left(x_{f}(t,{\bf x},{\bf r'}),
\overline{x}_{f}(t,{\bf x},{\bf r'}),
x_{i}({\bf r'}),
\overline{x}_{i}({\bf r'})
\right),$$ where the space-time coordinates depend on a space-time coordinate of the center of mass of the outgoing pair $(t,{\bf x})$ and a relative space coordinate of outgoing $(\bf r')$ and ingoing $(\bf r)$ pair. In the case of equal masses $$\begin{aligned}
\label{def r}
x_{f}=(t,{\bf x}+\frac{{\bf r'}}{2}),&\;&
\overline{x}_{f}=(t,{\bf x}-\frac{{\bf r'}}{2}),\nonumber \\
x_{i}=(0,\frac{{\bf r}}{2}),&\;&
\overline{x}_{i}=(0,-\frac{{\bf r}}{2}).\end{aligned}$$
$G_{(et)\beta}$ still contains an unnecessary piece of information — the dependence on the center of mass space coordinate. The natural way to remove it is to go over to momentum representation and put the center of mass momentum to zero. In coordinate representation, which is more convenient for gauge invariance check, we define the propagator $D_{\beta}$ of the fermion pair: $$\label{propDef}
G_{(et)\beta}(t,{\bf x},{\bf r'},{\bf r}) \equiv
D_{\beta}(t,{\bf r'},{\bf r})\delta({\bf x}) + \ldots,$$ where dots denote terms with derivatives of $\delta({\bf x})$. It is natural to consider $D_{\beta}$ as a time dependent kernel of an operator acting on wave-functions of relative coordinate. In what follows we will not make difference between a kernel and the corresponding operator. The naturalness of the above definition of the propagator is apparent in the nonrelativistic approximation: $$\label{NR}
{e^{i2mt}}D_{\beta}(t) \approx
\sum_{E_{0}} \theta(t)e^{-iE_{0}t} P(E_{0}),$$ where the summation runs over the spectrum of nonrelativistic Coulomb problem and $P(E_{0})$ are the projectors onto corresponding subspaces of the nonrelativistic state space. One can obtain (\[NR\]) keeping leading term in $\alpha$-expansion of the lhs if one will keep $t\propto 1/\alpha^{2}$ and ${\bf r'},{\bf r}\propto 1/\alpha$ (see [@Steinman; @Pivovarov]). The subscript on $E_{0}$ is to denote that it will get radiative corrections (see below). The exponential in the lhs is to make a natural shift in energy zero. In what follows we will include the energy shift in the definition of $
D_{\beta}(t)$.
The next step in calculation of radiative corrections to the energy levels is a crucial one: one should make an assumption about the general form of a deformation of the $t$-dependence of the rhs of (\[NR\]) caused by relativistic corrections. A naturall guess and the one which leads to the generally accepted rules of calculation of the relativistic corrections to the energy eigenvalues (see, for example [@Lepage78]) is to suppose that one can contrive oscillating part of the exact propagator $D_{\beta}$ from the rhs of (\[NR\]) just shifting energy levels and modifying the operator coefficiens $P(E_{0})$: $$\label{guess}
D_{\beta}(t) = \sum_{E_{0}+\Delta_{E_{0}}} \theta(t)
e^{-i\left(E_{0}+\Delta_{E_{0}}\right)t}
P_{\beta}(E_{0}+\Delta_{E_{0}}) + \ldots,$$ where dots denote terms which are slowly-varying in time (the natural time-scale here is $1/E_{0}$). The additional subscript $\beta$ on $P_{\beta}$ is to denote that oscillating part of $D_{\beta}(t)$ can acquire a gauge parameter dependence from relativistic corrections.
Let us see how one can use eq.(\[guess\]) in energy level calculations. It is quite sufficient to consider $D_{\beta}(t)$ on relatively short times when $\Delta_{E_{0}} t\ll 1,\, E_{0}t\sim 1$. For such times one can approximate $D_{\beta}$ expanding the rhs of eq.(\[guess\]) over $\Delta_{E_{0}}t$: $$\label{simple}
D_{\beta}(t) \approx \sum_{E_{0}} \theta(t)e^{-iE_{0}t}
\sum_{k}t^{k}A^{(k)}_{\beta}(E_{0}),$$ where $$\label{AE}
A^{(k)}_{\beta}(E_{0}) = \sum_{\Delta_{E_{0}}}
\frac{(-i\Delta_{E_{0}})^{k}}{k!}P_{\beta}(E_{0}+\Delta_{E_{0}}).$$ An extraction of these objects from the perturbation theory is an interim step in the level shift calculations. (Here we should mention that in calculational practice $A^{(k)}_{\beta}(E_{0})$ are exracted in momentum representation — i.e. not as coefficients near the powers of time but as the ones near the propagator-like singularities $(E-E_{0}+i\varepsilon)^{-(k+1)}$.) To come nerer to the level shift values, useful objects are $$\label{A}
A^{(k)}_{\beta} \equiv \sum_{E_{0}}A^{(k)}_{\beta}(E_{0})i^{k}k!.$$ Namely, as notations of (\[AE\]) suggest, eigenvalues of $A^{(0)}_{\beta}$ should be equal to normalizations of bound state wave functions which are driven from unit by relativistic corrections while the eigenvalues of $A^{(k)}_{\beta}$ should be energy shifts to the $k$-th power times corresponding normalizations. Thus, the eigenvalues of $$\label{Skdef}
S^{(k)}_{\beta} \equiv \frac{\left[A^{(0)}_{\beta}\right]^{-1}A^{(k)}_{\beta}
+ A^{(k)}_{\beta}\left[A^{(0)}_{\beta}\right]^{-1}}{2}$$ should be just energy shifts to the $k$-th power. Thus, we define $$\label{Sdef}
S_{\beta} \equiv S_{\beta}^{(0)}$$ to be the energy shift operator: its eigenvalues are the energy level shifts caused by relativistic corrections. Our aim is now to check $\beta$-independence of $S_{\beta}$ eigenvalues.
Some notes are in order: If the conjecture (\[guess\]) is true, $A^{(0)}_{\beta}$ should commute with $S^{(k)}_{\beta}$ and the following relation should hold: $$\label{powerrel}
S^{(k)}_{\beta} = \left[S_{\beta}\right]^{k}$$ This relation was suggested as a check of the cojecture (\[guess\]) in [@Steinman] and, to our knowlege, has never been checked. Another thing to note is that relativistic corrections affects the form of the scalar product of wave functions and, thus, one shoud add a definition of operator products to the formal expressions (\[Skdef\]),(\[powerrel\]). But the level of accuracy to which we will operate permits us not to go into this complication and use the operator products as they are in the nonrelativistic approximation — i.e. as the convolution of the corresponding kernels.
The way to the gauge invariance check of the energy shift calculations is clear now: Using the gauge evolution relation (\[answer\]) one should find the $\beta$-dependence of $S_{\beta}$ and then of its eigenvalues. As $S_{\beta}$ is defined in (\[Sdef\]),(\[Skdef\]) through $A^{(k)}_{\beta}$’s which are, in turn, defined in (\[simple\]) through the propagator $D_{\beta}$, the first step is to simplify (\[answer\]) to the reduced case of zero relative time and total momentum of the fermion pair: $$\begin{aligned}
\label{reduced}
D_{\beta}(t,{\bf r'},{\bf r})&=&\left[
\frac{\left(1-({\bf r'}-{\bf r})^{2}/(4t^{2})
\right)}
{\left(1-(({\bf r'}+{\bf r})^{2}/(4t^{2})
\right)}
\right]^
{\frac{\alpha}{2\pi}(\beta-\beta_{0})}\times\nonumber \\
& &\left[
\frac{Z}
{m^{2}{\bf r'}^{2}m^{2}{\bf r}^{2}}
\right]^{\frac{\alpha}{4\pi}(\beta-\beta_{0})}
D_{\beta_{0}}(t,{\bf r'},{\bf r}).\end{aligned}$$ The factor in the square brackets of the second line is time-independent and futher factorizible on factors depending on either ingoing or outgoing pair parameters. This reduce the influence of this factor to a change in the normalization of states. Being interested in gauge invariance of energy shifts, we omit this factor in what follows. Let us turn to the analysis of the influence of the factor in the first line of (\[reduced\]).
This factor is close to unit in the atomic scale ${\bf r'},{\bf r}\sim 1/\alpha,\,t\sim1/\alpha^{2}$. We will use its approximate form: $$\label{approx}
Factor \approx 1 + \frac{\alpha}{2\pi}(\beta-\beta_{0})
\frac{{\bf r'}{\bf r}}{t^2} + O(\alpha^{5}).$$
One can read the dependence of $A^{(k)}_{\beta}$ on $\beta$ from (\[simple\]),(\[reduced\]),(\[approx\]) as $$\label{betadep}
A^{(k)}_{\beta} \approx A^{(k)}_{\beta_{0}} -
\frac{\alpha}{2\pi}\frac{(\beta-\beta_{0})}{(k+1)(k+2)}
{\bf r}A^{(k+2)}_{\beta_{0}}{\bf r},$$ where $\bf r$ is the vector operator of relative position of interacting particles. The mixing of different $A^{(k)}_{\beta}$’s with a change in the gauge parameter is due to the presence of $1/t^{2}$ in the rhs of (\[approx\]). Finally, using the definition (\[Sdef\]), relations (\[powerrel\]) and the fact that $$\label{unit}
A^{(0)} \approx 1$$ in the nonrelativistic approximation one can derive the following $\beta$-dependence of $S_{\beta}$: $$\begin{aligned}
\label{Sanswer}
S_{\beta}&\approx&S_{\beta_{0}} -\nonumber \\
& &\frac{\alpha}{2\pi}(\beta-\beta_{0})
\left(\frac{1}{6}{\bf r}S_{\beta_{0}}^{3}{\bf r} -
\frac{1}{4}S_{\beta_{0}}{\bf r}S_{\beta_{0}}^{2}{\bf r} -
\frac{1}{4}{\bf r}S_{\beta_{0}}^{2}{\bf r}S_{\beta_{0}}
\right).\end{aligned}$$ Treating the term in the last line of the rhs of the above relation as a perturbation, one can get an approximate value of the $\beta$-dependent piece of the energy shift just averaging the perturbation with respect to the corresponding eigenstate of $S_{\beta_{0}}$.
Thus, we get for the leading order of $\beta $-derivative of an energy shift the following representation: $$\label{leading}
\left(\frac{\partial}{\partial\beta}\Delta_{\beta}\right)_{L}=
-\frac{\alpha}{2\pi}
\left(\frac{1}{6}\left\langle
{\bf r}S_{L}^{3}{\bf r}\right\rangle -
\frac{1}{4}\left\langle S_{L}{\bf r}S_{L}^{2}{\bf r}\right\rangle -
\frac{1}{4}\left\langle{\bf r}S_{L}^{2}{\bf r}S_{L}\right\rangle
\right),$$ where $\langle\ldots\rangle$ means averaging with respect to the corresponding nonrelativistic eigenstate and the subscript $L$ means the leading order in $\alpha$-expansion.
Eq.(\[leading\]) is sufficient to define an order in $\alpha$ in which the energy shifts become gauge dependent: $$\label{order}
\left(\frac{\partial}{\partial\beta}\Delta_{\beta}\right)_{L}
\sim \alpha^{11}.$$ Here we have taken into account that ${\bf r}\sim1/\alpha$ and $S_{L}\sim\alpha^{4}$.
To have a gauge dependence in any observable is clearly unacceptable. In the next section we will see how one should correct the above procedure of energy shift extraction from the QED Green function to get rid of the gauge dependence of energy shifts.
A Way Out
=========
The procedure recalled in the previous section is based on the conjecture (\[guess\]). A consequence of this conjecture is the gauge dependence of energy shifts of (\[leading\]). One can conclude that the conjecture is wrong. In particular, as one can infer from eq.(\[reduced\]), the operator coefficients near the oscillating exponentials in (\[guess\]) shoud get a time dependence from relativistic corrections. Even if in some gauge they are time independent, the gauge parameter evolution should generate a dependence which in the leading order in $\alpha$ reduce itself to the following replacement in (\[guess\]): $$\label{replacement}
P_{\beta}(E_{0}+\Delta_{E_{0}})\rightarrow
P_{\beta}(E_{0}+\Delta_{E_{0}}) + \frac{\Sigma_{\beta}(E_{0})}{t^{2}}.$$ That $\Sigma_{\beta}(E_{0})$ has nothing to do with energy shifts but will give contributions to $A^{(k)}_{\beta}(E_{0})$’s from eq.(\[simple\]). Being gauge dependent these contributions lead to the gauge dependence of energy shifts.
The way to the correct procedure is to through away terms like $\Sigma_{\beta}(E_{0})/t^{2}$ prior to the definition of the energy shift operator. Thus, a necessary step in the process of extracting energy shifts from the QED Green function (and the one which necessity is not recognized in the stanard procedure) is to calculate and subtract contributions like the last term in the rhs of (\[replacement\]) from the propagator of the fermion pair.
Below we report on a calculation of $\Sigma_{\beta}(E_{0})$ from (\[replacement\]). The most economical way to calculate it is to note that the energy dependence of the Fourier transform of the corresponding contribution to the propagator is $$\label{fourier}
(E-E_{0})\ln(-(E-E_{0}+i\varepsilon))$$ and that it comes from diagrams describing radiation and subsequent absorption of a soft photon with no change in the level $E_{0}$ of the radiating and absorbing bound state. Similar contributions (with another power of energy before the $log$) are well known for the propagator of a charged fermion [@Lifshits]
The first step in our calculation is to present the pair propagator in the following form: $$\label{soft}
D_{\beta}(t)\approx\left(e^{L_{s}}e^{ie{\bf rA}(t)}D_{inv}(t,A)
e^{-ie{\bf rA}(0)}
\right)_{A=0},$$ where $L_{s}$ is the same as in (\[connection\]) except a restriction on the momentum of photon propagator — the range of its variation is restricted to the soft region which border is of order of atomic binding energies; the exponentials with gauge potential are originated from the ones in (\[hot\]); $D_{inv}$ is a descendant of $G_{inv}$ from (\[hot\]): to go over from $G_{inv}$ to $D_{inv}$ one should make all pairing of non-soft photons in $G_{inv}$ and all the reductions of space-time coordinats which was involved in going over from the $G_{\beta}$ of (\[Gf\]) to the $D_{\beta}$ of (\[propDef\]); at last, all gauge potentials in (\[soft\]) are taken at zero of space coordinate in accord with the $\delta({\bf x})$ of eq.(\[propDef\]). The difference between the lhs and the rhs of eq.(\[soft\]) does not conribute to the term under the calculation.
The leading in the nonrelativistic approximation contribution to $D_{inv}$ is the same as for $D_{\beta}$ — it is just the propagator of the nonrelativistic Coulomb problem. We explicitly calculate the leading contribution to the dependence of $D_{inv}(t,A)$ on the gauge potential in its expansion over soft momenta of the external photons. Not surprisingly, the dipole interaction of the pair with the external photon field arise in this approximation: $$\label{Adef}
D_{inv}(t,A) \approx \left(i\frac{\partial}{\partial t} - H_{c}
+ e{\bf r}{\cal E}(t)
\right)^{-1},$$ where $H_{c}$ is the hamiltonian of the nonrelativistic Coulomb problem and $\cal E$ is the strength of the electric field: $$\label{Edef}
{\cal E}(t)\equiv -\dot{{\bf A}}(t) + \nabla A_{0}(t).$$
Substituting (\[Adef\]) in (\[soft\]) and keeping terms with only one soft photon propagator we get expressions which sum contains the term under calculation: $$\label{r1}
e^{2}\left(L_{s}
{\bf rA}(t)D_{nr}(t){\bf rA}(0)\right)_{A=0},$$ $$\label{r2}
e^{2}\left(L_{s}
\int d\tau_{1}d\tau_{2}\,
D_{nr}(t-\tau_{1}){\bf r}{\cal E}(\tau_{1})
D_{nr}(\tau_{1}-\tau_{2}){\bf r}{\cal E}(\tau_{2})
D_{nr}(\tau_{2})\right)_{A=0} ,$$ $$\begin{aligned}
\label{r3}
ie^{2}\biggl(L_{s}
\int d\tau\,\bigl(
D_{nr}(t-\tau){\bf r}{\cal E}(\tau)D_{nr}(\tau){\bf rA}(0)&-&
\\
& & {\bf rA}(t)D_{nr}(t-\tau){\bf r}{\cal E}(\tau)D_{nr}(\tau)
\bigr)
\biggr)_{A=0},\nonumber\end{aligned}$$ where $D_{nr}(t)$ is the propagator of the nonrelativistic Coulomb problem from the rhs of eq.(\[NR\]).
The next step is to pick out a contribution of a level $E_{0}$ in (\[r1\]),(\[r2\]),(\[r3\]). That is achievable by the replacement $$\label{repl}
D_{nr}(t)\rightarrow e^{-iE_{0}t}\theta(t)P(E_{0}).$$
The last ingredient that one needs to calculate (\[r1\]),(\[r2\]),(\[r3\]) is the time dependence of the soft photon propagators. It can be deduced from (\[gfix\]) as $$\begin{aligned}
\label{time}
\left(L_{s}A_{i}(t_{1})A_{j}(t_{2})\right)&=&
\theta\left((t_{1}-t_{2})^{2}>t_{c}^{2}\right)
\frac{\delta_{ij}\left(-1+\frac{\beta}{2}\right)}{4\pi^{2}(t_{1}-t_{2})^{2}},
\nonumber \\
\left(L_{s}A_{i}(t_{1}){\cal E}_{j}(t_{2})\right)&=&
\theta\left((t_{1}-t_{2})^{2}>t_{c}^{2}\right)
\frac{\delta_{ij}}{2\pi^{2}(t_{1}-t_{2})^{3}},\nonumber \\
\left(L_{s}{\cal E}_{i}(t_{1}){\cal E}_{j}(t_{2})\right)&=&
\theta\left((t_{1}-t_{2})^{2}>t_{c}^{2}\right)
\frac{\delta_{ij}}{\pi^{2}(t_{1}-t_{2})^{4}}.\end{aligned}$$ Here the $\theta$-functions are to account for the softness of the participating photons ($t_{c}\sim 1/E_{0}$).
Taking (\[time\]) into account we get the following contributions from (\[r1\]),(\[r2\]),(\[r3\]): $$\begin{aligned}
\label{contr}
(\ref{r1})&\rightarrow& \frac{1}{t^{2}}\theta(t)e^{-iE_{0}t}
\frac{\alpha}{\pi}\left(-1+\frac{\beta}{2}\right)
{\bf r}P(E_{0}){\bf r},\nonumber \\
(\ref{r2})&\rightarrow& \frac{1}{t^{2}}\theta(t)e^{-iE_{0}t}
\frac{\alpha}{\pi}\frac{2}{3}P(E_{0}){\bf r}P(E_{0}){\bf r}P(E_{0}),
\nonumber \\
(\ref{r3})&\rightarrow& \frac{1}{t^{2}}\theta(t)e^{-iE_{0}t}
\frac{\alpha}{\pi}i\left(P(E_{0}){\bf r}P(E_{0}){\bf r} -
{\bf r}P(E_{0}){\bf r}P(E_{0})\right).\end{aligned}$$
The sum of the above terms yields the result of our calculation: $$\begin{aligned}
\label{sigmansw}
\Sigma_{\beta}(E_{0})&=&\frac{\alpha}{\pi}
\biggl( \frac{2}{3}P(E_{0}){\bf r}P(E_{0}){\bf r}P(E_{0}) +
(-1+\frac{\beta}{2}){\bf r}P(E_{0}){\bf r} +\nonumber\\
& & i(P(E_{0}){\bf r}P(E_{0}){\bf r} - {\bf r}P(E_{0}){\bf r}P(E_{0}))
\biggr) .\end{aligned}$$
One can explicitly check that $\beta$-dependence of $\Sigma_{\beta}(E_{0})$ is the right one — i.e. if one subtracts the $\Sigma$-term from the propagator before the definition of the energy shift operator, the latter becomes gauge independent. Another observation is that the $\Sigma$-term cannot be killed by any choice of the gauge (in contrast to the case of charged fermion propagator where an analogous term is equal to zero in the Yennie gauge).
Summing up, in this paper we derived a relation between QED Green functions of different gauges. We used it to check the gauge invariance of the energy shift operator. It turns out to be gauge dependent. This fact forced us to recognize that energy shifts are not one, and the only one, source for the positive powers of time near the oscillating exponentials in the propagator of the pair. We found a particular additional source of the positive powers of time which is responsible for the gauge dependence of the naive energy shift operator. We conclude by an observation that at the moment we have not a clear definition of the energy shift operator — to get it one needs a criterion for picking out contributions to the positive powers of time originating from the energy shifts.
The author is grateful to A. Kataev, E. Kuraev, V. Kuzmin, A. Kuznetsov, S. Larin, Kh. Nirov, V. Rubakov, D. Son, P. Tinyakov for helpful discussions. This work was supported in part by The Fund for Fundamental Research of Russia under grant 94-02-14428.
[99]{} C. I. Westbrook [*et al*]{}., [*Phys.Rev.Lett.*]{} [**58**]{}(1987)1328 P. Labelle, G. P. Lepage, U. Magnea, [*Order $m\alpha^{8}$ contribution to the decay rate of Orthopositronium*]{}, preprint CLNS/93/1199, 1993, hep-ph 9310208 I. B. Khriplovich, A. I. Milstein, [*JETP*]{} [**79**]{}(1994)379 W. E. Caswell, G. P. Lepage, [*Phys. Rev.*]{} [**A18**]{}(1978)810 R. Barbieri, E. Remiddi, [*Nucl. Phys.*]{} [**B141**]{}(1978)417 O. Steinman, [*Nucl. Phys.*]{} [**B119**]{}(1982)394 G. S. Adkins, [*Ann. Phys. (N.Y.)*]{} [**146**]{}(1983)78 A. N. Vassiliev, [*Functional Methods in Quantum Field Theory and Statistical Mechanics*]{} (L.S.U., Leningrad, 1976);in Russian G. B. Pivovarov,[*Improved Nonrelativistic QED and other Effective Field Theories*]{}, in ”Quarks-94”, eds. V. A. Matveev [*et al.*]{} (World Scientific), in press V. B. Beresteskii, E. M. Lifshits, L. P. Pitaevskii, [*Quantum Electrodinamics*]{} (Nauka, Moscow, 1980); in Russian
[^1]: e-mail address: gbpivo@ms2.inr.ac.ru
|
---
author:
- 'Yositake [Takane]{}'
title: Spin Injection and Detection in a Mesoscopic Superconductor at Low Temperatures
---
Introduction
============
Experimental studies on spin injection and detection in a normal metal have attracted considerable attention recently in the field of spintronics. [@rf:johnson1; @rf:jedema1; @rf:jedema2; @rf:kimura; @rf:takahashi] More than two decades ago, Johnson and Silsbee [@rf:johnson1] performed the first experiment on this subject by using a large normal metal sample with two electrodes made of a ferromagnetic metal, where each electrode serves as a spin injector or detector. Spin-polarized electrons created near the injector diffuse in the normal metal, and spin imbalance is transmitted to the detector if spin-flip scattering does not suppress it. They found an evidence of spin imbalance by measuring an open-circuit voltage induced at the detector. Several experiments using devices in the mesoscopic regime have been reported to date. [@rf:jedema1; @rf:jedema2; @rf:kimura] The most popular device consists of a thin normal metal wire connected to a few ferromagnetic metal wires. In this system, we supply injection current $I_{\rm inj}$ with spin polarization $P_{\rm spin}$ into the normal metal from one of ferromagnetic metals and measure an open-circuit voltage between another ferromagnetic metal and the normal metal. Let $V_{\rm p}$ ($V_{\rm ap}$) be the open-circuit voltage when the magnetizations of the two ferromagnetic metals are parallel (antiparallel). We are interested in the nonlocal spin signal defined by $$\begin{aligned}
\label{eq:def-Rs}
R_{\rm spin} = \frac{V_{\rm p}-V_{\rm ap}}{I_{\rm inj}} ,\end{aligned}$$ which crucially depends on the spin diffusion length $\lambda_{\rm sf}$ and the distance $d$ between the injection and detection points. In the case where the normal metal and the two ferromagnetic metals are connected by tunnel junctions, the spin signal is given by [@rf:jedema2; @rf:takahashi] $$\begin{aligned}
R_{\rm spin} = P_{\rm spin}^{2} R_{\rm N}
{\rm e}^{-\frac{d}{\lambda_{\rm sf}}} ,\end{aligned}$$ where $R_{\rm N} \equiv \rho_{\rm N}\lambda_{\rm sf}/A_{\rm N}$ with $\rho_{\rm N}$ and $A_{\rm N}$ being the resistivity and the cross-sectional area of the normal metal, respectively.
Spin injection and detection in a superconductor is also attracted considerable attention. [@rf:takahashi; @rf:johnson2; @rf:gu; @rf:shin; @rf:miura; @rf:urech; @rf:poli] Our primary interest focuses on how the spin signal is modified by the transition to the superconducting state. Takahashi and Maekawa [@rf:takahashi] studied this problem and predicted that $$\begin{aligned}
\label{eq:TM}
R_{\rm spin} = \frac{1}{2f_{0}(\Delta)}P_{\rm spin}^{2} R_{\rm N}
{\rm e}^{-\frac{d}{\lambda_{\rm sf}}} ,\end{aligned}$$ where $f_{0}(\Delta) = 1/({\rm exp}(\Delta/T)+1)$ with the superconducting energy gap $\Delta$ and temperature $T$. This indicates that $R_{\rm spin}$ exponentially increases with decreasing $T$. They claimed that this modification is caused by the increase of spin resistivity due to the opening of the energy gap $\Delta$. The increase of $R_{\rm spin}$ with decreasing $T$ has been successfully observed in the recent experiment by Poli *et al*. [@rf:poli] However, there remain a few points to be clarified. We focus on the following two points. Firstly, Takahashi and Maekawa implicitly assume in their derivation of eq. (\[eq:TM\]) that spin imbalance in a superconductor can be described by a shift of spin-dependent chemical potential. This assumption cannot be justified at low temperatures, where energy relaxation due to phonon scattering is not strong. Secondly, Poli *et al*. observed convergence of $R_{\rm spin}$ with decreasing temperature. This behavior cannot be explained by eq. (\[eq:TM\]).
In this paper, we theoretically study nonequilibrium spin transport in a hybrid system consisting of a superconducting wire and two ferromagnetic metal wires. Each ferromagnetic metal is connected by a tunnel junction to the superconductor, and serves as an injector or detector of spin-polarized quasiparticles. We present a set of Boltzmann equations governing nonequilibrium quasiparticles in this system. We focus on the case of small injection current at low temperatures, and obtain not only the quasiparticle distribution in the superconducting wire but also that in the ferromagnetic metal wire for detection. On the basis of the resulting nonequilibrium distributions, we derive an analytical expression for the nonlocal spin signal. It is shown that although the spin signal originates from spin imbalance transmitted to the detection junction, its magnitude is not solely determined by the spin imbalance but is strongly affected by the quasiparticle distribution in the ferromagnetic metal. We observe that when $T$ is higher than a crossover temperature $T_{\rm cross}$, the spin signal exponentially increases with decreasing $T$ reflecting the reduction of thermally excited quasiparticles in the ferromagnetic metal. At low temperatures below $T_{\rm cross}$, however, the magnitude of the spin signal is determined by nonequilibrium quasiparticles created by the tunneling from the superconductor instead of thermally excited ones, and the spin signal becomes independent of $T$. This explains the convergence of the spin signal with decreasing $T$ observed by Poli *et al*. [@rf:poli]
In the next section, we present a set of Boltzmann equations to describe nonequilibrium quasiparticle distributions in the hybrid system consisting of a superconducting wire and two ferromagnetic metal wires. In §3, we obtain nonequilibrium quasiparticle distributions in this system by solving the set of Boltzmann equations, and derive an analytical expression of the spin signal on the basis of the resulting quasiparticle distributions. In §4, we compare our theoretical result with the recent experimental result. We set $\hbar = k_{\rm B} = 1$ throughout this paper.
Formulation
===========
Let us consider the hybrid system consisting of a superconducting wire and two ferromagnetic metal wires (see Fig. 1).
{height="6cm"}
We assume that the superconductor is connected by a tunnel junction to each ferromagnetic metal. The left and right junctions serve as spin injector and detector, respectively. We adopt a simple one-dimensional model for this device assuming that the superconductor and the ferromagnetic metals are very thin. We introduce the $x$ axis in the superconductor on which the left and right junctions are located at $x = x_{\rm inj}$ and $x = x_{\rm det}$, respectively, and the $y$ axis in the left (right) ferromagnetic metal on which the injection (detection) junction is located at $y = y_{\rm inj}$ ($y_{\rm det}$). We denote by $d$ the separation between the two junctions. That is, $d \equiv x_{\rm det}-x_{\rm inj}$. We inject spin-polarized current into the superconductor by applying a bias voltage $V_{\rm inj}$ across the injection junction, and measure an induced open-circuit voltage $V_{\rm det}$ across the detection junction under the condition that net current flow vanishes between the superconductor and the ferromagnetic metal for detection. We simply assume that the spin polarization $P_{\rm spin}$ of the injection current is proportional to the difference between the density of states $N_{{\rm F}\uparrow}$ for up-spin electrons and $N_{{\rm F}\downarrow}$ for down-spin electrons. The spin polarization is expressed as $$\begin{aligned}
P_{\rm spin} = \frac{N_{{\rm F}\uparrow}-N_{{\rm F}\downarrow}}
{N_{{\rm F}\uparrow}+N_{{\rm F}\downarrow}} .\end{aligned}$$ We assume that spin relaxation in the superconductor is caused by spin-flip scattering due to spin-orbit interaction as well as magnetic impurities.
To present an expression for the tunneling current across each junction, we consider nonequilibrium quasiparticle distributions in the superconductor and the ferromagnetic metals. We first introduce the quasiparticle distribution function $g_{{\rm FL}\sigma}$ in the left ferromagnetic metal for injection, where $\sigma = \uparrow, \downarrow$ is the spin variable. We assume $g_{{\rm FL}\sigma}(y,\epsilon) = f_{0}(\epsilon-eV_{\rm inj})$ with the Fermi-Dirac distribution function $f_{0}(\epsilon)$. Here and hereafter, we measure quasiparticle energy from the chemical potential of the superconductor not only in the superconductor but also in the ferromagnetic metals. We next introduce the quasiparticle distribution function $g_{{\rm S}\sigma}$ in the superconductor. In terms of four distribution functions $f_{{\rm L}+}$, $f_{{\rm L}-}$, $f_{{\rm T}+}$ and $f_{{\rm T}-}$ for nonequilibrium quasiparticles, we express it as [@rf:schmid; @rf:hu; @rf:morten1; @rf:morten2; @rf:takane1] $$\begin{aligned}
\label{eq:f-up}
g_{{\rm S}\uparrow}(x,\epsilon)
& = f_{0}(\epsilon)
+ f_{{\rm L}+}(x,\epsilon) + f_{{\rm L}-}(x,\epsilon)
\nonumber \\
& \hspace{20mm}
+ f_{{\rm T}+}(x,\epsilon) + f_{{\rm T}-}(x,\epsilon) ,
\\
\label{eq:f-down}
g_{{\rm S}\downarrow}(x,\epsilon)
& = f_{0}(\epsilon)
- f_{{\rm L}+}(x,\epsilon) + f_{{\rm L}-}(x,\epsilon)
\nonumber \\
& \hspace{20mm}
+ f_{{\rm T}+}(x,\epsilon) - f_{{\rm T}-}(x,\epsilon) .\end{aligned}$$ The four distribution functions satisfy $$\begin{aligned}
f_{{\rm L,T}+}(x,-\epsilon) & = f_{{\rm L,T}+}(x,\epsilon) ,
\\
f_{{\rm L,T}-}(x,-\epsilon) & = - f_{{\rm L,T}-}(x,\epsilon) .\end{aligned}$$ Note that $f_{{\rm L}+}$ describes spin imbalance, while $f_{{\rm T}+}$ describes charge imbalance. [@rf:clarke; @rf:tinkham] The other two functions $f_{{\rm L}-}$ and $f_{{\rm T}-}$ describe total energy imbalance and energy imbalance between up-spin and down-spin quasiparticles, respectively. Finally, we introduce the distribution function $g_{{\rm FR}\sigma}$ in the right ferromagnetic metal in which nonequilibrium quasiparticles appear due to quasiparticle tunneling from the superconductor. We express it as $$\begin{aligned}
g_{{\rm FR}\sigma}(y,\epsilon)
= f_{0}(\epsilon-eV_{\rm det}) + f_{{\rm F}\sigma}(y,\epsilon) .\end{aligned}$$ We hereafter assume that the magnitude of the energy gap $\Delta$ is unaffected by spin injection everywhere in the superconductor. This allows us to consider $f_{{\rm L}\pm}(x,\epsilon)$ and $f_{{\rm T}\pm}(x,\epsilon)$ only for $|\epsilon| > \Delta$. The nonequilibrium distribution functions $f_{{\rm L}\pm}$, $f_{{\rm T}\pm}$ and $f_{{\rm F}\sigma}$ are governed by Boltzmann equations which we present below.
The tunneling current at the injection junction is given by $$\begin{aligned}
\label{eq:I_inj}
I_{\rm inj}(V_{\rm inj})
= \frac{\Delta}{eR_{\rm inj}} J_{1}(V_{\rm inj},T) ,\end{aligned}$$ where $R_{\rm inj}$ is the tunnel resistance of the injection junction and $$\begin{aligned}
J_{1}(V,T)
& = \frac{1}{\Delta}
\int_{0}^{\infty}{\rm d}\epsilon N_{1}(\epsilon)
\bigl( f_{0} \left(\epsilon-eV\right)
- f_{0} \left(\epsilon+eV\right)
\bigr) \end{aligned}$$ with $N_{1}$ being the normalized density of states in the superconductor, given by $N_{1}(\epsilon) = |\epsilon|/\sqrt{\epsilon^{2}-\Delta^{2}}$ for $|\epsilon| > \Delta$ in the BCS limit. In deriving eq. (\[eq:I\_inj\]), we have ignored small contributions arising from nonequilibrium quasiparticles in the superconductor. The tunneling current between the superconductor and the right ferromagnetic metal for detection is expressed as $$\begin{aligned}
I_{\rm det}(V_{\rm det})
= I_{\rm q}(V_{\rm det}) + I_{\rm F}(V_{\rm det}) - I_{\rm S}(V_{\rm det}) ,\end{aligned}$$ where $I_{\rm q}$ is the ordinary tunneling current arising from thermally excited quasiparticles, while $I_{\rm F}$ represents the contribution from nonequilibrium quasiparticles induced in the ferromagnetic metal. The third term $I_{\rm S}$ represents the contribution from spin and charge imbalances. They are expressed as $$\begin{aligned}
I_{\rm q}(V_{\rm det})
& = \frac{\Delta}{eR_{\rm det}} J_{1}(V_{\rm det},T) ,
\\
\label{eq:I_F}
I_{\rm F}(V_{\rm det})
& = \frac{1}{eR_{\rm det}} \int_{0}^{\infty} {\rm d}\epsilon \hspace{1mm}
N_{1}(\epsilon)
\nonumber \\
& \hspace{-5mm}
\times
\biggl( \frac{1+P_{\rm spin}}{2}
\left( f_{{\rm F}\uparrow}(y_{\rm det},\epsilon)
+ f_{{\rm F}\uparrow}(y_{\rm det},-\epsilon) \right)
\nonumber \\
& \hspace{0mm}
+ \frac{1-P_{\rm spin}}{2}
\left( f_{{\rm F}\downarrow}(y_{\rm det},\epsilon)
+ f_{{\rm F}\downarrow}(y_{\rm det},-\epsilon) \right)
\biggr) ,
\\
\label{eq:I_S}
I_{\rm S}(V_{\rm det})
& = \frac{2}{eR_{\rm det}} \int_{0}^{\infty} {\rm d}\epsilon \hspace{1mm}
N_{1}(\epsilon)
\nonumber \\
& \hspace{5mm}
\times
\bigl( P_{\rm spin} f_{{\rm L}+}(x_{\rm det}, \epsilon)
+ f_{{\rm T}+}(x_{\rm det}, \epsilon)
\bigr) ,\end{aligned}$$ where $R_{\rm det}$ is the tunnel resistance of the detection junction. In eq. (\[eq:I\_S\]), the first term with $f_{{\rm L}+}$ represents the contribution from spin imbalance and is the origin of the spin signal, while the second term with $f_{{\rm T}+}$ represents that from charge imbalance. In deriving eqs. (\[eq:I\_F\]) and (\[eq:I\_S\]), we have assumed the parallel alignment of magnetizations. The corresponding expressions for the antiparallel alignment is obtained by reversing the sign of $P_{\rm spin}$.
To present Boltzmann equations for $f_{{\rm L}\pm}$ and $f_{{\rm T}\pm}$, we introduce the Usadel equation [@rf:usadel] for the quasiclassical retarded Green’s functions $g^{R}$ and $f^{R}$, $$\begin{aligned}
\label{eq:usadel}
{\rm i} \epsilon f^{R}(\epsilon) + \Delta g^{R}(\epsilon)
- \frac{1}{\tau_{\rm m}} g^{R}(\epsilon)f^{R}(\epsilon) = 0 ,\end{aligned}$$ where $\tau_{\rm m}$ represents the magnetic impurity scattering time and we have assumed that the superconductor is spatially homogeneous. The spectral functions $N_{1}$, $N_{2}$, $R_{1}$ and $R_{2}$ are defined as $$\begin{aligned}
g^{R}(\epsilon) & = N_{1}(\epsilon) + {\rm i} R_{1}(\epsilon) ,
\\
f^{R}(\epsilon) & = N_{2}(\epsilon) + {\rm i} R_{2}(\epsilon) .\end{aligned}$$ In terms of the spectral functions, the Boltzmann equations are expressed as [@rf:schmid; @rf:hu; @rf:morten1; @rf:morten2; @rf:takane1] $$\begin{aligned}
\label{eq:fL+}
& D_{\rm S} \left( N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon) \right)
\partial_{x}^{2} f_{{\rm L}+}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
- \frac{4}{3\tau_{\rm so}} \left( N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon)
\right) f_{{\rm L}+}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
- \frac{4}{3\tau_{\rm m}} \left( N_{1}^{2}(\epsilon)+R_{2}^{2}(\epsilon)
\right) f_{{\rm L}+}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
+ P_{{\rm L}+} (x,\epsilon) = 0 ,
\\
\label{eq:fL-}
& D_{\rm S} \left( N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon) \right)
\partial_{x}^{2} f_{{\rm L}-}(x,\epsilon)
+ P_{{\rm L}-} (x,\epsilon) = 0 ,
\\
\label{eq:fT+}
& D_{\rm S} \left( N_{1}^{2}(\epsilon)+N_{2}^{2}(\epsilon) \right)
\partial_{x}^{2} f_{{\rm T}+}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
- \frac{1}{\tau_{\rm conv}(\epsilon)} f_{{\rm T}+}(x,\epsilon)
+ P_{{\rm T}+} (x,\epsilon) = 0 ,
\\
\label{eq:fT-}
& D_{\rm S} \left( N_{1}^{2}(\epsilon)+N_{2}^{2}(\epsilon) \right)
\partial_{x}^{2} f_{{\rm T}-}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
- \frac{4}{3\tau_{\rm so}} \left( N_{1}^{2}(\epsilon)+N_{2}^{2}(\epsilon)
\right) f_{{\rm T}-}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
- \frac{4}{3\tau_{\rm m}} \left( N_{1}^{2}(\epsilon)-N_{2}^{2}(\epsilon)
\right) f_{{\rm T}-}(x,\epsilon)
\nonumber \\
& \hspace{10mm}
- \frac{1}{\tau_{\rm conv}(\epsilon)} f_{{\rm T}-}(x,\epsilon)
+ P_{{\rm T}-} (x,\epsilon) = 0 ,\end{aligned}$$ where $D_{\rm S}$ is the diffusion constant, $\tau_{\rm so}$ and $\tau_{\rm conv}$ are the spin-orbit scattering time and the charge imbalance conversion time, respectively, and $P_{{\rm L}\pm}$ and $P_{{\rm T}\pm}$ are the injection terms which represent quasiparticle tunneling between the superconductor and the left ferromagnetic metal. The injection terms are given as [@rf:takane1; @rf:takane2] $$\begin{aligned}
\label{eq:PL+}
P_{{\rm L}+}(x,\epsilon)
& = \frac{\delta(x-x_{\rm inj})N_{1}(\epsilon)}
{4e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}}
\nonumber \\
& \hspace{-5mm}
\times
\Bigl[ P_{\rm spin}
\bigl( f_{0} \left(\epsilon-eV_{\rm inj}\right)
- f_{0} \left(\epsilon+eV_{\rm inj}\right)
\bigr)
\nonumber \\
& \hspace{0mm}
- 2 \bigl( f_{{\rm L}+}(x_{\rm inj},\epsilon)
+ P_{\rm spin} f_{{\rm T}+}(x_{\rm inj},\epsilon)
\bigr)
\Bigr] ,
\\
\label{eq:PL-}
P_{{\rm L}-}(x,\epsilon)
& = \frac{\delta(x-x_{\rm inj})N_{1}(\epsilon)}
{4e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}}
\nonumber \\
& \hspace{-5mm}
\times
\Bigl[ f_{0} \left(\epsilon+eV_{\rm inj}\right)
+ f_{0} \left(\epsilon-eV_{\rm inj}\right)
- 2 f_{0} \left(\epsilon\right)
\nonumber \\
& \hspace{0mm}
- 2 \bigl( f_{{\rm L}-}(x_{\rm inj},\epsilon)
+ P_{\rm spin} f_{{\rm T}-}(x_{\rm inj},\epsilon)
\bigr)
\Bigr] ,
\\
\label{eq:PT+}
P_{{\rm T}+}(x,\epsilon)
& = \frac{\delta(x-x_{\rm inj})N_{1}(\epsilon)}
{4e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}}
\nonumber \\
& \hspace{-5mm}
\times
\Bigl[ f_{0} \left(\epsilon-eV_{\rm inj}\right)
- f_{0} \left(\epsilon+eV_{\rm inj}\right)
\nonumber \\
& \hspace{0mm}
- 2 \bigl( P_{\rm spin} f_{{\rm L}+}(x_{\rm inj},\epsilon)
+ f_{{\rm T}+}(x_{\rm inj},\epsilon)
\bigr)
\Bigr] ,
\\
\label{eq:PT-}
P_{{\rm T}-}(x,\epsilon)
& = \frac{\delta(x-x_{\rm inj})N_{1}(x,\epsilon)}
{4e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}}
\nonumber \\
& \hspace{-5mm}
\times
\Bigl[ P_{\rm spin}
\bigl( f_{0} \left(\epsilon+eV_{\rm inj}\right)
+ f_{0} \left(\epsilon-eV_{\rm inj}\right)
- 2 f_{0} \left(\epsilon\right)
\bigr)
\nonumber \\
& \hspace{0mm}
- 2
\bigl( P_{\rm spin} f_{{\rm L}-}(x_{\rm inj},\epsilon)
+ f_{{\rm T}-}(x_{\rm inj},\epsilon)
\bigr)
\Bigr] ,\end{aligned}$$ where $N_{\rm S}$ and $A_{\rm S}$ are the density of states at the Fermi level in the normal state and the cross-sectional area of the superconducting wire, respectively. We can ignore $f_{{\rm L}\pm}(x_{\rm inj},\epsilon)$ and $f_{{\rm T}\pm}(x_{\rm inj},\epsilon)$ in these injection terms when the injection current is small. It should be noted that inelastic phonon scattering has been ignored in the Boltzmann equations because its role is not relevant at low temperatures, in which we are interested. We have also ignored very small contributions to $f_{{\rm L}\pm}$ and $f_{{\rm T}\pm}$ arising from the coupling with the right ferromagnetic metal for detection.
We turn to quasiparticle distributions in the right ferromagnetic metal for detection. We note that in obtaining $I_{\rm F}$, the spin-dependence of $f_{{\rm F}\sigma}$ is not important as long as the spin polarization is small. This indicates that we need not consider complicated spin-dependent dynamics of nonequilibrium quasiparticles. We thus define $$\begin{aligned}
f_{{\rm F}+}(y,\epsilon)
= \frac{ f_{{\rm F}\uparrow}(y,\epsilon)
+ f_{{\rm F}\downarrow}(y,\epsilon)}
{2} ,\end{aligned}$$ and approximate the expression of $I_{\rm F}$ as $$\begin{aligned}
\label{eq:I_F_mod}
I_{\rm F}(V_{\rm det})
& = \frac{1}{eR_{\rm det}} \int_{0}^{\infty} {\rm d}\epsilon \hspace{1mm}
N_{1}(\epsilon)
\nonumber \\
& \hspace{5mm}
\times
\bigl( f_{{\rm F}+}(y_{\rm det},\epsilon)
+ f_{{\rm F}+}(y_{\rm det},-\epsilon)
\bigr) .\end{aligned}$$ We present an appropriate Boltzmann equation for $f_{{\rm F}+}$. We ignore roles of spin-flip scattering since the spin-dependence is not important for our argument. However, we must consider the energy relaxation process due to phonon scattering. The reason for this is as follows. Since quasiparticles in the ferromagnetic metal are induced by the tunneling from the superconductor with the energy gap $\Delta$, their excitation energy is of the order of $\Delta$ and no quasiparticle is directly created in the subgap region. Quasiparticles in such a nonequilibrium situation inevitably experience the energy relaxation. We thus assume that $f_{{\rm F}+}$ obeys $$\begin{aligned}
\label{eq:fF+}
D_{\rm F} \partial_{y}^{2} f_{{\rm F}+}(y,\epsilon)
- \frac{1}{\tau_{\rm e}(\epsilon-eV_{\rm det})} f_{{\rm F}+}(y,\epsilon)
+ P_{{\rm F}+} (y,\epsilon) = 0 ,\end{aligned}$$ where $D_{\rm F}$ is the diffusion constant averaged over spin directions and $\tau_{\rm e}$ is the energy relaxation time with $\epsilon-eV_{\rm det}$ being the quasiparticle energy measured from the chemical potential of the ferromagnetic metal. The source term $P_{{\rm F}+}$ describing quasiparticle tunneling from the superconductor is given by $$\begin{aligned}
P_{{\rm F}+}(y,\epsilon)
& = \frac{\delta(y-y_{\rm det})N_{1}(\epsilon)}
{2e^{2}N_{\rm F}A_{\rm F}R_{\rm det}}
\bigl( f_{0}\left(\epsilon\right)
- f_{0} \left(\epsilon-eV_{\rm det}\right)
\nonumber \\
& \hspace{0mm}
- f_{{\rm F}+}(y_{\rm det},\epsilon)
+ f_{{\rm L}-}(x_{\rm det},\epsilon)
+ f_{{\rm T}+}(x_{\rm det},\epsilon)
\bigr) ,\end{aligned}$$ where $N_{\rm F} \equiv (N_{{\rm F}\uparrow}+N_{{\rm F}\downarrow})/2$ and $A_{\rm F}$ is the cross-sectional area of the ferromagnetic metal. For the expression of $\tau_{\rm e}$, we adopt $$\begin{aligned}
\frac{1}{\tau_{\rm e}(\epsilon)}
& = 2 \int_{-\infty}^{\infty} {\rm d}\epsilon'
\sigma_{\rm F}(\epsilon, \epsilon')
\nonumber \\
& \hspace{5mm}
\times
\left( \coth \left(\frac{\epsilon'-\epsilon}{2T}\right)
- \tanh \left(\frac{\epsilon'}{2T}\right)
\right)\end{aligned}$$ with $$\begin{aligned}
\label{eq:sigma_F}
\sigma_{\rm F}(\epsilon, \epsilon')
= \frac{\alpha_{\rm F}}{4} {\rm sign}(\epsilon'-\epsilon)
\times (\epsilon'-\epsilon)^{2} ,\end{aligned}$$ where $\alpha_{\rm F}$ characterizes the strength of electron-phonon coupling. For $|\epsilon| \gg T$, we approximately obtain $$\begin{aligned}
\frac{1}{\tau_{\rm e}(\epsilon)}
= \frac{\alpha_{\rm F}}{3} |\epsilon|^{3} .\end{aligned}$$
Spin Signal
===========
In this section, we solve the Boltzmann equations and obtain the spin signal defined in eq. (\[eq:def-Rs\]) by evaluating $V_{\rm p}$ and $V_{\rm ap}$. Note that $V_{\rm p}$ ($V_{\rm ap}$) is the open-circuit voltage induced across the detection junction when the magnetizations of the injector and detector are in the parallel (antiparallel) alignment. We determine $V_{\rm p}$ and $V_{\rm ap}$ by the condition of $I_{\rm det} = 0$. We assume that the magnitude of $V_{\rm p}$ and $V_{\rm ap}$ is much smaller than $\Delta/e$. However, we do not assume $V_{\rm inj} \ll \Delta/e$. We focus on the case where the injection current is so small that injected quasiparticles are populated only near the gap edge (i.e., $|\epsilon| \approx \Delta$). In this case, $f_{{\rm T}+}$ and $f_{{\rm T}-}$ quickly relaxes because the conversion time becomes very short near the gap edge. [@rf:tinkham; @rf:schmid; @rf:takane3; @rf:takane4] Therefore, we ignore $f_{{\rm T}+}$ and $f_{{\rm T}-}$ in the following argument. Furthermore, the smallness of the injection current also allows us to ignore $f_{{\rm L}+}(x_{\rm inj},\epsilon)$ and $f_{{\rm L}-}(x_{\rm inj},\epsilon)$ in the injection terms given in eqs. (\[eq:PL+\]) and (\[eq:PL-\]).
We first assume that magnetic impurities are absent (i.e., $\tau_{\rm m}^{-1} = 0$) and define the spin-flip scattering time $\tau_{\rm sf}$ as $$\begin{aligned}
\frac{1}{\tau_{\rm sf}} = \frac{4}{3\tau_{\rm so}} .\end{aligned}$$ In this case, the spectral functions for $|\epsilon| > \Delta$ are simply given by $$\begin{aligned}
\label{eq:N1-0}
N_{1}(\epsilon) & = \frac{|\epsilon|}
{\sqrt{\epsilon^{2}-\Delta^{2}}} ,
\\
\label{eq:R2-0}
R_{2}(\epsilon) & = \frac{{\rm sign} (\epsilon) \Delta}
{\sqrt{\epsilon^{2}-\Delta^{2}}} ,\end{aligned}$$ and $N_{2}(\epsilon) = R_{1}(\epsilon) = 0$. This indicates that $N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon) = 1$. We first obtain $f_{{\rm L}+}(x_{\rm det},\epsilon)$ by solving eq. (\[eq:fL+\]). Note that $f_{{\rm L}+}(x,\epsilon)$ decays exponentially as a function of $|x-x_{\rm inj}|$ and this decay is characterized by the spin-diffusion length given by $\lambda_{\rm sf} = \sqrt{D_{\rm S}\tau_{\rm sf}}$. We obtain $$\begin{aligned}
\label{eq:f_+det}
f_{{\rm L}+}(x_{\rm det},\epsilon)
= \frac{P_{\rm spin}\lambda_{\rm sf}}
{8e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}D_{\rm S}}
\Sigma_{+}(\epsilon,V_{\rm inj}) {\rm e}^{-\frac{d}{\lambda_{\rm sf}}}\end{aligned}$$ with $$\begin{aligned}
\Sigma_{+}(\epsilon,V_{\rm inj})
= N_{1}(\epsilon) \bigl( f_{0} \left(\epsilon-eV_{\rm inj}\right)
- f_{0} \left(\epsilon+eV_{\rm inj}\right)
\bigr) .\end{aligned}$$ Next, we obtain $f_{{\rm L}-}(x_{\rm det},\epsilon)$ which is necessary to obtain $f_{{\rm F}+}(y_{\rm det},\epsilon)$. A special care must be paid in solving eq. (\[eq:fL-\]) since no relaxation process is included in this equation. The relaxation of $f_{{\rm L}-}$ is mainly caused by the phonon-mediated recombination process, which is described by adding the following nonlinear term [@rf:takane2] $$\begin{aligned}
I_{{\rm L}-}(x,\epsilon)
& = - 4 \int {\rm d} \epsilon' \sigma_{\rm S}(\epsilon,\epsilon')
\nonumber \\
& \hspace{-5mm}
\times
\left( N_{1}(\epsilon)N_{1}(\epsilon')
- R_{2}(\epsilon)R_{2}(\epsilon') \right)
f_{{\rm L}-}(x,\epsilon)f_{{\rm L}-}(x,\epsilon')\end{aligned}$$ to eq. (\[eq:fL-\]). Here, $\sigma_{\rm S}(\epsilon,\epsilon')$ is identical to $\sigma_{\rm F}(\epsilon,\epsilon')$ in eq. (\[eq:sigma\_F\]) if $\alpha_{\rm F}$ is replaced by $\alpha_{\rm S}$. From this expression, we observe that the corresponding decay length $L_{\rm c}$ becomes very long when the injection current is small and therefore $|f_{{\rm L}-}(x,\epsilon)| \ll 1$. We thus assume that $L_{\rm c}$ is longer than, or at least of the order of, the length of the superconducting wire, and adopt the boundary condition that $f_{{\rm L}-}$ vanishes at each end of the superconducting wire. We further assume that the distance between the injection junction and each end of the superconductor is nearly equal to $L$, and $L \gg d$. Under this assumption, we approximately obtain $$\begin{aligned}
\label{eq:f_-det}
f_{{\rm L}-}(x_{\rm det},\epsilon)
= \frac{L}{8e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}D_{\rm S}}
\Sigma_{-}(\epsilon,V_{\rm inj})\end{aligned}$$ with $$\begin{aligned}
\Sigma_{-}(\epsilon,V_{\rm inj})
& = N_{1}(\epsilon) \bigl( f_{0} \left(\epsilon-eV_{\rm inj}\right)
+ f_{0} \left(\epsilon+eV_{\rm inj}\right)
\nonumber \\
& \hspace{30mm}
- 2 f_{0} \left(\epsilon\right)
\bigr) .\end{aligned}$$ If $L \gg L_{\rm c}$, we must replace $L$ in eq. (\[eq:f\_-det\]) with $L_{\rm c}$. Finally, we obtain $f_{{\rm F}+}(y_{\rm det},\epsilon)$. It should be emphasized that $I_{\rm F}$ containing $f_{{\rm F}+}$ becomes relevant in the low temperature regime where the ordinary contribution $I_{\rm q}$ is exponentially suppressed due to the opening of the energy gap. In this regime, the term with $f_{0}(\epsilon)-f_{0}(\epsilon-eV_{\rm det})$ in $P_{{\rm F}+}$ can be neglected. Furthermore, since $f_{{\rm T}+}$ can be ignored, the dominant contribution to $P_{{\rm F}+}$ arises from the term with $f_{{\rm L}-}$. This indicates that nonequilibrium quasiparticles are created by energy imbalance in the superconductor. We thus approximate $P_{{\rm F}+}$ as $$\begin{aligned}
P_{{\rm F}+}(y,\epsilon)
= \frac{\delta(y-y_{\rm det})N_{1}(\epsilon)}
{2e^{2}N_{\rm F}A_{\rm F}R_{\rm det}}
f_{{\rm L}-}(x_{\rm det},\epsilon) .\end{aligned}$$ Solving eq. (\[eq:fF+\]), we obtain $$\begin{aligned}
f_{{\rm F}+}(y_{\rm det},\epsilon)
= \frac{\lambda_{\rm e}(\epsilon-eV_{\rm det})}
{4e^{2}N_{\rm F}A_{\rm F}R_{\rm det}D_{\rm F}}
N_{1}(\epsilon) f_{{\rm L}-}(x_{\rm det},\epsilon) ,\end{aligned}$$ where the energy relaxation length $\lambda_{\rm e}$ is given by $\lambda_{\rm e}(\epsilon) = \sqrt{D_{\rm F}\tau_{\rm e}(\epsilon)}$. Combining this and eq. (\[eq:f\_-det\]) and noting that quasiparticles are populated near the gap edge (i.e., $|\epsilon| \approx \Delta$), we approximately obtain $$\begin{aligned}
\label{eq:f(+)+f(-)}
& f_{{\rm F}+}(y_{\rm det},\epsilon) + f_{{\rm F}+}(y_{\rm det},-\epsilon)
\nonumber \\
& \hspace{5mm}
= \frac{L}{8e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}D_{\rm S}}
\frac{\lambda_{\rm e}(\Delta)}
{4e^{2}N_{\rm F}A_{\rm F}R_{\rm det}D_{\rm F}}
\nonumber \\
& \hspace{10mm} \times
N_{1}(\epsilon) \Sigma_{-}(\epsilon,V_{\rm inj})
\frac{3eV_{\rm det}}{\Delta} .\end{aligned}$$ From eqs. (\[eq:I\_F\_mod\]) and (\[eq:f(+)+f(-)\]), we observe that $I_{\rm F}= 0$ at $V_{\rm det} = 0$. This reflects the fact that the quasiparticle distribution $f_{{\rm F}+}$ created by the tunneling of energy-imbalanced quasiparticles can contribute to the tunneling current only when the energy relaxation time for $f_{{\rm F}+}(y,\epsilon)$ is different from that for $f_{{\rm F}+}(y,-\epsilon)$. [@rf:takane2] That is, the energy relaxation process is essential in obtaining a nonzero $I_{\rm F}$.
We obtain $I_{\rm F}$ and $I_{\rm S}$ by substituting the resulting quasiparticle distributions into eqs. (\[eq:I\_S\]) and (\[eq:I\_F\_mod\]). The three terms are given as follows: $$\begin{aligned}
I_{\rm q} & = \chi(T) \frac{V_{\rm det}}{R_{\rm det}} ,
\\
I_{\rm F} & = \frac{3R_{\rm S}R_{\rm F}}{8R_{\rm inj}R_{\rm det}}
\frac{L\lambda_{\rm e}(\Delta)}{\lambda_{\rm sf}^{2}}
J_{3}(V_{\rm inj},T) \frac{V_{\rm det}}{R_{\rm det}} ,
\\
I_{\rm S} & = \eta
\frac{R_{\rm S}}{2R_{\rm inj}}
P_{\rm spin}^{2}{\rm e}^{-\frac{d}{\lambda_{\rm sf}}}
J_{2}(V_{\rm inj},T) \frac{\Delta}{eR_{\rm det}} ,\end{aligned}$$ where $\eta = 1 (-1)$ for the parallel (antiparallel) alignment and $$\begin{aligned}
\chi(T) & = \int_{0}^{\infty}{\rm d}\epsilon N_{1}(\epsilon)
\left(-2\frac{\partial f_{0}(\epsilon)}{\partial \epsilon}
\right) ,
\\
J_{3}(V,T)
& = \frac{1}{\Delta}
\int_{0}^{\infty}{\rm d}\epsilon N_{1}^{3}(\epsilon)
\bigl( f_{0} \left(\epsilon-eV\right)
+ f_{0} \left(\epsilon+eV\right)
\nonumber \\
& \hspace{40mm}
- 2 f_{0} \left(\epsilon\right)
\bigr) ,
\\
J_{2}(V,T)
& = \frac{1}{\Delta}
\int_{0}^{\infty}{\rm d}\epsilon N_{1}^{2}(\epsilon)
\bigl( f_{0} \left(\epsilon-eV\right)
- f_{0} \left(\epsilon+eV\right)
\bigr) .\end{aligned}$$ The resistances $R_{\rm S}$ and $R_{\rm F}$ are defined by $R_{\rm S} \equiv \lambda_{\rm sf}\rho_{\rm S}/A_{\rm S}$ and $R_{\rm F} \equiv \lambda_{\rm sf}\rho_{\rm F}/A_{\rm F}$ with the resistivities $\rho_{\rm S} = (2e^{2}N_{\rm S}D_{\rm S})^{-1}$ and $\rho_{\rm F} = (2e^{2}N_{\rm F}D_{\rm F})^{-1}$. It should be noted that $J_{3}(V,T)$ and $J_{2}(V,T)$ diverge if eq. (\[eq:N1-0\]) is adopted as the expression of $N_{1}(\epsilon)$. This unphysical divergence does not arise if we adopt a more realistic expression of $N_{1}(\epsilon)$, which does not diverges at the gap edge. Indeed, the divergence of $N_{1}(\epsilon)$ is actually removed if we take account of gap anisotropy, inelastic electron scattering or magnetic impurity scattering. We obtain $V_{\rm p}$ and $V_{\rm ap}$ by solving $I_{\rm det}(V_{\rm p}) = 0$ for $\eta = 1$ and $I_{\rm det}(V_{\rm ap}) = 0$ for $\eta = - 1$, respectively. Substituting the resulting expressions and eq. (\[eq:I\_inj\]) into eq. (\[eq:def-Rs\]), we finally obtain $$\begin{aligned}
\label{eq:R_spin-result}
R_{\rm spin} = \gamma
P_{\rm spin}^{2}R_{\rm S}{\rm e}^{-\frac{d}{\lambda_{\rm sf}}}\end{aligned}$$ with $$\begin{aligned}
\label{eq:gamma-result}
\gamma = \frac{J_{2}(V_{\rm inj},T)}
{J_{1}(V_{\rm inj},T)
\left( \chi(T)+\frac{3R_{\rm S}R_{\rm F}}
{8R_{\rm inj}R_{\rm det}}
\frac{L\lambda_{\rm e}(\Delta)}
{\lambda_{\rm sf}^{2}} J_{3}(V_{\rm inj},T)
\right)} .\end{aligned}$$ Note that $\gamma$ represents the renormalization of the spin signal induced by the transition to the superconducting state, and $\gamma = 1$ corresponds to the normal state.
In the remaining of this section, we briefly consider the influence of magnetic impurities. We redefine $\tau_{\rm sf}$ as $$\begin{aligned}
\frac{1}{\tau_{\rm sf}}
= \frac{4}{3\tau_{\rm so}} + \frac{4}{3\tau_{\rm m}} ,\end{aligned}$$ and introduce the parameter [@rf:poli] $$\begin{aligned}
\beta
= \frac{\tau_{\rm so}-\tau_{\rm m}}{\tau_{\rm so}+\tau_{\rm m}}\end{aligned}$$ which characterizes the relative strength of spin-orbit scattering and magnetic impurity scattering. Here, $\tau_{\rm sf}$ should be regarded as the spin-flip scattering time in the normal state. We observe that $\beta = - 1$ in the absence of magnetic impurities and $\beta = 1$ when spin-orbit scattering does not occur. If $\beta \neq - 1$, we must solve eq. (\[eq:usadel\]) to obtain the spectral functions. Strictly speaking, eqs. (\[eq:N1-0\]) and (\[eq:R2-0\]) are not justified in the presence of magnetic impurities and the relation $N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon) = 1$ no longer holds exactly. Consequently, $f_{{\rm L}+}(x_{\rm det},\epsilon)$ and $f_{{\rm L}-}(x_{\rm det},\epsilon)$ are modified as $$\begin{aligned}
\label{eq:f_+det-mag}
f_{{\rm L}+}(x_{\rm det},\epsilon)
& = \frac{P_{\rm spin}\alpha(\epsilon)\lambda_{\rm sf}}
{8e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}D_{\rm S}}
\frac{\Sigma_{+}(\epsilon,V_{\rm inj})}
{(N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon))}
\nonumber \\
& \hspace{30mm} \times
{\rm e}^{-\frac{d}{\alpha(\epsilon)\lambda_{\rm sf}}} ,
\\
\label{eq:f_-det-mag}
f_{{\rm L}-}(x_{\rm det},\epsilon)
& = \frac{L}{8e^{2}N_{\rm S}A_{\rm S}R_{\rm inj}D_{\rm S}}
\frac{\Sigma_{-}(\epsilon,V_{\rm inj})}
{(N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon))} ,\end{aligned}$$ where $$\begin{aligned}
\alpha(\epsilon)
= \sqrt{\frac{N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon)}
{N_{1}^{2}(\epsilon)+\beta R_{2}^{2}(\epsilon)}} .\end{aligned}$$ The parameter $\alpha(\epsilon)$ represents the renormalization of the spin-flip scattering time on transition to the superconducting state. [@rf:morten2] Using eqs. (\[eq:f\_+det-mag\]) and (\[eq:f\_-det-mag\]), we can show that eqs. (\[eq:R\_spin-result\]) and (\[eq:gamma-result\]) are applicable to this case if $J_{3}$ and $J_{2}$ are replaced by the following expressions, $$\begin{aligned}
J_{3}(V,T)
& = \frac{1}{\Delta}
\int_{0}^{\infty}{\rm d}\epsilon
\frac{N_{1}^{3}(\epsilon)}
{N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon)}
\bigl( f_{0} \left(\epsilon-eV\right)
\nonumber \\
& \hspace{20mm}
+ f_{0} \left(\epsilon+eV\right)
- 2 f_{0} \left(\epsilon\right)
\bigr) ,
\\
J_{2}(V,T)
& = \frac{1}{\Delta}
\int_{0}^{\infty}{\rm d}\epsilon
\frac{\alpha(\epsilon)N_{1}^{2}(\epsilon)}
{N_{1}^{2}(\epsilon)-R_{2}^{2}(\epsilon)}
{\rm e}^{-(\alpha(\epsilon)^{-1}-1)\frac{d}{\lambda_{\rm sf}}}
\nonumber \\
& \hspace{10mm} \times
\bigl( f_{0} \left(\epsilon-eV\right)
- f_{0} \left(\epsilon+eV\right)
\bigr) .\end{aligned}$$
We here comment on the expression of the spin signal presented by Poli *et al*. [@rf:poli] We note that they ignore the influence of nonequilibrium quasiparticles in the ferromagnetic metal for detection and therefore the corresponding term is lacking. This is the significant difference between their expression and ours. In addition, they assume $V_{\rm inj} \ll \Delta/e$. Finally, we point out that $\alpha(\epsilon)$-dependence is slightly different between them. Indeed, if the factor $2\alpha + N(E)R_{N}/R_{I}$ in eq. (4) of ref. is replaced by $2\alpha^{-1}$, their expression becomes nearly identical to ours in the case of $J_{3}(V_{\rm inj},T) = 0$ and $V_{\rm inj} \ll \Delta/e$. The reason for this difference is not clear.
Discussion
==========
Let us consider the temperature dependence of the renormalization factor $\gamma$ under the condition that the injection current $I_{\rm inj}$ is kept constant. We adjust $V_{\rm inj}$ to supply a constant injection current. This means that $V_{\rm inj}$ is determined as a function of $T$ for a given $I_{\rm inj}$, so we rewrite $J_{i}(V_{\rm inj},T)$ as $J_{i}(I_{\rm inj},T)$ ($i = 1,2,3$). It should be noted here that even though $I_{\rm inj}$ is very small, $V_{\rm inj}$ approaches to $\Delta/e$ as $T \to 0$.
We focus on the low temperature regime where the $T$-dependence of $\Delta$ can be neglected. In this regime, $\chi(T)$ behaves as $$\begin{aligned}
\label{eq:chai-LT}
\chi(T) = \sqrt{\frac{2\pi\Delta}{T}} {\rm e}^{-\frac{\Delta}{T}} .\end{aligned}$$ When $T$ is not very low and $\chi(T)$ is much greater than the term with $J_{3}(I_{\rm inj},T)$ in the denominator of eq. (\[eq:gamma-result\]), the renormalization factor is reduced to $$\begin{aligned}
\label{eq:gamma-result-HT}
\gamma = \frac{J_{2}(I_{\rm inj},T)}
{J_{1}(I_{\rm inj},T) \chi(T)} .\end{aligned}$$ Because the $T$-dependence of $J_{2}(I_{\rm inj},T)/J_{1}(I_{\rm inj},T)$ is weak, we obtain $\gamma \propto \chi(T)^{-1}$. This indicates that $\gamma$ behaves as $\gamma \propto {\rm e}^{\Delta/T}$. However, because $\chi(T)$ is exponentially suppressed with decreasing $T$, the term with $J_{3}(I_{\rm inj},T)$ eventually dominates $\chi(T)$ below a crossover temperature $T_{\rm cross}$. Below $T_{\rm cross}$, we can ignore $\chi(T)$ in eq. (\[eq:gamma-result\]) and the renormalization factor is reduced to $$\begin{aligned}
\label{eq:gamma-result-LT}
\gamma = \frac{8R_{\rm inj}R_{\rm det}}{3R_{\rm S}R_{\rm F}}
\frac{\lambda_{\rm sf}^{2}}{L\lambda_{\rm e}(\Delta)}
\frac{J_{2}(I_{\rm inj},T)J_{3}(I_{\rm inj},T)}
{J_{1}(I_{\rm inj},T)} .\end{aligned}$$ The crossover temperature is determined by $$\begin{aligned}
\label{eq:T_cross}
\chi(T_{\rm cross})
= \frac{3R_{\rm S}R_{\rm F}}
{8R_{\rm inj}R_{\rm det}}
\frac{L\lambda_{\rm e}(\Delta)}
{\lambda_{\rm sf}^{2}} J_{3}(I_{\rm inj},T_{\rm cross}) .\end{aligned}$$ As $T$ is lowered below $T_{\rm cross}$, the injection voltage $V_{\rm inj}$ approaches to $\Delta/e$. In this situation, the $T$-dependence of $J_{3}(I_{\rm inj},T)$ becomes weak. Furthermore, we can neglect the weak $T$-dependence of $J_{2}(I_{\rm inj},T)/J_{1}(I_{\rm inj},T)$. Thus, we conclude that below $T_{\rm cross}$, the renormalization factor $\gamma$ rapidly converges to the value given by $\gamma_{0} \equiv \lim_{T \to 0} \gamma$. We can obtain $\gamma_{0}$ from eq. (\[eq:gamma-result-LT\]) with $T = 0$.
From the above argument, we observe the qualitative behavior of $R_{\rm spin}$ as follows. In the regime of $T \gg T_{\rm cross}$, the spin signal exponentially increases with decreasing $T$ as $R_{\rm spin} \propto {\rm e}^{\Delta/T}$. Below $T_{\rm cross}$, however, the spin signal converges as $R_{\rm spin} \to \gamma_{0}
P_{\rm spin}^{2}R_{\rm S}{\rm e}^{-d/\lambda_{\rm sf}}$. We here point out that the behavior of $R_{\rm spin}$ in the regime of $T \gg T_{\rm cross}$ is qualitatively equivalent to the previous result, eq. (\[eq:TM\]), reported by Takahashi and Maekawa. [@rf:takahashi] However, our argument indicates that the exponential increase of $R_{\rm spin}$ should not be attributed to the increase of spin resistivity. [@rf:takahashi] We simply understand that $R_{\rm spin}$ increases reflecting the suppression of thermally excited quasiparticles in the detection junction.
Let us consider the experimental result reported by Poli *et al*. [@rf:poli] on the basis of our theoretical framework. Particularly, we focus on the convergence of $R_{\rm spin}$ observed at low temperatures. They employed the device consisting of a superconducting wire of Al and ferromagnetic metal wires of Co. Since it has been believed that spin-flip scattering in Al is mainly caused by spin-orbit interaction, we assume that magnetic impurity scattering is much less relevant than spin-orbit scattering and set $\tau_{\rm m}^{-1} = 0$. We estimate the limiting value $\gamma_{0}$ of the renormalization factor from eq. (\[eq:gamma-result-LT\]) with $T = 0$ and compare it with their experimental value. Following refs. and , we employ the parameters: $I_{\rm inj} = 1 \ {\rm nA}$, $\lambda_{\rm sf} = 1 \ \mu {\rm m}$, $\Delta = 200 \ \mu {\rm eV}$, $R_{\rm inj} = R_{\rm det} = 100 \ {\rm k}\Omega$, $\rho_{\rm S} = 10 \ \mu \Omega {\rm cm}$, $A_{\rm S} = 10 \times 150 \ {\rm nm}^{2}$, $A_{\rm F} = 50 \times 130 \ {\rm nm}^{2}$. For the other parameters, we assume $L = 10 \ \mu {\rm m}$, $D_{\rm F} = 3 \times 10^{-3} \ {\rm m^{2}s^{-1}}$, $\rho_{\rm F} = 14 \ \mu \Omega {\rm cm}$, $\alpha_{\rm F} = 9 \times 10^{3} \ {\rm eV}^{-2}$. The value of $\alpha_{\rm F}$ is estimated by using the relation [@rf:tinkham] $\alpha_{\rm F} \sim 2/\tau_{\rm D}T_{\rm D}^{3}$ with $T_{\rm D} = 385 \ {\rm K}$ and $\tau_{\rm D} = 0.4 \times 10^{-14} \ {\rm s}$, where $T_{\rm D}$ and $\tau_{\rm D}$ are the Debye temperature and the phonon scattering time at $T_{\rm D}$, respectively. From these parameters, we obtain $\lambda_{\rm e}(\Delta) = 9 \ \mu {\rm m}$, $R_{\rm S} = 67 \ \Omega$ and $R_{\rm F} = 22 \ \Omega$. The integral $J_{1}(I_{\rm inj},T)$ does not depend on $T$ and is obtained from eq. (\[eq:I\_inj\]) as $J_{1}(I_{\rm inj},T) = eR_{\rm inj}I_{\rm inj}/\Delta = 0.5$. We finally consider $J_{2}(I_{\rm inj},T)$ and $J_{3}(I_{\rm inj},T)$ in the limit of $T \to 0$. The evaluation of these integrals is not simple, so we roughly approximate them as $J_{2}(I_{\rm inj},0) = J_{3}(I_{\rm inj},0) = J_{1}(I_{\rm inj},T)$. Substituting these parameters into eq. (\[eq:gamma-result-LT\]), we approximately obtain $\gamma_{0} \sim 10^{5}$. This indicates that $R_{\rm spin}$ below $T_{\rm cross}$ is a factor of $10^{5}$ larger than that in the normal state. This is consistent with the experimental result which indicates the enhancement of $4$ or $5$ orders of magnitude. We estimate the crossover temperature by solving eq. (\[eq:T\_cross\]) with eq. (\[eq:chai-LT\]) and obtain $T_{\rm cross} \sim 0.1 \ {\rm K}$. This is also consistent with the experimental value of $T_{\rm cross} \sim 0.16 \ {\rm K}$.
We have shown that nonequilibrium quasiparticles with $|\epsilon| \approx \Delta$ are created in the ferromagnetic metal for detection by the tunneling of energy-imbalanced quasiparticles, and that these quasiparticles contribute to $I_{\rm det}$ in combination with the energy relaxation process due to phonon scattering. It should be noted that the energy relaxation of quasiparticles excites phonons near the detection junction, leading to the increase of effective temperature $T_{\rm eff}$ for quasiparticles. If $T_{\rm eff}$ becomes greater than $T_{\rm cross}$, the convergence of the spin signal is determined by this heating effect instead of the convergence mechanism which we discussed above. The separation of these two mechanisms is a future problem for experiments.
In addition to the heating effect, we have ignored charge imbalance. If the injection current is not small, we must consider its influences. Charge imbalance provides a nearly constant contribution $I_{\rm Q}$ to $I_{\rm S}$ regardless of the alignment of magnetizations. Since $I_{\rm Q}$ must be cancelled by $I_{\rm q}$ and $I_{\rm F}$ to ensure $I_{\rm det} = 0$, we expect that both $V_{\rm p}$ and $V_{\rm ap}$ increases with increasing $I_{\rm Q}$. However, if $I_{\rm Q}$ is sufficiently small, the increase of $V_{\rm p}$ is equivalent to that of $V_{\rm ap}$ because both $I_{\rm q}(V_{\rm det})$ and $I_{\rm F}(V_{\rm det})$ linearly depends on $V_{\rm det}$ when $|V_{\rm det}| \ll \Delta/e$. Therefore, we expect that no qualitative change of the spin signal appears as long as charge imbalance is not very large.
In summary, we have studied the transport of spin-polarized nonequilibrium quasiparticles in a superconducting wire connected by tunnel junctions to two ferromagnetic metal wires, each of which serves as a spin injector or detector. We have presented a basic formalism to determine spin-polarized quasiparticle distributions in this system, and obtained an analytical expression for the nonlocal spin signal. We have taken account of nonequilibrium quasiparticles in the ferromagnetic metal for detection, which are created by the tunneling of energy-imbalanced quasiparticles in the superconductor. We have shown that they induce the convergence of the spin signal at low temperatures.
[99]{}
M. Johnson and R. H. Silsbee: Phys. Rev. Lett. [**55**]{} (1985) 1790.
F. J. Jedema, A. T. Filip, and B. J. van Wees: Nature (London) [**410**]{} (2001) 345.
F. J. Jedema, H. B. Heersche, A. T. Filip, J.J.A. Baselmans, and B.J. van Wees: Nature (London) [**416**]{} (2002) 713.
T. Kimura, J. Hamrle, Y. Otani, K. Tsukagoshi, and Y. Aoyagi: Appl. Phys. Lett. [**85**]{} (2004) 3501.
S. Takahashi and S. Maekawa: Phys. Rev. B [**67**]{} (2003) 052409.
M. Johnson: Appl. Phys. Lett. [**65**]{} (1994) 1460.
J. Y. Gu, J. A. Caballero, R. D. Slater, R. Loloee, and W. P. Pratt, Jr.: Phys. Rev. B [**66**]{} (2002) 140507.
Y.-S. Shin, H.-J. Lee, and H.-W. Lee: Phys. Rev. B [**71**]{} (2005) 144513.
K. Miura, S. Kasai, K. Kobayashi, and T. Ono: Jpn. J. Appl. Phys. [**45**]{} (2006) 2888.
M. Urech, J. Johansson, N. Poli, V. Korenivski, and D.B. Haviland: J. Appl. Phys. [**99**]{} (2006) 08M513.
N. Poli, J. P. Morten, M. Urech, A. Brataas, D. B. Haviland, and V. Korenivski: Phys. Rev. Lett. [**100**]{} (2008) 136601.
A. Schmid and G. Sch[ö]{}n: J. Low Tem. Phys. [**20**]{} (1975) 207.
C.-R. Hu: Phys. Rev. B [**21**]{} (1980) 2775.
J. P. Morten, A. Brataas, and W. Belzig: Phys. Rev. B [**70**]{} (2004) 212508.
J. P. Morten, A. Brataas, and W. Belzig: Phys. Rev. B [**72**]{} (2005) 014510.
Y. Takane: J. Phys. Soc. Jpn. [**75**]{} (2006) 074711.
J. Clarke: Phys. Rev. Lett. [**28**]{} (1972) 1363.
M. Tinkham: Phys. Rev. B [**6**]{} (1972) 1747.
K. D. Usadel: Phys. Rev. Lett. [**25**]{} (1970) 507.
Y. Takane: J. Phys. Soc. Jpn. [**76**]{} (2007) 043701.
Y. Takane: J. Phys. Soc. Jpn. [**75**]{} (2006) 023706.
Y. Takane and Y. Nagato: J. Phys. Soc. Jpn. [**77**]{} (2008) 093713.
|
---
abstract: 'We prove that the complete $L$-function associated to any cuspidal automorphic representation of $\operatorname{GL}_2({\mathbb{A}}_{\mathbb{Q}})$ has infinitely many simple zeros.'
address:
- 'School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom'
- 'Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, Korea'
- 'Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, Korea'
author:
- 'Andrew R. Booker'
- 'Peter J. Cho'
- Myoungil Kim
bibliography:
- 'BCK.bib'
title: 'Simple zeros of automorphic $L$-functions'
---
[^1]
[^2]
Introduction
============
In [@Booker], the first author showed that the complete $L$-functions associated to classical holomorphic newforms have infinitely many simple zeros. The purpose of this paper is to extend that result to the remaining degree $2$ automorphic $L$-functions over ${\mathbb{Q}}$, i.e.those associated to cuspidal Maass newforms. This also extends work of the second author [@Cho] which established a quantitative estimate for the first few Maass forms of level $1$. When combined with the holomorphic case from [@Booker], we obtain the following:
\[thm:main\] Let ${\mathbb{A}}_{\mathbb{Q}}$ denote the adèle ring of ${\mathbb{Q}}$, and let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2({\mathbb{A}}_{\mathbb{Q}})$. Then the associated complete $L$-function $\Lambda(s,\pi)$ has infinitely many simple zeros.
The basic idea of the proof is the same as in [@Booker], which is in turn based on the method of Conrey and Ghosh [@CG]. Let $f$ be a primitive Maass cuspform of weight $k\in\{0,1\}$ for $\Gamma_0(N)$ with nebentypus character $\xi$, and let $L_f(s)$ be the finite $L$-function attached to $f$: $$L_f(s)=\sum_{n=1}^\infty\lambda_f(n) n^{-s}.$$ We define $$D_f(s)=L_f(s) \frac{d^2}{ds^2} \log{L_f(s)}=\sum_{n=1}^\infty c_f(n)n^{-s}.$$ Then it is easy to see that $D_f(s)$ has a pole at some point if and only if $L_f(s)$ has a simple zero there.
For $\alpha\in{\mathbb{Q}}$ and $j\ge0$ we define the additive twists $$L_f(s,\alpha,\cos^{(j)})=\sum_{n=1}^\infty\lambda_f(n)
\cos^{(j)}(2\pi n\alpha)n^{-s},\quad
D_f(s,\alpha,\cos^{(j)})=\sum_{n=1}^\infty c_f(n)
\cos^{(j)}(2\pi n\alpha)n^{-s},$$ where $\cos^{(j)}$ denotes the $j$th derivative of the cosine function. Let $q\nmid N$ be a prime and $\chi_0$ the principal character mod $q$. Then we have the following expansions of the trigonometric functions in terms of Dirichlet characters: $$\begin{aligned}
\cos\!\left(\frac{2\pi n}q\right)
&=1-\frac{q}{q-1}\chi_0(n) + \frac{\sqrt{q}}{q-1}
\sum_{\substack{\chi\pmod*{q}\\\chi(-1)=1\\\chi\neq\chi_0}}
\overline{\epsilon_\chi}\chi(n),\\
\sin\!\left(\frac{2\pi n}q\right)
&=\frac{\sqrt{q}}{q-1}
\sum_{\substack{\chi\pmod*{q}\\\chi(-1)=-1}}
\overline{\epsilon_\chi}\chi(n),\\\end{aligned}$$ where $\epsilon_\chi$ denotes the root number of the Dirichlet $L$-function $L(s,\chi)$. In particular, we have $$D_f(s,\tfrac1q,\cos)=D_f(s)-\frac{q}{q-1}D_f(s,\chi_0) + \frac{\sqrt{q}}{q-1}
\sum_{\substack{\chi\pmod*{q}\\\chi(-1)=1\\\chi\neq\chi_0}}
\overline{\epsilon_\chi}D_f(s,\chi),$$ where $$D_f(s,\chi)=\sum_{n=1}^\infty c_f(n)\chi(n)n^{-s}$$ is the corresponding multiplicative twist.
By the non-vanishing results for automorphic $L$-functions [@JS], all non-trivial poles of $D_f(s)$ and $D_f(s,\chi)$ for $\chi\ne\chi_0$ are located in the critical strip $\{s\in{\mathbb{C}}:0<\Re(s)<1\}$. However, for the case of the principal character, since $$L_f(s,\chi_0)=\sum_{n=1}^\infty\lambda_f(n)\chi_0(n) n^{-s}
=(1-\lambda_f(q)q^{-s} + \xi(q) q^{-2s})L_f(s),$$ $D_f(s,\chi_0)$ has a pole at every simple zero of the local Euler factor polynomial, $1-\lambda_f(q)q^{-s}+\xi(q)q^{-2s}$, at which $L_f(s)$ does not vanish.
Since $f$ is cuspidal, the Rankin–Selberg method implies that the average of $|\lambda_f(q)|^2$ over primes $q$ is $1$, i.e.$$\label{eq:RS}
\lim_{x\to\infty}
\frac{\sum_{\substack{q\text{ prime}\\q\le x}}|\lambda_f(q)|^2}
{\#\{q\text{ prime}:q\le x\}}
=1.$$ To see this, write $$-\frac{L_f'}{L_f}(s)=\sum_{n=1}^\infty\Lambda(n)a_nn^{-s},$$ where $\Lambda$ is the von Mangoldt function and $a_n=0$ unless $n$ is prime or a prime power. Then by [@LY Lemma 5.2], we have $$\label{eq:RS2}
\sum_{n\le x}\Lambda(n)|a_n|^2\sim x
\quad\text{as }x\to\infty.$$ By the estimate of Kim and Sarnak [@Kim], we have $|a_n|\le n^{7/64}+n^{-7/64}$, so the contribution of composite $n$ to is $O(x^{\frac{23}{32}})$. Since $a_q=\lambda_f(q)$ for primes $q$, this implies that $$\sum_{\substack{q\text{ prime}\\q\le x}}(\log{q})|\lambda_f(q)|^2\sim x,$$ and follows by partial summation and the prime number theorem.
In particular, there are infinitely many $q\nmid N$ such that $|\lambda_f(q)|<2$. For any such $q$, it follows that $D_f(s,\chi_0)$ has infinitely many poles on the line $\Re(s)=0$. In view of the above, $D_f(s,1/q,\cos)$ inherits these poles when they occur. On the other hand, under the assumption that $L_f(s)$ has at most finitely many non-trivial simple zeros, we will show that $D_f(s,1/q,\cos)$ is holomorphic apart from possible poles along two horizontal lines. The contradiction between these two implies the main theorem.
Overview
--------
We begin with an overview of the proof. First, by [@DFI (4.36)], $f$ has the Fourier–Whittaker expansion $$f(x+iy)=\sum_{n=1}^\infty
\left(\rho(n)W_{\frac k2,\nu}(4\pi ny)e(nx)
+\rho(-n)W_{-\frac k2,\nu}(4\pi ny)e(-nx)\right),$$ where $W_{\alpha,\beta}$ is the Whittaker function defined in [@DFI (4.20)], and $\nu=\sqrt{\frac14-\lambda}$, where $\lambda$ is the eigenvalue of $f$ with respect to the weight $k$ Laplace operator. When $k=1$, the Selberg eigenvalue conjecture holds, so that $\nu\in i[0,\infty)$. When $k=0$ the conjecture remains open, but we have the partial result of Kim–Sarnak [@Kim] that $\nu\in(0,\frac{7}{64}]\cup i[0,\infty)$.
Since $f$ is primitive, it is an eigenfunction of the operator $Q_{sk}$ defined in [@DFI (4.65)], so that $$\rho(-n)=\epsilon\frac{\Gamma(\frac{1+k}2+\nu)}{\Gamma(\frac{1-k}2+\nu)}\rho(n)
=\epsilon\nu^k\rho(n)$$ for some $\epsilon\in\{\pm1\}$. Further, we have $\rho(n)=\rho(1)\lambda_f(n)/\sqrt{n}$. Choosing the normalization $\rho(1)=\pi^{-\frac{k}2}$ and writing $e(\pm nx)=\cos(2\pi nx)\pm i\sin(2\pi nx)$, we obtain the expansion $$\label{eq:fseries}
f(x+iy)=\sum_{n=1}^\infty \frac{\lambda_f(n)}{\sqrt{n}}
\bigl(V_f^+(ny)\cos(2\pi nx)
+iV_f^-(ny)\sin(2\pi nx)\bigr),$$ where $$\label{eq:Vdef}
V_f^{\pm}(y)=\pi^{-\frac{k}2}\left(
W_{\frac{k}2,\nu}(4\pi y)\pm\epsilon\nu^kW_{-\frac{k}2,\nu}(4\pi y)\right)
=\begin{cases}
4\sqrt{y}K_{\nu}(2\pi y)
&\text{if }k=0\text{ and }\epsilon=\pm1,\\
0&\text{if }k=0\text{ and }\epsilon=\mp1,\\
4yK_{\nu\pm\frac{\epsilon}2}(2\pi y)
&\text{if }k=1.
\end{cases}$$
Let $\bar{f}(z):=\overline{f(-\bar{z})}$ denote the dual of $f$. Since $f$ is primitive, it is also an eigenfunction of the operator $\overline{W}_k$ defined in [@DFI (6.10)], so we have $$\label{eq:fmod}
f(z)=\eta\left(i\frac{|z|}{z}\right)^k
\bar{f}\!\left(-\frac1{Nz}\right)$$ for some $\eta\in{\mathbb{C}}$ with $|\eta|=1$.
Next we define a formal Fourier series $F(z)$ associated to $D_f(s)$ by replacing $\lambda_f(n)$ in the above by $c_f(n)$: $$\begin{aligned}
F(x+iy)=\sum_{n=1}^\infty \frac{c_f(n)}{\sqrt{n}}
\bigl(V_f^+(ny)\cos(2\pi nx)
+iV_f^-(ny)\sin(2\pi nx)\bigr).\end{aligned}$$ We expect $F(z)$ to satisfy a relation similar to the modularity relation . To make this precise, we first recall the functional equation for $L_f(s)$. Define $$\label{eq:gammadef}
\gamma_f^{\pm}(s)
=\Gamma_{\mathbb{R}}\!\left(s+\frac{1\mp(-1)^k\epsilon}2+\nu\right)
\Gamma_{\mathbb{R}}\!\left(s+\frac{1\mp\epsilon}2-\nu\right).$$ Then the complete $L$-function $\Lambda_f(s):=\gamma_f^+(s)L_f(s)$ satisfies $$\label{eq:FE}
\Lambda_f(s)=\eta\epsilon^{1-k}N^{\frac12-s}\Lambda_{\bar{f}}(1-s),$$ with $\eta$ as above.
We define a completed version of $D_f(s)$ by multiplying by the same $\Gamma$-factor: $\Delta_f(s):=\gamma_f^+(s)D_f(s)$. Then, differentiating the functional equation , we obtain $$\label{eq:FEofD}
\Delta_f(s)+\bigl(\psi_f'(s)-\psi_{\bar{f}}'(1-s)\bigr)\Lambda_f(s)
=\eta\epsilon^{1-k}N^{\frac12-s}\Delta_{\bar{f}}(1-s),$$ where $\psi_f(s):=\frac{d}{ds}\log\gamma_f^+(s)$. In Section \[sec:AFE\], we take a suitable inverse Mellin transform of . Under the assumption that $\Lambda_f(s)$ has at most finitely many simple zeros, this yields a pseudo-modularity relation for $F$ of the form $$\label{eq:FEofF}
F(z)+A(z) =\eta\left(i\frac{|z|}{z}\right)^k
\overline{F}\!\left(-\frac1{Nz}\right)+B(z),$$ for certain auxiliary functions $A$ and $B$, where $\overline{F}(z):=\overline{F(-\bar{z})}$. Roughly speaking, $A$ is the contribution from the correction term $\bigl(\psi_f'(s)-\psi_{\bar{f}}'(1-s)\bigr)\Lambda_f(s)$ in , and $B$ comes from the non-trivial poles of $\Delta_f(s)$.
The main technical ingredient needed to carry this out is the following pair of Mellin transforms involving the $K$-Bessel function and trigonometric functions [@GR 6.699(3) and 6.699(4)]: $$\label{eq:Ksin}
\int_0^\infty x^{\lambda+1} K_\mu (ax) \sin(bx) \frac{dx}{x}
=2^\lambda b \Gamma\!\left( \frac{2+\lambda+\mu}{2}\right)
\Gamma\!\left( \frac{2+\lambda - \mu}{2}\right)
\operatorname{{}_2F_1}\!\left(\frac{2+\lambda+\mu}{2},\frac{2+\lambda-\mu}{2};\frac{3}{2};-\frac{b^2}{a^2}\right)$$ and $$\label{eq:Kcos}
\int_0^\infty x^{\lambda+1} K_\mu (ax) \cos(bx) \frac{dx}{x}
=\frac{2^{\lambda-1}}{a^{\lambda+1}}
\Gamma\!\left( \frac{1+\lambda+\mu}{2}\right)
\Gamma\!\left( \frac{1+\lambda - \mu}{2}\right)
\operatorname{{}_2F_1}\!\left( \frac{1+\lambda+\mu}{2},\frac{1+\lambda - \mu}{2};\frac{1}{2};-\frac{b^2}{a^2}\right),$$ where $$\label{eq:2F1def}
\operatorname{{}_2F_1}(a,b;c;z)=
\sum_{j=1}^\infty\frac{a(a+1)\cdots(a+j-1)
\cdot b(b+1)\cdots(b+j-1)}{c(c+1)\cdots(c+j-1)}
\frac{z^j}{j!}$$ is the Gauss hypergeometric function. The origin of these hypergeometric factors is explained in the introduction to [@BT], and the need to analyze them is the main difference between this paper and the holomorphic case from [@Booker] (for which corresponding factors are elementary functions).
Specializing to $z=\alpha+iy$ for $\alpha\in{\mathbb{Q}}^\times$, we have $$\label{eq:mainid}
F(\alpha+iy)+A(\alpha+iy)=\eta\left(i\frac{|\alpha+iy|}{\alpha+iy}\right)^k
\overline{F}\!\left(-\frac1{N(\alpha+iy)}\right)+B(\alpha+iy).$$ We will take the Mellin transform of . Without difficulty the reader can guess that the transform of $F(\alpha+iy)$ will be a combination of $D_f(s,\alpha,\cos)$ and $D_f(s,\alpha,\sin)$. The calculation of the other terms is non-trivial, but ultimately we obtain the following proposition, which will play the role of Proposition 2.1 in [@Booker]:
\[prop:main\] Suppose that $\Lambda_f(s)$ has at most finitely many simple zeros. Then, for every $M\in{\mathbb{Z}}_{\ge0}$ and $a\in\{0,1\}$, $$\begin{aligned}
&P_f(s;a,0)\Delta_f(s,\alpha, \cos^{(a+k)})\\
&-\eta(-\operatorname{sgn}\alpha)^k(N\alpha^2)^{s-\frac12}
\sum_{m=0}^{M-1}\frac{(2\pi N\alpha)^m}{m!}
P_f(s;a,m)\Delta_{\bar{f}}\!\left(
s+m,-\frac1{N\alpha},\cos^{(a+m)}\right)\end{aligned}$$ is holomorphic for $\Re(s)>\frac32-M$ except for possible poles for $s\pm\nu\in{\mathbb{Z}}$, where $$P_f(s;a,m)
=\frac{\gamma_{f}^{(-)^a}(1-s) }{\gamma_{f}^{(-)^a}(1-s-2\lfloor m/2\rfloor)}
\begin{cases}
\frac{s+2\lfloor m/2\rfloor-(-1)^a\epsilon\nu}{2\pi}
&\text{if }k=1\text{ and }2\nmid m,\\
0&\text{if }k=0\text{ and }(-1)^a=-\epsilon,\\
1&\text{otherwise}
\end{cases}$$ and $$\Delta_{f}(s,\alpha,\cos^{(a)})=
\gamma_{f}^{(-)^a}(s)D_{f}(s,\alpha,\cos^{(a)}).$$
Proof of Theorem \[thm:main\]
-----------------------------
Assuming Proposition \[prop:main\] for the moment, we can complete the proof of Theorem \[thm:main\] for the case of $\pi$ corresponding to a Maass cusp form, $f$. First, as noted above, we may choose a prime $q\nmid N$ for which $D_f(s,1/q,\cos)$ has infinitely many poles on the line $\Re(s)=0$. Then, by Dirichlet’s theorem on primes in an arithmetic progression, for any $M\in{\mathbb{Z}}_{>0}$ there are distinct primes $q_0,q_1,\ldots,q_{M-1}$ such that $q_j\equiv q\pmod*{N}$ and $D_{\bar{f}}(s,-q_j/N,\cos^{(a)})
=D_{\bar{f}}(s,-q/N,\cos^{(a)})$ for all $j$, $a$.
Let $m_0$ be an integer with $0\leq m_0\leq M-1$. By the Vandermonde determinant, there exist rational numbers $c_0,c_1,\ldots,c_{M-1}$ such that $$\sum_{j=0}^{M-1}c_jq_j^{-m}=
\begin{cases}
1&\text{if }m=m_0,\\
0&\text{if }m\ne m_0
\end{cases}
\quad\text{for all }m\in\{0,1,\ldots,M-1\}.$$ We fix $\delta\in\{0,1\}$ and apply Proposition \[prop:main\] with $a\equiv\delta+m_0\pmod*{2}$ and $\alpha=1/q_j$ for $j=0,1,\ldots,M-1$. Multiplying by $(-1)^kc_j(q_j^2/N)^{s-\frac12}$, summing over $j$ and replacing $s$ by $s-m_0$, we find that $$\begin{aligned}
\sum_{j=0}^{M-1}&(-1)^kc_j\left(\frac{q_j^2}{N}\right)^{s-m_0-\frac12}
P_f(s-m_0;\delta+m_0,0)
\Delta_f\!\left(s-m_0,\frac1{q_j},\cos^{(\delta+m_0+k)}\right)\\
&-\eta\frac{(-2\pi N)^{m_0}}{m_0!}P_f(s-m_0;\delta+m_0,m_0)
\Delta_{\bar{f}}\!\left(s,-\frac{q}{N},\cos^{(\delta)}\right) \end{aligned}$$ is holomorphic on $\{s\in\Omega:\Re(s)>\frac32+m_0-M\}$, where we set $$\Omega=\{s\in{\mathbb{C}}:s\pm\nu\notin{\mathbb{Z}}\}.$$ Since $D_f(s-m_0,1/q_j,\cos^{(\delta+m_0+k)})$ is holomorphic on $\{s\in\Omega:\Re(s)<m_0-\frac12\}$, choosing $m_0=2+\delta+\frac{1-\epsilon}2$ and $M$ arbitrarily large, we conclude that $D_{\bar{f}}(s,-q/N,\cos^{(\delta)})$ is holomorphic on $\Omega$.
Next we apply Proposition \[prop:main\] again with $a=k$, $\alpha=1/q$ and $M=2$. When $k=1$ or $k=0$ and $\epsilon=1$, we see that $D_f(s,1/q,\cos)$ is holomorphic on $\{s\in\Omega:\Re(s)=0\}$. This is a contradiction, and Theorem \[thm:main\] follows in these cases.
The remaining case is that of odd Maass forms of weight $0$. The above argument with $\delta=1$ shows that $D_f(s,-q/N,\sin)$ is entire apart from possible poles for $s\pm\nu\in{\mathbb{Z}}$. Applying Proposition \[prop:main\] with $a=1$, $\alpha=-q/N$ and $M=3$, we find that $$\begin{aligned}
-\Delta_f\!\left(s,-\frac{q}{N},\sin\right)
+\eta\left(\frac{q^2}N\right)^{s-\frac12}\biggl[
&\Delta_{\bar{f}}\!\left(s,\frac1q,\sin\right)
-2\pi q\Delta_{\bar{f}}\!\left(s+1,\frac1q,\cos\right)\\
&-\frac{(2\pi q)^2}{2!}P_f(s;1,2)
\Delta_{\bar{f}}\!\left(s+2,\frac1q,\sin\right)
\biggr]\end{aligned}$$ is holomorphic on $\{s\in\Omega:\Re(s)>-\frac52\}$. Since $D_{\bar{f}}(s,1/q,\sin)$ is holomorphic on the lines $\Re(s)=-1$ and $\Re(s)=1$, we see that $D_{\bar{f}}(s,1/q,\cos)$ is holomorphic on $\{s\in\Omega:\Re(s)=0\}$. This is again a contradiction, and concludes the proof.
Proof of Proposition \[prop:main\] {#sec:AFE}
==================================
Using the expansion , we take the Mellin transform of along the line $z=({\omega}+i)y$. First, the left-hand side becomes, for $\Re(s)\gg1$, $$\label{eq:LHSMellin}
\begin{aligned}
\int_0^\infty f({\omega}y+iy)y^{s-\frac12}\frac{dy}{y}
&=\sum_{n=1}^\infty\frac{\lambda_f(n)}{\sqrt{n}}
\int_0^\infty
\bigl(V_f^+(ny)\cos(2\pi n{\omega}y)
+iV_f^-(ny)\sin(2\pi n{\omega}y)\bigr)
y^{s-\frac12}\frac{dy}{y}\\
&=G_f(s,{\omega})L_f(s),
\end{aligned}$$ where, by , and , $$\label{eq:Gdef}
\begin{aligned}
&G_f(s,{\omega})=\int_0^\infty
\bigl(V_f^+(y)\cos(2\pi{\omega}y)+iV_f^-(y)\sin(2\pi{\omega}y)\bigr)
y^{s-\frac12}\frac{dy}{y}\\
&=\begin{cases}
(2\pi i{\omega})^{\frac{1-\epsilon}2}\gamma_f^+(s)
\operatorname{{}_2F_1}\!\left(\frac{s+\frac{1-\epsilon}2+\nu}2,\frac{s+\frac{1-\epsilon}2-\nu}2;
1-\frac{\epsilon}2;-{\omega}^2\right)
&\text{if }k=0,\\
\gamma_f^+(s)\operatorname{{}_2F_1}\!\left(\frac{s+\frac{1+\epsilon}2+\nu}{2},
\frac{s+\frac{1-\epsilon}2-\nu}{2};\frac12;-{\omega}^2\right)
+2\pi i{\omega}\gamma_f^-(s+1)
\operatorname{{}_2F_1}\!\left(\frac{s+\frac{3-\epsilon}2+\nu}{2} ,
\frac{s+\frac{3+\epsilon}2-\nu}{2};\frac32;-{\omega}^2\right)
&\text{if }k=1.
\end{cases}
\end{aligned}$$ Note that we have $G_{\bar{f}}(s,{\omega})=\overline{G_f(\bar{s},-{\omega})}$.
On the other hand, the Mellin transform of the right-hand side of is, for $-\Re(s)\gg1$, $$\eta\left(i\frac{|{\omega}+i|}{{\omega}+i}\right)^k\int_0^\infty
\bar{f}\!\left(-\frac{{\omega}}{N({\omega}^2+1)y}+\frac{i}{N({\omega}^2+1)y}\right)
y^{s-\frac12}\frac{dy}{y}.$$ Making the substitution $y\mapsto(N({\omega}^2+1)y)^{-1}$, this becomes $$\label{eq:RHSMellin}
\eta\left(i\frac{|{\omega}+i|}{{\omega}+i}\right)^k\bigl(N(1+{\omega}^2)\bigr)^{\frac12-s}
\int_0^\infty\bar{f}(-{\omega}y+iy)y^{\frac12-s}\frac{dy}{y}
=\eta\left(i\frac{|{\omega}+i|}{{\omega}+i}\right)^k
\bigl(N(1+{\omega}^2)\bigr)^{\frac12-s}
G_{\bar{f}}(1-s,-{\omega})L_{\bar{f}}(1-s).$$
By , and must continue to entire functions and equal each other. In particular, taking ${\omega}\to0$, we recover the functional equation . Equating with and dividing by , we discover the functional equation for the hypergeometric factor $H_f(s,{\omega}):=G_f(s,{\omega})/\gamma_f^+(s)$: $$\label{eq:FEofG}
H_f(s,{\omega})=
\epsilon^{1-k}\left(i\frac{|{\omega}+i|}{{\omega}+i}\right)^k
(1+{\omega}^2)^{\frac12-s}H_{\bar{f}}(1-s,-{\omega}).$$
Next, for $z=x+iy\in{\mathbb{H}}$, define $$A(z)=
\frac1{2\pi i}\int_{\Re(s)=\frac12}
\bigl(\psi'(s+\nu)+\psi'(s-\nu)\bigr)H_f(s,x/y)\Lambda_f(s)
y^{\frac12-s}\,ds$$ and $$\label{eq:Bdef}
B(z)=
\frac1{2\pi i}\int_{\Re(s)=\frac12}X_f(s)\Lambda_f(s)
H_f(s,x/y)y^{\frac12-s}\,ds
-\sum_\rho\Lambda_f'(\rho)H_f(\rho,x/y)y^{\frac12-\rho},$$ where the sum runs over all simple zeros of $\Lambda_f(s)$, and $$X_f(s)=\frac{\pi^2}{4}\left[
\csc^2\!\left(\frac{\pi}{2}\left[s+\frac{1+(-1)^k\epsilon}2+\nu\right]\right)
+\csc^2\!\left(\frac{\pi}{2}\left[s+\frac{1+\epsilon}2-\nu\right]\right)
\right].$$
\[lem:FEofF\] $$F(z)+A(z)
=\eta\left(i\frac{|z|}{z}\right)^k\overline{F}\!\left(-\frac1{Nz}\right)+B(z)
\quad\text{for all }z\in{\mathbb{H}}.$$
Fix $z=x+iy\in{\mathbb{H}}$, and put ${\omega}=x/y$. Applying Mellin inversion as in , we have $$F(z)=\frac1{2\pi i}\int_{\Re(s)=2}D_f(s)G_f(s,{\omega})y^{\frac12-s}\,ds$$ and $$\begin{aligned}
\eta\left(i\frac{|z|}{z}\right)^k
\overline{F}\!\left(-\frac1{Nz}\right)
&=\eta\left(i\frac{|{\omega}+i|}{{\omega}+i}\right)^k
\cdot\frac1{2\pi i}\int_{\Re(s)=2}
G_{\bar{f}}(s,-{\omega})D_{\bar{f}}(s)
\bigl(N(1+{\omega}^2)y\bigr)^{s-\frac12}\,ds\\
&=\eta\left(i\frac{|{\omega}+i|}{{\omega}+i}\right)^k
\cdot\frac1{2\pi i}\int_{\Re(s)=-1}
H_{\bar{f}}(1-s,-{\omega})\Delta_{\bar{f}}(1-s)
\bigl(N(1+{\omega}^2)y\bigr)^{\frac12-s}\,ds.\end{aligned}$$ Applying \[eq:FEofG\] and , and using the fact that $\psi_{\bar{f}}'(1-s)$ is holomorphic for $\Re(s)\le\frac12$, the last line becomes $$\begin{aligned}
\frac1{2\pi i}&\int_{\Re(s)=-1}
\eta\epsilon^{1-k}H_f(s,{\omega})\Delta_{\bar{f}}(1-s)(Ny)^{\frac12-s}\,ds\\
&=\frac1{2\pi i}\int_{\Re(s)=-1}H_f(s,{\omega})
\Bigl[\Delta_f(s)+\bigl(\psi_f'(s)-\psi_{\bar{f}}'(1-s)\bigr)\Lambda_f(s)\Bigr]
y^{\frac12-s}\,ds\\
&=\frac1{2\pi i}\int_{\Re(s)=-1}
H_f(s,{\omega})\Bigl[\Delta_f(s)+\psi_f'(s)\Lambda_f(s)\Bigr]y^{\frac12-s}\,ds
-\frac1{2\pi i}\int_{\Re(s)=\frac12}H_f(s,{\omega})
\psi_{\bar{f}}'(1-s)\Lambda_f(s)y^{\frac12-s}\,ds.\end{aligned}$$ Shifting the contour of the first integral to the right and using that $\psi_f'(s)$ is holomorphic for $\Re(s)\ge\frac12$, we get $$\begin{aligned}
\frac1{2\pi i}\int_{\Re(s)=2}&
H_f(s,{\omega})\Delta_f(s)y^{\frac12-s}\,ds
-\frac1{2\pi i}\int_{\mathcal{C}}H_f(s,{\omega})
\bigl(\Delta_f(s)+\psi_f'(s)\Lambda_f(s)\bigr)y^{\frac12-s}\,ds\\
&+\frac1{2\pi i}\int_{\Re(s)=\frac12}
\bigl(\psi_f'(s)-\psi_{\bar{f}}'(1-s)\bigr)H_f(s,{\omega})\Lambda_f(s)
y^{\frac12-s}\,ds,\end{aligned}$$ where $\mathcal{C}$ is the contour running from $2-i\infty$ to $2+i\infty$ and from $-1+i\infty$ to $-1-i\infty$. Note that $$\Delta_f(s)+\psi_f'(s)\Lambda_f(s)
=\Lambda_f(s)\frac{d^2}{ds^2}\log\Lambda_f(s),$$ which has a pole at every simple zero $\rho$ of $\Lambda_f(s)$, with residue $-\Lambda_f'(\rho)$. Hence, $$-\frac1{2\pi i}\int_{\mathcal{C}}
H_f(s,{\omega})\bigl(\Delta_f(s)+\psi_f'(s)\Lambda_f(s)\bigr)y^{\frac12-s}\,ds
=\sum_\rho\Lambda_f'(\rho)H_f(\rho,{\omega})y^{\frac12-\rho}.$$
Next, writing $\psi_{\mathbb{R}}(s)=\frac{\Gamma_{\mathbb{R}}'}{\Gamma_{\mathbb{R}}}(s)$, we have $$\psi_f(s)=\psi_{\mathbb{R}}\!\left(s+\frac{1-(-1)^k\epsilon}2+\nu\right)
+\psi_{\mathbb{R}}\!\left(s+\frac{1-\epsilon}2-\nu\right).$$ Applying the reflection formula and Legendre duplication formula in the form $$\psi_{\mathbb{R}}'(s)=\frac{\pi^2}{4}\csc^2\!\left(\frac{\pi s}{2}\right)-\psi_{\mathbb{R}}'(2-s)
\quad\text{and}\quad
\psi_{\mathbb{R}}'(s)+\psi_{\mathbb{R}}'(s+1)=\psi'(s),$$ we derive $$\begin{aligned}
\psi_f'(s)-\psi_{\bar{f}}'(1-s)=\psi'(s+\nu)+\psi'(s-\nu)-X_f(s).\end{aligned}$$ Thus, $$\frac1{2\pi i}\int_{\Re(s)=\frac12}
\bigl(\psi_f'(s)-\psi_{\bar{f}}'(1-s)\bigr)H_f(s,{\omega})\Lambda_f(s)
y^{\frac12-s}\,ds
=A(z)-\frac1{2\pi i}\int_{\Re(s)=\frac12}X_f(s)
H_f(s,{\omega})\Lambda_f(s)y^{\frac12-s}\,ds.$$ Rearranging terms completes the proof.
\[lem:AMellin\] For any $\alpha\in{\mathbb{Q}}^\times$, $$\frac1{\Gamma(s+\nu)\Gamma(s-\nu)}
\int_0^\infty A(\alpha+iy)y^{s-\frac 12}\frac{dy}{y}$$ continues to an entire function of $s$.
Define $\Phi(s)=\psi'(s+\nu)+\psi'(s-\nu)$. Then we have $\Phi(s)=\int_1^\infty\phi(x)x^{\frac12-s}\,dx$, where $\phi(x)=\frac{\cosh(\nu\log x)\log x}{\sinh(\frac12\log{x})}$. Applying and the change of variables $y\mapsto xt$, we have $$\begin{aligned}
\Phi(s)G_f(s,{\omega})&=\int_1^\infty\int_0^\infty
\phi(x)\bigl(V_f^+(y)\cos(2\pi{\omega}y)
+iV_f^-(y)\sin(2\pi{\omega}y)\bigr)
\left(\frac{y}{x}\right)^{s-\frac12}
\frac{dy}{y}\,dx\\
&=\int_0^\infty\left(\int_1^\infty
\phi(x)\bigl(V_f^+(tx)\cos(2\pi{\omega}tx)
+iV_f^-(tx)\sin(2\pi{\omega}tx)\bigr)
\,dx\right)t^{s-\frac12}\frac{dt}{t}.\end{aligned}$$ Hence, writing ${\omega}=\alpha/y$, we have $$\begin{aligned}
A(\alpha+iy)&=\frac1{2\pi i}\int_{\Re(s)=2}
\Lambda_f(s)\Phi(s)H_f(s,{\omega})y^{\frac12-s}\,ds
=\sum_{n=1}^\infty\frac{\lambda_f(n)}{\sqrt{n}}
\frac1{2\pi i}\int_{\Re(s)=2}\Phi(s)G_f(s,{\omega})(ny)^{\frac12-s}\,ds\\
&=\sum_{n=1}^\infty\frac{\lambda_f(n)}{\sqrt{n}}\int_1^\infty
\phi(x)\bigl(V_f^+(nxy)\cos(2\pi\alpha nx)
+iV_f^-(nxy)\sin(2\pi\alpha nx)\bigr)\,dx,\end{aligned}$$ so that $$\begin{aligned}
\int_0^\infty A(\alpha+iy)y^{s-\frac12}\frac{dy}{y}
&=\sum_{n=1}^\infty\frac{\lambda_f(n)}{\sqrt{n}}\int_1^\infty
\phi(x)\int_0^\infty\bigl(V_f^+(nxy)\cos(2\pi\alpha nx)
+iV_f^-(nxy)\sin(2\pi\alpha nx)\bigr)
y^{s-\frac12}\frac{dy}{y}\,dx\\
&=\sum_{n=1}^\infty \lambda_f(n)n^{-s}\int_1^\infty\phi(x)x^{\frac12-s}
\Bigl(\widetilde{V}_f^+(s)\cos(2\pi\alpha nx)
+i\widetilde{V}_f^-(s)\sin(2\pi\alpha nx)\Bigr)
\,dx,\end{aligned}$$ where $$\label{eq:tVdef}
\widetilde{V}_f^\pm(s)=
\int_0^\infty V_f^\pm(y)y^{s-\frac12}\frac{dy}y
=\begin{cases}
\gamma_f^\pm(s)&\text{if }k=1\text{ or }\epsilon=\pm1,\\
0&\text{otherwise}.
\end{cases}$$ A case-by-case inspection of shows that $\widetilde{V}_f^\pm(s)/(\Gamma(s+\nu)\Gamma(s-\nu))$ is entire for both choices of sign.
Define $\phi_j=\phi_j(x,s)$ for $j\ge0$ by $$\phi_0=\phi,
\quad\text{and}\quad
\phi_{j+1}=x\frac{\partial \phi_j}{\partial x}
-(s+j-\tfrac12)\phi_j.$$ Then, applying integration by parts $m$ times, we see that $$\begin{aligned}
\int_1^\infty\phi(x)\cos(2\pi\alpha nx)x^{\frac12-s}\,dx
=\sum_{j=0}^{m-1}\frac{\cos^{(j+1)}(2\pi\alpha n)}{(2\pi\alpha n)^{j+1}}
\phi_j(1,s)
+\int_1^\infty\frac{\cos^{(m)}(2\pi\alpha nx)}{(2\pi\alpha n)^m}\phi_k(x,s)
x^{\frac12-m-s}\,dx\end{aligned}$$ and $$\begin{aligned}
\int_1^\infty\phi(x)\sin(2\pi\alpha nx)x^{\frac12-s}\,dx
=\sum_{j=0}^{m-1}\frac{\sin^{(j+1)}(2\pi\alpha n)}{(2\pi\alpha n)^{j+1}}
\phi_j(1,s)
+\int_1^\infty\frac{\sin^{(m)}(2\pi\alpha nx)}{(2\pi\alpha n)^m}\phi_k(x,s)
x^{\frac12-m-s}\,dx.\end{aligned}$$ Thus, $$\begin{aligned}
&\int_0^\infty A(\alpha+iy)y^{s-\frac12}\frac{dy}{y}\\
&=\widetilde{V}_f^+(s)\left[
\sum_{j=0}^{m-1}\frac{\phi_j(1,s)L(f,s+j+1,\alpha,\cos^{(j+1)})}
{(2\pi\alpha)^{j+1}}
+\frac1{(2\pi\alpha)^m}\sum_{n=1}^\infty\frac{a_f(n)}{n^{s+m}}
\int_1^\infty\cos^{(m)}(2\pi\alpha nx)\phi_m(x,s)x^{\frac12-m-s}\,dx\right]\\
&+i\widetilde{V}_f^-(s)\left[
\sum_{j=0}^{m-1}\frac{\phi_j(1,s)L(f,s+j+1,\alpha,\sin^{(j+1)})}
{(2\pi\alpha)^{j+1}}
+\frac1{(2\pi\alpha)^m}\sum_{n=1}^\infty\frac{a_f(n)}{n^{s+m}}
\int_1^\infty\sin^{(m)}(2\pi\alpha nx)\phi_m(x,s)x^{\frac12-m-s}\,dx\right].\end{aligned}$$ It follows from [@BK Prop. 3.1] that $L_f(s,\alpha,\cos)$ and $L_f(s,\alpha,\sin)$ continue to entire functions. We see by induction that $\phi_m(x,s)\ll_m\bigl((1+|s|)(1+|\nu|)\bigr)^mx^{-1}$ uniformly for $x\ge1$, and thus the integral terms above are holomorphic for $\Re(s)>\frac12-m$. Choosing $m$ arbitrarily large, the lemma follows.
\[lem:Kerror\] For any $\sigma\ge0$ and any $l\in{\mathbb{Z}}_{\ge0}$, we have $$\frac{y^l}{l!}(V_{\bar{f}}^\pm)^{(l)}(y)\ll_\sigma 2^l y^{-\sigma}
\quad\text{for }y>0.$$
In view of , since $|\Re(\nu)|<\frac12$, for any $\sigma\ge0$ we have the integral representation $$V_{\bar{f}}^\pm(y)=\frac1{2\pi i}\int_{\Re(s)=\sigma+\frac12}
\widetilde{V}_{\bar{f}}^\pm(s)y^{\frac12-s}\,ds.$$ Differentiating $l$ times, we obtain $$\frac{y^l}{l!}(V_{\bar{f}}^\pm)^{(l)}(y)=\frac1{2\pi i}\int_{\Re(s)=\sigma+\frac12}
{{\frac12-s}\choose{l}}
\widetilde{V}_{\bar{f}}^\pm(s)y^{\frac12-s}\,ds.$$ Using the estimate $$\left|{{\frac12-s}\choose{l}}\right|
=\left|{{s-\frac12+l}\choose{l}}\right|
\le2^{|s-\frac12|+l},$$ we have $$\frac{y^l}{l!}(V_{\bar{f}}^\pm)^{(l)}(y)\le2^ly^{-\sigma}\cdot
\frac1{2\pi}\int_{\Re(s)=\sigma+\frac12}
2^{|s-\frac12|}\bigl|\widetilde{V}_{\bar{f}}^\pm(s)\,ds\bigr|
\ll_\sigma 2^ly^{-\sigma},$$ where the last inequality is justified by Stirling’s formula.
\[lem:dualside\] Let $\alpha\in{\mathbb{Q}}^\times$ and $z=\alpha+iy$ for some $y\in(0,|\alpha|/2]$. Then, for any integer $T\ge0$, we have $$\label{eq:taylor}
\begin{aligned}
\left(i\frac{|z|}{z}\right)^k\overline{F}\!\left(-\frac1{Nz}\right)
=O_{\alpha,T}(y^{T-1})&+(i\operatorname{sgn}(\alpha))^k
\sum_{t=0}^{T-1}\frac{(2\pi iN\alpha)^t}{t!}
\\
&\cdot \sum_{a\in\{0,1\}}
\frac{i^{-a}}{2\pi i}\int_{\Re(s)=2}
P_f(s;a+t,t)
\Delta_{\bar{f}}\!\left(s+t,-\frac1{N\alpha},\cos^{(a)}\right)
\left( \frac{y}{N\alpha^2} \right)^{\frac12-s}\,ds.
\end{aligned}$$
Let $z=\alpha+iy$, $\beta=-1/N\alpha$ and $u=y/\alpha$. Then $$-\frac1{Nz}=\frac{\beta}{1+u^2}+i\frac{|\beta u|}{1+u^2},$$ so that $$\begin{aligned}
&\left(i\frac{|z|}{z}\right)^k\overline{F}\!\left(-\frac1{Nz}\right)
=\left(i\operatorname{sgn}(\alpha)\frac{|1+iu|}{1+iu}\right)^k
\overline{F}\!\left(\frac{\beta}{1+u^2}+i\frac{|\beta u|}{1+u^2}\right)\\
&=\left(i\operatorname{sgn}(\alpha)\frac{|1+iu|}{1+iu}\right)^k
\sum_{n=1}^\infty\frac{c_{\bar{f}}(n)}{\sqrt{n}}
\left(V_{\bar{f}}^+\!\left(\frac{|\beta nu|}{1+u^2}\right)
\cos\!\left(\frac{2\pi\beta n}{1+u^2}\right)
+iV_{\bar{f}}^-\!\left(\frac{|\beta nu|}{1+u^2}\right)
\sin\!\left(\frac{2\pi\beta n}{1+u^2}\right)\right).\end{aligned}$$
By Lemma \[lem:Kerror\], for any $\sigma\ge0$ and any $l_0\in{\mathbb{Z}}_{\ge0}$, we have $$\begin{aligned}
V_{\bar{f}}^\pm\!\left(\frac{|\beta nu|}{1+u^2}\right)
&=\sum_{l=0}^\infty
\frac1{l!}(V_{\bar{f}}^\pm)^{(l)}(|\beta nu|)
\left(\frac{\beta nu^3}{1+u^2}\right)^l\\
&=\sum_{l=0}^{l_0-1}\frac1{l!}(V_{\bar{f}}^\pm)^{(l)}(|\beta nu|)
\left(\frac{\beta nu^3}{1+u^2}\right)^l
+O_\sigma\!\left(|\beta nu|^{-\sigma}
\sum_{l=l_0}^\infty\left(\frac{2u^2}{1+u^2}\right)^l
\right)\\
&=\sum_{l=0}^{l_0-1}\frac1{l!}(V_{\bar{f}}^\pm)^{(l)}(|\beta nu|)
\left(\frac{\beta nu^3}{1+u^2}\right)^l
+O_{\alpha,\sigma,l_0}\!\left(|nu|^{-\sigma}u^{2l_0}\right).\end{aligned}$$ Similarly, for any $a\in\{0,1\}$, we have $$\begin{aligned}
\cos^{(a)}\!\left(\frac{2\pi\beta n}{1+u^2}\right)
&=\sum_{j=0}^\infty\frac1{j!}\cos^{(j+a)}(2\pi\beta n)
\left(-\frac{2\pi\beta nu^2}{1+u^2}\right)^j\\
&=\sum_{j=0}^{j_0-1}\frac1{j!}\cos^{(j+a)}(2\pi\beta n)
\left(-\frac{2\pi\beta nu^2}{1+u^2}\right)^j
+O\!\left(\frac1{j_0!}\left|\frac{2\pi\beta nu^2}{1+u^2}\right|^{j_0}\right)\\
&=\sum_{j=0}^{j_0-1}\frac1{j!}\cos^{(j+a)}(2\pi\beta n)
\left(-\frac{2\pi\beta nu^2}{1+u^2}\right)^j
+O_{\alpha,j_0}\bigl((nu^2)^{j_0}\bigr),\end{aligned}$$ by the Lagrange form of the error in Taylor’s theorem. Taking $j_0=2(l_0-l)$ and applying Lemma \[lem:Kerror\] with $\sigma$ replaced by $\sigma+2(l_0-l)$, we obtain $$\begin{aligned}
V_{\bar{f}}^{(-)^a}&\!\left(\frac{|\beta nu|}{1+u^2}\right)
\cos^{(a)}\!\left(\frac{2\pi\beta n}{1+u^2}\right)\\
&=\sum_{j+2l<2l_0}\frac{(-2\pi)^j}{j!l!}
(V_{\bar{f}}^{(-)^a})^{(l)}(|\beta nu|)\cos^{(j+a)}(2\pi\beta n)
u^l\left(\frac{\beta nu^2}{1+u^2}\right)^{j+l}
+O_{\alpha,\sigma,l_0}\bigl(|nu|^{-\sigma}u^{2l_0}\bigr).\end{aligned}$$
Next, defining $$b_{j,k,l,m}=
\begin{cases}
{{j+l-1+\lfloor\frac{m}2\rfloor + \frac{k}2}\choose\lfloor\frac{m}2\rfloor}
&\text{if }k=1\text{ or }k=0\text{ and }2\mid m,\\
0&\text{otherwise},
\end{cases}$$ we have $$\begin{aligned}
\left(\frac{|1+iu|}{1+iu}\right)^k(1+u^2)^{-j-l}
&=(1-iu)^k(1+u^2)^{-j-l-\frac{k}2}
=\sum_{m=0}^\infty b_{j,k,l,m}(-iu)^m\\
&=\sum_{m=0}^{m_0-1}b_{j,k,l,m}(-iu)^m
+O\!\left(\sum_{m=m_0}^\infty 2^{j+l+\frac{m}2}|u|^m\right)\\
&=\sum_{m=0}^{m_0-1}b_{j,k,l,m}(-iu)^m
+O_{j,l,m_0}(|u|^{m_0}).\end{aligned}$$ Taking $m_0=2l_0-j-2l$ and applying Lemma \[lem:Kerror\] with $\sigma$ replaced by $\sigma+j$, we obtain $$\begin{aligned}
&\left(i\operatorname{sgn}(\alpha)\frac{|1+iu|}{1+iu}\right)^k
V_{\bar{f}}^{(-)^a}\!\left(\frac{|\beta nu|}{1+u^2}\right)
\cos^{(a)}\!\left(\frac{2\pi\beta n}{1+u^2}\right)\\
&=(i\operatorname{sgn}(\alpha))^k\sum_{j+2l+m<2l_0}\frac{(-2\pi)^j(-i)^m}{j!l!}b_{j,k,l,m}
(\beta nu)^{j+l}\bigl(V_{\bar{f}}^{(-)^a}\bigr)^{(l)}(|\beta nu|)
\cos^{(j+a)}(2\pi\beta n)u^{j+2l+m}\\
&\quad+O_{\alpha,\sigma,l_0}\bigl(|nu|^{-\sigma}u^{2l_0}\bigr).\end{aligned}$$
Recalling the definition of $u$, multplying by $c_{\bar{f}}(n)/\sqrt{n}$ and summing over $n$ and both choices of $a$, the error term converges if $\sigma\ge1$, to give $$\begin{aligned}
\sum_{a\in\{0,1\}}i^{-a}
\left(i\frac{|\alpha+iy|}{\alpha+iy}\right)^k
\sum_{n=1}^\infty\frac{c_{\bar{f}}(n)}{\sqrt{n}}
V_{\bar{f}}^{(-)^a}&\!\left(\frac{ny}{N(\alpha^2+y^2)}\right)
\cos^{(a)}\!\left(\frac{2\pi\beta n}{1+(y/\alpha)^2}\right)\\
=\sum_{j+2l+m<2l_0}
(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}i^{-a}
\sum_{n=1}^\infty&\frac{c_{\bar{f}}(n)}{\sqrt{n}}
\frac{(2\pi i)^j}{j!l!}b_{j,k,l,m}
\left(\frac{ny}{N\alpha^2}\right)^{j+l}\\
&\cdot\bigl(V_{\bar{f}}^{(-)^a}\bigr)^{(l)}\!\left(\frac{ny}{N\alpha^2}\right)
\cos^{(j+a)}(2\pi\beta n)\left(\frac{y}{i\alpha}\right)^{j+2l+m}
+O_{\alpha,\sigma,l_0}\bigl(y^{2l_0-\sigma}\bigr)\\
=\sum_{j+2l+m<2l_0}
(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}i^{-a}
\sum_{n=1}^\infty&\frac{c_{\bar{f}}(n)}{\sqrt{n}}
\frac{(-2\pi)^j}{j!l!}b_{j,k,l,m}
\left(\frac{ny}{N\alpha^2}\right)^{j+l}\\
&\cdot\bigl(V_{\bar{f}}^{(-)^{a+j}}\bigr)^{(l)}\!\left(\frac{ny}{N\alpha^2}\right)
\cos^{(a)}(2\pi\beta n)\left(\frac{y}{i\alpha}\right)^{j+2l+m}
+O_{\alpha,\sigma,l_0}\bigl(y^{2l_0-\sigma}\bigr).\end{aligned}$$
Taking the Mellin transform of a single term of the sum over $j,l,m$ and making the change of variables $y\mapsto N\alpha^2y/n$, we get $$\begin{aligned}
(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}i^{-a}
\int_0^\infty
\sum_{n=1}^\infty&\frac{c_{\bar{f}}(n)}{\sqrt{n}}
\frac{(-2\pi)^j}{j!l!}b_{j,k,l,m}
\left(\frac{ny}{N\alpha^2}\right)^{j+l}\\
&\cdot\bigl(V_{\bar{f}}^{(-)^{a+j}}\bigr)^{(l)}\!\left(\frac{ny}{N\alpha^2}\right)
\cos^{(a)}(2\pi\beta n)\left(\frac{y}{i\alpha}\right)^{j+2l+m}
y^{s-\frac12}\frac{dy}{y}\\
=(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}&i^{-a}
(N\alpha^2)^{s-\frac12}(-iN\alpha)^{j+2l+m}
\frac{(-2\pi)^j}{j!}b_{j,k,l,m}\\
&\cdot\sum_{n=1}^\infty\frac{c_{\bar{f}}(n)\cos^{(a)}(2\pi\beta n)}{n^{s+j+2l+m}}
\int_0^\infty\frac{y^l}{l!}(V_{\bar{f}}^{(-)^{a+j}})^{(l)}(y)y^{s+2j+2l+m-\frac12}\frac{dy}{y}\\
=(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}&i^{-a}
(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\frac{(-2\pi)^j}{j!}b_{j,k,l,m}\\
&\cdot D_{\bar{f}}(s+t,\beta,\cos^{(a)})
{{\frac12-s-t-j}\choose{l}}
\widetilde{V}_{\bar{f}}^{(-)^{a+j}}(s+t+j),\end{aligned}$$ where $t=j+2l+m$.
Next we fix $t\in{\mathbb{Z}}_{\ge0}$ and sum over all $(j,l,m)$ satisfying $j+2l+m=t$. When $k=0$, $b_{j,k,l,m}$ vanishes unless $m$ is even. Hence, defining $$I_k(m)=\begin{cases}
1&\text{if }k=1\text{ or }2\mid m,\\
0&\text{otherwise},
\end{cases}$$ we get $$\begin{aligned}
(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}i^{-a}
(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{j+2l+m=t}&I_k(t-j)
\frac{(-2\pi)^j}{j!}
{{j+l-1+\lfloor\frac{m}2\rfloor+\frac{k}2}\choose\lfloor\frac{m}2\rfloor}\\
&\cdot D_{\bar{f}}(s+t,\beta,\cos^{(a)})
{{\frac12-s-t-j}\choose{l}}
\widetilde{V}_{\bar{f}}^{(-)^{a+j}}(s+t+j)\\
=(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}i^{-a}
(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{j=0}^t&I_k(t-j)
\frac{(-2\pi)^j}{j!}
D_{\bar{f}}(s+t,\beta,\cos^{(a)})
\widetilde{V}_{\bar{f}}^{(-)^{a+j}}(s+t+j)\\
&\cdot\sum_{l=0}^{\lfloor\frac{t-j}2\rfloor}
{{j+\lfloor\frac{t-j}2\rfloor+ \frac{k}2-1}\choose\lfloor\frac{t-j}2\rfloor-l}
{{\frac12-s-t-j}\choose{l}}\\
=(i\operatorname{sgn}(\alpha))^k\sum_{a\in\{0,1\}}i^{-a}
(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{j=0}^t&I_k(t-j)
\frac{(-2\pi)^j}{j!}
D_{\bar{f}}(s+t,\beta,\cos^{(a)})
\widetilde{V}_{\bar{f}}^{(-)^{a+j}}(s+t+j) \\
&\cdot{{\lfloor\frac{t-j}2\rfloor+\frac{k-1}2-s-t}\choose\lfloor\frac{t-j}2\rfloor},\end{aligned}$$ by the Chu–Vandermonde identity.
We now break into cases according to the weight, $k$. When $k=0$, the inner sum vanishes identically when $(-1)^{a+t}=-\epsilon$, so we may assume that $(-1)^{a+t}=\epsilon$. Thus, in this case, we have $$\begin{aligned}
(N\alpha^2)^{s-\frac12}(iN\alpha)^ti^{-a}
D_{\bar{f}}(s+t,\beta,\cos^{(a)})
\sum_{\substack{j\le t\\j\equiv t\pmod*{2}}}
\frac{(2\pi)^j}{j!}
\gamma_{\bar{f}}^{(-)^{a+t}}(s+t+j)
{{\frac{t-j}2-\frac12-s-t}\choose\frac{t-j}2}.\end{aligned}$$ Put $t=2n+b$, with $b\in\{0,1\}$. Then, writing $j=2r+b$, the above becomes $$\begin{aligned}
&(N\alpha^2)^{s-\frac12}(iN\alpha)^ti^{-a}
\Delta_{\bar{f}}(s+t,\beta,\cos^{(a)})\\
&\qquad\cdot\sum_{r=0}^n
\frac{(2\pi)^{2r+b}}{(2r+b)!}
\frac{\Gamma_{\mathbb{R}}(s+t+2r+b+\nu)\Gamma_{\mathbb{R}}(s+t+2r+b-\nu)}
{\Gamma_{\mathbb{R}}(s+t+b+\nu)\Gamma_{\mathbb{R}}(s+t+b-\nu)}
{{n-r-\frac12-s-t}\choose{n-r}}\\
&=(N\alpha^2)^{s-\frac12}(iN\alpha)^ti^{-a}
\Delta_{\bar{f}}(s+t,\beta,\cos^{(a)})(-1)^n \\
&\qquad\cdot\sum_{r=0}^n
\left(\frac{2\pi}{2r+1}\right)^b
\frac{(-4)^rr!^2}{(2r)!}
{{-(s+t+b+\nu)/2}\choose{r}}
{{-(s+t+b-\nu)/2}\choose{r}}
{{s+t-\frac12}\choose{n-r}}.\end{aligned}$$ Applying [@BK Lemma A.1(ii)–(iii)], we get $$\begin{aligned}
&(N\alpha^2)^{s-\frac12}(iN\alpha)^ti^{-a}
\Delta_{\bar{f}}(s+t,\beta,\cos^{(a)})\\
&\qquad\cdot\left(\frac{2\pi}{2n+1}\right)^b
\frac{4^nn!^2}{(2n)!}
{{(s+t-1-b+\nu)/2}\choose{n}}
{{(s+t-1-b-\nu)/2}\choose{n}}\\
&\quad=(N\alpha^2)^{s-\frac12}\frac{(2\pi iN\alpha)^t}{t!}i^{-a}
\frac{\gamma_{f}^{(-)^{a+t}}(1-s)}{\gamma_{f}^{(-)^{a+t}}(1-s-2n)}
\Delta_{\bar{f}}(s+t,\beta,\cos^{(a)}).\end{aligned}$$
Turning to $k=1$, we have $$\begin{aligned}
i\operatorname{sgn}(\alpha)
(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{a\in\{0,1\}}i^{-a}
\sum_{j=0}^t
&\frac{(-2\pi)^j}{j!}
D_{\bar{f}}(s+t,\beta,\cos^{(a)})\\
&\cdot\gamma_{\bar{f}}^{(-)^{a+j}}(s+t+j)
{{\lfloor\frac{t-j}2\rfloor -s-t}\choose\lfloor\frac{t-j}2\rfloor}\\
= i\operatorname{sgn}(\alpha)
(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{a\in\{0,1\}}i^{-a}
&D_{\bar{f}}(s+t,\beta,\cos^{(a)})\\
&\cdot\sum_{j=0}^t\frac{(-2\pi)^j}{j!}
\gamma_{\bar{f}}^{(-1)^{a+j}}(s+t+j)
{{\lfloor\frac{t-j}2\rfloor-s-t}\choose\lfloor\frac{t-j}2\rfloor}.\end{aligned}$$ Writing $j=2r-c$ with $c\in\{0,1\}$, this is $$\begin{aligned}
&i\operatorname{sgn}(\alpha)(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{a\in\{0,1\}}i^{-a}\Delta_{\bar{f}}(s+t,\beta,\cos^{(a)})
\sum_{c\in\{0,1\}}\sum_{2r-c\le t}\frac{(-2\pi)^{2r-c}}{(2r-c)!}
{{n-r+\lfloor\frac{b+c}2\rfloor-s-t}\choose n-r+\lfloor\frac{b+c}2\rfloor}\\
&\quad\cdot\frac{\Gamma_{\mathbb{R}}\!\left(s+t+2r-c+\frac{1-(-1)^{a+c}\epsilon}2+\nu\right)
\Gamma_{\mathbb{R}}\!\left(s+t+2r-c+\frac{1+(-1)^{a+c}\epsilon}2-\nu\right)}
{\Gamma_{\mathbb{R}}\!\left(s+t+\frac{1-(-1)^a\epsilon}2+\nu\right)
\Gamma_{\mathbb{R}}\!\left(s+t+\frac{1+(-1)^a\epsilon}2-\nu\right)}\\
&=i\operatorname{sgn}(\alpha)(N\alpha^2)^{s-\frac12}(-iN\alpha)^t
\sum_{a\in\{0,1\}}i^{-a}\Delta_{\bar{f}}(s+t,\beta,\cos^{(a)})
\sum_{c\in\{0,1\}}(-1)^{n+bc}\\
&\quad\cdot\sum_{2r-c \leq t} \frac{(-4)^rr!^2}{(2r)!}
{{-(s+t+\frac{1-(-1)^a\epsilon}2+\nu)/2}\choose
{r-c\frac{1-(-1)^a\epsilon}2}}
{{-(s+t+\frac{1+(-1)^a\epsilon}2-\nu)/2}\choose
{r-c\frac{1+(-1)^a\epsilon}2}}
{{s+t-1}\choose n+bc-r}.\end{aligned}$$ For $b=0$, applying [@BK Lemma A.1(ii)], the sum over $c$ becomes $$\begin{aligned}
(-1)^n&\sum_{r=0}^n\frac{(-4)^rr!^2}{(2r)!}
{{-(s+t-1+\frac{1-(-1)^a\epsilon}2-\nu)/2}
\choose{r}}
{{-(s+t-1+\frac{1+(-1)^a\epsilon}2+\nu)/2}
\choose{r}}
{{s+t-1}\choose n-r}\\
&=\frac{4^nn!^2}{(2n)!}
{{(s+2n-2+\frac{1-(-1)^a\epsilon}2-\nu)/2}
\choose{n}}
{{(s+2n-2+\frac{1+(-1)^a\epsilon}2+\nu)/2}
\choose{n}}\\
&=\frac{(-2\pi)^{2n}}{(2n)!}
\frac{\Gamma_{\mathbb{R}}(1-s+\frac{1+(-1)^a\epsilon}2+\nu)}
{\Gamma_{\mathbb{R}}(1-s-2n+\frac{1+(-1)^a\epsilon}2+\nu)}
\frac{\Gamma_{\mathbb{R}}(1-s+\frac{1-(-1)^a\epsilon}2-\nu)}
{\Gamma_{\mathbb{R}}(1-s-2n+\frac{1-(-1)^a\epsilon}2-\nu)}\\
&=\frac{(-2\pi)^t}{t!}
\frac{\gamma_{f}^{(-)^{a}}(1-s)}{\gamma_{f}^{(-)^a}(1-s-2n)}
=\frac{(-2\pi)^t}{t!}
\frac{\gamma_{f}^{(-)^{a+t}}(1-s)}{\gamma_{f}^{(-)^{a+t}}(1-s-2\lfloor t/2\rfloor)}.\end{aligned}$$ For $b=1$ and $c=0$, the inner sum is $$(-1)^n\sum_{r=0}^n\frac{(-4)^r r!^2}{(2r)!}
{{-(s+t+\frac{1-(-1)^a\epsilon}{2}+\nu)/2}\choose{r}}
{{-(s+t+\frac{1+(-1)^a\epsilon}{2}-\nu)/2}\choose{r}}
{{s+t-1}\choose n-r}.$$ Writing ${{s+t-1}\choose n-r}={{s+t} \choose{n-r+1}} -{{s+t-1}\choose{n-r+1}}$ and applying [@BK Lemma A.1(ii)], we get $$\begin{aligned}
&(-1)^n\left[ \frac{(-4)^{n+1}(n+1)!^2}{(2n+2)!}
{{(s+t-\frac{1+(-1)^a\epsilon}{2}+\nu)/2}\choose{n+1}}{{(s+t-\frac{1-(-1)^a\epsilon}{2}-\nu)/2}\choose{n+1}} \right. \\
&\qquad\qquad\left.-\frac{(-4)^{n+1}(n+1)!^2}{(2n+2)!}
{{-(s+t+\frac{1-(-1)^a\epsilon}{2}+\nu)/2}\choose{n+1}}
{{-(s+t+\frac{1+(-1)^a\epsilon}{2}-\nu)/2}\choose{n+1}}\right]\\
&\quad+(-1)^{n+1}\left[\sum_{r=0}^{n+1}\frac{(-4)^rr!^2}{(2r)!}
{{-(s+t+\frac{1-(-1)^a\epsilon}{2}+\nu)/2}\choose{r}}
{{-(s+t+\frac{1+(-1)^a\epsilon}{2}-\nu)/2}\choose{r}}
{{s+t-1}\choose{n-r+1}}\right.\\
&\qquad\qquad\left.-\frac{(-4)^{n+1}(n+1)!^2}{(2n+2)!}
{{-(s+t+\frac{1-(-1)^a\epsilon}{2}+\nu)/2}\choose{n+1}}
{{-(s+t+\frac{1+(-1)^a\epsilon}{2}-\nu)/2}\choose{n+1}}\right]\\
&=(-1)^n\frac{(-4)^{n+1}(n+1)!^2}{(2n+2)!}
{{(s+t-1+(-1)^a\epsilon\nu)/2}\choose{n+1}}
{{(s+t-(-1)^a\epsilon\nu)/2}\choose{n+1}}\\
&\quad+(-1)^{n+1}\sum_{r=0}^{n+1}\frac{(-4)^rr!^2}{(2r)!}
{{-(s+t-(-1)^a\epsilon\nu+1)/2}\choose{r}}
{{-(s+t+(-1)^a\epsilon\nu)/2}\choose{r}}
{{s+t-1} \choose{n-r+1}}.\end{aligned}$$ For $b=1$ and $c=1$ the inner sum is $$\begin{aligned}
(-1)^{n+1}&
\sum_{r=1}^{n+1}\frac{(-4)^rr!^2}{(2r)!}
{{-(s+t-(-1)^a\epsilon\nu +1)/2}\choose{r-1}}
{{-(s+t +(-1)^a\epsilon\nu)/2}\choose{r}}
{{s+t-1}\choose n+1-r},\end{aligned}$$ and adding this to the contribution from $c=0$, for $b=1$ we obtain $$\begin{aligned}
&(-1)^n\frac{(-4)^{n+1}(n+1)!^2}{(2n+2)!}
{{(s+t-1+(-1)^a\epsilon\nu)/2}\choose{n+1}}
{{(s+t-(-1)^a\epsilon\nu)/2}\choose{n+1}}\\
&\quad+(-1)^{n+1}\left[{{s+t-1}\choose{n+1}}
+\sum_{r=1}^{n+1}\frac{(-4)^rr!^2}{(2r)!}
{{1-(s+t-(-1)^a\epsilon\nu+1)/2}\choose{r}}
{{-(s+t+(-1)^a\epsilon \nu )/2}\choose{r}}
{{s+t-1}\choose{n-r+1}}\right]\\
&=(-1)^n\frac{(-4)^{n+1}(n+1)!^2}{(2n+2)!}
{{(s+t-1+(-1)^a\epsilon\nu)/2}\choose{n+1}}
{{(s+t-(-1)^a\epsilon\nu)/2}\choose{n+1}}\\
&\quad+(-1)^{n+1}\sum_{r=0}^{n+1}\frac{(-4)^rr!^2}{(2r)!}
{{1-(s+t-(-1)^a\epsilon\nu+1)/2}\choose{r}}
{{-(s+t+(-1)^a\epsilon\nu)/2}\choose{r}}
{{s+t-1}\choose{n-r+1}}.\end{aligned}$$ Applying [@BK Lemma A.1(ii)], this is $$\begin{aligned}
&-\frac{4^{n+1}(n+1)!^2}{(2n+2)!}
{{(s+t-1+(-1)^a\epsilon\nu)/2}\choose{n+1}}
\left[{{(s+t-(-1)^a\epsilon\nu)/2}\choose{n+1}}
-{{(s+t-(-1)^a\epsilon\nu)/2-1}\choose{n+1}}\right]\\
&=-\frac{4^{n+1}(n+1)!^2}{(2n+2)!}
\frac{(s+(-1)^a\epsilon\nu+2n)/2}{n+1}
{{(s+2n-2+\frac{1+(-1)^a\epsilon}{2}-\nu)/2}\choose{n}}
{{(s+2n-2+\frac{1-(-1)^a\epsilon}{2}+\nu)/2}\choose{n}}\\
&=\frac{s+2\lfloor t/2\rfloor-(-1)^{a+t}\epsilon\nu}{2\pi}\frac{(-2\pi)^t}{t!}
\frac{\gamma_{f}^{(-)^{a+t}}(1-s)}
{\gamma_{f}^{(-)^{a+t}}(1-s-2\lfloor t/2\rfloor)}.\end{aligned}$$
In all cases, the result matches the formula for $P_f(s;a+t,t)$. Taking $l_0=\lceil{T/2}\rceil$, $\sigma=1$ and applying Mellin inversion, we get , with $T+1$ in place of $T$ when $T$ is odd. In that case, we estimate the final term of the sum by shifting the contour to $\Re(s)=\frac32-T$, which yields $O(y^{T-1})$.
\[lem:Bseries\] Assume that $\Lambda_f(s)$ has at most finitely many simple zeros, and let $\alpha\in{\mathbb{Q}}^\times$ and $z=\alpha+iy$ for some $y\in(0,|\alpha|/4]$. Then there are numbers $a_j(\alpha),b_j(\alpha)\in{\mathbb{C}}$ such that, for any integer $M\geq0$, we have $$\label{eq:Bseries}
B(\alpha+iy)=
O_{\alpha,f,M}(y^M)
+\sum_{j=0}^{M-1}y^{j+\frac12}
\begin{cases}
a_j(\alpha)+b_j(\alpha)\log{y}&\text{if }\nu=k=0,\\
a_j(\alpha)y^\nu+b_j(\alpha)y^{-\nu}&\text{otherwise}.
\end{cases}$$
Let $s\in{\mathbb{C}}$ with $\Re(s)\in(0,1)$, and set ${\omega}=\alpha/y$. We will show that there are numbers $a_j(\alpha,s),b_j(\alpha,s)\in{\mathbb{C}}$ satisfying $$\label{eq:Gseries}
H_f(s,{\omega})y^{\frac12-s}
=\sum_{j=0}^\infty y^{j+\frac12}
\begin{cases}
a_j(\alpha,s)+b_j(\alpha,s)\log{y}&\text{if }\nu=k=0,\\
a_j(\alpha,s)y^\nu+b_j(\alpha,s)y^{-\nu}&\text{otherwise}
\end{cases}$$ and $$\label{eq:abestimate}
a_j(\alpha,s),b_j(\alpha,s)
\ll_{f,\alpha,\varepsilon}(2e^{\pi/2})^{(1+\varepsilon)|s|}
|2/\alpha|^{j+\frac12}\sqrt{j+1},
\quad\text{for all }\varepsilon>0.$$
Let us assume this for now. Then, since $y\le|\alpha|/4$, we have $$\sum_{j=M}^\infty\left(\frac{2y}{|\alpha|}\right)^{j+\frac12}\sqrt{j+1}
\ll_{\alpha,M}y^{M+\frac12},$$ so that (by the trivial estimate $|\Re(\nu)|<\frac12$), $$\label{eq:GseriesM}
H_f(s,{\omega})y^{\frac12-s}
=O_{f,\alpha,M,\varepsilon}((2e^{\pi/2})^{(1+\varepsilon)|s|}y^M)
+\sum_{j=0}^{M-1}y^{j+\frac12}
\begin{cases}
a_j(\alpha,s)+b_j(\alpha,s)\log{y}&\text{if }\nu=k=0,\\
a_j(\alpha,s)y^\nu+b_j(\alpha,s)y^{-\nu}&\text{otherwise}.
\end{cases}$$ We substitute this expansion into . By hypothesis, $\Lambda_f(s)$ has at most finitely many simple zeros, so the sum over $\rho$ in is a finite linear combination of the series with $s=\rho$, which yields an expansion of the shape . As for the integral term in , by the convexity bound and Stirling’s formula, we have $$X_f(s)\Lambda_f(s)\ll_{f,\varepsilon}e^{-(3\pi/2-\varepsilon)|s|}
\quad\text{for }\Re(s)=\tfrac12, \varepsilon>0.$$ Since $2<e^\pi$, the integral converges absolutely and again yields something of the shape .
It remains to show and . First suppose that $k=0$. Then, by , we have $$H_f(s,{\omega})y^{\frac12-s}=|\alpha/{\omega}|^{\frac12-s}
(2\pi i{\omega})^{\frac{1-\epsilon}2}
\operatorname{{}_2F_1}\!\left(\frac{s+\frac{1-\epsilon}2+\nu}2,\frac{s+\frac{1-\epsilon}2-\nu}2;
1-\frac{\epsilon}2;-{\omega}^2\right).$$ Applying the hypergeometric transformation [@GR 9.132(2)] and the defining series , this is $$\label{eq:euler}
\begin{aligned}
&(\pi i\operatorname{sgn}(\alpha))^{\frac{1-\epsilon}2}
|\alpha|^{\frac12-s}\pi^{\frac12}\sum_\pm
\frac{|y/\alpha|^{\frac12\pm\nu}\Gamma(\mp\nu)}
{\Gamma\!\left(1-\frac{s+\frac{1+\epsilon}{2}\pm\nu}{2}\right)
\Gamma\!\left(\frac{s+\frac{1-\epsilon}{2}\mp\nu}{2}\right)}
\operatorname{{}_2F_1}\!\left(\frac{s+\frac{1-\epsilon}{2}\pm\nu}{2},
\frac{s+\frac{1+\epsilon}{2}\pm\nu}{2};
1\pm\nu;-\left(\frac{y}{\alpha}\right)^2\right)\\
&=(\pi i\operatorname{sgn}(\alpha))^{\frac{1-\epsilon}2}
|\alpha|^{\frac12-s}\pi^{\frac12}
\sum_{j=0}^\infty
\sum_\pm
\frac{\Gamma(\mp\nu)}
{\Gamma\!\left(1-\frac{s+\frac{1+\epsilon}{2}\pm\nu}{2}\right)
\Gamma\!\left(\frac{s+\frac{1-\epsilon}{2}\mp\nu}{2}\right)}
\frac{{{-\frac{s+\frac{1-\epsilon}{2}\pm\nu}{2}}\choose{j}}
{{-\frac{s+\frac{1+\epsilon}{2}\pm\nu}{2}}\choose{j}}}
{{{-1\mp\nu}\choose{j}}}
\left|\frac{y}{\alpha}\right|^{2j+\frac12\pm\nu}.
\end{aligned}$$ To pass from this to , we replace $2j$ by $j$ and set $a_j=b_j=0$ when $j$ is odd.
When $\nu\ne0$ we use the estimates $$\left|{{-\frac{s+a\pm\nu}{2}}\choose{j}}\right|
=\left|{{\frac{s+a\pm\nu}{2}+j-1}\choose{j}}\right|
\le 2^{|s+a\pm\nu|/2+j}\ll_f 2^{|s|/2+j}
\quad\text{for }a\in\{0,1\},$$ $$\left|{{-1\mp\nu}\choose{j}}\right|
=\prod_{l=1}^j\left|1\pm\frac{\nu}{l}\right|
\ge\prod_{l=1}^j\left|1-\frac1{2l}\right|
=\left|{{-\frac12}\choose{j}}\right|\gg\frac1{\sqrt{2j+1}}$$ and $$(\pi i\operatorname{sgn}(\alpha))^{\frac{1-\epsilon}2}
|\alpha|^{\frac12-s}\pi^{\frac12}
\frac{\Gamma(\mp\nu)}
{\Gamma\!\left(1-\frac{s+\frac{1+\epsilon}{2}\pm\nu}{2}\right)
\Gamma\!\left(\frac{s+\frac{1-\epsilon}{2}\mp\nu}{2}\right)}
\ll_{f,\varepsilon}e^{(\pi/2+\varepsilon)|s|}
\quad\text{for all }\varepsilon>0$$ to obtain .
When $\nu=0$, has a singularity arising from the $\Gamma(\pm\nu)$ factors, but we can still understand the formula by analytic continuation. To remove the singularity, we replace $y^{\pm\nu}$ by $(y^{\pm\nu}-1)+1$. Since $$\lim_{\nu\to0}\Gamma(\pm\nu)(y^{\pm\nu}-1)=\log{y},$$ in the terms with $y^{\pm\nu}-1$ we can simply take the limit and estimate the remaining factors as before; this gives the $b_j$ terms in and . The terms with $1$ can be written in the form $y^{2j+\frac12}(h_j(\nu)+h_j(-\nu))$, where $h_j$ is meromorphic with a simple pole at $\nu=0$, and independent of $y$. Then $h_j(\nu)+h_j(-\nu)$ is even, so it has a removable singularity at $\nu=0$. By the Cauchy integral formula, we have $$\lim_{\nu\to0}(h_j(\nu)+h_j(-\nu))
=\frac1{2\pi i}\int_{|\nu|=\frac12}\frac{h_j(\nu)+h_j(-\nu)}{\nu}\,d\nu.$$ Since the above estimates hold uniformly for $\nu\in{\mathbb{C}}$ with $|\nu|=\frac12$, they also hold for $\lim_{\nu\to0}(h_j(\nu)+h_j(-\nu))$. This concludes the proof of and when $k=0$.
Turning to $k=1$, by we have $$\begin{aligned}
H_f(s,{\omega})y^{\frac12-s}=
\sum_{\delta\in\{0,1\}}
&\left|\frac{\alpha}{{\omega}}\right|^{\frac12-s}(i{\omega}(s-\epsilon\nu))^\delta\\
&\cdot\operatorname{{}_2F_1}\!\left(\frac{s+(-1)^\delta\frac{1+\epsilon}2+\nu}{2}+\delta,
\frac{s+(-1)^\delta\frac{1-\epsilon}2-\nu}{2}+\delta;\frac12+\delta;
-{\omega}^2\right),\end{aligned}$$ and applying [@GR 9.132(2)], this becomes $$\begin{aligned}
\pi^{\frac12}|\alpha|^{\frac12-s}\sum_{\delta\in\{0,1\}}
&\left(\frac{i\operatorname{sgn}(\alpha)(s-\epsilon\nu)}2\right)^\delta\sum_\pm
\left|\frac{y}{\alpha}\right|^{\frac12+\frac{1\pm(-1)^\delta\epsilon}2\pm\nu}
\frac{\Gamma\bigl(\mp(\nu+(-1)^\delta\frac{\epsilon}2)\bigr)}
{\Gamma\bigl(\frac{s+(-1)^\delta\frac{1\mp\epsilon}2\mp\nu}2+\delta\bigr)
\Gamma\bigl(\frac12-\frac{s+(-1)^\delta\frac{1\pm\epsilon}2\pm\nu}2\bigr)}\\
&\cdot\operatorname{{}_2F_1}\!\left(
\frac{s+(-1)^\delta\frac{1\pm\epsilon}2\pm\nu}2+\delta,
\frac{s+(-1)^\delta\frac{1\pm\epsilon}2\pm\nu}2+\frac12;
1\pm\left(\nu+(-1)^\delta\frac{\epsilon}2\right);
-\left(\frac{y}{\alpha}\right)^2\right).\end{aligned}$$ In this case no singularity arises from the $\Gamma$-factor in the numerator, so expanding the final $\operatorname{{}_2F_1}$ as a series and applying a similar analysis to the above, we arrive at and .
With the lemmas in place, we can now complete the proof of Proposition \[prop:main\]. Let $$\chi_{(0,\frac{|\alpha|}4]}(y)=
\begin{cases}
1&\text{if }y\le\frac{|\alpha|}4,\\
0&\text{if }y>\frac{|\alpha|}4,
\end{cases}$$ and define $$\begin{aligned}
g(y)&=F(\alpha+iy)+A(\alpha+iy)
-\chi_{(0,\frac{|\alpha|}{4}]}(y)\sum_{j=0}^{M-1}y^{j+\frac12}
\begin{cases}
a_j(\alpha)+b_j(\alpha)\log{y}&\text{if }\nu=k=0,\\
a_j(\alpha)y^\nu+b_j(\alpha)y^{-\nu}&\text{otherwise }
\end{cases}\\
&-\eta(i\operatorname{sgn}(\alpha))^k\sum_{t=0}^{M-1}
\frac{(2\pi iN\alpha)^t}{t!}\sum_{a\in\{0,1\}}\frac{i^{-a}}{2\pi i}
\int_{\Re(s)=2}P_f(s;a+t,t)\Delta_{\bar{f}}\!\left(
s+t,-\frac1{N\alpha},\cos^{(a)}\right)
\left(\frac{y}{N\alpha^2}\right)^{\frac12-s}\,ds.\end{aligned}$$ By Lemmas \[lem:FEofF\], \[lem:dualside\] and \[lem:Bseries\], we have $g(y)=O_{\alpha,M}(y^{M-1})$ for $y\le|\alpha|/4$. On the other hand, shifting the contour of the above to the right, we see that $g$ decays rapidly as $y\to\infty$. Hence, $\int_0^\infty g(y)y^{s-\frac12}\frac{dy}{y}$ converges absolutely and defines a holomorphic function for $\Re(s)>\frac52-M$.
We have $$\begin{aligned}
\int_0^\infty F(\alpha+iy)y^{s-\frac 12}\frac{dy}{y}=
\sum_{a\in\{0,1\}}i^{-a}\Delta_f\bigl(s,\alpha,\cos^{(a)}\bigr)
\begin{cases}
1&\text{if }k=1\text{ or }(-1)^a=\epsilon,\\
0&\text{otherwise.}
\end{cases}\end{aligned}$$ By Lemma \[lem:AMellin\], $\int_0^\infty A(\alpha+iy)y^{s-\frac 12}\frac{dy}y$ continues to a holomorphic function on $\Omega$. Similarly, $$\begin{aligned}
\int_0^\infty&y^{s-\frac12}\frac{dy}{y}\cdot
\chi_{(0,\frac{|\alpha|}{4}]}(y)\sum_{j=0}^{M-1}y^{j+\frac12}
\begin{cases}
a_j(\alpha)+b_j(\alpha)\log{y}&\text{if }\nu=k=0,\\
a_j(\alpha)y^\nu+b_j(\alpha)y^{-\nu}&\text{otherwise}
\end{cases}\\
&=\sum_{j=0}^{M-1}
\begin{cases}
\frac{|\alpha/4|^{s+j}}{s+j}\left[a_j(\alpha)
+b_j(\alpha)\left(\log|\alpha/4|-\frac1{s+j}\right)\right]
&\text{if }\nu=k=0,\\
a_j(\alpha)\frac{|\alpha/4|^{s+j+\nu}}{s+j+\nu}
+b_j(\alpha)\frac{|\alpha/4|^{s+j-\nu}}{s+j-\nu}
&\text{otherwise}
\end{cases}\end{aligned}$$ is holomorphic on $\Omega$. Hence, by Mellin inversion, $$\label{eq:gmellin}
\begin{aligned}
&\sum_{a\in\{0,1\}}i^{-a}\Delta_f\bigl(s,\alpha,\cos^{(a)}\bigr)
\begin{cases}
1&\text{if }k=1\text{ or }(-1)^a=\epsilon,\\
0&\text{otherwise}
\end{cases}\\
&-\eta(i\operatorname{sgn}(\alpha))^k(N\alpha^2)^{s-\frac12}
\sum_{t=0}^{M-1}
\frac{(2\pi iN\alpha)^t}{t!}\sum_{a\in\{0,1\}}i^{-a}
P_f(s;a+t,t)\Delta_{\bar{f}}\!\left(
s+t,-\frac1{N\alpha},\cos^{(a)}\right)
\end{aligned}$$ is holomorphic on $\{s\in\Omega:\Re(s)>\frac52-M\}$.
Denoting by $h(\alpha)$, we consider the combination $\frac12(i^{k+a_0}h(\alpha)+i^{-k-a_0}h(-\alpha))$ for some $a_0\in\{0,1\}$. This picks out the term with $a\equiv k+a_0\pmod*{2}$ in the first sum over $a$, and $a\equiv t+a_0\pmod*{2}$ in the second. Therefore, since $$P_f(s;a_0,0)=\begin{cases}
1&\text{if }k=1\text{ or }(-1)^{a_0}=\epsilon,\\
0&\text{otherwise},
\end{cases}$$ we find that $$\label{eq:diff}
\begin{aligned}
P_f(s;a_0,0)&\Delta_f\bigl(s,\alpha,\cos^{(k+a_0)}\bigr)\\
&-\eta(-\operatorname{sgn}(\alpha))^k(N\alpha^2)^{s-\frac12}
\sum_{t=0}^{M-1}
\frac{(2\pi N\alpha)^t}{t!}
P_f(s;a_0,t)\Delta_{\bar{f}}\!\left(
s+t,-\frac1{N\alpha},\cos^{(t+a_0)}\right)
\end{aligned}$$ is holomorphic on $\{s\in\Omega:\Re(s)>\frac52-M\}$. Finally, replacing $M$ by $M+1$ and discarding the final term of the sum, we see that is holomorphic on $\{s\in\Omega:\Re(s)>\frac32-M\}$, as required.
[^1]: A. R. Booker was partially supported by EPSRC Grant `EP/K034383/1`. No data were created in the course of this study.
[^2]: P. J. Cho was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A1B03935186).
|
---
abstract: 'In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton–Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton–Jacobi equations is also obtained.'
address:
- ' Institute for Sustainable Sciences and Development, Hiroshima University 1-4-1 Kagamiyama, Higashi-Hiroshima-shi 739-8527, Japan'
- ' Department of Mathematics, University of Wisconsin Madison, 480 Lincoln Drive, Madison, WI 53706, USA'
author:
- 'Hiroyoshi Mitake and Hung V. Tran'
title: On uniqueness sets of additive eigenvalue problems and applications
---
[^1]
Introduction
============
Let ${\mathbb{T}}^n$ be the usual $n$-dimensional torus. Let the Hamiltonian $H = H(x,p) \in C^2({\mathbb{T}}^n \times {\mathbb{R}}^n)$ be such that
- for every $x\in {\mathbb{T}}^n$, $p \mapsto H(x,p)$ is convex,
- uniformly for $x \in {\mathbb{T}}^n$, $$\lim_{|p| \to \infty} \frac{H(x,p)}{|p|}=+\infty
\quad \text{and} \quad
\lim_{|p| \to \infty} \left( \frac{1}{2} H(x,p)^2 + D_x H(x,p)\cdot p \right)= +\infty.$$
The first order additive eigenvalue (ergodic) problem corresponding to $H$ is $${\rm (E)} \qquad H(x,Dw) = c \qquad \text{ in } {\mathbb{T}}^n.$$ Here, $(w,c) \in C({\mathbb{T}}^n) \times {\mathbb{R}}$ is a pair of unknowns. It was shown in [@LPV] that there exists a unique constant $c\in {\mathbb{R}}$ such that (E) has a viscosity solution $w\in C({\mathbb{T}}^n)$. We denote by $c$ the ergodic constant of (E). Without loss of generality, we normalize the ergodic constant $c$ to be zero henceforth.
One of the most interesting points to study (E) is that (E) is not monotone in $w$, and in general, (E) has many viscosity solutions of different types (see examples in [@LMT Chapter 6] for instance). It is therefore fundamental to understand why this *nonuniqueness phenomenon* appears, and in particular, to find a *uniqueness set* for (E). It turns out that this has deep relations to Hamiltonian dynamics and weak KAM theory. In fact, a uniqueness set for (E) has already been studied in [@FaB; @FS] in the context of weak KAM theory.
In this short paper, we provide a new and simple way to look at this phenomenon by using PDE techniques. Some applications and generalizations are also provided.
Settings and main results
-------------------------
We first recall the definition of Mather measures. Consider the following minimization problem $$\label{M-min}
\min_{\mu \in {\mathcal{F}}} \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} L(x,v) \,d\mu(x,v),$$ where $L$ is the Legendre transform of $H$, that is, $$L(x,v)=\sup_{ p \in {\mathbb{R}}^n} \left( p\cdot v - H(x,p) \right)
\quad\text{for} \ (x,v) \in {\mathbb{T}}^n \times {\mathbb{R}}^n,$$ and $${\mathcal{F}}=\left\{ \mu \in \mathcal{P}({\mathbb{T}}^n \times {\mathbb{R}}^n)\,:\, \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} v\cdot D\phi(x) \,d\mu(x,v) = 0 \ \text{ for all } \phi \in C^1({\mathbb{T}}^n) \right\}.$$ Here, $\mathcal{P}({\mathbb{T}}^n \times {\mathbb{R}}^n)$ is the set of all Radon probability measures on ${\mathbb{T}}^n \times {\mathbb{R}}^n$. Measures belong to ${\mathcal{F}}$ are called *holonomic measures* associated with (E).
Let $\widetilde {\mathcal{M}}\subset{\mathcal{F}}$ be the set of all minimizers of . Each measure in $\widetilde {\mathcal{M}}$ is called a Mather measure.
As we normalize $c=0$, we actually have that (see [@M; @Man; @FaB; @FS] for instance) $$\label{min-0}
\min_{\mu \in {\mathcal{F}}} \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} L(x,v) \,d\mu(x,v)=-c=0.$$ See [@MT6], [@LMT Lemma 6.12] for a proof of a more general version this fact. Here is our first main result.
\[thm:uniqueness\] Assume [(H1)–(H2)]{}. Let $w_1, w_2$ be two viscosity solutions of ergodic problem [(E)]{}. Assume further that $$\label{con-unique}
\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} w_1(x) \,d\mu(x,v) \leq \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} w_2(x)\,d\mu(x,v) \quad \text{ for all } \mu\in \widetilde {\mathcal{M}}.$$ Then $w_1 \leq w_2$ in ${\mathbb{T}}^n$.
Let ${\mathcal{M}}$ be the projected Mather set on ${\mathbb{T}}^n$, that is, $${\mathcal{M}}= \overline{\bigcup_{\mu \in \widetilde {\mathcal{M}}} \text{supp} \left(\text{proj}_{{\mathbb{T}}^n} \mu\right)}.$$ Theorem \[thm:uniqueness\] gives the following straightforward result.
\[cor:uniqueness\] Assume [(H1)–(H2)]{}. Let $w_1, w_2$ be two viscosity solutions of ergodic problem [(E)]{}. Assume further that $w_1 = w_2$ on ${\mathcal{M}}$. Then $w_1 = w_2$ in ${\mathbb{T}}^n$.
Corollary \[cor:uniqueness\] was derived in [@FaB; @FS] much earlier. We provide a simple proof for Theorem \[thm:uniqueness\] in Section \[sec:first\], which is a new application of the nonlinear adjoint method introduced in [@Ev1] (see also [@T1]). A generalization of Theorem \[thm:uniqueness\] to the second order (degenerate viscous) setting, Theorem \[thm:uniqueness2\], is given in Section \[sec:degenerate\]. It is worth mentioning that the result of Theorem \[thm:uniqueness2\] is new in the literature.
Application
-----------
We provide here an application in large time behavior. In this context, we need to strengthen the convexity of $H$ in (H1).
- There exists ${\gamma}>0$ such that $$D^2_{pp}H(x,p) \geq {\gamma}I_n \quad \text{ for all } (x,p) \in {\mathbb{T}}^n \times {\mathbb{R}}^n.$$
Here, $I_n$ is the identity matrix of size $n$.
Under assumptions [(H1’)]{}, [(H2)]{} and that the ergodic constant $c=0$, for given $u_0 \in {{\rm Lip\,}}({\mathbb{T}}^n)$, the viscosity solution $u \in C({\mathbb{T}}^n \times [0,\infty))$ of the Cauchy problem $${\rm (C)} \qquad
\begin{cases}
u_t + H(x,Du)=0 \quad &\text{ in } {\mathbb{T}}^n \times (0,\infty),\\
u(x,0)= u_0(x) \quad &\text{ on } {\mathbb{T}}^n.
\end{cases}$$ has the following large time behavior $$\label{thm:large-time}
\lim_{t \to \infty} \|u(\cdot,t) - v\|_{L^\infty({\mathbb{T}}^n)} =0,$$ where $v \in {{\rm Lip\,}}({\mathbb{T}}^n)$ is a viscosity solution of [(E)]{}. This result was first proved in [@F1]. Notice that there are various different ways to prove it (see [@BS; @CGMT; @LMT] and the references therein). We say that $v$ is the *asymptotic profile* of $u$, and denote it by $u^\infty$, or $u^\infty[u_0]$ to display the clear dependence on the initial data $u_0$.
We now give a representation formula for $u^\infty[u_0]$.
\[thm:profile\] Assume that [(H1’)]{} and [(H2)]{} hold, and the ergodic constant $c=0$. For given $u_0 \in {{\rm Lip\,}}({\mathbb{T}}^n)$, let $u^\infty[u_0]$ be the corresponding asymptotic profile. Then, we have
- $u^\infty[u_0](y) = u_0^-(y)$ for all $y \in {\mathcal{M}}$,
- $u^\infty[u_0](x) = \min \left\{ d(x,y) + u_0^-(y)\,:\, y \in {\mathcal{M}}\right\}$ for all $x\in {\mathbb{T}}^n$.
Here, $$\begin{aligned}
&u_0^-(x)
= \sup \left\{ v(x)\,:\, v \leq u_0 \ \text{on} \ {\mathbb{T}}^n, \ \text{and $v$ is a subsolution to {\rm(E)}} \right\}, \\
&
d(x,y) = \sup\left\{v(x)-v(y)\,:\, v \text{ is a subsolution to {\rm(E)}}\right\}.\end{aligned}$$
Theorem \[thm:profile\] was first proved in [@DS Theorem 3.1], and our purpose is to give a different proof in Section \[sec:app\], which seems to be simpler.
Uniqueness set of the ergodic problem {#sec:first}
=====================================
We present in this section the proof of Theorem \[thm:uniqueness\].
We use ideas introduced in [@CGMT].
For each $i=1,2$ and each ${\varepsilon}>0$, let $u_i^{\varepsilon}$ be the viscosity solution to the Cauchy problem $$\label{C-ep}
\begin{cases}
{\varepsilon}(u^{{\varepsilon}}_i)_t + H(x,Du^{{\varepsilon}}_i) = {\varepsilon}^4{\Delta}u^{{\varepsilon}}_i \qquad &\text { in } {\mathbb{T}}^n \times (0,1),\\
u^{{\varepsilon}}_i(x,0) = w_i(x) \qquad &\text{ on } {\mathbb{T}}^n.
\end{cases}$$ Without the viscosity term, becomes $$\label{C-0}
\begin{cases}
{\varepsilon}(u_i)_t + H(x,Du_i) = 0 \qquad &\text { in } {\mathbb{T}}^n \times (0,1),\\
u_i(x,0) = w_i(x) \qquad &\text{ on } {\mathbb{T}}^n.
\end{cases}$$ It is clear that the unique viscosity solution to is $u_i(x,t) = w_i(x)$ for all $(x,t) \in {\mathbb{T}}^n \times [0,1)$ because of the fact that $w_i$ is a viscosity solution to (E). Thanks to (H2), by a standard argument, there exists $C>0$ independent of ${\varepsilon}$ such that $$\label{grad-bound0}
\|Du^{\varepsilon}_i\|_{{L^{\infty}}({\mathbb{T}}^n \times (0,1))} \leq C$$ and $$\label{eqn-error0}
\|u_i^{{\varepsilon}}- w_i\|_{{L^{\infty}}({\mathbb{T}}^n\times(0,1))}\le C{\varepsilon}.$$ See [@LMT Propositions 4.15 and 5.5] for the proofs of similar versions of and for instance. Our plan is to use $u_1^{\varepsilon}, u_2^{\varepsilon}$ to deduce the conclusion as ${\varepsilon}\to 0$.
For any $x_0 \in {\mathbb{T}}^n$, let ${\sigma}^{\varepsilon}$ be the solution to $$\begin{cases}
-{\varepsilon}{\sigma}^{\varepsilon}_t -\text{div}(D_p H(x,Du^{\varepsilon}_2) {\sigma}^{\varepsilon}) = {\varepsilon}^4{\Delta}{\sigma}^{{\varepsilon}} \qquad &\text { in } {\mathbb{T}}^n \times (0,1),\\
{\sigma}^{\varepsilon}(x,1)= {\delta}_{x_0} \qquad &\text{ on } {\mathbb{T}}^n.
\end{cases}$$ Here ${\delta}_{x_0}$ is the Dirac delta mass at $x_0$.
By convexity of $H$ in (H1), we have $${\varepsilon}(u_1^{\varepsilon}- u_2^{\varepsilon})_t + D_pH(x,Du_2^{\varepsilon})\cdot D(u_1^{\varepsilon}-u_2^{\varepsilon}) \leq {\varepsilon}^4{\Delta}(u_1^{\varepsilon}-u_2^{\varepsilon}).$$ Multiply this by ${\sigma}^{\varepsilon}$, integrate on ${\mathbb{T}}^n$, and note that $$\begin{aligned}
&
\int_{{\mathbb{T}}^n}
\left(-D_pH(x,Du_2^{\varepsilon})\cdot D(u_1^{\varepsilon}-u_2^{\varepsilon})+{\varepsilon}^4{\Delta}(u_1^{\varepsilon}-u_2^{\varepsilon})\right){\sigma}^{\varepsilon}\,dx\\
=&\,
\int_{{\mathbb{T}}^n}
\left(\text{div}(D_p H(x,Du^{\varepsilon}_2) {\sigma}^{\varepsilon})+{\varepsilon}^4{\Delta}{\sigma}^{{\varepsilon}} \right)(u_1^{\varepsilon}-u_2^{\varepsilon})\,dx
=-\int_{{\mathbb{T}}^n}
{\varepsilon}{\sigma}_{t}^{{\varepsilon}}(u_1^{\varepsilon}-u_2^{\varepsilon})\,dx. \end{aligned}$$ Thus, $$\frac{d}{dt} \int_{{\mathbb{T}}^n} (u_1^{\varepsilon}- u_2^{\varepsilon}) {\sigma}^{\varepsilon}\,dx \leq 0,$$ which yields $$\label{u-ineq0}
(u_1^{\varepsilon}- u_2^{\varepsilon})(x_0,1) \leq \int_0^1 \int_{{\mathbb{T}}^n} (u_1^{\varepsilon}- u_2^{\varepsilon}) {\sigma}^{\varepsilon}\,dxdt.$$
In light of the Riesz theorem, there exists $\nu^{{\varepsilon}}\in{\mathcal{P}}({\mathbb{T}}^n\times{\mathbb{R}}^n)$ such that $$\label{def-mu0}
\iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n}\varphi(x,p)\,d\nu^{{\varepsilon}}(x,p)
=
\int_{0}^{1}\int_{{\mathbb{T}}^n}\varphi(x,Du_2^{\varepsilon}){\sigma}^{{\varepsilon}}\,dxdt\quad
\text{for all} \ \varphi\in C_{c}({\mathbb{T}}^n\times{\mathbb{R}}^n).$$ Then, becomes $$\label{ineq-1-0}
(u_1^{\varepsilon}- u_2^{\varepsilon})(x_0,1) \leq \iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n} (u_1^{\varepsilon}- u_2^{\varepsilon}) \,d\nu^{{\varepsilon}}(x,p).$$
Thanks to , we have that $\text{supp}(\nu^{\varepsilon}) \subset {\mathbb{T}}^n \times {\overline}{B}(0,C)$. There exists $\{{\varepsilon}_{j}\}\to 0$ such that $\nu^{{\varepsilon}_j}\rightharpoonup \nu\in{\mathcal{P}}({\mathbb{T}}^n\times{\mathbb{R}}^n)$ as $j\to\infty$ weakly in the sense of measures. We set $\mu \in {\mathcal{P}}({\mathbb{T}}^n \times {\mathbb{R}}^n)$ be such that $$\label{def-mu-nu-0}
\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} \varphi(x,p)\, d\nu(x,p) = \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} \varphi(x,D_v L(x,v))\,d\mu(x,v).$$ We provide a proof that $\mu$ is a Mather measure in Lemma \[lem:mu\] below for completeness (see also [@MT6 Proposition 2.3], [@LMT Proposition 6.11]).
Sending $j \to \infty$ in and using to yield $$w_1(x_0) -w_2(x_0) \leq \iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n} (w_1 - w_2)\,d\mu(x,v) \leq 0.
\qedhere$$
\[lem:mu\] For each ${\varepsilon}>0$, let $\nu^{\varepsilon}$ be the measure defined in . Assume that there exists a sequence $\{{\varepsilon}_j\} \to 0$ such that $\nu^{{\varepsilon}_j} \rightharpoonup \nu \in {\mathcal{P}}({\mathbb{T}}^n \times {\mathbb{R}}^n)$ as $j \to \infty$ weakly in the sense of measures. Let $\mu$ be a measure defined through $\nu$ by . Then $\mu$ is a Mather measure.
Fix any $\phi \in C^1({\mathbb{T}}^n)$, and consider a family $\left\{\phi^m \right\} \subset C^\infty({\mathbb{T}}^n)$ such that $\phi^m\to\phi$ in $C^1({\mathbb{T}}^n)$ as $m \to \infty$.
Multiply the adjoint equation with $\phi^m$ and integrate on ${\mathbb{T}}^n \times [0,1]$ to imply $$\begin{gathered}
{\varepsilon}\int_{{\mathbb{T}}^n} \phi^m(x) {\sigma}^{\varepsilon}(x,0)\,dx - {\varepsilon}\phi^m(x_0) +\int_0^1 \int_{{\mathbb{T}}^n} D_p H(x,Du_2^{\varepsilon}) \cdot D\phi^m(x) {\sigma}^{\varepsilon}(x,t)\,dxdt \\
= {\varepsilon}^4 \int_0^1 \int_{{\mathbb{T}}^n} {\Delta}\phi^m(x) {\sigma}^{\varepsilon}(x,t)\,dxdt.\end{gathered}$$ Let ${\varepsilon}={\varepsilon}_j \to 0$ and $m\to\infty$ in this order to get $$\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} D_p H(x,p)\cdot D\phi(x) \, d\nu(x,p) =
\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} v \cdot D\phi(x) \, d\mu(x,v) = 0.$$ Thus, $\mu \in {\mathcal{F}}$.
We rewrite as $${\varepsilon}(u_2^{\varepsilon})_t + D_p H(x,Du_2^{\varepsilon})\cdot Du_2^{\varepsilon}- {\varepsilon}^4 {\Delta}u_2^{\varepsilon}= D_p H(x,Du_2^{\varepsilon})\cdot Du_2^{\varepsilon}- H(x,Du_2^{\varepsilon}).$$ Multiply this by ${\sigma}^{\varepsilon}$ and integrate on ${\mathbb{T}}^n \times [0,1]$ to yield $${\varepsilon}u_2^{\varepsilon}(x_0,1) - {\varepsilon}\int_{{\mathbb{T}}^n} u_2^{\varepsilon}(x,0) {\sigma}^{\varepsilon}(x,0)\,dx
=\int_0^1 \int_{{\mathbb{T}}^n} (D_p H(x,Du_2^{\varepsilon})\cdot Du_2^{\varepsilon}- H(x,Du_2^{\varepsilon})){\sigma}^{\varepsilon}\,dxdt.$$ We again let ${\varepsilon}={\varepsilon}_j \to 0$ to achieve that $$0=\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} (D_p H(x,p)\cdot p - H(x,p))\, d\nu(x,p)
=\iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} L(x,v)\, d\mu(x,v).$$ Also, note that we have $$\label{ge-0}
\iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n}L(x,v)\,d\mu\ge0
\qquad\text{for all} \ \mu\in{\mathcal{F}},$$ which, together with , completes the proof. See [@LMT Lemma 6.12] for a proof of .
Application {#sec:app}
===========
In this section, we always assume that (H1’)–(H2) hold and that the ergodic constant $c=0$.
\[lem:sub\] Assume that $u_0$ is a viscosity subsolution of [(E)]{}. Then, $$u^\infty[u_0] = u_0 \quad \text{ on } {\mathcal{M}}.$$
We write $u^\infty$ for $u^\infty[u_0]$ in the proof for simplicity.
By the usual comparison principle, we have $u(x,t) \geq u_0(x)$ for all $(x,t) \in {\mathbb{T}}^n \times [0,\infty)$. Hence, $u^\infty \geq u_0$ on ${\mathbb{T}}^n$.
Next, let $\rho$ be a standard mollifier in ${\mathbb{R}}^n$. For each ${\delta}>0$, let $\rho^{\delta}(x) = {\delta}^{-n} \rho({\delta}^{-1}x)$ for all $x\in {\mathbb{R}}^n$. Let $u^{\delta}= \rho^{\delta}*u$. Then due to the convexity of $H$ in $p$, $u^{\delta}$ is a subsolution to $$u^{\delta}_t + H(x,Du^{\delta}) \leq C {\delta}\quad \text{ in }{\mathbb{T}}^n \times (0,\infty).$$ For any Mather measure $\mu\in\widetilde{{\mathcal{M}}}$, by the holonomic and minimizing properties, we have $$\begin{aligned}
\frac{d}{dt} \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} u^{\delta}(x,t) \, d\mu
&= \iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} (u^{\delta}_t + v \cdot Du^{\delta}- L(x,v))\, d\mu\\
& \le \iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} u^{\delta}_t + H(x,Du^{\delta})\, d\mu
\leq C{\delta}.\end{aligned}$$ Therefore, for any $T>0$, $$\iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} u^{\delta}(x,T)\,d\mu
\leq \iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} (u_0)^{\delta}(x)\,d\mu + C{\delta}T.$$ Let ${\delta}\to 0$ and $T \to \infty$ in this order to yield $$\iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} u^\infty \,d\mu \leq \iint_{{\mathbb{T}}^n\times {\mathbb{R}}^n} u_0\,d\mu.$$ Combined with $u^\infty \geq u_0$ on ${\mathbb{T}}^n$, we obtain $u^\infty= u_0$ on ${\mathcal{M}}$, which completes the proof.
Notice that we get $$u(x,t) = u_0(x) \quad \text{ for all } x\in {\mathcal{M}},\ t \in [0,\infty),$$ in the above proof.
We present next the proof of Theorem \[thm:profile\]. Before proceeding to the proof, it is important noticing that $d$ has the following representation formula $$d(x,y) =\inf \left\{ \int_0^t L({\gamma}(s),-\dot {\gamma}(s))\,ds\,:\, t>0, {\gamma}\in {{\rm AC\,}}([0,t],{\mathbb{T}}^n), {\gamma}(0)=x, {\gamma}(t)=y\right\}.$$
It is enough to give only the proof of (i). The second claim (ii) follows immediately from Corollary \[cor:uniqueness\], claim (i) and the representation formulas of $d$ as well as of solutions to (E).
By the definition of $u_0^-$, we have $u_0^-\le u_0$ on ${\mathbb{T}}^n$. In light of the comparison principle, $u_0^-\le u$ on ${\mathbb{T}}^n\times[0,\infty)$, which implies $u_0^- \leq u^\infty$ on ${\mathbb{T}}^n$.
We prove the reverse inequality holds on ${\mathcal{M}}$. Fix $y\in {\mathcal{M}}$ and $z\in {\mathbb{T}}^n$. Set $w_0^z(x) = u_0(z) + d(x,z)$ for $x\in {\mathbb{T}}^n$. Then, note that $w_0^z$ is a viscosity subsolution to (E). Let $w$ be the solution to (C) with initial data $w_0^z$. Thanks to Lemma \[lem:sub\], we get $$\label{eq:w1}
w(y,t) = w_0^z(y) = u_0(z) + d(y,z) \quad \text{ for all } t\in [0,\infty).$$ For a large $t >1$, pick ${\gamma}:[0,t] \to {\mathbb{T}}^n$ to be an optimal path such that ${\gamma}(0)=y$ and $$w(y,t) = w_0^z({\gamma}(t)) + \int_0^t L({\gamma}(s),-\dot {\gamma}(s))\,ds= u_0(z) + d({\gamma}(t),z) + \int_0^t L({\gamma}(s),-\dot {\gamma}(s))\,ds.$$
On the other hand, for any ${\varepsilon}>0$, there exists $t_{\varepsilon}>0$ and a path ${\gamma}: [t,t+t_{\varepsilon}] \to {\mathbb{T}}^n$ with ${\gamma}(t+t_{\varepsilon})=z$ satisfying $$d({\gamma}(t),z) \geq \int_t^{t+t_{\varepsilon}} L({\gamma}(s),-\dot {\gamma}(s))\,ds - {\varepsilon}.$$ Combine the two relations above to imply $$\label{eq:w2}
w_0^z(y) +{\varepsilon}\geq u_0(z) +\int_0^{t+t_{\varepsilon}} L({\gamma}(s),-\dot {\gamma}(s))\,ds \geq u(y,t+t_{\varepsilon}).$$ By letting $t \to \infty$ in , one gets $$w_0^z(y) + {\varepsilon}\geq u^\infty(y).$$ Next, let ${\varepsilon}\to 0$ to conclude that $u_0(z) + d(y,z) \geq u^\infty(y)$. Vary $z$ to yield $$u^\infty(y) \leq \min_{z\in {\mathbb{T}}^n} (u_0(z) + d(y,z)).$$ Notice here that in view of the inf-stability of viscosity subsolutions to convex first order Hamilton–Jacobi equations, we have $\min_{z\in {\mathbb{T}}^n} (u_0(z) + d(y,z))=u_0^-(y)$, which finishes the proof.
Generalization: degenerate viscous cases {#sec:degenerate}
========================================
In this section, we present a generalization of Theorem \[thm:uniqueness\] to the second order (degenerate viscous) setting. In this setting, the ergodic problem is $${\rm (VE)} \qquad
H(x,Dw) = {{\rm tr}\,}\left(A(x)D^2 w\right) + c \quad \text{ in } {\mathbb{T}}^n.$$ As above, $(w,c) \in C({\mathbb{T}}^n) \times {\mathbb{R}}$ is a pair of unknowns. Here $A:{\mathbb{T}}^n \to \mathbb M^{n \times n}_{\text{sym}}$ is the diffusion matrix, where $ \mathbb M^{n \times n}_{\text{sym}}$ is the set of all $n\times n$ real symmetric matrices. We need the following assumptions.
- There exist ${\gamma}>1$ and $C>0$ such that, for all $(x,p) \in {\mathbb{T}}^n \times {\mathbb{R}}^n$, $$\begin{cases}
\displaystyle \frac{1}{C}|p|^{\gamma}- C \leq H(x,p) \leq C(|p|^{\gamma}+1),\\
|D_x H(x,p)| \leq C(1+|p|^{\gamma}),\\
|D_p H(x,p)| \leq C(1 + |p|^{{\gamma}-1}).
\end{cases}$$
- $A(x)=(a_{ij}(x))_{1\leq i,j \leq n} \in \mathbb M^{n \times n}_{\text{sym}}$ with $A\ge0$, and $a_{ij}\in C^2({\mathbb{T}}^n)$ for all $1\leq i,j \leq n$.
By normalization, we always assume that $c=0$ in this section. In fact, under assumptions (H1), (H2’) and (H3), for any $w\in C({\mathbb{T}}^n)$ solving (VE), $w\in {{\rm Lip\,}}({\mathbb{T}}^n)$ (see [@AT Theorem 3.1]).
\[def:generalized\] Let $\widetilde {\mathcal{M}}_V$ be the set of all minimizers of the minimizing problem $$\label{Mv-min}
\min_{\mu \in {\mathcal{F}}} \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} L(x,v) \,d\mu(x,v),$$ where $${\mathcal{F}}_V=\left\{ \mu \in \mathcal{P}({\mathbb{T}}^n \times {\mathbb{R}}^n)\,:\,
\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} v\cdot D\phi -a_{ij}\phi_{x_ix_j}\,d\mu(x,v) = 0 \
\text{ for all } \phi \in C^2({\mathbb{T}}^n) \right\}.$$ Each measure in $\widetilde {\mathcal{M}}_V$ is called a generalized Mather measure.
Because of normalization that $c=0$, as in the first order case, one has that $$\label{min-viscous}
\min_{\mu \in {\mathcal{F}}_V} \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} L(x,v) \,d\mu(x,v)=0.$$ The proof of this claim follows [@LMT Lemma 6.12]. To be more precise, [@LMT Lemma 6.12] deals with the special case $A(x) = a(x) I_n$ where $a\in C^2({\mathbb{T}}^n,[0,\infty))$ and $I_n$ is the identity matrix of size $n$. For general diffusion matrix $A$ satisfying (H3), we perform first inf-sup convolutions, and then normal convolution of a solution $w$ of (VE). See also [@IMT1] for a form of in fully nonlinear, degenerate elliptic PDE settings.
\[thm:uniqueness2\] Assume [(H1), (H2’), (H3)]{}. Let $w_1, w_2$ be two continuous viscosity solutions of ergodic problem [(E)]{}. Assume further that $$\label{con-unique}
\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} w_1(x) \,d\mu(x,v) \leq \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} w_2(x)\,d\mu(x,v) \quad \text{ for all } \mu\in \widetilde {\mathcal{M}}_V.$$ Then $w_1 \leq w_2$ in ${\mathbb{T}}^n$.
We basically repeat the proof of Theorem \[thm:uniqueness\].
For each $k=1,2$ and each ${\varepsilon}>0$, let $u_k^{\varepsilon}$ be the solution to the Cauchy problem $$\label{VC-ep}
\begin{cases}
{\varepsilon}(u^{{\varepsilon}}_k)_t + H(x,Du^{{\varepsilon}}_k) = a_{ij}(u_k^{\varepsilon})_{x_ix_j}+{\varepsilon}^4{\Delta}u^{{\varepsilon}}_k \qquad &\text { in } {\mathbb{T}}^n \times (0,1),\\
u^{{\varepsilon}}_k(x,0) = w_k(x) \qquad &\text{ on } {\mathbb{T}}^n.
\end{cases}$$ Without the viscosity ${\varepsilon}^4 {\Delta}u^{{\varepsilon}}_k $, becomes $$\label{VC-0}
\begin{cases}
{\varepsilon}(u_k)_t + H(x,Du_k) = a_{ij}(u_k)_{x_ix_j} \qquad &\text { in } {\mathbb{T}}^n \times (0,1),\\
u_k(x,0) = w_k(x) \qquad &\text{ on } {\mathbb{T}}^n,
\end{cases}$$ It is clear that the unique viscosity solution to is $u_k(x,t) = w_k(x)$ for all $(x,t) \in {\mathbb{T}}^n \times [0,1)$ because of the fact that $w_k$ is a solution to (VE). Thanks to (H2’) (see [@LMT Theorem 4.5] for instance), there exists $C>0$ independent of ${\varepsilon}$ such that $$\label{grad-bound}
\|Du^{\varepsilon}_i\|_{{L^{\infty}}({\mathbb{T}}^n \times (0,1))} \leq C \quad \text{and} \quad
\|u_i^{{\varepsilon}}- w_i\|_{{L^{\infty}}({\mathbb{T}}^n\times(0,1))}\le C{\varepsilon}.$$ As above, we use $u_1^{\varepsilon}, u_2^{\varepsilon}$ to deduce the conclusion as ${\varepsilon}\to 0$.
For any $x_0 \in {\mathbb{T}}^n$, let ${\sigma}^{\varepsilon}$ be the solution to $$\begin{cases}
-{\varepsilon}{\sigma}^{\varepsilon}_t -\text{div}(D_p H(x,Du^{\varepsilon}_2) {\sigma}^{\varepsilon}) =
(a_{ij}{\sigma}^{{\varepsilon}})_{x_ix_j}+{\varepsilon}^4{\Delta}{\sigma}^{{\varepsilon}} \qquad &\text { in } {\mathbb{T}}^n \times (0,1),\\
{\sigma}^{\varepsilon}(x,1)= {\delta}_{x_0} \qquad &\text{ on } {\mathbb{T}}^n.
\end{cases}$$ Here ${\delta}_{x_0}$ is the Dirac delta mass at $x_0$.
By convexity of $H$, we have $${\varepsilon}(u_1^{\varepsilon}- u_2^{\varepsilon})_t + D_pH(x,Du_2^{\varepsilon})\cdot D(u_1^{\varepsilon}-u_2^{\varepsilon}) \leq
a_{ij}(u_1^{\varepsilon}-u_2^{\varepsilon})_{x_ix_j}+{\varepsilon}^4{\Delta}(u_1^{\varepsilon}-u_2^{\varepsilon}).$$ Multiply this by ${\sigma}^{\varepsilon}$ and integrate on ${\mathbb{T}}^n$ to yield $$\frac{d}{dt} \int_{{\mathbb{T}}^n} (u_1^{\varepsilon}- u_2^{\varepsilon}) {\sigma}^{\varepsilon}\,dx \leq 0.$$ Hence, $$\label{u-ineq}
(u_1^{\varepsilon}- u_2^{\varepsilon})(x_0,1) \leq \int_0^1 \int_{{\mathbb{T}}^n} (u_1^{\varepsilon}- u_2^{\varepsilon}) {\sigma}^{\varepsilon}\,dxdt.$$
Let $\nu^{{\varepsilon}}\in{\mathcal{P}}({\mathbb{T}}^n\times{\mathbb{R}}^n)$ be the measure satisfying $$\label{def-mu}
\iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n}\varphi(x,p)\,d\nu^{{\varepsilon}}(x,p)
=
\int_{0}^{1}\int_{{\mathbb{T}}^n}\varphi(x,Du_2^{\varepsilon}){\sigma}^{{\varepsilon}}\,dxdt\quad
\text{for all} \ \varphi\in C_{c}({\mathbb{T}}^n\times{\mathbb{R}}^n).$$ Then, becomes $$\label{ineq-1}
(u_1^{\varepsilon}- u_2^{\varepsilon})(x_0,1) \leq \iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n} (u_1^{\varepsilon}- u_2^{\varepsilon}) \,d\nu^{{\varepsilon}}(x,p).$$
Thanks to , we have that $\text{supp}(\nu^{\varepsilon}) \subset {\mathbb{T}}^n \times {\overline}{B}(0,C)$. There exists $\{{\varepsilon}_{j}\}\to 0$ such that $\nu^{{\varepsilon}_j}\rightharpoonup \nu\in{\mathcal{P}}({\mathbb{T}}^n\times{\mathbb{R}}^n)$ as $j\to\infty$ weakly in the sense of measures. We set $\mu \in {\mathcal{P}}({\mathbb{T}}^n \times {\mathbb{R}}^n)$ be such that $$\label{def-mu-nu}
\iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} \varphi(x,p) d\nu(x,p) = \iint_{{\mathbb{T}}^n \times {\mathbb{R}}^n} \varphi(x,D_v L(x,v))\,d\mu(x,v).$$ Note that $\mu$ is a generalized Mather measure defined in Definition \[def:generalized\]. We refer to [@MT6 Proposition 2.3] or [@LMT Proposition 6.11] for the details.
Sending $j \to \infty$ in and using to yield $$w_1(x_0) -w_2(x_0) \leq \iint_{{\mathbb{T}}^n\times{\mathbb{R}}^n} (w_1 - w_2)\,d\mu(x,v) \leq 0.
\qedhere$$
Let ${\mathcal{M}}_V$ be the generalized projected Mather set on ${\mathbb{T}}^n$, that is, $${\mathcal{M}}_V = \overline{\bigcup_{\mu \in \widetilde {\mathcal{M}}_V} \text{supp} \left(\text{proj}_{{\mathbb{T}}^n} \mu\right)}.$$ Theorem \[thm:uniqueness2\] gives the following straightforward result.
Assume [(H1), (H2’), (H3)]{}. Let $w_1, w_2$ be two continuous viscosity solutions of ergodic problem [(VE)]{}. Assume further that $w_1 \leq w_2$ on ${\mathcal{M}}_V$. Then $w_1 \leq w_2$ in ${\mathbb{T}}^n$.
[30]{}
S. N. Armstrong, H. V. Tran, *Viscosity solutions of general viscous Hamilton-Jacobi equations*, Math. Ann. 361, 2015, no. 3-4, 647–687.
G. Barles, P. E. Souganidis, *On the large time behavior of solutions of Hamilton–Jacobi equations*, SIAM J. Math. Anal. [**31**]{} (2000), no. 4, 925–939.
F. Cagnetti, D. Gomes, H. Mitake, H. V. Tran, *A new method for large time behavior of convex Hamilton-Jacobi equations: degenerate equations and weakly coupled systems*, Ann. Inst. H. Poincare Anal. Non Lineaire. 32 (2015), 183–200.
A. Davini, A. Siconolfi, *A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations*, SIAM J. Math. Anal. 38 (2006), no. 2, 478–502.
L. C. Evans, *Adjoint and compensated compactness methods for Hamilton-Jacobi PDE*, Arch. Rat. Mech. Anal. [**197**]{} (2010), 1053–1088.
A. Fathi, *Sur la convergence du semi-groupe de Lax-Oleinik*, C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 3, 267–270. A. Fathi, Weak KAM Theorem in Lagrangian Dynamics.
A. Fathi, A. Siconolfi, *PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians*, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185–228.
H. Ishii, H. Mitake, H. V. Tran, *The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus*, [J. Math. Pures Appl. (9)]{}, 108 (2017), no. 2, 125–149.
N. Q. Le, H. Mitake, H.V. Tran, Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations, Lecture Notes in Mathematics 2183, Springer.
P.-L. Lions, G. Papanicolaou, S. R. S. Varadhan, *Homogenization of Hamilton-Jacobi equations*, unpublished work (1987).
R. Mañé, *Generic properties and problems of minimizing measures of Lagrangian systems*. Nonlinearity 9 (1996), no. 2, 273–310.
J. N. Mather, *Action minimizing invariant measures for positive definite Lagrangian systems*, Math. Z. 207 (1991), no. 2, 169–207.
H. Mitake, H. V. Tran, *Selection problems for a discount degenerate viscous Hamilton–Jacobi equation* , Adv. Math., [**306**]{} (2017), 684–703.
H. V. Tran, *Adjoint methods for static Hamilton-Jacobi equations*, Calculus of Variations and PDE [**41**]{} (2011), 301–319.
[^1]: The work of HM was partially supported by the JSPS grants: KAKENHI \#15K17574, \#26287024, \#16H03948, and the work of HT was partially supported by NSF grants DMS-1615944 and DMS-1664424.
|
---
abstract: 'The meaning of “linear expansion” is explained. Particularly accurate [*relative*]{} distances are compiled and homogenized a) for 246 SNeIa and 35 clusters with $v<30,000{{\,\rm km\,s}^{-1}}$, and b) for relatively nearby galaxies with 176 TRGB and 30 Cepheid distances. The 487 objects define a tight Hubble diagram from $300-30,000{{\,\rm km\,s}^{-1}}$ implying individual distance errors of $\la7.5\%$. Here the velocities are corrected for Virgocentric steaming (locally $220{{\,\rm km\,s}^{-1}}$) and – if $v_{220}>3500{{\,\rm km\,s}^{-1}}$ – for a $495{{\,\rm km\,s}^{-1}}$ motion of the Local Supercluster towards the warm CMB pole at $l=275$, $b=12$; local peculiar motions are averaged out by large numbers. A test for linear expansion shows that the corrected velocities increase with distance as predicted by a standard model with $q_{0}=-0.55$ \[corresponding to $(\Omega_{\rm M},\Omega_{\Lambda})=(0.3,0.7)$\], but the same holds – due to the distance limitation of the present sample – for a range of models with $q_{0}$ between $\sim\!0.00$ and $-1.00$. For these models $H_{0}$ does not vary systematically by more than $\pm2.3\%$ over the entire range. Local, distance-dependent variations are equally limited to 2.3% on average. In particular the proposed Hubble Bubble of Zehavi et al. and Jha et al. is rejected at the $4\sigma$ level. – Velocity residuals in function of the angle from the CMB pole yield a satisfactory apex velocity of $448\pm73{{\,\rm km\,s}^{-1}}$ and a coherence radius of the Local Supercluster of $\sim3500{{\,\rm km\,s}^{-1}}$ ($\sim56\;$Mpc), beyond which galaxies are seen on average at rest in co-moving coordinates with respect to the CMB. Since no obvious single accelerator of the Local Supercluster exists in the direction of the CMB dipole its motion must be due to the integral gravitational force of all surrounding structures. Most of the gravitational dipole comes probably from within $5000{{\,\rm km\,s}^{-1}}$.'
author:
- 'A. Sandage'
- 'B. Reindl and G. A. Tammann'
title: ' The Linearity of the Cosmic Expansion Field from 300 to $30,000{{\,\rm km\,s}^{-1}}$ and the Bulk Motion of the Local Supercluster with Respect to the CMB'
---
{#sec:1}
Preliminaries {#sec:1:1}
-------------
The basic prediction of all models of ideal universes that are homogeneous and isotropic is that, if the expansion is real, there must necessarily be a linear relation between redshift and distance. There are many proofs, but among the earliest are those by @Lemaitre:27 [@Lemaitre:31] and @Robertson:28 even before the observational announcement by @Hubble:29 in which he, in the final sentence of his paper, suggested linearity may be only local.
Modern theoretical proofs for global linearity are set out in many of the standard text books at various levels of mathematical sophistication. Popular among the simpler proofs is that a linear relation is the only one that preserves relative shapes of geometrical shapes in an expanding manifold, and which is the same from all vantage points. Deeper proofs, based on properties of the metric of homogeneous, isotropic models, follow @Robertson:29 [@Robertson:33] and @Walker:36, and lead to the Robertson-Walker line element of the metric.
A unique property of a linear velocity field is that every vantage point in the field appears to be the center of the expansion and has the same ratio of velocity to distance over the entire field. Because of this important property, much effort over the past 80 years has been made by the observers to prove, or disprove, linearity, either locally or globally. The first comprehensive result was that by @Hubble:Humason:31, enlarging the observational data available to Hubble in [-@Hubble:29].
Many summaries of the theoretical expectation of linearity and observational verification for the standard model exist. From the theoretical side the classic text books include those by @Heckmann:42, @Bondi:60, @Robertson:Noonan:68, @Harrison:81, @Narlikar:83, @Peebles:93, @Peacock:99. @Bowers:Deeming:84, and @Carroll:Ostlie:96 are exemplary at the high end of the intermediate level. On the observational side, text book-like chapters by @Gunn:Oke:75 and @Sandage:75 [@Sandage:88; @Sandage:95] are useful.
Deviations from a linear velocity-distance relation divide into two categories. (1) The relation is non-linear everywhere, i.e. the deviation is global. (2) The deviation is local, going over into a linear relation at large distances.
The first category is the most fundamental because it denies the standard homogeneous model everywhere. The second contains models where either the local Hubble constant changes up to a certain distance such as increasing or decreasing outward, or where the local irregularities are due to streaming motions due, presumably, to a local inhomogeneous distribution of matter.
The most radical of the first category of global non-linear models can easily be disproved observationally, even as early as in @Hubble:Humason:31, by noting that the slope of the local log(redshift) - magnitude relation is close to 0.2 as expected in the case of linear expansion. This conclusion was much strengthened by the tight Hubble diagram from brightest clusters galaxies in @Humason:etal:56 and then, from 1972 to 1975, by a series of papers on the velocity-distance relation based on new redshifts and apparent magnitudes of galaxy clusters measured at Mount Wilson and Palomar [@Sandage:99 Table 1 for a summary].
Non-linear models in the second category are more difficult to disprove because the deviations from the pure Hubble linear flow are much smaller than in the radical first group and can easily be mocked by systematic distance errors due to bias effects. The literature is large on deviations from a pure Hubble flow, either due to local streaming motions or to a variation of the Hubble constant outward, or to a local bulk motion relative to a globally significant distant kinematic frame. Some are easier to disprove than others.
One such non-linear model assumes that the local global distribution of matter is hierarchical. Following @Charlier:08 [@Charlier:22] and using a suggestion by @Carpenter:38 of a density-size relation for all objects in the universe, @deVaucouleurs:70 [@deVaucouleurs:71] postulated a universe made of hierarchies of decreasing mean density with increasing volume size up to some limiting distance. The velocity-distance relation in such a universe was formulated by @Haggerty:70, and @Haggerty:Wertz:71, following a prediction of @Wertz:70, and was found to be nearly quadratic locally, but becoming linear at large distances. The model was shown, however, to be irreconcilable with observations [@STH:72].
It was early demonstrated [e.g. @ST:75; @Teerikorpi:75a; @Teerikorpi:75b; @Teerikorpi:84; @Teerikorpi:87; @Fall:Jones:76; @Bottinelli:etal:86; @Bottinelli:etal:87; @Bottinelli:etal:88; @Sandage:88; @Sandage:94; @FST:94] that the claimed changes of the Hubble constant outward were often caused by uncorrected observational selection bias [see also @Sandage:94; @Sandage:95 Chapter 10].[^1] It must also be stated that in the early papers some of the claimed distortions of the velocity field were due to the omission of velocity corrections for Virgocentric infall and – for the more distant galaxies – for CMB dipole motion.
Hence, despite many papers and conferences on proposed streaming motions and searches for variations of the Hubble constant with distance, the problem is still open. What has been needed are distance indicators that have very small intrinsic dispersion so as to eliminate, or greatly reduce, the effect of distance-dependent incompleteness selection bias. At small distances the increasing number of Cepheid and TRGB distances becomes most helpful here, whereas for large distances Type Ia supernovae at maximum light have emerged as the best standard candles, just as @Zwicky:62 had proposed.
The Complications: The Invisible World Map Must be Inferred from the Visible World Picture {#sec:1:2}
------------------------------------------------------------------------------------------
The concept of linearity of a velocity-distance velocity field, so simple to describe, is most complicated to define properly in an expanding universe. @Milne:35 proposed language that clarifies the problem. @Robertson:55 also uses the same language.
The problems are these.
\(1) Because we do not observe the universe at large redshift at the same cosmic time as at low redshift, we must distinguish between what Milne called the world map and the world picture. The world map is the state of the universe at a given cosmic time. The world picture is what appears to us. We cut the series of world maps at different cosmic times as we look to different redshifts.
By a linear “velocity”-“distance” relation is meant that at a particular cosmic time there is a linear relation in the world map between the coordinate distance as measured by the metric, and redshift. But the metric distance, $R(t)r$, where $R(t)$ is the time dependent expansion scale factor and $r$ is the invariant (constant for all time) comoving radial metric “distance”. It is not the same as a distance measured by an astronomer using observational data. There are many kinds of such “astronomical” distances, depending on the method by which they are measured. There is the distance when light left a galaxy, the distance when light is received, the luminosity distance by apparent magnitude, the distance by angular size, the round-trip distance by radar signals. All differ at large redshifts, and are the same only in the zero redshift limit. @McVittie:74 discussion is useful here. Which distance do we use in formulating a velocity-“distance” relation and proving that it is “linear”?
There is also the difficulty with “velocity”. What we measure is redshift, not velocity. The two are not the same except, again, in the zero redshift limit [@Harrison:93].
The problem is solved by not using the complicated concepts of velocity and distance but by transforming the world map into the observer’s world picture [*using only observables*]{}. Before 1958 this mapping was done by making a Taylor series expansion of the $R(t)$ expansion factor about the present cosmic time so as to sample the world map at the earlier cosmic times [@Robertson:55]. This could be called a tangent mapping, and is not very useful for redshifts larger than those at moderately local distances. What was needed was a general transformation mapping that is valid for all redshifts.
@Mattig:58 solved the problem with his famous equation that relates the metric distance of a galaxy, $R_{0}r$, at the time of light reception with the redshift, no matter how large, in models with zero cosmological constant. Metric distance is further replaced by apparent magnitude by the @Robertson:38 equation that connects the two. In addition to the apparent magnitude and the redshift the equation contains the @Robertson:55 deceleration parameter, $q_{0}=-R_{0}\ddot{R}/\dot{R}^{2}$, which, in the simplest models with no cosmological constant, is determined by the mean matter density of the model. In that case the values of $H_{0}$ and $q_{0}$ observed at the present epoch are used to parametrize the past and the future of the world map.
The Mattig equation, or more complicated versions for models that incorporate the cosmological constant [e.g. @Mattig:59; @Carroll:etal:92] which we use in the next section, provides the connection between the world map and the world picture. It has properly been said that the Mattig equation is “one of the single most useful equations in cosmology as far as observers are concerned” [@Peacock:99 p. 89]. Mattig’s solution began the modern era of practical observational cosmology and has been used for many auxiliary and related problems concerning such observational data as galaxy counts, angular diameters, and others throughout the past 50 years.
Because the world map defined in the Robertson-Walker metric has a well defined linear relation between coordinate (metric) distance and redshift, and because a Mattig-like equation transforms the map in to the observed picture, the test for linearity becomes one of comparing the observed redshifts at a given apparent magnitude with the predicted redshifts from a model. The linearity test becomes, then, a search for residuals in the subtraction of the redshifts predicted [*from the model*]{} from the observed redshifts at a given photometric distance obtained from the observations as $(m-M)$. Streaming motions and/or variations of the Hubble constant outward are searched for as correlations (or not) of the residuals with direction and photometric distance.
The test for linearity is, then, model-dependent: we must compare the observed redshift-magnitude relation with the prediction for some adopted world model as it is transformed into the world picture by a Mattig-like equation of $(m-M)=f(z,q_{0},\Lambda)$. We can only test linearity relative to the adopted model. We test for the sensitivity to the adopted model in § \[sec:3\].
This Paper {#sec:1:3}
----------
The purpose of this paper is: (1) To test the linearity of the local expansion field within $z<0.1$. Beyond this limit the linearity of the expansion of space is well documented out to $z=0.4$ by the SNeIa of @Hicken:etal:09, which have ample overlap with the present data, and which can be tightly fit to the SNeIa compiled by @Kessler:etal:09 extending to $z>1$. The high-$z$ data are essential to optimize the determination of the world model for which linear expansion is valid. But at smaller distances the character of the expansion field is still poorly known because so far the number of objects with sufficiently accurate distances has been small. Yet the test is important to understand the effect of the observed clumping of visible matter on the local dynamics. It is also important to find the minimum distance at which the cosmic value of $H_{0}$ can be found independent of peculiar velocities. Strong local deviations from linearity have in fact frequently been claimed up to the recent past by @Zehavi:etal:98 and @Riess:etal:09. (2) To determine the size of the comoving volume that partakes of our observed velocity relative to the dipole of the CMB.
The plan of the paper is this. In § \[sec:2\] the data are specified. In § \[sec:2:1\] accurate [*relative*]{} distances of 246 SNeIa and 35 clusters within the adopted distance range are compiled from eight different sources. The distances have demonstrably rms errors of $\le0.18{\mbox{$\;$mag}}$; only relative distances are needed in the present context. The distances on an arbitrary zero point, reduced to the barycenter of the Local Group, are listed in Table \[tab:ml\] together with the appropriate velocities $v_{\rm hel}$, $v_{220}$ (corrected for Virgocentric infall), and $v_{\rm CMB}$ (corrected for motion with respect of the CMB dipole $A_{\rm corr}$ if $v_{220}>3500{{\,\rm km\,s}^{-1}}$). In § \[sec:2:2\] 30 Cepheid and 176 TRGB distances which are equally accurate as those of SNeIa are added in order to extend the sample down to $300{{\,\rm km\,s}^{-1}}$. They are reduced to the barycenter of the Local Group and corrected for the Virgocentric flow as in § \[sec:2:1\]. We justify in § \[sec:2:3\] why other distance indicators are not considered here.
In § \[sec:3\] the problem of linear expansion is set out. The actual test in § \[sec:3:1\] uses a Hubble diagram of [*all*]{} objects in the sample and analyzes the residuals from a Hubble line for a standard $\Lambda$CDM model with $\Omega_{\rm M}=0.3$ and $\Omega_{\Lambda}=0.7$. (Note that this model implies a deceleration parameter of $q_{0}=-0.55$ because it holds that $q_{0}=0.5(\Omega_{\rm M}-2\Omega_{\Lambda})$ for all Friedmann models \[e.g. @Sahni:Starobinsky:00\]). § \[sec:3:2\] explores the dependence of linearity on the adopted model in terms of $q_{0}$.
An analysis of the velocity residuals of objects with $v_{220}<7000{{\,\rm km\,s}^{-1}}$ (the limit is set to avoid an overwhelming effect of distance errors) in function of direction is in § \[sec:4\], showing the size of the Local Supercluster and its motion toward $A_{\rm corr}$, which is the direction of the warm pole of the CMB after correction for our Virgocentric velocity vector.
Results and conclusions are in § \[sec:5\].
{#sec:2}
In order to trace the local expansion field the [*most accurate relative*]{} distances are compiled of objects, comprising SNeIa, clusters, Cepheids, and tip of the red-giant branch (TRGB) distances, with velocities from $300$ to $30,000{{\,\rm km\,s}^{-1}}$.
All quoted distances are reduced to the barycenter of the Local Group (LG) which is assumed to lie on the line between the Galaxy and M31 and at a distance of one third of $(m-M)_{\rm M31}$. The small random error of the distance moduli ($0.15{\mbox{$\;$mag}}$ on average as shown below) makes the samples unusually insensitive to incompleteness bias.
Heliocentric velocities $v_{\rm hel}$ are available for all objects. They are reduced here to the barycenter of the LG following @Yahil:etal:77. The resulting velocities, $v_{\rm LG}$, are then corrected for a self-consistent Virgocentric infall model by $\Delta v_{220}$ which is the vector sum of the $220 (\pm\sim\!30){{\,\rm km\,s}^{-1}}$ infall vector at the position of the LG, [@Peebles:76; @Peebles:80; @Yahil:etal:80a; @Hoffman:etal:80; @Tonry:Davis:81; @Dressler:84; @Yahil:85; @Tammann:Sandage:85; @Kraan-Korteweg:86; @deFreitasPacheco:86; @Giraud:90; @Jerjen:Tammann:93] and the infall velocity of a particular object, both projected onto the line of sight between the observer and the object. The center of the Virgo cluster is taken to coincide with NGC4486 at $l=283.8$, $b=74.5$. The infall vectors scale with Virgocentric distance $r_{\rm Virgo}$ like $1/r_{\rm Virgo}$ if a density profile of the Local Supercluster of $\rho\sim r^{-2}$ is assumed [@Yahil:etal:80b]. Since good (relative) distances are known for all objects including the Virgo cluster, the corrections $\Delta v_{220}$ can be calculated from equation (5) in @STS:06. The velocities $v_{220}=v_{\rm LG}-\Delta v_{220}$ would be observed in the absence of any streaming towards the Virgo cluster center and if the infall model were exact, but deviation from the model [see e.g. @Klypin:etal:03] will add to the true peculiar velocities of local galaxies. The velocities $v_{\rm CMB}$ of only the objects with $v_{220}>3500{{\,\rm km\,s}^{-1}}$ are in addition corrected by $\Delta v_{\rm CMB}=v_{220}-495\cos\alpha{{\,\rm km\,s}^{-1}}$ ($\alpha$ being the angle from the CMB apex $A_{\rm corr}$ to compensate for the reflex of the Local Supercluster motion relative to the CMB \[see § \[sec:4\]\]).
SNIa and Cluster Distances {#sec:2:1}
--------------------------
Eight published sets of accurate relative distances that contain more than 10 SNeIa or clusters are considered here.
Six sets are based on SNeIa. Since SNeIa are treated as standard candles it makes no difference whether their distance measures are published as corrected apparent magnitudes at maximum or as distance moduli. In either case the data are converted to “true” distance moduli by forcing each data set to give the same fixed mean value of $H_{0}$. We have chosen here $H_{0}=62.3$ \[km$\;$s$^{-1}\;$Mpc$^{-1}$\] as found from Cepheid-calibrated SNeIa [@STS:06; @TSR:08b]. As mentioned before the absolute calibration is not necessary for the present investigation, but a realistic calibration will simplify some of the following discussions.
Two sets of relative cluster distances expressed in ${{\,\rm km\,s}^{-1}}$ yield equally tight Hubble diagrams as SNeIa. This is thanks to the large number of galaxy distances determined in each cluster. The mean cluster distances are normalized to the same fiducial value of $H_{0}$.
The objects in each data set are separately plotted in Hubble diagrams $\log v_{\rm 220/{\rm CMB}}$ vs. $(m-M)$ in Figure \[fig:01\]a-g. The Hubble line shown holds for the adopted $\Lambda$CDM model with $\Omega_{\rm M}=0.3$, $\Omega_{\Lambda}=0.7$ and is defined by [@Carroll:etal:92] $$\label{eq:carroll}
(m-M) = 5\log c(1+z_{1})\int_{0}^{z_{1}}[(1+z)^{2}(1+\Omega_{\rm
M}z)-\Omega_{\Lambda}z(2+z)]^{-1/2}dz + 25 -5\log H_{0}.$$ The scatter about the Hubble line ($0.13-0.18{\mbox{$\;$mag}}$) is shown in each panel. The values refer to the velocity range of $3000<v_{220/{\rm CMB}}<20,000{{\,\rm km\,s}^{-1}}$ where they are least affected by peculiar motions, $K$-corrections, and photometry at faint levels.
The sources of the eight samples – all normalized here to the fiducial value of $H_{0}$ – are detailed in the following:
\(1) Maximum magnitudes $m^{\rm corr}_{V}$ are available for 105 Type Ia supernovae (SNIa) (62 of which fall into the fiducial range $3000<v_{220/{\rm CMB}}<20,000{{\,\rm km\,s}^{-1}}$) from @Reindl:etal:05, excluding spectroscopically peculiar objects of type SN1991T and 1991bg. They have been corrected for Galactic and internal absorption and homogenized as to decline rate and intrinsic color. The corresponding distances yield the Hubble diagram in Figure \[fig:01\]a.
\(2) @Wang:etal:06 give corrected distance moduli based on maximum $UBVI$ magnitudes and some adopted standard luminosity of 98 SNeIa (55 of which fall into the fiducial velocity range). The magnitudes are corrected for absorption and homogenized as to decline rate and color index with somewhat different precepts as under (1). Their published SN magnitudes are brighter than in (1) because the intrinsic color of SNeIa was assumed to be improbably blue causing large absorption corrections, but this does not disturb the internal consistency of the data. The data give the Hubble diagram in Figure \[fig:01\]b.
\(3) @Jha:etal:07 have derived homogenized, absorption-corrected maximum $V$ magnitudes of 95 SNeIa (72 of which fall into the fiducial velocity range) by fitting multi-color light curves to templates. The SNeIa are plotted in Figure \[fig:01\]c.
\(4) Maximum $H$ magnitudes of 33 SNeIa (of which 26 fall into the fiducial velocity range) have been published by @Wood-Vasey:etal:08. They provide, [*without*]{} any homogenization for decline rate or color and with only insignificant absorption corrections, the Hubble diagram shown in Figure \[fig:01\]d.
\(5) Maximum $I$ magnitudes, corrected for absorption and decline rate, for 21 SNeIa (of which 19 fall into the fiducial velocity range) have been published by @Freedman:etal:09. The data have been added in Figure \[fig:01\]d.
\(6) @Hicken:etal:09 have reduced SN data for a very large sample of SNeIa in four different ways. Two methods follow @Guy:etal:05 [@Guy:etal:07] and two methods follow the multi-color light curve fitting of @Jha:etal:07 with two different assumptions on the absorption-to-reddening ratio for SNeIa ($R_{V}=3.1$ and 1.7). The authors give distance moduli for all four cases on the assumption of $H_{0}=65$. We have selected the 91 SNeIa of 2001 and later in the range of $3000 < \log v_{220/{\rm CMB}} < 20,000{{\,\rm km\,s}^{-1}}$ for which we could find the parent galaxy, position, and a sufficiently accurate redshift in the NASA/IPAC Extragalactic Database (NED).[^2] For two SNeIa (2002es and 2007qe) the available redshifts are discrepant; they are left out as well as the two deviating SNeIa (2002jy and 2007bz) leaving 87 SNeIa. Their Hubble diagrams based on the reduction methods of @Guy:etal:07 have significantly larger scatter ($\sigma_{(m-M)}=0.23$) than those following the method of @Jha:etal:07 We have adopted therefore the latter taking the mean of the moduli from $R_{V}=3.1$ and 1.7. These means, normalized to the present distance scale, are used to construct the Hubble diagram in Figure \[fig:01\]e.
\(7) @Masters:etal:06 have derived relative mean cluster distances (in ${{\,\rm km\,s}^{-1}}$) from 21cm line width distances of 26 member galaxies per cluster on average. Their sample of 31 clusters defines a tight Hubble diagram with the exception of the three nearest clusters (not shown) whose relative distances have been measured too large [see @TSR:08a]. The reason for the discrepancy, seen already in the preceding work of @Giovanelli:etal:99, is not understood; it is apparently not possible to apply the same selection criteria for cluster members in nearby and distant clusters. The remaining 28 cluster distances, normalized as before, are shown in Figure \[fig:01\]f.
\(8) @Jorgensen:etal:96 have determined relative distances (in ${{\,\rm km\,s}^{-1}}$) of 10 clusters by averaging Fundamental Plane (FP) distances of about 23 galaxies per cluster. The sample is small, but it provides one of the few (relative) distance determinations of the Coma cluster. The Hubble diagram with the normalized cluster distances is in Figure \[fig:01\]g.
The sources (1)–(8) contain 502 entries for 246 different SNeIa and 35 clusters and groups. The distance of objects with more than one entry have been averaged. The relevant parameters of the 281 objects are listed in Table \[tab:ml\].
The columns that need explanation are these: Columns (4) and (5) are the Galactic coordinates. Columns (6), (7), and (8) are the velocities relative to the Sun taken from the NED (Col. 6), the velocities $v_{220}$ corrected for the self-consistent Virgocentric infall model (Col. 7), and – in case $v_{220}>3500{{\,\rm km\,s}^{-1}}$ – the velocities $v_{\rm CMB}$ corrected to the inertial frame of the CMB as explained above (Col. 8). Column (9) is the distance modulus with the zero-point set as before by $H_{0}=62.3$. Note that the moduli are given relative to the barycenter of the LG. Column (10) is the angle in degrees from the adopted CMB apex $A_{\rm corr}$. Column (11) is the difference $\Delta v_{220}$ between the velocity $v_{220}$ and the model velocity $v_{\rm model}$ predicted by equation (\[eq:carroll\]) using the adopted distance moduli from Column (9). Column (12) gives the key to the original sources. 1: @Reindl:etal:05; 2: @Wang:etal:06; 3: @Jha:etal:07; 4a: @Wood-Vasey:etal:08; 4b: @Freedman:etal:09; 5: @Hicken:etal:09; 6: @Masters:etal:06; 7: @Jorgensen:etal:96.
The 281 [*mean*]{} distances in Table \[tab:ml\] define the Hubble diagram in Figure \[fig:01\]h. The subset of 218 objects which fall into the interval $3000 < v_{220/{\rm CMB}} < 20,000{{\,\rm km\,s}^{-1}}$ scatter about the Hubble line by $\sigma_{(m-M)}=0.15{\mbox{$\;$mag}}$, which must be due in part to distance errors and to peculiar velocities.
The systematic distance difference of those SNeIa occurring in at least two of the above sources is zero by construction. Their mean random differences are given in Table \[tab:02\]. The overall average difference is $0.15{\mbox{$\;$mag}}$ which implies a mean error of a single distance determination of $\sim\!0.10{\mbox{$\;$mag}}$. If this value is subtracted in quadrature from the scatter of $0.15{\mbox{$\;$mag}}$ observed in Figure \[fig:01\]h one is left with an error, read in velocity, of $\sigma_{\log\Delta v}=0.022$ or 5%, which corresponds to a radial velocity dispersion of $250{{\,\rm km\,s}^{-1}}$ at $v_{\rm CMB}=5000{{\,\rm km\,s}^{-1}}$ and $430{{\,\rm km\,s}^{-1}}$ in three dimensions. The value compares well with the peculiar velocity of the Local Supercluster of $495{{\,\rm km\,s}^{-1}}$ toward the CMB apex $A_{\rm corr}$ (see § \[sec:4\]).
It is central to this paper to note that the free-fit [*linear*]{} regressions in the Hubble diagrams of Figure \[fig:01\]a-h in the well occupied range of $3000< v_{220/{\rm CMB}}<20,000{{\,\rm km\,s}^{-1}}$, with the slopes set out in Table \[tab:03\], are flatter than the canonical value of 0.2 in six out of seven cases; the mean slope is flatter by $3.5\sigma$. The reason concerns the difference between the world picture seen in the data and the world map as derived by the @Mattig:58 transformation between them to be described in § \[sec:3\].
The Extension of the Hubble Diagram to Smaller Distances {#sec:2:2}
--------------------------------------------------------
### Cepheids {#sec:2:2:1}
The uniformly reduced Cepheid distances of 30 galaxies with $(m-M)>28.2$ were compiled in @TSR:08b. They are, after exclusion of the deviating NGC3627, plotted in Figure \[fig:02\] where also the objects of Figure \[fig:01\]h are repeated.
The Cepheids can be added directly to the Hubble diagram of SNeIa and clusters, because they all share the same zero point of the distance scale. This is by construction, because 10 of the Cepheid distances were used in @STS:06 as the only luminosity calibrators of SNeIa. The agreement is confirmed by the statistical equality of $H_{0}$ from only the 29 Cepheids ($63.4\pm1.8$) by @TSR:08b and the value derived from the SNeIa ($62.3\pm1.3$) by @STS:06.
### TRGB distances {#sec:2:2:2}
The 78, through RR Lyr stars [*independently*]{} calibrated TRGB distances with $(m-M)>28.2$ yield a quite local value of $H_{0}=62.9\pm1.6$ [@TSR:08b]. The close agreement of this value with the fiducial value of 62.3 is taken as justification to plot all 176 TRGB distances outside the LG into Figure \[fig:02\] without any additional normalization. The combination of the SNeIa and clusters in § \[sec:2:1\] with the Cepheids and TRGB distances here defines the Hubble line in Figure \[fig:02\] from the lowest recession velocities up to $30,000{{\,\rm km\,s}^{-1}}$.
It could be objected that the fit of the galaxies with TRGB distances to the more distant objects depended solely on the similarity of the respective values of $H_{0}$, but that the agreement was only the product of chance. Proponents of a high large-scale value of $H_{0}>70$ are actually forced to argue in this way. To answer this objection we have fitted separately the Hubble lines of the TRGB objects and that of the Cepheids and SNeIa in the region of overlap, i.e. $28.2< (m-M)<31.5$. (The lower limit is chosen as elsewhere in this paper to avoid a dominant effect of peculiar velocities). The 29 Cepheids and 19 SNe in this interval fix the intercept of the Hubble line to within $\Delta \log v = \pm0.012$ and the 78 TRGB distances in the same interval to within $\pm0.011$. Joining the two samples leaves therefore an uncertainty of $\Delta \log v = \pm0.016$, corresponding to a distance margin of $\Delta(m-M)=\pm0.08$ (or 4% in linear distance). Without taken regress to the actual value of $H_{0}$, the change of $H_{0}$ between the very local value from TRGB distances and the value on larger scales from Cepheids and SNeIa can therefore be limited to $\pm4\%$ $(1\sigma)$.
Other Hubble Diagrams of Field Galaxies and Clusters {#sec:2:3}
----------------------------------------------------
Different methods to determine the distances of field galaxies have been applied to map the velocity field. They are struck by random distance errors of $>0.3{\mbox{$\;$mag}}$ (except field galaxies with SNeIa) which causes severe statistical problems unless one has complete (or fair) [*distance-limited*]{} samples. Since it is not possible to define such samples much beyond $1000{{\,\rm km\,s}^{-1}}$ the methods have been applied to [*apparent-magnitude-limited samples*]{}. In this case the less luminous galaxies are increasingly discriminated against as one progresses to larger distances. The result is that the mean luminosity of the catalogued galaxies increases with distance (incompleteness bias). If this effect is not allowed for one derives a compressed distance scale with $H_{0}$ seemingly increasing with distance. This then can lead to spurious peculiar velocities and galaxy streamings. Of course other reasons for distance-dependent biases exist, e.g. if the distance indicator depends on resolution or if it depends on galaxy size. In all cases deviations from linear expansion increasing systematically with distance are a sign of some kind of bias.
The problem of bias in the presence of large scatter is discussed for the following four examples.
### The Hubble diagram from 21cm and optical line width distances of spirals {#sec:2:3:1}
A Hubble diagram of a complete [*distance*]{}-limited sample of 104 field galaxies, yet extending out to only $v_{220}=1000{{\,\rm km\,s}^{-1}}$, has been constructed with 21cm line width distances [@TSR:08b Fig. 3a]. The slope of the Hubble line is consistent with 0.2, but the very large scatter of $\sigma_{(m-M)}=0.69{\mbox{$\;$mag}}$ – much too large to be caused by peculiar velocities – prevents a rigorous test for linear expansion. Also an early [*magnitude*]{}-limited sample of 217 field galaxies has been presented by @Aaronson:etal:82. Combining 21cm line widths with $H$ magnitudes they have derived distances which – if cut at $v_{220}<2500{{\,\rm km\,s}^{-1}}$ – define a Hubble line consistent with a slope of 0.2, but the error is substantial due to the large scatter.
@FST:94 have analyzed a magnitude-limited sample of 1355 galaxies out to $\sim10,000{{\,\rm km\,s}^{-1}}$ for which @Mathewson:etal:92 have collected 21cm [*and*]{} optical line widths and corrected $I$ magnitudes. The galaxies, lying in two fields towards and perpendicular to the Great Attractor, define Hubble diagrams with dispersions of $0.4-0.7{\mbox{$\;$mag}}$ depending on line width, and which clearly reveal incompleteness bias by producing slopes significantly larger than 0.2. On the assumption of linear expansion the bias has been corrected out by means of so-called Spaenhauer-diagrams. The corrected distances show a local velocity anomaly in the direction of the Great Attractor of $500{{\,\rm km\,s}^{-1}}$ which, however, levels off at a distance of $4000{{\,\rm km\,s}^{-1}}$. The conclusion was that there is no streaming extending to the Great Attractor at $v\sim4500{{\,\rm km\,s}^{-1}}$.
In order to beat the large intrinsic dispersion of line width distances and the accompanying incompleteness bias very large all-sky samples of several thousand field and cluster galaxies have been studied [@Springob:etal:07; @Springob:etal:09; @Theureau:etal:07] and will be further increased in the future[@Masters:08].
### Hubble diagrams from D$_{n}-\sigma$ and FP distances of early-type galaxies {#sec:2:3:2}
@Faber:etal:89 have derived D$_{n}-\sigma$ distances (in ${{\,\rm km\,s}^{-1}}$) of 317 E and S0 galaxies out to $\sim\!10,000{{\,\rm km\,s}^{-1}}$. The corresponding Hubble diagram has a scatter of $\sigma_{(m-M)}=0.69{\mbox{$\;$mag}}$, which is reduced to $\sigma_{(m-M)}=0.50{\mbox{$\;$mag}}$ by correcting for what the authors call “Malmquist correction”, but which is actually a correction for the population size of the aggregate from which a galaxy is drawn. In their Table 4 they list [*mean*]{} distances of 59 clusters and groups and distances of 58 single galaxies which define the Hubble diagram shown in Figure \[fig:DnSigma\]. The scatter is still large ($0.48{\mbox{$\;$mag}}$) and the slope ($0.187\pm0.07$) is flatter than the expected value of $0.2$. This implies that if $H_{0}$ is assumed to be $62.3$ at $(m-M)=31.5$ it decreases to $H_{0}=56.1$ at $(m-M)=35.0$. The decrease of $H_{0}$ is contrary to what is expected from an incompleteness bias; it may be that the specific “Malmquist correction” of the authors leads to an overcorrection of the distances.
All of @Faber:etal:89 galaxy distances were analyzed in a number of publications some of which preceded the paper by @Faber:etal:89 [e.g. @Dressler:87; @Faber:Burstein:88; @Lynden-Bell:etal:88; @Faber:etal:88; @Burstein:90]. The authors, taking the distances at face value, concluded that a coherent large-scale influx existed into a group of galaxy clusters centered on Abell 3627, called the “Great Attractor”, at a distance of $\sim\!4500{{\,\rm km\,s}^{-1}}$, whose dominant attraction would be the main cause for the Local Group’s motion with respect to the CMB. – First doubts about the r[ô]{}le of the “Centaurus Concentration” (synonymous for Great Attractor) came from @Lynden-Bell:etal:89, who concluded from the optical dipole from galaxy catalogs that most of the Local Group’s motion is caused from within $3500{{\,\rm km\,s}^{-1}}$ and that the more distant cluster concentration about Abell 3627 was only a minor contributor. Additional evidence for this view is given in § \[sec:5\].
Mean Fundamental Plane (FP) distances of 85 clusters with $v_{\rm CMB}=5000-20,000{{\,\rm km\,s}^{-1}}$ have also been derived by @Colless:etal:01, but they are based on average on only $\sim\!5$ cluster members. The resulting Hubble diagram has a large scatter of $\sigma_{(m-M)}=0.34{\mbox{$\;$mag}}$ and the Hubble line has too flat a slope ($0.165$). For these reasons the data have not been used for the linearity test.
Forthcoming bias-corrected FP distances of up to $\ga10,000$ galaxies in the Southern sky are announced [@Smith:etal:04; @Jones:etal:04; @Springob:etal:10]; they shall serve to map the velocity field of the local universe out to $\la20,000{{\,\rm km\,s}^{-1}}$.
### The Hubble diagram from surface brightness fluctuation (SBF) distances {#sec:2:3:3}
SBF distances of 124 mainly E and S0 galaxies have been published by @Tonry:etal:01. Their Hubble diagram is shown in Figure \[fig:SBF\]. The scatter of $\sigma_{(m-M)}=0.43{\mbox{$\;$mag}}$ is large; even for this relatively nearby sample ($v_{220}(\mbox{median}) = 1626{{\,\rm km\,s}^{-1}}$) only part of the scatter can be attributed to peculiar motions. Yet the most striking is the steepness of the Hubble line with slope 0.233; it implies values of $H_{0}=62.5$ at $(m-M)=30.00$ and of $H_{0}=78.7$ at $(m-M)=33.00$. The seeming non-linearity of the expansion field is impossible and must be due to some bias of the data, which may be caused by a selection effect depending on apparent-magnitude (incompleteness bias) or some other distance-dependent bias of this sensitive method (perhaps due to incomplete removal of non-stellar images). The consequence is that the method should not be used to test the character of the expansion field nor for the determination of reliable galaxy distances.
### The Hubble diagram of planetary nebula (PNLF) distances {#sec:2:3:4}
Galaxy distances derived from the bright tail of the luminosity function of the shells of planetary nebulae in the light of the $\lambda\,5007\,$[Å]{} line have been compiled from @Ciardullo:etal:02, @Feldmeier:etal:07, @Herrmann:etal:08, and the NED. Of the resulting galaxies 46 lie outside the LG. They define a Hubble diagram as shown in Figure \[fig:PNLF\]. A free fit through the 36 points beyond $(m-M)=28.2$ (omitting NGC524 with only a lower limit to its distance) leads to a Hubble line with large scatter ($\sigma_{(m-M)}=0.43{\mbox{$\;$mag}}$) and very steep slope. The non-linearity is so pronounced that $H_{0}$ increases from 62.9 to 82.1 as one goes from $(m-M)=29.0$ to 31.5! The reason may be that the brightness of the brightest planetary nebulae depends on galaxy size [@Bottinelli:etal:91; @Tammann:93]. Whatever the reason, the luminosity function of planetary nebulae is not a useful distance indicator.
{#sec:3}
As said at the end of § \[sec:1:2\], the test for linearity – the test if $H_{0}$ is or is not a function of distance in the world map – is model-dependent because it depends on transforming the world picture into the world map. This can only be done by using the $R(t)$ scale factor which depends on the model, and by taking account of the streaming motions where they exist.
The problem and its evident degeneracy is illustrated by the deviation of the slopes in Table \[tab:03\] from $0.200$ which is the value if the world [*map*]{} describes a homogeneous and isotropic universe in the mean. The naive interpretation of the mean slope of $0.193\pm0.002$ would be that the Hubble constant [*decreases*]{} outward such that if $H_{0}=62.3$ locally at say $v=1000{{\,\rm km\,s}^{-1}}$, then the value at $10,000{{\,\rm km\,s}^{-1}}$ would be 58.0.
However, such a conclusion is incorrect. It ignores the transformation of the world picture into the world map via a Mattig-like equation where a second parameter is required in the theoretical, model dependent equation for the Hubble diagram.
We test first in § \[sec:3:1\] for linearity in case of a fixed value of $q_{0}$ and then investigate in § \[sec:3:2\] the range of $q_{0}$ for which linearity cannot be excluded using only the local data with $z<0.1$.
The Test for Linearity Using a World Model with $\Omega_{\rm M}=0.3, \Omega_{\Lambda}=0.7$ {#sec:3:1}
------------------------------------------------------------------------------------------
In a first step of the linearity test we adopt the values of the “concordance model” of $\Omega_{\rm M}=0.3$, $\Omega_{\Lambda}=0.7$, which corresponds to $q_{0}=-0.55$.
The velocity residuals $\Delta\log v$ of the objects in Figure \[fig:02\] from the Hubble line calculated with equation (\[eq:carroll\]) are plotted against the adopted distance moduli in Figure \[fig:06\]a. The scatter of the residuals increases towards smaller distances due to the relatively larger contribution of peculiar motions to the recession velocities. The distribution of the points is as flat as can be expected, giving a slope of $0.000\pm0.001$. This implies that the value of $H_{0}$ remains constant to within $\pm2.3\%$ over a modulus interval of $\Delta(m-M)=10$, corresponding to a distance factor of 100.
In order to test for [*local*]{} distance-dependent variations of $H_{0}$ the data of Figure \[fig:06\]a are averaged per bins of width $\Delta(m-M)=0.5{\mbox{$\;$mag}}$ and shown in Figure \[fig:06\]b. The 21 averaged residuals between $(m-M)=28.0$ and 38.0 are about normally distributed and have an average rms deviation of $\sigma_{\Delta\log v}=0.012$ from the adopted horizontal line, which restricts the local variations of $H_{0}$ to $2.8\%$ on average. Individual intervals with 10 or less objects deviate by up to 7%. Shifting the boundaries of the bins does not change this result.
Earlier suggestions of $H_{0}$ to decrease gradually by 5% out to $18,000{{\,\rm km\,s}^{-1}}$ [@Tammann:99] cannot be maintained in the light of the present, much increased data, which also deny the so-called “Hubble Bubble”. This feature was proposed as a drop of $H_{0}$ by $6.5\pm2.0\%$ beyond a distance of $\sim\!7200{{\,\rm km\,s}^{-1}}$ (corresponding to $(m-M)=35.3$ in the adopted distance scale) by @Zehavi:etal:98 and @Jha:etal:07. This drop has been taken as support for a local overdensity and for a dark-energy-free cosmology [@Wiltshire:08]. Yet @Giovanelli:etal:99, @Conley:etal:07, and @Hicken:etal:09 have questioned the result as we do here. Indeed comparing the bin in Figure \[fig:06\]b just preceding the break with the one following the break – either bin containing more than 30 objects – shows the more distant bin to lie [*higher*]{} by $\delta(\Delta\log v)=0.005\pm0.008$ or $H_{0}$ to be [*larger*]{} by $1.3\pm1.7\%$. This rejects the Hubble Bubble at the level of 4 sigma. – It is also noted that the five intervals within $32.5<(m-M)<35.0$ and the five intervals within $35.5<(m-M)<38.0$, embracing the break point and comprising a total of more than 100 objects each, give an increase of $H_{0}$ with distance of $0.5\pm1.2\%$ instead of the putative decrease of $\sim\!6.5\%$. This rejects a persistent decrease of $H_{0}$ at an even higher level of significance.
The Model Dependence of the Linearity Test {#sec:3:2}
------------------------------------------
It is interesting to ask for the sensitivity of the test for linearity for a range of world models.
### The linearity test for different values of $\Omega_{\rm M}, \Omega_{\Lambda}$ {#sec:3:2:1}
We first consider two [*flat*]{} $\Lambda$CDM models with rather extreme values of ($\Omega_{\rm M}, \Omega_{\Lambda}$) = (0.6, 0.4) and (0.1, 0.9) corresponding to $q_{0}=-0.10$ and $-0.85$, respectively. The differences between the moduli $(m\!-\!M)_{\rm
model}$ for the two trial models from equation (\[eq:carroll\]) and the moduli for the standard model under § \[sec:3:1\] are listed in Table \[tab:q0\], Columns (3) & (4) for various values of $z$. The differences are shown as smooth curves in Figure \[fig:07\], where the differences $\Delta(m\!-\!M)$ are plotted against $\log z$. Also shown are the mean values of $\Delta(m\!-\!M)=(m\!-\!M)_{\rm obs}-(m\!-\!M)_{\rm standard\,model}$ of all objects in Figure \[fig:06\] averaged over intervals of $\Delta\log z=0.3$ and plotted in steps of $\Delta\log z=0.1$. The conclusion is that the averaged data points at larger distances fit reasonably within the boundaries of the two trial models. This means that linearity holds for all flat models with parameters between the two trial models. The weak restriction on the parameter space is no surprise in view of the short leverage in $z$ of the data used.
### The linearity test for different values of $q_{0}$ {#sec:3:2:2}
We now relax the condition of a flat model and specify the trial models only by $q_{0}$. For this purpose we recall the derivation of the theoretical Hubble diagram of $\log z$ vs. apparent magnitude using a standard candle in various homogeneous, isotropic universes.
@Robertson:38 equation that relates the apparent luminosity, $l$, with the metric distance at the present instant of cosmic time is $$l = L[(4\pi R_{0}r)^{-2} (1+z)^{-2}] \label{eq:01}$$ where $L$ is the absolute luminosity and $R_{0}r$ is the metric distance where $r$ is the constant (for all time) dimensionless metric distance, and $R_{0}$ is given by the Mattig solution (for $\Lambda=0$) as $$R_{0}r = \frac{c}{H_{0}q_{0}^{2}(1+z)}
[zq_{0}+(q_{0}-1)\{-1 + (2q_{0}z+1)^{1/2}\}]. \label{eq:02}$$
Combining equations (\[eq:01\]) and (\[eq:02\]) gives the equation for the theoretical Hubble diagram to be $$(m\!-\!M) = 5\log q_{0}^{-2}[zq_{0} +
(q_{0}^{-1})\{-1 +(2q_{0} +1)^{1/2}\}] + C, \label{eq:03}$$ where $C$ contains the Hubble constant and the intrinsic mean absolute magnitude of the distance indicator.
The series expansion of equation (\[eq:03\]), is $$(m\!-\!M) = 5\log z + 1.086(1-q_{0})z
+1/6(2-q_{0}-3q_{0}^{2})z^{2} + O(z^{3}) + C. \label{eq:04}$$ For small values of $z$ equation (\[eq:04\]) is also valid if $\Lambda\ne0$. The effect of the third term is $\le0.003$ for $q_{0}>-1$ and $z<0.1$.
The values of $(m\!-\!M)$ are calculated from equation (\[eq:04\]) for four trial values of $q_{0}=0.00, -0.20, -0.80$, and $-1.00$ and for various values of $z$. Subtracting from these values the modulus $(m\!-\!M)$ resulting from $q_{0}=-0.55$ and the corresponding values of $z$, one obtains the values $\Delta(m\!-\!M)_{\rm model}$ listed in Table \[tab:q0\] (Col. 5-8) and shown in Figure \[fig:07\] as smooth dashed lines. As in § \[sec:3:2:1\] the more distant data points lie within the boundaries for $q_{0}=0.00$ and $-1.00$. The conclusion is that $q_{0}$ is probably negative irrespective of the value of $\Lambda$.
The Linearity of the Expansion at Large Scales {#sec:3:3}
----------------------------------------------
If linear expansion – albeit for a considerable range of parameters – was found in the foregoing from a sample arbitrarily cut at $z<0.1$ ($30,000{{\,\rm km\,s}^{-1}}$), there is sufficient overlap with other SNIa samples extending to higher redshifts to carry the linearity test to $z>1$ (§ \[sec:1:3\]).
The main conclusion of § \[sec:3\] is that the cosmic expansion is linear on all scales. This holds surprisingly, if corrections are applied for the local Virgocentric flow and for the bulk motion of the Local Supercluster toward the warm pole of the CMB (see § \[sec:4\]), down to distances of $300{{\,\rm km\,s}^{-1}}$ ($\sim5\;$Mpc).
{#sec:4}
As evidenced by the dipole of the CMB radiation the LG partakes of a bulk motion in addition to the Virgocentric vector. The dipole, first received with disbelief, was discovered by @Henry:71, @Corey:Wilkinson:76, and @Smoot:etal:77. After attempts to explain the dipole as a primordial effect [@Gunn:88; @Paczynski:Piran:90] it soon became clear that it could be caused only by a local motion which, seen from the Sun, amounts to $369\pm2{{\,\rm km\,s}^{-1}}$ towards the observed apex $A_{\rm obs}$ at $l=263.86\pm0.04$, $b=48.24\pm0.10$ [@Bennett:etal:96; @Hinshaw:etal:07] or, if translated to the barycenter of the LG following @Yahil:etal:77, to $v_{\rm LG}=626\pm30{{\,\rm km\,s}^{-1}}$ towards $l=276\pm2$, $b=30\pm2$. Subtracting the Virgocentric infall vector from $v_{\rm LG}$ one obtains a velocity of the LG of $v_{\rm CMB}=495\pm25{{\,\rm km\,s}^{-1}}$ towards the corrected CMB apex $A_{\rm corr}$ at $l=275\pm2$, $b=12\pm4$ [cf. also @Sandage:Tammann:84]. It was clear from the beginning that a velocity of such size must comprise a very large volume including the Virgo cluster and extending to at least $1500{{\,\rm km\,s}^{-1}}$ because otherwise two galaxies at roughly equal distances in the apex and antapex direction would differ in redshift by $990{{\,\rm km\,s}^{-1}}$! The conclusion is that the whole Local Supercluster moves more or less coherently towards the CMB apex.
Figure \[fig:hypoHD\] shows the aspect of the Hubble diagram if the Galaxy were the only object moving towards the CMB pole with a velocity of $495{{\,\rm km\,s}^{-1}}$. In that case all other nearby galaxies, being at rest in co-moving coordinates with respect to the CMB, would exhibit large peculiar motions as seen from the LG and fill the Hubble diagram within the wide curved envelopes as shown in the Figure. The assumption, besides being unphysical, is clearly contradicted by the concentration of the observed galaxies (repeated from Fig. \[fig:02\]) towards the center line of the diagram. Most galaxies within say $1000{{\,\rm km\,s}^{-1}}$ must share the CMB motion of the Galaxy. Beyond this point the diagram loses its diagnostic power because the vector of $495{{\,\rm km\,s}^{-1}}$ is drowned in the recession velocities of more distant galaxies and in the natural scatter.
The convergence radius of the Local Supercluster, i.e. the distance where field galaxies merge into the inertial frame of the CMB has been determined by several authors, some of which are compiled in Table \[tab:LSconvergence\]. The size and direction of the bulk motion from the different authors are not listed in Table \[tab:LSconvergence\], because some authors postulate additional velocity vectors on larger scales to explain the full CMB dipole motion. One of the reasons – besides systematic distance errors which always lead to large-scale motions – may be that they have not allowed for Virgocentric infall vectors and that they have compared their results with the CMB apex $A_{\rm obs}$ instead of $A_{\rm corr}$ (i.e. after correction of the local Virgocentric infall vector of $220{{\,\rm km\,s}^{-1}}$).
A direct determination of the size of the Local Supercluster and its bulk motion is obtained when one plots the velocity residuals from the Hubble line $\Delta v_{220}= v_{220} - v_{\rm model}$ against the angle $\alpha$ from the apex $A_{\rm corr}$. Galaxies from Figure \[fig:02\] are divided into two bins, $500<v_{220}<3500$ and $3500<v_{220}<7000{{\,\rm km\,s}^{-1}}$ and their apex diagrams are shown in Figure \[fig:apex\]a,b. The result is striking. The distribution of the nearer sample is flat and the more distant sample reflects a local peculiar velocity of $448\pm73{{\,\rm km\,s}^{-1}}$ in satisfactory agreement with the corrected CMB dipole motion of $495\pm25{{\,\rm km\,s}^{-1}}$. Clearly the majority of galaxies within $\sim\!3500{{\,\rm km\,s}^{-1}}$ share the coherent bulk motion of the Local Supercluster, whereas the galaxies beyond this limit are in rest with respect to the inertial frame of the CMB.
The scatter in Figure \[fig:apex\]b is significantly larger than in Figure \[fig:apex\]a. This is because the distance errors of $\ga6\%$ predict velocities with errors increasing with distances such as to overwhelm eventually the signal from streaming velocities. For this reason the apex diagrams should not be carried much beyond $7000{{\,\rm km\,s}^{-1}}$.
Aiming for a still better determination of the convergence length, the objects of Figure \[fig:apex\] in the range $2400<v_{220}<4800{{\,\rm km\,s}^{-1}}$ have been re-binned in $1000{{\,\rm km\,s}^{-1}}$ intervals and are plotted in steps of $200{{\,\rm km\,s}^{-1}}$ in Figure \[fig:apex2\]. The nearer samples in panel a) and b) show already a marginal slope suggesting that the Local Supercluster begins to peter out at even smaller velocities (distances) than $3500{{\,\rm km\,s}^{-1}}$. Panel c) centered at $3300{{\,\rm km\,s}^{-1}}$ support that conclusion at the $1.6\sigma$ level, while panel d) centered at $3500{{\,\rm km\,s}^{-1}}$ shows a highly significant apex motion. Finally the more distant panels e)$-$h) are fully consistent with the expected asymptotic apex motion of $495\pm25{{\,\rm km\,s}^{-1}}$.
The conclusion is that the convergence distance is reached close to $3500{{\,\rm km\,s}^{-1}}$. The exact convergence distance depends presumably on direction, but for a corresponding test the large, yet still restricted sample is not adequate.
For visualization the 90 objects in Figure \[fig:apex\]b (i.e.with $3500<v_{\rm 220}<7000{{\,\rm km\,s}^{-1}}$) are plotted in an Aitoff projection (Fig. \[fig:aitoff\]). The association of the objects with positive and negative velocity residuals with their respective apices is striking. The eccentric position of the apices within their associated objects is due to the paucity of objects in the zone of avoidance. The lopsided distribution of the available objects about the apices is also the reason why we have not independently solved for the direction of the apex, but have taken the CMB dipole direction as given.
The flow pattern of the 144 sample objects with $500<v_{220}<10,000{{\,\rm km\,s}^{-1}}$ and $|\beta|<45^{\circ}$ is shown in Figure \[fig:flow\]. Their distances and velocity residuals $\Delta v_{220}=v_{220}-v_{\rm
model}$ are projected on the “apex plane” which is tilted against the Galactic plane by $12^{\circ}$ and rotated such that the CMB apex $A_{\rm corr}$ has the new coordinates $\lambda=270^{\circ}$, $\beta=0^{\circ}$. The objects within $3500{{\,\rm km\,s}^{-1}}$ have small velocity residuals, whereas the more distant objects have larger velocity residuals on average. The distribution of the available objects beyond $3500{{\,\rm km\,s}^{-1}}$, although not forming a complete nor objectively selected sample, is far from random. Of the 18 objects in the $45^{\circ}$ sector about the apex direction 12 have negative and 6 have positive velocity residuals. The chance probability for this distribution is $12\%$. Of the 29 objects in the corresponding sector about the antapex direction 25 have positive and 4 have negative velocity residuals. In this case the chance probability is only $P=0.01\%$. The bulk motion of the Local Supercluster is hence reflected in Figure \[fig:flow\] at a high level of significance. Consistent with this is the nearly random distribution of the velocity residuals in the two perpendicular sectors about $\lambda=0^{\circ}$ and $\lambda=180^{\circ}$.
{#sec:5}
Exceptionally accurate relative distances are compiled from eight sources in § \[sec:2:1\]. They are based on SNIa magnitudes and on mean 21cm line widths and the Fundamental Plane in case of clusters. They are normalized to an arbitrary value of $H_{0}$ of 62.3 without loss of generality. Accurate TRGB and Cepheid distances were added in § \[sec:2:2\]. The final Hubble diagram with 480 objects extends from 300 to $30,000{{\,\rm km\,s}^{-1}}$ and has a scatter of only $0.15{\mbox{$\;$mag}}$ (8% in linear distance) beyond $3000{{\,\rm km\,s}^{-1}}$; the increase of the scatter at shorter distances, due to peculiar motions, is compensated by large-number statistics.
The expansion of space is as linear as can be measured after allowance is made for a Virgocentric flow model and for the CMB motion of the Local Supercluster. Over the range of 300 to $30,000{{\,\rm km\,s}^{-1}}$ the value of $H_{0}$ does not change systematically by more than $\pm2.3\%$. At poorly populated distances $H_{0}$ may locally vary by 7%. But the so-called Hubble Bubble suggesting a sudden decrease of $H_{0}$ by 6.5% – be it a local dip or a persistent feature beyond – at the particularly well occupied distance around $7200{{\,\rm km\,s}^{-1}}$ is excluded by $4\sigma$ or more.
The question at which distance the cosmic value of $H_{0}$ can be found is answered by Figure \[fig:06\]: beyond $(m-M)=28.0$ ($\sim\!4\;$Mpc) and out to at least $20,000{{\,\rm km\,s}^{-1}}$ any systematic deviation from linear expansion is limited to a few percent. At $20,000{{\,\rm km\,s}^{-1}}$ the expansion field is well tied to the equally linear large-scale expansion field [e.g. @Kessler:etal:09]. The cosmic value of $H_{0}$ can therefore be found quite locally if sufficient calibrators are available to compensate the locally important peculiar velocities. The available 78 TRGB distances outside $(m-M)=28.0$ yield a well determined value of $H_{0}=62.9\pm1.6$ (statistical error) in good agreement with local Cepheids and SNeIa [@TSR:08a]. Allowing for a $0.10{\mbox{$\;$mag}}$ error of the TRGB zero point, based on RR Lyr stars and other distance determinations, and for a generous variation of $H_{0}$ with distance of $0.10{\mbox{$\;$mag}}$ gives a compounded error of $H_{0}$ of $\pm4.7$ (8%), including the statistical error. The discrepancy with values of $H_{0}>70$ [e.g. @Riess:etal:09] is not a subject of the present paper.
The Local Supercluster emerges as a comoving entity of radius $3500{{\,\rm km\,s}^{-1}}$, corresponding to a diameter of $\sim\!110\;$Mpc, with the Virgo cluster at its center and several additional clusters (UMa, Fornax, Eridanus) and a large number of groups which must induce a network of small peculiar motions not considered in this paper [see @Klypin:etal:03 for a map]. Inside the Supercluster the expansion is decelerated about inversely to the distance from the Virgo cluster. The galaxies of the Supercluster are concentrated toward the supergalactic plane [@deVaucouleurs:56] which extends to at least $4500{{\,\rm km\,s}^{-1}}$ [@Lahav:etal:00]. It is to be noted that the CMB apex $A_{\rm corr}$ lies at supergalactic latitude $-39^{\circ}$, which shows that objects near to the plane (like the Great Attractor and the Shapley Concentration) can contribute only a fraction of the acceleration of the Local Supercluster.
The Supercluster’s reflex motion with respect to galaxies beyond $3500{{\,\rm km\,s}^{-1}}$ amounts to $448\pm73{{\,\rm km\,s}^{-1}}$ in good agreement with the velocity of $495\pm25{{\,\rm km\,s}^{-1}}$ toward an apex in the zone of avoidance ($l=275\pm2$, $b=12\pm4$) as inferred from the CMB dipole after correction for the local Virgocentric infall vector. The infall vector at the position of the LG amounts to $220\pm30{{\,\rm km\,s}^{-1}}$ and diminishes for larger Virgocentric velocities according to the mass profile of the Supercluster. The question as to the exact shape of the Local Supercluster is very complex as evidenced for instance by the near component of the Centaurus cluster [@Lucey:etal:91; @Stein:etal:97], and the Pegasus and Hydra clusters all three of which lie near to its outer boundary.
The Local Supercluster is the largest volume to our knowledge for which an unambiguous peculiar motion has been found. Claims for coherent streaming motions over still larger scales are either disproven or remain very doubtful in view of possible systematic errors of the applied distance indicators with large intrinsic scatter. In any case any large-scale streaming motion of which the Local Supercluster could partake is limited by the agreement between the absolute CMB dipole motion of $495\pm25{{\,\rm km\,s}^{-1}}$ and the observed bulk motion of $448\pm73{{\,\rm km\,s}^{-1}}$ with respect to galaxies beyond $3500{{\,\rm km\,s}^{-1}}$. The question as to the relation between volume size and corresponding peculiar velocity is decisive for the theory of structure evolution which predicts decreasing peculiar velocities with increasing structure size. $\Lambda$CDM models with $\Omega_{\rm M}=0.3$, $\Omega_{\Lambda}=0.7$ predict quite generally that a bulk motion of $\la500{{\,\rm km\,s}^{-1}}$ lies in the upper, yet permissible range for a volume of radius $3500{{\,\rm km\,s}^{-1}}$ (@Dekel:00, Fig. 1; @Colless:etal:01, Fig. 14).
The bulk motions of galaxy concentrations outside the Local Supercluster are still difficult to determine in view of the remaining distance errors. The one-dimensional bulk velocities in the inertial frame of the CMB of the 28 (super-) clusters and groups with $3500 < v_{\rm CMB} < 10,000{{\,\rm km\,s}^{-1}}$ contained in Table \[tab:ml\] are compiled in Table \[tab:cluster\] (in order of RA). Columns (1)$-$(3) give the name of the cluster, its adopted distance modulus $\mu^{0}$ from SNeIa, D$_{n}-\sigma$ and/or fundamental plane distances, and the number of distance determinations, respectively. The observed mean cluster velocity, expressed as $v_{\rm CMB}$ is in column (4). The expected velocity $v_{\rm model}$ in column (5) is calculated for $H_{0}=62.3$, $\Omega_{\rm M}=0.3$, and $\Omega_{\Lambda}=0.7$ as throughout in this paper. The bulk velocity $v_{\rm CMB}-v_{\rm model}$ and its error follow in column (6). The $1\sigma$ error is estimated by assuming here a distance error of 7.5% for one distance determination and of 5% for two or three distance determinations. The average (absolute) bulk velocity of the 28 aggregates is $353{{\,\rm km\,s}^{-1}}$. However the signal of $353{{\,\rm km\,s}^{-1}}$ is smaller than the average error of $387{{\,\rm km\,s}^{-1}}$, questioning whether any of the bulk velocities are significant. If one considers instead only the 11 clusters and groups with $3500<v_{\rm CMB}\le5000{{\,\rm km\,s}^{-1}}$, for which the distance errors are less effective, one obtains an average bulk motion of $175\pm250{{\,\rm km\,s}^{-1}}$, or $303\pm432{{\,\rm km\,s}^{-1}}$ in three dimensions. In the sample of 11 only the Hydra cluster has a bulk radial velocity with a significance of $1.9\sigma$, i.e. $-349\pm199{{\,\rm km\,s}^{-1}}$ toward the Local Supercluster. Peculiar velocities of large aggregates of some hundred ${{\,\rm km\,s}^{-1}}$ appear to be compatible with the data. The one-dimensional bulk velocity of the Local Supercluster of $480:\sqrt{3}=280{{\,\rm km\,s}^{-1}}$, as seen from a random external observer, is therefore not exeptional.
It is unphysical to seek for a single attractor accelerating the Local Supercluster as has been proposed in the case of the Great Attractor (see § \[sec:2:3:1\]) or the Shapley Concentration [@Mathewson:etal:92; @Saunders:etal:00; @Kocevski:Ebeling:06; @Basilakos:Plionis:06], because they are 53 and 34 degrees away from the apex direction $A_{\rm corr}$. It is clear that instead the acceleration must be caused by the integrated force exerted by all surrounding mass concentrations and voids. A lower distance limit of the dominant accelerators is set by the radius of the Local Supercluster of $\nobreak{\sim\!3500{{\,\rm km\,s}^{-1}}}$. A direct determination of the convergence depth is afforded by integrating the light of all-sky samples of galaxies up to the point where the light dipole – which is the same as the dipole of the gravitational pull if the mass-to-light ratio is assumed to be constant – agrees with the CMB dipole [see @Yahil:etal:80a; @Davis:Huchra:82; @Lahav:87; @Lynden-Bell:etal:89; @Strauss:etal:92; @Maller:etal:03]. Particularly suited for this purpose is the relatively absorption-free $2\;\mu$m all-sky 2MASS Redshift Survey from which @Erdogdu:etal:06a have derived – by excluding a few nearby galaxies and integrating out to $3000{{\,\rm km\,s}^{-1}}$ – an apex at $l=269\pm9$, $b=37\pm9$. This agrees well with the CMB apex at $l=276\pm2$, $b=30\pm2$ as seen from the Local Group (see § \[sec:4\]).[^3] The agreement is only marginally improved if the integration is carried out to $5000{{\,\rm km\,s}^{-1}}$. The conclusion is that most of the acceleration of the Local Supercluster comes from a distance of $\sim\!4000\pm1000{{\,\rm km\,s}^{-1}}$. A map of the dominant overdensities (in $2\;\mu$m) and voids within a shell centered at 4000 kms-1 is given by @Erdogdu:etal:06b [their Fig. 4]. The structures seen in this shell are probably the main players accelerating the Local Supercluster.
A.S. thanks the Observatories of the Carnegie Institution for post-retirement facilities. G.A.T. thanks Prof. Norbert Straumann for stimulating discussions. The authors thank the Referee for helpful comments.
Aaronson, M., et al. 1982, ApJS, 50, 241 Aaronson, M., Bothun, G., Mould, J., Schommer, R. A., & Cornell, M. E. 1986, ApJ, 302, 536 Basilakos, S., & Plionis, M. 2006, MNRAS, 373, 1112 Bennett, C. L., et al. 1996, ApJ, 464, L1 Bondi, H. 1960, Cosmology (Cambridge: Cambridge Univ. Press) Bottinelli, L., Gouguenheim, L., Fouqu[é]{}, P., Paturel, G., & Teerikorpi, P. 1987, A&A, 181, 1 Bottinelli, L., Gouguenheim, L., Paturel, G., & Teerikorpi, P. 1986, A&A, 156, 157 Bottinelli, L., Gouguenheim, L., Paturel, G., & Teerikorpi, P. 1988, ApJ, 328, 4 Bottinelli, L., Gouguenheim, L., Paturel, G., & Teerikorpi, P. 1991, A&A, 252, 550 Bowers, R., & Deeming, T. 1984, Astrophysics II: Interstellar Matter and Galaxies (Boston: Jones & Bartlett) Burstein, D. 1990, Rep. Prog. Phys., 53, 421 Carpenter, E. F. 1938, ApJ, 88, 344 Carroll, S. M., Press, W. H., & Turner, E. L. 1992, ARA&A, 30, 499 Carroll, B. W., & Ostlie, D. A. 1996, An Introduction to Modern Astrophysics (New York: Addison) Charlier, C. V. L. 1908, Ark. Mat. Astron. Fys. 4, 1 Charlier, C. V. L. 1922, Ark. Mat. Astron. Fys. 16, 1; Medd. Lund. Obs. No. 98 Ciardullo, R., Feldmeier, J. J., Jacoby, G. H., Kuzio de Naray, R., Laychak, M. B., & Durrell, P. R. 2002, ApJ, 577, 31 Colless, M., Saglia, R. P., Burstein, D., Davies, R. L., McMahan, R. K., & Wegner, G. 2001, MNRAS, 321, 277 Conley, A., Carlberg, R. G., Guy, J., Howell, D. A., Jha, S., Riess, A. G., & Sullivan, M. 2007, ApJ, 664, 13 Corey, B. E., & Wilkinson, D. T. 1976, BAAS, 8, 351 Davis, M., & Huchra, J. 1976, ApJ, 254, 437 Dale, D. A., Giovanelli, R., Haynes, M. P., Campusano, L. E., Hardy, E., & Borgani, S. 2000, ApJ, 510, L11 de Freitas Pacheco, J. A. 1986, Rev. Mex. AA, 12, 74 Dekel, A. 2000, in ASP Conf. Ser. 201, Cosmic Flows Workshop, ed. S. Courteau & J. Willick (San Francisco: ASP), 420 de Vaucouleurs, G. 1956, Vistas in Astronomy, 2, 1584 de Vaucouleurs, G. 1970, Science, 167, 1203 de Vaucouleurs, G. 1971, PASP, 83, 113 Dressler, A. 1984, ApJ, 281, 512 Dressler, A. 1987, Sci. Am., 257, 46 Erdoğdu, P., et al. 2006a, MNRAS, 368, 1515 Erdoğdu, P., et al. 2006b, MNRAS, 373, 45 Faber, S. M., & Burstein, D. 1988, in Large-scale motions in the universe (Princeton: Princeton Univ. Press), 115 Faber, S. M., Burstein, D., Davies, R. L., Dressler, A., Lynden-Bell, D., Terlevich, R., & Wegner, G. 1988, in IAU Symp. 130, Large Scale Structures of the Universe, ed. J. Audouze, M.-C. Pelletan, & S. Szalay (Dordrecht: Kluwer), 169 Faber, S. M., Wegner, G., Burstein, D., Davies, R. L., Dressler, A., Lynden-Bell, D., & Terlevich, R. J. 1989, ApJS, 69, 763 Fall, S. M., & Jones, B. J. T. 1976, Nature, 262, 457 Federspiel, M., Sandage, A., & Tammann, G. A. 1994, ApJ, 430, 29 Feldmeier, J. J., Jacoby, G. H., & Philips, M. M. 2007, ApJ, 657, 76 Freedman, W. L., et al. 2009, ApJ, 704, 1036 Giovanelli, R., Haynes, M. P., Salzer, J. J., Wegner, G., da Costa, L. N., & Freudling, W. 1998, AJ, 116, 2632 Giovanelli, R., Dale, D. A., Haynes, M. P., Hardy, E., & Campusano, L. E. 1999, ApJ, 525, 25 Giraud, E. 1990, A&A, 231, 1 Gunn, J. E. 1988, in ASP Conf. Ser. 4, The Extragalactic Distance Scale, ed. S. van den Bergh & C. J. Pritchet (San Francisco: ASP), 344 Gunn, J. E., & Oke, J. B. 1975, ApJ, 195, 255 Guy, J., Astier, P., Nobili, S., Regnault, N., & Pain, R. 2005, A&A, 443, 781 Guy, J., et al. 2007, A&A, 466, 11 Haggerty, M. J. 1970, Physica, 50, 391 Haggerty, M. J. 1971, MNRAS, 155, 495 Harrison, E. R. 1981, 2000, Cosmology: The Science of the Universe (Cambridge: Cambridge Univ. Press) Harrison, E. R. 1993, ApJ, 403, 28 Heckmann, O. 1942, Theorien der Kosmologie (Berlin: Springer) Henry, P. S. 1971, Nature, 231, 516 Herrmann, K. A., Ciardullo, R., Feldmeier, J. J., & Vinciguerra, M. 2007, ApJ, 683, 630 Hicken, M., Wood-Vasey, W. M., Blondin, S., Challis, P., Jha, S., Kelly, P. L., Rest, A., & Kirshner, R. P. 2009, ApJ, 700, 1097 Hinshaw, G., et al. 2007, ApJS, 170, 288 Hoffman, J. H., Olson, D. W., & Salpeter, E. E. 1980, ApJ, 242, 861 Hoffman, Y., Eldar, A., Zaroubi, S., & Dekel, A. 2001, preprint (astro-ph/0102190) Hubble, E. 1929, Proc. Nat. Acad. Sci., 15, 168 Hubble, E., & Humason, M. L. 1931, ApJ, 74, 43 Humason, M. L., Mayall, N. U., & Sandage, A. R. 1956, AJ, 61, 97 Jha, S., Riess, A. G., & Kirshner, R. P. 2007, ApJ, 659, 122 Jerjen, H., & Tammann, G. A. 1993, A&A, 276, 1 Jones, D. H., et al. 2004, MNRAS, 355, 747 J[ø]{}rgensen, I., Franx, M., & Kjaergaard, P. 1996, MNRAS, 280, 167 Kessler, R., et al. 2009, ApJS, 185, 32 Klypin, A., Hoffman, Y., Kravtsov, A. V., & Gottl[ö]{}ber, S. 2003, ApJ, 596, 19 Kocevski, D. D., & Ebeling, H. 2006, ApJ, 645, 1043 Komatsu, E., et al. 2009, ApJS, 180, 330 Kraan-Korteweg, R. C. 1986, A&AS, 66, 255 Lahav, O. 1987, MNRAS, 225, 213 Lahav, O., Santiago, B. X., Webster, A. M., Strauss, M. A., Davis, M., Dressler, A., & Huchra, J. P. 2000, MNRAS, 312, 166 Lema[î]{}tre, G. 1927, Ann. Soc. Sci. Bruxelles A., 47, 49 Lema[î]{}tre, G. 1931, MNRAS, 91, 483 Lilje, P. B., Yahil, A., Jones, B. J. T. 1986, ApJ, 307, 91 Lucey, J. R., Lahav, O., Lynden-Bell, D., Terlevich, R. J., & Melnick, J. 1991, in ASP Conf. Ser. 15, Large-Scale Structures and Peculiar Motions in the Universe, ed. D. W. Latham & L. N. da Costa (San Francisco: ASP), 31 Lynden-Bell, D., Faber, S. M., Burstein, D., Davies, R. L., Dressler, A., Terlevich, R. J., & Wegner, G. 1988, ApJ, 326, 19 Lynden-Bell, D., Lahav, O., & Burstein, D. 1989, MNRAS, 241, 325 Maller, A. H., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003, ApJ, 598, 1 Masters, K. L., Springob, C. M., Haynes, M. P., & Giovanelli, R. 2006, ApJ, 653, 861 Masters, K. L. 2008, ASP Conf. Ser. 395, Frontiers of Astrophysics: Celebration of NRAO 50th Anniversary Science Symposium, ed. A. H. Bridle, J. J. Condon, & G. C. Hunt (San Francisco: ASP), 137 Mathewson, D. S., Ford, V. L., & Buchhorn, M. 1992, ApJS, 81, 413 Mattig, W. 1958, Astron. Nachr., 284, 109 Mattig, W. 1959, Astron. Nachr., 285, 1 McVittie, G. C. 1974, QJRAS, 15, 246 Milne, A. E. 1935, Relativity, Gravitation and World Structure (Oxford: Clarendon) Narlikar, J. V. 1983, 1993, 2002, An Introduction to Cosmology (Cambridge: Cambridge Univ. Press), Three Editions Paczynski, B., & Piran, T. 1990, ApJ, 364, 341 Peacock, J. A. 1999, Cosmological Physics (Cambridge: Cambridge Univ. Press) Peebles, P. J. E. 1976, ApJ, 205, 318 Peebles, P. J. E. 1980, The Large-Scale Structure of the Universe (Princeton: Princeton Univ. Press) Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton: Princeton Univ. Press) Reindl, B., Tammann, G. A., Sandage, A., & Saha, A. 2005, ApJ, 624, 532 Riess, A. G., Press, W. H., & Kirshner, R. P. 1995, ApJ, 445, L91 Riess, A. G., et al. 2009, ApJ, 699, 53
Robertson, H. P. 1928, Phil. Mag., 5, 835 Robertson, H. P. 1929, Proc. Nat. Acad. Sci., 15, 822 Robertson, H. P. 1933, Rev. Mod. Phys., 5, 62 Robertson, H. P. 1938, Z. Astrophys., 15, 69 Robertson, H. P. 1955, PASP, 67, 82 Robertson, H. P., & Noonan, T. W. 1968, Relativity and Cosmology (Philadelphia: Saunders) Sahni, V., & Starobinsky, A. 2000, Int. J. Mod. Phys. D, 9, 373 Sandage, A. 1975, in Galaxies and the Universe, Vol. 9 of Stars and Stellar Systems, ed. A. Sandage, M. Sandage, & J. Kristian (Chicago: Chicago Univ. Press), Chapter 19 Sandage, A. 1988, ARA&A, 26, 561 Sandage, A. 1994, ApJ, 430, 1 Sandage, A. 1995, in The Deep Universe, Saas-Fee Advance Lecture Course 23, (Berlin: Springer), Chapters 5 and 10 Sandage, A. 1999, ARA&A, 37, 445 Sandage, A., & Tammann, G. A. 1975, ApJ, 197, 265 (appendix on the bias proof against the Rubin-Ford effect) Sandage, A., & Tammann, G. A. 1984, in Large-Scale Structure of the Universe, Cosmology and Fundamental Physics, ed. G. Setti & L. Van Hove (Garching: ESO), 127 Sandage, A., Tammann, G. A., & Hardy, E. 1972, ApJ, 172, 253 Sandage, A., Tammann, G. A., Saha, A., Reindl, B., Macchetto, F. D., & Panagia, N. 2006, ApJ, 653, 843 Saunders, W., et al. 2000, in ASP Conf. Ser. 218, Mapping the Hidden Universe: The Universe behind the Milky Way – The Universe in HI, ed. R. C. Kraan-Korteweg, P. A. Henning, & H. Andernach (San Francisco: ASP), 141 Smith, R. J., et al. 2004, AJ, 128, 1558 Smoot, G. F., Gorenstein, M. V., & Muller, R. A. 1977, Phys. Rev. Lett., 39, 898 Spergel, D.N., et al. 2007, ApJS, 170, 377 Springob, C. M., Masters, K. L., Haynes, M. P., Giovanelli, R., & Marinoni, C. 2007, ApJS, 172, 599 Springob, C. M., Masters, K. L., Haynes, M. P., Giovanelli, R., & Marinoni, C. 2007, ApJS, 182, 474 Springob, C. M., et al. 2010, BAAS, 41, 551 Stein, P., Jerjen, H., & Federspiel, M. 1997, A&A, 327, 952 Strauss, M. A., Yahil, A., Davis, M., Huchra, J. P., & Fisher, K. 1992, ApJ, 397, 395 Tammann, G. A. 1993, in IAU Symp. 155, Planetary nebulae, ed. R. Weinberger & A. Acker (Dordrecht: Kluwer), 515 Tammann G. A. 1999, in Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, ed. T. Piran & R. Ruffini (Singapore: World Scientific), 243 Tammann, G. A., & Sandage, A. 1985, ApJ, 294, 81 Tammann, G. A., Sandage, A., & Reindl, B. 2008a, ApJ, 679, 52 Tammann, G. A., Sandage, A., & Reindl, B. 2008b, A&ARv, 15, 289 Teerikorpi, P. 1975a, A&A, 45, 117 Teerikorpi, P. 1975b, Observatory, 95, 105 Teerikorpi, P. 1984, A&A, 141, 407 Teerikorpi, P. 1987, A&A, 173, 39 Theureau, G., Hanski, M. O., Coudreau, N., Hallet, N., & Martin, J.-M. 2007, A&A, 465, 71 Tonry, J. L., & Davis, M. 1981, ApJ, 246, 680 Tonry, J. L., Dressler, A., Blakeslee, J. P., Ajhar, E. A., Fletcher, A. B., Luppino, G. A., Metzger, M. R., & Moore, C. B. 2001, ApJ, 546, 681 Walker, A. G. 1936, Proc. London Math. Soc., 42, 90 Wang, X., Wang, L., Pain, R., Zhou, X., & Li, Z. 2006, ApJ, 645, 488 Watkins, R., Feldman, H. A., Hudson, M. J. 2009, MNRAS, 392, 743 Wertz, J. R. 1970, Ph.D. thesis, Univ. Texas at Austin, Pub. Dept. Astr. Ser II, Vol. 3 No. 4 Wiltshire, D. L. 2008, Int. J. Mod. Phys. D., 17, 641 Wood-Vasey, W. M., et al. 2008, ApJ, 689, 377 Yahil, A. 1985, in The Virgo Cluster of Galaxies, ed. O.-G. Richter & B. Binggeli (Garching: ESO), 359 Yahil, A., Sandage, A., & Tammann, G. A. 1980a, ApJ, 242, 448 Yahil, A., Sandage, A., & Tammann, G. A. 1980b, in Physical Cosmology, ed. E. Balian, J. Audouze, & D. N. Schramm (Amsterdam: North-Holland), 127 Yahil, A., Tammann, G. A., & Sandage, A. 1977, ApJ, 217, 903 Zehavi, I., Riess, A. G., Kirshner, R. P., & Dekel, A. 1998, ApJ, 503, 483 Zwicky, F. 1962, in IAU Symp. 15, Problems of Extragalactic Research, ed. G. C. McVittie (New York: Macmillan), 347
[lllrrrrrcrrl]{} UGC 14 & 2006sr & & 108.87 & -38.36 & 7237 & 7352 & 7069 & 35.33 & 151 & 230 & 5\
UGC 40 & 2003it & & 110.66 & -34.33 & 7531 & 7664 & 7362 & 35.47 & 153 & 76 & 5\
UGC 52 & 2002hw & & 104.56 & -52.64 & 5257 & 5309 & 5103 & 34.66 & 138 & 53 & 5\
Anon 0010-49 & 1992au & & 319.12 & -65.88 & 18287 & 18061 & 18267 & 37.43 & 84 & -173 & 1,3\
UGC 139 & 1998dk & & 102.85 & -62.16 & 3963 & 3981 & 3826 & 33.86 & 129 & 330 & 1,3\
NGC 105 & 1997cw & & 113.09 & -49.48 & 5290 & 5369 & 5140 & 34.09 & 139 & 1314 & 1,3\
Anon 0036+11 & 1996bl & & 116.99 & -51.30 & 10793 & 10830 & 10651 & 36.16 & 137 & 476 & 1,2,3\
NGC 191A & 2006ej & & 113.14 & -71.65 & 6131 & 6079 & 6021 & 35.00 & 119 & -54 & 5\
A2806 & & A2806 & 306.16 & -60.90 & 8304 & 8075 & 8310 & 35.47 & 77 & 487 & 6\
MCG -02-02-86 & 2003ic & A0085 & 115.24 & -72.03 & 16690 & 16616 & 16583 & 36.86 & 118 & 2455 & 5\
NGC 232 & 2006et & & 93.67 & -85.93 & 6647 & 6528 & 6570 & 35.04 & 106 & 283 & 5\
MCG +06-02-17 & 2006mo & & 121.85 & -26.53 & 11093 & 11245 & 10941 & 36.41 & 150 & -339 & 5\
2MASX J0056-01 & 2006gt & & 125.75 & -64.47 & 13422 & 13382 & 13309 & 36.74 & 123 & -43 & 4b\
Anon 0056-01 & 2006nz & & 125.82 & -64.07 & 11468 & 11431 & 11355 & 36.51 & 123 & -682 & 5\
UGC 607 & 1999ef & & 125.71 & -50.08 & 11733 & 11764 & 11602 & 36.74 & 134 & -1660 & 1,3\
UGC 646 & 1998ef & & 125.88 & -30.57 & 5319 & 5473 & 5176 & 34.37 & 146 & 867 & 1,3\
NGC 382 & 2000dk & N383Gr & 126.84 & -30.35 & 5229 & 5380 & 5088 & 34.56 & 145 & 357 & 1,2,3\
NGC 383 Gr & & N383Gr & 126.84 & -30.34 & 5098 & 5250 & 4957 & 34.52 & 145 & 319 & 6\
A2877 & & A2877 & 293.13 & -70.88 & 7405 & 7194 & 7396 & 35.46 & 84 & -359 & 6\
Anon 0116+01 & 2005ir & & 136.24 & -61.42 & 22886 & 22842 & 22782 & 38.12 & 122 & -1840 & 4b,5\
ESO 352-57 & 1992bo & & 261.87 & -80.35 & 5549 & 5380 & 5518 & 34.87 & 92 & -402 & 1,2,3\
NGC 507 Gr & & N507Gr & 130.64 & -29.13 & 4934 & 5089 & 4802 & 34.35 & 142 & 525 & 6\
NGC 0523 & 2001en & N507Gr & 130.91 & -28.32 & 4758 & 4919 & 4627 & 34.27 & 142 & 518 & 5\
A0194 & & A0194 & 142.06 & -63.10 & 5396 & 5370 & 5302 & 34.60 & 119 & 256 & 6,7\
MCG -01-04-44 & 1998dm & & 145.97 & -67.40 & 1959 & 1940 & & 33.21 & 115 & -773 & 1,3\
Anon & 2005hj & & 142.55 & -62.76 & 17388 & 17336 & 17294 & 37.35 & 119 & -265 & 4b,5\
2MASX J0127+19 & 2005hf & A0195 & 134.47 & -42.95 & 12924 & 12980 & 12804 & 36.64 & 134 & 141 & 5\
IC 126 & 1993ae & & 144.62 & -63.23 & 5712 & 5682 & 5621 & 34.70 & 118 & 330 & 1,3\
UGC 1087 & 1999dk & & 137.35 & -47.47 & 4485 & 4548 & 4372 & 34.22 & 130 & 246 & 1,2,3\
NGC 632 & 1998es & & 143.19 & -55.18 & 3168 & 3204 & & 33.29 & 123 & 390 & 1,2,3\
UGC 1162 & 2001eh & & 132.24 & -20.37 & 11103 & 11269 & 10976 & 36.18 & 143 & 821 & 5\
Anon 0145-56 & 1992br & & 288.01 & -59.43 & 26382 & 26119 & 26411 & 38.23 & 72 & 226 & 1,2,3\
NGC 673 & 1996bo & & 144.46 & -48.96 & 5182 & 5224 & 5084 & 34.24 & 125 & 883 & 1,2,3\
UGC 1333 & 2006ob & & 153.30 & -59.00 & 17759 & 17708 & 17680 & 37.33 & 116 & 261 & 5\
A0262 & & A0262 & 136.59 & -25.09 & 4887 & 5048 & 4770 & 34.44 & 138 & 293 & 6\
MCG +00-06-03 & 2005hc & & 155.87 & -58.84 & 13771 & 13719 & 13696 & 36.85 & 115 & -380 & 4b,5\
2MASX J0158+36 & 2006td & & 137.68 & -24.60 & 4761 & 4919 & 4647 & 34.65 & 137 & -313 & 5\
NGC 0809 & 2006ef & & 169.46 & -64.80 & 5361 & 5288 & 5306 & 34.87 & 107 & -494 & 5\
MCG +06-06-12 & 2002hu & & 141.44 & -22.27 & 8994:& 9132:& 8891:& 36.28 & 134 & -1796:& 2\
Anon 0227+28 & 2005eu & & 147.48 & -30.06 & 10463 & 10558 & 10373 & 35.89 & 128 & 1387 & 4a\
UGC 1993 & 1999gp & & 143.25 & -19.51 & 8018 & 8172 & 7922 & 35.57 & 133 & 234 & 1,2,3,4a\
NGC 976 & 1999dq & & 152.84 & -35.87 & 4295 & 4387 & 4218 & 33.79 & 123 & 851 & 1,2,3\
IC 1844 & 1995ak & & 169.66 & -48.98 & 6811 & 6785 & 6767 & 35.04 & 109 & 540 & 1,2,3\
UGC 2320 & 2003iv & & 162.50 & -40.73 & 10285 & 10293 & 10230 & 36.29 & 114 & -684 & 5\
CGCG 539-121 & 2005ls & & 145.75 & -14.64 & 6331 & 6508 & 6245 & 34.82 & 130 & 856 & 5\
UGC 2384 & 2006os & A0397 & 161.38 & -37.47 & 9836 & 9864 & 9780 & 35.86 & 116 & 817 & 5\
A0397 & & A0397 & 161.91 & -37.24 & 9803 & 9830 & 9749 & 35.87 & 115 & 742 & 6\
A0400 & & A0400 & 170.24 & -44.93 & 7315 & 7299 & 7276 & 35.19 & 108 & 615 & 6\
ESO 300-09 & 1992bc & & 245.70 & -59.64 & 5996 & 5791 & 6036 & 35.03 & 75 & -426 & 1,2,3\
MCG -01-09-06 & 2005eq & & 187.64 & -51.74 & 8687 & 8598 & 8677 & 35.78 & 97 & -129 & 4a,b\
NGC 1259 & 2008L & A0426 & 150.24 & -13.62 & 5816 & 5988 & 5745 & 34.63 & 126 & 803 & 5\
NGC 1275 & 2005mz & A0426 & 150.58 & -13.26 & 5264 & 5435 & 5194 & 34.72 & 125 & 34 & 5\
NGC 1316 & 1981D & Fornax & 240.16 & -56.69 & 1760 & 1371 & & 31.23 & 74 & & 1,3\
NGC 1316 & 1980N & Fornax & 240.16 & -56.69 & 1760 & 1371 & & 31.71 & 74 & & 1,3\
Anon 0329-37 & 1992bs & & 240.03 & -55.34 & 18887 & 18659 & 18938 & 37.68 & 73 & -1699 & 1,2,3\
NGC 1380 & 1992A & Fornax & 235.93 & -54.06 & 1877 & 1371 & & 31.82 & 74 & -64 & 1,2,3\
Anon 0336-18 & 1992bp & & 208.83 & -51.09 & 23684 & 23525 & 23713 & 37.79 & 85 & 2160 & 1,2,3\
Anon 0335-33 & 1990Y & & 232.64 & -53.85 & 11702 & 11496 & 11753 & 36.38 & 75 & 68 & 1,3\
UGC 2829 & 2006kf & & 178.55 & -35.71 & 6386 & 6378 & 6378 & 35.12 & 102 & -98 & 5\
ESO 156-08 & 1992bk & A3158 & 265.05 & -48.93 & 17598 & 17333 & 17675 & 37.27 & 61 & 343 & 1,2,3\
NGC 1448 & 2001el & & 251.52 & -51.39 & 1168 & 1015 & & 31.73 & 66 & -362 & 1,2,3\
Grm13 & & Grm13 & 266.04 & -43.47 & 929 & 754 & & 31.74 & 56 & -629 & 7\
A0496 & & A0496 & 209.57 & -36.48 & 9863 & 9736 & 9928 & 36.03 & 78 & -31 & 6\
UGC 3108 & 2006lf & & 159.92 & -1.93 & 3959 & 4150 & 3930 & 34.21 & 115 & -132 & 4a\
NGC 1699 & 2001ep & & 203.60 & -27.55 & 3901 & 3850 & 3971 & 33.95 & 79 & 46 & 5\
Anon 0459-58 & 1992bh & & 267.85 & -37.33 & 13491 & 13220 & 13602 & 36.88 & 50 & -1069 & 1,2,3\
UGC 3218 & 2006le & & 147.89 & 12.18 & 5226 & 5479 & 5176 & 35.03 & 122 & -738 & 4a\
NGC 1819 & 2005el & & 196.17 & -19.40 & 4470 & 4466 & 4535 & 34.21 & 83 & 183 & 4a,b\
A0539 & & A0539 & 195.70 & -17.72 & 8514 & 8486 & 8580 & 35.95 & 83 & -936 & 7\
UGC 3329 & 1999ek & & 189.40 & -8.23 & 5253 & 5304 & 5314 & 34.59 & 87 & 213 & 1,2,3,4a\
PGC 17787 & 1993ac & & 149.72 & 17.21 & 14690 & 14923 & 14652 & 37.00 & 118 & -149 & 1,2,3\
CGCG 308-09 & 2006N & & 149.44 & 19.97 & 4280 & 4561 & 4245 & 34.36 & 117 & -24 & 4a\
A3381 & & A3381 & 240.29 & -22.70 & 10763 & 10561 & 10907 & 36.32 & 48 & -565 & 7\
UGC 3432 & 1996bv & & 157.34 & 17.97 & 4998 & 5247 & 4984 & 34.44 & 111 & 491 & 1,2,3\
2MASX J0627-35 & 1999ao & & 243.83 & -20.02 & 16189 & 15974 & 16342 & 37.21 & 44 & -570 & 5\
CGCG 233-023 & 2002kf & & 165.59 & 18.25 & 5786 & 5999 & 5799 & 34.98 & 104 & -79 & 5\
ESO 427-06 & 2004S & & 240.79 & -14.79 & 2806 & 2672 & & 33.47 & 43 & -383 & 2\
NGC 2258 & 1997E & & 140.22 & 25.82 & 4059 & 4374 & 4007 & 34.26 & 122 & -7 & 1,2,3\
UGC 3576 & 1998ec & & 166.30 & 20.71 & 5966 & 6180 & 5984 & 35.16 & 102 & -415 & 1,3\
UGC 3634 & 2005na & & 201.40 & 8.61 & 7891 & 7923 & 8005 & 35.57 & 72 & -16 & 4a,b\
NGC 2320 & 2000B & & 166.36 & 22.79 & 5944 & 6169 & 5964 & 34.76 & 102 & 669 & 1,3\
A0569 & & A0569 & 168.57 & 22.81 & 6026 & 6237 & 6053 & 35.03 & 100 & 20 & 6\
UGC 3725 & 2007au & A0569 & 167.38 & 23.53 & 6171 & 6390 & 6195 & 34.98 & 101 & 312 & 5\
NGC 2268 & 1982B & & 129.24 & 27.55 & 2222 & 2610 & & 32.48 & 128 & 668 & 1\
UGC 3770 & 2000fa & & 194.17 & 15.48 & 6378 & 6469 & 6477 & 35.23 & 78 & -338 & 1,2,3\
UGC 3787 & 2003ch & & 207.22 & 10.26 & 7495 & 7504 & 7626 & 35.97 & 66 & -2002 & 5\
UGC 3845 & 1997do & & 171.00 & 25.27 & 3034 & 3279 & & 33.69 & 97 & -99 & 1,2,3\
Anon 0741-62 & 1992bg & & 274.61 & -18.35 & 10793 & 10537 & 10947 & 36.28 & 30 & -391 & 1,2,3\
MCG +08-14-43 & 2007R & & 174.35 & 28.16 & 9258 & 9443 & 9307 & 36.12 & 93 & -727 & 5\
NGC 2441 & 1995E & & 141.99 & 30.26 & 3470 & 3804 & 3430 & 33.66 & 118 & 472 & 1,2,3\
UGC 4133 & 2006qo & & 161.38 & 31.68 & 9130 & 9368 & 9145 & 35.80 & 103 & 561 & 5\
UGC 4195 & 2000ce & & 149.10 & 32.00 & 4888 & 5180 & 4869 & 34.69 & 112 & -148 & 1,2,3,4a\
CGCG 207-42 & 2006te & & 178.98 & 32.08 & 9471 & 9648 & 9536 & 36.08 & 88 & -341 & 5\
A0634 & & A0634 & 159.40 & 33.64 & 7945 & 8192 & 7957 & 35.72 & 104 & -303 & 6\
UGC 4322 & 2002he & & 153.60 & 33.98 & 7364 & 7631 & 7360 & 35.51 & 108 & -95 & 5\
Cancer & & Cancer & 202.55 & 28.69 & 4497 & 4613 & 4623 & 34.49 & 69 & -252 & 6\
NGC 4414 & 1974G & & 174.54 & 83.18 & 716 & 788 & & 31.47 & 79 & -433 & 1,2\
UGC 4414 & 2005mc & & 202.50 & 30.23 & 7561 & 7656 & 7687 & 35.69 & 69 & -724 & 5\
NGC 2595 & 1999aa & Cancer & 202.73 & 30.31 & 4330 & 4452 & 4457 & 34.52 & 69 & -479 & 1,2,3\
UGC 4455 & 2007bd & & 226.07 & 21.52 & 9299 & 9277 & 9474 & 35.89 & 47 & 107 & 5\
NGC 2623 & 1999gd & & 198.84 & 33.97 & 5549 & 5682 & 5666 & 34.90 & 72 & -179 & 1,2,3\
MCG +03-22-20 & 2004gs & & 207.96 & 31.32 & 7988 & 8065 & 8126 & 35.78 & 64 & -662 & 4b\
UGC 4614 & 2005ms & & 186.90 & 38.49 & 7556 & 7727 & 7644 & 35.60 & 81 & -319 & 5\
MCG -01-23-08 & 2002hd & & 234.77 & 23.16 & 10493 & 10449 & 10680 & 35.93 & 40 & 1112 & 5\
CGCG 180-22 & 1999X & & 186.59 & 39.59 & 7546 & 7726 & 7634 & 35.49 & 81 & 70 & 1,3\
MCG +08-17-43 & 2001G & & 168.32 & 42.31 & 5028 & 5292 & 5072 & 34.37 & 94 & 686 & 5\
A0779 & & A0779 & 191.07 & 44.41 & 6742 & 6913 & 6840 & 35.52 & 77 & -848 & 6\
NGC 2935 & 1996Z & & 253.59 & 22.57 & 2271 & 2280 & & 32.76 & 23 & 72 & 1,2,3\
NGC 2930 & 2005M & & 206.89 & 46.22 & 6599 & 6733 & 6728 & 35.35 & 66 & -454 & 4b\
UGC 5129 & 2001fe & & 203.70 & 46.88 & 4059 & 4240 & 4182 & 34.17 & 68 & 35 & 5\
NGC 2962 & 1995D & & 230.00 & 39.67 & 1966 & 2120 & & 32.85 & 48 & -180 & 1,2,3\
NGC 2986 & 1999gh & & 255.04 & 23.72 & 2302 & 2300 & & 32.94 & 22 & -97 & 1,2,3\
UGC 5234 & 2003W & & 217.68 & 45.93 & 6017 & 6132 & 6164 & 34.89 & 59 & 297 & 5\
NGC 3021 & 1995al & & 192.18 & 50.84 & 1541 & 1841 & & 32.71 & 76 & -317 & 1,2,3\
UGC 5378 & 2007S & & 234.37 & 43.38 & 4161 & 4246 & 4332 & 34.26 & 47 & -135 & 5\
UGC 5542 & 2001ie & & 150.36 & 47.78 & 9215 & 9490 & 9223 & 35.89 & 103 & 319 & 5\
Anon 1003-35 & 1993ag & & 268.44 & 15.93 & 14700 & 14542 & 14898 & 37.03 & 7 & -732 & 1,2,3\
Anon 1009-26 & 1992J & & 263.55 & 23.54 & 13491 & 13375 & 13691 & 36.62 & 16 & 651 & 1,3\
CGCG 266-31 & 2002bf & & 156.46 & 50.08 & 7254 & 7521 & 7278 & 35.60 & 98 & -525 & 3\
NGC 3147 & 1997bq & & 136.29 & 39.46 & 2820 & 3188 & & 33.45 & 116 & 161 & 1,2,3\
NGC 3190 & 2002bo & & 213.04 & 54.85 & 1271 & 1573 & & 32.20 & 64 & -135 & 1,2,3,4a\
MCG +03-27-38 & 2004L & & 223.79 & 54.79 & 9686 & 9790 & 9829 & 36.08 & 58 & -199 & 5\
UGC 5691 & 1991S & & 214.07 & 57.42 & 16489 & 16610 & 16617 & 37.28 & 64 & -455 & 1,3\
AS636 & & Antlia & 272.95 & 19.19 & 2608 & 2541 & & 33.40 & 8 & -418 & 6\
Anon 1034-34 & 1993B & & 273.33 & 20.46 & 20686 & 20538 & 20881 & 37.75 & 9 & -456 & 1,2,3\
NGC 3294 & 1992G & & 184.62 & -59.84 & 1586 & 1542 & & 32.70 & 100 & -606 & 3\
A1060 & & Hydra & 269.63 & 26.51 & 3777 & 3717 & 3973 & 34.23 & 15 & -604 & 6,7\
MCG +11-13-36 & 2006ar & & 142.76 & 46.62 & 6757 & 7054 & 6747 & 35.46 & 107 & -499 & 5\
Anon 1039+05 & 2006al & A1066 & 241.92 & 51.69 & 20341 & 20380 & 20505 & 37.72 & 48 & -339 & 5\
NGC 3327 & 2001N & & 211.35 & 60.29 & 6303 & 6471 & 6424 & 35.11 & 67 & 24 & 5\
NGC 3332 & 2005ki & & 236.83 & 54.27 & 5758 & 5858 & 5914 & 34.96 & 52 & -165 & 4b,5\
AS639 & & AS639 & 280.54 & 10.91 & 6326 & 6171 & 6512 & 34.89 & 6 & 337 & 7\
NGC 3368 & 1998bu & & 234.44 & 57.01 & 897 & 719 & & 30.45 & 55 & -45 & 1,2,3,4a\
NGC 3370 & 1994ae & & 225.35 & 59.67 & 1279 & 1603 & & 32.63 & 60 & -477 & 1,2,3\
UGC 6015 & 2006cf & & 165.96 & 60.09 & 12457 & 12690 & 12509 & 36.55 & 89 & 357 & 5\
Anon 1101-06 & 1999aw & & 260.24 & 47.45 & 11992 & 11994 & 12168 & 36.59 & 38 & -560 & 1,2,3\
2MASX J1109+28 & 2006ak & & 203.15 & 67.48 & 11422 & 11595 & 11523 & 36.42 & 72 & -41 & 5\
NGC 3557 Gr & & N3557Gr & 281.58 & 21.09 & 3056 & 2990 & & 33.51 & 11 & -121 & 6\
UGC 6211 & 2001ah & & 149.13 & 56.52 & 17315 & 17573 & 17334 & 37.21 & 98 & 1028 & 5\
UGC 6332 & 2007bc & & 224.67 & 68.06 & 6227 & 6394 & 6346 & 35.13 & 65 & -111 & 5\
NGC 3627 & 1989B & & 241.96 & 64.42 & 727 & 431 & & 30.35 & 57 & -299 & 1,2,3\
UGC 6363 & 2004bg & & 223.72 & 68.62 & 6306 & 6477 & 6423 & 35.10 & 65 & 60 & 5\
NGC 3663 & 2006ax & & 271.83 & 45.23 & 5018 & 5041 & 5194 & 34.62 & 34 & -120 & 4a,b\
HOLM 254B & 2004as & & 220.68 & 70.12 & 9300 & 9457 & 9412 & 36.13 & 67 & -759 & 5\
A1314 & & A1314 & 151.83 & 63.57 & 10043 & 10302 & 10076 & 36.05 & 93 & 446 & 6\
ESO 439-18 & 2001ba & & 285.38 & 28.03 & 8861 & 8759 & 9041 & 35.97 & 19 & -748 & 2,3,4a\
EROS J1139-08 & 1999bp & & 274.68 & 50.04 & 23084 & 23075 & 23250 & 38.06 & 38 & -970 & 5\
A1367 & & A1367 & 234.80 & 73.03 & 6595 & 6765 & 6707 & 35.17 & 65 & 140 & 6\
NGC 3873 & 2007ci & A1367 & 235.50 & 73.26 & 5434 & 5618 & 5546 & 34.78 & 66 & 68 & 5\
NGC 3978 & 2003cq & & 134.85 & 55.31 & 9978 & 10270 & 9971 & 36.07 & 105 & 326 & 5\
NGC 3982 & 1998aq & & 138.83 & 60.27 & 1109 & 1517 & & 32.07 & 100 & -92 & 1,2,3\
NGC 3987 & 2001V & & 218.95 & 77.72 & 4502 & 4752 & 4596 & 34.18 & 72 & 527 & 1,2,3\
NGC 4172 & 2006az & & 133.89 & 60.11 & 9274 & 9563 & 9276 & 35.91 & 102 & 309 & 5\
UGC 7357 & 2006cp & & 243.90 & 81.31 & 6682 & 6873 & 6773 & 35.31 & 71 & -185 & 4a\
ESO 573-14 & 2000bh & & 293.74 & 40.34 & 6838 & 6812 & 6997 & 35.35 & 33 & -375 & 2,3,4a\
NGC 4321 & 2006X & Virgo & 271.14 & 76.90 & 1571 & 1152 & & 31.42 & 65 & -42 & 4a\
NGC 4419 & 1984A & Virgo & 276.45 & 76.64 & -261 & 1152 & & 31.20 & 65 & & 1\
NGC 4493 & 1994M & & 291.69 & 63.04 & 6943 & 7034 & 7072 & 35.39 & 53 & -284 & 1,2,3\
NGC 4495 & 1994S & & 187.34 & 85.14 & 4550 & 4804 & 4619 & 34.47 & 78 & -17 & 1,2,3\
NGC 4496A & 1960F & VirW & 290.56 & 66.33 & 1730 & 1168 & & 30.81 & 56 & 266 & 1\
NGC 4501 & 1999cl & Virgo & 282.33 & 76.51 & 2281 & 1152 & & 31.33 & 65 & & 4a\
NGC 4520 & 2000bk & & 295.26 & 55.23 & 7628 & 7671 & 7767 & 35.70 & 46 & -747 & 1,3,4a\
NGC 4526 & 1994D & Virgo & 290.16 & 70.14 & 448 & 1152 & & 31.36 & 59 & & 1,2,3\
NGC 4536 & 1981B & VirW & 292.95 & 64.73 & 1808 & 1407 & & 31.26 & 54 & & 1,2,3\
NGC 4619 & 2006ac & & 136.98 & 81.80 & 6927 & 7174 & 6976 & 35.31 & 84 & 116 & 4a\
Cen30 & & Cen30 & 300.97 & 22.15 & 3041 & 2991 & & 33.44 & 27 & -22 & 6\
IC 3690 & 1992P & & 295.62 & 73.11 & 7615 & 7751 & 7720 & 35.71 & 63 & -705 & 1,2,3\
NGC 4639 & 1990N & Virgo & 294.29 & 75.99 & 1018 & 1152 & & 32.16 & 65 & & 1,2,3\
NGC 4675 & 1997Y & & 124.77 & 62.37 & 4757 & 5083 & 4753 & 34.60 & 102 & -32 & 1,2,3\
UGC 7934 & 2006S & & 131.43 & 81.95 & 9624 & 9856 & 9671 & 36.15 & 85 & -453 & 5\
2MASX J1246+12 & 2004gu & & 298.25 & 74.78 & 13748 & 13875 & 13848 & 36.78 & 64 & 209 & 4b\
NGC 4680 & 1997bp & & 301.16 & 51.22 & 2492 & 2675 & & 32.92 & 45 & 299 & 1,2,3\
NGC 4679 & 2001cz & & 302.11 & 23.29 & 4643 & 4563 & 4794 & 34.37 & 29 & -43 & 2,3,4a\
NGC 4704 & 1998ab & & 124.87 & 75.20 & 8134 & 8407 & 8162 & 35.37 & 91 & 1154 & 1,2,3\
NGC 4753 & 1983G & & 303.42 & 61.67 & 1239 & 1264 & & 31.45 & 54 & 53 & 1\
MGC -01-33-34 & 2006D & & 303.40 & 53.09 & 2556 & 2736 & & 33.06 & 47 & 203 & 4a\
2MASX J1259+28 & 2006cj & & 68.06 & 87.86 & 20241 & 20433 & 20297 & 37.96 & 80 & -2583 & 5\
A1656 & & Coma & 58.08 & 87.96 & 6925 & 7152 & 6982 & 35.37 & 80 & -100 & 6,7\
IC 4042A & 2006bz & Coma & 55.57 & 87.78 & 8366 & 8583 & 8423 & 35.73 & 80 & 50 & 5\
UGC 8162 & 2007F & & 118.24 & 66.40 & 7072 & 7367 & 7072 & 35.44 & 100 & -118 & 5\
2MASX J1305+28 & 2006cg & & 61.91 & 86.59 & 8413 & 8645 & 8466 & 35.26 & 81 & 1745 & 5\
IC 4182 & 1937C & & 107.70 & 79.09 & 321 & 298 & & 28.51 & 89 & -15 & 1,2\
MCG +06-29-43 & 2002G & & 96.96 & 82.19 & 10114 & 10347 & 10152 & 36.21 & 86 & -243 & 5\
ESO 508 Gr & & E508 Gr & 307.98 & 39.08 & 3196 & 3252 & & 33.41 & 40 & 279 & 6\
NGC 5018 & 2002dj & & 309.90 & 43.06 & 2816 & 2928 & & 33.05 & 44 & 407 & 5\
NGC 5061 & 1996X & & 310.25 & 35.66 & 2065 & 2157 & & 32.45 & 40 & 242 & 1,2,3\
PGC 46640 & 1994T & & 318.02 & 59.84 & 10390 & 10465 & 10493 & 36.20 & 58 & -78 & 1,2,3\
IC 4232 & 1991U & & 311.82 & 36.21 & 9426 & 9389 & 9554 & 35.74 & 41 & 818 & 1,3\
ESO 508-67 & 1992ag & & 312.49 & 38.39 & 7795 & 7783 & 7921 & 35.33 & 43 & 661 & 1,2,3\
IC 4239 & 2006cq & & 62.88 & 81.84 & 14491 & 14707 & 14528 & 36.98 & 85 & -231 & 5\
ESO 508-75 & 2007cg & & 312.72 & 37.57 & 9952 & 9919 & 10078 & 35.93 & 43 & 582 & 5\
MCG -02-34-61 & 2007ca & & 316.95 & 46.69 & 4217 & 4264 & 4331 & 34.73 & 50 & -1162 & 5\
Anon 1331-33 & 1993O & & 312.42 & 28.92 & 15589 & 15502 & 15717 & 37.18 & 39 & -824 & 1,2,3\
ESO 383-32 & 2000ca & & 313.20 & 27.83 & 7080 & 7013 & 7206 & 35.31 & 39 & -45 & 2,3,4a\
NGC 5253 & 1972E & & 314.86 & 30.11 & 407 & 171 & & 27.94 & 42 & -69 & 1,2,3\
NGC 5283 & 2005dv & & 115.81 & 48.76 & 3119 & 3489 & & 33.93 & 117 & -281 & 5\
NGC 5308 & 1996bk & & 111.25 & 54.88 & 2041 & 2459 & & 32.62 & 112 & 389 & 1,2,3\
A3574 & & A3574 & 317.46 & 30.94 & 4797 & 4775 & 4913 & 34.43 & 44 & 41 & 6,7\
NGC 5304 & 2005al & A3574 & 317.59 & 30.62 & 3718 & 3699 & 3833 & 34.39 & 44 & -949 & 4b\
MCG +08-25-47 & 1996C & & 99.62 & 65.04 & 8094 & 8384 & 8077 & 36.05 & 103 & -1472 & 1,2,3\
ESO 445-66 & 1993H & & 318.22 & 30.33 & 7257 & 7211 & 7371 & 35.33 & 44 & 89 & 1,2,3\
AS753 & & AS753 & 319.63 & 26.55 & 4197 & 4167 & 4306 & 34.27 & 45 & -234 & 7\
NGC 5468 & 1999cp & & 334.87 & 52.70 & 2842 & 3010 & & 33.46 & 63 & -32 & 2,3,4a\
NGC 5468 & 2002cr & & 334.87 & 52.70 & 2842 & 3006 & & 33.58 & 63 & -207 & 5\
MCG +05-34-33 & 2002bz & & 36.89 & 69.23 & 11138 & 11357 & 11146 & 36.47 & 90 & -542 & 5\
IC 4423 & 2001ay & & 35.97 & 68.82 & 9067 & 9290 & 9075 & 36.09 & 90 & -744 & 2,3\
Anon 1433+03 & 2006bw & & 353.79 & 56.19 & 8994 & 9123 & 9036 & 35.87 & 74 & 35 & 5\
UGC 9391 & 2003du & & 101.18 & 53.21 & 1914 & 2306 & & 33.24 & 115 & -445 & 2,3,4a\
UGC 9612 & 2007O & & 77.66 & 59.25 & 10856 & 11144 & 10813 & 36.16 & 108 & 789 & 5\
Anon & 2005ag & & 7.85 & 55.50 & 23807 & 23945 & 23822 & 37.94 & 82 & 1128 & 4b\
UGC 9640 & 2008af & & 20.11 & 58.62 & 10045 & 10236 & 10046 & 35.99 & 88 & 643 & 5\
MCG -01-39-03 & 2005cf & & 354.81 & 39.85 & 1937 & 2136 & & 32.38 & 75 & 282 & 4\
UGC 10030 & 2002ck & & 6.55 & 39.30 & 8953 & 9068 & 8946 & 35.85 & 84 & 61 & 5\
MCG +11-19-25 & 2000cf & & 99.88 & 42.17 & 10920 & 11241 & 10831 & 36.42 & 126 & -395 & 1,2,3\
MCG +03-41-03 & 2007ap & & 28.86 & 46.04 & 4742 & 4964 & 4701 & 34.77 & 98 & -561 & 5\
CGCG 108-13 & 2006bt & & 33.70 & 47.24 & 9640 & 9850 & 9593 & 36.03 & 100 & 83 & 5\
IC 1151 & 1991M & & 30.36 & 45.90 & 2169 & 2436 & & 33.64 & 99 & -865 & 3\
NGC 6038 & 1999cc & & 59.67 & 48.75 & 9392 & 9665 & 9320 & 36.01 & 112 & -15 & 1,2,3\
NGC 6063 & 1999ac & & 19.89 & 39.95 & 2848 & 3077 & & 33.38 & 94 & 145 & 1,2,3\
UGC 10244 & 2006cc & & 68.18 & 47.10 & 9752 & 10037 & 9670 & 36.31 & 117 & -1038 & 5\
ESO 584-07 & 2007ai & & 353.04 & 21.09 & 9492 & 9500 & 9499 & 36.00 & 75 & -137 & 5\
NGC 6104 & 2002de & & 57.37 & 45.91 & 8429 & 8702 & 8350 & 35.74 & 113 & 131 & 5\
A2199 & & A2199 & 62.70 & 43.70 & 9039 & 9322 & 8949 & 35.76 & 117 & 673 & 6\
UGC 10483 & 2001az & & 108.95 & 34.29 & 12200 & 12520 & 12100 & 36.58 & 132 & 21 & 5\
PGC 59076 & 1994Q & A2199 & 64.39 & 39.68 & 8863 & 9148 & 8760 & 35.87 & 121 & 59 & 1,3\
UGC 10704 & 2007ae & & 111.49 & 31.71 & 19303 & 19615 & 19201 & 37.33 & 134 & 2168 & 5\
UGC 10738 & 2001cp & & 26.50 & 24.95 & 6716 & 6874 & 6634 & 35.29 & 104 & -120 & 5\
UGC 10743 & 2002er & & 28.67 & 25.83 & 2569 & 2796 & & 33.15 & 106 & 157 & 1,2,3\
MCG +03-44-03 & 1990O & & 37.65 & 28.37 & 9193 & 9389 & 9095 & 35.87 & 112 & 301 & 1,2,3\
NGC 6365A & 2003U & & 91.47 & 34.06 & 8496 & 8823 & 8377 & 35.64 & 134 & 630 & 5\
NGC 6462 & 2005ao & & 90.99 & 31.49 & 11514 & 11828 & 11388 & 36.52 & 137 & -340 & 4a\
UGC 11064 & 2000cn & & 53.45 & 23.32 & 7043 & 7285 & 6908 & 35.41 & 127 & -99 & 1,2,3\
MCG +04-42-22 & 2001bf & & 52.16 & 21.97 & 4647 & 4911 & 4511 & 34.26 & 126 & 530 & 5\
UGC 11149 & 1998dx & & 77.68 & 26.67 & 16256 & 16546 & 16113 & 37.02 & 138 & 1339 & 1,2,3\
NGC 6627 & 1998V & & 43.94 & 13.34 & 5272 & 5480 & 5132 & 34.56 & 124 & 458 & 1,2,3\
MCG +05-43-16 & 2007co & & 57.59 & 18.82 & 8083 & 8329 & 7933 & 35.57 & 132 & 391 & 5\
NGC 6685 & 2006bq & & 68.85 & 19.09 & 6567 & 6851 & 6409 & 35.15 & 140 & 286 & 5\
IC 4758 & 2001cn & PavoII & 329.65 & -24.05 & 4647 & 4475 & 4676 & 34.28 & 65 & 54 & 2,3,4a\
AS805 & & PavoII & 332.25 & -23.59 & 4167 & 4002 & 4189 & 34.28 & 67 & -419 & 6\
2MASX J1911+77 & 2003hu & & 109.56 & 25.44 & 22484 & 22790 & 22364 & 37.61 & 140 & 3050 & 5\
IC 4830 & 2001bt & & 337.32 & -25.87 & 4388 & 4240 & 4392 & 34.13 & 72 & 111 & 2,3,4a\
MCG +07-41-01 & 2002do & & 75.61 & 6.13 & 4761 & 5041 & 4578 & 34.54 & 154 & 65 & 5\
PGC 63925 & 1990T & & 341.50 & -31.52 & 11992 & 11809 & 11977 & 36.63 & 77 & -972 & 1,3\
IC 4919 & 1991ag & Grm15 & 342.55 & -31.64 & 4264 & 4124 & 4246 & 34.07 & 78 & 106 & 1,2,3\
Grm15 & & Grm15 & 341.85 & -32.45 & 4286 & 4134 & 4269 & 34.36 & 78 & -451 & 7\
PavoI & & PavoI & 324.10 & -32.58 & 4107 & 3916 & 4138 & 33.98 & 65 & 60 & 6\
NGC 6928 & 2004eo & & 54.16 & -17.26 & 4707 & 4855 & 4521 & 34.15 & 141 & 688 & 4a,b\
NGC 6951 & 2000E & & 100.90 & 14.85 & 1424 & 1814 & & 31.96 & 153 & 284 & 1,2,3\
ESO 234-69 & 1992al & & 347.34 & -38.49 & 4381 & 4242 & 4343 & 34.19 & 84 & -2 & 1,2,3\
NGC 6962 & 2002ha & & 47.41 & -25.37 & 4211 & 4311 & 4035 & 34.10 & 134 & 238 & 5\
NGC 6986 & 2002el & & 28.77 & -35.67 & 8612 & 8591 & 8473 & 35.46 & 116 & 1038 & 2\
2MASX J2120+44 & 2001fh & & 88.22 & -3.81 & 3894 & 4185 & 3702 & 33.55 & 170 & 1016 & 5\
Anon 2123-00 & 2006oa & & 51.74 & -33.74 & 17988 & 18027 & 17810 & 37.35 & 135 & 425 & 5\
Anon 2128-61 & 1992ae & & 332.70 & -41.99 & 22484 & 22260 & 22479 & 37.76 & 76 & 1174 & 1,2,3\
Anon 2135-62 & 1990af & & 330.82 & -42.24 & 15080 & 14856 & 15079 & 36.90 & 75 & 440 & 1,2,3\
NGC 7131 & 1998co & & 41.52 & -44.94 & 5418 & 5419 & 5264 & 34.55 & 124 & 419 & 3\
UGC 11816 & 2004ey & & 57.47 & -38.27 & 4733 & 4804 & 4553 & 34.33 & 138 & 282 & 4b\
Anon 2155-01 & 2006on & & 57.18 & -40.55 & 20985 & 21009 & 20808 & 37.68 & 136 & 651 & 5\
MDL59 & & MDL59 & 14.95 & 24.39 & 2567 & 2740 & & 33.12 & 94 & 137 & 6\
IC 5179 & 1999ee & & 6.50 & -55.93 & 3422 & 3325 & & 33.57 & 101 & 127 & 1,2,3,4a\
UGC 12071 & 2006gr & & 90.75 & 38.36 & 10372 & 10689 & 10265 & 36.39 & 130 & -792 & 4a\
NGC 7311 & 2005kc & & 72.34 & -43.41 & 4533 & 4612 & 4351 & 34.23 & 143 & 291 & 4b,5\
UGC 12133 & 1998eg & & 76.48 & -42.06 & 7423 & 7497 & 7238 & 35.46 & 146 & -56 & 1,2,3\
NGC 7329 & 2006bh & & 320.97 & -45.79 & 3252 & 3058 & & 33.63 & 71 & -228 & 4b\
Anon J2241-00 & 2006py & & 68.44 & -48.79 & 17358 & 17372 & 17186 & 37.14 & 137 & 1333 & 4b\
UGC 12158 & 2004ef & & 85.92 & -33.43 & 9289 & 9413 & 9096 & 35.80 & 157 & 606 & 4b\
IC 5270 & 1993L & & 5.94 & -64.39 & 1983 & 1915 & & 32.03 & 101 & 335 & 1,3\
NGC 7448 & 1997dt & & 87.57 & -39.12 & 2194 & 2338 & & 32.99 & 152 & -115 & 1,3\
Anon 2304-37 & 1992aq & & 1.78 & -65.31 & 30279 & 30107 & 30202 & 38.60 & 100 & -285 & 1,2,3\
MCG +05-54-41 & 2006en & & 98.23 & -27.74 & 9575 & 9733 & 9387 & 36.00 & 164 & 97 & 5\
NGC 7541 & 1998dh & & 82.84 & -50.65 & 2689 & 2776 & & 32.94 & 140 & 378 & 1,2,3\
ESO 291-11 & 1992bl & & 344.15 & -63.93 & 12891 & 12701 & 12839 & 36.64 & 92 & -138 & 1,2,3\
Pegasus & & Pegasus & 87.78 & -48.38 & 3554 & 3633 & 3381 & 33.96 & 143 & -188 & 6\
NGC 7634 & 1972J & Pegasus & 88.69 & -47.93 & 3225 & 3317 & & 33.53 & 144 & 177 & 1\
NGC 7678 & 2002dp & & 98.88 & -36.55 & 3489 & 3642 & 3308 & 33.65 & 155 & 325 & 5\
A2634 & & A2634 & 103.45 & -33.06 & 9409 & 9546 & 9230 & 35.72 & 157 & 1051 & 6\
Anon 2340+26 & 1997dg & A2634 & 103.62 & -33.98 & 10193 & 10321 & 10014 & 36.22 & 156 & -317 & 1,2,3\
A4038 & & A4038 & 25.19 & -75.84 & 8994 & 8864 & 8904 & 35.82 & 106 & -22 & 6\
ESO 471-27 & 1993ah & A4038 & 25.87 & -76.77 & 8803 & 8674 & 8714 & 35.70 & 106 & 256 & 1,3\
NGC 7780 & 2001da & & 99.20 & -52.06 & 5155 & 5214 & 4995 & 34.37 & 140 & 608 & 5\
ESO 538-13 & 2005iq & & 64.83 & -75.21 & 10206 & 10110 & 10097 & 36.15 & 114 & -199 & 4a,b\
[ccccccccc]{} $\sigma_{(m\!-\!M)}$: & 0.09 & 0.14 & 0.16 & 0.12 & 0.20 & 0.20 & 0.14 & 0.18\
N: & 81 & 101 & 7 & 88 & 15 & 15 & 6 & 14\
[lcccccr]{} Reindl & 62 & 0.188 & 0.004 & 0.16\
Wang & 55 & 0.190 & 0.003 & 0.13\
Jha & 72 & 0.183 & 0.003 & 0.18\
Wood-Vasey/Freedman & 45 & 0.189 & 0.005 & 0.16\
Hicken & 88 & 0.194 & 0.004 & 0.16\
Masters & 27 & 0.208 & 0.007 & 0.14\
J[ø]{}rgensen & 10 & 0.179 & 0.014 & 0.18\
\
all & 218 & 0.193 & 0.002 & 0.15\
[cccccccccc]{} $0.01$ & $-2.000$ & & $ 0.00$ & $-0.00$ & & $ 0.01$ & $ 0.00$ & $-0.00$ & $-0.00$\
$0.02$ & $-1.699$ & & $ 0.01$ & $-0.01$ & & $ 0.01$ & $ 0.01$ & $-0.01$ & $-0.01$\
$0.03$ & $-1.523$ & & $ 0.01$ & $-0.01$ & & $ 0.02$ & $ 0.01$ & $-0.01$ & $-0.01$\
$0.04$ & $-1.398$ & & $ 0.02$ & $-0.01$ & & $ 0.02$ & $ 0.02$ & $-0.01$ & $-0.02$\
$0.05$ & $-1.301$ & & $ 0.02$ & $-0.02$ & & $ 0.03$ & $ 0.02$ & $-0.01$ & $-0.02$\
$0.06$ & $-1.222$ & & $ 0.03$ & $-0.02$ & & $ 0.04$ & $ 0.02$ & $-0.02$ & $-0.03$\
$0.07$ & $-1.155$ & & $ 0.03$ & $-0.02$ & & $ 0.04$ & $ 0.03$ & $-0.02$ & $-0.03$\
$0.08$ & $-1.097$ & & $ 0.04$ & $-0.03$ & & $ 0.05$ & $ 0.03$ & $-0.02$ & $-0.04$\
$0.09$ & $-1.046$ & & $ 0.04$ & $-0.03$ & & $ 0.05$ & $ 0.03$ & $-0.02$ & $-0.04$\
$0.10$ & $-1.000$ & & $ 0.05$ & $-0.03$ & & $ 0.06$ & $ 0.04$ & $-0.03$ & $-0.05$\
[llll]{} @Aaronson:etal:86 & clusters (21cm) & 10 & $<10,000$\
@Lilje:etal:86 & 21cm & field galaxies & $\sim4000$\
@Jerjen:Tammann:93 & clusters & 15 & $<6400$\
@FST:94 & line widths & field galaxies & $4000$\
@Riess:etal:95 & SNeIa & 13 & $<7000$\
@Giovanelli:etal:98 & clusters (21cm) & 24 & $<6000$\
@Dale:etal:99 & clusters (21cm) & 52 & $4000$-$6000$\
@Hoffman:etal:01 & various & field galaxies & $<6000$\
@Watkins:etal:09 & various & field galaxies & $5000$\
present paper & SNeIa, clusters & 170 & $3500\pm300$\
[lccccr]{} A2806 & 35.47 & 1 & 8310 & 7588 & $ 722\pm623$\
N383Gr & 34.54 & 2 & 4957 & 4977 & $ -20\pm248$\
A2877 & 35.46 & 1 & 7396 & 7554 & $-158\pm555$\
N507Gr & 34.31 & 2 & 4802 & 4482 & $ 319\pm240$\
A0194 & 34.60 & 2 & 5302 & 5115 & $ 187\pm265$\
A0262 & 34.44 & 1 & 4770 & 4756 & $ 14\pm358$\
A0397 & 35.87 & 2 & 9749 & 9089 & $ 660\pm487$\
A0400 & 35.19 & 1 & 7276 & 6685 & $ 591\pm546$\
A0426 & 34.68 & 2 & 5745 & 5304 & $ 441\pm287$\
A0496 & 36.03 & 1 & 9928 & 9767 & $ 160\pm745$\
A0539 & 35.95 & 1 & 8580 & 9422 & $-842\pm643$\
A0569 & 34.92 & 3 & 6053 & 5915 & $ 138\pm303$\
A0634 & 35.72 & 1 & 7957 & 8495 & $-538\pm597$\
Cancer & 34.51 & 2 & 4623 & 4910 & $-286\pm231$\
A0779 & 35.52 & 1 & 6840 & 7761 & $-921\pm513$\
Hydra & 34.23 & 2 & 3973 & 4322 & $-349\pm199$\
AS639 & 34.89 & 1 & 6512 & 5835 & $ 677\pm488$\
A1367 & 34.98 & 2 & 6707 & 6078 & $ 629\pm335$\
Coma & 35.49 & 3 & 6982 & 7657 & $-675\pm349$\
A3574 & 34.42 & 3 & 4913 & 4713 & $ 200\pm246$\
AS753 & 34.27 & 1 & 4306 & 4401 & $ -95\pm323$\
A2199 & 35.81 & 2 & 8949 & 8847 & $ 103\pm447$\
PavoII & 34.28 & 2 & 4189 & 4422 & $-233\pm209$\
Grm15 & 34.22 & 2 & 4269 & 4302 & $ -34\pm213$\
PavoI & 33.98 & 1 & 4138 & 3857 & $ 281\pm310$\
Pegasus & 33.75 & 2 & 3381 & 3472 & $ -91\pm169$\
A2634 & 35.97 & 2 & 9230 & 9507 & $-278\pm461$\
A4038 & 35.76 & 2 & 8904 & 8649 & $ 255\pm445$\
[^1]: It is important to emphasize that bias of this nature is [*not*]{} the famous Malmquist bias, but rather is an incompleteness bias that increases with distance. In contrast, the Malmquist bias is distance independent. It only concerns the error made in assigning a number to an absolute magnitude calibration of a distance indicator when using a sample that is magnitude, rather than, distance limited. On the other hand, an incompleteness bias causes an error that increases with distance, which, if uncorrected, makes the Hubble constant appear to increase outward with distance. The error of calling the incompleteness bias as Malmquist is wide spread in the literature.
[^2]: See http://nedwww.ipac.caltech.edu.
[^3]: In this case the dipole of the integrated light must be compared with the CMB apex as seen from the Local Group and not with the apex $A_{\rm corr}$, i.e. corrected for Virgocentric infall, because the integration starts at small velocities and includes the Virgo cluster.
|
---
author:
- 'M. van Leeuwen [^1] [*for the ALICE collaboration*]{}'
bibliography:
- 'alice\_highpt\_mvanleeuwen\_hcp.bib'
title: 'High-[$p_{\mathrm{T}}$]{} results from ALICE'
---
Introduction {#intro}
============
High-energy collisions of heavy nuclei are used to study the high-temperature states of strongly interacting matter and the expected transition from confined matter to a deconfined Quark-Gluon Plasma. Partons with high transverse momentum [$p_{\mathrm{T}}$]{} are formed in hard scatterings which happen early in the collision. The produced partons then propagate through the hot and dense medium and lose energy through interactions with the medium. Measurements of high-[$p_{\mathrm{T}}$]{} particle production are used to study the interactions between fast partons and the medium and to determine the medium properties using these interactions. Here we report a number of recent high-[$p_{\mathrm{T}}$]{} results from ALICE, the dedicated heavy-ion experiment at the Large Hadron Collider (LHC), from Pb–Pb collisons with a centre-of-mass energy $\sqrt{s_{NN}}=2.76$ TeV recorded during the heavy ion run of the LHC in November 2010.
Nuclear modification factor
===========================
One of the most basic measurements that is sensitive to parton energy loss in the hot and dense QCD medium are the charged particle production spectra at high [$p_{\mathrm{T}}$]{}. The measured transverse momentum spectra of primary charged particles in Pb–Pb collisions with three different centrality selections are shown in the left panel of Fig. \[fig:spec\_RAA\]. The collision centrality is determined using the total multiplicity detected in the forward VZERO detectors and reported as a fraction of the total hadronic cross section, with 0% labeling the most central events. The dashed lines in the Figure indicate a parametrisation of the spectrum measured in pp collisions, scaled by the total number of binary nucleon-nucleon collisions $\langle N_\mathrm{coll} \rangle$ as determined from a Glauber model [@Miller:2007ri; @Aamodt:2010cz]. It can be seen in the figure that for central collisions, the shape of the [$p_{\mathrm{T}}$]{}-spectra in Pb–Pb collisions is different from pp collisions, with a large suppression for ${\ensuremath{p_{\mathrm{T}}}}=4-10$ GeV/$c$.
![\[fig:spec\_RAA\]Left panel: Transverse momentum distributions of primary charged particles in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with three different centrality selections. Right panel: nuclear modification factor $R_{AA}$ for charged particles in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with three different centrality selections.](2011-Sep-13-pT_PbPb_3centr.eps "fig:"){width="49.00000%"} ![\[fig:spec\_RAA\]Left panel: Transverse momentum distributions of primary charged particles in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with three different centrality selections. Right panel: nuclear modification factor $R_{AA}$ for charged particles in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with three different centrality selections.](2011-Sep-13-raa_3centr.eps "fig:"){width="49.00000%"}
To quantify the differences, the nuclear modification factor $$R_{AA} = \frac{dN/d{\ensuremath{p_{\mathrm{T}}}}|_\mathrm{PbPb}}{\langle N_\mathrm{coll} \rangle dN/d{\ensuremath{p_{\mathrm{T}}}}|_\mathrm{pp}},$$ i.e. the ratio between the [$p_{\mathrm{T}}$]{}-distributions in Pb–Pb collisions $dN/d{\ensuremath{p_{\mathrm{T}}}}|_\mathrm{PbPb}$ and in pp collisions $dN/d{\ensuremath{p_{\mathrm{T}}}}|_\mathrm{pp}$, scaled with the number of binary collisions, is calculated. The nuclear modification factor for charged particles in Pb–Pb collisions with three different centrality selections is shown in the right panel of Fig. \[fig:spec\_RAA\]. The figure clearly shows a significant suppression $R_{AA} < 1$. The effect is largest for the most central bin $0-5\%$, where the medium density and the average path length through the medium are the largest. The strongest suppression is seen for ${\ensuremath{p_{\mathrm{T}}}}\approx 7$ GeV/$c$, with a gradual rise of $R_{AA}$ towards larger [$p_{\mathrm{T}}$]{}. The increase of $R_{AA}$ with [$p_{\mathrm{T}}$]{} is qualitatively consistent with the expectation that parton energy loss $\Delta E$ is only weakly dependent on the parton energy $E$, leading to a decrease of the relative energy loss $\Delta E/E$ with increasing [$p_{\mathrm{T}}$]{}.
![Nuclear modification factor $R_{AA}$ for identified hadrons in the 0-5% most central Pb–Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV.[]{data-label="fig:RAA_lamka"}](2011-Sep-17-raa_central_lin_ch_K0_L_chPion.eps){width="50.00000%"}
Figure \[fig:RAA\_lamka\] shows a comparison of the nuclear modification factor for $\Lambda$ and $K_0^s$, measured using reconstruction of the weak-decay topology, to the result for unidentified charged particles and identified pions for the most central events. It is interesting to see that the $R_{AA}$ for identified mesons shows a similar [$p_{\mathrm{T}}$]{}-dependence to the charged particles, while the $\Lambda$ show much smaller suppression at intermediate ${\ensuremath{p_{\mathrm{T}}}}< 6$ GeV/$c$. The enhancement of baryon production compared to meson production at intermediate [$p_{\mathrm{T}}$]{}, might be due to a large contribution of hadron formation by coalescence of quarks from the hot and dense medium [@Lin:2002rw; @Hwa:2002tu; @Fries:2003vb].
At higher [$p_{\mathrm{T}}$]{}, all hadrons show the same suppression, which suggests that the dominant energy loss mechanism is at work at the partonic level. If hadronic energy loss would be important, one would expect to see that different hadrons would have different cross sections for the relevant energy loss mechanism and thus be affected differently.
Di-hadron measurements
======================
![\[fig:alice\_iaa\]Ratios of measured charged hadron yield associated with a high-[$p_{\mathrm{T}}$]{} [*trigger*]{} particle with $8 < {\ensuremath{p_{\mathrm{T}}}}<
15$ GeV/$c$ in Pb–Pb and pp collisions on the near ($|\Delta\phi|<0.7$, left panel) and away ($|\Delta\phi-\pi|<0.7$, right panel) sides, as a function of asociated particle [$p_{\mathrm{T}}$]{}. Results are shown for two different centrality selections: peripheral 60-90% (solid red data points) and central 0-5% (open data points). Three different background subtraction methods are shown. The grey bars indicate systematic uncertainties from tracking efficiency and secondary particles [@ALICE:2011vg].](fig2a_iaa.eps){width="80.00000%"}
Using di-hadron correlation techniques, the yield of charged particles produced in association with high-[$p_{\mathrm{T}}$]{} hadrons can be determined. In these measurements, we distinguish the [*near-side*]{} yield of particles produced in the same jet as the high-[$p_{\mathrm{T}}$]{} [*trigger*]{} hadron, and the [*away-side*]{} or recoil yield of particles in the recoiling jet. This measurement has been performed in pp and Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV using a trigger particle selection of $8 < {\ensuremath{p_{\mathrm{T}}}}^\mathrm{trig} < 15$ GeV/$c$ and the ratio $I_{AA}$ of the associated yield per trigger particle in Pb–Pb and pp collisions is shown in Fig. \[fig:alice\_iaa\]. The results for peripheral collisions (red points in Fig. \[fig:alice\_iaa\]) are very similar to pp ($I_{AA} \approx 1$), while for central collisions (grey points) a slight enhancement of the yield is seen on the near side and a suppression on the away side. Both effects are qualitatively consistent with expectations from parton energy loss in combination with a trigger bias which cause the parton on the away-side to typically have a larger energy loss than the one on the near side. The enhancement on the near side suggests that the trigger particle selects hard scattered partons with higher energy in the Pb–Pb collisions than in pp collisions, due to energy loss of the leading parton.
Constraining theoretical models
===============================
![\[fig:RAA\_IAA\_renk\]Nuclear modification factor $R_{AA}$ and per-trigger associated yield modification factor $I_{AA}$ for the 0-5% most central Pb–Pb collisions compared to model calculations (see text).](RAA_ALICE_prel_renk.eps "fig:"){width="44.00000%"} ![\[fig:RAA\_IAA\_renk\]Nuclear modification factor $R_{AA}$ and per-trigger associated yield modification factor $I_{AA}$ for the 0-5% most central Pb–Pb collisions compared to model calculations (see text).](IAA_ALICE_Renk.eps "fig:"){width="55.00000%"}
The single-inclusive nuclear modification factor $R_{AA}$ and the di-hadron modification $I_{AA}$ sample the geometry of the collision zone with different weights. A simultaneous comparison of both measurements with theoretical calculations can be used to infer the path-length dependence of energy loss [@Renk:2007id]. Fig. \[fig:RAA\_IAA\_renk\] shows a comparison of the measured $R_{AA}$ and $I_{AA}$ to model calculations by Renk [@Renk:2011wp; @Renk:2011gj].
The blue line in Figure \[fig:RAA\_IAA\_renk\] labeled ‘Renk ASW’ indicates the expected energy loss using the ‘quenching weights’ calculation for the multiple soft scattering approximation by Armesto, Salgado and Wiedemann [@Salgado:2003gb] in a realistic medium-density profile, based on hydrodynamical simulations. One overall scaling parameter was used to relate the local transport coefficient $\hat{q}$ to the medium density in the hydrodynamical model. This scaling parameter was tuned using measurements from the Relativistic Heavy Ion Collider (RHIC) at $\sqrt{s_{NN}}=200$ GeV. The fact that the blue line in the left panel of Fig. \[fig:RAA\_IAA\_renk\] is below the measured data points indicates that the single hadron suppression increases less from RHIC to LHC than expected based on the density inferred from multiplicity measurements which are used to tune the hydrodynamical evolution. The agreement of the ASW calculation with the di-hadron correlations is better.
The other two lines in Figure \[fig:RAA\_IAA\_renk\] represent energy loss calculations based on a Monte Carlo shower model YaJEM (‘Yet another Jet Energy-loss Model’) [@Renk:2011wp; @Renk:2011gj] in which medium-induced radiation is generated by increasing the virtuality as the parton propagates through the medium. Default YaJEM (orange dotted lines in Fig. \[fig:RAA\_IAA\_renk\]) agrees rather well with the $R_{AA}$ measurement, while the di-hadron suppression is too weak, due to the approximately linear dependence of the energy loss on the path length $L$. YaJEM-D (green dashed lines in Fig. \[fig:RAA\_IAA\_renk\]) is identical to YaJEM, but introduces a minimum virtuality of the parton $Q_0=\sqrt{E/L}$, related to the requirement that the formation time of each in-medium shower is shorter than its path length through the medium. This causes a stronger path length dependence of the energy loss, similar to the expected $L^2$ dependence due to the Landau-Pomeranchuk-Migdal effect, which leads to a stronger suppression of the di-hadron recoil yield. However, it should be noted that for the current setting of the model, the inclusive hadron suppression is also smaller ($R_{AA}$ larger) than measured. Increasing the medium density in YaJEM-D would improve the agreement with the measurements.
All of the models presented in Figure \[fig:RAA\_IAA\_renk\] show deviations from the measured values in several places. A more systematic comparison of models with the measurements will be needed to quantify deviations of the models from the data and to disentangle effects from the medium geometry and the path-length dependence of the energy loss process itself.
Heavy flavour
=============
![\[fig:RAAcharm\]$R_{AA}$ for $D$ mesons in 0-20% central Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. The $D$ mesons are reconstructed from their hadronic decays: $D^0 \rightarrow K\pi$, $D^{\pm} \rightarrow K\pi\pi$ and $D^{*\pm} \rightarrow \pi^{\pm} +
D^0 \rightarrow \pi^{\pm} + K\pi$.](2011-Sep-15-DmesonRaa-020CC-GlobalSyst-ChargedPions-120911.eps){width="80.00000%"}
A specific expectation for radiative parton energy loss is that the effect will be smaller for heavy quarks at lower [$p_{\mathrm{T}}$]{}, when the quarks travel in the medium at speeds significanty below the speed of light, due to the so-called ‘dead cone’ effect [@Dokshitzer:2001zm; @Armesto:2003jh]. To test this expectation, ALICE has measured the nuclear modification factor of $D$ mesons, as shown in Fig. \[fig:RAAcharm\]. The measured $D$ meson suppression is slightly smaller than the values seen for pions, but the difference is within the current statistical and systematic uncertainties. Related measurements of muon and electron production from heavy flavour decay are reported in [@suire].
A more precise measurement using a larger data sample and a careful comparison to theoretical expectations are needed to determine whether the dead-cone effect is really observed in experimental data. In addition, measurements of B mesons are planned, which will have a larger discriminating power, because the dead-cone effect is larger for the heavier $b$ quarks.
Jets
====
![\[fig:jets\]Left panel: Transverse momentum distributions of reconstructed jet spectra in Pb–Pb collisions with different centralities, using only charged particle tracks. The energy of the uncorrelated background has been subtracted, but no corrections are applied for background fluctuations and detector effects. Right panel: Background energy distribution in central 0-10% events, determined using random cones and jet and track embedding. The curves show Gaussian fits to the left-hand side (LHS) of the distributions.](2011-May-19-notePlot_CentSpectraPbPb_Jets_15_0.eps "fig:"){width="54.00000%"} ![\[fig:jets\]Left panel: Transverse momentum distributions of reconstructed jet spectra in Pb–Pb collisions with different centralities, using only charged particle tracks. The energy of the uncorrelated background has been subtracted, but no corrections are applied for background fluctuations and detector effects. Right panel: Background energy distribution in central 0-10% events, determined using random cones and jet and track embedding. The curves show Gaussian fits to the left-hand side (LHS) of the distributions.](2011-May-18-notePlot_deltaPt_pT060_100_B2_cen00_409.eps "fig:"){width="45.00000%"}
Measurements of inclusive hadrons and di-hadrons at high [$p_{\mathrm{T}}$]{} are mostly sensitive to leading jet fragments, because the steeply falling [$p_{\mathrm{T}}$]{}-spectrum causes subleading fragments to be overwhelmed by the much larger yields of fragments of lower momentum partons. In addition, the measurements integrate over a large range of energies of the orginal partons. Jet reconstruction in Pb–Pb collisons has the potential to largely overcome both drawbacks: if the jet cone radius is large compared to the typical angles of gluon emission, most the radiated energy is recovered by the jet-finding algorithm, which then provides a measure of the energy of the parton from the hard scattering (before energy loss).
However, jet reconstruction in Pb–Pb collisions at the LHC is challenging, due to the large underlying event energy. The left panel of Fig. \[fig:jets\] shows the reconstructed momentum distribution of charged jets from the anti-[$k_{\mathrm{T}}$]{} algorithm with $R=0.4$, after subtraction of the uncorrelated background, which is measured on event-by-event basis using the [$k_{\mathrm{T}}$]{} algorithm from the FastJet package [@Cacciari:2011ma]. At low [$p_{\mathrm{T}}$]{}, a clear excess is visible in the jet spectrum of central events compared to peripheral events due to background fluctuations, which lead to ‘fake jets’. Judging from the curvature of the jet spectrum, fluctuations/fake jets dominate the jet spectrum up to ${\ensuremath{p_{\mathrm{T}}}}\approx 70$ GeV/$c$ for central (0-10%) events.
The right panel of Fig. \[fig:jets\] shows the background fluctuations as measured directly using random cones and two types of embedding [@ALICE:2012ej]. The random cone technique places ‘jet’ cones in the event at random location and then calculates the background-subtracted transverse momentum in the cone to measure the fluctuations. The embedding technique adds a track or several tracks from a jet to the event and then compares the reconstructed transverse momentum to the transverse momentum of the input track or jet to measure the fluctuations. The three methods give very similar results, indicating that the background measurement is mostly sensitive to the track density (and correlations) in the event and not to details of the jet fragmentation or the placement of the cone.
The jet results can only be interpreted in conjunction with the background fluctuation measurement. ALICE is currently pursuing unfolding techniques to remove the effect of the background fluctuations from the reconstructed jet spectrum. Given the large [$p_{\mathrm{T}}$]{}-reach of the fluctuations, a larger data set is likely needed to extend the measured jet spectrum and allow unfolding of the background fluctuations.
Outlook
=======
In these proceedings, we have reported first results on high-[$p_{\mathrm{T}}$]{} measurements of Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV performed by ALICE. The effects of the energy loss of partons propagating through the hot and dense medium are clearly seen in the suppression the inclusive yields of charged particles and di-hadrons, as well as for heavy mesons.
The goal of this research is to develop a quantitative understanding of the interactions between energetic partons and the medium and to use this understanding to determine properties of the medium such as the energy density or transport coefficient(s). A careful comparison of multiple measurements with theoretical expectations is needed to develop our understanding of parton energy loss. A first attempt of such a comparison for the LHC energy was shown in Fig. \[fig:RAA\_IAA\_renk\], but a more systematic approach will be pursued in the near future.
The results in these proceedings are based on the data sample of about 20M hadronic interactions collected in the heavy ion run of the LHC in 2010. A much larger data sample has been collected in November 2011, which will allow to improve the precision of the heavy flavour and jet measurements and extend the [$p_{\mathrm{T}}$]{}-range over which the various measurements can be performed. These improvements will be essential to further constrain the theory of parton energy loss.
[^1]:
|
---
abstract: 'We give an introduction to Joyce’s construction of the motivic Hall algebra of coherent sheaves on a variety $M$. When $M$ is a Calabi-Yau threefold we define a semi-classical integration map from a Poisson subalgebra of this Hall algebra to the ring of functions on a symplectic torus. This material will be used in [@forth] to prove some basic properties of Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds.'
author:
- Tom Bridgeland
title: An introduction to motivic Hall algebras
---
Introduction
============
This paper is a gentle introduction to part of Joyce’s theory of motivic Hall algebras [@Jo0; @Jo1; @Jo2; @JS]. It started life as the first half of the author’s paper [@forth] in which this theory is used to prove some properties of Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds. Eventually it became clear that there were enough points at which our presentation differs from Joyce’s to justify a separate paper. Nonetheless, most of the basic ideas can be found in Joyce’s work. The application of Hall algebras to the study of invariants of moduli spaces originated with Reineke’s computation of the Betti numbers of the spaces of stable quiver representations [@rei]. His technique was to translate categorical statements into identities in a suitable Hall algebra, and to then apply a ring homomorphism into a completed skew-polynomial ring, thus obtaining identities involving generating functions for the invariants of interest. The relevant category in Reineke’s paper is the category of representations of a finite quiver without relations. Such categories can be defined over any field, and Reineke worked with a Hall algebra based on counting points over $\mathbb{F}_q$. In [@Jo2] Joyce used Grothendieck rings of Artin stacks to construct a motivic version of the Hall algebra defined in arbitrary characteristic. This can be applied, for example, to categories of coherent sheaves on complex varieties.
The interesting part of the theory is the construction of a homomorphism from the Hall algebra to a skew polynomial ring, often viewed as a ring of functions on a quantum torus. Such maps go under the general name of integration maps, since they involve integrating an element of the Hall algebra over the moduli stack. In Reineke’s case the existence of such a map relied on the fact that the relevant categories of representations were of homological dimension one. Remarkably it seems that integration maps also exist when the underlying abelian category is Calabi-Yau of dimension 3.
In the CY$_3$ case Joyce’s integration map is a homomorphism of Lie algebras defined on a Lie subalgebra of the Hall algebra. Kontsevich and Soibelman [@KS] suggested that incorporating motivic vanishing cycles should enable one to construct an algebra morphism from the full Hall algebra. Unfortunately the details of the constructions in their paper are currently rather sketchy and rely on some unproved conjectures. Joyce and Song [@JS] went on to use some of the ideas from [@KS] to prove an important property of Behrend functions (stated here as Theorem \[see\]) and so incorporate such functions into Joyce’s Lie algebra integration map.
The main result of this paper (Theorem \[se\]) is the existence of an integration map in the CY$_3$ case that is a homomorphism of Poisson algebras. It can be viewed as the semi-classical limit of the ring homomorphism envisaged by Kontsevich and Soibelman. It relies on the same property of the Behrend function proved by Joyce and Song. Combined with a difficult no-poles result of Joyce it can be used to prove non-trivial results on Donaldson-Thomas invariants [@forth].
Acknowledgements {#acknowledgements .unnumbered}
----------------
Thanks most of all to Dominic Joyce who patiently explained many things about motivic Hall algebras. Thanks also to Arend Bayer, Andrew Kresch and Max Lieblich for useful conversations. Finally, I’m very grateful to Maxim Kontsevich and Yan Soibelman for sharing a preliminary version of their paper [@KS].
Grothendieck rings of varieties and schemes
===========================================
Here we review some basic definitions concerning Grothendieck rings of varieties. Some good references are [@bitt; @loo]. For us a complex variety is a reduced, separated scheme of finite type over ${\mathbb C}$. We denote by $$\label{blbl}{\operatorname{Var}}/{\mathbb C}\subset {\operatorname{Sch}}/{\mathbb C}\subset{\operatorname{Sp}}/{\mathbb C}$$ the categories of varieties over ${\mathbb C}$, of schemes of finite type over ${\mathbb C}$, and of algebraic spaces of finite type over ${\mathbb C}$ respectively.
Grothendieck ring of varieties {#fir}
------------------------------
Recall first the definition of the Grothendieck ring of varieties.
\[scc\]Let ${K({\operatorname{Var/{{\mathbb C}}}})}$ denote the free abelian group on isomorphism classes of complex varieties, modulo relations $$\label{scissor}
[X]=[Z]+[U]$$ for $Z\subset X$ a closed subvariety with complementary open subvariety $U$.
The relations are called the scissor relations, since they involve cutting a variety up into pieces. We can equip ${K({\operatorname{Var/{{\mathbb C}}}})}$ with the structure of a commutative ring by setting $$[X]\cdot [Y]=[X\times Y].$$ The class of a point $1=[{\operatorname{Spec}}({\mathbb C})]$ is then a unit. We write $${\mathbb{L}}=[\mathbb{A}^1] \in {K({\operatorname{Var/{{\mathbb C}}}})}$$ for the class of the affine line. By a stratification of a variety $X$ we mean a collection of disjoint locally-closed subsets $X_i\subset X$ which together cover $X$.
\[silk\] If a variety $X$ is stratified by subvarieties $X_i$ then only finitely many of the $X_i$ are non-empty and $$[X]=\sum_i [X_i]\in {K({\operatorname{Var/{{\mathbb C}}}})}.$$
The result is clear for varieties of dimension 0, so let us use induction on the dimension $d$ of $X$, and assume the result known for varieties of dimension $<d$.
First consider the case when $X$ is irreducible. Then one of the $X_i=U$ contains the generic point and is therefore open. The complement $Z=X\setminus
U$ is of smaller dimension and is stratified by the other subvarieties $X_i$. Since $$[X]=[Z]+[U]$$ the result follows by induction.
In the case when $X$ is reducible we can take an irreducible subvariety and remove the intersections with the other irreducible components. This gives an irreducible open subset $U\subset X$ with complement a closed subvariety $Z$ having fewer irreducible components than $X$. By induction on this number one can therefore conclude that $$[Z]=\sum_i[Z\cap X_i], \quad [U]=\sum_i [U\cap X_i],$$ with finitely many non-empty terms appearing in each sum. Since $$[X_i]=[Z\cap X_i]+[U\cap X_i],$$ the result then follows from the scissor relations.
There is a ring homomorphism $\chi\colon {K({\operatorname{Var/{{\mathbb C}}}})}\to
{\mathbb{Z}}$ defined by sending the class of a variety $X$ to its topological Euler characteristic $$\chi(X)=\sum_{i=0}^{2d} (-1)^i \dim H^i(X_{{\operatorname{an}}},{\mathbb C}),$$ where $X_{{\operatorname{an}}}$ denotes $X$ equipped with the analytic topology, and $H^i$ denotes singular cohomology.
\[grimer\] Suppose $x\in{K({\operatorname{Var/{{\mathbb C}}}})}$ satisfies $${\mathbb{L}}^m\cdot ({\mathbb{L}}^n-1) \cdot x =0$$ for some $m,n{\geqslant}1$. Then $\chi(x)=0$.
There is a ring homomorphism $\chi_t\colon {K({\operatorname{Var/{{\mathbb C}}}})}\to {\mathbb{Z}}[t]$ that sends the class of a smooth complete variety $X$ to the Poincar[é]{} polynomial $$\chi_t(X)=\sum_{i=0}^{2d} t^i \cdot \dim H^i(X_{an},{\mathbb C}).$$ It specializes at $t=-1$ to the Euler characteristic. Now $$\chi_t({\mathbb{L}})=\chi_t({\operatorname{\mathbb P}}^1)-\chi_t(1)=t^2.$$ Since ${\mathbb{Z}}[t]$ is an integral domain one therefore has $\chi_t(x)=0$. Setting $t=-1$ gives the result.
Zariski fibrations
------------------
There is a useful identity in ${K({\operatorname{Var/{{\mathbb C}}}})}$ relating to fibrations.
A morphism of schemes $f\colon X\to Y$ will be called a Zariski fibration if there is an open cover $Y=\bigcup_{i\in I} U_i$ and diagrams $$\xymatrix@C=1em{
f^{-1}(U_i)\ar[rr]^{g_i} \ar[dr]_{f}&& U_i\times
F_i\ar[dl]^{\pi_1}\\ &U_i }$$with each $g_i$ an isomorphism.
Of course if $Y$ is connected then all the fibres $F_i$ are isomorphic, but this will often not be the case. We say that two Zariski fibrations $$f_1 \colon X_1 \to Y\text{ and }f_2 \colon X_2 \to Y$$ have the same fibres, if for any point $y\in Y({\mathbb C})$ the fibres of $f_1$ and $f_2$ over $y$ are isomorphic.
\[fib\] Suppose $f_1\colon X_1\to Y$ and $f_2\colon X_2\to Y$ are Zariski fibrations of varieties with the same fibres. Then $$[X_1]=[X_2]\in
{K({\operatorname{Var/{{\mathbb C}}}})}.$$
We can stratify $Y$ by a finite collection of connected, locally-closed subvarieties $Y_i\subset Y$ such that $f_1$ and $f_2$ are trivial fibrations over each $Y_i$. Then $$[X_1]=\sum_i [f_1^{-1}(Y_i)]=\sum_i [F_i]\cdot [Y_i]=\sum_i [f_2^{-1}(Y_i)]=[X_2],$$ where $F_i$ is the common fibre of $f_1$ and $f_2$ over $Y_i$.
The following application of Lemma \[fib\] will be important later.
There is an identity $$[{\operatorname{GL}}_d]={\mathbb{L}}^{ \frac{1}{2} d (d-1)}\cdot \prod_{k=1}^{d} ({\mathbb{L}}^{k} -1)\in
{K({\operatorname{Var/{{\mathbb C}}}})}.$$
Let $B\subset {\operatorname{GL}}_d$ be the stabilizer of a nonzero vector $x\in {\mathbb C}^d$. The assignment $g\mapsto g(x)$ defines a morphism $$\pi\colon {\operatorname{GL}}_d\to{\mathbb C}^d\setminus\{0\}$$ which is easily seen to be a Zariski fibration with fibre $B$. But there is also an isomorphism $$B{\cong}{\operatorname{GL}}_{d-1}\times{\mathbb C}^{d-1}.$$ Thus by Lemma \[fib\] $$[{\operatorname{GL}}_d]=({\mathbb{L}}^{d}
-1)\cdot {\mathbb{L}}^{d-1} \cdot [{\operatorname{GL}}_{d-1}],$$ and the result follows by induction.
Geometric bijections
--------------------
We will base our treatment of Grothendieck groups on the following class of maps.
\[gb\] A morphism $f\colon X \to Y$ in the category ${\operatorname{Sch}}/{\mathbb C}$ is a geometric bijection if it induces a bijection $$f({\mathbb C}) \colon
X({\mathbb C}) \to Y({\mathbb C})$$ between the sets of ${\mathbb C}$-valued points.
Using Lemma \[bij\] below, it is not difficult to prove that the condition of Definition \[gb\] is equivalent to $f$ being a bijection, or a universal bijection, but for various reasons we prefer to introduce new terminology.
By a stratification of a scheme $X$ we mean a collection of disjoint locally-closed subschemes $X_i \subset X$ which together cover $X$. If the scheme $X$ is of finite type over ${\mathbb C}$ then the argument of Lemma \[silk\] shows that only finitely many of the $X_i$ can be non-empty.
\[bij\] A morphism $f\colon X\to Y$ in the category ${\operatorname{Sch}}/{\mathbb C}$ is a geometric bijection precisely if there are stratifications $$X_i\subset X, \quad Y_i\subset Y,$$ such that $f$ induces isomorphisms $f_i\colon X_i \to Y_i$.
A more precise statement of the condition is that there should be isomorphisms $f_i\colon X_i \to Y_i$ and commuting diagrams $$\begin{CD} X_i &@>f_i>> &Y_i \\
@Vj_iVV && @VVk_iV \\
X &@>f>> &Y \\\end{CD}$$ where the morphisms $j_i$ and $k_i$ are the embeddings of the given locally-closed subschemes. This condition is clearly sufficient since every ${\mathbb C}$-valued point of $X$ or $Y$ factors through a unique one of the given subschemes.
For the converse we may as well assume that $X$ and $Y$ are reduced, since a stratification of $X_{{{\operatorname{red}}}}$ also gives a stratification of $X$, and similarly for $Y$. We claim that there is an open subscheme $Y_1\subset Y$ such that $f$ induces an isomorphism $$f\colon X_1\to Y_1,$$ where $X_1=f^{-1}(Y_1)$. This will be enough since we can then replace $X$ and $Y$ by the complements of $X_1$ and $Y_1$ and repeat.
To prove the claim we can pass to an open subset of $Y$ and hence assume that $Y$ is an irreducible variety. By generic flatness, we can also assume that $f$ is flat, and hence open. Now we can replace $X$ by an irreducible open subvariety, and so $f$ becomes a map of irreducible varieties. The claim then holds by [@mum Prop. 3.17].
Using Lemma \[bij\] we can give an alternative definition of the Grothendieck ring in terms of bijections. This is sometimes useful, particularly when considering Grothendieck rings of schemes and stacks of possibly infinite type.
\[sccc\]The group ${K({\operatorname{Var/{{\mathbb C}}}})}$ is the free abelian group on isomorphism classes of the category ${\operatorname{Var}}/{\mathbb C}$, modulo relations
- $[X_1\amalg X_2]=[X_1]+[X_2]$ for every pair of varieties $X_1$ and $X_2$,
- $[X]=[Y]$ for every geometric bijection $f\colon X\to Y$.
Given a geometric bijection of varieties $f\colon X \to Y$, we can take stratifications of $X$ and $Y$ as in Lemma \[bij\]. We can always assume that the subschemes $X_i$ and $Y_i$ are reduced and hence subvarieties. Then, by Lemma \[silk\], $$[X]=\sum_i [X_i]=\sum_i [Y_i] = [Y] \in {K({\operatorname{Var/{{\mathbb C}}}})}.$$ Thus relation (b) is a consequence of the scissor relations, and clearly relation (a) is a special case of them. Conversely, given a decomposition as in Definition \[scc\], the obvious morphism $Z\amalg U\to X$ is a geometric bijection, so the scissor relations of Definition \[scc\] are a consequence of the relations (a) and (b) in the statement of the Lemma.
Grothendieck rings of schemes and algebraic spaces {#inf}
--------------------------------------------------
Before considering stacks in the next section, it is worth briefly considering the case of schemes and algebraic spaces, always of finite type over ${\mathbb C}$.
\[mantle\] Let ${K({\operatorname{Sch/{{\mathbb C}}}})}$ be the free abelian group on isomorphism classes of the category ${\operatorname{Sch}}/{\mathbb C}$, modulo relations
- $[X_1\amalg X_2]=[X_1]+[X_2]$ for every pair of schemes $X_1$ and $X_2$,
- $[X]=[Y]$ for every geometric bijection $f\colon X\to Y$.
The product in ${\operatorname{Sch}}/{\mathbb C}$ gives the group ${K({\operatorname{Sch/{{\mathbb C}}}})}$ the structure of a commutative ring. One could alternatively define ${K({\operatorname{Sch/{{\mathbb C}}}})}$ via scissor relations as in Definition \[scc\]; the argument of Lemma \[sccc\] shows that this would give the same ring.
In the case of algebraic spaces we define geometric bijections in the category ${\operatorname{Sp}}/{\mathbb C}$ exactly as in Definition \[gb\]. We can also define the notion of a stratification of an algebraic space in the obvious way.
\[bijspace\] A morphism $f\colon X\to Y$ in the category ${\operatorname{Sp}}/{\mathbb C}$ is a geometric bijection precisely if there are stratifications $$X_i\subset X, \quad Y_i\subset Y,$$ such that $f$ induces isomorphisms $f_i\colon X_i \to Y_i$.
This follows the same lines as that of Lemma \[bij\]. The extra argument needed is the following. Suppose given a morphism $f\colon X\to Y$ in ${\operatorname{Sp}}/{\mathbb C}$ with $X$ and $Y$ reduced. We must show that there is an open subspace $Y_1\subset Y$ such that $f$ induces an isomorphism $$f\colon X_1\to Y_1,$$ where $X_1=f^{-1}(Y_1)$. By [@Knutson Prop. II.6.6] we can pass to an open subset and so assume that $Y$ is a scheme, and even an irreducible variety. By generic flatness we can also assume that $f$ is flat. Then, using the same result from [@Knutson] again, there is an open subset $X_0\subset X$ representable by an irreducible variety. Since the induced map $f\colon X_0\to Y$ is flat, its image is an open subvariety $Y_0\subset Y$. We can then apply [@mum Prop. 3.17] as in the proof of Lemma \[bij\].
The Grothendieck group ${K({\operatorname{Sp/{{\mathbb C}}}})}$ is defined as in Definition \[mantle\], replacing the category ${\operatorname{Sch}}/{\mathbb C}$ by ${\operatorname{Sp}}/{\mathbb C}$. The following result shows that from the point of view of Grothendieck rings, providing we stick to objects of finite type, the distinction between varieties, schemes and algebraic spaces disappears.
\[proofabove\] The embeddings of categories induce isomorphisms of rings $${K({\operatorname{Var/{{\mathbb C}}}})}{\cong}{K({\operatorname{Sch/{{\mathbb C}}}})}{\cong}{K({\operatorname{Sp/{{\mathbb C}}}})}.$$
The basic point is that if $Y\in {\operatorname{Sp}}/{\mathbb C}$ there is a geometric bijection $$f\colon X\to Y$$ with $X$ a variety. Indeed, by [@Knutson Prop. II.6.6] there is an open subspace $U\subset Y$ that is representable by an affine scheme. Taking the complement and repeating, we can stratify $Y$ by affine schemes $Y_i \subset Y$. The inclusion map from the disjoint union of these strata then defines a geometric bijection $$X=\coprod Y_i \to Y$$ with $X$ an affine scheme of finite type over ${\mathbb C}$. We can assume that $X$ is reduced and hence a variety since the inclusion of its reduced subscheme is another geometric bijection.
Now consider the homomorphism $$I\colon {K({\operatorname{Var/{{\mathbb C}}}})} \to {K({\operatorname{Sp/{{\mathbb C}}}})}$$ induced by the inclusion of varieties in algebraic spaces. By the above it is surjective, so it will be enough to construct a left inverse $$P \colon {K({\operatorname{Sp/{{\mathbb C}}}})} \to {K({\operatorname{Var/{{\mathbb C}}}})}$$ Given an object $Y\in {\operatorname{Sp}}/{\mathbb C}$, take a geometric bijection $f\colon
X\to Y$ with $X$ a variety, and set $$P([Y])=[X].$$ This is well-defined, because if $W$ is another variety with a geometric bijection $g\colon W\to Y$, then we can form the fibre square $$\begin{CD} Z &@>p>> &W \\
@VqVV && @VVgV \\
X &@>f>> &Y \end{CD}$$ and taking a variety $T$ with a geometric bijection $g\colon T\to
Z$, the composite morphisms $p\circ g$ and $q\circ g$ are geometric bijections, so $$[X]=[W]\in {K({\operatorname{Var/{{\mathbb C}}}})}.$$ It is easy to check that $P$ preserves the relations and hence defines the required inverse.
Grothendieck rings of stacks
============================
In this section we consider Grothendieck rings of algebraic stacks. The author learnt most of the material in this section from the papers of Joyce [@Jo0] and To[ë]{}n [@Toen]. We refer to [@G] for a readable introduction to stacks, and to [@LMB] for a more detailed treatment. All stacks will be Artin stacks and will be assumed to be locally of finite type over ${\mathbb C}$. We denote by ${\operatorname{St}}/{\mathbb C}$ the 2-category of algebraic stacks of finite type over ${\mathbb C}$.
Given a scheme $S$ and a stack $X$ we denote by $X(S)$ the groupoid of $S$-valued points of $X$. We will be fairly sloppy with 2-terminology: by a commutative (resp. Cartesian) diagram of stacks we mean one that is 2-commutative (resp. 2-Cartesian). By an isomorphism of stacks we mean what a category-theorist would call an equivalence.
Geometric bijections and Zariski fibrations
-------------------------------------------
In the case of stacks the appropriate analogue of Definition \[gb\] is as follows.
\[blblbl\] A morphism $f\colon X\to Y$ in the category ${\operatorname{St}}/{\mathbb C}$ will be called a geometric bijection if it is representable and the induced functor on groupoids of ${\mathbb C}$-valued points $$f({\mathbb C})\colon X({\mathbb C})\to Y({\mathbb C})$$ is an equivalence of categories.
In fact, it is easy enough to show that the assumption that $f$ be representable in Definition \[blblbl\] follows from the other condition. This comes down to showing that if $f\colon X \to Y$ is a group scheme in ${\operatorname{Sch}}/{\mathbb C}$ such that for each point $y\in Y({\mathbb C})$ the fibre $X_y$ is the trivial group, then $f$ is an isomorphism.
By a stratification of a stack $X$ we mean a collection of locally-closed substacks $X_i\subset X$ that are disjoint and together cover $X$. If $X$ is of finite type over ${\mathbb C}$ it follows by pulling back to an atlas that only finitely many of the $X_i$ can be non-empty.
\[stratastack\] A morphism $f\colon X\to Y$ in ${\operatorname{St}}/{\mathbb C}$ is a geometric bijection precisely if there are stratifications $$X_i\subset X, \quad Y_i\subset Y,$$ such that $f$ induces isomorphisms $f_i\colon X_i \to Y_i$.
As in the proof of Lemma \[bij\] it is easy to see that the condition is sufficient. For the converse, assume that $f$ is a geometric bijection. Replacing $X$ and $Y$ by their reduced substacks we can assume that $X$ and $Y$ are reduced. It will be enough to show that there is a non-empty open substack $Y_1
\subset Y$ such that $f$ induces an isomorphism $$f\colon X_1 \to Y_1,$$ where $X_1=f^{-1}(Y_1)$. We can then take complements and repeat.
Pulling $f$ back to an atlas $\pi\colon T\to Y$ we obtain a diagram $$\begin{CD} S &@>g>> &T \\
@VVV && @VV\pi V \\
X &@>f>> &Y \\\end{CD}$$with $S$ an algebraic space and $g$ a geometric bijection.
Let $T_1\subset T$ be the largest open subset over which $g$ is an isomorphism. By Lemma \[bijspace\] this subset is non-empty. That it descends to $Y$ follows from the following statement. Suppose given a Cartesian diagram in ${\operatorname{Sp}}/{\mathbb C}$ $$\label{diaa}\begin{CD} S' &@>g'>> &T'\\
@VqVV && @VVpV \\
S &@>g>> &T \\\end{CD}$$ with $p$ faithfully flat. Then $g'$ is an isomorphism over an open subset $U'\subset T'$ precisely if $g$ is an isomorphism over the open subset $U=p(U')\subset
T$.
Replacing $T$ by $U$ and $T'$ by $U'$ we can reduce further to the statement that given a diagram with $p$ faithfully flat and $g'$ an isomorphism then $g$ is also an isomorphism. This is the well-known statement that isomorphisms are stable in the faithfully flat topology [@Knutson Prop. 3.5].
A morphism of stacks $f\colon X\to Y$ is a Zariski fibration if its pullback to any scheme is a Zariski fibration of schemes.
In particular a Zariski fibration of stacks is representable. Note however that there need not be a cover of the stack $Y$ by open substacks over which $f$ is trivial. For this reason, in the Grothendieck group of stacks, the fibration identity of Lemma \[fib\] is not a consequence of the other relations, and we will impose it by hand.
Grothendieck ring of stacks
---------------------------
To make the comparison result in the next section true, it will be necessary to restrict slightly the class of stacks we allow in our Grothendieck ring.
A stack $X$ locally of finite type over ${\mathbb C}$ has affine stabilizers if for every ${\mathbb C}$-valued point $x \in X({\mathbb C})$ the group ${\operatorname{Isom}}_{\mathbb C}(x,x)$ is affine.
The importance of this notion lies in the following corollary of a result of Kresch.
\[kresch\] A stack $X\in {\operatorname{St}}/{\mathbb C}$ has affine stabilizers precisely if there is a variety $Y$ with an action of $G={\operatorname{GL}}_d$ and a geometric bijection $$f\colon Y/G\to X.$$
The condition is clearly sufficient. For the converse, suppose first that the automorphism groups of all geometric points of $X$ are affine. Kresch then shows [@K Prop. 3.5.2, Prop. 3.5.9] that the associated reduced stack $X_{{{\operatorname{red}}}}$ has a stratification by locally-closed substacks of the form $$X_i=Y_i/G_i,$$ where by [@K Lemma 3.5.1] we can take each group $G_i$ to be of the form $G_i={\operatorname{GL}}_{d_i}$. The obvious map $f\colon \amalg X_i \to X$ is then a geometric bijection, and the result follows from the isomorphism $$(X_1/G_1) \amalg (X_2/G_2) {\cong}[(X_1\times G_2) \sqcup (X_2\times
G_1)]/(G_1\times G_2),$$ which shows that the disjoint union of quotient stacks is another quotient stack.
To finish the proof we must check that if the automorphism groups of all ${\mathbb C}$-valued points of $X$ are affine, then the same is true for all geometric points. To prove this,[^1] suppose $f\colon G \to S$ is a group scheme, with $G$ and $S$ of finite type over ${\mathbb C}$, and suppose that there is a geometric point ${\operatorname{Spec}}(K)\to S$ such that the corresponding geometric fibre of $f$ is non-affine. We must prove that there is a ${\mathbb C}$-valued point of $S$ such that the corresponding fibre of $f$ is non-affine.
Since being affine is invariant under field extension \[EGA IV.2.7.1\] we can assume that $K$ is the algebraic closure of the residue field $k(s)$ of a point $s\in S$. Chevalley’s Theorem [@Chev] implies that an algebraic group $G$ over a field $K$ of characteristic zero is non-affine precisely if there is an epimorphism $G_0 {\twoheadrightarrow}A$ where $G_0\subset G$ is the connected component of the identity, and $A$ is a positive-dimensional abelian variety. For a modern proof of this see [@Con].
Applying this to our situation, the epimorphism $G_0{\twoheadrightarrow}A$ is defined over a finite extension of $k(s)$, and hence over a finite type scheme with a morphism $T\to S$ dominating the closure of the point $s$. Restricting to a ${\mathbb C}$-valued point and applying Chevalley’s theorem again completes the proof.
We can now give the following definition.
\[mainy\]Let ${K({\operatorname{St/{{\mathbb C}}}})}$ be the free abelian group spanned by isomorphism classes of stacks of finite type over ${\mathbb C}$ with affine stabilizers, modulo relations
- $[X_1\amalg X_2]=[X_1]+[X_2]$ for every pair of stacks $X_1$ and $X_2$,
- $[X]=[Y]$ for every geometric bijection $f\colon X\to Y$,
- $[X_1]=[X_2]$ for every pair of Zariski fibrations $f_i\colon X_i \to Y$ with the same fibres.
Fibre product of stacks over ${\mathbb C}$ gives ${K({\operatorname{St/{{\mathbb C}}}})}$ the structure of a commutative ring. There is an obvious homomorphism of commutative rings $$\label{obvious}{K({\operatorname{Var/{{\mathbb C}}}})}\to{K({\operatorname{St/{{\mathbb C}}}})}$$ obtained by considering a variety as a representable stack.
Comparison Lemma
----------------
In this section, following To[ë]{}n [@Toen], we show that if one localises ${K({\operatorname{Var/{{\mathbb C}}}})}$ at the classes of all special algebraic groups, the map becomes an isomorphism.
An algebraic group is special if any map of schemes $$f\colon X\to Y$$ which is a principal $G$-bundle in the [é]{}tale topology is a Zariski fibration.
Examples of special groups include the general linear groups ${\operatorname{GL}}_d$. All special groups are affine and connected [@Chow].
Localizing the ring ${K({\operatorname{Var/{{\mathbb C}}}})}$ with respect to any of the following three sets of elements gives the same result:
- the classes $[G]$ for $G$ a special algebraic group,
- the classes $[{\operatorname{GL}}_d]$ for $d{\geqslant}1$,
- the elements ${\mathbb{L}}$ and ${\mathbb{L}}^i-1$ for $i{\geqslant}1$.
If $G\subset {\operatorname{GL}}_d$ is a closed subgroup, then the quotient map $$\pi\colon {\operatorname{GL}}_d \to {\operatorname{GL}}_d/G$$ is a principal $G$-bundle. It is locally trivial in the smooth topology (equivalently, in the [é]{}tale topology [@vis Example 2.51]) because it is smooth, and becomes trivial when pulled back to itself. Thus if $G$ is special we can conclude that $$[{\operatorname{GL}}_d]=[G]\cdot [{\operatorname{GL}}_d/G].$$ Since ${\operatorname{GL}}_d$ is itself special this proves the equivalence of (a) and (b). The equivalence of (b) and (c) follows from Lemma \[fib\].
If an algebraic group $G$ acts on a variety $X$ then the map to the quotient stack $$\pi\colon X \to X/G$$ is a principal $G$-bundle when pulled back to any scheme. If $G$ is special it follows that $\pi$ is a Zariski fibration with fibre $G$ and hence $$\label{matrix}[X]=[G]\cdot
[X/G]\in{K({\operatorname{St/{{\mathbb C}}}})}.$$ In particular taking $X$ to be a point we see that $[G]$ is invertible in ${K({\operatorname{St/{{\mathbb C}}}})}$. The following result is due to To[ë]{}n [@Toen Theorem 3.10] (who also considered higher stacks). A slightly weaker version was proved independently by Joyce [@Jo0 Theorem 4.10].
\[toen\] The homomorphism induces an isomorphism of commutative rings $$Q\colon {K({\operatorname{Var/{{\mathbb C}}}})}[[{\operatorname{GL}}(d)]^{-1}:d{\geqslant}1] {\longrightarrow}{K({\operatorname{St/{{\mathbb C}}}})}.$$
The expression shows that the map $Q$ is well-defined and satisfies $$Q([X]/[G])=[X/G]\in{K({\operatorname{St/{{\mathbb C}}}})}.$$ We shall construct an inverse to $Q$. Suppose $Z$ is a stack with affine stabilizers. By Proposition \[kresch\] there is a geometric bijection $g\colon X/G\to Z$ with $G$ a special algebraic group acting on a variety $X$. Set $$R(Z)=[X]/[G] \in {K({\operatorname{Var/{{\mathbb C}}}})}[[{\operatorname{GL}}(d)]^{-1}:d{\geqslant}1].$$ To check that this is well-defined, suppose given another bijection $h\colon Y/H\to Z$. Consider the diagram of Cartesian squares $$\begin{CD}
R@>p>> &Q @>j>> &Y \\
@VqVV & @VVV &@VVV \\
P @>>> &W @>>> &Y/H\\
@VkVV & @VVV &@VVh V \\
X @>\pi>> &X/G @>g>> &Z \\ \end{CD}$$ Then $Q$ is an algebraic space and $j$ is a geometric bijection, so $[Q]=[Y]\in {K({\operatorname{Sp/{{\mathbb C}}}})}$. The map $\pi$, and hence also $p$, is a Zariski fibration with fibre $G$; pulling it back to a bijection $Q'\to Q$ with $Q'$ a scheme, it follows that $[R]=[G]\cdot [Q]\in{K({\operatorname{Sp/{{\mathbb C}}}})}$. Hence we obtain $[R]=[G]\cdot [Y]$ and, by symmetry, $[R]=[H]\cdot [X]$. Thus, using Lemma \[proofabove\] $$[G]\cdot [Y]=[H]\cdot [X] \in {K({\operatorname{Var/{{\mathbb C}}}})}.$$
To show that $R$ descends to the level of the Grothendieck group we must check that the relations in Definition \[mainy\] are mapped to zero. This is very easy for (a) and (b). For (c) suppose $f_i\colon X_i\to Y$ are Zariski fibrations with the same fibres. Take a geometric bijection $W/G\to Y$ and form the diagrams $$\begin{CD}
S_i @>k_i>> &T_i @>h_i>> &X_i\\
@Vp_iVV & @VVV &@VVf_iV \\
W @>\pi>> &W/G @>g>> &Y \end{CD}$$ Then there are induced actions of $G$ on the varieties $S_i$ such that $T_i{\cong}S_i/G$. Since $h_i$ is a geometric bijection, $R(X_i)=[S_i]/[G]$. On the other hand, by pullback, the morphisms $p_i\colon S_i\to W$ are Zariski fibrations of schemes with the same fibres, and hence by the argument of Lemma \[fib\], $[S_1]=[S_2]\in {K({\operatorname{Sch/{{\mathbb C}}}})}$.
Relative Grothendieck groups
----------------------------
Let $S$ be a fixed algebraic stack, locally of finite type over ${\mathbb C}$. We shall always assume that $S$ has affine stabilizers. There is a 2-category of algebraic stacks over $S$. Let $\operatorname{St/S}$ denote the full subcategory consisting of objects $$\label{obvious2}f \colon X \to S$$ for which $X$ is of finite type over ${\mathbb C}$. Such an object will be said to have affine stabilizers if the stack $X$ has. Repeating Definition \[mainy\] in this relative context gives the following.
\[rel\] Let ${K({\operatorname{St/{S}}})}$ be the free abelian group spanned by isomorphism classes of objects of $\operatorname{St/S}$, with affine stabilizers, modulo relations
- for every pair of objects $X_1$ and $X_2$ a relation$$[X_1\amalg X_2{\xrightarrow{\ f_1\sqcup f_2\ }} S]=[X_1{\xrightarrow{\ f_1\ }}
S]+[X_2{\xrightarrow{\ f_2\ }} S],$$
- for every commutative diagram $$\xymatrix@C=.8em{
X_1\ar[dr]_{f_1}\ar[rr]^{g} && X_2\ar[dl]^{f_2}\\
&S }$$with $g$ a geometric bijection, a relation $$[X_1{\xrightarrow{\ f_1\ }} S]=[X_2{\xrightarrow{\ f_2\ }} S],$$
- for every pair of Zariski fibrations $$h_1\colon X_1 \to Y, \quad h_2\colon X_2\to Y$$ with the same fibres, and every morphism $g\colon Y\to S$, a relation $$[X_1{\xrightarrow{\ g\circ h_1\ }} S]=[X_2{\xrightarrow{\ g\circ h_2\ }} S].$$
The group ${K({\operatorname{St/{S}}})}$ has the structure of a ${K({\operatorname{St/{{\mathbb C}}}})}$-module, defined by setting $$[X]\cdot [Y{\xrightarrow{\ f\ }} S]=[X\times Y{\xrightarrow{\ f\circ\pi_2\ }} S]$$ and extending linearly.
Suppose that $\Lambda$ is a ${\mathbb{Q}}$-algebra and $$\Upsilon \colon {K({\operatorname{St/{{\mathbb C}}}})}{\longrightarrow}\Lambda$$ is a ring homomorphism. Then for each stack $S$ with affine stabilizers Joyce defines [@Jo2 Section 4.3] a $\Lambda$-module $\operatorname{SF}(S,\Upsilon,\Lambda)$ whose elements he calls stack functions. It is easy to see that there is an isomorphism of $\Lambda$-modules $$\operatorname{SF}(S,\Upsilon,\Lambda){\cong}{K({\operatorname{St/{S}}})}{\otimes}_{{K({\operatorname{St/{{\mathbb C}}}})}}
\Lambda.$$ We leave the proof to the reader.
Functoriality {#funct}
-------------
The following statements are are all easy consequences of the basic properties of fibre products of stacks, and we leave the proofs to the reader. We assume that all stacks appearing have affine stabilizers.
- A morphism of stacks $a\colon S \to T$ induces a map of ${K({\operatorname{St/{{\mathbb C}}}})}$-modules $$a_* \colon {K({\operatorname{St/{S}}})} {\longrightarrow}{K({\operatorname{St/{T}}})}$$ sending $[X{\xrightarrow{\ f\ }} S]$ to $[X{\xrightarrow{\ a\circ f\ }} T]$.
- A morphism of stacks $a\colon S \to T$ of finite type induces a map of ${K({\operatorname{St/{{\mathbb C}}}})}$-modules $$a^*\colon {K({\operatorname{St/{T}}})}{\longrightarrow}{K({\operatorname{St/{S}}})}$$ sending $[ Y{\xrightarrow{\ g\ }} T]$ to $[X{\xrightarrow{\ f\ }} S]$ where $f$ is the map appearing in the Cartesian square $$\begin{CD}
X& @>>> &Y\\
@VfVV &&@VVg V \\
S &@>a>> &T\end{CD}$$
- The above assignments are functorial, in that $$(b\circ a)_* = b_* \circ a_*, \quad (b\circ a)^* = a^*\circ b^*,$$ whenever $a$ and $b$ are composable morphisms of stacks with the required properties.
- Given a Cartesian square of maps $$\begin{CD} U &@>c>>& V\\
@VdVV && @VVbV \\
S& @>a>> &T\end{CD}$$ one has the base-change property $$b^*\circ
a_* = c_*\circ d^*\colon {K({\operatorname{St/{S}}})}{\longrightarrow}{K({\operatorname{St/{V}}})}.$$
- For every pair of stacks $(S_1,S_2)$ there is a K[ü]{}nneth map $$K\colon {K({\operatorname{St/{S_1}}})} {\otimes}{K({\operatorname{St/{S_2}}})} \to {K({\operatorname{St/{S_1 \times S_2}}})}$$ given by $$[X_1{\xrightarrow{\ f_1\ }} S_1] {\otimes}[X_2{\xrightarrow{\ f_2\ }} S_2] \mapsto
[X_1\times X_2 {\xrightarrow{\ f_1\times f_2\ }} S_1\times S_2].$$ It is a morphism of ${K({\operatorname{St/{{\mathbb C}}}})}$-modules.
We can view the functor ${K({\operatorname{St/{--}}})}$ as defining a primitive cohomology theory for stacks.
Motivic Hall algebra {#hall}
====================
Let $M$ be a smooth projective variety and $${\mathcal A}={\operatorname{Coh}}(M)$$ its category of coherent sheaves. In this section, following Joyce [@Jo2], we introduce the motivic Hall algebra of the category ${\mathcal A}$. Much of what we do here would apply with minor modifications to other abelian categories, but we make no attempt at maximal generality.
We use the following abuse of notation throughout: if $f\colon T\to S$ is a morphism of schemes, and $E$ is a sheaf on $S\times M$, we use the shorthand $f^*(E)$ for the pullback to $T\times M$, rather than the more correct $(f\times 1_M)^*(E)$.
Stacks of flags
---------------
Let ${\mathcal{M}}^{(n)}$ denote the moduli stack of $n$-flags of coherent sheaves on $M$. The objects over a scheme $S$ are chains of monomorphisms of coherent sheaves on $S\times M$ of the form $$\label{star} 0=E_0{\hookrightarrow}E_1 {\hookrightarrow}\cdots {\hookrightarrow}E_n=E$$ such that each factor $F_i=E_i/E_{i-1}$ is $S$-flat. It follows that each sheaf $E_i$ is also $S$-flat. If $$0=E'_0{\hookrightarrow}E'_1 {\hookrightarrow}\cdots {\hookrightarrow}E'_n=E$$ is another such object over a scheme $T$, then a morphism in ${\mathcal{M}}^{(n)}$ lying over a morphism of schemes $f\colon T\to S$ is a collection of isomorphisms of sheaves $$\theta_i \colon f^*(E_i) \to E'_i$$ such that each diagram $$\begin{CD} f^*(E_i) &@>>> &f^*(E_{i+1})\\
@V\theta_iVV && @VV\theta_{i+1}V \\
E'_i& @>>> &E'_{i+1}\end{CD}$$ commutes. Here we have taken the usual step of choosing, for each morphism of sheaves, a pullback of every coherent sheaf on its target. The stack property for ${\mathcal{M}}^{(n)}$ follows easily from the corresponding property of the stack ${\mathcal{M}}={\mathcal{M}}^{(1)}$. There are morphisms of stacks $$a_i\colon {\mathcal{M}}^{(n)}\to {\mathcal{M}}, \quad 1{\leqslant}i{\leqslant}n,$$ sending a flag to its $i$th factor $F_i=E_i/E_{i-1}$. To define these it is first necessary to choose a cokernel for each monomorphism $E_{i-1} \to E_i$. There is another morphism $$b\colon {\mathcal{M}}^{(n)}\to {\mathcal{M}}$$ sending a flag to the sheaf $E_n=E$. Note that the functors defining all these morphisms of stacks have the iso-fibration property of Lemma \[headache\]. Using this it is immediate that for $n>1$ there is a Cartesian square $$\label{first}\begin{CD}
{\mathcal{M}}^{(n)} & @>t>> &{\mathcal{M}}^{(2)} \\
@VsVV & &@VVa_1V \\
{\mathcal{M}}^{(n-1)} &@>b>> &{\mathcal{M}}\end{CD}$$ where $s$ and $t$ send a flag to the flags $$E_1{\hookrightarrow}\cdots{\hookrightarrow}E_{n-1}\text{ and }E_{n-1}{\hookrightarrow}E_n$$ respectively. There is a kind of duality around here, which is the basic reason for the associativity of the Hall algebra. Instead of considering flags of the form we could instead consider flags $$\label{dagger} E=E^0{\twoheadrightarrow}E^{1}{\twoheadrightarrow}\cdots {\twoheadrightarrow}E^{n-1}{\twoheadrightarrow}E^n=0.$$ Setting $E^i=E/E_i$ shows that this gives an isomorphic stack. This dual approach leads to Cartesian diagrams $$\label{second}\begin{CD}
{\mathcal{M}}^{(n)} & @>v>> &{\mathcal{M}}^{(2)} \\
@VuVV & &@VVa_2V \\
{\mathcal{M}}^{(n-1)} &@>b>> &{\mathcal{M}}\end{CD}$$ where $u$ and $v$ send a flag to the flags $$E^1{\twoheadrightarrow}\cdots {\twoheadrightarrow}E^{n-1}\text{ and }E^{0}{\twoheadrightarrow}E^1$$ respectively.
The stack ${\mathcal{M}}^{(2)}$ can be thought of as the stack of short exact sequences in ${\mathcal A}$. There is a diagram $$\label{blib}\begin{CD}
{\mathcal{M}}^{(2)} &@>b>> {\mathcal{M}}\\
@V(a_1,a_2)VV \\
{\mathcal{M}}\times{\mathcal{M}}\end{CD}$$ where, as above, the morphisms $a_1,a_2$ and $b$ send a short exact sequence $$0{\longrightarrow}A_1{\longrightarrow}B{\longrightarrow}A_2{\longrightarrow}0$$ to the sheaves $A_1$, $A_2$ and $B$ respectively.
The stacks ${\mathcal{M}}^{(n)}$ are algebraic.
Suppose $f\colon S \to {\mathcal{M}}$ is a morphism of stacks corresponding to a flat family of sheaves $B$ on $S\times M$. Forming the Cartesian square $$\begin{CD}
Z & @>>> &{\mathcal{M}}^{(2)} \\
@VVV & &@VVbV \\
S &@>f>> &{\mathcal{M}}\end{CD}$$ it is easy to see that $Z$ is represented by the relative Quot scheme parameterising quotients of $B$ over $S$. Thus $b$ is representable, and pulling back an atlas for ${\mathcal{M}}$ gives an atlas for ${\mathcal{M}}^{(2)}$. Since fibre products of algebraic stacks are algebraic the result then follows by induction and the existence of the squares .
The morphism $(a_1,a_2)$ is not representable. The fibre over a point of ${\mathcal{M}}\times {\mathcal{M}}$ corresponding to a pair of sheaves $(A_1,A_2)$ is the quotient stack $$[{\operatorname{Ext}}^1(A_2,A_1)/{\operatorname{Hom}}(A_2,A_1)],$$ with the action of the vector space ${\operatorname{Hom}}_{\mathcal A}(A_2,A_1)$ being the trivial one. This statement follows from Proposition \[tom\] below.
The morphism $(a_1,a_2)$ is of finite type.
Fix a projective embedding of $M$. For each integer $m$ and each polynomial $P$ there is a finite type open substack ${\mathcal{M}}_m(P)\subset {\mathcal{M}}$ parameterising $m$-regular sheaves with Hilbert polynomial $P$. Define an open substack $$Y=(a_1,a_2)^{-1}({\mathcal{M}}_m(P_1)\times {\mathcal{M}}_m(P_2))\subset {\mathcal{M}}^{(2)}.$$ It will be enough to show that $Y$ is of finite type. But the morphism $b$ restricts to a map $$b\colon Y \to {\mathcal{M}}_m(P_1+P_2)$$ since an extension of two $m$-regular sheaves is also $m$-regular. This map is of finite type, because once one fixes the Hilbert polynomials involved, the relative Quot scheme is of finite type.
The Hall algebra
----------------
Let us set $${\operatorname{H}}({\mathcal A})={K({\operatorname{St/{{\mathcal{M}}}}})}.$$ Applying the results of Section \[funct\] to the diagram gives a morphism of ${K({\operatorname{St/{{\mathbb C}}}})}$-modules $$m=b_*\circ (a_1,a_2)^* \colon {\operatorname{H}}({\mathcal A}){\otimes}{\operatorname{H}}({\mathcal A}) {\longrightarrow}{\operatorname{H}}({\mathcal A})$$ which we call the convolution product.[^2] Explicitly this is given by the rule $$[X_1{\xrightarrow{\ f_1\ }}{\mathcal{M}}] * [X_2{\xrightarrow{\ f_2\ }} {\mathcal{M}}] = [Z{\xrightarrow{\ b\circ h\ }}{\mathcal{M}}],$$ where $h$ is defined by the following Cartesian square $$\begin{CD}
Z & @>h>> &{\mathcal{M}}^{(2)} &@>b>> {\mathcal{M}}\\
@VVV &&@VV(a_1,a_2)V \\
X_1\times X_2 &@>f_1\times f_2>> &{\mathcal{M}}\times{\mathcal{M}}\end{CD}$$
The following result is due to Joyce [@Jo2 Theorem 5.2], although the basic idea is of course the same as for previous incarnations of the Hall algebra.
The product $m$ gives ${\operatorname{H}}({\mathcal A})$ the structure of an associative unital algebra over ${K({\operatorname{St/{{\mathbb C}}}})}$. The unit element is $$1=[{\mathcal{M}}_{0}\subset {\mathcal{M}}],$$ where ${\mathcal{M}}_{0}{\cong}{\operatorname{Spec}}({\mathbb C})$ is the substack of zero objects in ${\mathcal A}$.
Consider the composition $$\begin{CD}
{\operatorname{H}}({\mathcal A}){\otimes}{\operatorname{H}}({\mathcal A}){\otimes}{\operatorname{H}}({\mathcal A})@>m{\otimes}{\operatorname{id}}>>
{\operatorname{H}}({\mathcal A}){\otimes}{\operatorname{H}}({\mathcal A}) @>m>> {\operatorname{H}}({\mathcal A})\end{CD}.$$ It is induced by the diagram $$\begin{CD}
&&&& {\mathcal{M}}^{(2)} &@>b>> &{\mathcal{M}}\\
&&&& @V(a_1,a_2)VV \\
{\mathcal{M}}^{(2)}\times {\mathcal{M}}&@> (b,{\operatorname{id}}) >> &{\mathcal{M}}\times {\mathcal{M}}\\
@V (a_1,a_2,{\operatorname{id}}) VV \\
{\mathcal{M}}\times{\mathcal{M}}\times {\mathcal{M}}\end{CD}$$ There is a bigger commutative diagram obtained by filling in the top left square with $$\begin{CD}
{\mathcal{M}}^{(3)} & @>t>> &{\mathcal{M}}^{(2)} \\
@V(s,a_2\circ t)VV && @VV(a_1,a_2)V \\
{\mathcal{M}}^{(2)} \times {\mathcal{M}}&@>(b,{\operatorname{id}})>> &{\mathcal{M}}\times{\mathcal{M}}\end{CD}$$ where $s$ sends a flag $E_1{\hookrightarrow}E_2{\hookrightarrow}E_3$ to the flag $E_1{\hookrightarrow}E_2$, and $t$ sends it to $E_2{\hookrightarrow}E_3$. This square is Cartesian because of the square $(\ref{first})$ and Lemma \[nonsense\], so by the base-change property of Section \[funct\] $$m\circ (m{\otimes}{\operatorname{id}}) = b_* \circ (a_1,a_2,a_3)^*$$ is induced by the diagram $$\begin{CD} {\mathcal{M}}^{(3)} @>b>> {\mathcal{M}}\\
@V(a_1,a_2, a_3)VV \\
{\mathcal{M}}^3 \end{CD}$$
A similar argument using the square $(\ref{second})$ shows that the other composition is induced by the same diagram. The multiplication is therefore associative. We leave the reader to check the unit property.
The $n$-fold product $$m_n\colon {\operatorname{H}}({\mathcal A})^{{\otimes}n} \to {\operatorname{H}}({\mathcal A})$$ is induced by the diagram $$\begin{CD} {\mathcal{M}}^{(n)} @>b>> {\mathcal{M}}\\
@V(a_1,\cdots, a_n)VV \\
{\mathcal{M}}^n \end{CD}$$ in the sense that $$m_n=b_*\circ (a_1,\cdots,a_n)^* \colon {\operatorname{H}}({\mathcal A})^{{\otimes}n}{\longrightarrow}{\operatorname{H}}({\mathcal A}).$$
This follows by induction and a similar argument to the one above, but using the Cartesian diagram $$\begin{CD} {\mathcal{M}}^{(n)} & @>t>> &{\mathcal{M}}^{(2)} \\
@V(s,a_2\circ t)VV & &@VV(a_1,a_2)V \\
{\mathcal{M}}^{(n-1)}\times{\mathcal{M}}&@>(b,{\operatorname{id}})>> &{\mathcal{M}}\times{\mathcal{M}}\end{CD}$$ given by and Lemma \[nonsense\].
Grading
-------
Let $K(M)=K({\mathcal A})$ denote the Grothendieck group of the category ${\mathcal A}$. Given two coherent sheaves $E$ and $F$ we can define $$\chi(E,F)=\sum_i (-1)^i \dim_{{\mathbb C}}\,{\operatorname{Ext}}^i(E,F).$$ This defines a bilinear form $\chi(-,-)$ on $K(M)$ called the Euler form. Serre duality implies that the left and right radicals $^{\perp}K(M)$ and $K(M)^{\perp}$ are equal. The numerical Grothendieck group is the quotient $$N(M)=K(M)/K(M)^{\perp}.$$ Let $\Gamma\subset N(M)$ denote the monoid of effective classes, which is to say classes of the form $[E]$ with $E$ a sheaf.
\[stan\] Suppose $S$ is a connected scheme and $F$ is an $S$-flat coherent sheaf on $S\times M$. For each point $s\in S({\mathbb C})$ let $$F_s=F|_{\{s\} \times M}$$ be the corresponding sheaf on $M$. Then the class $[F_s]\in N(M)$ is independent of the point $s$.
For any locally-free sheaf $E$, the integer $$\chi(E,F_s)=\chi(E^{{\vee}}{\otimes}F_s)$$ is locally constant on $S$. Since $M$ is smooth, the Grothendieck group $K(M)$ is spanned by the classes of locally-free sheaves, so the class $[F_s]\in N(M)$ is also locally constant.
It follows from Lemma \[stan\] that the stack ${\mathcal{M}}$ splits as a disjoint union of open and closed substacks $${\mathcal{M}}=\bigsqcup_{\alpha\in \Gamma} {\mathcal{M}}_\alpha,$$ where ${\mathcal{M}}_\alpha\subset {\mathcal{M}}$ is the stack of objects of class $\alpha\in \Gamma$. The inclusion ${\mathcal{M}}_\alpha\subset {\mathcal{M}}$ induces an embeddding $${K({\operatorname{St/{{\mathcal{M}}_\alpha}}})}\subset {K({\operatorname{St/{{\mathcal{M}}}}})}.$$ There is thus a direct sum decomposition $$\label{grading} {\operatorname{H}}({\mathcal A})=\bigoplus_{\alpha\in \Gamma}
{\operatorname{H}}({\mathcal A})_\alpha,$$ and ${\operatorname{H}}({\mathcal A})$ with the convolution product becomes a $\Gamma$-graded algebra.
Sheaves supported in dimension ${\leqslant}d$
---------------------------------------------
For any integer $d$ there is a full abelian subcategory $${\mathcal A}_{{\leqslant}d}={\operatorname{Coh}}_{{\leqslant}d}(M)\subset{\mathcal A}={\operatorname{Coh}}(M)$$ closed under extensions, consisting of coherent sheaves on $M$ whose support has dimension ${\leqslant}d$. There is a subgroup $$N_{{\leqslant}d}(M) \subset N(M)$$ spanned by classes of objects of ${\operatorname{Coh}}_{{\leqslant}d}(M),$ and a corresponding positive cone$$\Gamma_{{\leqslant}d}=N_{{\leqslant}d}(M) \cap
\Gamma.$$ One can define a Hall algebra ${\operatorname{H}}({\mathcal A}_{{\leqslant}d})$ by replacing the moduli stack ${\mathcal{M}}$ in the above discussion with the substack ${\mathcal{M}}_{{\leqslant}d}$ of objects of ${\mathcal A}_{{\leqslant}d}$.
In fact it is easy to see that a coherent sheaf $E$ lies in ${\operatorname{Coh}}_{{\leqslant}d}(M)$ precisely if its class $[E]\in N(M)$ lies in the subgroup $N_{{\leqslant}d}(M)$. Thus $${\mathcal{M}}_{{\leqslant}d} = \bigsqcup_{\alpha\in \Gamma_{{\leqslant}d}} {\mathcal{M}}_\alpha,$$ and there is an identification $${\operatorname{H}}({\mathcal A}_{{\leqslant}d}) = \bigoplus_{\alpha\in \Gamma_{{\leqslant}d}}
{\operatorname{H}}_\alpha({\mathcal A}).$$ We will make heavy use of the $d=1$ version of this construction in [@forth].
Integration map
===============
In this section we construct a homomorphism of Poisson algebras from a semi-classical limit of the Hall algebra to an algebra of functions on a symplectic torus. It can be viewed as the semi-classical limit of the ring homomorphism envisaged by Kontsevich and Soibelman [@KS]. We assume throughout that $M$ is a smooth projective Calabi-Yau threefold over ${\mathbb C}$. We include in this the condition that $$H^1(M,{\mathcal O}_M)=0.$$ There are two versions of the story depending on a choice of sign $\sigma\in\{\pm 1\}$ which we fix throughout. The choice $\sigma=+1$ will lead to naive Euler characteristic invariants, while $\sigma=-1$ leads to Donaldson-Thomas invariants.
Regular elements
----------------
Consider the maps of commutative rings $${K({\operatorname{Var/{{\mathbb C}}}})} \to
{K({\operatorname{Var/{{\mathbb C}}}})}[{\mathbb{L}}^{-1}] \to {K({\operatorname{St/{{\mathbb C}}}})},$$ and recall that ${\operatorname{H}}({\mathcal A})$ is an algebra over ${K({\operatorname{St/{{\mathbb C}}}})}$. Define a ${K({\operatorname{Var/{{\mathbb C}}}})}[{\mathbb{L}}^{-1}]$-module $${\operatorname{H_{{reg}}}}({\mathcal A}) \subset {\operatorname{H}}({\mathcal A})$$ to be the span of classes of maps $[X {\xrightarrow{\ f\ }} {\mathcal{M}}]$ with $X$ a variety. We call an element of ${\operatorname{H}}({\mathcal A})$ regular if it lies in this submodule. The following result will be proved in Section \[firstproof\] below.
\[fi\] The submodule of regular elements is closed under the convolution product: $${\operatorname{H_{{reg}}}}({\mathcal A}) * {\operatorname{H_{{reg}}}}({\mathcal A}) \subset {\operatorname{H_{{reg}}}}({\mathcal A}),$$ and is therefore a ${K({\operatorname{Var/{{\mathbb C}}}})}[{\mathbb{L}}^{-1}]$-algebra. Moreover the quotient $${\operatorname{H_{sc}}}({\mathcal A})={\operatorname{H_{{reg}}}}({\mathcal A})/({\mathbb{L}}-1) {\operatorname{H_{{reg}}}}({\mathcal A})$$ is a commutative ${K({\operatorname{Var/{{\mathbb C}}}})}$-algebra.
We call the algebra ${\operatorname{H_{sc}}}({\mathcal A})$ the semi-classical Hall algebra. Since $$[{\mathbb C}^*]={\mathbb{L}}-1$$ is invertible in ${K({\operatorname{St/{{\mathbb C}}}})}$, there is a Poisson bracket on ${\operatorname{H}}({\mathcal A})$ given by the formula $$\{f,g\}=\frac{f*g-g*f}{{\mathbb{L}}-1} .$$ This bracket preserves the subalgebra ${\operatorname{H_{{reg}}}}({\mathcal A})$ because the multiplication in ${\operatorname{H_{{reg}}}}({\mathcal A})$ is commutative modulo the ideal $({\mathbb{L}}-1)$. The induced bracket on ${\operatorname{H_{{reg}}}}({\mathcal A})$ then descends to give a Poisson bracket on the commutative algebra ${\operatorname{H_{sc}}}({\mathcal A})$.
The integration map
-------------------
Define a ring$${\mathbb{Z}}_\sigma[\Gamma]=\bigoplus_{\alpha\in \Gamma}
{\mathbb{Z}}\cdot x^{\alpha}$$ by taking the free abelian group spanned by symbols $x^\alpha$ for $\alpha\in\Gamma$ and setting $$x^\alpha * x^\beta = \sigma^{\chi(\alpha,\beta)} \cdot x^{\alpha +\beta}.$$ Since the Euler form is skew-symmetric this ring is commutative. Equip it with a Poisson structure by defining $$\{x^\alpha,x^\beta\}= \sigma^{\chi(\alpha,\beta)}\cdot
\chi(\alpha,\beta) \cdot x^{\alpha+\beta}.$$ Take a locally constructible function $$\lambda\colon{\mathcal{M}}\to{\mathbb{Z}}.$$ For definitions of constructible functions on stacks see [@Jo-1]. Associated to every sheaf $E\in {\mathcal A}$ is an integral weight $$\lambda(E)\in{\mathbb{Z}}.$$ If $X$ is a variety with a map $f \colon X \to {\mathcal{M}}$, there is an induced constructible function $$f^*(\lambda)\colon X \to {\mathbb{Z}},$$ and a weighted Euler characteristic $$\chi(X,f^*(\lambda))=\sum_{n\in {\mathbb{Z}}} n \cdot \chi((\lambda\circ f)^{-1}(n)).$$ We prove the following result in Section \[secondproof\] below.
\[se\]Given a locally constructible function $\lambda\colon{\mathcal{M}}\to{\mathbb{Z}}$, there is a unique group homomorphism $$I \colon {\operatorname{H_{sc}}}({\mathcal A}) {\longrightarrow}{\mathbb{Z}}_\sigma[\Gamma],$$such that if $X$ is a variety and $f\colon X\to {\mathcal{M}}$ factors via ${\mathcal{M}}_\alpha\subset {\mathcal{M}}$ then $$I\big([X{\xrightarrow{\ f\ }}{\mathcal{M}}]\big)=\chi(X,f^*(\lambda)) \cdot x^\alpha.$$ Moreover, $I$ is a homomorphism of commutative algebras if for all $E,F\in{\mathcal A}$ $$\label{firstt}\lambda(E\oplus
F)={\sigma}^{\chi(E,F)} \cdot \lambda(E)\cdot
\lambda(F),$$ and a homomorphism of Poisson algebras if, in addition, the expression $$\label{secondd}m(E,F)=\chi({\operatorname{\mathbb P}}{\operatorname{Ext}}^1(F,E),
\lambda(G_\theta)-\lambda(G_0))$$ is symmetric in $E$ and $F$.
This statement may need a little explanation. We have written $G_\theta$ for the sheaf $$0{\longrightarrow}E{\longrightarrow}G_\theta {\longrightarrow}F {\longrightarrow}0$$ corresponding to a class $\theta\in {\operatorname{Ext}}^1(F,E)$. In particular $G_0=E\oplus F$. For nonzero $\theta$ the isomorphism class of the object $G_\theta$ depends only on the class $$[\theta]\in {\operatorname{\mathbb P}}{\operatorname{Ext}}^1(F,E),$$ and it is easy to see that the map $[\theta]\mapsto \lambda(G_\theta)$ is a constructible function on the projective space ${\operatorname{\mathbb P}}{\operatorname{Ext}}^1(F,E)$.
Behrend function
----------------
Recall [@Be] that Behrend associates to any scheme $S$ of finite type over ${\mathbb C}$ a constructible function $$\nu_S\colon S\to{\mathbb{Z}}$$ with the property that if $S$ is a proper moduli scheme with a symmetric obstruction theory then the associated Donaldson-Thomas virtual count coincides with the weighted Euler characteristic: $$\#_{{\operatorname{vir}}}(S) := \int_{[S]^{{\operatorname{vir}}}} 1=\chi(S,\nu_S).$$ These functions satisfy the relation $$\label{mum} f^*(\nu_S)=(-1)^d \nu_T$$ whenever $f\colon T\to S$ is a smooth morphism of relative dimension $d$. It follows easily from this that every stack $S$, locally of finite type over ${\mathbb C}$, has an associated locally constructible function $$\nu_S\colon S\to{\mathbb{Z}}$$ defined uniquely by the condition that also holds for smooth morphisms of stacks.
It is clear that the constant function $$\mathbf{1}\colon {\mathcal{M}}\to {\mathbb{Z}}$$ satisfies the conditions of Theorem \[se\] with the sign $\sigma=+1$. Regarding the Behrend function, Joyce and Song [@JS Theorem 5.9] proved the following wonderful result. For their proof (which uses gauge-theoretic methods) it is essential that our base variety $M$ is proper and that we are working over ${\mathbb C}$.
\[see\] The Behrend function $$\nu_{\mathcal{M}}\colon {\mathcal{M}}\to {\mathbb{Z}}$$ for the moduli stack ${\mathcal{M}}$ satisfies the conditions of Theorem \[se\] with the sign $\sigma=-1$.
Thus we have (at least) two integration maps: taking $\lambda=\mathbf{1}$ leads to invariants defined by unweighted Euler characteristics, whereas taking $\lambda=\nu_{\mathcal{M}}$ leads to Donaldson-Thomas invariants.
Fibres of the convolution map
=============================
In this section we give a careful analysis of the fibres of the morphism of stacks $(a_1,a_2)$ appearing in the definition of the convolution product. This will be the main tool in the proofs of Theorems \[fi\] and \[se\] we give in the next section. We again adopt the abuse of notation for pullbacks of families of sheaves explained in the preamble to Section 4.
Universal extensions
--------------------
Suppose $S$ is an affine scheme and $E_1$ and $E_2$ are coherent sheaves on $S\times M$, flat over $S$. Let ${\operatorname{Aff/{S}}}$ denote the category of affine schemes over $S$. Define a functor $$\Phi^k_S(E_1,E_2) \colon {\operatorname{Aff/{S}}} \to \operatorname{Ab}$$ by sending an object $f\colon T\to S$ to the abelian group $$\Phi^k_S(E_1,E_2)(f)={\operatorname{Ext}}^k_{T\times M} (f^*(E_1),f^*(E_2)).$$ The image of a morphism$$\xymatrix@C=.8em{
U\ar[dr]_{g}\ar[rr]^{h} && T\ar[dl]^{f}\\
&S }$$ in ${\operatorname{Aff/{S}}}$ is defined using the canonical map $$h^*\colon {\operatorname{Ext}}^k_{T\times M} (f^*(E_1),f^*(E_2))\to
{\operatorname{Ext}}^k_{U\times M}(h^*f^*(E_1),h^*f^*(E_2))$$ together with the canonical isomorphisms $${\operatorname{can}}\colon g^*(E_i) {\cong}h^* f^* (E_i).$$ To check that this does indeed define a functor one needs to apply the uniqueness properties of pullback in the usual way (see for example [@vis Section 3.2.1]). Consider the object $${\operatorname{\mathbf{R}\mathcal{H} om}}_{{\mathcal O}_S} (E_1,E_2)={\mathbf R}\pi_{S,*}{\operatorname{\mathbf{R}\mathcal{H} om}}_{{\mathcal O}_{S\times M}}(E_1,E_2) \in {{D}}{\operatorname{Coh}}(S).$$ For each $k{\geqslant}0$ we set $${\operatorname{Ext}}^k_{{\mathcal O}_S} (E_1,E_2):=H^k({\operatorname{\mathbf{R}\mathcal{H} om}}_{{\mathcal O}_S} (E_1,E_2))\in {\operatorname{Coh}}(S).$$ We shall say that $E_1$ and $E_2$ have constant ${\operatorname{Ext}}$ groups if these sheaves are all locally-free.
\[stanage\] Suppose $E_1$ and $E_2$ are $S$-flat coherent sheaves on $S\times M$ with constant ${\operatorname{Ext}}$ groups. Then the functor $$\Phi^k_S(E_1,E_2) \colon {\operatorname{Aff/{S}}} \to
\operatorname{Ab}$$ defined above is represented by the vector bundle $V^k(S)$ over $S$ corresponding to the locally-free sheaf ${\operatorname{Ext}}^k_{{\mathcal O}_S}
(E_1,E_2)$.
If $f\colon T \to S$ is a morphism of schemes then by flat base-change, $${\operatorname{\mathbf{R}\mathcal{H} om}}_{{\mathcal O}_T}
(f^*(E_1),f^*(E_2)){\cong}\mathbf{L} f^* \circ {\operatorname{\mathbf{R}\mathcal{H} om}}_{{\mathcal O}_S}
(E_1,E_2).$$ If $E_1$ and $E_2$ have constant ${\operatorname{Ext}}$ groups it follows that $${\operatorname{Ext}}^k_{{\mathcal O}_T} (f^*(E_1),f^*(E_2)){\cong}f^*{\operatorname{Ext}}^k_{{\mathcal O}_S} (E_1,E_2).$$ There is also an identity $${\operatorname{\mathbf{R} Hom}}_{T\times M}(f^*(E_1),f^*(E_2)){\cong}{\mathbf R}\Gamma
\circ {\operatorname{\mathbf{R} Hom}}_{{\mathcal O}_T}(f^*(E_1),f^*(E_2)).$$ If $T$ is affine it follows that $$\Phi^k_S(E_1,E_2)(f){\cong}\Gamma(T,f^* ({\operatorname{Ext}}^k_{{\mathcal O}_S}
(E_1,E_2))){\cong}{\operatorname{Map}}_S(T,V^k(S)).$$ These isomorphisms commute with pullback and hence define an isomorphism of functors.
The convolution morphism
------------------------
The main result of this section is as follows.
\[tom\] Let $X_1$ and $X_2$ be varieties with morphisms $$f_1\colon X_1 \to {\mathcal{M}},\qquad f_2\colon X_2\to{\mathcal{M}},$$ and let $E_i\in {\operatorname{Coh}}(X_i\times M)$ be the corresponding families of sheaves on $M$. Then we can stratify $X_1\times X_2$ by locally-closed affine subvarieties $$S\subset X_1\times X_2$$ with the following property. For each point $s\in S({\mathbb C})$ the space $$V^k(s)={\operatorname{Ext}}^k_M(E_2|_{\{s\}\times M}, E_1|_{\{s\}\times
M})$$ has a fixed dimension $d_k(S)$, and if we form the Cartesian squares $$\begin{CD}
Z_S &@>>> &Z & @>h>> &{\mathcal{M}}^{(2)} \\
@VuVV && @VtVV &&@VV(a_1,a_2)V \\
S &@>>> &X_1\times X_2 &@>f_1\times f_2>>
&{\mathcal{M}}\times{\mathcal{M}}\end{CD}$$then $$Z_S{\cong}S\times [{\mathbb C}^{d_1(S)}/{\mathbb C}^{d_0(S)}],$$ where the vector space ${\mathbb C}^{d_0(S)}$ acts trivially, and the map $u$ is the obvious projection.
By the existence of flattening stratifications, we can stratify $X_1\times X_2$ by locally-closed subschemes $S$ such that the restrictions of the families $E_1$ and $E_2$ have constant ${\operatorname{Ext}}$ groups. To ease the notation set $A=E_1$, $B=E_2$ and $$V^k(S)={\operatorname{Ext}}^k_{{\mathcal O}_S}(B,A)$$ considered as a vector bundle over $S$. The projection morphism $$p\colon V^k(S)\to S$$ defines an abelian group scheme over $S$. We will show that $$\label{object} Z_S{\cong}[V^1(S)/V^0(S)],$$ where the action of $V^0(S)$ on $V^1(S)$ is the trivial one. This will be enough because refining the stratification if necessary we can assume that each of the bundles $V^i(S)$ is trivial with fibre ${\mathbb C}^{d_i(S)}$. Then there are obvious isomorphisms $$[V^1(S)/V^0(S)] {\cong}[S\times
{\mathbb C}^{d_1(S)}/ S \times {\mathbb C}^{d_0(S)}]{\cong}S \times
[{\mathbb C}^{d_1(S)}/{\mathbb C}^{d_0(S)}].$$
Using Lemma \[headache\] one derives the following description of the stack $Z_S$. The objects over a scheme $T$ consist of a morphism $f\colon T \to S$ and a short exact sequence of $T$-flat sheaves on $T\times M$ of the form $$\label{formm} 0{\longrightarrow}f^*(A) {\xrightarrow{\ \alpha\ }} E
{\xrightarrow{\ \beta\ }} f^*(B){\longrightarrow}0.$$ Suppose given another such map $g\colon U\to S$ and a sequence $$0{\longrightarrow}g^*(A) {\xrightarrow{\ \gamma\ }} F {\xrightarrow{\ \delta\ }} g^*(B){\longrightarrow}0.$$ Then a morphism between these two objects in $Z_S$ is a commuting diagram of schemes $$\xymatrix@C=.8em{
U\ar[dr]_{g}\ar[rr]^{h} && T\ar[dl]^{f}\\
&S }$$ and an isomorphism of sheaves $\theta\colon h^*(E) \to F$ such that the diagram $$\begin{CD}\label{c}
0 &@>>>& h^*f^*(A)& @>h^*(\alpha)>> &h^*(E) &@>h^*(\beta)>> &h^*f^*(B) &@>>> &0 \\
&&&&@V{{\operatorname{can}}}VV && @V{\theta}VV && @V{{\operatorname{can}}}VV \\
0& @>>>& g^*(A) &@>\gamma>> &F &@>\delta>> &g^*(B) &@>>>
&0\end{CD}$$ commutes. Here ${\operatorname{can}}$ denotes the canonical isomorphism. Lemma \[stanage\] implies that there is a universal extension class $$\eta\in {\operatorname{Ext}}^1_{V^1(S)\times M} (p^*(B),p^*(A)).$$ Choose a corresponding short exact sequence $$\label{morekids}0{\longrightarrow}p^*(A) {\xrightarrow{\ \gamma\ }} F {\xrightarrow{\ \delta\ }} p^*(B)
{\longrightarrow}0.$$ This defines an object of $Z_S(V^1(S))$ and hence a morphism of stacks $$q\colon V^1(S) \to Z_S.$$ Consider the fibre product $$W=V^1(S)\times_{Z_S}
V^1(S).$$ For each scheme $T$ the groupoid $W(T)$ is a set, so we can identify $W$ with the corresponding functor $$W \colon {\operatorname{Sch}}/{\mathbb C}\to \operatorname{Set}.$$ It will be enough to show that the functor $W$ is isomorphic to the functor $$\Phi^0_{V^1(S)} (p^*(B),p^*(A))$$ and that there is a Cartesian diagram of stacks $$\begin{CD} V^1(S)\times_S V^0(S) &@>\pi_1>>& V^1(S) \\
@V\pi_1VV &&@VVqV \\
V^1(S) &@>q>> &Z_S\end{CD}$$ Then $Z_S$ is isomorphic to the quotient stack corresponding to the trivial action of $V^0(S)$ on $V^1(S)$ as claimed. By the universal property of $V^1(S)$ a morphism $a\colon T \to V^1(S)$ corresponds to a morphism $f \colon T \to S$ together with an an extension class $$\zeta\in {\operatorname{Ext}}^1_{S\times M} (f^*(B),f^*(A)).$$ Under this correspondence $f=p\circ a$. The composite morphism $$q\circ a\colon T \to Z_S$$ then corresponds to the object of $Z_S(T)$ defined by the morphism $f$ and the short exact sequence $$0{\longrightarrow}f^*(A) {\longrightarrow}a^*(F) {\longrightarrow}f^*(B) {\longrightarrow}0$$ obtained by applying $a^*$ to and composing with the canonical isomorphisms. Suppose $b\colon T \to V^1(S)$ is another morphism corresponding to a morphism $g\colon T \to S$ and an extension class $$\eta\in
{\operatorname{Ext}}^1_{S\times M} (g^*(B),g^*(A)).$$ Then there is an isomorphism of the corresponding objects of $Z_S(T)$ lying over the identity of $T$ precisely if $f=g$ and there is an isomorphism of short exact sequences $$\begin{CD}
0 &@>>>& f^*(A)& @>>> &a^*(F) &@>>> &f^*(B) &@>>> &0 \\
&&&&@V{=}VV && @VVV && @V{=}VV \\
0& @>>>& f^*(A) &@>>> &b^*(F) &@>>> &f^*(B) &@>>>
&0.\end{CD}$$ In particular, it follows that $\zeta=\eta$, and hence by the universal property of $V^1(S)$ one has $a=b$. Moreover the set of possible isomorphisms is in bijection with $${\operatorname{Hom}}_{T\times M} (f^*(B),f^*(A) ).$$ Thus the elements of the set $W(T)$ consist of a morphism $a\colon
T \to V^1(S)$ and an element of $$\Phi^0_{V^1(S)} (p^*(B),p^*(A))(a).$$ We leave it to the reader to check that this correspondence commutes with pullback and hence defines an isomorphism of functors.
Proofs of Theorems \[fi\] and \[se\]
====================================
Using Proposition \[tom\] we can now give the proofs of Theorems \[fi\] and \[se\].
Proof of Theorem \[fi\] {#firstproof}
-----------------------
Consider two elements $$a_i= [X_i{\xrightarrow{\ f_i\ }} {\mathcal{M}}]\in {\operatorname{H_{{reg}}}}({\mathcal A}),
\quad i=1,2,$$ with $X_1$ and $X_2$ varieties. Let $E_i$ be the family of coherent sheaves on $X_i$ corresponding to the map $f_i$. Stratify $X_1\times X_2$ by locally-closed subvarieties $S_j$ as in Proposition \[tom\]. In particular, the vector spaces $$V^k(x_1,x_2)={\operatorname{Ext}}^k_M\big(E_2|_{\{x_2\}\times M}, E_1|_{\{x_1\}\times M}\big),\qquad (x_1,x_2)\in S_j,$$ have constant dimension $d_k(S_j)$. Consider the diagram $$\begin{CD}
Z_j &@>>> &Z &@>q>> {\mathcal{M}}^{(2)} @>h>> {\mathcal{M}}\\
@Vt_j VV && @VtVV &@VV(a_1,a_2)V \\
S_j &@>>> &X_1\times X_2 &@>f_1\times f_2>> {\mathcal{M}}\times{\mathcal{M}}\end{CD}$$ According to Proposition \[tom\] one has $$Z_j{\cong}[Q_j/{\mathbb C}^{d_0(S_j)}]$$ where $Q_j=V^1(S_j)$ is the total space of a trivial vector bundle over $S_j$ with fibre $V^1(x_1,x_2)$ over a point $(x_1,x_2)$. Since the $Z_j$ stratify $Z$ it follows that $$a_1*a_2=[Z {\xrightarrow{\ b\circ h\ }} {\mathcal{M}}]= \sum_j\, {\mathbb{L}}^{-d_0(S_j)} [Q_j
{\xrightarrow{\ g_j\ }} {\mathcal{M}}],$$ which is regular. Here the morphism $g_j$ is induced by the universal extension of the families $E_1$ and $E_2$ over $S_j$. For the second claim split $Q_j$ into the zero-section and its complement. The latter is a ${\mathbb C}^*$ bundle over the associated projective bundle, and it is easy to see that the morphism to ${\mathcal{M}}$ factors via this map. Thus $$\label{domo} a_1*a_2=\sum_j
{\mathbb{L}}^{-d_0(S_j)}\bigg([S_j{\xrightarrow{\ k\ }}{\mathcal{M}}] + ({\mathbb{L}}-1) [{\operatorname{\mathbb P}}( Q_j){\xrightarrow{\ g_j\ }}
{\mathcal{M}}]\bigg),$$where the morphism $k$ is induced by the direct sum of the families $E_1$ and $E_2$. We therefore obtain $$\label{bo}
a_1*a_2 = \sum_j [S_j{\xrightarrow{\ k\ }} {\mathcal{M}}] = [X_1 \times X_2{\xrightarrow{\ k\ }} {\mathcal{M}}] \mod
({\mathbb{L}}-1).$$ Clearly we would get the same answer if we calculated $a_2*a_1$.
Proof of Theorem \[se\] {#secondproof}
-----------------------
We first check that the map $I$ is well-defined; it is then clearly unique. Stratify ${\mathcal{M}}$ by locally-closed substacks ${\mathcal{M}}_\tau$ such that $\lambda$ has constant value $\lambda(\tau)$ on ${\mathcal{M}}_\tau$. There are projection maps $$\pi_\tau\colon {K({\operatorname{St/{{\mathcal{M}}}}})}\to {K({\operatorname{St/{{\mathcal{M}}}}})}$$ defined by taking the fibre product with the inclusion ${\mathcal{M}}_\tau\subset {\mathcal{M}}$. For any $a\in{K({\operatorname{St/{{\mathcal{M}}}}})}$ there is a canonical decomposition $$a=\sum_{i} \pi_\tau(a),$$ where only finitely many of the terms are nonzero. If $a\in{K({\operatorname{St/{{\mathcal{M}}}}})}$ is regular so are each of the $\pi_\tau(a)$. On the other hand if $b\in{K({\operatorname{St/{{\mathcal{M}}}}})}$ is regular, we can project to an element of ${K({\operatorname{St/{{\mathbb C}}}})}$, and using Lemma \[grimer\] and Lemma \[toen\] obtain a well-defined Euler characteristic $\chi(b)\in{\mathbb{Z}}$. Thus we can define a group homomorphism $I$ by the formula $$I(a)=\sum_i \lambda(\tau) \chi(\pi_\tau(a))$$ and this will clearly have the property stated in the Theorem.
Now take notation as in the proof of Theorem \[fi\]. By Serre duality, we have $$\label{serre} V^k(x_1,x_2)=V^{3-k}(x_2,x_1)^*.$$ Let $\hat{Q}_j=V^2(S_j)$ be the bundle over $S_j$ whose fibre at $(x_1,x_2)$ is $V^1(x_1,x_2)$. Let $$g_j\colon Q_j \to {\mathcal{M}}, \quad
\hat{g}_j\colon \hat{Q}_j\to {\mathcal{M}},$$ be the morphisms induced by taking the universal extensions of the families $E_1$ and $E_2$ over $S_j$.
We can assume that $f_i$ maps into ${\mathcal{M}}_{\alpha_i}\subset {\mathcal{M}}$ and that $f_i^*(\lambda)$ is equal to the constant function with value $n_i$. Then $$I(a_i)=n_i \cdot \chi(X_i)\cdot x^{\alpha_i}.$$ Since $\chi({\mathbb{L}})=1$, the expression shows that $$I( a_1*a_2)= \chi(X_1\times X_2,k^*(\lambda))\cdot x^{\alpha_1+\alpha_2}.$$ Using the first assumption , we therefore obtain $$I(a_1* a_2)=\sigma^{\chi(\alpha_1,\alpha_2)}\cdot n_1 n_2 \cdot \chi(X_1\times
X_2)= I(a_1)
* I(a_2).$$
To compute the Poisson bracket we use again. Applying with $k=0$, and noting that $$\frac{{\mathbb{L}}^{n}-{\mathbb{L}}^{m}}{{\mathbb{L}}-1}= n-m \mod ({\mathbb{L}}-1),$$ we obtain $$\begin{aligned}
\{a_1,a_2\}=\sum_j \bigg(
(d_3(S_j)-d_0(S_j))&\cdot [S_j{\xrightarrow{\ k\ }}{\mathcal{M}}] \\&+ [{\operatorname{\mathbb P}}(
Q_j){\xrightarrow{\ g_j\ }}
{\mathcal{M}}]-[{\operatorname{\mathbb P}}(\hat{Q}_j){\xrightarrow{\ \hat{g}_j\ }}{\mathcal{M}}]\bigg).\end{aligned}$$ To compute the Euler characteristic of a constructible function over ${\operatorname{\mathbb P}}(Q_j)$ it follows from [@mac Prop. 1] (see also [@verdier Cor. 5.1]) that we can first integrate over the fibres of the projection $${\operatorname{\mathbb P}}(Q_j)\to Q_j$$ and then integrate the resulting constructible function on the base $Q_j$. The second assumption together with $\chi({\operatorname{\mathbb P}}({\mathbb C}^n))=n$ therefore gives $$\chi({\operatorname{\mathbb P}}(Q_j),g_j^*(\lambda)) - \chi({\operatorname{\mathbb P}}(\hat{Q}_j),
\hat{g}_j^*(\lambda))
=d_1(S_j)\cdot\chi(S_j, k^*(\lambda)) -d_2(S_j)\cdot\chi(S_j,
k^*(\lambda)),$$ and so $$\begin{aligned}
I(\{a_1,a_2\})&=\chi(\alpha_1,\alpha_2)\cdot\chi(X_1\times X_2,k^*(\lambda))
\cdot x^{\alpha_1+\alpha_2}\\
&=\sigma^{\chi(\alpha_1,\alpha_2)} \cdot n_1 n_2 \cdot
\chi(\alpha_1,\alpha_2)\cdot \chi(X_1\times X_2)\cdot
x^{\alpha_1+\alpha_2} =\{I(a_1),I(a_2)\}\end{aligned}$$ as required.
Fibre products of stacks
========================
Here we collect some well known material on fibre products of stacks.
Fibre product
-------------
Suppose given morphisms of stacks $$f\colon X\to Z,\quad g\colon Y\to Z.$$ Recall the definition of the fibre product stack $X\times_Z Y$ and the 2-commutative diagram $$\begin{CD} X\times_Z Y &@>\pi_Y>> &Y \\
@V\pi_XVV && @VVgV \\
X &@>f>> &Z \\\end{CD}$$ The objects of $X\times_Z Y$ are triples $(x,y,\theta)$, where $x$ and $y$ are objects of $X$ and $Y$ over the same scheme $S$, and $\theta\colon f(x)\to g(y)$ is an isomorphism in the groupoid $Z(S)$. A morphism $$(\alpha,\beta)\colon (x,y,\theta)\to (x',y',\theta')$$ consists of morphisms $\alpha\colon x\to x'$ in $X$ and $\beta\colon y\to y'$ in $Y$ such that the diagram $$\begin{CD} f(x) &@>\theta>> &g(y) \\
@Vf(\alpha)VV && @VVg(\beta)V \\
f(x') &@>\theta'>> &g(y') \\\end{CD}$$ commutes. The morphisms $\pi_X$ and $\pi_Y$ are defined in the obvious way.
We will call a morphism of stacks $f\colon X \to Z$ an iso-fibration if the following property holds. Suppose $S$ is a scheme and $$\theta\colon a \to b$$ is an isomorphism in the groupoid $Z(S)$. Suppose that there is an $a'\in X(S)$ such that $f(a')=a$. Then there is an isomorphism $$\theta'\colon a'\to b'$$ in $X(S)$ such that $f(\theta')=\theta$. The following easy Lemma will simplify many computations of such fibre products.
\[headache\] With notation as above, define a full subcategory $W\subset X\times_Z Y$ whose objects are triples $(x,y,\theta)$ as above for which there is an object $z\in Z$ with $$f(x)=z=g(y), \quad \theta={\operatorname{id}}_z.$$ Suppose one of the morphisms $f$ or $g$ is an iso-fibration. Then the inclusion functor $W\to X\times_Z Y$ is an equivalence of categories.
For definiteness suppose that it is $f$ that is an iso-fibration. Given an object $$(x,y,\theta)\in X\times_Z Y$$ take an object $x'\in X$ such that $f(x')=g(y)$ and a morphism $\alpha\colon x\to x'$ such that $f(\alpha)=\theta$. Then $$(\alpha,{\operatorname{id}}_y)\colon (x,y,\theta)\to (x',y,{\operatorname{id}})$$ defines an isomorphism. Thus $(x,y,\theta)$ is isomorphic to an object of the subcategory $W$.
Cartesian diagrams
------------------
A diagram of stacks $$\begin{CD} W &@>h>> &Y \\
@VjVV && @VVgV \\
X &@>f>> &Z \\\end{CD}$$ is called 2-Cartesian if there is an equivalence of stacks $$t\colon W \to X \times_Z Y$$ such that $j=\pi_X\circ t$ and $h=\pi_Y \circ t$.
\[nonse\] Consider a 2-commutative diagram of the form $$\begin{CD} V &@>>> &W &@>>> &Y\\
@VVV && @VVV && @VVV\\
U &@>>> &X &@>>> &Z\\\end{CD}$$ and assume the right-hand small square to be 2-Cartesian. Then the left hand small square is 2-Cartesian iff the big square is 2-Cartesian.
This is a standard fact, and we leave the proof to the reader.
We used the following easy consequence many times in Section 4.
\[nonsense\] Consider the following two diagrams of morphisms of stacks $$\begin{CD}
W @>f>> {Y} &\qquad\qquad\qquad& {W} @>f>> Y \\
@VgVV @VVhV @V(g,k\circ f)VV @VV(h,k)V \\
X @>j>> {Z} &\qquad\qquad\qquad& {{X} \times {T}} @>(j,1)>> Z\times T \\
\end{CD}$$ Then if one is 2-Cartesian, so is the other.
This follows from Lemma \[nonse\] once one knows that the square $$\begin{CD} X\times T &@>(f,1)>> &Z\times T \\
@V\pi_XVV && @VV\pi_ZV \\
X &@>f>> &Z \\\end{CD}$$ is 2-Cartesian. This follows from the diagram $$\begin{CD} X \times T&@>f\times 1>> &Z\times T &@>\pi_T>> &T\\
@V\pi_XVV && @V\pi_ZVV && @VVV\\
X &@>f>> &Z&@>>> &{{\scriptscriptstyle\bullet}}\\\end{CD}$$by another application of Lemma \[nonse\].
[101]{} K. Behrend, Donaldson-Thomas invariants via microlocal geometry, preprint arXiv math 0507523.
F. Bittner, The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140 (2004), no. 4, 1011–1032.
T. Bridgeland, Hall algebras and curve-counting invariants, preprint arXiv 1002.4374.
C. Chevalley, A. Grothendieck and J.P. Serre, Anneaux de Chow et applications, S[é]{}minaire C. Chevalley, 2e ann[é]{}e, Ecole Normale Sup[é]{}rieure, Secr[é]{}tariat math[é]{}matique, Paris 1958.
C. Chevalley, Une d[é]{}monstration d’un th[é]{}or[è]{}me sur les groupes alg[é]{}briques, J. Math. Pures Appl. (9) 39 (1960) 307–317.
B. Conrad, A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1–18.
T. Gomez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 1, 1–31.
D. Joyce, Constructible functions on Artin stacks, J. London Math. Soc. (2) 74 (2006), no. 3, 583–606. D. Joyce, Motivic invariants of Artin stacks and stack functions, Q. J. Math. 58 (2007), no. 3, 345–392. D. Joyce, Configurations in abelian categories. I Basic properties and moduli stacks. Adv. Math. 203 (2006), no. 1, 194–255. D. Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635–706.
D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, preprint arXiv 0810.5645.
D. Knutson, Algebraic spaces, Lect. Notes in Math., Vol. 203. Springer-Verlag, Berlin-New York, 1971. vi+261 pp.
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint arxiv 0811.2435.
A. Kresch, Cycle groups for Artin stacks. Invent. Math. 138 (1999), no. 3, 495–536.
G. Laumon and L. Moret-Bailly, Champs alg[é]{}briques, Ergegnisse der Math. und ihrer Grenzgebiete. 3. Folge, 39 (Springer Verlag) (2000)
E. Looijenga, Motivic measures. S[é]{}minaire Bourbaki, Vol. 1999/2000. Astérisque No. 276 (2002), 267–297.
R. MacPherson, Chern classes for singular algebraic varieties. Ann. of Math. (2) 100 (1974), 423–432.
D. Mumford, Complex projective varieties, Algebraic geometry. I. Complex projective varieties. Grundlehren der Math, No. 221. Springer-Verlag, Berlin-New York, 1976. x+186 pp.
M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), no. 2, 349–368.
B. To[ë]{}n, Grothendieck rings of Artin $n$-stacks, preprint arXiv math/0509098.
J.-L. Verdier, Stratifications de Whitney et th[é]{}or[è]{}me de Bertini-Sard. Invent. Math. 36 (1976), 295–312.
A. Vistoli, Grothendieck topologies, fibered categories and descent theory. Fundamental algebraic geometry, 1–104, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005.
[^1]: The author is grateful to Andrew Kresch for explaining this argument.
[^2]: Here and in the rest of this section we suppress an application of the K[ü]{}nneth map $${K({\operatorname{St/{{\mathcal{M}}}}})}{\otimes}{K({\operatorname{St/{{\mathcal{M}}}}})} \to {K({\operatorname{St/{{\mathcal{M}}\times{\mathcal{M}}}}})}$$ from the notation.
|
---
abstract: 'A new approach to the dynamics of the universe based on work by Ó Murchadha, Foster, Anderson and the author is presented. The only kinematics presupposed is the spatial geometry needed to define configuration spaces in purely relational terms. A new formulation of the relativity principle based on Poincaré’s analysis of the problem of absolute and relative motion (Mach’s principle) is given. The entire dynamics is based on shape and nothing else. It leads to much stronger predictions than standard Newtonian theory. For the dynamics of Riemannian 3-geometries on which matter fields also evolve, implementation of the new relativity principle establishes unexpected links between special relativity, general relativity and the gauge principle. They all emerge together as a self-consistent complex from a unified and completely relational approach to dynamics. A connection between time and scale invariance is established. In particular, the representation of general relativity as evolution of the shape of space leads to a unique dynamical definition of simultaneity. This opens up the prospect of a solution of the problem of time in quantum gravity on the basis of a fundamental dynamical principle.'
---
**[DYNAMICS OF PURE SHAPE, RELATIVITY AND THE PROBLEM OF TIME]{}\
\
**
$^1$ College Farm, South Newington, Banbury, Oxon, OX15 4JG, UK
Electronic address:[$^1$ julian@platonia.com]{}
Introduction
============
In this paper, I wish to discuss the foundations of cosmology and our notion of time. The goal is to contribute to the creation of a quantum theory of the universe. I shall draw attention to some conceptual issues that in my opinion have not hitherto been adequately discussed. Since this is a contribution to an interdisciplinary workshop, I shall keep the discussion as simple as possible. In fact, the model that I shall present, which treats non-relativistic particles in Euclidean space, hardly seems realistic. However, it contains the simplest implementation of a dynamical variational principle that can also be applied in general relativity, as I shall outline in more qualitative terms. It is the principle that I wish to explain. Moreover, the problems to be addressed are so challenging, there is much to be said for attacking them initially in the simplest possible situations. I also happen to believe that anyone competent in quantum mechanics should be interested in the issues raised by the model independently of the quantum cosmological significance. The question is this: Can one do meaningful quantum mechanics with significantly less external kinematic structure than is currently employed in quantum theory? This is a topic ideally suited to this interdisciplinary volume.
Some historical background will be helpful. In order to formulate his laws of motion, Newton introduced a rigid external framework: absolute space and time. This framework was simply taken over in its entirety by the creators of quantum mechanics. It was only somewhat modified to accommodate quantum field theory, and a fixed framework is still deeply embedded in that theory. However, ten years before the creation of quantum mechanics, Einstein had created the general theory of relativity. There is universal agreement that Einstein to a very large degree abolished Newton’s absolute framework. There is less agreement about what, if anything, replaced it in the classical theory and what kind of framework is appropriate for a quantum theory of gravity. This, I believe, is the main reason why people are still seeking the foundations of quantum cosmology. I want to argue for a new and clearer formulation of the relativity principle. This will lead me to the notion of *dynamics of pure shape*.
It will be helpful to distinguish two different notions of relativity. There is, first, the intuitive idea that position can only be defined relative to observable objects. Suppose we were to contemplate a universe consisting of $N$ point particles in Euclidean space with separations $r_{ij}$ between them. The opponents of Newton’s absolute space and time, above all Leibniz [@Leibniz] and Mach [@Mach], argued that dynamics should be directly formulated in terms of the $r_{ij}$. Let me call this *kinematic relativity*. In 1902, Poincaré [@Poincare1] pointed out that in fact Newtonian point-particle dynamics can always be formulated in terms of the $r_{ij}$, but that then the structure of its initial-value problem is changed compared with the formulation in absolute space (or, in modern terms, an inertial system). This is such an important issue that it needs to be spelled out. If we are given initial positions $\textbf{x}_i$ and initial velocities $\dot\textbf{x}_i$ in an inertial system (together with the masses $m_{i}$ and the force law), then Newton’s laws determine the past and future motions uniquely. The situation is characteristically different if one is given $r_{ij}$ and $\dot r_{ij}$. The fact is that such data contain no information at all about the overall rotation of the system. One cannot determine the angular momentum $\textbf M$ of the system, and for different $\textbf M$ very different evolutions result. Initial data that are identical from the point of view of kinematic relativity give rise to different evolutions. Moreover the defect has an odd structure. Suppose $N$ is large. In three-dimensional Euclidean space, the $r_{ij}$ contain $3N-6$ independent data, and their time derivatives contain the same number. This number is the order of a million for a globular cluster. Just three more numbers are needed if the evolution is to be uniquely determined. They could be three of the second time derivatives. But why does one need three and not $3n-6$? Poincaré argued, persuasively in my view, that the only valid objection to Newton’s use of absolute space and time resided in this curious need to specify a small number of extra data. He said that if one could formulate a relational dynamics (i.e., one containing only $r_{ij}$ and its time derivatives) free of this defect, the problem of absolute motion would be solved. Poincaré made no attempt to find such a dynamics. Moreover, he pointed out that the empirical evidence seemed to show conclusively that nature did not work in this way.
I shall argue that Poincaré’s analysis is very sound and that empirically adequate theories satisfying a criterion along the lines he proposed can be formulated. However, it will be helpful to push his analysis somewhat further, which I shall do in the next section. For the moment, let me merely note that if one takes kinematic relativity seriously one will wish to formulate a dynamics whose initial-value problem satisfies a well-defined criterion. For reasons that will soon become apparent, let me call this the *constructive approach* or *Poincaré’s relativity principle*. It can be contrasted with the approach that was adopted by Einstein in creating both special and general relativity and is known as the *principle approach*. It grew out of a generalization of *Galilean relativity*, which Einstein transformed into the restricted relativity principle. In order to construct a theory in which uniform motion could not be detected, Einstein made no attempt to formulate a complete theory of particles interacting with Maxwell’s electromagnetic field. Instead, he postulated the existence of a family of distinguished (inertial) frames all in uniform translational motion relative to each other and required the laws of nature to take the identical form in all of them. The dramatic results that he obtained came from combination of this relativity principle with his postulate about the behavior of light. As he explained in his Autobiographical Notes [@AutoNotes] (see also [@Barbour], on which this discussion is based), he was encouraged to adopt this approach because of the success of phenomenological thermodynamics based based on ‘impotence’ principles: the impossibility of constructing perpetual motion machines of the first and second kind. The impotence in his case was the impossibility of detecting uniform motion through the ether (or absolute space) by means of processes that unfold in a closed system.
In 1907, Einstein realized that the equivalence principle enabled him to extend the restricted relativity principle to include another ‘impotence’ – the inability to detect uniform acceleration. This insight was decisive. Some years earlier, Einstein had read Mach’s critique of Newton’s absolute space and time and was extremely keen to reformulate dynamics along the broad lines advocated by Mach. He wanted to show that absolute space did not correspond to anything in reality – that it could not be revealed by any experiment. Special relativity had shown that uniform motion – relative to the ether or to absolute space – could not be detected by any physical process. The equivalence principle suggested to him that it might be possible to extend his relativity principle further. If he could extend it so far as to show that the laws of nature could be expressed in identical form in all conceivable frames of reference, this requirement of general covariance would “\[take\] away from space and time the last remnant of physical objectivity”[@Einstein16]. He would have achieved his Machian aim. Unfortunately, within two years Einstein had been forced by a critique of Kretschmann [@Kretschmann] to acknowledge that any physical theory must, if it is to have any content, be expressible in generally covariant form. He argued [@Einstein18] that the principle nevertheless had great heuristic value. One should seek only those theories that are *simple* when expressed in generally covariant form. However, Einstein gave no definition of simplicity. Since then, and especially as a result of quantization attempts, there has been a vast amount of inconclusive discussion about the significance of general covariance [@BarPlanck] and its implications for quantization. A point worth noting is that Einstein treated space and time as a single unit and considered general coordinate transformations on a four-dimensional manifold. The distinction between space and time and the manner in which they are to be treated is very largely erased.
In several recent papers [@BOF; @AB; @SIG; @ABFO], my collaborators and I have developed what we call *the 3-space approach*. It is based on a generalization of Poincaré’s relativity principle and casts new light on this issue. Above all, it replaces Einstein’s vague simplicity requirement by a well-defined constructive principle based on the amount of data needed to formulate initial-value problems. In addition, space and time are treated in completely different ways. This might seem to be a retrogressive step, but actually the approach not only achieves everything that Einstein did by presupposing a four-dimensional unity of space and time but even more. In a real sense it explains why there is a universal light cone (which Einstein presupposed) and why the gauge principle holds. The details of this work goes beyond the scope of the present paper, though I will indicate how these results are obtained in the final section. Readers wishing for full details are referred to the original papers. In this paper, I merely wish to get across the basic ideas and draw attention to some of the interesting possibilities that arise.
Basic Ideas
===========
The 3-space approach is a natural modification of the basic scheme employed in the variational principles of mechanics [@Lanczos]. The main difference is that the 3-space approach uses less kinematic structure. Let us first consider the notion of configuration space. In the Newtonian $N$-body problem, this is defined relative to an inertial system, which serves two purposes. First, it provides a definition of *equilocality* at different instants of time: relative to the system, one can say that a particle is at the same position at two different times. Displacements are then well defined. This is essentially the reason why Newton introduced absolute space. Second, an inertial system brings with it a notion of time difference. Given a notion of simultaneity, so that one can say that the system has some instantaneous configuration at a given instant, that still does not say ‘how much time there is’ between two different configurations in a history. This extra information is supplied with an inertial system.
Let us now see how much of this structure can be shed without making dynamics impossible. Suppose a universe consisting of $N$ point particles in Euclidean space. It is rather natural to assume that only the relative configuration counts as physical reality. One would like to say that all configurations that can be carried into exact congruence by Euclidean translations and rotations are the same physical configuration. This amounts to quotienting the Newtonian configuration space Q of $3N$ dimensions by these six symmetries (three translations, three rotations) to obtain the *relative configuration space* (RCS) Q$_{\scriptsize\textrm{RCS}}$ of $3N-6$ dimensions [@BB]. It is also very natural to go one step further and say that size is relative. Then two configurations that can be made congruent by the action of translations, rotations and dilatations are to be regarded as identical. The corresponding quotient space has $3N-7$ dimensions and may be called *shape space*: Q$_{\scriptsize\textrm{SS}}$.
A Newtonian dynamical history is a curve in Q traversed at a certain speed relative to some external measure of time. Now time must always be deduced from the motion of some object in the universe that is itself subject to the laws of nature. Therefore, if the system that we are considering is the entire universe, the information from which we deduce time must in fact already be encoded in the curve in Q. We shall see how this is done shortly. But this observation itself already suggests that, if we are considering the dynamics of the entire universe, it ought to be sufficient to set up a theory of *curves* in the configuration space and dispense with the idea that they are traversed at some speed. The simplest curves are those whose determining law is such that an initial point and initial direction suffice to determine the entire curve. They are geodesics. The key difference from Newtonian curves is that only the initial direction, not the initial direction and the speed in that direction, needs to be specified. This is a natural extension to Poincaré’s relativity principle. We seek laws that determine histories with the minimum number of initial data. Such laws are *maximally predictive*.
Going from curves traversed at speed to geodesics is one way of the two ways in which we can reduce the number of initial data. The second is to formulate Poincaré’s relativity principle on the smaller quotient spaces Q$_{\scriptsize\textrm{RCS}}$ and Q$_{\scriptsize\textrm{SS}}$. One of the main points I want to make in this paper is that there is a surprising interconnection between these two ways of reducing initial data. The first may be called the elimination of time and the second elimination of potentially redundant geometrical structure. We shall see that the elimination of time is not truly effective unless the employed geometrical structure is pared down to the absolute minimum. There is an unexpected connection between time and geometry, specifically scale invariance.
An overall characterization of the 3-space approach is here appropriate. The basic idea is to postulate the geometrical structure of space, which is assumed here to have three dimensions but in principle any dimension is possible. This paper will mainly be concerned with Euclidean space, though much more interesting possibilities arise if space is Riemannian. Because of length restrictions, I shall only be able to discuss them briefly and will concentrate on the Euclidean case in order to get the idea across. We assume that the space we consider is occupied by geometrical objects, which may either be point particles or fields. These objects and the space in which they reside define configurations and associated configuration spaces. The key point, as already explained, is that one obtains a hierarchy of configuration spaces by quotienting with respect to the symmetries inherent in the space that is presupposed. The mathematical existence of these quotient spaces seems to reflect a deep property of the world. They seem to be a necessary concomitant of the existence of spatial order. The results so far obtained in the 3-space approach suggest that the structure of the quotient spaces determines much more of the fundamental laws of classical physics than has hitherto been supposed. The point is that it is not easy to construct geodesic principles on quotient spaces. If they are to be consistent, they impose strong restrictions.
There exists more than one way to construct geodesic principles on quotient spaces, but one of them seems to be clearly distinguished compared with all the others on account of its geometrical nature. This is based on a principle that we call *best matching*. If we are to define geodesics on any space, we need to define a metric on it. We need to define a distance between any two pairs of neighboring points of the space. The points in our case are complete configurations of the system that we are considering. They necessarily have some small intrinsic difference. Now the basis of geometry is congruence. However, two intrinsically different configurations cannot be brought to exact congruence. Best matching of two such configurations is based on the idea of bringing them, in a well-defined sense, as close as possible to exact congruence and using the ‘mismatch’ from it to define the distance between them. Geodesics can then be defined with respect to this best-matching metric. As we shall see, the consistent application of this idea leads to interesting restrictions. Before proceeding with the formal development, let me give an idea of the nature of these restrictions.
The benchmark for considering them is standard Newtonian theory. Suppose we take a generic solution of the Newtonian $N$-body problem of celestial mechanics and successively ‘throw away information’. First, instead of giving the time at which the successive configurations are realized we can simply give their sequence. Next, we can omit the information, characterized by six numbers, that specify the overall position and orientation of the system. We can do this by giving only the inter-particle separations $r_{ij}$. Finally, we can omit the scale information (one number) contained in the $r_{ij}$. This is most conveniently done by normalizing them by the square root of the moment of inertia $I$ about the center of mass: $$I=\sum_i m_i \textbf{x}_i^2={1\over M}\sum_{i<j}m_im_jr_{ij}^2, M=\sum_i m_i.
\label{MOI}$$
The resulting information can be plotted as a curve in shape space. If this curve were a geodesic, it would be determined by specification of an initial point and initial direction in shape space. For the 3-body problem, shape space has two dimensions, so in this case one would need three numbers: two to specify the initial point, one to specify the initial direction. This is the ideal that must be met by a dynamics of pure shape. It turns out that a generic Newtonian solution needs no less than five further numbers to be fully specified. It is illuminating to consider what they are.
First, it should be noted that in shape space all dimensional information is lost. We have no knowledge of length scales or clock rates. Now the fundamental dynamical quantities in Newtonian theory such as energy, momentum, and angular momentum, as well as Newton’s gravitational constant G, depend on these scales, but only scale-invariant quantities can affect the form of the Newtonian curves projected down to shape space. Let us consider what they are. First, at any instant the angular momentum vector has a certain direction relative to the instantaneous configuration. Two numbers are associated with this information. Next, the Newtonian kinetic energy can be decomposed into a part associated with overall rotation, a part associated with change of shape, and a part associated with change of size. Two independent scale-invariant ratios can be formed from them. These are four of the five numbers. The final number is in many ways the most enigmatic. It is the instantaneous ratio $H=T/V$ of the kinetic energy $T$ to the potential energy $V$. Intuitively it exists in Newtonian theory because the external time makes it possible to convert displacements into velocities.
We shall see that the transition from Q to Q$_{\scriptsize\textrm{RCS}}$ ensures that the rotational motion associated with angular momentum no longer plays a role. This eliminates three of the above five numbers. However, two still remain. Rather remarkably, both are eliminated when we take the further step to a dynamics of pure shape on Q$_{\scriptsize\textrm{SS}}$. One of them measures the kinetic energy associated with change of size, and it is no surprise that this is eliminated in a dynamics of pure shape. But the other, related to the energy, seems intuitively to have something to do with time. After all, time and energy form a canonical pair in Hamiltonian dynamics [@Lanczos]. It is therefore surprising that a scaling requirement appears to have a bearing on time. I shall come back to this later.
In the next section, I shall discuss the formulation of geodesic principles in a way that highlights the difference from Newtonian theory. In Sec. 4, I shall explain the technique of best matching and in Sec. 5 show how Newtonian theory can be recovered to excellent accuracy from a scale-invariant theory. In Sec. 6, I shall indicate how these ideas can be applied to Riemannian geometry and fields and yield a new perspective on general relativity.
Jacobi’s Principle
==================
The first step to a dynamics of pure shape is the elimination of time by Jacobi’s principle [@Lanczos], which describes all Newtonian motions of one value $E$ of the total energy as geodesics on configuration space. Further discussion of the implications of Jacobi’s principle can be found in [@BOF; @CQG94; @EOT].
For $N$ particles of masses $m_{i}$ with potential $U(\textbf{x}_{1}, \dots , \textbf{x}_{N})$ and energy $E$, the Jacobi action is [@Lanczos] $$I_{\scriptsize\textrm{Jacobi}} = 2\int\sqrt{E -
U}\sqrt{\tilde T} \textrm{d}\lambda,
\label{Jacobi}$$ where $\lambda$ labels the points on trial curves and $ \tilde{T}
= \sum {m_{i} \over
2}{\textrm{d}{\textbf{x}}_{i}\over\textrm{d}\lambda}\cdot
{\textrm{d}{\textbf{x}}_{i}\over\textrm{d}\lambda}$ is the parametrized kinetic energy. The action (\[Jacobi\]) is timeless since the label $\lambda$ could be omitted and the mere displacements $d\textbf{x}_{i}$ employed, as is reflected in the invariance of $I_\textrm{\scriptsize{Jacobi}}$ under the reparametrization $$\lambda \rightarrow f(\lambda).\label{rep}$$
In fact, it is much more illuminating to write the Jacobi action in the form $$I_{\scriptsize\textrm{Jacobi}} = 2\int\sqrt{E -
U}\sqrt{T^*},\hspace{.5cm}T^*=\sum {m_{i} \over
2}\textrm{d}{\textbf{x}}_{i}\cdot
{\textrm{d}{\textbf{x}}_{i}},
\label{Jacobi*}$$ which makes its timeless nature obvious and dispenses with the label $\lambda$.
The characteristic square roots of $I_{\scriptsize\textrm{Jacobi}}$ fix the structure of the canonical momenta: $$\textbf{p}_{i} =
{\partial {\cal L} \over \partial(\textrm{d}\textbf{x}_{i}/\textrm{d}\lambda)} =
m_{i}\sqrt{E - U \over {\tilde T}} {d \textbf{x}_{i}
\over d \lambda},
\label{CanMom}$$ which, being homogeneous of degree zero in the velocities, satisfy the constraint [@Dirac] $$\sum{{{\textbf{p}}_{i}}\cdot
{{\textbf{p}}_{i}}\over 2m_{i}}-E+U=0.
\label{QuadCon}$$
The Euler–Lagrange equations are $${\textrm{d}\textbf{p}^{i} \over \textrm{d} \lambda}= {\partial
{\cal L} \over \partial \textbf{x}_{i}} = -\sqrt{{\tilde T} \over
E - U}{\partial U\over
\partial \textbf{x}_{i}} ,
\label{JacobiEL}$$ where $\lambda$ is still arbitrary. If we choose it such that $${{\tilde T} \over E - U} = 1 \Rightarrow {\tilde T} = E - U
\label{EnCon}$$ then (\[CanMom\]) and (\[JacobiEL\]) become $$\textbf{p}_{i} = m_{i}{{\textrm{d}\textbf{x}_{i}} \over \textrm{d}
\lambda},\hspace{1.0cm} {\textrm{d} \textbf{p}_{i} \over
\textrm{d}\lambda} = -{\partial U\over\partial \textbf{x}_{i}},$$ and we recover Newton’s second law w.r.t this special $\lambda$. However, (\[EnCon\]), which is usually taken to express energy conservation, becomes the *definition of time*. Indeed, this emergent time, chosen to make the equations of motion take their simplest form [@Poincare2], is the astronomers’ operational ephemeris time [@Clemence]. It is helpful to see how ‘change creates time’. The increment $\delta t$ generated by displacements $\delta \textbf{x}_{i}$ is $$\delta t={\sqrt{\sum
m_{i}\delta\textbf{x}_{i}\cdot\delta\textbf{x}_{i}}\over\sqrt{2(E-U)}}
\equiv{\delta s\over\sqrt{2(E-U)}}.
\label{clem}$$ Each particle ‘advances time’ in proportion to the square root of its mass and to its displacement, the total contribution $\delta s$ being weighted by $\sqrt{2(E-U)}$.
In the previous section, I discussed the role of the energy in determining the curves of generic Newtonian solutions when projected down to shape space. The Jacobi action (\[Jacobi\*\]) illuminates this issue. Considered purely mathematically, $T^*$ by itself already defines a (Riemannian) metric on Q. It is the kinetic metric [@Lanczos]. The function $(E-U)$ multiplying $T^*$ is a conformal factor that transforms the original kinetic metric, which describes pure inertial motion, into a conformally related metric. It is this conformal factor that introduces forces and the effect of the energy into Newtonian mechanics. The decomposition of the conformal factor into the constant $E$ and the conventional Newtonian potential $-U$, which is a function of the inter-particle separations, is artificial from this point of view. In the development of best matching in the next section, it will be best to start by allowing the conformal factor to be an arbitrary function on Q.
Best Matching
=============
The idea of best matching is simple and arises from a very natural problem: How can one quantify the difference between two nearly identical configurations in an intrinsic manner? No additional structure like an inertial system is to be used. In addition, a universally applicable method is required. This is explained in detail in [@SIG; @ABFO]. Here I will explain the gist of the method for the case of the $N$-body problem. Represent the two configurations in the same coordinate grid. In configuration 1, particle $i$ will have coordinates $\textbf{x}_i$. In configuration 2, it will have coordinates $\textbf{x}_i+\textrm d\textbf{x}_i$. Now consider the quadratic form $$F\sum_i m_i \textrm d\textbf{x}_i\cdot\textrm d\textbf{x}_i,
\label{a}$$ where the conformal factor $F$, assumed positive since we are going to take the square root of (\[a\]) in order to obtain a Jacobi-type action, can in principle be an arbitrary function of the coordinates $\textbf x_i$.[^1] We can now use the Euclidean generators of translations, rotations and dilatations separately on each of the configurations, generating different ‘placings’ of them, and calculate (\[a\]) for each change made to the pair of configurations. The points they define in shape space will be unchanged by these operations, which merely affect their mathematical representation. The idea of best matching is to seek the minimum of (\[a\]) with respect to all possible placings and to declare this to be the metric distance between the configurations. If a consistent scheme is to be obtained, interesting restrictions arise. It is easier to visualize best matching in the finite-difference form just described. However, calculations are more readily done with continuous variations, which correspond to a Jacobi action of the form $$I_{\scriptsize\textrm{BM}}=2\int\textrm d\lambda\sqrt F\sqrt T, \hspace{.5cm}T=\sqrt{{1\over 2}\sum_i m_i\left ({\textrm d\textbf x_i\over \textrm d\lambda}-\textbf c_i\right )\cdot \left ({\textrm d\textbf x_i\over \textrm d\lambda}-\textbf c_i\right )},
\label{b}$$where $\textbf c_i$, which has the dimensions of a velocity, is the correction that arises from $\lambda$-dependent transformations on the instantaneous configuration generated by translations, rotations and dilatations. For example, consider a $\lambda$-dependent translation $\textbf x_i\rightarrow \textbf x_i+\textbf b(\lambda).$ It generates the velocity transformation $\dot\textbf x_i\rightarrow \dot\textbf x_i+\dot\textbf b(\lambda)$, where $\dot\textbf x_i=\textrm d\textbf x_i/\textrm d\lambda$. If we make such transformations, (\[b\]) without the correction terms will be changed in an arbitrary manner by the arbitrary vector function $\dot\textbf b$. To counteract this effect of translations, we take the correction $\textbf c_i$ to be an arbitrary vector function $\textbf a$ and vary the action (\[b\]) with respect to it as a Lagrange multiplier. To counteract the effect of simultaneous arbitrary translations, rotations and dilatations, we take the correction to be $$\textbf c_i=\textbf a+\omega \times\textbf x_i+D\textbf x_i
\label{c}$$and vary with respect to the vector functions $\textbf a$ and $\omega$ and the scalar function $D$ as Lagrange multipliers. This variation leads to constraints satisfied by the canonical momenta $\textbf p_i$, $$\textbf p_i=\sqrt{F\over T}m_i\left ({\textrm d\textbf x_i\over \textrm d\lambda}-\textbf c_i\right ),
\label{d}$$ of the physical variables $\textbf x_i$. The constraints that arise from the translations, rotations and dilatations are, respectively, $$\textbf P\equiv\sum_i\textbf p_i=0,
\label{e}$$ $$\textbf M\equiv\sum_i\textbf x_i\times\textbf p_i=0,
\label{f}$$ $$v\equiv\sum_i\textbf x_i\cdot\textbf p_i.
\label{g}$$
Now comes a crucial point. Do the Euler–Lagrange equations, $${\textrm d\textbf p_i\over\textrm d\lambda}=\sqrt{T\over F}{\partial F\over\partial \textbf x_i},
\label{h}$$propagate the constraints? The simple calculation shows that (\[e\]) will propagate only if $F$ is translationally invariant, (\[f\]) will propagate only if $F$ is rotationally invariant, and (\[g\]) will propagate only if $F$ is homogeneous of degree -2.[^2] Note that, as is described in detail in [@SIG], the linear constraints (\[e\]), (\[f\]), and (\[g\]) owe their existence to the fact that (\[b\]) is invariant under $\lambda$-dependent translations, rotations, and dilatations provided one defines the transformation law of the three correction terms in $\textbf c_i$ to be the same as that of the velocities but with the opposite sign. Besides the three linear constraints, there is also a quadratic constraint analogous to (\[QuadCon\]): $$\sum{{{\textbf{p}}_{i}}\cdot
{{\textbf{p}}_{i}}\over 2m_{i}}-F=0.
\label{i}$$
This model, with linear constraints that are uniquely determined by the symmetries of space and a quadratic constraint that follows directly from the idea that time is redundant if the dynamics of the universe (as opposed to subsystems of it) is considered, is interesting from several points of view. Before we discuss them, some preparatory remarks are in order. First, the constraints apply only to the complete system of particles treated as an ‘island universe’. Subsystems are not constrained. It is only necessary that the contributions of all subsystems sum to zero. Second, it is always possible to employ a coordinate frame in which the corrections terms $c_i$ vanish and a special ‘time’ label $\lambda$ for which $F/T=const$. Then the Euler–Lagrange equations are identical to Newton’s equations in an inertial system. Third, the first two linear constraints tell us that in this preferred system the Newtonian momentum and angular momentum of the universe vanish. Because of the Galilean invariance of Newtonian mechanics, the vanishing of the momentum is not a new physical prediction. It is however derived within the logic of best matching. The vanishing of the angular momentum is not enforced by Newton’s equations, the rotational symmetry of which only ensures conservation of angular momentum. Best matching enforces both the symmetry and the exact vanishing of the conserved quantity.
Now we come to consider the third linear constraint. This is by far the most drastic in its consequences and also impacts on the quadratic constraint (\[i\]). It introduces a new conserved quantity in Newtonian dynamics, which however, from the point of view of Newtonian dynamics, occurs only under very special circumstances. It has long been recognized by $N$-body specialists that potentials homogenous of degree -2 in the inter-particle separations represent an interesting special case. This follows from the so-called Lagrange–Jacobi relation [@SIG], which gives an universal expression for the time variation of the moment of inertia (\[MOI\]) in any case in which the the potential is homogenous of degree k: $$\ddot{I}=4(E-U)-2kU.
\label{Iddh}$$ Consider Newtonian celestial mechanics, for which $k=-1$. Then $\ddot{I}=4E-2U$, from which Lagrange deduced the first qualitative result in dynamics. Since $U<0$ for gravity, $E\geq 0$ implies $ \ddot I>0.$ Thus $I$ is concave upwards and must tend to infinity as $t\longrightarrow +\infty$ and $t\longrightarrow -\infty$. In turn, this means that at least one of the interparticle distances must increase unboundedly, so that any system with $E\geq 0$ is unstable.
Another consequence of (\[Iddh\]) is the virial theorem. For suppose that the system has virialized, so that $I\approx0$. Then $4E=(2k+4)U.$
For our purposes, the most interesting consequence of (\[Iddh\]) arises when $k=-2$. For then $$\ddot I=4E.
\label{Ih-2}$$
Thus, $I$ has the parabolic dependence $I=2Et^{2}+bt+c$ on the time and will tend rapidly to zero or infinity. Such a system is extremely unstable, either imploding or exploding.
However, suppose $E=0$. Then $\ddot I=0$ by (\[Ih-2\]), so that $$\dot I=2\sum m_{i}\dot\textbf x_{i}\cdot \textbf x_{i}=2\sum
\textbf p_{i}\cdot \textbf x_{i}=\textrm{constant}.
\label{j}$$ Thus, $v=\sum \textbf p_{i}\cdot \textbf x_{i}$ is a *new conserved quantity*. I am not aware that it has been given any definite name in the literature (or even that its potential significance in dynamics has been recognized). Since it has the same dimensions (action) as angular momentum and is closely analogous to it, I have called it in [@SIG] the *expansive momentum*. It is precisely the quantity that we have found must vanish if best matching with respect to dilatations is applied. It is especially interesting that vanishing of the energy is simultaneously enforced. The reason for this is the drastic consequence of scale invariance. The point is that the kinetic energy has dimensions of length squared, and even under a $\lambda$-independent dilatation $\textbf x_i\rightarrow D\textbf x_i$ changes by a factor $D^2$. This already means that the structure of the Jacobi metric, with kinetic metric that describes pure inertial motion and a conformally related metric that describes inertial motion modified by forces, is not possible on shape space Q$_{\scriptsize\textrm{SS}}$ (though it is still possible on Q$_{\scriptsize\textrm{RCS}}$). One cannot construct a metric on shape space without a compensating potential term $F$ that is homogeneous of degree -2 and therefore transforms as $D^{-2}$, thereby compensating the $D^2$ of the kinetic term. This means that, in contrast to the Jacobi action, one cannot have a conformal factor of the form $E-U$ made up of the constant total energy $E$ and a ‘proper’ potential that depends on the inter-particle separations.
Hidden Scale Invariance
=======================
If we are to take scale invariance seriously, we must now confront the problem that the standard potentials in Newtonian dynamics, for gravity and electrostatics, derive from potentials homogenous of degree -1, not -2. Nature would appear to be sending us a strong signal that it is not scale invariant. There is, however, a possibility that scale invariance is realized but hidden remarkably effectively. We have seen above in (\[j\]) that the time derivative of the moment of inertia (\[MOI\]) is the expansive momentum $v$. Therefore, if $v$ vanishes the moment of inertia becomes a conserved quantity. Within standard Newtonian theory, this is an exceptional case, requiring the simultaneous vanishing of the energy and the expansive momentum. It is, however, a necessary consequence of scale invariance as defined here. Let us therefore exploit this fact by converting given Newtonian potentials into scale-invariant analogues that have the necessary homogeneity of degree -2. To do this, we shall use $I$, or rather $\sqrt {MI}$: $$\mu=\sqrt{\sum_{i<j}m_{i}m_{j}r_{ij}^{2}}.
\label{Rho}$$
Just as one passes from special to general relativity (with gravity minimally coupled to matter) by replacing ordinary derivatives in the matter Lagrangians by covariant derivatives, Newtonian potentials can be converted into potentials that respect scale invariance. One simply multiplies by an appropriate power of $\mu$, which has the dimensions of length. This is a rather obvious mechanism. What is perhaps unexpected is that the modified potentials lead to forces $\textit{identical}$ to the originals accompanied by a universal cosmological force with minute local effects. The scale invariance is hidden because $\mu$ is conserved.
Let some standard Newtonian potential $U$ consist of a sum of potentials $U_{k}$ each homogeneous of degree $k$: $$U=\sum_{k=-\infty}^{\infty}a_{k}U_{k}.
\label{NewtPot}$$
The $a_{k}$ are freely disposable strength constants. The energy $E$ in the Jacobi action (\[Jacobi\]) will be treated as a constant potential ($k=0$). (It plays a role like the cosmological constant $\Lambda$ in GR).
Now replace (\[NewtPot\]) by $$\tilde{U}=\sum_{k=-\infty}^{\infty}b_{k}U_{k}\mu^{-(2+k)}.
\label{ScaledPot}$$
The equations of motion for (\[NewtPot\]) are $${\textrm{d}\textbf{p}^{i}\over\textrm{d}t}=
-\sum_{k=-\infty}^{\infty}a_{k}{\partial
U_{k}\over\partial\textbf{x}^{i}};$$ for (\[ScaledPot\]) they are $${\textrm{d}\textbf{p}^{i}\over\textrm{d}t}=
-\sum_{k=-\infty}^{\infty}b_{k}\mu^{-(2+k)}
{\partial U_{k}\over\partial\textbf{x}^{i}}+
\sum_{k=-\infty}^{\infty}(2+k)b_{k}\mu^{-(2+k)}U_{k}
{1\over\mu}{\partial\mu\over\partial\textbf{x}^{i}}.$$
Since $\mu$ is constant ‘on shell’, we can define new strength constants that are truly constant: $$b_{k}=a_{k}\mu^{2+k}.
\label{DefB}$$ The equations for the modified potential become $${\textrm{d}\textbf{p}^{i}\over\textrm{d}t}=
-\sum_{k=-\infty}^{\infty}a_{k}
{\partial U_{k}\over\partial\textbf{x}^{i}}+
\sum_{k=-\infty}^{\infty}(2+k)a_{k}U_{k}
{1\over\mu}{\partial\mu\over\partial\textbf{x}^{i}}.
\label{ModEq}$$ If we define $$C(t)={\sum_{k=-\infty}^{\infty}(2+k)a_{k}U_{k}\over
2\sum_{i<j}m_{i}m_{j}r_{ij}^{2}}
\label{DefC}$$ and express $\mu$ in terms of $r_{ij}$, then equations (\[ModEq\]) become $${\textrm{d}\textbf{p}^{i}\over\textrm{d}t}=
-\sum_{k=-\infty}^{\infty}a_{k}
{\partial U_{k}\over\partial\textbf{x}^{i}}+
C(t)\sum_{j}m_{i}m_{j}{\partial{r_{ij}^{2}}\over\partial\textbf{x}^{i}}.
\label{AbbModEq}$$
We recover the original forces exactly together with a universal force. It has an epoch-dependent strength constant $C(t)$ and gives rise to forces between all pairs of particles that, like gravitational forces, are proportional to the inertial mass but increase in strength linearly with the distance.[^3] The universal force will be attractive or repulsive depending on the sign of $C(t)$, which is an explicit function of the $r_{ij}$’s. For small enough $r_{ij}$, the force will be negligible compared with Newtonian gravity. However, on cosmological scales it will be significant. I refer the reader to [@SIG] for a discussion of the possible cosmological implications. Since this paper is primarily concerned with an alternative formulation of the relativity principle and the problem of time, let me conclude this section with some comments about these two issues and then, in the next section, describe what happens in the context of Riemannian geometry and field theory.
The main significance of the model presented here is, I believe, methodological. It shows that one can formulate a powerful constructive relativity principle and implement it universally. One postulates a spatial geometry and the nature of the objects it. Then the geodesic principle and best matching lead to a highly predictive theoretical framework that is completely relational. Above all, observable effects have genuine observable causes. This is because the kinematic structure that is presupposed is pared down to the bare minimum: spatial geometrical relationships quotiented with respect to the spatial symmetries. No extra kinematics associated with time and inertial systems is assumed. Nevertheless, Newton’s laws with extra restrictions that do not appear to be in conflict with observations are recovered. The theory clearly cannot fix everything despite the strong restrictions. The potential can still contain several independent terms with arbitrary relative strengths. However, effects without observable material causes are completely eliminated. Specifically, there are no observable effects that one could attribute to translation, rotation or change of size of the complete universe. These results were to be expected, but, very interestingly, the imposition of scale invariance also eliminates the last vestige of what one might call time kinematics: the possibility of including the constant total energy $E$ in Jacobi’s principle. In Newtonian theory, different values of $E$ are possible because the external absolute time allows different kinetic energies for a given spatial configuration. Absolute time does seem to be abolished from Jacobi’s principle, but its effect is exactly reproduced by the freely specifiable constant $E$ that is not associated with observable sources. There is still a free constant $E$ in the scale-invariant theory, but it is now the coefficient of a genuine potential.
Scale invariance is a highly intuitive and theoretically desirable attribute of any dynamics of the universe. I find it remarkable that it also ‘kills time.’ It has even more striking consequences in the context of Riemannian geometry and field theory, to which we now turn.
Riemannian Geometry and Fields
==============================
This section merely serves as a summary of the content of the papers [@BOF; @AB; @ABFO; @Variations] with emphasis on the connection between time and scale invariance. I shall start by reviewing the manner in which general relativity, which was, of course, originally formulated as a description of four-dimensional pseudo-Riemannian spacetime, can be interpreted as a dynamical theory of the evolution of three-dimensional Riemannian geometry.
A given Einsteinian spacetime, which for simplicity I shall assume has compact spatial sections (closed universe), can, provided it is globally hyperbolic, be foliated by spacelike hypersurfaces (‘leaves’), on which a Riemannian geometry is induced. Such foliations can be generated by laying down some system of coordinates on the original spacetime. Then the surfaces of constant value of the time coordinate are the hypersurfaces of the foliation, on which the induced Riemannian geometry is represented by a Riemannian 3-metric $g_{ij}$ defined relative to the spatial coordinates on the given leaf. Such 3-metrics constitute the dynamical variables when general relativity is treated as Hamiltonian theory. The space of all Riemannian 3-metrics $g_{ij}$ on a given 3-manifold $\cal M$ is called Riem($\cal M$). It is the analogue of the Newtonian configuration space. All 3-metrics related to each other by (spatial) coordinate transformations, or equivalently 3-diffeomorphisms, correspond to a given 3-geometry. The space of all 3-geometries (on a given manifold) is called *superspace*. Mathematically, superspace is the quotient of Riem with respect to 3-diffeomorphisms. There is an obvious direct analogy between 3-diffeomorphisms, which ‘drag the contents of the universe’ around on $\cal M$, and translations and rotations on Euclidean space. Thus, superspace is the analogue of the relative configuration space (RCS) of the particle model.
Given the spacetime and the foliation, each induced 3-metric will be a point in Riem, and the spacetime will be a curve in Riem. Keeping the foliation unchanged (i.e., the t=constant surfaces the same), but changing the spatial coordinates freely on the different slices, one obtains many different curves in Riem that all represent the same spacetime. On superspace, for the given foliation, there is just one curve. However, if one changes the foliation, one obtains a whole family of curves in superspace, one for each foliation. This multiplicity of curves, all representing the same spacetime, reflects the relativity of time in Einstein’s theory and has hitherto presented insuperable obstacles to the attempts at a canonical quantization of general relativity. This is *the problem of time* [@Kuchar; @Isham]. It is closely related to the question of the true degrees of freedom of the gravitational field. It is generally accepted that in Einstein’s theory there are two at each space point. The question is: can one identify them? When general relativity is treated as a dynamical theory, the $g_{0\nu}, \nu=0,1,2,3,$ components of the spacetime metric turn out to be Lagrange multipliers and are not proper degrees of freedom. The remaining six components $g_{ij}$ in the spatial part of the metric contain three arbitrary functions, corresponding to the possibility of making arbitrary coordinate transformations. This gauge freedom is quotiented out in the passage from Riem to superspace. At this level, one is left with three degrees of freedom per space point. However, the freedom to choose the time coordinate arbitrarily, changing thereby the foliation, represents a further gauge function per space point. But what then is evolving? If one wishes to maintain full general covariance, one cannot quotient away that freedom as one does in the passage from Riem to superspace.
However, if one is prepared to sacrifice general covariance, an obvious step is to perform a further quotienting like the passage from the RCS to shape space. The analogue of shape space is *conformal superspace* (CS), which is obtained by quotienting Riem not only by 3-diffeomorphisms but also by conformal transformations: $$g_{ij}\rightarrow\phi^4g_{ij},
\label{k}$$ where the fourth power of the positive function $\phi$ is chosen for mathematical convenience.
Many years ago, York [@York] showed that, indeed, one can parametrize the solutions of the initial-value constraints of general relativity by the two degrees of freedom per space point that reside in conformal superspace. Furthermore, he made effective use of the conformal transformations (\[k\]) to find such solutions. This important piece of work involved a *constant-mean-curvature* (CMC) foliation. This is defined as follows. At each point on any leaf of a foliation, the leaf has a certain extrinsic curvature tensor $K_{ij}$ (second fundamental form). This measures the manner in which the leaf is curved in the spacetime in which it is embedded. A CMC foliation is one for which the trace of $K_{ij}$, $K=g_{ij}K^{ij}$, where indices are raised and lowered by means of $g_{ij}$ and its inverse, is constant on each leaf. For spatially compact solutions, a CMC foliation is unique, and $K$ varies monotonically with the cosmic time. CMC foliations have many useful properties and are uniquely helpful in York’s method for finding solutions to the initial-value constraints of general relativity. However, because York did not arrive at his technique through a fundamental variational principle but merely exploited what has proved to be very convenient mathematics, the use of CMC foliations is regarded as a gauge-fixing condition that breaks four-dimensional general covariance. It ‘fixes time’, introducing a definition of simultaneity, which is anathema to many relativists. However, the 3-space approach suggests an interesting alternative interpretation in which the CMC foliation has a deep physical significance.
The stimulus to the development of the 3-space approach was the Lagrangian reformulation of the Dirac–ADM Hamiltonian representation of general relativity [@DiracHam; @ADM] found by Baierlein, Sharp and Wheeler [@BSW]. The key concepts in the ADM formalism are the 3-metrics $g_{ij}$, the lapse $N$ and the shift $N^{i}$. The lapse measures the rate of change of proper time w.r.t. the label time, while the shift determines how the coordinates are laid down on the successive 3-geometries. Prior to the transition to the Hamiltonian, the standard Hilbert–Einstein action for matter-free GR is rewritten, after divergence terms have been omitted, in the 3+1 form $$I= \int \textrm dt\int\sqrt{g}N\left[R + K^{ij}K_{ij} - K^2\right]\textrm d^3x. \label{first}$$
Here $R$ is the three-dimensional scalar curvature, and $K_{ij} =
-(1/2N)(\partial g_{ij}/
\partial t - N_{i;j} - N_{j;i})$ is the extrinsic curvature with trace $K$. From here the transition made by BSW [@BSW] is trivial. They first replaced $K_{ij}$ in the action by $k_{ij} = \partial g_{ij}/
\partial t - N_{i;j} - N_{j;i}$, the unnormalised normal derivative, to give $$I= \int \textrm dt\int\sqrt{g}\left[NR + {1 \over 4N}\left(k^{ij}k_{ij} - k^2\right)\right]\textrm d^3x. \label{second}$$ They varied this action with respect to the lapse and found an algebraic expression for it, $$N =
\sqrt{{k^{ij}k_{ij} - k^2 \over 4R}}. \label{third}$$ This expression for $N$ is substituted back into Eq.(\[first\]) to obtain the BSW Lagrangian $$I_{\scriptsize\textrm{BSW}} = \int\textrm dt\int {\cal L}\textrm d^3x= \int\textrm dt\int\sqrt{g}\sqrt{R}\sqrt{k^{ij}k_{ij} -
k^2}\textrm d^3x. \label{fourth}$$
This action is closely analogous to the Jacobi action (\[Jacobi\]): it has a geodesic-type square root and the Lagrange multiplier $N_i$ generates 3-diffeomorphisms in the same way that the multipliers in the particle model generate the Euclidean symmetry transformations. These properties lead, respectively, to the quadratic and linear constraints $$-p^{ij}p_{ij} + {1 \over 2}p^2 + gR = 0,\hspace{.5cm}p^{ij}_{~~;i} = 0,\hspace{.5cm}p = g_{ab}p^{ab}, \label{2.3}$$ where the canonical momenta are $$p^{ij} = {\delta {\cal L} \over \delta\left({\partial g_{ij} \over
\partial \lambda}\right)}, \label{2.2}$$ and $p=g_{ij}p^{ij}$ is their trace. It measures the expansion of space and is the analogue of the expansive momentum in the particle model.
The constraints (\[2.3\]), which are the fundamental ADM Hamiltonian and momentum constraints, are evidently like the particle constraints but with a crucial difference: instead of being global constraints that hold for the complete universe, these are infinitely many constraints, one per space point. Note, in particular, that the quadratic constraint is an identity that follows from the mere form of the Lagrangian. It is important that the square root is ‘local’, i.e., it is taken before the integration over space. This ensures that there is a constraint per space point.
In [@BOF; @AB; @SIG; @ABFO], my collaborators and I studied systematically gravity–matter-field Lagrangians that are natural generalizations of the BSW Lagrangian and have the form $$I = \int\textrm dt\int\sqrt{g}\sqrt{U_{\scriptsize\textrm g}+U_{\scriptsize\textrm m}}\sqrt{T_{\scriptsize\textrm g}+T_{\scriptsize\textrm m}}\textrm d^3x.
\label{l}$$ Here, the gravitational potential term $U_{\scriptsize\textrm g}$ depends only on the 3-metric $g_{ij}$, while the matter potential terms $U_{\scriptsize\textrm m}$ for the considered scalar and 3-vector fields depend on $g_{ij}$, the matter fields and their spatial derivatives. The kinetic terms are quadratic in the velocities, as in conventional Hamiltonian field theory, but with the all-important difference that the velocities are ‘corrected’ to take into account the effect of time-dependent diffeomorphisms and conformal transformations. The corrections are uniquely determined by the symmetry transformation and introduce corresponding Lagrange multipliers and linear momentum constraints, just as in the particle model. However, we included a free coefficient $A$ in $T_{\scriptsize\textrm g}=k^{ij}k_{ij} -Ak^2$ to reflect that, a priori, two independent scalars can contribute to the kinetic term. I will first discuss our results for the matter-free case and best matching with respect to diffeomorphisms.
They revealed an important difference from the particle case, in which propagation of the constraint linear in the momentum imposes conditions on the potential but the quadratic constraint propagates with any potential. In the case of the Riemannian symmetry and the local square root of the BSW action, the two constraints are ‘intertwined’ and the simultaneous propagation of them imposes strong restrictions. We found that the only consistent Lagrangians of the form (\[l\]) must have the free coefficient $A$ equal to unity, as in general relativity, while the gravitational potential term must have the form $U_{\scriptsize\textrm g}=\Lambda+sR, s=0, 1, -1$. The freely specifiable constant $\Lambda$ corresponds to the energy $E$ in Jacobi’s principle and is Einstein’s cosmological constant. The three possible values of $s$ correspond respectively to so-called strong gravity and general relativity with spacetime signatures -+++ and ++++. This meant that we had found a completely new derivation of general relativity by consistent application of the timeless generalization of Poincaré’s relativity principle applied in the case of Riemannian 3-geometries. Spacetime was in no way presupposed. It was derived.
Even more remarkable results came when we tried to couple scalar and 3-vector fields to gravity in the case of the Minkowskian signature -+++. In the standard spacetime approach, the fact that such fields must respect the same light cone as the gravitational field is put in the form of the assumption that spacetime in the small is Minkowskian. A universal light cone is presupposed. However, we found that it is enforced by consistent propagation of the two constraints, both of which are modified by the addition of matter terms. A key point is that the form of the momentum constraint is always completely determined by the tensorial nature of the fields and the fact that one is best matching with respect to diffeomorphisms, which affect all fields in a uniquely determined way. In contrast, the form of the quadratic constraint reflects the particular ansatz made for the Lagrangian. Only very special Lagrangians lead to consistent constraint propagation. Thus, our first result for matter was that there must be a universal light cone. Equally striking is the fact that in the case of 3-vector fields the very same requirement of constraint propagation forces the 3-vector fields to be gauge fields. In fact, the universal light cone and gauge theory are shown to have essentially the same origin. Of course, all derivations in theoretical physics include simplicity assumptions, either explicitly or implicitly. These are discussed by Anderson in [@Variations]. I think it is correct to say that the 3-space approach assumes less and derives more than the standard approach based on Einstein’s general principle of covariance.
So far, I have described only the effect of the local square root and best matching with respect to diffeomorphisms. In the more recent paper [@ABFO], we constructed Lagrangians in which best matching with respect to conformal transformations is also performed. We obtained two main results. First, if one best matches the BSW Lagrangian as it stands with respect to conformal transformations that preserve the spatial volume $V=\int\sqrt g\textrm d^3x$, then the standard Dirac–ADM constraints are augmented by the constraint $${p\over\sqrt g}=constant,
\label{m}$$ and one also obtains a condition on the lapse that ensures the propagation of this constraint by the Euler–Lagrange equations. When expressed in terms of the extrinsic curvature $K$, the constraint (\[m\]) is precisely York’s CMC slicing condition. (Note that $p/\sqrt g$ is constant on each leaf, but its value changes under the evolution.) We therefore have the striking result that such conformal transformations, which change all local scales completely freely subject to the single global restriction on the volume, lead us to general relativity in a distinguished foliation, i.e., to a distinguished definition of simultaneity. Once again, we find a strong connection between time and scale invariance. I believe that this result should be taken seriously. In general relativity, there are four gauge freedoms. Three are associated with 3-diffeomorphisms and the fourth with arbitrary transformations of the time coordinate. In our best-matching approach, there are also four gauge freedoms. The 3-diffeomorphisms are still present, but the freedom in the time gauge is replaced by freedom in the scale gauge. All the four gauge freedoms are now expressed through constraints linear in the canonical momenta. Moreover, all have their origin in the geometry of space.
However, from the point of view of scale invariance, general relativity is frustratingly not quite perfect. I find it extremely puzzling that the solitary volume-preserving restriction on full scale invariance is imposed on the conformal transformations. In fact, this is what permits volume to be a physical degree of freedom in general relativity and allows the expansion of the universe. I already mentioned that the trace $p$ of the canonical momenta measures the expansion of space. We see from the constraint (\[m\]) that $p$ does not vanish but that $p/\sqrt g$ is equal to an evolving spatial constant. When we best match with respect to all conformal transformations, dropping the volume-preserving restriction, we get $$p=0.
\label{n}$$
There is now full agreement with the scale-invariant particle model, for which the vanishing of the expansive momentum ensures constancy of the moment of inertia (the ‘size’ of the $N$-particle universe). Here, the vanishing of $p$ means that the volume of the universe cannot change. Because the BSW Lagrangian (\[fourth\]) allows the volume to change, it has to be modified if the constraint $p=0$ is to propagate. Just as constancy of the moment of inertia was achieved by dividing the potential by a power of the moment of inertia, we achieved propagation of $p=0$ by dividing the gravitational potential by an appropriate power of the volume. This led us to a consistent fully scale-invariant theory that we call *conformal gravity*. It is a remarkably small modification of general relativity. The single global variable that permits the volume of the universe to change is excised and one obtains a dynamics of the geometry of pure shape. Unfortunately, although conformal gravity should describe the solar-system and binary-pulsar data just as well as general relativity, the cosmology must be quite different. At the time of writing, it seems hard to believe that conformal gravity will be able to supplant general relativity as a cosmological theory. The difficulties are spelled out in [@SIG; @ABFO].
However, this probable failure of conformal gravity does not change the fact that the basic idea of using best-matching geodesics to implement Mach’s principle on the basis of Poincaré’s relativity principle establishes unexpected links between special relativity, general relativity and the gauge principle. They all emerge together as a self-consistent complex from a unified and completely relational approach to dynamics. We see that all of currently dynamics can be understood in terms of purely spatial geometry. Finally, a deep connection between time and shape is established.
It was a great pleasure to participate in DICE2002, and I hope this contribution to the proceedings will foster further interdisciplinary workshops.
[99]{}
Alexander H G (ed.) 1956 *The Leibniz–Clarke Correspondence*, Sec. 47 (Barnes and Noble, New York) Mach E 1883 *Die Mechanik in ihrer Entwicklung historisch-kritsch dargestellt* (Leipzig: Barth); 1893 *The Science of Mechanics* (Chicago: Open Court) Poincar$\acute{\textrm e}$ H 1905 *Science and Hypothesis* (London, translated from the French edition of 1902) Einstein A 1949 “Autobiographical Notes,” in *Albert Einstein – Philosopher – Scientist*, ed. P A Schilpp (The Library of Living Philosophers: Evanston, Illinois) Barbour J 1990 “The part played by Mach’s Principle in the genesis of relativistic cosmology” in *Modern Cosmology in Retrospect*, eds. B. Bertotti, R. Balbinot, S. Bergia, and A. Messina (Cambridge University Press: Cambridge) (see also: Barbour J 1999 “The development of Machian themes in the twentieth century” in *The Arguments of Time*, ed. J Butterfield (Oxford University Press: Oxford)). Einstein A 1916 *Annalen der Physik* **49**, 769 Kretschmann E 1917 *Annalen der Physik* **53**, 575 Einstein A 1918 *Annalen der Physik* **55**, 241 Barbour J 2001 “On general covariance and best matching” in *Physics Meets Philosophy at the Planck Length*, eds. Callender C and Huggett N (Cambridge: Cambridge University Press) Barbour J, Foster B Z and Ó Murchadha N 2002 *Class. Quantum Grav.* **19** 3217; “Relativity without relativity”, arXiv:gr-qc/0012089 Anderson E and Barbour J 2002 *Class. Quantum Grav.* **19** 3249; “Interacting vector fields in relativity without relativity”, arXiv:gr-qc/0201092 Barbour J 2002 “Scale-invariant gravity: particle dynamics”, arXiv:gr-qc/0211021 (to be published in *Class. Quantum Grav.*
Anderson E, Barbour J, Foster B Z and Ó Murchadha N 2002 “Scale-Invariant Gravity: Geometrodynamics", arXiv:gr-qc/0211022 (to be published in *Class. Quantum Grav.* Lanczos C 1949 *The Variational Principles of Mechanics* (University of Toronto Press, Toronto)(also available from Dover, New York) Barbour J and Bertotti B 1982 *Proc. R. Soc. A* **382** 295 Barbour J 1994 *Class. Quantum Grav.* **11** 2875 Barbour J 1999 *The End of Time* (London: Weidenfeld and Nicolson; New York: Oxford University Press) Dirac P A M 1964 *Lectures on Quantum Mechanics* (New York: Yeshiva University) Poincaré H 1898 *Rev. Métaphys. Morale* **6** 1 (English translation 1913: “The Measure of Time” in *The Value of Science* (New York: Science Press) Clemence G 1957 *Rev. Mod. Phys.* **29** 2 Anderson E 2003 “Variations on the seventh route to relativity,” arXiv:gr-qc/0302035 Kuchař K 1992 “Time and interpretations of quantum gravity” in *Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics* eds. G Kunstatter, D Vincent and J Williams (World Scientific: Singapore) Isham C 1992 “Canonical quantum gravity and the problem of time” in *Integrable Systems, Quantum Groups, and Quantum Field Theories* eds. L Ibart and M Rodriguez (Kluwer: Amsterdam) York J 1971 *Phys. Rev. Letters* [**26**]{}, 1656; York J 1972 *Phys. Rev. Letters* [**28**]{}, 1082 Dirac P 1958 *Proc. Roy. Soc. Lond.* [**A246**]{}, 333
Arnowitt R, Deser S, Misner C 1972 “The dynamics of general relativity” in *Gravitation: an Introduction to Current Research*, ed. L Witten (Wiley: New York) Baierlein R, Sharp D and Wheeler J 1962 *Phys. Rev.* [**126**]{}, 1864
[^1]: Clearly, a more general, non-diagonal form could be assumed instead of (\[a\]). This is one of several issues within the 3-space approach that are currently being studied [@Variations].
[^2]: These conditions, derived here as consistency requirements, are necessary consequences of the fact that best matching is being used to define a metric on a quotient space. In fact, as is shown in [@SIG], each symmetry with respect to which best matching is performed leads to two conditions: a linear constraint on canonical momenta and a condition on the potential $F$ that ensures its propagation.
[^3]: Although $C(t)$ is epoch dependent, this does not mean that the theory contains any fundamental coupling constants with such a dependence. The epoch dependence is an artefact of the decomposition of the forces into Newtonian-type forces and a residue, which is the cosmological force.
|
---
abstract: 'We present three newly discovered globular clusters (GCs) in the Local Group dwarf irregular NGC 6822. Two are luminous and compact, while the third is a very low luminosity diffuse cluster. We report the integrated optical photometry of the clusters, drawing on archival CFHT/Megacam data. The spatial positions of the new GCs are consistent with the linear alignment of the already-known clusters. The most luminous of the new GCs is also highly elliptical, which we speculate may be due to the low tidal field in its environment.'
author:
- |
A. P. Huxor$^{1}$[^1], A. M. N. Ferguson$^{2}$, J. Veljanoski$^{2}$, A. D. Mackey$^{3}$, N. R. Tanvir$^{4}$\
$^{1}$Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchstra[ß]{}e 12 - 14,\
69120 Heidelberg, Germany.\
$^{2}$SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\
$^{3}$Research School of Astronomy & Astrophysics, Australian National University, Mt. Stromlo Observatory, Cotter Road,\
Weston Creek, ACT 2611, Australia\
$^{4}$Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK\
title: Three newly discovered globular clusters in NGC 6822
---
\[firstpage\]
galaxies: star clusters – galaxies: individual (NGC 6822)
Introduction
============
In $\Lambda$CDM cosmology, large galaxies are assembled as the result of the accretion and merger of smaller galaxies. If the globular cluster (GC) systems of large galaxies are also formed, at least in part, from the accretion of GCs from the smaller systems, then the GCs themselves can act as beacons of this process. Indeed, the seminal study of the Galactic GCs by @SearleZinn78 was crucial for understanding the history of our own Milky Way (MW). Being compact and luminous, GCs are excellent probes when field star populations cannot be resolved. In our previous work [@Huxoretal11; @Mackeyetal10], we have shown that the GC population of the outer halo of M31 arises largely from the accretion of dwarf galaxies. In many cases, one can identify the remnants of dwarf galaxies in the process of delivering their clusters into M31, where they are becoming part of its retinue of GCs. A similar process is also taking place in the MW, where the Sagittarius dwarf is most likely contributing its GCs to the MW halo [@Bellazzinietal03; @law10]. Recent work by @Kelleretal12 also supports the view that many GCs found in the outer halo of the MW have been accreted alongside their (now disrupted) dwarf galaxy hosts. They conclude that the MW halo has experienced the accretion of some three Magellanic-like or equally up to 30 Sculptor-like dwarf galaxies, or some intermediate mix of both types. Knowledge of the characteristics of the types of GCs found in a range of dwarf galaxies will assist in determining which scenario may have occurred.
To use GCs as probes, it is essential to understand the properties of the GC systems of dwarf galaxies, the relationship of these properties to their host galaxies, and whether we can still identify these after they have been accreted into a more massive galaxy. Dwarf irregulars are particularly interesting in this regard as they are usually found in the field. Their relative isolation makes them ideal laboratories for studying the pristine properties GC systems. This contrasts with dwarf spheroidal and elliptical galaxies which are usually found close to more massive galaxies, and thus they (and their GCs) will likely have been influenced by them. Motivated by our previous work in which wide-area searches yielded the discovery of many new clusters in M31 [@Huxoretal08; @Martinetal06], M33 [@Huxoretal09; @Cockcroftetal11], and the M31 satellite galaxies NGC 185 and NGC 147 (Veljanoski et al. in prep), we decided to investigate the outer regions of the dwarf irregular NGC 6822 which benefits from extensive archival CFHT/MegaCam imaging, and in which @Hwangetal11 have recently discovered four new extended star clusters.
NGC 6822 is a member of the Local Group, and is not associated with either the MW or M31. We use an adopted distance of 472 kpc determined as the average of published values for which the error of the distance modulus is $<$ 0.2 mags [@Gorskietal11]. It has an absolute magnitude $M_{V}$ of –15.2 [@Mateo98] and an R$_{25}$ of 465 arcsec (RC3.9 value, reported by NED[^2]) equal to $\sim$ 1 kpc at our adopted distance. NGC 6822 possesses a number of interesting features including a ring of gas and stars which is almost perpendicular to the main body of the galaxy [@deBlokWalter00]. The galaxy has also been found to have an extended stellar spheroid [@Battinellietal06] (see Figure \[Fi:dss\]). The central regions of NGC 6822 contain many young massive clusters [@Chandaretal00]; however, until very recently, it was believed that there was only one truly old globular cluster in the system [@Grebel02], known as Hubble VII – one of his original list of “nebulae"[^3] in NGC 6822 [@Hubble25]. @CohenBlakeslee98 undertook a spectroscopic study of this object, reporting an age of 11 Gyr and \[Fe/H\] = –1.95 dex. Another of Hubble’s candidates, Hubble VIII, has also been subjected to detailed study and appears to be a massive intermediate-age cluster. Using HST/WFPC2 observations, @Wyderetal00 derived an age of 1.5 Gyr however a spectroscopic study by @Straderetal03 found it to be somewhat older at 3–4 Gyr. Both @Chandaretal00 and @CohenBlakeslee98 derive spectroscopic \[Fe/H\] estimates for Hubble VIII, finding a value of about –2.0 dex.
In addition, NGC 6822 possesses four extended star clusters (SC1–SC4, shown in Figure \[Fi:dss\]) that have been found beyond the main body of the galaxy [@Hwangetal11]. These were discovered in a wide-field CFHT/Megacam survey of NGC 6822 that covered a region of 3$^{\circ}$ $\times$ 3$^{\circ}$. The clusters have half-light radii of 7.5 – 14 pc, and colour-magnitude diagrams that are consistent with a wide range of ages (2 – 10 Gyr) and metallicities (Z = 0.0001 – 0.004). These clusters are very similar to the extended clusters found in M31 [@Huxoretal05] and M33 [@Stonkuteetal08], with a couple of the clusters (SC1 and SC4) being very distant from NGC 6822 itself. @Hwangetal11 also noted that the extended clusters project on a line that is consistent with the major axis of the old stellar halo.
![Image from the DSS of NGC 6822 with locations of the new and previously-known clusters shown, in filter GSS bandpass number 36. The image is 2.57 x 2.57 degrees. The ellipse shows the extent of the RGB stars of the spheroid where the halo is detected above the noise [@Battinellietal06], with a semi-major axis of 36 and an ellipticity of 0.36. Also shown are contours from the HI map of @deBlokWalter00.[]{data-label="Fi:dss"}](n6822_dss_image_HI_small.ps){width="70mm"}
Drawing partly on the data from the Hubble clusters, @Straderetal03 suggest a view of NGC 6822 in which there has been a relatively constant star formation rate over time, with occasional stochastic outbursts that result in the formation of star clusters. A similar scenario has also been outlined by @ColucciBernstein11.
The Search for NGC 6822 Clusters
================================
An initial study of the archives found that NGC 6822 had considerable and contiguous coverage in CFHT/Megacam imaging (see Figure \[Fi:fields\]). This is a wide-field camera at the Canada-France-Hawaii Telescope (CFHT) with a 1$^{\circ}$ $\times$ 1$^{\circ}$ field of view and a pixel scale of 0$\arcsec$.187. We naturally use the same imaging as that of @Hwangetal11, but also include additional fields that extended the coverage and fill the gaps between the CCDs in their survey. In total, we searched 15 CFHT/Megacam fields from the programs 2003BK03, 2004AC02, 2004AQ98, and 2005AK08, with observations taken over the period August 2003–August 2006. Science exposures ranged from 660 – 1200 seconds in the g-band, 360 – 1000 seconds in the r-band, and 150 – 460 seconds in the i-band.
The images were visually inspected since star clusters at NGC 6822’s distance are easily resolved in CFHT/Megacam imaging, and indeed this is the optimal way to identify any additional examples of the extended clusters. We are only concerned with the outer regions of NGC 6822, and do not study the main body of the galaxy where many young clusters have already been documented [@Krienkeetal04].
We also examined archival Subaru/Suprime-Cam imaging of NGC 6822. This instrument has a $\sim$ 0.5$^{\circ}$ $\times$ 0.5$^{\circ}$ field of view and a pixel scale of 0$\arcsec$.20, and the imaging was mainly concentrated on the inner regions of the galaxy. We utilised only those images for which the exposure was greater than 200 seconds. Those available in the archive were obtained in B,V,R or I-band filters, and were taken for proposals o01422, o00005, o02419,o03147, o99005, o04151, and o05226. Although these pointings did not extend much beyond the main body of NGC 6822, many of the images were deeper than those from CFHT/Megacam and they proved useful to confirm, or otherwise, candidate clusters found in the CFHT/Megacam imaging.
![Locations of the CFHT/Megacam fields studied. The centre of NGC 6822 is represented by the small solid square, and the ellipse is the same as that in Figure \[Fi:dss\][]{data-label="Fi:fields"}](n6822_layout.ps){width="85mm"}
The New Clusters
=================
The search for new GCs found a total of three new clusters, in addition to rediscovering all those of @Hwangetal11. Two are luminous compact classical clusters, and one is very faint and appears extended in form. We continue the naming convention used by @Hwangetal11, and denote them as SC5, SC6 and SC7 (in order of right ascension). The coordinates of these objects and their projected distance from the centre of NGC 6822 are listed in Table \[tab:positions\]. The two luminous clusters (SC6 and SC7) are clear examples of GCs (see Figure \[Fi:image\_AB\]). The new faint cluster (SC5) is, in contrast, much more diffuse. Although barely detected in a single exposure from the CFHT archive, SC5 can be seen more clearly in a stacked image available through the CFHT archive (Figure \[Fi:image\_D\], left panel), and also in a deep archival Subaru/Suprime-Cam image (Figure \[Fi:image\_D\], right panel). SC5 resolves into stars while SC6 and SC7 only do so in their peripheries.
$
\begin{array}{cc}
\includegraphics[angle=0,width=40mm]{n6822_GC_B_figure.ps} &
\includegraphics[angle=0,width=40mm]{n6822_GC_A_figure.ps} \\
\end{array}$
$
\begin{array}{cc}
\includegraphics[angle=0,width=40mm]{n6822_GC_D_MC_figure.ps} &
\includegraphics[angle=0,width=40mm]{n6822_GC_D_SP_figure.ps}
\end{array}$
ID RA(J2000) Dec(J2000) R$_{proj}$ (kpc)
----- ------------------------------- ----------------------------------------- ------------------
SC5 19$^{h}$ 43$^{m}$ 42.30$^{s}$ –14$^{\circ}$ 41$\arcmin$ 59.7$\arcsec$ 2.6
SC6 19$^{h}$ 45$^{m}$ 37.02$^{s}$ –14$^{\circ}$ 41$\arcmin$ 10.8$\arcsec$ 1.6
SC7 19$^{h}$ 46$^{m}$ 00.85$^{s}$ –14$^{\circ}$ 32$\arcmin$ 35.4$\arcsec$ 3.0
: Locations of the new clusters, and their projected distances (R$_{proj}$) from the centre of NGC 6822 (RA = 19$^{h}$ 44$^{m}$ 57.7$^{s}$, Dec = –14$^{\circ}$ 48$\arcmin$ 12$\arcsec$).[]{data-label="tab:positions"}
Integrated Photometry
---------------------
Integrated photometry was undertaken for the two most luminous new clusters, using the archival imaging data available. The results are reported in CFHT/Megacam filter magnitudes (which are similar but not identical to standard Sloan filters) in Table \[tab:properties\].
In our photometry we used large apertures that enclose the full extent of the cluster for the total magnitudes. As there is no evidence that GCs have strong colour gradients, we employed smaller apertures to obtain more reliable colours. Photometric calibration of the CFHT data was undertaken using the magnitudes derived for the one pointing taken in photometric conditions, and cross-calibrating the other data using stars common to both.
Photometry for the brightest cluster, SC7, proved problematic. In the archival CFHT/MegaCam data, the cluster is saturated in the g- and r-bands and photometry can only be undertaken in the i-band. However, shorter exposures of cluster SC7 were also in the CFHT archive, taken for the purposes of photometric calibration. In these exposures, SC7 unfortunately lands on the edge of a CCD making measurements of the full cluster impossible. Hence, we use the central region of the shorter exposures to obtain the colours using an aperture radius of 1.5$\arcsec$. We then estimate the total magnitudes by using an aperture of radius 6$\arcsec$ on the long i-band image and applying the colour measurements to obtain total g- and r-band magnitudes. For cluster SC6, no such problem arose: the apertures employed for the deriving the color and total magnitude had radii of 2$\arcsec$ and 4.7$\arcsec$ respectively.
The very faint cluster SC5 was also difficult to photometer. This object is visible in a long CFHT/Megacam r-band stack (11000 seconds) but the g and i-band data, even when stacked, are too shallow to detect the cluster. The r-band stack was photometered with an aperture radius of 10$\arcsec$. SC5 was also found in archival Subaru data, confirming its status as a cluster.
Photometry in the CFHT/Megacam filter set was also converted to Johnson-Cousins V and I for SC6 and SC7. This was achieved by using the colour transform equations given on the SDSS web-pages[^4]. This was not possible for SC5 as we require photometry in more than the one filter for the transform equations.
Extinction is known to be a major issue with NGC 6822 due to its low Galactic latitude. @Battinellietal06 use the stellar population of NGC 6822 to estimate the foreground reddening across the area discussed in this paper, and find it is not only significant, but also patchy. Specifically, E(B-V) ranges from 0.19 to 0.30 (their Figure 2). We correct for this using the extinction maps – interpolated to the position of the new clusters – and relative extinction for the Sloan band-passes from @Schlegeletal98. NGC 6822 also has internal reddening and @Masseyetal95 find values of up to E(B-V) of 0.45 mags in the centre of the galaxy. However the new clusters lie far from the centre of NGC 6822 and should be minimally impacted by internal reddening. We note, however, that patchy Galactic extinction may limit the accuracy our final photometry.
ID g$_{0}$ r$_{0}$ i$_{0}$ V$_{0}$ (V-I)$_{0}$ M${_{V}}_{0}$ E(B-V)
----- --------- --------- --------- --------- ------------- --------------- --------
SC5 – 19.43 – – – – 0.219
SC6 15.55 15.01 14.79 15.28 0.84 –8.09 0.190
SC7 15.02 14.43 14.10 14.77 1.05 –8.60 0.207
: Photometric properties of the new clusters. Extinction corrections use values estimated from the extinction map of @Schlegeletal98. The g, r and i-magnitudes are in the CFHT/Megacam filter system. Photometric errors on the colours (derived for a inner aperture for SC6 and SC7 – see text) are estimated at $\pm$0.01 magnitudes. The major source of error for the total magnitudes is the uncertainty of the memberships of cluster stars within the aperture, which are estimated at $\pm$0.03 magnitudes.[]{data-label="tab:properties"}
![Ellipticity and PA as derived from IRAF/ELLIPSE for SC7. The PA is $\sim$50$\deg$ over radii of $\sim$10–30 pixels (c.f. circle on left panel of Fig.\[Fi:image\_AB\]). The ellipticity has a value of $\sim$0.25 over a large range of radii. The FWHM of the image is 4.7 pixels (dashed vertical line).[]{data-label="Fi:A_pa_ellipticity"}](n6822_A_ellipse.ps){width="85mm"}
Ellipticity of SC7
------------------
Visual inspection reveals that cluster SC7 is significantly elongated. We used IRAF/ELLIPSE[^5] to derive the ellipticity and position angle (PA) of the major axis of SC7 using a fixed centre and the results are shown in Figure \[Fi:A\_pa\_ellipticity\]. The PA beyond $\sim$ 12 pixels is 50$^{\circ}$ and the ellipticity has a value of $\sim$0.25 over the main body of the cluster. This high ellipticity is unusual for a GC and makes SC7 a clear outlier in a plot of M${_{V}}_{0}$ versus ellipticity (Figure \[Fi:ellipticities\]).
![The location of SC7 (filled circle) in a plot of M${_V}$-ellipticity. Also shown are MW GCs (diamonds) taken from the McMaster MW GC catalogue [@Harris96], M31 clusters taken from @Barmbyetal07 (crosses) and G1 [@Maetal07] (square), WLM-1 [@Stephensetal06] (triangle), cluster 77 from @Annibalietal11 (asterisk,) and NGC 121 [@Glattetal09] (thick diamond), where the magnitude for NGC 121 is the mean of the values they derive for a King and EFF fit to the profile. The most elliptical GC in the MW is M19, but this is known to be a result of differential reddening [@vandenBergh08]. []{data-label="Fi:ellipticities"}](n6822_A_ellipt_M31_MW_etc.ps){width="85mm"}
Discussion and Summary
=======================
The new clusters reported here substantially increase the number of classical GCs found in NGC 6822. If all the new massive clusters (excluding SC5, which is too faint to be found in comparable studies of local dwarfs) prove to be genuinely “old" GCs, then the four clusters in @Hwangetal11 and the two in this work would increase the specific frequency of NGC 6822 to $S_{N}$ $\sim$ 7, comparable to the newly-enlarged GC systems of NGC 147 and NGC 185 (Veljanoski et al, in prep.). This value is also consistent with values found for dwarf irregulars in the Virgo and Fornax galaxy clusters [@Sethetal04].
The cluster SC7 is of relatively high luminosity and SC6 is almost as luminous, with M${_{V}}_{0}$ $\sim$ –8 mags. The GC systems of M31 and the Milky Way have median values of M$_{V}$ are –7.9 and -7.3 respectively [@Huxoretal11], so both SC6 and SC7 are brighter than the turnover of the globular cluster luminosity function for these galaxies. Previously, in @Mackeyetal07 and @Huxoretal11, we found that M31 possesses a number of luminous GCs in its outer stellar halo, for which no counterparts exist in the Milky Way (excepting the very unusual cluster NGC 2419). If, as seems likely, the accretion history of M31 was different from that of the MW, we may have a natural source of M31’s luminous halo GCs in the accretion of systems such as NGC 6822. However such events would have had to happen at an early epoch since there is no evidence for young populations - which dominate in galaxies like NGC 6822 - in the M31 halo today.
The origin of high ellipticities in GCs, such as that of SC7, has been the source of some debate. @Kontizasetal90 find that the ellipticity for young SMC star clusters is greater than for the clusters in the somewhat more massive LMC, and similar results lead @Georgievetal08 to argue that the tidal field of the host galaxy is likely to be an important factor in determining cluster ellipticity. A scenario in which the SC7’s ellipticity is a consequence of it being formed in a dwarf galaxy host is also consistent with the presence of the extended clusters in NGC 6822. Indeed, @HurleyMackey10 argue that the formation and survival of extended clusters is a consequence of the more benign tidal fields in dwarf galaxies and the outer regions of massive galaxies. The origin of the high natal ellipticity, which a low tidal field preserves, may arise from a number of sources: rotation, galactic tides or anisotropy in the velocity dispersion between the major and minor axes. The latter was found to be the best explanation for the high ellipticity of the only GC known thus far in the dwarf galaxy WLM [@Stephensetal06].
It should be noted that an alternative scenario for the formation of luminous, elliptical GCs has also been proposed. In a study the star cluster system of the Magellanic-type starburst galaxy NGC 4449, @Annibalietal11 found that the brighter clusters tend to be more elliptical. One of their clusters (cluster 77) is old, massive and highly elliptical (see figure \[Fi:ellipticities\]), leading them to suggest that it may be the nucleus of a satellite galaxy that is currently being stripped [@Annibalietal12]. A similar picture has also been proposed for G1, the most luminous GC in M31[@Meylanetal01], and the anomalous Galactic GC $\omega$ Cen [e.g. @Romanoetal07].
One last notable aspect of the newly discovered GCs is that they lie in the linear arrangement noted by @Hwangetal11. As we have surveyed the full area in Figure \[Fi:fields\], this distribution cannot be a result of incomplete areal coverage. Such a disk alignment would not be unusual – the cluster population of the LMC exhibits disk-kinematics [@Schommeretal92; @Grocholskietal09] – but it would raise new questions about the formation of NGC 6822. If the GC system is found to exhibit disk-like kinematics, it might be hard to reconcile with a scenario where the galaxy formed via the merger of two similar mass gas-rich dwarfs [@Bekki08]. We have spectroscopic data for SC6 and SC7, and are currently obtaining data for other clusters in NGC 6822 to study the kinematics of the cluster system and address this question.
To summarise, we have presented the discovery of three new star clusters in the outskirts of NGC 6822 based on searches conducted with archival datasets. Two of these objects are massive compact GCs, very distinct from the extended clusters found by @Hwangetal11. The third is a very low-luminosity diffuse cluster. We have measured integrated photometry for these objects, but additional characterisation (e.g. structural parameters, stellar populations) will require deep high resolution data. SC6 and SC7 are so compact that R$_{h}$ is comparable to the FWHM in the data presented here. One of the clusters, SC7, is highly elliptical which we speculate could be due to the low tidal field it has experienced in the outer regions of a dwarf galaxy.
We note in closing that it is remarkable that SC6 and SC7 were not discovered earlier. These are high luminosity GCs in a Local Group galaxy that has been studied very extensively. This underscores yet again how the outer regions of galaxies have the ability to surprise and provide important clues about their histories.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partly supported by Sonderforschungsbereich SFB 881 “The Milky Way System” of the German Research Foundation (DFG). ADM is grateful for support by an Australian Research Fellowship (DP1093431) from the Australian Research Council. AHP, AMNF and ADM acknowledge support by a Marie Curie Excellence Grant from the European Commission under contract MCEXT-CT-2005-025869 during the early stages of this work. We also would like to thank Erwin de Blok for kindly providing the data for the HI map used in figure \[Fi:dss\].
Based on observations obtained with CFHT/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K.
[99]{}
Annibali F., Tosi M., Aloisi A., van der Marel R. P., Martinez-Delgado D., 2012, ApJ, 745, L1
Annibali F., Tosi M., Aloisi A., van der Marel R. P., 2011, AJ, 142, 129
Annibali F., Tosi M., Aloisi A., van der Marel R. P., 2011, AJ, 142, 129
Barmby P., McLaughlin D. E., Harris W. E., Harris G. L. H., Forbes D. A., 2007, AJ, 133, 2764
Battinelli P., Demers S., Kunkel W. E., 2006, A&A, 451, 99
Bekki K., 2008, ApJ, 680, L29
Bellazzini M., Ferraro F. R., Ibata R., 2003, AJ, 125, 188
Brook C. B., Governato F., Quinn T., Wadsley J., Brooks A. M., Willman B., Stilp A., Jonsson P., 2008, ApJ, 689, 678
Chandar R., Bianchi L., Ford H. C., 2000, AJ, 120, 3088
Cioni M.-R. L., et al., 2008, A&A, 487, 131
Cockcroft R., et al., 2011, ApJ, 730, 112
Cohen J. G., Blakeslee J. P., 1998, AJ, 115, 2356
Colucci J. E., Bernstein R. A., 2011, EAS, 48, 275
de Blok W. J. G., Walter F., 2000, ApJ, 537, L95
Gallart C., Aparicio A., Bertelli G., Chiosi C., 1996, AJ, 112, 195
Georgiev I. Y., Goudfrooij P., Puzia T. H., Hilker M., 2008, AJ, 135, 1858
Glatt K., et al., 2009, AJ, 138, 1403
G[ó]{}rski M., Pietrzy[ń]{}ski G., Gieren W., 2011, AJ, 141, 194
Grebel E. K., 2002, IAUS, 207, 94
Grocholski A. J., Parisi M. C., Geisler D., Sarajedini A., Cole A. A., Clari[á]{} J. J., Smith V. V., 2009, IAUS, 256, 287
Harris W. E., 1996, AJ, 112, 148
Hubble E. P., 1925, ApJ, 62, 409
Hurley J. R., Mackey A. D., 2010, MNRAS, 408, 2353
Huxor A. P., et al., 2011, MNRAS, 414, 770
Huxor A., Ferguson A. M. N., Barker M. K., Tanvir N. R., Irwin M. J., Chapman S. C., Ibata R., Lewis G., 2009, ApJ, 698, L77
Huxor A. P., Tanvir N. R., Ferguson A. M. N., Irwin M. J., Ibata R., Bridges T., Lewis G. F., 2008, MNRAS, 385, 1989
Huxor A. P., Tanvir N. R., Irwin M. J., Ibata R., Collett J. L., Ferguson A. M. N., Bridges T., Lewis G. F., 2005, MNRAS, 360, 1007
Huxor A., Tanvir N. R., Irwin M., Ferguson A., Ibata R., Lewis G., Bridges T., 2004, ASPC, 327, 118
Hwang N., Lee M. G., Lee J. C., Park W.-K., Park H. S., Kim S. C., Park J.-H., 2011, ApJ, 738, 58
Hwang N., et al., 2005, Near-field cosmology with dwarf elliptical galaxies, IAU Coll. Proc. Edited by Jerjen, H.; Binggeli, B. Cambridge: Cambridge University Press, p257
Keller S. C., Mackey D., Da Costa G. S., 2012, ApJ, 744, 57
Kontizas E., Kontizas M., Sedmak G., Smareglia R., Dapergolas A., 1990, AJ, 100, 425
Krienke K., Hodge P., 2004, PASP, 116, 497
Law, D. R., & Majewski, S. R., 2010, ApJ, 718, 1128
Ma J., et al., 2007, MNRAS, 376, 1621
Mackey A. D., et al., 2007, ApJ, 655, L85
Mackey A. D., et al., 2010, ApJ, 717, L11
Martin N. F., Ibata R. A., Irwin M. J., Chapman S., Lewis G. F., Ferguson A. M. N., Tanvir N., McConnachie A. W., 2006, MNRAS, 371, 198
Massey P., Armandroff T. E., Pyke R., Patel K., Wilson C. D., 1995, AJ, 110, 2715
Mateo M. L., 1998, ARA&A, 36, 435
Meylan G., Sarajedini A., Jablonka P., Djorgovski S. G., Bridges T., Rich R. M., 2001, AJ, 122, 830
Romano D., Matteucci F., Tosi M., Pancino E., Bellazzini M., Ferraro F. R., Limongi M., Sollima A., 2007, MNRAS, 376, 405
Schlegel D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525
Schommer R. A., Suntzeff N. B., Olszewski E. W., Harris H. C., 1992, AJ, 103, 447
Searle L., Zinn R., 1978, ApJ, 225, 357
Seth A., Olsen K., Miller B., Lotz J., Telford R., 2004, AJ, 127, 798
Stephens A. W., Catelan M., Contreras R. P., 2006, AJ, 131, 1426
Stonkut[ė]{} R., et al., 2008, AJ, 135, 1482
Strader J., Brodie J. P., Huchra J. P., 2003, MNRAS, 339, 707
van den Bergh S., 2008, AJ, 135, 1731
Wyder T. K., Hodge P. W., Zucker D. B., 2000, PASP, 112, 1162
[^1]: Email:avon@ari.uni-heidelberg.de
[^2]: http://ned.ipac.caltech.edu/
[^3]: The majority of these are HII regions. Hubble VI is a young cluster and the nature of Hubble IX is still unclear.
[^4]: http://www.sdss.org/dr4/algorithms/sdssUBVRITransform.html
[^5]: 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.
|
---
abstract: |
This thesis reports the $K$-band polarizations of a representative sample of nine radio galaxies: seven 3C objects at $0.7
< z < 1.3$, and two other distinctive sources. Careful consideration is given to the accurate measurement and ‘debiasing’ of faint polarizations, with recommendations for the function of polarimetric software.
3C 22
: has 3% polarization perpendicular to its radio structure, consistent with suggestions that it may be an obscured quasar.
3C 41
: also has 3% polarization perpendicular to its radio and may also be an obscured quasar; its scattering medium is probably dust rather than electrons.
3C 54
: is polarized at 6%, parallel to its radio structure.
3C 65
: is faint: its noisy measurements give no firm evidence for polarization.
3C 114
: has a complex structure of four bright knots, one offset from the radio structure and three along the axis. There is strong evidence for polarization in the source as a whole (12%) and the brightest knot (5%).
3C 356
: is faint: we do not detect any $K$-band continuation of the known visible/near-ultraviolet polarization.
3C 441
: lies in a rich field; one of its companions appears to be 18% polarized. The identification of the knot containing the active nucleus has been disputed, and is discussed.
LBDS 53W091
: was controversially reported to have a 40% $H$-band polarization. No firm evidence is found for non-zero $K$-band polarization in 53W091, though there is some evidence for its companion being polarized. The object is discussed in the context of other radio-weak galaxies.
MRC 0156$-$252
: at $z \sim 2$ is found to be unpolarized in $K$.
Simple spectral and spatial models for polarization in radio galaxies are discussed and used to interpret the measurements. The important cosmological question of the fraction of $K$-band light arising in radio galaxy nuclei is considered: in particular, the contribution of scattered nuclear light to the total $K$-band emission is estimated to be of order 7% in 3C 22 and 3C 41, 26% in 3C 114, and tentatively 25% or more in 3C 356.
author:
- Gareth James Leyshon
nocite: '[@*]'
title: 'The infrared polarizations of high-redshift radio galaxies'
---
\[2\][[|[\#1]{}\_[\#2%]{}]{}]{}
A three year project in astronomy relies on many factors to come to fruition: the guidance of one’s supervisor; chance remarks from colleagues; the tedious but very necessary work of those who mount archives on the World Wide Web; and most importantly, the availability of observatories, software and funding which makes it all possible!
Firstly, I would like to thank Steve Eales for his guidance over the last four years, and for his philosophy that ‘you don’t need to do lot of work to get a PhD’ – provided that [*sufficient*]{} work has been done! Also a big thank-you to Steve Rawlings at Oxford: had I not spent a month working efficiently on his radio galaxies in 1993 \[now published at long last! [@Lacy+99a]\], I might never have come to Cardiff.
Many thanks to all who gave constructive comments and advice throughout the last three years: Bob Thomson, Jim Hough, Chris Packham, Mike Disney and Mike Edmunds; Bryn Jones, Neal Jackson, Arjun Dey, Clive Tadhunter and Patrick Leahy. Thanks especially to Buell Jannuzi and Richard Elston for sharing their polarimetry results, Mark Dickinson for some optical magnitudes, Megan Urry for allowing me to reproduce a complicated diagram, and Mark Neeser for having his thesis in the right place at the right time.
Particular thanks to Jim Dunlop for help on my second observing trip and with the 53W091 data; and to Colin Aspin, Antonio Chrysostomou and Tim Carroll for help with making observations and data reduction. A special mention with many thanks to my A-level Statistics teacher, Eric Lewis!
This research has made use of the [nasa/ipac]{} extragalactic database ([ned]{}) which is operated by the Jet Propulsion Laboratory, CalTech, under contract with the National Aeronautics and Space Administration. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the U.K. Particle Physics and Astronomy Research Council. Thanks to the Department of Physical Sciences, University of Hertfordshire for providing IRPOL2 for the UKIRT.
Data reduction was performed with [starlink]{} and [iraf]{} routines. Thanks to Rodney Smith and Philip Fayers for their ceaseless efforts to keep Cardiff’s computers functional! The use of NASA’s [*SkyView*]{} facility ([http://skyview.gsfc.nasa.gov]{}) located at NASA Goddard Space Flight Center is acknowledged; as is that of the ADS abstract service at Harvard. This work was funded by a [pparc]{} postgraduate student research award.
Please note that certain conventions are adopted throughout this thesis:
- The results in this thesis are often quoted in the form of percentages (for polarizations, proportions of light from different sources, etc.). Whenever measurements are presented in the form $a \pm
b\%$, this should be read as $b$ being the absolute error on $a$ with both variables having the ‘units’ in percentages. The format of a percentage error on an absolute quantity is [**never**]{} used.
- Occasionally it has been necessary to use the same mathematical symbols in different ways in different chapters. Usage is always consistent within a chapter and the most mathematical ones conclude with a glossary of all symbols used.
- Position angles are always denoted $\phi$; the symbol $\theta$ is only used in polarization vector phase space.
- Assumptions about the cosmological parameters of the Universe are always explicitly stated where required; $h_0$ denotes the Hubble constant in units of 100 kms$^{-1}$Mpc$^{-1}$. Angular to linear scale conversion factors, when required, are taken from Peterson .
- Throughout this thesis, the term ‘optical’ is used to encompass the near infrared, visible light and the near ultraviolet, as opposed to ‘visible’, which explicitly means the region of the spectrum covered by the $R$, $V$ and $B$ bands.
- The different classes of active galaxies are defined in Chapter \[defineAGN\]. The term ‘quasar’ is used to cover both radio-quiet and radio-loud quasi-stellar objects.
- Each chapter is self-contained in abbreviations for papers cited. Any abbreviations used in a chapter are defined in the introduction to that chapter.
- The work is written in the first person plural, the scientific ‘we’, throughout. This does not imply collaboration in authorship except where explicitly noted by footnotes.
\[oddtest\]
Active Galaxies and the Unification Hypothesis {#defineAGN}
==============================================
> Meddle first, understand later. You had to meddle a bit before you had anything to try to understand. And the thing was never, ever, to go back and hide in the Lavatory of Unreason. You have to try to get your mind around the Universe before you can give it a twist.
>
> ------------------------------------------------------------------------
>
> — Ponder Stibbons, [*Interesting Times*]{}.
The study of the most distant objects in the Universe is a demanding task. The maximum amount of information must be gleaned from the minimal flux of photons reaching Earth. When a target is so faint that our best image consists of a few bright pixels on an infrared array, there seems little hope of probing the structure hidden within. Yet even the faintest light, limited by diffraction or seeing, carries with it a hidden property: polarization. This is the tool which has been investigated and used in this work, to reveal new data on nine radio galaxies.
Active Galaxies {#intunimod}
---------------
Humanity’s understanding of the Universe has developed radically since Immanuel Kant first speculated about the existence of ‘island universes’ in the eighteenth century. In 1845, Lord Rosse completed construction of his great reflecting telescope, and subsequently discovered spiral structure in many of Messier’s nebulae. By 1920, it was seriously argued that the spiral nebulae were in fact galaxies external to our own - epitomised by the ‘Great Debate’ of Astronomy between Curtis and Shapley that year [@Hoskin-76a]. Hubble’s determination of the distance to the spiral nebulae resolved the debate, and in the years that followed, the vast majority of galaxies were found to be of elliptical or spiral formation.
The advent of radio astronomy opened up a second waveband through which the Universe could be studied, and by the late 1960s, radio astrometry was sufficiently accurate that radio sources could be identified with their optical[^1] counterparts. It became apparent that many elliptical galaxies were strong radio sources – forming the class of [*radio galaxies*]{} [@McCarthy-93a]. Most of the sources consisted of double radio lobes spanning a distance 5-10 times the size of the parent galaxy at the centre.
At the same time, numerous other classes of unusual galaxies or intense radio sources were revealing themselves to new instruments. Earliest to become apparent was the class of Seyferts, galaxies (normally spirals) with unusually bright nuclei whose spectra included narrow ($\sim 1,000$ km/s) permitted and forbidden emission lines. Some Seyferts also exhibited broader ($\sim 10,000$ km/s) permitted emission lines, and were branded ‘Type 1’, while those without were spectroscopic ‘Type 2’. Seyferts exhibit radio emission, but this is usually weak. The lines were accompanied by a ‘featureless continuum’ whose profile was flat rather than the curve characteristic of blackbody thermal emission [@Robson-96a and references therein].
Meanwhile, radio surveys had also identified sources whose optical counterparts were found to be brilliant and pointlike: these were named quasars, the quasi-stellar radio sources. Like Type 1 Seyferts, quasars exhibited a flat spectrum optical continuum with strong emission lines, both narrow and broad. In 1963, quasar emission lines were first identified with an element: 3C 273’s emission lines were found to be characteristic of hydrogen at high redshift, $z=0.158$ [@Peterson-97a and references therein].
The radio-loud quasars were found to be excessively luminous in the $U$-band compared with stars and normal galaxies, which prompted optical surveys to hunt for more objects with ultraviolet excesses. These surveys discovered many more quasi-stellar objects with similar spectra, and 90-95% of all quasars[^2] are now thought to be radio-quiet.
Collectively, Seyferts, quasars and radio galaxies became known as ‘active galaxies’, the [*Collins Dictionary of Astronomy*]{} definition [@Illingworth-94a] being ‘galaxies that are emitting unusually large amounts of energy from a very compact central source — hence the alternative name of [*active galactic nuclei*]{} or [*AGN*]{}’. Classification of an object as an AGN may be made because the active nucleus has been observed directly, or be inferred from the presence of radio lobes. Certain extremely energetic AGN clearly dominated by an optically bright nucleus became known as ‘blazars’ [@Antonucci-93a §3.1].
At first it was unclear whether the wide-ranging class of ‘AGN’ was simply phenomenological, or whether the different types of AGN were linked by an underlying physical mechanism. All species of AGN demanded a mechanism whereby a much greater energy output might be obtained from the heart of a galaxy than could be accounted for by stellar nuclear fusion; the mechanism would have to be capable of giving rise to a flat spectrum in both radio and optical wavelengths, provide for the presence of hot clouds of gas emitting radiation at particular wavelengths, and allow for the presence or absence of radio jets.
Now it is generally accepted that the underlying mechanism in all these objects is the accretion of matter on to a black hole [@Antonucci-93a; @Urry+95a]. Infalling matter forms an accretion disk, heated by viscous and/or turbulent processes, which glows in the ultraviolet and possibly soft X-rays. Hard X-rays are emitted in the innermost part of the disk. Clouds of gas close to the black hole move rapidly in its gravitational potential, and produce line emission at visible and ultraviolet wavelengths — these form and occupy the Broad Line Region (BLR). Well beyond the accretion disk, gas and dust forms a second, warped, disk or torus. This torus screens the BLR from view in those AGN whose line of sight to the Earth is not close to the axis of the torus. Energetic particles escape in well-collimated jets at the poles of the torus. Gas clouds further from the active nucleus travel at lower velocities: not obscured by the torus, such clouds emit light whose emission lines suffer less Doppler broadening. Hence narrow lines are seen in all forms of AGN, whereas in those AGN oriented so our line of sight is ‘down the jet’ we see the otherwise obscured broad line regions and/or continuum light from the central engine. This model mechanism, illustrated in Figure \[unifig\], is generally known as the [*Unification Model*]{} of AGN. At present, this stands as the ‘best buy’ model for AGN, but is not universally accepted — especially in the case of quasars.
[The diagram below shows the postulated structure of an active nucleus according to the Unification Hypothesis. The molecular torus is cut away at the front to show the broad line region clouds and core. The black hole at the centre is surrounded by an accretion disc. This figure is reproduced from Urry & Padovani , © PASP, reprinted with permission of the authors. ]{}
Drawing on the spectroscopic classification of Seyferts, AGN generally are now classified ‘Type 1’ and ‘Type 2’. Type 2 objects are those with no evidence of a direct view of their central engines: radio galaxies exhibiting only narrow emission lines (NLRGs) join Seyfert 2s in this category. Type 1 objects are those which do seem to include radiation from the central engine, and Seyfert 1s are joined by BLRGs (broad-lined radio galaxies – which also show the narrow lines), and by quasars.
One important prediction of the Unification Model is that Type 2 objects, with their torus axis being aligned roughly in the plane of the sky, ought to include broad line regions (BLRs) whose light, though obscured from Earth, escapes into the plane of the sky. Dust particles or electrons in the host galaxy or in the clouds responsible for the narrow lines should scatter some of this light into our line of sight. When light becomes scattered, it becomes linearly polarized in the sense perpendicular to the pre- and post-scattering flight axes of the photon; hence linearly polarised imaging or spectroscopy of AGN should reveal light from BLRs polarised perpendicular to the direction of the radio jet (presumed to be aligned with the opening of the torus).
The motivation of this thesis is to search for evidence of such polarization in $K$-band infrared light from radio galaxies – a waveband thought, but not proven, to be dominated by light from the host galaxy’s stars rather than the active nucleus. Findings of $K$-band polarization would set important constraints on the relative strengths of the nuclear and stellar components. Accordingly, the next chapter presents a review of the [*status quo*]{}[^3] in our knowledge of the relationship between radio galaxies’ radio structure and their morphological and polarization properties. First, however, we must look at the ‘big picture’ of the different classes of AGN known to exist, and how they might be related to one another if the best-buy Unification Hypothesis is correct.
The Unification Hypothesis for AGN
----------------------------------
Contemporary authors embrace the Unification Model as the accepted model for AGN with various degrees of enthusiasm: Antonucci rushes to set it up as the ‘straw person model’ against which he reviews the current observational evidence; for Robson it is a solid foundation, while Peterson’s approach is more cautious. There is no other serious contender to explain the wide range of AGN phenomena, although in certain individual objects [@Peterson-97a §§2.4, 3.4] starbursts rather than black hole accretion may form the hidden engine driving the radiation output.
A fundamental division between the various classes of AGN is their radio strength. Radio galaxies, by definition, are radio loud. Seyferts are empirically found to be radio quiet. All blazars, without known exception, are radio loud; 90–95% of quasars are radio quiet. The various classifications of AGN stem historically from the (often extreme) prototypes of each class first discovered, and are not always helpful in classifying less extreme examples: for instance, a radio-loud galaxy with an obvious active nucleus would now be classified a radio galaxy rather than a Seyfert, so the absence of radio-loud Seyferts is a consequence of taxonomy rather than physics. While some attempt has been made to define the different AGN classes more rigorously [@Peterson-97a ch.2], the older literature and human nature militate against the use of clear-cut terms to distinguish different classes of object – objects which are hypothesised to lie on a continuum of classes in any case!
### Seyfert Galaxies {#polinSeyferts}
As summarised earlier, the first key to unification came from studies of Seyfert galaxy spectra. Seyferts are now defined as low luminosity AGN with absolute magnitudes $M_B > -21.5 +
\log h_0$ [@Peterson-97a §2.1]; it follows naturally, therefore, that all known Seyferts are at low redshift and their morphology is open for study. Nearly all Seyfert AGN are found to lie within spiral galaxies [@Peterson-97a §8.1.1], often of type Sa or Sb, and the host galaxies are more likely than normal spirals to be barred and/or deformed. Robson notes that in the rare cases where Seyferts are radio-loud, they often have other peculiar characteristics.
Spectral studies of Seyferts led to the Type 1 / Type 2 classification based on the presence or absence of broad spectral lines. The discovery of broad lines in Type 2 Seyfert NGC 1068 in polarised light [@Antonucci+85a] prompted the realization that the Broad Line emitting Region (BLR) must lie within some geometrical feature which screened it from direct view. This screening feature – the postulated molecular torus – is typically of diameter $\sim$ 100 pc, and should not be confused with the accretion disk in the central engine, measuring perhaps 0.03 pc. A typical schematic diagram is Figure 7.1 of Peterson ; an excellent cartoon sketching the structure of an AGN at eight different scales ($10^{-4}$–$10^{+6}$ pc) is borrowed from Blandford by Robson .
Recent studies of the near-infrared ($H$-band) polarization of NGC 1068 [@Young+96a; @Packham+97a] have been found to be consistent with a scattered light hypothesis and have even allowed the likely position and orientation of the molecular torus to be identified: the torus in this case has a diameter greater than 200 pc.
Miller & Goodrich studied eight further Seyfert 2s to see if spectropolarimetry would reveal Seyfert 1 features, choosing objects already known to have high broadband polarizations. Four gave definite positive results, two gave definite negatives, and two failed to produce reliable signal-to-noise. The four polarised galaxies revealed polarized features consistent with Seyfert 1 properties, albeit with a degree of ‘bluing’ in the spectrum indicative that dust scattering must be a contributing mechanism. Three of these, and possibly one of the two low signal-to-noise sources, exhibited perpendicular alignment between the polarization orientation and the radio axis. Similar perpendicular alignments had also been detected in Seyfert 2s by Antonucci . Weak [*parallel*]{} polarizations have been observed in a few Seyfert 1s [@Antonucci+90a and references therein].
The [*nuclear*]{} polarization levels obtained by Miller & Goodrich were only of the order of a few percent — rather low if the underlying mechanism is the scattering implied by the perpendicular alignment. Antonucci speculates that they may have underestimated the contribution of host galaxy starlight, and that the true polarization may be closer to the 16% level observed in NGC 1068.
Further tests for the Unification Model in Seyferts are reviewed by Antonucci — some may indicate refinements that need to take account of additional parameters in the model (e.g. the opening angle of the molecular torus) but none fatally wound the principle of Unification. Claims that some Seyfert 2s contain no BLR emission [@Antonucci-93a §2.6.2] call for more sensitive spectropolarimetry before they can seriously question Unification: Robson and Peterson note that this is a hot area of current research. Until proven otherwise, it can be safely stated that Seyfert galaxies fall into two distinct classes: broad line (Type 1) objects which sometimes exhibit weak polarizations parallel to the radio structure, and narrow line (Type 2) objects which often display strong perpendicular polarizations.
### Blazars
The precise definition of a blazar seems to depend on which source is consulted. I will follow Robson’s helpful advice that the term refers to a phenomenon rather than a simple class of object: specifically, the phenomenon of a relativistic jet beamed roughly in the direction of terrestrial observers, dominating the radio thru infrared spectrum with its non-thermal synchrotron emission. The blazar phenomenon is exhibited by three classes of object: BL Lacertae objects (BLLs), optically violent variable sources (OVVs), and highly polarized quasars (HPQs).
BLLs are distinguished and defined by the lack of emission or absorption features in their spectra. They often exhibit variability (changing their output by several magnitudes in the space of a few weeks) and usually lie in elliptical galaxies, though spiral hosts are also known. OVVs are AGN (with spectra including broad emission lines) which exhibit short timescale luminosity variations ($\ga 0.1$ mag) over timespans as short as a day [@Illingworth-94a; @Peterson-97a § 2.5]. No radio-quiet OVVs or BLLs are known [@Jannuzi+93a].
Antonucci argues strongly that the distinction between BLLs and OVVs is ill-founded – especially given that the sources’ very variability can switch them between the two categories – and that in fact all radio-loud AGN with radio structures dominated by emission from the core are part of the same family of objects. The optical components of core-dominated radio-loud AGN tend to be quite red, highly variable, and polarized: this is proposed to be the high-frequency tail of the core synchrotron emission. Only emission from the core can vary coherently over timescales of weeks or days. He proposes that BLLs are simply the extreme cases where the synchrotron emission utterly dominates other components of the optical output, and predicts that more sensitive spectropolarimetry of BL Lacertae objects would reveal faint unpolarised broad emission lines from the BLR clouds basking in the synchrotron jets. Conversely, he also suggests that those core-dominated radio-loud AGN not classed as blazars would reveal a faint red optical core under careful scrutiny.
Robson also includes HPQs in his phenomenological class of blazars. Scarpa & Falomo recently compared the optical properties of a sample of HPQs and BLLs, finding that optical properties of radio-selected BL Lacertae objects were very similar to those of highly polarised quasars. An earlier survey comparing high- and low-polarization quasars [@Moore+84a] found that all but two of their HPQs were radio-loud, and the two radio-quiet quasars had their own peculiarities. The orientation angles of the HPQ linear polarizations seemed randomly distributed with respect to the radio axes; this bears out the core emission hypothesis, as the synchrotron mechanism produces light whose polarisation orientation has no relationship with the radio jet geometry. It seems eminently reasonable to accommodate HPQs between less extreme core-dominated radio-loud AGN and the OVVs on Antonucci’s unified blazar scheme.
### Quasars
Complementing the definition of Seyferts, above, quasars are now defined as AGN with absolute magnitudes $M_B < -21.5 +
\log h_0$ [@Peterson-97a §2.2]. A small proportion (5-10%) are known to be radio-loud; spectroscopically, quasars exhibit spectra similar to Type 1 Seyferts [@Peterson-97a §7.4.1]. Since Moore & Stockman found HPQs to be quite distinct from low polarization quasars (LPQs) we have already dealt with HPQs as blazars; and Stockman, Moore & Angel undertook a specific study of the LPQs. The cut-off cannot be precisely defined, but 3% polarization is normally taken as an effective working threshold in the literature.
Stockman, Moore & Angel found that the typical LPQ polarization was around 0.6% and tended to be aligned [*parallel*]{} with the radio axis. There was no strong evidence for temporal variability in the degree or orientation of polarization, with upper limits of $\Delta p /
p \leq 0.16$ and $\Delta \phi \leq 8 \degr$. In a sub-sample of LPQs mostly selected in the radio, the $B$-band polarization was typically 50% greater than the $R$-band value; the equivalent test was not performed on optically-selected LPQs. The lack of variability and the tendency for polarization to increase at shorter wavelengths rules out a blazar origin for the polarized light in LPQs: models invoking scattering off dust grains or electrons in a disk or oblate cloud could account for such polarization but the mechanism is still very unclear. Stockman, Angel & Miley , Antonucci and Moore & Stockman provide evidence for a bimodal (parallel/perpendicular) distribution of scattering angles; perpendicular alignments can be easily accounted for by the usual mechanism. Antonucci & Barvainis suggest that the parallel LPQs and the few Seyfert 1s that exhibit weak parallel polarization may contain disks or tori with very large opening angles, which could produce parallel polarization by scattering.
The Stockman, Moore & Angel survey covered bright objects from a variety of catalogues, and was not statistically complete in any meaningful sense. To complement it, Fugmann & Meisenheimer studied a sample of faint 5GHz radio sources, and more recently, Impey, Lawrence & Tapia studied the optical polarization of a complete sample of radio sources, also selected at 5GHz.
Impey, Lawrence & Tapia’s complete 5GHz sample included both radio galaxies and quasars. Since HPQs are known to exhibit strong variability, it is possible that they may sometimes drop below the 3% threshold and are at risk of being labelled LPQs on the strength of a single measurement. Discovering a trend of polarization increasing with radio compactness, Impey, Lawrence & Tapia note that they ‘cannot exclude the possibility that [*all*]{} quasars with compact radio emission have $p_{\mathrm
max}>3\%$, at least some of the time’. Again, this would support a division of radio-loud quasars into those which are part of the blazar family, aligned such that their radio core would appear compact, and those whose beaming axis is not so closely aligned with the line of sight to Earth. Fugmann & Meisenheimer’s results also suggest that many compact radio objects not otherwise noted for optical variability exhibit polarization properties characteristic of blazars. Robson notes that the polarization properties of [*radio-quiet*]{} quasars have not been well measured, but are tightly constrained in the optical as being very low — low enough to be attributed to thermal emission.
One unanswered question for the Unification Model is why Type 2 spectra are not seen in quasars. Peterson offers two suggestions: that the molecular torus surrounding such a powerful central engine is thinned to the point of ineffectiveness; or that ‘Quasar 2s’ exist but have been misidentified as something else, perhaps the ultraluminous far infrared galaxies [@Sanders+88a].
Robson pursues the latter hypothesis in the shape of IRAS galaxy IRAS FSC 10214+4724. This remarkable object, at $z=2.286$, appears to be gravitationally lensed, to be undergoing a starburst phase, [*and*]{} to contain an active nucleus! Images [@Lawrence+93a] taken through polarizing filters reveal a polarization of about 16% regardless of aperture, but ambiguous indications of any Alignment Effect. Polarized spectra [@Goodrich+96a] reveal broad quasar-like emission lines. Dust scattering from an active nucleus is proposed as the most likely source of the polarization, but scattered light from a blanketed starburst might also provide an explanation. IRAS 09104+4109 [@Hines+93a] is also notable as an IRAS galaxy containing a powerful radio source and with a constant nuclear polarization of $\sim 18 \%$, although the polarization is misaligned with its radio structure (possibly due to the geometry of thin regions in its blanketing dust). Both Antonucci and Robson speculate that future analyses of the most luminous IRAS galaxies will reveal some (perhaps ten percent) of them to be hiding the missing Type 2 quasars.
Radio Galaxies
--------------
Radio Galaxies form the remaining category of AGN. Most of the AGN sources considered so far have been radio-quiet, except for the blazars which are dominated by strongly beamed radiation over many decades of their spectrum. Radio galaxies join the 5-10% of quasars in the distinct class of radio-loud AGN. Reviewing the status of high redshift radio galaxies, McCarthy notes that the distinction between radio galaxies and radio quasars is becoming blurred as the host galaxies of quasars have been identified for quasars with $z \la 0.5$. Classically, the distinction had been that a powerful radio source was a ‘quasar’ if the host galaxy could not be seen beneath the active nucleus; an ‘N galaxy’ if the nucleus was exceptionally bright but did not wash out all traces of the starlight; or an ordinary radio galaxy otherwise.
### The spectra of radio galaxies {#RGspectrum}
Today, radio galaxies are classified on the basis of two distinct sets of properties: their optical emission lines and their radio structure. McCarthy notes that Broad Line Radio Galaxies (BLRGs), i.e. those with H[i]{} lines having widths over 2,000 km/s, tend to have the morphological classification of ‘N galaxies’, and their broad line spectra are similar to those exhibited by Seyfert 1s. Narrow Line Radio Galaxy (NLRG) spectra have only narrow lines for both permitted and forbidden transitions; BLRGs have narrow forbidden line spectra similar to those of NLRGs.
A distinction is often made between ‘nuclear’ and ‘extended’ emission, but isolating the nucleus from any extended emission region is not trivial when a 2 slit encompasses more than 10 kpc of an object at $z>1$. If spectroscopy can be performed on distinct regions of a radio galaxy (an operation not possible with unresolved quasars), the properties of the spectra would enable the composition of the different parts of the galaxy to be identified. Similarly, imaging polarimetry has the potential to be an invaluable tool to determine the properties of different parts of the emission. But both techniques are limited in practice by the faintness of the galaxies [@Cimatti+96a].
The optical radiation emitted by radio galaxies is thought to be a combination of starlight and nebular emission from the host galaxy, and a quasar-like (power-law) component originating in the active nucleus hidden in the heart of the galaxy. \[Note that at this stage we need make no assumptions about the [*reason*]{} for the shape of the quasar spectrum, but only utilise its profile. It has been suggested [@Binette+88a] that the quasar spectrum could be synthesised from a suitable combination of blackbody curves.\]
Manzini & di Serego Alighieri tested this three-component hypothesis by modelling radio galaxy spectra at rest frame wavelengths from 0.2[$\umu$m]{} to 1.0[$\umu$m]{}. Starlight from the host galaxy was modelled as the synthetic spectrum of Bruzual & Charlot for a galaxy with an initial burst of star formation and no subsequent formation. Nebular emission was modelled with a spectrum selected from Aller . The active nuclear component was modelled by the composite radio-loud quasar spectrum of Cristiani & Vio , and attenuated according to different distributions of dust grains which might be present to scatter nuclear light into the line of sight to Earth.
Manzini & di Serego Alighieri applied their modelling to a small sample of radio galaxies at redshifts ranging from 0.11 to 2.63, and have demonstrated that their observed magnitudes (by multiwaveband photometry) are consistent with artificial spectra synthesised from three such components. The contribution of the starlight becomes greater to longer wavelengths, while the nuclear component decreases. For five out of their six galaxies, the stellar component of the light has become dominant by a rest-frame wavelength of 0.5[$\umu$m]{}; in 3C 277.2 $(z=0.766)$ the starlight only exceeds the nuclear component at about 0.85[$\umu$m]{}. Hence the ‘galaxy plus quasar’ model predicts that starlight should dominate the infrared output of radio galaxies, while nuclear emission is predominant in the ultraviolet. Hammer, LeFèvre & Angonin confirm that the ultraviolet $\lambda < 400$nm light from $z \sim 1$ 3C radio galaxies is dominated by the presence of an active nucleus.
We have already noted (§\[polinSeyferts\]) that the distinction between BLRGs and NLRGs can be interpreted as a Type 1 / Type 2 orientation effect, with the BLR obscured by an assumed molecular torus in those galaxies classed NLRGs. Quasars and radio galaxies have been shown to have comparable emission line luminosities, arising in emission line regions less than 1 kpc in diameter [@Spinrad-82a]. If the Unification Model is the correct model to apply to radio galaxies, then Manzini & di Serego Alighieri are correct to model their ‘quasar’ component as scattered into the line of sight by dust; and their results show that radio galaxies can be accurately modelled as containing quasar cores (with molecular tori of dimensions less than 1 kpc), with core light scattered by plausible (albeit idealized) distributions of dust.
### The Hubble diagram: radio galaxy evolution {#HubbleK}
If the infrared emission of radio galaxies is dominated by starlight, then studies of the variation of their $K$-band magnitudes with redshift should tell us something about galactic evolution. Lilly & Longair produced a $K-z$ plot, or [*Hubble diagram*]{}, for 3CR radio galaxies, i.e. for those radio galaxies with the most powerful radio emissions. Two features were evident in the resulting Hubble diagram: the dispersion of the $K$-magnitudes about the average value remained constant up to $z \ga 1$, but the average magnitude evolved with redshift such that galaxies at $z \sim 1$ were about 1 mag more luminous than at $z = 0$.
Eales & Rawlings summarise the natural interpretation of these findings: low dispersion implies that the radio galaxies were not passing through any transient phase in their evolution (which would have caused wider variation in their luminosity) over the span of redshifts covered. The declining luminosity to lower redshifts is consistent with a period of star formation at $z > 5$, followed by passive evolution as stars of decreasing mass reach the end of their lives. This seems perfectly reasonable since nearby radio galaxies are known to lie within giant ellipticals with a small spread of absolute magnitude [@Laing+83a], and a similar evolutionary model has been suggested for radio-quiet elliptical galaxies [@Eggen+62a]. Imaging of the rest-frame visible structure of radio galaxies out to $z \sim 2$ shows many of them to have dynamically relaxed structures, suggesting that these are active elliptical galaxies, too [@McCarthy+92a; @Rigler+92a; @Cimatti+94a].
It should be noted, however, that many distant radio galaxies at $z>0.6$ [@McCarthy-93a; @Cimatti+94a see below] do not have elliptical morphologies: an evolving elliptical model alone cannot explain the disturbed morphology of these objects, so at best a modified evolving elliptical model is needed. \[It has also been suggested that the radio galaxies are in fact young objects which pass through a radio-loud phase only a few hundred Myr after a rapid star-forming phase itself lasting of order 100 Myr [@Chambers+90a].\]
Lilly & Longair’s Hubble diagram suffers from the unavoidable selection effect that 3CR galaxies contain the most powerful radio sources. Eales et al. therefore analysed a 90% complete set of radio galaxies selected at lower radio luminosities in the B2 and 6C catalogues, and created a Hubble diagram allowing the $K$-magnitudes of 3CR galaxies to be compared to those whose radio output was only one-sixth as strong. Analyzing the diagram above and below the natural threshold of $z=0.6$, they found that the low redshift B2/6C galaxies had $K$-magnitudes statistically identical to the 3CR sample, but at $0.6<z<1.8$, the 3CR sample was brighter by a median 0.6 magnitudes. Demonstrating that sources of bias have either been eliminated or would make their result stronger, Eales et al. argue that the $K$-band emission of the brightest radio galaxies must be contaminated by light from a source whose luminosity is correlated with the radio strength of the galaxy – presumably direct or indirect $K$-band emission from the active nucleus itself (but see below).
Eales & Rawlings note that Hubble diagram for the 6C/B2 galaxies (assumed to be unpolluted by nuclear emission) follows a curve for [*no*]{} stellar evolution; the 3C Hubble diagram which had previously been interpreted as indicating stellar evolution rather represents a series of galaxies showing no evolutionary effects, hosting nuclear sources which tend to be brighter at higher redshift by the selection effect of a flux-limited radio sample. It has been argued [@Best+98a] that the apparent ‘no evolution’ result occurs because of a cosmic conspiracy: the host galaxies of the radio sources evolve in the same way as radio-quiet Brightest Cluster Galaxies (BCGs). The $K-z$ Hubble diagram for BCGs also suggests an unphysical no-evolution scenario, which must be accounted for by postulating evolution in the galactic structure whose net effect counteracts that of stellar evolution. The most likely explanation is ongoing formation according to hierarchical clustering models. Radio galaxies are preferentially found in clusters at high redshift, so it would not be surprising for their behaviour to follow that of BCGs; while the fact that low-redshift radio galaxies are not preferentially found in clusters should make us suspicious of accepting Lilly & Longair’s continuous Hubble diagram across the $z=0.6$ morphology break.
Best, Longair & Röttgering argue that $K$-band emission from the active nucleus cannot contribute more than 15% (typically 4%) to the brightness of a 3CR radio galaxy, nor cause more than 0.3 mag of brightening in 3CR objects over 6C objects. Ruling out the possibility that 3CR objects contain more young stars, they suggest rather that 3CR objects simply contain greater masses of stars, and cite evidence [@Kormendy+95a] that if the central engine is a black hole whose accretion rate depends on the material available in the host galaxy, then the mass of stars in the galaxy will be correlated with the radio, and hence the optical [@Willott+98a; @Serjeant+98a].
### The radio morphology of radio-loud AGN
The radio classification of radio-loud AGN distinguishes those which are ‘lobe-dominated’ and those which are ‘core-dominated’ [@Robson-96a §3.7]. A more detailed discussion of the different radio structures observed in radio galaxies is given by Miley .
Measurements of radio flux density, $S$, at different frequencies allow a power law spectrum to be fitted to the source, characterised by a spectral index $\alpha$, such that $S \propto \nu^{-\alpha}$. Core-dominated radio sources tend to have flat spectra ($\alpha \sim 0$) and often show a single kpc-scale jet; in fact, most core-dominated radio sources would fall into the category of blazars rather than radio galaxies [@Robson-96a §3.7.2] — and there is recent evidence that radio galaxies which are core-dominated exhibit variable optical polarization due to synchrotron emission [@Cohen+97a; @Tran+98a].
Lobe-dominated radio structures emit radio waves from a locus of space which can span many tens of kiloparsecs, up to 3 Mpc in the case of 3C 236 [@Robson-96a §3.7.1]. The extended lobes of radiation are zones of synchrotron emission, and are fed by a stream of relativistic electrons flowing out of the poles of the central engine. The radio spectra of lobe-dominated sources tend to be steep, with $\alpha \sim 1$.
A further division is made according to the criteria of Fanaroff & Riley , whence sources with spectral luminosity density $P_{\mathrm 178\,MHz} > 5 \times
10^{25}$ WHz$^{-1}$ are class FR II, and those less luminous are class FR I. The class FR I sources are usually associated with radio galaxies alone, and the most prominent parts of the the radio structure (‘hot spots’) lie closer to the core than to the edge of the radio structure.
Both quasars and radio galaxies can exhibit class FR II structure, and their hotspots lie closer to the edge of the emission region than to the core [@Illingworth-94a]. Such sources usually have a compact radio core co-located with the optical nucleus of the galaxy. All radio-loud quasars and many class FR II radio galaxies are asymmetric, with a kpc-scale jet only visible between the core and one of the lobes [@Robson-96a §8.1]. The most likely explanation of the asymmetry again invokes an orientation argument, with those jets travelling towards us closest to our line of sight being most Doppler-brightened and the counterjets similarly Doppler-supressed [@Robson-96a §8.2.1]; hence quasars, where we are thought to be looking close to ‘down the jet’, are always asymmetric, while radio galaxies are viewed closer to ‘sideways on’ and the two jets can appear to be of comparable brightness. This has been borne out by the discovery of the Laing-Garrington effect [@Antonucci-93a §3.4], where the radio emission from the far side of such a source suffers depolarization by its passage through the interstellar material in the host galaxy and hence appears less polarized than the radio emission from the lobe associated with the jet.
For nearby radio galaxies, class FR II sources are normally found in (otherwise normal) giant elliptical galaxies, although not in galaxies forming part of rich clusters. Nearby FR I sources, however, are very likely to be hosted by more luminous ellipticals, often the type D or cD galaxies which dominate rich clusters [@Robson-96a §3.7.1]. At high redshift, though, there is no clear evidence for a distinction between the richness of clusters hosting FR I and FR II classes; and the morphology of the host galaxy is often distorted by the presence of knots. We will return to this in the next chapter, in a discussion of the Alignment Effect.
### Radio source unification revisited
The presence or absence of strong radio emission seems to be a fundamental characteristic of AGN, and is strongly linked with optical morphology: Peterson notes that it is ‘true in general’ that radio-quiet AGN (Seyferts, most quasars) are found in spiral galaxies \[but see Ridgway & Stockton and references therein for evidence of elliptical hosts being common in radio-quiet quasars\], while strong radio sources (radio quasars, radio galaxies and blazars) tend to have elliptical hosts. Further, the host galaxies of radio quasars are, on average, 0.7 mag brighter in absolute $B$-magnitude than radio-quiet quasar hosts. The absence of FR I class quasars may follow from the strength of the central engine: if the nucleus is powerful enough to be optically classified a quasar, then its jets may [*de facto*]{} be strong enough to produce FR II class radio structure.
The first indications that orientation effects may be important in understanding the nature of radio sources came with the discovery of apparent superluminal motion in four of the brightest radio sources [@Cohen+77a]. Superluminal observations can be understood if the source is ejecting matter at relativistic velocities along a path close to the line of sight to the observer [@Rees-66a; @Blandford+79a]; in which case superluminal sources must be the beamed subset of some parent population. This model is now generally accepted as the explanation of superluminal radio sources and is consistent with other observed features of the superluminal sources – including blazar properties and asymmetric jets [@Barthel-89a].
The Unification Model predicts that radio galaxies should not exhibit superluminal motion. Of Cohen et al.’s four original superluminal sources, three were quasars but the fourth was classed as a radio galaxy. This object, 3C 120, is a nearby $(z=0.033)$ galaxy which has been variously classified as a core-dominated broad-line radio galaxy, and as a radio-loud Seyfert 1 with disturbed morphology [@Grandi+97a]. If we set aside 3C 120 as an anomalous object, what of other AGN? A survey of relativistic motion in all sources of known VLBI core size, appearing in the literature 1986–1992, was assembled by Ghisellini et al. . Of the sources for which speeds were recorded, definite superluminal velocities were observed in all 11 BLLs, 23 out of 29 quasars (with 5 more having superluminal upper limits), and none of 6 radio galaxies. One radio galaxy (0108+388) displayed an apparent speed of $1.0c$, and another (0710+439) had an upper limit of $2.5c$ quoted. Neither of these findings are strong enough to prove the existence of superluminal motion in a radio galaxy.
The Unification Model hypothesises that radio-loud quasars are an oriented subset of some intrinsically radio-loud parent population. Barthel poses the question, ‘Is every quasar beamed?’, and reviews the evidence. As discussed above, those quasars which show superluminal motion are likely to be beamed, and we have also considered the evidence of the Laing-Garrington effect. A beaming hypothesis can explain several statistical properties, including the correlation between brightness and component motion, and more limited statistics showing that lobe-dominated sources display lower superluminal velocities than core-dominated (more closely aligned?) sources. Against that must be set the problem of why certain radio-loud quasars have very large extended structures, whose deprojected linear size would be enormous; and the statistical finding that the asymmetric brightness of jets over counterjets is, on average, larger than can be accounted for by beaming effects alone if the parent population is of [*randomly oriented quasars*]{}.
Barthel goes on to demonstrate that by assuming the parent population is of powerful radio galaxies, with quasars as the beamed subset, the statistics of jet/counterjet asymmetries can be justified; and the sizes of the largest radio galaxies are such that the largest superluminal quasars are not too large in the context of the parent population. Radio-loud quasars and FR II radio galaxies at $0.5 < z
<1$ in the 3CRR catalogue were compared; after the exclusion of unsuitable candidates, 12 quasars and 30 radio galaxies remained, with the linear dimensions of the galaxy radio structure averaging about twice that of the quasar mean. The relative numbers suggest that a source will appear as a quasar if viewed within $44.4\degr$ of its axis, and as a radio galaxy otherwise. This being the case, the quasars should be foreshortened 1.8 times as much as the radio galaxies, consistent with the observed factor of 2. \[If the radio-quiet unbeamed counterparts of the quasars really are the far infrared galaxies, Barthel notes that the number count statistics are consistent with such unification: there are 11 quasars for every 49 far infrared galaxies.\]
Further evidence for radio galaxy / radio quasar Unification comes from other wavebands [@Barthel-89a]: Blazars and quasars are known to be strong in their X-ray emission, while radio galaxies are weak; Seyfert 1s are luminous in X-rays while Seyfert 2s are not, suggesting that the molecular torus is an effective shield of X-ray radiation. On the other hand, the torus viewed from any angle ought to be bright in the far infrared because of its own heat, and detection statistics of both radio galaxies and quasars in this waveband bear this out. Two pieces of evidence oppose Unification, however: claims that the host galaxies of radio quasars differ significantly from radio galaxies [@Hutchings-87a]; and that extended radio sources lie in denser environments than compact sources [@Prestage+88a].
Robson and Antonucci draw differing conclusions on how blazars fit into the unification picture. We have already seen how Antonucci proposes that BLLs and OVVs are attributed to different luminosities in otherwise similar synchrotron cores. Recall that under this scheme, BLLs are postulated to be blazars with the most powerful synchrotron sources, whose emission effectively drowns out the broad lines from the BLR which lies in the line of sight. Robson suggests that BL Lac objects are not end-on quasars, claiming rather than BLLs are beamed FR I objects and OVVs are beamed FR II objects — and noting differences in the radio polarization properties of BLLs and OVVs which hint that BLLs are more likely to have shocks in their jets. The absence of emission lines in BLLs would be related to the weakness of the output of the line-emitting clouds rather than the overpowering strength of the optical synchrotron core emission. Ghisellini et al. find that their statistics from a survey of 105 radio-loud AGN support this idea.
Antonucci warns against the automatic identification of the BLL/OVV classification with the Fanaroff-Riley class, noting that some famous BL Lacs are FR class II [@Kollgaard+92a]. What is clear, is that many blazars have sufficient radio output in their diffuse emission alone to make it into the 3C or 4C catalogues; and so ‘misaligned blazars’ not beamed towards the Earth must be part of a unified continuum with some other classes of AGN which are [*already known*]{}.
One modification to the standard Unification Model is that some putative radio galaxies may be quasars obscured not by their molecular torus, but by other obscuring material. For instance, infrared spectroscopy revealed a broad H$\alpha$ line in ‘NLRG’ 3C 22 [@Rawlings+95a], suggesting that this source may be in the quasar orientation, but with opaque material obscuring much of the light from the active nucleus. Similarly polarization measurements of 3C 109 [@Goodrich+92a] can be understood if there is a hidden quasar core whose light suffers polarization by transmission through dust. If such obscured quasars are common [@Tran+98a 3C 234 could be another example], then many putative radio galaxies may have their axis closer to their line of sight to Earth than hitherto thought, and orientation statistics will be affected accordingly.
Two major questions remained unanswered by current Unification models. Why do some galaxies and quasars produce jets strong enough to drive radio emission, and others not? Why is radio emission is only found in elliptical galaxies? We shall not pursue these questions or debate the nature of blazars here, as such matters lie outside the scope of this thesis, but pause to note that work is still very much in progress on the refining of the Unification Model.
The Alignment Effect and Polarization in Radio Galaxies {#reviewRG}
=======================================================
> There comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.
>
> ------------------------------------------------------------------------
>
> — Sherlock Holmes, [*A Study in Scarlet*]{}.
The current paradigm within which radio galaxies are explored is that of the [*Unification Hypothesis*]{}. We have already explored how this hypothesis can be used to account for the wide range of AGN phenomena described in the previous chapter; now we look more specifically at radio galaxies. Distinctively among AGN, radio galaxies often display significant alignments between their radio structures and the orientation of their polarization and/or optical structure. Study of these alignments can help confirm or refute the appropriateness of the Unification Hypothesis to describe individual radio galaxies, and statistically, the class as a whole.
Polarization in AGN {#rguni}
-------------------
Since active nuclei lie within host galaxies whose stars emit unpolarized blackbody radiation, the measured polarization of any active galaxy will be that of the active nucleus diluted by starlight. It is important to distinguish whether polarization figures quoted in a given case are those of the raw measurement, or corrected for removing the unpolarized stellar intensity to yield the polarization of the nucleus. The contribution of starlight diminishes in the near ultraviolet and at shorter wavelengths; rest-frame ultraviolet measurements can be presumed to give a good indication of the nuclear polarization.
The 1980 review paper of Angel & Stockman summarised what was then known of the visible and infrared polarization of extragalactic objects: the three classes of active galaxies known to produce polarised light were blazars, quasars and Seyfert 2s. Although the spectra of BLRGs suggested that they were related to quasars and Seyfert 2s, their relative faintness meant that polarimetric studies of low redshift radio galaxies did not appear in the literature until the early 1980s [@Antonucci-82a; @Rudy+83a], and high redshift studies a decade later [@Cimatti+93a].
Angel & Stockman wrote before the Unification Hypothesis had become popular, and reviewed numerous mechanisms which might account for the low visible/infrared polarizations observed in many Seyferts. Originally, Seyfert polarizations (of order 1%) were attributed to the visible high-frequency tail of synchrotron radiation. Multicolour and spectroscopic studies of Seyfert polarization, however, showed that in most Seyferts, both the core continuum light and the emission lines were polarised, and the polarization (corrected for the stellar contribution) was stronger in the blue than in the red, but with little rotation of position angle. These facts suggested that the total light emerging from the central engine was being polarised by some subsequent interaction, most probably with dust. \[We dealt with this in some detail in §\[polinSeyferts\].\]
The discovery of Type 1 features in the polarised spectra of Seyfert 2s was the key to the first stage of Unification [@Peterson-97a § 7.1], the realization that orientation alone might be the distinguishing feature between the two classes of Seyferts. Antonucci’s review paper describes how the prototypical Seyfert NGC 1068 was investigated and found to be generating visible/ultraviolet light polarised at 16% in its nucleus. Since electrons scatter light equally strongly at all wavelengths whereas dust (via Rayleigh scattering and similar mechanisms) preferentially scatters blue light, it is implied that the scattering medium in NGC 1068 may be free electrons. Other Seyferts show evidence of higher nuclear polarizations at shorter wavelengths, characteristic of dust. Most yield a polarization orientation perpendicular to the radio structure, as would be expected for scattering.
Similarly, by the time of Antonucci’s paper, evidence was accumulating that BLRGs and NLRGs were the Type 1 and Type 2 classes for radio galaxies analogous to the classification of Seyferts. The picture seemed to be clearest for radio galaxies at redshifts $z>0.6$, where light measured in the $V$-filter on Earth corresponded to rest frame ultraviolet ($\lambda <330$nm) emissions in the AGN, uncontaminated by significant starlight. The first measured high redshift radio galaxy polarization was reported in [*Nature*]{} by di Serego Alighieri et al. ; and since then, mounting evidence [@Jannuzi+91a; @Tadhunter+92a; @Alighieri+93a; @Cimatti+97a] has generally borne out the empirical rule of thumb [@Cimatti+93a] that distant radio galaxies $(z \ga 0.6)$ should display diluted polarizations of 5% or more, oriented roughly perpendicular to the radio lobe structure.
In addition to this polarimetric evidence (reviewed in more detail below, §\[polevi\]), the apparent alignment of the knotted optical structures of high redshift radio galaxies with the radio axes [@Chambers+87a; @McCarthy+87a; @McCarthy-93a §5] lent weight to the concept of the torus and central engine proposed by the Unification Model. As we shall see in §\[aligneff\], this so-called ‘Alignment Effect’ is manifested most strongly in the AGN with the most powerful radio emission, and is intimately linked to the Unification Model and the presumed scattering mechanism for polarization.
While not proven – and inevitably suffering from a number of pathological cases which fit poorly – the Unification Model is now generally accepted, to the extent of being the foundation of the first textbooks on AGN to become available [@Robson-96a; @Peterson-97a]. The model must stand or fall, however, according to objective tests, not by indications of its popularity among astronomers. As new technology becomes available to the astronomical community, it is naturally the Unification Model which experiments are designed to test, and reinforce or falsify. With the recent availability of infrared arrays and polarisers [@McLean-97a], it has become possible to extend imaging polarimetry into the $K$-band. The work contained in this thesis represents the first studies of linear polarization in high redshift radio galaxies in this waveband.
The Observational Challenge of Distant Radio Galaxies
-----------------------------------------------------
### Observational techniques
Modern astronomical detectors [@McLean-97a] make a range of observational techniques available. Light from distant objects can be imaged on a detector array, and the light intensity measured in a synthetic aperture covering any part of the image. The light can be dispersed to form a spectrum or passed through an analyser which separates orthogonally polarised components.
Both spectroscopy and polarimetry are time-consuming procedures for faint objects: the former disperses the minimal available light into its component wavelengths, and the latter requires an accurate measurement of the intensity difference between the two orthogonal components of the light. Only recently has technology made it possible to employ both techniques simultaneously and perform spectropolarimetry of high-redshift radio galaxies, and our capabilities are limited: even with the light-gathering capacity of the 10m Keck telescope, the resulting spectra must be rebinned at low resolution to extract a meaningful signal [@Cimatti+96a]. Alternatively, if the orientation of the polarization is already known, a 4m class telescope can obtain a spectrum of light polarized in the known direction in a reasonable time [@Antonucci+94a].
The unique value of spectropolarimetry lies in its ability to identify the spectrum of scattered light present in the total signal, and so trace the emission properties of whatever hidden component is illuminating the scattering medium. (The spectrum will also, of course, give us an indication of whether polarization is attributable to a mechanism other than scattering.)
Images taken through a polarizing filter have their own value; photopolarimetry (i.e. photometry of polarized light) can be performed in synthetic apertures, yielding a polarization map or a study of the polarization of individual structures in an object of complex morphology. But again, the faintest sources are not amenable to a pixel-by-pixel polarimetric analysis; regions several pixels square may need to be binned together to obtain an acceptable signal-to-noise; and many of the radio galaxy figures given in the literature are simply polarizations integrated over the whole structure.
Where photopolarimetry is available in multiple wavebands, models of the polarized spectra of radio galaxies can be fitted against these broadband measurements: Manzini & di Serego Alighieri used this technique to establish their result (§\[RGspectrum\]) that if the radio galaxies sampled consist of stars, nebular emission and an excess component in the form of a power-law, then starlight is still their dominant component in the near infrared.
### The importance of the $K$-band
While observations of the rest frame ultraviolet have been important in polarimetric studies of AGN, on the assumption that the host galaxy contribution to the ultraviolet emission is negligible, the infrared is important for the opposite reason. Radio galaxies, as a species of AGN distinguished by their radio properties but \[ideally\] unremarkable in their optical emission, could be used as examples of ‘normal’ galaxies at high redshift (hence having experienced less cosmological evolution than nearby galaxies). They could be detected at high redshifts by virtue of their radio emission, and then studied optically in the hope that the active nucleus has not had too great an influence on their evolution (compared to ‘inactive’ galaxies), and is not polluting the light from the host galaxy. At high redshifts, of course, light originating in the visible or near infrared arrives at the Earth shifted into longer infrared wavelengths.
The atmospheric $K$-band window lies at a convenient wavelength for studying the near infrared properties of objects at $z \sim 1$, and the previous chapter reviewed how the $K-z$ Hubble diagram for 3CR radio galaxies [@Lilly+84a] suggested that the observed $K$-band light was essentially stellar emission from passively evolving elliptical galaxies. So in the late 1980s, $K$-band studies of radio galaxies were thought to be revealing the properties of young elliptical galaxies. As we have seen (§\[HubbleK\]), this has now been called into question by studies of fainter radio sources [@Eales+97a]; and it seems that a substantial fraction of the observed $K$-band light in 3CR galaxies must come from the active nucleus after all.
While the Hubble diagram is a useful tool to analyse the statistical properties of a set of galaxies, it tells us nothing about individual galaxies. Studies of the polarization, morphology and spectra of 3CR galaxies are needed to determine the properties of their $K$-band excess; an understanding of the influence of the excess in individual radio galaxies is essential if we are to salvage their role as probes of galactic evolution.
The new observations presented in this thesis are polarized $K$-band images of nine radio galaxies, including seven from the 3CR catalogue. The signal-to-noise limit prevents the meaningful analysis of any structure finer than lobes of individual bright objects. In some cases, polarimetry in other optical bands is available and can be used together with our findings for comparison with synthetic spectra. In all cases, an upper limit can be assigned to the maximum contribution of any scattered component to the $K$-band light, providing an independent means of determining the influence of the active nucleus on the apparent luminosity of 3CR sources.
In the light of our measurements, and those in the literature, it is then possible to model the most likely mechanism giving rise to the observed polarizations. If the strength of the $K$-band emission is related to the power output of the central engine, the simplest explanation of this result is that a significant fraction of a radio galaxy’s infrared light emerges from the active nucleus in a restricted cone, and enters our line of sight after scattering by dust or electrons. In this case, as for the visible and ultraviolet light, the scattered infrared light should be polarised perpendicular to the direction of the radio jet. As we shall see in the rest of this chapter, the relationship between the polarization and structure of radio galaxies measured at radio, visible and infrared wavelengths is already well documented, and we shall discuss the properties of our representative sample of radio galaxies in this context.
The motivation for performing studies of the $K$-band polarizations of high redshift radio galaxies, therefore, is to probe the origin of their $K$-band emission. A finding of no polarization would suggest that infrared light could be used as a safe indicator of the properties of the host galaxy (but would make it hard to explain the suspected infrared Alignment Effect). A finding of infrared light polarised perpendicular to the radio axis would suggest scattered nuclear light; and where polarimetry exists in other wavebands, would provide a longer baseline to test the likely origin of the polarisation — electron scattering, dust scattering and direct sight of a synchrotron source each have a distinct dependence on wavelength. Any other finding would be an invitation to further scientific study!
### The need for rigorous statistics
As in any scientific investigation, a thorough error analysis is required to give the final data their due weight. Polarimetry, however, is more demanding than other forms of photometry. Measuring a polarization is akin to determining the magnitude of a vector, a definite positive quantity. While the measurements of the vector’s components may fluctuate about zero for an unpolarised object, the magnitude stubbornly remains greater than zero and must be ‘debiased’ accordingly.
The astronomical literature contains not a few papers by statisticians [@Simmons+85a] and careful polarimetrists accusing astronomers of failing to debias their work adequately, although most 1990s papers on radio galaxy polarizations do address this issue. Given the need for debiasing polarization figures and the low signal-to-noise inevitable when studying objects at high redshift, a great deal of work in this thesis has been devoted to the accurate debiasing and error estimation of the data available. Much of the work has been published in the form of a step-by-step guide to polarimetry [@Leyshon-98a]; the format has been retained, though the work has been refined and updated, in Chapter \[stoch\] of this thesis.
Observational Evidence for Orientation Correlations in Radio Galaxies
---------------------------------------------------------------------
### Evidence for the Alignment Effect {#aligneff}
CCD technology of the mid-1980s allowed the optical structures of high redshift radio galaxies to be investigated for the first time. Radio galaxies at $z \sim 1$ were found to look nothing like the giant ellipticals associated with lower redshift radio galaxies; rather, the high redshift galaxies were often elongated and contained two or more bright ‘knots’ rather than a single identifiable nucleus [@Lilly+84a; @Spinrad+84a; @Spinrad+84b; @McCarthy+87b]. After further studies of the most powerful radio sources, it was found that the major axis of the optical elongated or knotted structures was usually aligned within a few tens of degrees of the radio axis [@Chambers+87a; @McCarthy+87a] – an association which has become known as the ‘Alignment Effect’.
Subsequent investigations with detectors sensitive to visible light revealed that the Alignment Effect cuts in at redshifts $z \geq 0.6$ [@McCarthy-93a], and that the knotted optical structures are known to be emitting [*continuum*]{} radiation, ruling out theories that the aligned structures are attributable to line-emitting gas clouds. More recent observations [@Longair+95a; @Best+97a; @Ridgway+97a] confirm the alignment of the rest-frame ultraviolet emitting regions with the radio structure.
There is no consensus at present about the mechanism which gives rise to the knotted structure; current observations continue to investigate the extent to which the Alignment Effect is associated with emission from the active nucleus. Two key tests of the relationship with the nuclear emission are whether the effect becomes weaker to longer wavelengths, and whether it becomes less prominent in less powerful radio galaxies.
The $z=0.6$ cut-off suggests that either there is an evolutionary process at work, or that we are observing a property of the rest-frame ultraviolet which does not extend to the rest-frame visible. Infrared observations of twenty 1 Jy galaxies at $z>1.5$ [@McCarthy-93a] show little evidence for extended structure at rest-frame visible wavelengths. Conversely, $U$-band images of the low-redshift $(z=0.1)$ radio galaxy 3C 195 reveal a distinct aligned ultraviolet structure [@Cimatti+95a], and a bipolar aligned structure is also seen in ultraviolet images of Cygnus A [@Hurt+99a]. Recent $U$-band imaging of the nearby radio galaxy NGC 6251 [@Crane+97a] has revealed several extended regions of emission: the most prominent feature of this radiation lies interior to a dust ring, is nearly [*perpendicular*]{} to the radio jet axis, and has a polarization below 10%.
The first $U$-band survey of low redshift radio galaxies (15 3CR objects at $0 < z < 0.6$) [@Roche+99a] found evidence for $\Delta \phi < 12\degr$ alignment between the radio axis and the $U$-band structure in 6 objects (two such alignments would be expected by chance alone). Two different mechanisms seemed to be at work: three sources showed alignment in the optical structure surrounding the radio nucleus, while the other three appeared to have a merging companion galaxy close to the radio axis. Of the three sources with an elongated nucleus, 3C 348 was also elongated in $V$ but gave no evidence for knots in either band, while the other two examples displayed knots in $U$ but no elongation or knots in $V$. The most radio-luminous radio galaxies, therefore, including those with no obvious aligned structure in the $V$-band, are now known to be able to display alignment in their near-ultraviolet structure at low redshifts, too.
The first $K$-band images of 3CR galaxies [@Chambers+88a; @Eales+90a; @Eisenhardt+90a] revealed that the near-infrared emissions of the most luminous radio galaxies displayed structure as knotted and complex as the ultraviolet emissions. As with visible light detectors, efforts were made to obtain infrared images of radio galaxies with lower radio luminosity. Many of these programmes used the $K$-band, in which observations of radio galaxies at $z \sim 1$ trace emissions at $\lambda \sim 1.1$[$\umu$m]{} in the rest frame.
Dunlop & Peacock compared $K$-band images of 3CR and PSR (Parkes Selected Regions) radio galaxies in a narrow bin of redshifts. The 3CR galaxies were selected at $0.8 < z < 1.3$, and the PSR galaxies ($S_{\mathrm 2.7\,GHz} > 0.1$Jy) were known or estimated to be in a similar redshift range. A definite infrared-radio alignment effect was determined in the sample of 19 3CR galaxies, although the $K$-band structure was, on average, less extended than the optical structure. In some cases, the infrared structure seemed to be significantly more closely aligned with the radio axis than structure observed at visible wavelengths. These findings are consistent with the smaller 3C sample of Rigler et al. , which suggested that an infrared alignment effect was present, but weak. Best, Longair & Röttgering imaged 28 3CR galaxies at $0.6<z<1.2$ and again found distinct alignments at visible wavelengths, with less complex structure and a weaker Alignment Effect in the infrared. On average, only about 10% of the $K$-band flux of 3CR galaxies at $z \sim 1$ is associated with aligned structures [@Rigler+92a; @Best+98a].
Lacy et al. investigated the Alignment Effect in a sample of $0.5 < z < 0.82$ 7C radio galaxies, with radio luminosities of order one-twentieth those of 3C galaxies. The Effect was still present, albeit very weakly above 400nm, but only over small scales. 3C radio galaxies exhibit alignment in structures of order 15kpc and 50kpc; in the 7C sample, the effect seen at 15 kpc did not extend to structure at 50kpc. Dunlop & Peacock’s PSR sample was tested in the $K$, $J$, $B$ and $R$-bands, with no evidence for alignment being found in the red or infrared, and only a possible marginal effect in the $B$-band. Wieringa & Katgert also found that the optical morphology of less luminous radio galaxies was more rounded.
Eales et al. criticise Dunlop & Peacock’s selection technique for the PSR galaxies: given that nuclear light biases upwards the apparent brightness of 3CR galaxies, the $K-z$ relation for 3CR galaxies cannot be used to estimate redshifts for PSR galaxies unless a correction is made for the nuclear component of the brightness. Constructing their own sample of 6C/B2 galaxies at redshifts well matched to Dunlop & Peacock’s 3CR sample, and following the same position angle analysis technique, Eales et al. found no strong evidence of an Alignment Effect, but a limited statistical analysis showed a probability $< 20\%$ that the null hypothesis (no alignment effect whatsoever) was true.
Best, Longair & Röttgering analyzed a complete subsample of eight 3CR galaxies; all lay at $1.0 \la z \la 1.3$ and emitted radio emission at $S_{\mathrm 178\,MHz} \sim 10$Jy, so the set should be free of evolutionary or radio luminosity trends. The sample showed a clear trend, such that those galaxies with small radio structure had complex knotted structures, closely aligned with the radio axis, which $K$-band imaging showed to be on the same scale as the host galaxy. Galaxies with much larger radio structures showed only one or two bright knots, and the alignment, if present, was not so accurately matched with the radio hotspot axis.
The observational evidence to date, therefore, shows a clear infrared-radio alignment and a clear visible-radio alignment in 3CR galaxies at $z \sim 1$, with the possibility that the visible structures are slightly misaligned $(\Delta \phi \sim 10\degr)$ with and/or more extended than their infrared counterparts. \[misalign1\] Galaxies at lower radio luminosity show only marginal evidence for radio-optical alignment in any optical band, except for the near ultraviolet small-scale ($\la$ 15kpc) alignments of Lacy et al. . Evidence for an Alignment Effect in quasar host galaxies has also been reported recently [@Ridgway+97a].
We will consider the possible explanations of the Alignment Effect offered by the Unification Model and its alternatives in the following sections after reviewing the evidence of polarized optical radiation. It may be worth noting, however, the first glimpse of ordinary galaxies at high redshift, as provided by the [*Hubble Deep Field*]{}. This window on a younger universe has revealed many galaxies of disturbed morphology [@Naim+97a], including elongated objects which have become known as ‘chain galaxies’. The evolutionary relationship between these objects and the morphological classes of galaxies in today’s older universe remains to be resolved, but we cannot rule out the possibility of some common factor at work in these chain galaxies and the hosts of radio galaxies.
### Broadband polarization measurements {#polevi}
Motivated by the discovery of broad lines in the polarized spectra of Seyfert 2s [@Antonucci+85a], the late 1980s saw several NLRGs analysed by spectropolarimetry in the hope of revealing broad lines in their polarized flux. Bailey et al. and Hough et al. found that Centaurus A and IC 5063 respectively were $\ga 10\%$ polarized perpendicular to their radio structures in light at 2[$\umu$m]{}. They suggested that this polarized light might be direct emission from nuclei obscured at visible wavelengths, and that the high polarization was indicative of blazar activity. Antonucci & Barvainis agree that the nuclear light is more visible in the infrared, partly because kpc-scale dust lanes optically thick in the visible are more transparent to the near infrared; but they point out that the lobe-dominated radio structure and the strong perpendicular radio structure/infrared polarization alignment are not characteristic of blazars. They review the discovery of polarized broad lines in 3C 234, arguing that this object is an NLRG and very similar to NGC 1068; and both Centaurus A and IC 5063 could be objects of the same class. \[3C 234 is a $z=0.185$ object now known to have a spectrum similar to that of Seyfert 2s, and is sufficiently luminous to be harbouring a quasar nucleus [@Tran+95a].\]
More recently, detailed $K$-band imaging polarimetry studies have been performed on Centaurus A [@Packham+96a]. It was found that in the near-infrared, the polarization vectors mainly lie along the dust lane, with the polarization being produced by dichroic absorption of the radiation from stars embedded within it. But an additional larger polarized component was detected in the nucleus at 2.2[$\umu$m]{}, with the position angle of polarization perpendicular to the inner radio jet and the X-ray jet. Millimetre-wave observations at 0.8mm and 1.1mm found no evidence of polarization at these wavelengths. Centaurus A can hence be explained by the usual scattering model.
Other NLRG imaging polarization measurements rapidly followed 3C 234 in the literature. Polarization was found in regions distinct from the nucleus in PKS 2152-69 $(z=0.028)$ [@Alighieri+88a], and in the first high-redshift $(z=1.132)$ radio galaxy successfully analysed, 3C 368 [@Scarrott+90a; @Alighieri+89a]. Nuclear polarization was detected in 3C 277.2 $(z=0.766)$ [@Alighieri+88a]. Results published in other papers were as follows:
[Antonucci & Barvainis ]{} attempted to measure the polarization of several other NLRGs but obtained only large upper limits in most cases (they blame obscuring kpc-scale dust lanes for their failure to detect nuclear light in these cases). They obtained a significant result for 3C 223.1 $(z=0.108)$, with its 2.2[$\umu$m]{}polarization measured at $4.9 \pm 0.7 \%$ oriented at $116 \pm 4 \degr$, an offset from the radio structure of $80 \pm 6 \degr$. The visible-light polarization was found to be below 0.5%.
[Impey, Lawrence & Tapia ]{} took a complete sample of radio sources covering both radio galaxies and radio-loud quasars, to analyse their polarizations. Polarizations were successfully measured or obtained from the literature for 20 of the 30 radio galaxies forming the sample; those polarizations obtained by the authors themselves were unfiltered, with a nominal wavelength of 570nm defined by the properties of the GaAs phototubes used. Only two radio galaxies consistently yielded polarizations higher than 3%, [*viz.*]{} 3C 109 and 3C 234. Most had polarization values in the 1–2% range.
[Jannuzi & Elston ]{} investigated the radio galaxy 3C 265, discovering that the orientation of its polarization in both $B$ and $R$ passbands is roughly perpendicular to the axis of the radio emission and to the major axis of the structure seen in ultraviolet emission. The data show no evidence of wavelength dependence in the polarization between $B$ and $R$.
[Tadhunter et al. ]{} compared the polarization of medium and high redshift radio galaxies, imaging using either no filter, or standard or broadened $V$-band filters. Seven objects at $0.5<z<0.85$ were included in their sample, of which five exhibited raw polarizations in the range 5–20%, with generally perpendicular alignments to the radio axes. None of the five intermediate-redshift objects at $0.2<z<0.5$ showed polarizations over 5%.
[di Serego Alighieri, Cimatti & Fosbury ]{} measured six high-redshift radio galaxies in bands corresponding to rest-frame wavelengths around 300nm. Four of these were found to have polarizations of 4–18%, all oriented perpendicular to their optical structure.
[Shaw et al. ]{} analysed four southern radio galaxies at $0.3<z<0.7$ in the $B$-band. Two, PKS 1602+01 and PKS 2135-20, are BLRGs and have low polarization. PKS 1547-79 is also a BLRG, and seems to be polarized but may be contaminated by dust. PKS 2250-41 is a NLRG with high polarization.
[Cimatti & di Serego Alighieri ]{} collected data on eight 3C radio galaxies at $0.09 \leq z \leq 0.47$ in Johnson-Cousins filters selected individually to reveal the rest frame properties of each galaxy at around 300nm – the most notable result being findings of an ultraviolet alignment effect in the [*low*]{}-redshift radio galaxy 3C 195. The same group also analyzed the near ultraviolet properties of a $z=2.63$ object, MRC 2025-218 [@Cimatti+94a].
### Polarization trends {#poltrend}
Those various results listed above which had been published by 1992, together with a few other individual objects in the literature, were gathered together by Cimatti et al. . They took 42 radio galaxies at $z\geq0.1$ from the literature and their own observations, and looked for correlations between the optical polarization and other properties. In cases where the object was extended, an integrated polarization was taken. Trends were sought both with the observed polarization (debiased), and with the corrected nuclear polarization which would be present if the light was being diluted by an elliptical host galaxy. Because of the wide variety of filters used in collecting the data, and the range in redshifts of the objects observed, the polarizations represent rest-frame emissions at differing wavelengths between 200nm and 700nm. Five trends were discerned for observed polarizations, not all of which survive for the underlying nuclear values.
[[*Redshift, $z$.*]{}]{} High polarization is observed preferentially at high redshift. The observed values are almost perfectly bimodal: six out of the seven objects at $z>0.6$ are polarized above 8%, while all but one of the lower redshift objects are polarized below 7%. After dilution correction, the nuclear polarization still appears to increase with redshift, with $2\sigma$ significance.
[[*Rest-frame wavelength, $\lambda_r$.*]{}]{} High polarizations are preferentially observed in wavebands corresponding to emission bluewards of 400nm. Again, a $2\sigma$ significance correlation remains between the nuclear polarizations and the wavelength of emission.
[[*Radio power, $P_{\mathrm 178\,MHz}$.*]{}]{} The total radio power emitted at 178MHz is highest for objects with the greatest observed polarization; yet again, the nuclear polarizations retain this correlation with $2\sigma$ significance.
[[*Radio spectral index, $\alpha_r$.*]{}]{} Most of the objects surveyed possessed spectral indices between 0.5 and 1.0. The higher the observed polarization, the closer to 1.0 the spectral index tended to be. This correlation was weaker than the previous three, being significant at the $3\sigma$ level for observed polarizations, and dropping to $1.5\sigma$ for the corrected nuclear values.
[[*Radio $Q$-structure.*]{}]{} The $Q$-parameter [@McCarthy+91a] is a measure of the asymmetry of the radio structure, the ratio of the longer radio arm to the shorter. No clear correlation could be confirmed, but there was a noticeable absence of galaxies combining low $Q$-values (i.e. symmetric radio structure) with high observed polarization.
When the orientation of the polarization was assessed as well, the two major observational results were (A) that radio galaxies at $z>0.6$ and polarizations $p>5\%$ [*always*]{} show perpendicular polarization, with a tendency to be aligned more closely with the structures observed in the ultraviolet continuum than with the radio axis; and (B) that those radio galaxies exhibiting parallel polarization/structure alignments always had polarizations $p<5\%$.
----------------- ----- ------- ------ ------------------ ---------------------- -------------------------------- ---------------- ----- --
Source B/N Band $\lambda_r$ (nm) $P $\theta \pm \sigma_{\theta}$() $\delta_{o-r}$ Ref
\pm \sigma_P$(%)
PKS 2250-41 nuc N 0.310 $B$ 336 $\ell$ 4.4 $\pm$ 0.8 152 $\pm$ 5 58 S+
PKS 2250-41 W N 0.310 $B$ 336 $\ell$$(<4.5)$ 10 $\pm$ 8 93 S+
3C 313 ? 0.461 $B$ 303 $(<6.0)$ n/a n/a CA
PKS 1602+01 B 0.462 $B$ 301 $\ell$$(<4.8)$ 119 $\pm$ 22 5 S+
PKS 1547-79 B 0.483 $B$ 297 $\ell$2.9 $\pm$ 0.7 66 $\pm$ 2 42 S+
PKS 2135-20 B 0.635 $B$ 269 $\ell$$(<2.7)$ 180 $\pm$ 16 n/a S+
3C 277.2 N 0.766 $i$ 450 e9.9 $\pm$ 1.4 169 $\pm$ 7 108 A+
3C 226 N 0.818 $i$ 440 e2.9 $\pm$ 1.4 84 $\pm$ 13 120 A+
FSC 10214+4724 N 2.286 Ø 228 16.2 $\pm$ 1.8 75 $\pm$ 3 145 L+
MRC 2025-218 ? 2.627 $R$ 185 w8.3 $\pm$ 2.3 93 $\pm$ 8 63 C+
----------------- ----- ------- ------ ------------------ ---------------------- -------------------------------- ---------------- ----- --
Key: Source: Common name of object; B/N: BLRG / NLRG classification; $z$: redshift; Band: waveband of observation, Johnson-Cousins or Gunn designation \[Ø denotes no filter, effective passband 400–1000nm\]; $\lambda_r$: central wavelength of observed frame transformed into source’s rest-frame; $P\pm\sigma_P$: percentage polarization (debiased) with 1$\sigma$ error \[e indicates corrected for suspected emission line contamination; $\ell$ indicates largest of cited nucleocentric apertures taken; w indicates whole galaxy, not nucleus\]; $\theta\pm\sigma_\theta$: Electric vector orientation E of N (); $\delta_{o-r}$: Orientation offset of optical polarization minus radio position angle (); Ref: Data source, as follows: A+ [@Alighieri+94a]; C+ [@Cimatti+94a]; CA [@Cimatti+95a]; L+ [@Lawrence+93a]; S+ [@Shaw+95a]. \[litpol\]
Since Cimatti et al. compiled their paper, further broadband polarizations have appeared in the literature. Those for radio galaxies at $z>0.2$ are given in Table \[litpol\]. If the observed polarizations are compared with the trends in redshift and rest-frame wavelength seen in the 1993 data, most of the new data points lie within the scatter of the existing points. Notable exceptions are 3C 226, whose 3% polarization is very low for a $z=0.8$ object, and PKS 2135-20, also at low polarization and high redshift, whose upper limit of 2.7% is much lower than the polarization of any other object seen in rest-frame light emitted below 280nm.
Three sources show a distinct perpendicular alignment between polarization orientation and radio structure: PKS 2250-41, 3C 277.2, 3C 226. It should be noted, however, that the two 3C objects both occur already as perpendicular objects in Cimatti et al.’s data, but are observed here in different wavebands. PKS 1602+01 shows a distinct parallel alignment. Several objects (PKS 2250-41 nuc, PKS 1547-49, FSC 10214+4724, and MRC 2025-219) have polarization orientations which seem to be about 20 offset from perpendicular alignment with the radio structure — it would be instructive to determine whether these objects displayed a closer perpendicular alignment between polarization and their optical structure. Results A and B still stand, except in the case of the significant misalignment of FSC 10214+4724. \[misalign2\]
### Radio galaxy spectropolarimetry and extended imaging {#rgsspecpol}
Early data on nearby radio galaxies’ polarized spectra was compiled by Antonucci . Although the initial sample included 45 objects, several showed signs of variability in their optical polarization, and high quality radio maps were not available for many of the others. From the few objects for which it was possible to compare the radio structure and the optical polarization orientation, there was clear evidence for a class of galaxies polarized parallel to the radio structure, and weak evidence that the non-parallel galaxies might form a perpendicular class. The polarization of the [*radio*]{} emission from the core was also analyzed, and was found to have a tendency to be aligned perpendicular to the radio structure. Too few objects had both radio and optical polarization measurements to provide meaningful data on any correlation between the two orientations.
The first spectropolarimetric analysis of high-redshift radio galaxies was made on three targets for which the broadband polarization orientations were already known [@Alighieri+94a] — 3C 226, 3C 277.2, and 3C 324. This enabled spectra to be taken through a linear analyzer oriented to extract light polarized parallel and perpendicular to the broadband polarization angle, making the most efficient use of limited observing time, at the price of the impeded detection of any lines polarised at intermediate angles. All three galaxies yielded evidence of a polarized continuum, a broad Mg[ii]{} $\lambda$2798 line polarised to the same degree, and narrow lines which were consistent with zero polarization, but not compatible with polarization as high as that of the continuum.
The specific measurements suggested that 3C 226 has a polarization of about 11% in the range 210-370nm, possibly constant but possibly declining to the red; its $i$-band polarization is considerably lower (see Table \[litpol\]). 3C 277.2 exhibited more variation in its continuum polarization, with values between 11% and 24% seen between 200nm and 380nm. Again, a possible decreasing polarization to the red is reinforced by a much lower $i$-band measurement. Finally, 3C 324 also seems to have an 11% continuum. The built-in assumption about the polarization orientation means that these values are only true polarizations if the assumption is correct; and we cannot, of course derive polarization orientation angles from such observational data.
Cimatti et al. analyzed the same $z=1.206$ galaxy 3C 324 with the W. M. Keck telescope. The continuum emission polarization between 200nm and 400nm (rest-frame) remained fairly constant at about 12% $\pm$ 4%, 17 $\pm$ 2. The two most prominent emission lines in the polarized flux were \[O[ii]{}\] $\lambda$3727, which bore a much lower polarization than the continuum, and at a perpendicular angle; and Mg[ii]{} $\lambda$2800, which is more polarized than the continuum, but in the same orientation.
Keck observations of further 3C objects followed: firstly 3C 256 [@Dey+96a]; then 3C 13 $(z=1.351)$ and 3C 356 ($z=1.079$, one of the subjects of this thesis) [@Cimatti+97a], and later two powerful radio galaxies at $z \sim 2.5$, 4C 00.54 and 4C 23.56 [@Cimatti+98a using the Keck II]. In all these cases, linear polarization oriented perpendicular to the radio structure was found. One important exception to this trend was also discovered [@Dey+97a]: Keck spectropolarimetry of the $z=3.798$ galaxy 4C 41.17 showed no evidence for polarization, with a $2\sigma$ upper limit of 2.4%, with a number of strong absorption features. This anomalous galaxy is proposed as an example of a galaxy caught in the act of star formation.
Tran et al. used the Keck I to obtain both spectropolarimetry and extended imaging polarimetry: their targets were 3C 265 and further observations of 3C 277.2; 3C 324 (imaging only) and 3C 343.1 (spectropolarimetry only). \[The imaging data are not reproduced in Table \[litpol\] since there is no one overall figure; the results are published in the form of polarization vector maps.\] The three galaxies with imaging maps all displayed a bipolar fan of polarization vectors centred on the nucleus, perpendicular to the optical structure and misaligned by tens of degrees with the radio axis. The [*nuclear*]{} polarization of 3C 265 appears to be about 12%; the diluted polarization of the near ultraviolet emission of 3C 277.2 is rated at 29% $\pm$ 6%. The third source analyzed in its spectrum, 3C 343.1, was not found to be highly polarised but was contaminated by an object lying at an intermediate redshift.
Interpretations of Orientation Data
-----------------------------------
### Disentangling the spectra of radio galaxies
How are we to interpret these polarization and Alignment Effect observations? The most revealing findings, though historically the most recent, are the spectropolarimetry results. Invoking Occam’s Razor, it is safe to assume that any polarization which displays a constant position angle over a range of wavelengths is generated by a single mechanism [@Alighieri+94a; @Cimatti+96a]. Further, if two different ‘features’ (e.g. the continuum and emission lines) are polarized in the same direction and with comparable strength, the polarization mechanism is probably one which polarizes light in transit rather than anything intrinsic to the emission of light at the source. \[Were we to find polarization angles which systematically changed with wavelength, one interpretation might be a relativistic effect violating the Einstein Equivalence Principle; conversely, the observed constancy of the orientation angle in galaxies at various different redshifts provides further reinforcement for General Relativity, as noted by Cimatti et al. .\]
Two approaches can be used to disentangle a polarized spectrum — modelling the separate components thought to contribute to it, or separating the 100% polarized component from the unpolarized component by producing the product spectrum $p(\lambda) \times S_{\nu}(\lambda)$. Although the underlying polarised component will probably not be 100% polarized, the polarized flux will isolate features unique to the partially polarized component. Clearly if [*all*]{} the features in a spectrum with several components are polarised, the most likely mechanism would be external to the source object — most likely to be transmission through aligned dust grains, which might be near the source, in the intergalactic medium, or in our own Galaxy. In such a case polarization studies would tell us much about the extragalactic or Galactic dust, but nothing about the source galaxy.
The next assumption which can be made is to model the light from the radio galaxy as two components: evolved stellar blackbody emission, and a nuclear component to be explained. Since nearby radio galaxies are clearly ellipticals, more distant radio galaxies must surely also contain an evolved stellar population — although we must remember that part of the motivation for studying radio galaxies is to determine the evolutionary changes in this population at high redshift. A third, weaker, component may also be included in modelling [@Manzini+96a]: nebular continuum emission. Individual cases are known where the nebular contribution may be significant [@Cimatti+98a 10%] or even dominant [@Cimatti+97a 3C 368].
Neither nebular emission nor stellar blackbody emission is intrinsically polarized, so we expect these sources to contribute only to the unpolarized spectrum, while the nuclear component may be partially polarized. This nuclear contribution may itself consist of several features with different polarization strengths, indicative of the various mechanisms at work within the nucleus. The presence of an unpolarized stellar (and nebular) component together with a substantially polarized nuclear component is indicative of the fact that the polarizing mechanism is contained inside the host galaxy (otherwise the stellar emission would also become polarized) but does not consist of many independent cells of polarized emission [@Antonucci-84a] which, if independently oriented, would tend to cancel out one another’s polarization and produce a low overall figure.
A key feature of evolved galactic spectra is the so-called 4000Åspectral break [@Bruzual-83a]: the intensity of the blackbody radiation from the stars in the galaxy drops substantially bluewards of 400nm, due to the scarcity of short-lived massive stars which would be luminous in the near ultraviolet. Therefore, significant amounts of ultraviolet radiation are diagnostic of star-forming activity or non-blackbody processes at work.
The shape of the polarized component of the spectrum provides clues about the likely polarization mechanism [@Cimatti+93a]. Synchrotron radiation is most intense at long wavelengths and falls off to the blue – but will not result in polarized emission lines. Scattering by electrons (Thomson scattering) will not change the spectral profile of the light being scattered. Dust scattering (Rayleigh scattering) is most effective at short wavelengths, so the spectrum of light incident on dust clouds will become blued as the light scatters. Elongated dust grains aligned in a magnetic field can also polarize light passing through the dust cloud. Of course, in dust transmission or scattering scenarios, we must also bear in mind the possibility of the incident light being partially absorbed and reddened by the dust [@Alighieri+94a]. Orientation correlations between polarization angles, optical structure, and the positions of the radio jets can be used together with the spectral profile to identify the most likely candidate mechanism in each case.
The three galaxies observed by di Serego Alighieri et al., 3C 226, 3C 277.2, and 3C 324, all displayed constant polarization orientations, and Mg[ii]{} lines polarized at approximately the same level as the continuum. This, therefore, is indicative that some scattering/transmission process is modifying the light; and the decline to the red suggests scattering. The polarization strength of the continuum could differ from that of the magnesium line if the sources of emission were in slightly different positions and the geometry of the scattering process made the polarization of light from one source more efficient than that of the other. The narrow oxygen line observed at low polarization in 3C 324 by Cimatti et al. could be assumed to arise outside the nuclear scattering region, and they suggest that this galaxy also possesses a dusty region capable of producing polarization by transmission.
Similar findings were reported for 3C 256 [@Dey+96a] and 3C 356 [@Cimatti+97a]: broad magnesium lines were visible in both unpolarized and polarized flux, while narrow forbidden lines were only visible in unpolarized flux. In 3C 356, the ultraviolet continuum appeared to contribute about 80% of the total light at 280nm, and the remainder could be modelled by an evolved stellar population aged $\sim$ 1.5–2.0 Gyr. Both of the $z \sim$ 2.5 radio galaxies, 4C 00.54 and 4C 23.56 [@Cimatti+98a] were found to be dominated by non-stellar emission at 150nm, with young massive stars contributing no more than half the total continuum flux. For all the sources, their polarization orientations were approximately perpendicular to their major structural axes, implying that the scattered light was originally travelling parallel to the major axis.
Recent spectropolarimetry shows, therefore, that without making any assumptions about the nature of the central engine, some of the most powerful high redshift radio galaxies have an evolved stellar population, and contain continuum and broad line sources in a confined region such that only their emission parallel to the radio/optical structural axis is able to be scattered into our line of sight. It must be noted, of course, that those radio galaxies chosen for spectropolarimetry tend to be those known to have high polarization [*a priori*]{}, and which are bright enough for spectra to be taken in a reasonable time.
### Interpreting broadband polarizations
Although spectropolarimetry of high redshift radio galaxies has only recently become available, a similar analysis can be carried out by comparing broadband multiwavelength polarimetry with synthetic spectra modelled from stellar, nebular and power law components. In such cases, it is not possible to distinguish the presence or polarization states of broad or narrow lines, but the wavelength dependence of the continuum polarization will be apparent.
We have already seen how Manzini & di Serego Alighieri used a synthesis technique to distinguish components in unpolarised spectra (§\[RGspectrum\]); they were also able to simulate the effects of dust scattering by different species of dust grains and so synthesise polarized spectra which they then fitted against the photopolarimetry available from the literature. By doing so, they could estimate not only the fraction of light present in each component, but also the age of the host galaxy and the most likely properties of the scattering dust. Cimatti et al. note how in most cases, broadband fitting shows that radio galaxies probably contain evolved stellar populations — it is also possible to interpret findings in some galaxies, though, as due to young stellar populations born with non-standard IMFs (initial mass functions) [@Bithell+90a].
In most cases, however, only one or two measurements of polarization will be available, and then the best analysis which can be carried out is that of orientation correlations with other properties, as in the trends analysis of Cimatti et al. , reviewed above (§\[poltrend\]). We saw that polarization tended to be highest at the shortest rest-frame wavelengths and for the most distant radio galaxies; it is not clear which of these correlations is primary, and which is a consequence of the other.
One further finding which we must review is that the optical and radio structures seem to be misaligned by $\sim 15\degr$ in many cases; and the optical polarization tends to be perpendicular to the extended optical structure rather than the radio structure [@Cimatti+94a]. Tran et al. note how a similar effect has been observed in Seyfert galaxies. This might be attributed to rotation in transit of light polarised perpendicular to the radio structure; but it seems most likely that the extended optical emission region is the scattering zone for the polarised radiation, yielding the natural perpendicular result. It must then be explained why the Alignment Effect is not perfectly parallel to the radio structure; and why the infrared Alignment Effect [@Rigler+92a; @Dunlop+93a] traces the radio structure more closely than the ultraviolet extended emission.
### Further interpretation of spectral features
One further property which is of note is the classification of radio galaxies as BLRGs or NLRGs. Now that we know that some distant radio galaxies contain broad emission lines visible in polarized light, we must ask what makes a galaxy fall into the BLRG class. Broad lines will become visible in the unpolarized spectrum if the scattered BLR emission is sufficiently strong, or if the geometry permits a direct view of part of the broad line region. \[There are NLRGs known where the broad lines are not totally obscured, while the narrow lines appear to be partially extinguished; this may be explained by carefully selecting the geometry so the obscuring torus covers some narrow line clouds and not all of the broad line emission region [@Alighieri+94a].\]
Since Cimatti et al.’s sample contained only 8 BLRGs out of 42 objects, and the most distant lay at $z=0.306$, so it would be dangerous to draw conclusions about differences in the observed polarizations of BLRGs and NLRGs. Nevertheless, recalling that the broad-lined Seyfert 1s tend to have low parallel polarizations, it is noteworthy that three of the eight BLRGs have only upper limits to their polarizations, and three more have low ($\la 6\%)$ parallel polarizations. Only 3C 332 (3%) and 3C 234 (6%) are perpendicular. Three of the additional objects recorded in Table \[litpol\] above are BLRGs, all polarized at $<5\%$; and we note also that one of these three objects is a definite parallel polarization, a second is closer to parallel than perpendicular, and the radio axis of the third is undetermined. Against this is the earlier observation [@Antonucci-84a] that radio galaxies exhibiting parallel polarizations tended to be NLRGs; or else that those polarised parallel which do possess broad lines also have other anomalous features.
Other spectral properties, in particular comparisons of radio galaxy and Seyfert spectra, can be found in the literature: in particular, Seyferts only exhibit magnesium lines in polarised spectra while this emission line is sufficiently strong to appear in total spectra in radio galaxies [@Alighieri+94a]. Seyfert 1s are compared to BLRGs by Rudy et al. who find the BLRGs of similar luminosity but with weaker Fe[ii]{} lines, a steeper Balmer decrement and a larger \[O[iii]{}\]/H$\beta$ ratio.
### Unified models of radio galaxies
We have now gone as far as we can in interpreting our results in terms of the properties of an abstract power-law source and broad line region at the heart of an elliptical galaxy. Now we must ask: what physical mechanism can explain the Alignment Effect, Cimatti et al.’s polarization trends, and the spectropolarimetric evidence for scattered light from power-law and broad line components?
The obvious candidate is that radio galaxies contain the same kind of central engine as is postulated to exist in Seyfert galaxies and quasars. Since radio galaxies are radio-loud by definition, the mechanism must be closest to that at work in the radio-loud quasars. But radio galaxies are not quasars, and a plausible model must also explain the differences.
Let us assume that a radio galaxy consists of an evolved elliptical galaxy containing a black hole, accretion disk, BLR clouds, obscuring molecular torus, and narrow line emission clouds described in the previous chapter (§\[intunimod\]); as a radio galaxy, the central engine will possess a powerful bipolar outflow jet responsible for the radio structure. There is evidence [@Kormendy+95a; @Willott+98a; @Serjeant+98a] that the radio and optical luminosity of the active nucleus may be correlated with the mass of the galaxy, and hence its stellar luminosity; but we will assume that the relative contributions of the stellar and nuclear components can be varied freely with a wide dynamic range.
We postulate that dust and/or electrons may be present in the outer regions of the galaxy, and that these particles are capable of scattering light into the line of sight to the Earth. Particles illuminated by the central engine through the opening angle of the molecular torus may therefore scatter nuclear light towards Earth; such light will naturally become partially polarized orthogonal to the lines joining the scattering region to the central engine and to the observer. We assume that light is only scattered once, since multiple scattering would randomize the polarization angle of emerging light; and with such a scattering efficiency, much less than half the nuclear light travelling in a given direction can be scattered out of its original path.
In radio galaxies with relatively strong central engines, we will see dust clouds illuminated in a broad cone, and to a considerable distance from the nucleus. Where the central engine is weaker, or the dust clouds more tenuous, only the region closest to the nucleus will be discernibly illuminated and so at limited resolution, it will be harder to distinguish deviations from perfect alignment. This would be consistent with the small-scale (15kpc) Alignment Effect in 7C objects [@Lacy+98b], and also with the great radio strength/loose optical structure correlation [@Best+96a] if powerful radio jets tend to sweep space clear of scattering material in their path. The scattering efficiency of dust is much lower in the infrared than in the ultraviolet, so again, tighter and closer aligned structures might be expected before nuclear infrared light diverges below a detectable intensity, while visible light from a nucleus of similar luminosity might diverge further and create an impression of ‘misalignment’ before falling below the intensity threshold.
The tendency towards increasing observed polarizations at $z>0.6$ can be largely explained by the rest-frame wavelength of images taken in standard filters moving towards the blue; nuclear polarization is more apparent as the diluting effects of the host galaxy fall off shortward of the 4000Å spectral break [@Bruzual-83a]. Any residual tendency in the nuclear polarization [@Cimatti+93a] could be attributed to the increasing efficiency of Rayleigh scattering at shorter wavelengths. Similarly, the Alignment Effect, if due simply to the structure of the scattering regions compared to the stellar structure, will be most pronounced in sources with the strongest active nuclei \[i.e. visible even in the $K$-band for 3CR sources [@Dunlop+93a]\], and at rest frame wavelengths below 400nm where the stellar emission falls off rapidly — again explaining the $z<0.6$ cut-off for visible images, and the indications of $U$-band alignment in nearby radio galaxies.
Hammer, LeFèvre & Angonin find no evidence for the 4000Å spectral break in a composite spectrum of ten radio galaxies at $0.75 \le z \le 1.1$; but it has been pointed out [@Alighieri+94a; @Cimatti+94a] that the composite spectrum was of total light (stellar plus nuclear) in which the break can be masked by the nuclear contribution, and can be hard to measure accurately with several emission lines lying close to 400nm. In using the properties of the spectral break to estimate the age of the stellar population of the host galaxy, it must be remembered that other factors which can affect the solution for the age include the timescale of formation and the metallicity.
In some cases it may be possible to identify the most likely composition of the scattering medium, though in a real galaxy there are likely to be regions containing dust and regions of free electrons [@Tran+98a]. Rudy et al. attribute some of the features of their BLRG spectra to dust extinction effects. Electron scattering produces much stronger polarization \[20–50% as opposed to 10–20% in dust [@Alighieri+94a]\] and so it may be necessary to invoke an additional component of hot young stars to dilute the observed ultraviolet polarization if electrons rather than dust are suspected in particular cases — Cimatti et al. demonstrate how different combinations of old and young stellar populations, direct nuclear radiation and dust or electron scattered components may be combined to form models consistent with observed polarizations. On the other hand, certain distributions of dust grains may be able to produce polarizations comparably high to those due to free electrons [@Manzini+96a; @Cimatti+96a]. Cimatti et al. demonstrate how an upper limit can be set on the free electron temperature by the width of observed lines, which would be broadened beyond visibility [@Fabian-89a] by scattering in a too-hot plasma.
So far, a Unification Model can be made to fit the observed facts. But if the redshift dependence of polarization and the Alignment Effect is merely an artifact of our standard filters and the 4000Å spectral break, is there any evidence for evolution in radio galaxies? We have seen that the host galaxy may contain a naturally evolving stellar population whose aging process is masked by evolution in the structure of the whole galaxy [@Eales+96a; @Best+98a]; high redshift radio galaxies are akin to BCGs in this respect, and themselves tend to form in clusters. Why, then, are some low redshift radio galaxies – the luminous FR IIs – not in clusters? Have the FR IIs’ powerful outputs disrupted their clusters over time? Or does the formation of an active nucleus take longer in an isolated galaxy, and ignite at FR II luminosity – in which case we are only now seeing the birth of the first FR IIs outside clusters, and none have had time to decay into FR Is?
Clearly one simple model cannot tell the whole story, however; while the presence of perpendicular polarization lays out very strong evidence that scattering must be an important mechanism, other contributions are not ruled out. On the contrary, jet-induced star formation models can provide an explanation for the presence of the dust clouds needed to cause scattering [@Cimatti+98a]. In some cases this is clearly not the case [@Tran+98a], for example when imaging polarimetry shows polarization increasing with distance from the nucleus: clearly stellar dilution from the host galaxy is decreasing, and there are no young blue stars in the extended region to compensate. The Alignment Effect has been observed in some galaxies which are not strongly polarised, and alternative models are needed to account for such cases — especially where other observed factors do not correspond to those commonly observed in radio galaxies which fit the canonical scattering model.
Longair, Best & Röttgering , for example, invoke three very different models to account for three 3CR galaxies for which they obtained [*Hubble Space Telescope*]{} images. 3C 265 $(z=0.81)$ displays optical structure poorly aligned (25 offset) with the radio structure, and at one one tenth of the scale. The observed structure is most likely to be attributable to the interactions of two or more galaxies, they suggest, with a modicum of alignment possibly due to scattering or jet-induced star formation.
Their second example, 3C 324 $(z=1.21)$, appears as a giant elliptical in the $K$-band but exhibits a very knotted structure in their [*Hubble*]{} (690nm and 783nm) images. The structure is aligned with the axis along which relativistic material is believed to flow, although the line linking the radio hotspots is offset from this axis by about 30. If the optical knots are associated with companion galaxies, some theory is needed to account for their close alignment with the relativistic jet; possibly that of West which postulates the formation of a prolate galaxy and central black hole rotating about the axis of the large scale matter distribution. \[Such a theory predicts the presence of structure up to Mpc scales, however, while 3C 324 shows no aligned galaxies beyond 100 kpc; in general there is no evidence for the radio axis to be aligned with a particular axis of a triaxially symmetry galaxy [@Sansom+87a].\] Otherwise, this source could be a classic case with knots of scattered light explicable by the Unification Model, as argued by Cimatti et al. .
Finally, 3C 368 $(z=1.13)$ exhibits optical structure of the same scale as the radio structure and might best be explained by jet-induced star formation; star-forming regions on the radio axes would be rich in the dust needed to account for its high optical polarization, apparently scattered light from an AGN. The wavelength dependence of 3C 368’s polarization [@Cimatti+93a] rules out both electron scattering and synchrotron radiation as possible mechanisms, leaving dust scattering as the most likely hypothesis.
Similarly, Pentericci et al.’s study of very high redshift radio galaxies $(2.6<z<3.2)$ finds that the galaxies fall into several classes: those where the ultraviolet emission closely traces the radio structure (akin to 3C 368 above); those where there is a clear triangular emission region (presumably a cone of scattered light); those where there is a clear radio/ultraviolet alignment effect but no close relationship between the structures; and a couple of pathological misaligned cases which may be peculiar for other reasons.
### Alternatives to radio galaxy unification
The Unification Model with scattered light causing perpendicular polarization and the Alignment Effect is far from universally accepted as the explanation of the various observed radio galaxy phenomena. Simulations of galaxy formation based on hierarchical clustering models [@Baron+87a note fig. 3] produce knotted structures in young galaxies similar to that observed in radio galaxies. McCarthy notes that galactic objects with the morphology characteristic of the most luminous distant radio galaxies must undergo substantial orbit mixing in $\sim 100$ Myr [@Daly-90a], so any viable model must explain the radio and optical structures as a short-lived phenomenon within that timescale. Longair, Best & Röttgering point out that aligned structure cannot be explained by scattering alone, as the aligned regions often do not exhibit the conical shape which scattered light would illuminate [@Ridgway+97a].
Synchrotron emission – which is polarized, although not necessarily with the orientation alignments commonly observed in radio galaxy polarization – is another candidate mechanism. This can often be ruled out in individual cases, however, by showing that the extrapolation of the radio emission at the measured spectral index would not produce optical synchrotron emission at the luminosity required [@Cimatti+94a]. Among other models suggested we have the following — all of which are wanting, since none provide for an optical polarization mechanism:
The difference in the Hubble diagrams for 3CR and weaker radio galaxy populations must logically be ascribed to the presence of two or more components to their infrared light, a stellar component and one related to the active nucleus. One possibility for the nuclear-related component considered by Eales et al. is that emission lines — known to be directly correlated with radio luminosity [@Willott+99a] — are polluting the $K$-band light; but infrared spectroscopy of $z \sim 1$ galaxies [@Rawlings+91a] shows that emission lines do not contribute more than a quarter of the total light intensity. Nevertheless, warm emission nebulae excited by the active nucleus are known to contribute a significant fraction of the observed ultraviolet light in some cases [@Dickson+95a].
Another alternative [@Eales+97a] allows that quasars form the central engines of radio galaxies, and posits that the dust obscuring the visible light from the quasar nuclei is not thick enough to obscure the near infrared emission – as has already been proposed in the specific case of 3C 22 [@Rawlings+95a]. But if this mechanism were widespread, it would tend to concentrate light in a single nucleus (3C 22 is pointlike, showing only small, faint optical and infrared extended structure – see Chapter \[resulch\]) and would not provide any explanation for the knotted structures, aligned or otherwise. Quasars which do not exhibit the blazar phenomenon have low polarization, so if orientation-based theories are incorrect, radio galaxies with polarizations $p > 3\%$ ought to show signs of blazar activity, which would be distinctive. \[The presence of broad lines [@Alighieri+94a] is sufficient to show that even if a blazar component is present, the quasar component is dominant over it.\]
McCarthy and Dunlop & Peacock review models which attempt to explain the Alignment Effect as the result of a zone of star formation triggered by the passage of a radio source. If such models are correct then the newly-formed stars must be younger than the radio source, which would give them an age of only 10–100 Myr (although the generally accepted ages of radio sources are not indisputable). But such young ages are hard to reconcile with the colours observed by Dunlop & Peacock and with the low scatter [@Lilly+84a], indicative of a settled population, in the Hubble diagram. Since the alignments between radio and optical structure are good only to $\sim
10\degr$, it is also difficult to explain how an expanding radio source can cause star formation so far off-beam. Neither is there any evidence of star formation in more than one or two examples of the tens of low-redshift radio galaxies which have been studied in detail now [@Alighieri+89a]. Best, Longair & Röttgering’s findings, however, can best be interpreted in terms of such a model: as radio hotspots pass through the intergalactic medium of their host, they trigger bursts of star formation (hence the complex knots associated with small radio structures). Later, they have travelled well outside the visible region of the host galaxy and stellar formation activity ceases – explaining why the larger radio structures do not exhibit so many optical knots.
Several other models have been proposed to explain the Alignment Effect, but these again do not account for the observed polarization orientations. Suggested mechanisms include two-component blazar models [@Brindle+86a], thermal plasma emission [@Daly-92a] and the illumination pattern of a Doppler-beamed continuum as seen in blazars [@Tran+98a]. The consequences of selection effects following from increased luminosity of radio sources in the plane of a flattened disk of gas have also been suggested [@Eales-92a], but McCarthy suggests that the timescale for this would be too long for the 100 Myr transient phenomenon of radio galaxies.
For the purposes of this thesis, we need only consider models which are relevant to the interpretation of broadband infrared aperture polarimetry. We will not, therefore, review further these other models which need to be invoked to explain non-polarimetric features of radio galaxies, but turn instead to the matter of the statistical techniques applicable to aperture polarimetry.
Mathematical Glossary
---------------------
$h_0$
: The Hubble constant in units of 100 kms$^{-1}$Mpc$^{-1}$.
$P_\nu$
: The radiated power per unit bandwidth of a source as measured at frequency $\nu$.
$p$
: The degree of linear polarization.
$S_\nu$
: The flux density of a source as measured at frequency $\nu$; measured in jansky, such that 1 Jy = $10^{-26}$ Wm$^{-2}$Hz$^{-1}$. [@Illingworth-94a]
$\alpha_r$
: The spectral index of a radio spectrum, such that $S
\propto \nu^{-\alpha_r}$.
$\lambda_r$
: The rest-frame wavelength of light emitted by a distant galaxy.
$\nu$
: The frequency of (radio) emission.
$\phi$
: The orientation (relative to celestial North) of the [$E$]{}-vector of linearly polarized radiation.
The Measurement and Publication of Polarization {#stoch}
===============================================
> Although you can format an equation almost any way you want with LaTeX, you have to work harder to do it wrong.
>
> ------------------------------------------------------------------------
>
> — Leslie Lamport, [*The LaTeX Reference Manual*]{}.
At the start of this research project, the available literature seemed to provide no coherent and unified account of the best way to reduce, analyse and present data on the polarization of astronomical sources. Accordingly the best method as described here was submitted for publication in [*Experimental Astronomy*]{} [@Leyshon-98a]. The following chapter is based on that paper, updated in the light of the new book by Tinbergen and the work of Sánchez Almeida , and of Maronna, Feinstein & Clocchiatti . In an age when theses are becoming increasingly available via the World-Wide Web, it seems most useful to retain the format of a ‘how-to’ manual for this chapter, in the hope that it will prove useful and instructive to polarimetrists of the 21st Century.
The task of the polarimetrist
-----------------------------
When performing optical polarimetry of astronomical objects, we wish to answer several distinct, but related, physical questions.
Firstly, is the object polarized at all? Secondly, if it is, what is the best estimate of the polarization? And thirdly, what confidence can we give to this measure of polarization? It is also necessary to be able to test whether the polarization has changed from one epoch to another, or differs in neighbouring spectral bands.
In addition to these physical questions is a presentational one: in what format should the results be published, to be of most utility to the scientific community?
The questions of quantifying and presenting data on linear polarization have been discussed at length by Simmons & Stewart , who note that the traditional method used by optical astronomers, that of Serkowski , does not give the best estimate of the true polarization under most circumstances. More recently, Sánchez Almeida , Maronna, Feinstein & Clocchiatti and Clarke with colleagues [@Clarke+86a; @Clarke+93a; @Clarke+94a] have developed the statistical basis of how noise affects measurements of polarization. Using their recommendations, I present here a recipe for reducing polarimetric data.
Paradigm
--------
In this chapter, I will not consider the origin of the polarization of light. It may arise from intrinsic polarization of the source, from interaction with the interstellar medium, or within Earth’s atmosphere. Each of these sources represents a genuine polarization, which must be taken into account in explaining the measured polarization values. Some possible sources of such systematic polarization are discussed by Hsu & Breger .
Most modern optical polarimetry systems employ a two-channel system, normally a Wollaston prism. Such a prism splits the incoming light into two parallel beams (‘channels’) with orthogonal polarizations – it functions as a pair of co-located linear analyzers. The transmission axes of the analyzers can be changed either by placing a half-wave plate before the prism in the optical path, and rotating this, or by rotating the actual Wollaston prism. Such a system is incapable of distinguishing circularly polarized light from unpolarized light, and references to ‘unpolarized’ light in the remainder of this chapter strictly refer to light which is not linearly polarized; such light may be totally unpolarized (i.e. randomly polarized), or may include a circularly polarized component.
Where a half-wave plate is used, an anticlockwise rotation $\chi$ of the waveplate results in an anticlockwise rotation of $\eta = 2\chi$ of the transmission axes. \[For the theory of Wollaston prisms and wave plates, see, for instance, Chapter 8 in Hecht ; for a general survey of the theory and practice of astronomical polarimetry, see Tinbergen or the briefer accounts by Kitchin or McLean .\]
We will suppose that Channel 1 of the detector has a transmission axis which can be rotated by some angle $\eta$ anticlockwise on the celestial sphere, relative to a reference position $\eta_0$ east of north. (See Figure \[figeta\].)
(250,200)(-120,-100) (0,-75)[(0,1)[150]{}]{} (-3,80)[N $(x)$ (Declination)]{} (90,0)[(-1,0)[165]{}]{} (-105,-3)[E $(y)$]{} (-105,-20)[(Right Ascension)]{} (0,0)[(3,-2)[80]{}]{} (90,-63)[$\oplus\ (z)$]{} (40,-90)[(Propagation direction, to Earth)]{} (20,0)[(3,-2)[16.64]{}]{} (36.64,-11.09)[(-1,0)[20]{}]{} (15,-10)[(0,1)[18.03]{}]{} (15,8.03)[(-3,2)[15]{}]{} (0,0)[(-1,2)[30]{}]{} (-37,65)[R]{} (-13,32)[$\eta_0$]{} (0,50)[(-2,-1)[20]{}]{} (-23,18)[$\eta$]{} (-20,40)[(-1,-3)[8.57]{}]{} (0,0)[(-2,1)[70]{}]{} (-84,35)[T1]{} (0,0)[(-1,-2)[30]{}]{} (-41,-72)[T2]{}
The transmission axes T1, T2, of Channels 1 and 2 are hence at $\eta_0 + \eta$ and $\eta_0 + 90{\degr}+ \eta$ respectively.
The reference angle $\eta_0$ will depend on the construction of the polarizer, and will not, in general, be neatly due north. For mathematical convenience in the rest of this chapter, we will take $\eta_0$ to define a reference direction, ‘R’, in our instrumental co-ordinate system and relate all other angles to it. Such instrumental angles can then be mapped on to the Celestial Sphere by the addition of $\eta_0$.
Since the light emerging in the two beams has traversed identical paths until reaching the Wollaston prism, this method of polarimetry does not suffer from the systematic errors due to sky fluctuation which affect single-channel polarimetry (where a single beam polarimeter alternately samples the two orthogonal polarizations).
The two channels will each feed some sort of photometric array, e.g. a [ccd]{} or infrared array, which will record a photon count. Since such images are often built up by a process of shifting the image position on the array and combining the results, we will refer to a composite image taken in one transmission axis orientation, $\eta$, as a [*mosaic*]{}. We will denote the rate of arrival of photons recorded in Channel 1 and Channel 2 by $n_{1}(\eta)$ and $n_{2}(\eta)$ respectively. From these rates, we can calculate the total intensity ($I$) of the source, and the difference ($S$) between the two channels:
$$I(\eta) = n_{1}(\eta) + n_{2}(\eta),
\label{Idef}$$
$$S(\eta) = n_{1}(\eta) - n_{2}(\eta).
\label{Sdef}$$
We can also define a [*normalized*]{} difference: $$\label{normdiff}
s(\eta) = \frac{S(\eta)}{I(\eta)}.$$
The purpose of this chapter is to discuss how to interpret and present such data.
Curve Fitting for $p$
---------------------
Suppose we have a beam of light, which has a linearly polarized component of intensity $I_p$, whose electric vector points at an angle $\phi$ anticlockwise of R. Its (linearly) unpolarized component is of intensity $I_u$. When such a beam enters our detector, we can use Malus’ Law [@Hecht-87a §8.2.1] to deduce that $$n_1(\eta) = {{\frac{1}{2}}}I_u + I_p.\cos^2(\phi - \eta)$$ and $$n_2(\eta) = {{\frac{1}{2}}}I_u + I_p.\sin^2(\phi - \eta),$$ from which we find $$\label{Ipol}
I(\eta) = I_u + I_p,$$ and, less trivially, $$S(\eta) = I_p.\cos[2(\phi - \eta)].
\label{Spol}$$ The [*degree of linear polarization*]{}, $p$, is defined by $$\label{ppol}
p = \frac{I_p}{I_p + I_u}$$ and so we can obtain the normalized difference by substituting Equations \[Ipol\], \[Spol\] and \[ppol\] into \[normdiff\]: $$s(\eta) = p.\cos[2(\phi - \eta)].$$
Now, if observations have been made at a number of different angles, $\eta_j$, of the transmission axis, then a series of values for $\eta_j$ and $s_j(\eta_j)$ will be known, and $p$ and $\phi$ may be determined by fitting a sine curve to this data, weighted by errors $\sigma_{s_j}(\eta_j)$ as necessary. This method has been used, for example, by di Serego Alighieri et al. . (Their refinement of the method allowed for the correction of the $s_j(\eta_j)$ for instrumental polarization at each $\eta_j$, which was necessary as they were rotating the entire camera, their system having no half-wave plate.)
We note that if there is any systematic bias of Channel 1 compared to Channel 2, this will show up as an $\eta$-independent ([dc]{}) term added to the sinusoidal component when $s_j(\eta_j)$ is fitted to the data. Such bias could arise if an object appears close to the edge of the [ ccd]{} in one channel, for example.
The Stokes Parameters
---------------------
### Basic definitions
Polarized light is normally quantified using Stokes’ parameterisation. \[For basic definitions see, for example, Clarke, in Gehrels (ed.) .\] Where interference properties need not be treated, the intensity of polarized light can exhaustively be characterised by the four Stokes Parameters: one for the overall amplitude, two orthogonal linear components, and one circular component.
Various conventions are known in the literature for the four Stokes Parameters; this thesis uses the most common, the $I,Q,U,V$ notation. The $V$ parameter will not be considered here, as it parameterises circular polarization, which a system involving only half-wave plates and linear analyzers cannot measure. The total intensity, $I$, of the light is an absolute Stokes Parameter. The other two parameters are defined relative to some reference axis, which in our case will be R, the $\eta_0$ direction. Thus we define: $$Q = S\,(0{\degr}) = - S\,(90{\degr}),$$ and $$U = S\,(45{\degr}) = - S\,(135{\degr}).$$
[*Normalized*]{} Stokes Parameters are denoted by lower case letters ($q$,$u$,$v$), and are found by dividing the raw parameters by $I$. We note that $S$ and the normalized $s$ can be thought of as a Stokes Parameter like $Q$ or $U$, generalised to an arbitrary angle – and results which can be derived for $S$ (or $s$) will apply to $Q$ and $U$ (or $q$ and $u$) as special cases.
If the Stokes Parameters are known, then the degree and angle of linear polarization can be found: $$\label{defp}
p = \sqrt{q^2 + u^2};$$ $$\label{phidef}
\phi = {{\frac{1}{2}}}.\tan^{-1} \left( \frac{u}{q} \right) ,$$ where the signs of $q$ and $u$ must be inspected to determine the correct quadrant for the inverse tangent. Note that $S\,(\eta)$, $Q$ and $U$ [*must*]{} be defined as above to be consistent with the choice of R as Reference.
We must now distinguish between the true values of the Stokes Parameters for a source, and the values which we measure in the presence of noise. We will use the subscript $0$ to denote the underlying values, and the subscript $i$ for individual measured values. In the rest of this chapter, symbols such as $S_i$ and $\sigma_{S_i}$, where not followed by $(\eta)$, can be read as denoting ‘either $Q_i$ or $U_i$’, ‘either $\sigma_{Q_i}$ or $\sigma_{U_i}$’, etc.; arithmetic means are denoted in the usual way, by an overbar, hence for $\nu_S$ measurements $S_i$, $\bar{S}=\sum_{i=1}^{\nu_S} S_i / \nu_S$.
### The importance of Stokes Parameters
In particular, consider a source which is not polarized, so $q_0=u_0=0$, $p_0=0$, and $\phi_0$ is undefined. Since the $q_i$ and $u_i$ include noise, they will not, in general, be zero, and because of the quadrature form of Equation \[defp\], $p_i$ will be a definite-positive quantity. In short, $p_i$ is a [*biased*]{} estimator for $p_0$.
There is no known [*unbiased*]{} estimator for $p_0$, and Simmons & Stewart discuss at length the question of which estimator should be used. They conclude that the Stokes Parameters themselves are more useful than $p$ and $\phi$ in many applications. Since $p$ inevitably suffers from bias while estimators of the Stokes Parameters may, in principle, be unbiased, it is recommended that all published polarimetric data should include the values of the normalized Stokes Parameters. This would provide a standard format for further use by the scientific community, whereas tabulated values of $p$ and $\phi$ would always be sensitive to the debiasing scheme used to obtain them.
Given this preference for the Stokes Parameters it appears that one should eschew the curve fitting method in favour of direct evaluation of the parameters, at least when we only have data for the usual angles $\eta_j =
0{\degr}, 45{\degr}, 90{\degr}, 135{\degr}$. In practice, observers will take several observations of an object at each transmission angle. This raises the question of how best to combine all the measured values $q_i, u_i$ to yield a single pair of ‘best estimators’ for $q_0$ and $u_0$ – a question which is dealt with by Clarke et al. and developed by Maronna, Feinstein & Clocchiatti .
A full discussion of the optimal method of estimating the true value of a normalised Stokes Parameter based on such individual measurements can be found in the Appendix (sections \[optest\] and \[optbin\]). In accordance with the notation used there, $\tilde{s}=\bar{S}/\bar{I}$ represents the (approximate) optimal estimator of $s_0$. It is possible to determine $\bar{I}$ by pooling the $I_i$ values used to determine $Q$ and those used to determine $U$, which would reduce the error on the mean; but to avoid systematic effects it is safer to calculate two separate $I$ values and produce $\tilde{q}=\bar{Q}/\bar{I_Q}, \tilde{u}=\bar{U}/\bar{I_U}$.
Noise Affecting the Measurement of Stokes Parameters {#realnoise}
----------------------------------------------------
The raw numbers which our photometric system produces will be a set of count rates $n_{1i}(\eta)$ and $n_{2i}(\eta)$, together with their errors, $\sigma_{n_{1i}}(\eta)$ and $\sigma_{n_{2i}}(\eta)$. These errors arise from three sources: photon counting noise; pixel-to-pixel variations in the sky value superimposed on the target object; and imperfect estimation of the modal sky value to subtract from the image [@Sterken+92a; @IRAFphot].
The fundmental physical limitation on the measurement of any low intensity of light is the quantum nature of light itself: low intensity monochromatic light of frequency $f$ arrives in discrete photons of energy $hf$. For a beam of light whose average intensity is $\cal R$ photons per second, the probability of a given number of photons actually passing a point in the beam during time $\tau$ is distributed according to a Poisson distribution with mean ${\cal R}\tau$, and hence standard deviation $\sqrt{{\cal R}\tau}$. Now ${\cal R}\tau$ will not necessarily be an integer, but individual measurements must give integer results; and the fluctuation in the measured photon counts for repeated integrations of time $\tau$ is termed [*shot noise*]{}.
As shown in Appendix \[photapp\] (see Equation \[shotSN\]), a photon counting system registers one count for every $\Delta$ photons incident on it, and the shot noise in one channel is related to the count rate as follows: $$\label{shotnSN}
\sigma_{\mathsf{shot}}^2 = \bar{n}_{\times i} /\tau \Delta.$$
The modal value of a sky pixel, $n_{\mathsf{sky}}$ can be found by considering, say, the pixel values in an annulus of dark sky around the object in question, an annulus which contains ${{\mathcal{D}}}$ pixels altogether. The root-mean-square deviation of these pixels’ values about the mode can also be found, and we will label this, $\sigma_{\mathsf{sky}}$. Hence we can estimate the error on the mode, $\sigma_{\mathsf{sky}}/\sqrt{{{\mathcal{D}}}}$.
If we perform aperture-limited photometry on our target, with an aperture of area ${{\mathcal{A}}}$ in pixels, we must subtract the modal sky level, ${{\mathcal{A}}}.n_{\mathsf{sky}}$, which will introduce an error $\sigma_{\mathsf{skysub}} =
{{\mathcal{A}}}.\sigma_{\mathsf{sky}}/\sqrt{{{\mathcal{D}}}}$.
Each individual pixel in the aperture will be subject to a random sky fluctuation; adding these in quadrature for each of the ${{\mathcal{A}}}$ pixels, we obtain an error $\sigma_{\mathsf{skyfluc}} =
\sqrt{{{\mathcal{A}}}}.\sigma_{\mathsf{sky}}$.
Ultimately, the error on the measured, normalized, intensity, is the sum in quadrature of the three quantities, $\sigma_{\mathsf{shot}}$, $\sigma_{\mathsf{skysub}}$, and $\sigma_{\mathsf{skyfluc}}$. If the areas of the aperture and annulus are comparable, then both the second and third terms will be significant; in practice, for long exposure times, the first (shot) noise term will be much smaller and can be neglected. This is important as, unlike the sky noise, the shot noise depends on the magnitude of the target object itself. If its contribution to the error terms is negligible, then sky-dominated error terms can be compared between objects of different brightness on the same frame.
\[step\][Data Check]{}
\[smallshot\]
For each object observed in each channel of each mosaic, the photometry system will have produced a count rate $n_{\times i}$ with an error, $\sigma_{n_{\times i}}$. For each such measurement, calculate $\sqrt{n_{\times i}} / \Delta \tau$ and verify that it is much less than $\sigma_{n_{\times i}}$. Then one can be certain that the noise terms are dominated by sky noise rather than shot noise.
[*The treatment which follows in this chapter relies on sky noise being dominant. Other scenarios are possible: in particular, an investigation into the case where shot noise is dominant is presented in Appendix \[photapp\], and the effects of scintillation noise are considered by Clarke & Stewart . These cases become more relevant for brighter sources but are not useful for the faint AGN which form the subject of this thesis.*]{}
Testing for DC bias
-------------------
In practice, for each target object, we will have taken a number of mosaics at each angle $\eta_j$. We can immediately use each pair of intensities $n_{1i}, n_{2i}$ to find $I_{i}(\eta_j)$ and $S_{i}(\eta_j)$ using Equations \[Idef\] and \[Sdef\].
Since the errors on the two channels are independent, we can trivially find the errors on both $I_{i}(\eta_j)$ and $S_{i}(\eta_j)$; the errors turn out to be identical, and are given by: $$\label{erreq}
\sigma_{I_{i}} = \sigma_{S_{i}} = \sqrt{{\sigma_{n_{1i}}}^2 +
{\sigma_{n_{2i}}}^2 }.$$
\[cbias\] Take the mean value of all the $S_{i}(\eta_j)$ by summing over all the values $S_i$ at all angles $\eta_j$; and obtain an error on this mean by combining in quadrature the error on each $S_i$. If the mean value of $S_i(\eta_j)$, averaged over all the angles $\eta_j$, is significantly greater than the propagated error, then there may be some [dc]{} bias.
Check \[cbias\] uses $S_{i}(\eta_j)$ as a measure of excess intensity in Channel 1 over Channel 2, and relies on the fact that there are similar numbers of observations at $\eta_j=\eta$ and $\eta_j=\eta+90{\degr}$ to average away effects due to polarization. If, as may happen in real data gathering exercises, there are not [ *identical*]{} numbers of observations at $\eta_j=\eta$ and $\eta_j=\eta+90{\degr}$, this could show up as apparent ‘[dc]{} bias’ in a highly polarized object. In practice, however, we are unlikely to encounter this combination of events; testing for bias by the above method will either reveal a bias much greater than the error (where the cause should be obvious when the original sky images are examined); or a bias consistent with the random sky noise, in which case we can assume that there is no significant bias.
Obtaining the Stokes Parameters
-------------------------------
Once we are satisfied that our raw data are not biased, we can proceed. At this stage in our data reduction, we will find it convenient to divide our set of $S_{i}(\eta_j)$ values, together with their associated $I_{i}(\eta_j)$ values, into the named Stokes Parameters, $$Q_i = S_{i}(\eta_j=0{\degr}) = -S_{i}(\eta_j=90{\degr})$$ and $$U_i = S_{i}(\eta_j=45{\degr}) = -S_{i}(\eta_j=135{\degr}).$$
\[getthei\]
For each pair of data $n_{1i}(\eta_j), n_{2i}(\eta_j)$, produce the sum, $I_{i}$, and the difference, $Q_{i}$ or $U_{i}$ as appropriate. Using Equation \[erreq\], produce the error common to the sum and difference, $\sigma_{Q_i}$ or $\sigma_{U_i}$. Also find the normalized difference, $q_i$ or $u_i$.
In practice, for a given target object, we will have taken a small number of measurements of $Q_i$ and $U_i$ – say $\nu_Q$ and $\nu_U$ respectively – with individual errors obtained for each measurement. If the errors on the individual values are not comparable, but vary widely, we may need to consider taking a weighted mean.
\[maxbig\] For a set of measurements of $(S_i,\sigma_{S_i})$, take all the measured errors, $\sigma_{S_i}$; and so find the mean error (call this ${{{\mathcal E}_{\mathsf{phot}}}}$) and the maximum deviation of any individual error from ${{{\mathcal E}_{\mathsf{phot}}}}$. If the maximum deviation is large compared to the actual error, consider whether you need to weight the data.
If the deviations are large, we can weight each data point, $S_i$, by ${\sigma_{S_i}}^{-2}$; but we will not pursue the subject of statistical tests on weighted means here. In practice, one normally finds that the noise does not vary widely between measurements.
We have already checked (see Check \[smallshot\]) that the shot noise is negligible compared with the sky noise terms. Therefore, the main source of variation will be the sky noise. If the maximum deviation of the errors from ${{{\mathcal E}_{\mathsf{phot}}}}$ is small, then we can infer that the fluctuation in the sky pixel values is similar in all the mosaics.
\[assumenorm\] In order to carry the statistical treatment further, we must assume that the sky noise is normally distributed. This is standard astronomical practice.
\[getmean\]
From the sample of Stokes Parameters $I_i$, $Q_i$ and $U_i$, obtained in Step \[getthei\], find the two means, $\bar{Q}$ and $\bar{U}$, with their corresponding intensities $\bar{I}_Q$ and $\bar{I}_U$; and find the standard deviations of the two [[**[samples]{}**]{}]{}, $\psi_Q$ and $\psi_U$.
### Photometric and statistical errors
Since modern photometric systems can estimate the sky noise on each frame, we are faced throughout our data reduction sequence with a choice between two methods for handling errors. We can propagate the errors on individual measurements through our calculations; or we can use the standard deviation, $\psi_S$, of the set of sample values, $S_i$.
In this chapter, I use the symbol $\sigma_{S_i}$ to denote the measured (sky-dominated) error on $S_i$, and $\sigma_{\bar{S}}$ for the standard error on the estimated mean, $\bar{S}$. The standard deviation of the population, which is the expected error on a single measurement $S_i$, could be denoted $\sigma_{S}$, but above I used ${{{\mathcal E}_{\mathsf{phot}}}}$ to make its photometric derivation obvious.
Using statistical estimators discards the data present in the photometric noise figures and uses only the spread in the data points to estimate the errors. We would expect the statistical estimator to be of similar magnitude to the photometric error in each case; and a cautious approach will embrace the greater of the two errors as the better error to quote in each case.
Because we may be dealing with a small sample (size $\nu_S$) for some Stokes Parameter, $S$, the standard deviation of the sample, $\psi_S$, will not be the best estimator of the population standard deviation. The best estimator is [@Clarke+83a §10.5, for example]: $$\label{stateq}
{{{\mathcal E}_{\mathsf{stat}}}} = \sqrt{\frac{\nu_S}{\nu_S - 1}}.\psi_S.$$
In this special case of the [[**[population]{}**]{}]{} standard deviation, I have used the notation ${{{\mathcal E}_{\mathsf{stat}}}}$ for clarity. Conventionally, $s$ is used for the ‘best estimator’ standard deviation, but this symbol is already in use here for a general normalized Stokes Parameter, so in this chapter I will use the variant form of sigma, $\varsigma$, for errors derived from the sample standard deviation, whence $\varsigma_S = {{{\mathcal E}_{\mathsf{stat}}}}$, and the (statistical) standard error on the mean is $$\varsigma_{\bar{S}} = \frac{\psi_S}{\sqrt{\nu_S - 1}} =
\frac{{{{\mathcal E}_{\mathsf{stat}}}}}{\sqrt{\nu_S}}.$$
The mean value of our Stokes Parameter, $\bar{S}$, is the best estimate of the true value $(S_0)$ regardless of the size of $\nu_S$. Given a choice of errors between $\sigma_{\bar{S}}$ and $\varsigma_{\bar{S}}$, we will cautiously take the greater of the two to be the ‘best’ error, which we shall denote ${{\hat{\sigma}_{\bar{S}}}}$.
\[noiseOK\]
We now have two ways of estimating the noise on a single measurement of a Stokes Parameter:
[${\bullet}\hspace{1em}$]{}${{{\mathcal E}_{\mathsf{phot}}}}$ is the mean sky noise level obtained from our photometry system: Check \[maxbig\] obtains its value and verifies that the noise levels do not fluctuate greatly about this mean.
[${\bullet}\hspace{1em}$]{}Statistical fluctuations in the actual values of the Stokes Parameter in question are quantified by ${{{\mathcal E}_{\mathsf{stat}}}}$, obtained by applying Equation \[stateq\] to the data from Step \[getmean\].
We would expect the two noise figures to be comparable, and this can be checked in our data. We may also consider photometry of other objects on the same frame: Check \[smallshot\] shows us that the errors are dominated by sky noise, and $\sigma_{\mathsf{sky}}$ should be comparable between objects, correcting for the different apertures used: $$\sigma_{\mathsf{sky}} = {{{\mathcal E}_{\mathsf{X}}}}/\sqrt{2{{\mathcal{A}}}(1+{{\mathcal{A}}}/{{\mathcal{D}}})}.$$
We therefore take the best error, ${{\hat{\sigma}_{S}}}$, on a Stokes Parameter, $S$, to be the greater of ${{{\mathcal E}_{\mathsf{phot}}}}$ and ${{{\mathcal E}_{\mathsf{stat}}}}$.
If our data passes the above test, then we can be reasonably confident that the statistical tests we will outline in the next sections will not be invalidated by noise fluctuations.
Testing for Polarization
------------------------
The linear polarization of light can be thought of as a vector of length $p_0$ and phase angle $\theta_0 = 2\phi_0$. There are two independent components to the polarization. If either $Q_0$ or $U_0$ is non-zero, the light is said to be polarized. Conversely, if the light is to be described as unpolarized, both $Q_0$ and $U_0$ must be shown to be zero.
The simplest way to test whether or not our target object emits polarized light is to test whether the measured Stokes Parameters, $\bar{Q}$ and $\bar{U}$, are consistent with zero. If either parameter is inconsistent with zero, then the source can be said to be polarized.
To proceed, we must rely on our assumption (Step \[assumenorm\]) that the sky-dominated noise causes the raw Stokes Parameters, $Q_i, U_i,$ to be distributed normally. Then we can perform hypothesis testing [@Clarke+83a Chapters 12 and 16] for the null hypotheses that $Q_0$ and $U_0$ are zero. Here, noting that the number of samples is typically small ($\nu_Q \simeq
\nu_U < 30$) we face a choice:
[${\bullet}\hspace{1em}$]{}[Either:]{} assume that the sky fluctuations are normally distributed with standard deviation ${{{\mathcal E}_{\mathsf{phot}}}}$, and perform hypothesis testing on the standard normal distribution with the statistic: $$z = \frac{\bar{S} - S_0}{{{{\mathcal E}_{\mathsf{phot}}}}/\sqrt{\nu_S}};$$
[${\bullet}\hspace{1em}$]{}[Or:]{} use the variation in the $S_i$ values to estimate the population standard deviation ${{{\mathcal E}_{\mathsf{stat}}}}$, and perform hypothesis testing on the Student’s $t$ distribution with $\nu_S - 1$ degrees of freedom, using the statistic: $$t = \frac{\bar{S} - S_0}{{{{\mathcal E}_{\mathsf{stat}}}}/\sqrt{\nu_S}}.$$
In either case, we can perform the usual statistical test to determine whether we can reject the null hypothesis that ‘$S_0 = 0$’, at the $C_S.100\%$ confidence level. The confidence intervals for retaining the null hypothesis will be symmetrical, and will be of the forms $-z_0<z<z_0$ and $-t_0<t<t_0$.
The values of $z_0$ and $t_0$ can be obtained from tables, and we define ${{\bar{S}_{C_{S}}}}$ to be the greater of $z_0.{{{\mathcal E}_{\mathsf{phot}}}}/\sqrt{\nu_S}$ and $t_0.{{{\mathcal E}_{\mathsf{stat}}}}/\sqrt{\nu_S}$. Then the more conservative hypothesis test will reject that null hypothesis at the $C_S.100\%$ confidence level when ${{|\bar{S}|}}>{{\bar{S}_{C_{S}}}}$.
In such a confidence test, the probability of making a ‘Type I Error’, i.e. of identifying an [[**[unpolarized]{}**]{}]{} target as being polarized in [ *one*]{} polarization sense, is simply $1-C_S$. The probability of correctly retaining the ‘unpolarized’ hypothesis is $C_S$.
The probability of making a ‘Type II Error’ [@Clarke+83a §12.7] (i.e. not identifying a [[**[polarized]{}**]{}]{} target as being polarized in one polarization sense) is not trivial to calculate.
Now because there are two independent senses of linear polarization, we must consider how to combine the results of tests on the two independent Stokes Parameters. Suppose we have a source which has no linear polarization. We test the two Stokes Parameters, $\bar{Q}$ and $\bar{U}$, for consistency with zero at confidence levels $C_Q$ and $C_U$ respectively. The combined probability of correctly retaining the null hypothesis for both channels is $C_Q.C_U$, and that of making the Type I Error of rejecting the null hypothesis in either or both channels is $1-C_Q.C_U$. Hence the overall confidence of the combined test is $C_Q.C_U.100\%$.
Since the null hypothesis is that $p_0=0$ and $\phi_0$ is undefined, there is no preferred direction in the null system, and therefore the confidence test should not prefer one channel over the other. Hence the test must always take place with $C_Q=C_U$.
Even so, the test does not treat all angles equally; the probability of a Type II Error depends on the orientation of the polarization of the source. Clearly if its polarization is closely aligned with a transmission axis, there is a low chance of a polarization consistent with the null hypothesis being recorded on the aligned axis, but a much higher chance of this happening on the perpendicular axis. As the alignment worsens, changing $\phi_0$ while keeping $p_0$ constant, the probabilities for retaining the null hypothesis on the two measurement axes approach one another.
Consider the case where we have taken equal numbers of measurements in the two channels, so $\nu_Q = \nu_U = \nu$, and where the errors on the measurements are all of order ${{{\mathcal E}_{\mathsf{phot}}}}$. Hence we can calculate $z_0$ for the null hypothesis as above. Its value will be common to the $Q$ and $U$ channels, as the noise level and the number of measurements are the same in both channels.
Now suppose that the source has intensity $I_0$ and a true non-zero polarization $p_0$ oriented at position angle $\phi_0$. Then we can write $Q_0 = I_0 p_0 \cos(2\phi_0)$, and $U_0 = I_0
p_0 \sin(2\phi_0)$. To generate a Type II error, a false null result must be recorded on both axes. The probability of a false null can be calculated for specified $p_0$ and $\phi_0$: defining $z_1 =
\frac{I_0 p_0}{{{{\mathcal E}_{\mathsf{phot}}}}/\sqrt{\nu}}$ then the probability of such a Type II error is $$\label{IIprob}
P_{\rm II} = \frac{1}{2\pi}
\int_{x= z_1 \cos(2\phi_0) - z_0}^{x= z_1 \cos(2\phi_0) + z_0}
\int_{y= z_1 \sin(2\phi_0) - z_0}^{y= z_1 \sin(2\phi_0) + z_0}
\exp\left[ - {{\frac{1}{2}}}(x^2 + y^2) \right] dx\,dy.$$ Clearly this probability is not independent of $\phi_0$.
\[findconfr\] Find the 90% confidence region limits, $\cln{Q}{90}$ and $\cln{U}{90}$, and inspect whether ${{|\bar{Q}|}}<\cln{Q}{90}$ and ${{|\bar{U}|}}<\cln{U}{90}$.
[${\bullet}\hspace{1em}$]{}If both Stokes Parameters fall within the limits, then the target is not shown to be polarized at the 81% confidence level. In this case we can try to find polarization with some lower confidence, so repeat the test for $C_Q=C_U=85\%$. If the null hypothesis can be rejected in either channel, then we have a detection at the 72.25% confidence level. There is probably little merit in plumbing lower confidences than this.
[${\bullet}\hspace{1em}$]{}If, however, polarization is detected in one or both of the Stokes Parameters at the starting point of 90%, test the polarized parameters to see if the polarization remains at higher confidences, say 95% and 97.5%. The highest confidence with which we can reject the null (unpolarized) hypothesis for either Stokes Parameter should be squared to give the confidence with which we may claim to have detected an overall polarization.
It is worth noting, [*en passent*]{}, that there is also a statistical test which is applicable to test whether two polarization measurements taken at different epochs or in neighbouring spectral bands are likely to indicate a common underlying value or not. Details of this, the Welch test, are given in the review by Clarke & Stewart .
In our hypothesis testing, we have made the [*a priori*]{} assumption that all targets are to be assumed unpolarized until proven otherwise. This is a useful question, as we must ask whether our data are worth processing further – and we ask it using the raw Stokes Parameters, without resorting to complicated formulae. To publish useful results, however, we must produce the normalized Stokes Parameters, together with some sort of error estimate, and it is this matter which we will consider next.
The Normalized Stokes Parameters
--------------------------------
We have already derived an exact formula for the error on a normalised Stokes Parameter (Equation \[getsterr\]). In order to simplify the calculation, we recall that in Check \[maxbig\], we checked that the errors on all the $S_i$ (and hence $I_i$) were similar. Thus the mean error on [*one*]{} rate in [*one*]{} channel is ${{{\mathcal E}_{\mathsf{phot}}}}/\sqrt{2}$. Since the number of measurements made of $S$ is $\nu_S$, then $$\sigma_{\bar{n}_{1i}} \simeq \sigma_{\bar{n}_{2i}} \simeq
{{{\mathcal E}_{\mathsf{phot}}}}/\sqrt{2\nu_S}$$ and the error formula approximates to: $$\label{normerr}
\sigma_{\tilde{s}} =
\tilde{s}.{{{\mathcal E}_{\mathsf{phot}}}}.\sqrt{(\bar{S}^{-2}+\bar{I}^{-2})/\nu_S}.$$
In practice, we will be dealing with small polarizations, so $\bar{S} \ll
\bar{I}$, and knowing $\tilde{s}$ from Equation \[stilde\], then Equation \[normerr\] approximates to: $$\label{simerrs}
\sigma_{\tilde{s}} \simeq
\frac{\tilde{s}.{{{\mathcal E}_{\mathsf{phot}}}}}{\bar{S}.\sqrt{\nu_S}} =
\frac{{{{\mathcal E}_{\mathsf{phot}}}}}{\bar{I}.\sqrt{\nu_S}}$$
As we had before with ${{{\mathcal E}_{\mathsf{stat}}}}$ and ${{{\mathcal E}_{\mathsf{phot}}}}$, so now we have a choice of using sky photometry or the statistics to estimate errors. The above method gives us the photometric error on a normalized Stokes’ Parameter as ${{\varepsilon_{\mathsf{phot}}}} = {{{\mathcal E}_{\mathsf{phot}}}}/\bar{I} = \sigma_{\tilde{s}}.\sqrt{\nu_S}$; the statistical method would be to take the root-mean-square deviation of the measured $s_i$, obtained in Step \[getthei\], about Clarke et al.’s best estimator value, $\tilde{s}$: $$\label{Nstaterr}
{{\varepsilon_{\mathsf{stat}}}} = \varsigma_{\tilde{s}}.\sqrt{\nu_S} =
\frac{1}{\sqrt{\nu_S-1}}.\left[{\sum_{i=1}^{\nu_S}(s_i -
\tilde{s})^2}\right]^{{{\frac{1}{2}}}}$$
\[hereNSPs\] Following the method outlined for finding $\tilde{s}$ and $\sigma_{\tilde{s}}$, apply Equations \[stilde\] and \[simerrs\] to the data obtained in Step \[getmean\] to obtain $\tilde{q}$ with $\sigma_{\tilde{q}}$ and $\tilde{u}$ with $\sigma_{\tilde{u}}$.
\[stoeq\] Using $\tilde{q}$ and $\tilde{u}$, compute $\varsigma_{\tilde{q}}$ and $\varsigma_{\tilde{u}}$; find ${{\varepsilon_{\mathsf{stat}}}}$ for both normalized Stokes Parameters, and compare it with ${{\varepsilon_{\mathsf{phot}}}}$ in each case. Verify also that the errors, ${{\varepsilon_{\mathsf{X}}}}$, on the population standard deviations for the two Stokes Parameters are similar – this should follow from the $S$-independence of Equation \[simerrs\] for small $\tilde{q}$ and $\tilde{u}$.
So which error should one publish as the best estimate, ${{\hat{\sigma}_{\tilde{s}}}}$, on our final $\tilde{s}$ — $\sigma_{\tilde{s}}$ or $\varsigma_{\tilde{s}}$? Again, a conservative approach would be to take the greater of the two in each case.
\[gotnorm\] Choose the more conservative error on each normalized Stokes Parameter, and record the results as $\tilde{q} \pm {{\hat{\sigma}_{\tilde{q}}}}$ and $\tilde{u} \pm {{\hat{\sigma}_{\tilde{u}}}}$. Record also the best population standard deviations, ${{\hat{\sigma}_{q}}}$ and ${{\hat{\sigma}_{u}}}$.
The Degree of Linear Polarization
---------------------------------
### The distribution of the normalised Stokes Parameters
Having obtained estimated values for $q$ and $u$, with conservative errors, these values – together with the reference angle $\eta_0$ – can and should be published as the most convenient form of data for colleagues to work with. It is often desired, however, to express the polarization not in terms of $q$ and $u$, but of $p$ and $\phi$.
Simmons & Stewart discuss in detail the estimation of the degree of linear polarization. Their treatment makes a fundamental assumption that the [*normalized*]{} Stokes Parameters have a normal distribution [@Clarke+86a §4.2], and that the errors on $\tilde{q}$ and $\tilde{u}$ are similar. This latter condition is true for small polarizations (see Check \[stoeq\]), but before we can proceed, we must test whether the former condition is satisfied. (Maronna, Feinstein & Clocchiatti outline cases where $\tilde{s}$ approximates to the normal distribution, but the criteria are vague: that $\nu_S$ and/or $I$ should be ‘large’.)
If one assumes (Step \[assumenorm\]) that $n_{1}$ and $n_{2}$ are normally distributed, one can construct, following Clarke et al. , a joint distribution for $s$ whose parameters are the underlying [*population*]{} means $(n_{1_0}, n_{2_0})$ and standard deviations $(\sigma_1, \sigma_2)$ for the count rates $n_{1i}$ and $n_{2i}$. The algebra gets a little messy here, so we define three parameters, $A, B, C$:
$$\label{paralpha}
A = {{\frac{1}{2}}}\left[ \frac{1}{{\sigma_1}^2} +
\frac{1}{{\sigma_2}^2} \left( \frac{1-s}{1+s} \right) ^2 \right],$$
$$\label{parbeta}
B = {{\frac{1}{2}}}\left[ \frac{n_{1_0}}{{\sigma_1}^2} +
\frac{n_{2_0}}{{\sigma_2}^2} \left( \frac{1-s}{1+s} \right) \right],$$
$$\label{pargamma}
C = {{\frac{1}{2}}}\left[ \frac{{n_{1_0}}^2}{{\sigma_1}^2} +
\frac{{n_{2_0}}^2}{{\sigma_2}^2} \right].$$
Using these three equations, we can write the probability distribution for $s$ as: $$\label{sdist}
P(s) = \frac{B.\exp[\frac{B^2}{A} -
C]}{\sigma_1.\sigma_2.\sqrt{\pi.A^3}.(1+s)^2}.$$
This can be compared to the limiting case of the normal distribution whose mean $\tilde{s}_0$ and standard error $\sigma_0$ are obtained by propagating the underlying means $(n_{1_0}, n_{2_0})$ and standard deviations $(\sigma_1,\sigma_2)$ through Equations \[getstilde\] and \[getsterr\]: $$\label{snorm}
P_n(s) =
\frac{\exp[\frac{-(s -
\tilde{s}_0)^2}{2.{\sigma_0}^2}]}{\sigma_0.\sqrt{2\pi}};$$
We can derive an expression for the ratio $R(s) = P(s)/P_n(s)$, which should be close to unity if the normalized Stokes Parameter, $s$, is approximately normally distributed.
\[nearnormal\]
[${\bullet}\hspace{1em}$]{}Estimate $n_{1_0}$ and $n_{2_0}$ using Equations \[Idef\] and \[Sdef\], and the data from Step \[getmean\]. Estimate $\sigma_1
\simeq \sigma_2 \simeq {{\hat{\sigma}_{S}}}/\sqrt{2}$, where ${{\hat{\sigma}_{S}}}$ is obtained from Check \[noiseOK\].
[${\bullet}\hspace{1em}$]{}Use the values of $\tilde{s}$ and ${{\hat{\sigma}_{s}}}$ obtained in Step \[gotnorm\] as the best estimates of $\tilde{s}_0$ and $\sigma_{0}$.
[${\bullet}\hspace{1em}$]{}Hence use a computer program to calculate and plot $R(s)$ in the domain $-3{{\hat{\sigma}_{s}}} < s <
+3{{\hat{\sigma}_{s}}}$. If R(s) is close to unity throughout this domain, then we may treat the normalized Stokes Parameters as being normally distributed.
### Point estimation of $p$
If the data passes Checks \[stoeq\] and \[nearnormal\], then we can follow the method of Simmons & Stewart . They ‘normalize’ the intensity-normalized Stokes Parameters, $q$ and $u$, by dividing them by their common population standard deviation, $\sigma$. For clarity of notation, in a field where one can be discussing both probability and polarization, I will recast their formulae, such that the [*measured*]{} degree of polarization, normalized as required, is here given in the form $m = \tilde{p}/\sigma$; and the [*actual*]{} (underlying) degree of polarization, also normalized, is $a = p_0/\sigma$. It follows from the definition of $p$ (Equation \[defp\]) that $$\label{errp}
\sigma_p = \sqrt{\frac{q^2.{\sigma_q}^2 + u^2.{\sigma_u}^2}{q^2+u^2}}.$$ If ${\sigma_q} = {\sigma_u} = \sigma$, then $\sigma_p = \sigma$.
Now, Simmons & Stewart consider the case of a ‘single measurement’ of each of $q$ and $u$, whereas we have found our best estimate of these parameters following the method of Clarke et al. However, we can consider the whole process described by Clarke et al. as ‘a measurement’, and so the treatment holds when applied to our best estimate of the normalized Stokes Parameters, together with the error on that estimate.
\[findperr\] Find ${{\hat{\sigma}_{p}}}$, and hence $\sigma={{\hat{\sigma}_{p}}}$, by substituting our best estimates of $q$ and $u$ and their errors (Step \[gotnorm\]) into Equation \[errp\]. Hence calculate $m$: $$m = \sqrt{\tilde{q}^2+\tilde{u}^2}/\sigma.$$
The probability distribution $F(m,a)$ of obtaining a measured value, $m$, for some underlying value, $a$, is given by the Rice distribution [@Simmons+85a; @Wardle+74a], which is cast in the current notation using the modified Bessel function, $I_0$ [@Boas-83a as defined in Ch.12, §17]: $$\label{rice}
F(m,a) = m.\exp \left[ \frac{-(a^2+m^2)}{2} \right] .I_0(ma) \ldots
(m\geq 0)$$ $$F(m,a)=0 {\mathit{~otherwise}}.$$
Simmons & Stewart have tested various estimators ${{\hat{a}}_{\mathsf{X}}}$ for bias. They find that when ${{{{a\:}^{{<}}_{{\sim}}{\:0.7}}}}$, the best estimator is the ‘Maximum Likelihood Estimator’, ${{\hat{a}}_{\mathsf{ML}}}$, which maximises $F(m,a)$ with respect to $a$. So ${{\hat{a}}_{\mathsf{ML}}}$ is the solution for $a$ of: $$\label{MLest}
a.I_0(ma) - m.I_1(ma) = 0.$$
If $m<1.41$ then the solution of this equation is ${{\hat{a}}_{\mathsf{ML}}} = 0$.
When ${{{{a\:}^{{>}}_{{\sim}}{\:0.7}}}}$, the best estimator is that traditionally used by radio astronomers, e.g. Wardle & Kronberg . In this case, the best estimator, ${{\hat{a}}_{\mathsf{WK}}}$, is that which maximises $F(m,a)$ with respect to m, being the solution for $a$ of: $$\label{WKest}
(1-m^2).I_0(ma) + ma.I_1(ma) = 0.$$ If $m<1.00$ then the solution of this equation is ${{\hat{a}}_{\mathsf{WK}}} = 0$.
Simmons & Stewart graph $m(a)$ for both cases, and so show that $m$ is a monotonically increasing function of $a$, and that ${{\hat{a}}_{\mathsf{ML}}} <
{{\hat{a}}_{\mathsf{WK}}} < m$ $\forall m$. But which estimator should one use? Under their treatment, the selection of one of these estimators over the other depends on the underlying value of $a$; they point out that there may be good [*a priori*]{} reasons to assume greater or lesser polarizations depending upon the nature of the source.
If we do not make any such assumptions, we can use monotonicity of $m$ and the inequality ${{\hat{a}}_{\mathsf{ML}}} < {{\hat{a}}_{\mathsf{WK}}}$ $\forall m$, to find two limiting cases:
[${\bullet}\hspace{1em}$]{}Let ${{m}_{\mathsf{WKmin}}}$ be the solution of the Wardle & Kronberg Equation (\[WKest\]) for $m$ with $a=0.6$. Hence if $m<{{m}_{\mathsf{WKmin}}}$, then ${{\hat{a}}_{\mathsf{ML}}} < {{\hat{a}}_{\mathsf{WK}}}
< 0.7$ and the Maximum Likelihood estimator is certainly the most appropriate. Calculating, we find ${{m}_{\mathsf{WKmin}}} = 1.0982 \ll 1.41$ and so the Maximum Likelihood estimator will in fact be zero.
[${\bullet}\hspace{1em}$]{}Let ${{m}_{\mathsf{MLmax}}}$ be the solution of Maximum Likelihood Equation (\[MLest\]) for $m$ with $a=0.8$. We find ${{m}_{\mathsf{MLmax}}} = 1.5347$. Hence if $m>{{m}_{\mathsf{MLmax}}}$, then $0.7 < {{\hat{a}}_{\mathsf{ML}}} <
{{\hat{a}}_{\mathsf{WK}}}$, and Wardle & Kronberg’s estimator will clearly be the most appropriate.
Between these two extremes, we have ${{{{{{\hat{a}}_{\mathsf{ML}}}\:}^{{<}}_{{\sim}}{\:0}}}}.{{{{7\:}^{{<}}_{{\sim}}{\:{{\hat{a}}_{\mathsf{WK}}}}}}}$. This presents a problem, in that each estimator suggests that its estimate is more appropriate than that of the other estimator. If our measured value is ${{m}_{\mathsf{WKmin}}} < m < {{m}_{\mathsf{MLmax}}}$, what should we take as our best estimate? We could take the mean of the two estimators, but this would divide the codomain of $\hat{a}(m)$ into three discontinuous regions; there might be some possible polarization which this method could never predict! It would be better, then, to interpolate between the two extremes, such that in the range ${{m}_{\mathsf{WKmin}}} < m < {{m}_{\mathsf{MLmax}}}$, $$\label{interpa}
\hat{a} = \frac{m-{{m}_{\mathsf{WKmin}}}}{{{m}_{\mathsf{MLmax}}}-{{m}_{\mathsf{WKmin}}}}.{{\hat{a}}_{\mathsf{ML}}} +
\frac{{{m}_{\mathsf{MLmax}}}-m}{{{m}_{\mathsf{MLmax}}}-{{m}_{\mathsf{WKmin}}}}.{{\hat{a}}_{\mathsf{WK}}}.$$
If we do not know, [*a priori*]{}, whether a source is likely to be unpolarized, polarized to less than 1%, or with a greater polarization, then $\hat{a}$ would seem to be a reasonable estimator of the true noise-normalized polarization, and certainly better than the biased $m$.
\[esta\] Use the above criteria to find $\hat{a}$, and hence obtain the best estimate, $\hat{p} = \hat{a}.\sigma$, of the true polarization of the target.
### A confidence interval for $p$
As well as a point estimate for $p$, we would like error bars. The Rice distribution, Equation \[rice\], gives the probability of obtaining some $m$ given $a$, and can, therefore, be used to find a confidence interval for the likely values of $m$ given $a$. We can define two functions, ${{\mathcal{L}}(a)}$ and ${{\mathcal{U}}(a)}$, which give the lower and upper confidence limits for $m$, with some confidence $C_p$; integrating the Rice distribution, these will satisfy: $$\label{Ldef}
\int_{m=-\infty}^{m={{\mathcal{L}}(a)}} F(m,a).dm = p_1$$ and $$\label{Udef}
\int_{m={{\mathcal{U}}(a)}}^{m=+\infty} F(m,a).dm = p_2$$ such that $$\label{addprobs}
1 - C_p = p_1 + p_2.$$
Such confidence intervals are non-unique, and we need to impose an additional constraint. We could require that the tails outside the confidence region be equal, $p_1 = p_2$, but following Simmons & Stewart , we shall require that the confidence interval have the smallest possible width, in which case our additional constraint is: $$\label{aconst}
F[{{\mathcal{U}}(a)},a] = F[{{\mathcal{L}}(a)},a].$$
Figure adapted from Leyshon & Eales . \[ricefig\]
From the form of the Rice distribution, ${{\mathcal{L}}(a)}$ and ${{\mathcal{U}}(a)}$ will be monotonically increasing functions of $a$, as shown in Figure \[ricefig\]. Given a particular underlying polarization $a_0$, the $C_p$ confidence interval $(m_1,m_2)$ can be obtained by numerically solving Equations \[Ldef\] thru \[aconst\] to yield $m_1 = {{\mathcal{L}}(a_0)}$ and $m_2 = {{\mathcal{U}}(a_0)}$.
Now, it can be shown [@Mood+74a Ch.VIII, §4.2] that the process can also be inverted, i.e. if we have obtained some measured value $m_0$, then solving for $m_0 = {{\mathcal{U}}(a_1)} = {{\mathcal{L}}(a_2)}$ will yield a confidence interval $(a_1,a_2)$, such that the confidence of $a$ lying within this interval is $C_p$.
Since the contours for ${{\mathcal{U}}(a)}$ and ${{\mathcal{L}}(a)}$ cut the $m$-axis at non-zero values of $m$, we must distinguish three cases, depending on whether or not $m_0$ lies above one or both of the intercepts. The values of ${{\mathcal{L}}(0)}$ and ${{\mathcal{U}}(0)}$ depend only on the confidence interval chosen; substituting $a = 0$ into Equations \[Ldef\] thru \[aconst\] results in the pair of equations $$\label{cforz}
C_m = \exp \left[ -\frac{{{\mathcal{L}}(0)}^2}{2} \right] -
\exp \left[ -\frac{{{\mathcal{U}}(0)}^2}{2} \right]$$ and $$\label{transce}
{{\mathcal{L}}(0)}.\exp \left[ -\frac{{{\mathcal{L}}(0)}^2}{2} \right] =
{{\mathcal{U}}(0)}.\exp \left[ -\frac{{{\mathcal{U}}(0)}^2}{2} \right].$$
A numerical solution of this pair of equations can be found for any given confidence interval, $C_m$; we find that, in 67% $(1\sigma)$ interval, ${{\mathcal{L}}(0)} = 0.4438,\: {{\mathcal{U}}(0)} = 1.6968$, while in a 95% $(2\sigma)$ interval, ${{\mathcal{L}}(0)} = 0.1094,\: {{\mathcal{U}}(0)} = 2.5048$. Hence, knowing $m_0$, and having chosen our desired confidence level, we can determine the interval $(a_1,a_2)$ by the following criteria:
[${\bullet}\hspace{1em}$]{}[$m_0 \geq {{\mathcal{U}}(0)}$]{} There are non-zero solutions for both ${{\mathcal{U}}(a_1)}$ and ${{\mathcal{L}}(a_2)}$.
[${\bullet}\hspace{1em}$]{}[${{\mathcal{L}}(0)} < m_0 < {{\mathcal{U}}(0)}$]{} In this case, $a_1=0$, and we must solve $m_0 =
{{\mathcal{L}}(a_2)}$.
[${\bullet}\hspace{1em}$]{}[$m_0 \leq {{\mathcal{L}}(0)}$]{} Here, $a_1=a_2=0$.
Simmons & Stewart note that the third case is formally a confidence interval of zero width, and suggest that this is counter-intuitive; and they go on to suggest an [*ad hoc*]{} method of obtaining a non-zero interval. However, it is perfectly reasonable to find a finite probability that the degree of polarization is identically zero: the source may, after all, be unpolarized. This can be used as the basis of estimating the probability that there is a non-zero underlying polarization, as will be shown in the next section.
\[getint\] Knowing $m$ from Step \[findperr\], find the limits $(a_1,a_2)$ appropriate to confidence intervals of 67% and 95%. Hence, multiplying by $\sigma$, find the confidence intervals on the estimated degree of polarization. The 67% limits may be quoted as the ‘error’ on the best estimate.
### The probability of there being polarization
Consider the contour $m={{\mathcal{U}}(a)}$ on Figure \[ricefig\]. As defined by Equation \[Udef\] and the inversion of Mood et al. , it divides the domain into two regions, such that there is a probability $p_2$ of the underlying polarization being greater than $a={\mathcal U}^{-1}(m_0)$. There is clearly a limiting case where the contour cuts the $m$-axis at $m_0$, hence dividing the domain into the polarized region $a>0$ with probability $p_P$, and the unpolarized region with probability $1-p_P$.
Now we may substitute the Rice Distribution, Equation \[rice\], into Equation \[Udef\] and evaluate it analytically for the limiting case, $a=0$: $$\label{propol}
p_P = 1 - \exp(-{m_0}^2/2).$$ Equation \[propol\] hence yields the probability that a measured source actually has an underlying polarization.
\[estpolun\] Substitute $m$ from Step \[findperr\] into Equation \[propol\]. Hence quote the probability that the observed source is truly polarized.
A more powerful method, applicable to cases where $\nu_Q \neq \nu_U$ and $\sigma_Q \neq \sigma_U$, is given by Clarke & Stewart : they define and tabulate values for a statistic $Z_{\alpha-1}$ such that the $(\alpha-1).100\%$ confidence interval for $p$ is an ellipse in the $q,u$ plane centred on $(\bar{q},\bar{u})$ and with semi-axes given by $\sqrt{{\varsigma_q}^2.Z_{\alpha-1}}$ and $\sqrt{{\varsigma_u}^2.Z_{\alpha-1}}$. Values are only tabulated, however, for certain $\nu_S$, all multiples of 5 or 10. In the current notation, the statistic is $$\label{zstatdef}
Z_{\alpha-1} = \frac{(\bar{q}-q_0)^2}{{\varsigma_q}^2} + \frac{(\bar{u}-q_0)^2}{{\varsigma_u}^2}.$$
The Polarization Axis
---------------------
It remains to determine the axis of polarization, for which an unbiased estimate is given by Equation \[phidef\]. Once again, we have a choice of using the statistical or photometric errors — and, indeed, a choice of raw or normalized Stokes Parameters. Since $$\label{redf}
2\phi = \theta = \tan^{-1}(u/q) = \tan^{-1}(r),$$ our first problem is to obtain the best figure for $r = u/q$.
Now, as we saw in our discussion of the best normalized Stokes Parameter, it is better to ratio a pair of means than to take the mean of a set of ratios. We could take $r=\bar{U}/\bar{Q}$, but for a very small sample, there is the danger that the mean intensity of the $Q$ observations will differ from that of the $U$ values. Therefore, we should use the normalized Stokes Parameters, and the least error prone estimate of the required ratio will be $\tilde{r}=\tilde{u}/\tilde{q}$, yielding $\tilde{\phi}$.
Knowing the errors on $\tilde{q}$ and $\tilde{u}$, we can find the propagated error in $\tilde{r}$: $$\label{getrerr}
\sigma_{\tilde{r}} =
\tilde{r}.\sqrt{\left(\frac{\tilde{q}}{\sigma_{\tilde{q}}}\right)^2 +
\left(\frac{\tilde{u}}{\sigma_{\tilde{u}}}\right)^2};$$ given the non-linear nature of the tan function, the error on $\tilde{\phi}$ should be found by separately calculating $\sigma_+ = {{\frac{1}{2}}}\tan^{-1}(\tilde{r}+\sigma_{\tilde{r}}) - \tilde{\phi}$ and $\sigma_- = {{\frac{1}{2}}}\tan^{-1}(\tilde{r}-\sigma_{\tilde{r}}) - \tilde{\phi}$. Careful attention must be paid in the case where the error takes the phase angle across the boundary between the first and fourth quadrants, as the addition of $\pm \pi$ to the inverse tangent may be necessary to yield a sensible error in the phase angle.
\[propphi\] Obtain $\tilde{\phi}$, the best estimate of $\phi$, and the propagated error on it, $\sigma_{\tilde{\phi}} = {{\frac{1}{2}}}(|\sigma_+| +
|\sigma_-|)$, using Equations \[redf\] and \[getrerr\]. Add $\eta_0$ to $\tilde{\phi}$ and hence quote the best estimate of the polarization orientation in true celestial co-ordinates.
For the statistical error, we note that the probability distribution of observed [*phase*]{} angles, $\theta=2\phi$, calculated by Vinokur , and quoted elsewhere [@Wardle+74a; @Clarke+86a; @NaghizadehKhouei+93a], is: $$P(\theta) = \exp \left[ -\frac{a^2 \sin^2(\theta-\theta_0)}{2} \right]
.$$ $$\label{thetadis}
\left\{ \frac{1}{2\pi} \exp \left[ - \frac{a^2 \cos^2(\theta-\theta_0)}{2}
\right] + \frac{a \cos(\theta-\theta_0)}{\sqrt{2\pi}}.\left\{ {{\frac{1}{2}}}+ f[a
\cos(\theta-\theta_0)] \right\} \right\}$$ where $$\label{thetasup}
f(x) = \frac{{\mathrm sign}(x)}{\sqrt{2\pi}} \int_0^x
\exp \left(- \frac{z^2}{2} \right) \,dz
={\mathrm sign}(x).{\mathrm erf}(x)/\sqrt{8},$$ and ${\mathrm erf}(x)$ is the error function as defined by Boas . We do not know $a=p_0/\sigma$, and will have to use our best estimate, $\hat{a}$, as obtained from Step \[esta\]. The $C_\phi.100\%$ confidence interval on the measured angle, $(\theta_1,\theta_2)$, is given by numerically solving $$\label{thetaerr}
\int_{\theta_1}^{\theta_2} P(\theta).d\theta = C_\phi;$$ in this case we choose the symmetric interval, $\theta_2-\tilde{\theta} =
\tilde{\theta}-\theta_1$.
\[findangle\] Obtain the limiting values of $\phi=\theta/2$ for confidence intervals of 67% $(1\sigma)$ and 95% $(2\sigma)$. Quote the 67% limits as $\varsigma_{\tilde{\phi}} = (\phi_2-\phi_1)/2$. Choose the more conservative error from $\varsigma_{\tilde{\phi}}$ and $\sigma_{\tilde{\phi}}$ as the best error, ${{\hat{\sigma}_{\tilde{\phi}}}}$.
Comparison with Other Common Techniques
---------------------------------------
It may be instructive to note how the process of reducing polarimetric data outlined in this chapter compares with the methods commonly used in the existing literature. The paper by Simmons & Stewart gives a thorough review of five possible point estimators for the degree of polarisation. One of these methods is the trivial $m$ as an estimator of $a$. The other four methods all involve the calculation of thresholds ${{{\underline{m}}_{\mathsf{X}}}}$: if $m < {{{\underline{m}}_{\mathsf{X}}}}$ then ${{\hat{a}}_{\mathsf{X}}}=0$. These four methods are the following:
1. Maximum Likelihood: as defined above, ${{\hat{a}}_{\mathsf{ML}}}$ is the value of $a$ which maximises $F(m,a)$ with respect to $a$. Hence ${{\hat{a}}_{\mathsf{ML}}}$ is the solution for $a$ of Equation \[MLest\]. The limit ${{{\underline{m}}_{\mathsf{ML}}}}=1.41$ is found by a numerical method.
2. Median: ${{\hat{a}}_{\mathsf{med}}}$ fixes the distribution of possible measured values such that the actual measured value is the [*median*]{}, hence $\int_{m'=0}^{m'=m} F(m',{{\hat{a}}_{\mathsf{med}}}).dm' = 0.5$. The threshold is ${{{\underline{m}}_{\mathsf{med}}}}
= 1.18$, being the solution of $\int_{m'=0}^{m'={{{\underline{m}}_{\mathsf{med}}}}} F(m',0).dm'=0.5$.
3. Serkowski’s estimator: ${{\hat{a}}_{\mathsf{Serk}}}$ fixes the distribution of possible measured values such that the actual measured value is the [ *mean*]{}, hence $\int_{m'=0}^{m'=\infty} m'.F(m',{{\hat{a}}_{\mathsf{Serk}}}).dm' = m$. The threshold is ${{{\underline{m}}_{\mathsf{Serk}}}} = 1.25 = \int_{m'=0}^{m'=\infty} m'.F(m',0).dm'$.
4. Wardle & Kronberg’s method: as defined above, the estimator, ${{\hat{a}}_{\mathsf{WK}}}$, is that which maximises $F(m,a)$ with respect to $m$ (see Equation \[WKest\]), and ${{{\underline{m}}_{\mathsf{WK}}}}=1.00$.
Simmons & Stewart note that although widely used in the optical astronomy literature, Serkowski’s estimator is not the best for either high or low polarizations; they find that the Wardle & Kronberg method commonly used by radio astronomers is best when ${{{{a\:}^{{>}}_{{\sim}}{\:0.7}}}}$, i.e. when the underlying polarization is high and/or the measurement noise is very low. The Maximum Likelihood method, superior when ${{{{a\:}^{{<}}_{{\sim}}{\:0.7}}}}$ (i.e. in ‘difficult’ conditions of low polarization and/or high noise), appears to be unknown in the earlier literature. \[It seems to have been used independently shortly after Simmons & Stewart in Appendix B of the paper by Killeen, Bicknell & Ekers .\]
In this chapter, I have merely provided an interpolation scheme between the point estimators which they have shown to be appropriate to the ‘easy’ and ‘difficult’ measurement regimes. The construction of a confidence interval to estimate the error is actually independent of the choice of point estimator, although (as mentioned above) I believe that Simmons & Stewart’s unwillingness to ‘accept sets of zero interval as confidence intervals’ is unfounded, since physical intuition allows for the possibility of truly unpolarised sources (i.e. with identically zero polarizations), and their arbitrary method of avoiding zero-width intervals can be dispensed with.
Zero-Polarization Objects and a Residual Method {#zpos}
-----------------------------------------------
The data reduction process presented above has made no [*a priori*]{} assumptions about whether the target object has a high or low polarization, and is even general enough to cope with different numbers of observations of the $q$ and $u$ Stokes parameters if difficult observing conditions limit the data in this way.
Clarke et al. suggest a method which can be used to test whether the underlying polarization of a low polarization object is actually zero. For a zero polarization $(a=0)$ object, the Rice distribution simplifies to the Rayleigh distribution: $$\label{rayleigh}
F(m,0) = m.\exp \left[ \frac{-m^2}{2} \right] \ldots
(m\geq 0)$$ $$F(m,0)=0 {\mathit{~otherwise}}.$$
We can use the Rayleigh distribution to calculate the cumulative distribution function ${\Phi_{p}(m)}$ for the probability of obtaining a measurement $0<m_{i}<m$, and compare this to the actual fraction of measurements which lie between 0 and $m$ – the ‘empirical cumulative distribution’, ${\Xi_{p}(m)}$.
Integrating the Rayleigh distribution, we find $$\label{p-cdf}
{\Phi_{p}(m)} = 1 - \exp \left[ \frac{-m^2}{2} \right].$$ This equation gives us the probability that an [*unpolarised*]{} object might give a polarization measurement of $m$ or less, and is identical to Equation \[propol\] for the probability that an object yielding a measurement $m_0$ is actually unpolarised. (This follows from the inversion argument illustrated in Figure \[ricefig\].)
To obtain the ‘empirical cumulative distribution’, we must obtain and sort a set of $m_i$ based on pairs of individual measurements $q_i$, $u_i$. To calculate the $m_i$ we also need the relevant standard deviations. Now in theory, as long as the noise level is constant (not necessarily true if observations are pooled from different instruments) there should be a pair of population standard deviations $\sigma_{q}$, $\sigma_{u}$, which characterise the errors on any individual measurements of these normalised Stokes parameters. But Clarke et al. point out that the true errors are not known and must be estimated.
We estimated $\hat{\sigma}_{q}$ and $\hat{\sigma}_{u}$ at Step \[gotnorm\] based on the whole sample; these figures should be good estimates of the true value. Array-based photometry is also capable of giving an error $\sigma_{s_{i}}$ for each individual measurement: if Data Check \[maxbig\] verified that individual errors did not vary widely from the mean error, we can take $\sigma_{s_{i}} \simeq \hat{\sigma}_{s}$, but if there is wide variation, then the individual errors should be used, being calculated in analogy with Equation \[getsterr\] as: $$\label{nspi-err}
\sigma_{s_i} =
\frac{1}{n_{1}+n_{2}}.\sqrt{[(1-s_{i})\sigma_{{n}_{1}}]^2
+ [(1+s_{i})\sigma_{{n}_{2}}]^2}.$$
Ultimately, the $m_i$ can be calculated: $$\label{m-i}
m_{i} = \sqrt{ \left( \frac{q_i}{\sigma_{q_i}} \right)^{2} +
\left( \frac{u_i}{\sigma_{u_i}} \right)^{2} }.$$
A similar exercise can be conducted for the direction of polarization. An unpolarised object subject to random noise should not display any preferred direction of polarization, so the probability distribution function for the measured angle will be uniform between $\phi=0$ and $\phi=\pi$; the cumulative distribution function for angles will be ${\Phi_{\theta}(\phi)} = \phi/\pi$.
Again the measured Stokes parameters must be paired, to give the individual position angles $\phi_{i} = \tan^{-1}(u_{i}/q_{i})\div 2$ (compare Equation \[redf\]). In this case it is not necessary to consider the errors; the empirical cumulative distribution ${\Xi_{\theta}(\phi)}$ is simply the fraction of $\phi_{i}$ values in the range $0<\phi_{i}<\phi$.
Clarke et al. explain how the Kolmogorov(-Smirnov) test can be used to compare the theoretical and empirical distributions, and discuss systematic effects which might cause the empirical distributions to deviate from those expected for an unpolarised source.
Clarke & Naghizadeh-Khouei point out that if a good estimate $\tilde{s}$ is available for a normalized Stokes parameter of a polarized source, then the residuals $\breve{s}_{i} =
s_{i} - \tilde{s}$ should behave in the same way as the measured polarization of an unpolarised source. It would be possible, therefore, to proceed from Step \[gotnorm\] for an equal number, $\nu$, of measurements of the two Stokes parameters, as follows:
The normalised Stokes residuals $\breve{q}_{i}$, $\breve{u}_{i}$, may be calculated, and may be treated in the same way as the Stokes parameters of an unpolarised object; the empirical cumulative distributions of the residuals ${\Xi_{p}(\breve{m})}$ and ${\Xi_{\theta}(\breve{\phi})}$ can be obtained and tested for goodness-of-fit to the theoretical distributions for an unpolarised object. It might also be possible to iteratively refine the values of $\tilde{q}$ and $\tilde{u}$ to improve the fit.
Conclusion
----------
The reduction of polarimetric data can seem a daunting task to the neophyte in the field. In this chapter, I have attempted to bring together in one place the many recommendations made for the reduction and presentation of polarimetry, especially those of Simmons & Stewart , and of Clarke et al. . In addition, I have suggested that it is possible to develop the statistical technique used by Simmons & Stewart to obtain a simple probability that a measured object has non-zero underlying polarization. I have also suggested that there is a form of estimator for the overall degree of linear polarization which is more generally applicable than either the Maximum Likelihood or the Wardle & Kronberg estimators traditionally used, and which is especially relevant in cases where the measured data include degrees of polarization of order 0.7 times the estimated error.
Modern computer systems can estimate the noise on each individual mosaic of a sequence of images; this is useful information, and is not to be discarded in favour of a crude statistical analysis. A recurring theme in this chapter has been the comparison of the errors estimated from propagating the known sky noise, and from applying sampling theory to the measured intensities. Bearing this in mind, I have presented here a process for data reduction in the form of \[findangle\] rigorous steps and checks. The recipe might be used as the basis of an automated data reduction process, and I hope that it will be of particular use to the researcher – automated or otherwise – who is attempting polarimetry for the first time.
Mathematical Glossary
---------------------
Since this chapter uses a lot of mathematical terms common with Appendix \[photapp\], and a few which differ in definition, I have given both this chapter and that appendix a mathematical glossary defining the terms used. Latin symbols are listed in alphabetical order first, followed by Greek terms according to the Greek alphabet – except that terms of the form $\sigma_\aleph$ are listed under the entry for $\aleph$.
The true error-normalised polarization, $a = p_0/\sigma$.
- A generic estimator of $a$.
- A specified value of $a$ used to estimate a confidence region for $m_1, m_2$.
- A confidence region corresponding to a measured polarization $m_0$.
The number of pixels forming the aperture within which a source intensity is measured.
Analogue-to-Digital Unit: another name for DN ([*q.v.*]{}).
Variables used to parameterise the complicated expression for $P(s)$ ([*q.v.*]{}).
Generic symbol for $C_Q$ and $C_U$, respectively the confidence levels for rejecting null hypothesis when testing $\bar{Q}$ and $\bar{U}$ for consistency with zero.
The confidence specified for normalised polarization $m$ to lie within a given interval.
The confidence specified for polarization axis orientation $\phi$ to lie within a given interval.
The number of pixels forming the annulus surrounding a source, within which the dark sky intensity is measured.
Data Number: a photon counting system returns a count of 1 DN for every $\Delta$ photons incident.
The frequency of a beam of quasimonochromatic light.
The Rice distribution.
The modified Bessel Functions.
The intensity of a source in DN counts per second, $I=n_1 + n_2$. It takes the same annotations as $S$ ([*q.v.*]{}), and also:
- The linearly unpolarised and polarised components of a partially polarised beam of light.
- Two estimates of $I$ made by taking separately the means of those $I_i$ values obtained when determining $Q$ and those obtained in determining $U$.
Lower confidence limit for $m$.
The measured error-normalised polarization, $m =
\tilde{p}/\sigma$.
- A measured polarization used to estimate a confidence region for the true polarization, $a_1,
a_2$.
- A confidence region corresponding to likely measured values for a true polarization $a_0$.
- A lower limit: the lowest measurement likely to indicate that there is a true underlying polarization.
- The residual measured polarization calculated from the residual normalized Stokes Parameter measurements $\breve{s}_i$ ([*q.v.*]{}).
A count rate (in DN per unit time) measured in one channel of a two-channel polarimeter.
- The DN count rates measured in the two channels of a rotatable analyzer when set to orientation $\eta$.
- The individual DN count rates measured in the two channels of a two-channel polarimeter on the $i$th mosaic.
- The true values of $n_1, n_2$.
- Generic symbol for either of $n_{1i}, n_{2i}$.
- The mean values of a series of $\nu_S$ DN count rates.
- The errors on a pair of individual DN count rate measurements, based on the sky errors returned by the photometry system.
- The errors on a pair of mean DN count rate measurements.
This symbol is used both for probabilities, and for the degree of polarization of partly linearly polarised light.
- The true value of a polarization $p$.
- The best estimate of the error on a polarization $p$.
- Probabilities when estimating confidence intervals for polarization $p$.
- The probability that a source is not unpolarized.
The accurate distribution of a normalised Stokes Parameter, $s$, when the two contributing channel intensities can be treated as Gaussian.
The Gaussian approximation to $P(s)$.
Absolute and normalised linear Stokes Parameters. See $S, s$.
The ratio $U/Q$ used for finding the polarization axis.
- The best estimate $\tilde{u}/\tilde{q}$.
- The error on $\tilde{r}$.
Symbol for the reference direction corresponding to $\eta_0$.
The ratio of the true distribution $P(s)$ to its approximation $P_n(s)$.
The number of photons per second in a beam of light.
A generalised absolute Stokes Parameter $S=n_1 - n_2$ illustrating the generic properties of $Q$ and $U$.
- The true value of $S$.
- The $i$th measurement of $S$.
- The limiting value of $\bar{S}$ for accepting a null hypothesis at the $C_S.100\%$ confidence level.
- An estimate of $S_0$, the mean of the $S_i$ values.
- The error on an individual source intensity measurement, based on the sky errors returned by the photometry system.
- The standard error on the mean, indicating the accuracy with which $\bar{S}$ has been determined, based on the sky errors returned by the photometry system.
- The standard error on the mean, based on the spread of the $S_i$ values in the sample.
- The ‘best estimate’ of the standard error on the mean, taken as the greater of $\sigma_{\bar{S}}$ and $\varsigma_{\bar{S}}$.
- The ‘best estimate’ of the error on an individual measurement of $S$, taken as the greater of ${{{\mathcal E}_{\mathsf{phot}}}}$ and ${{{\mathcal E}_{\mathsf{stat}}}}$.
A general normalised Stokes Parameter $s=S/I$ illustrating the properties of $q$ and $u$.
- The true normalised Stokes Parameter of a source.
- The value of $s$ measured with a linear analyzer at the $j$th stepped position angle orientation $\eta_j$.
- The ratio of individual measurements of the absolute Stokes Parameters, $s_i = S_i / I_i$.
- The mean of the individual $s_i$, such that $\bar{s} = \sum_{1}^{\nu_S} s_i / \nu_S$.
- The optimal estimator which is the ratio of the mean Stokes Parameters, $\tilde{s} = \bar{S} / \bar{I}$.
- The residual Stokes Parameters when the optimal estimator is subtracted, $\breve{s}_i = s_i - \tilde{s}$.
- The error on $\tilde{s}$ estimated using ${{{\mathcal E}_{\mathsf{phot}}}}$, based on the sky errors returned by the photometry system.
- The error on $\tilde{s}$ estimated using ${{{\mathcal E}_{\mathsf{stat}}}}$, based on the spread of the $S_i$ values in the sample.
- The ‘best estimate’ of the standard error on the mean, taken as the greater of $\sigma_{\tilde{s}}$ and $\varsigma_{\tilde{s}}$.
- The ‘best estimate’ of the error on an individual measurement of $s$, taken as the greater of ${{\varepsilon_{\mathsf{phot}}}}$ and ${{\varepsilon_{\mathsf{stat}}}}$.
The statistic of the Student $t$ distribution.
Absolute and normalised linear Stokes Parameters. See $S, s$.
Upper confidence limit for $m$.
The statistic of the standard normal distribution.
A statistic listed here for compatibility with Clarke & Stewart .
The [*integer*]{} number of photons which must be detected to give a count of 1 DN.
A generic symbol for the error expected in making an individual measurement of $S_i$.
- The mean of the errors on individual measurements of $S_i$, such that ${{{\mathcal E}_{\mathsf{phot}}}} = \sum_{i=1}^{\nu_S} \sigma_{S_i}/\nu_S$. [*Hence ${{{\mathcal E}_{\mathsf{phot}}}}$ estimates the error on an individual measurement based on the error data from the photometry array.*]{}
- The estimated error on an individual measurement of $S_i$, based on the sample SD of all the $S_i$. [*Hence ${{{\mathcal E}_{\mathsf{stat}}}}$ estimates the error on an individual measurement without using the error data from the photometry array.*]{}
A generic symbol for the error expected in making an individual measurement of $s_i$.
- The expected error on an individual measurement of $s_i$, based on ${{{\mathcal E}_{\mathsf{phot}}}}$. [*Hence ${{\varepsilon_{\mathsf{phot}}}}$ makes use of the photometric error data.*]{}
- The expected error on an individual measurement of $s_i$, based on the spread of all the $s_i$ about $\tilde{s}$. [*Hence ${{\varepsilon_{\mathsf{stat}}}}$ estimates the error on an individual measurement without using the error data from the photometry array.*]{}
The position angle measured East of North on the celestial sphere of the transmission axis of a linear analyser.
- The position of the reference direction relative to which others are measured by a particular polarimeter.
- One specific position angle setting of a stepped rotating analyzer.
The phase angle of the polarization expressed as a vector in a phase space where $\theta = 2\phi$. See $\phi$.
The number of individual pairs of measurements made with a two-channel polarimeter in order to determine a set of $I_i$ and $S_i$. Note that in practical cases, often $\nu_Q \neq \nu_U$.
[$\Xi$]{} An empirical cumulative probability (ECD): the empirical probability that a measured quantity does not exceed a stated value.
- The empirical probability that the measured polarization is not greater than $m$.
- The empirical probability that the measured polarization axis orientation is not greater than $\phi$.
A standard deviation. Terms of the form $\sigma_\aleph$ are listed under the entry for $\aleph$, but note:
- Without annotation, $\sigma$ is the best estimate of the common error on $q$ and $u$, in the case ${{\hat{\sigma}_{\tilde{q}}}} \simeq {{\hat{\sigma}_{\tilde{u}}}}$.
- The expected errors on measurements of $n_1, n_2$ ([*q.v.*]{}).
- The approximate SD of an approximately normally distributed $s$.
The integration time for measuring light intensity.
The orientation position angle projected on the celestial plane of the electric field vector of partially linearly polarised light.
- The true values of $\phi$.
- The best estimate of $\phi$, derived from $\tilde{r}$.
- The error on the best estimate of $\phi$, derived from photometric errors.
- The error on the best estimate of $\phi$, derived from its theoretical distribution.
- The best estimate of the error on $\phi$.
- The residual measured polarization axis calculated from the residual normalized Stokes Parameter measurements $\breve{s}_i$ ([*q.v.*]{}).
[$\Phi$]{} A cumulative distribution function (CDF): the theoretical probability that a measured quantity will not exceed a stated value.
- The CDF for the probability that the measured polarization is not greater than $m$.
- The CDF for the probability that the measured polarization axis orientation is not greater than $\phi$.
The physical angle of rotation of a half-wave waveplate: $\eta = 2\chi$.
The standard deviation of the sample of $S_i$ values about their mean.
Observations, Reduction Procedure and Sample Selection {#mediuch}
======================================================
\[obsch\]
> When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
>
> ------------------------------------------------------------------------
>
> — Sherlock Holmes, [*The Sign of Four*]{}.
Observational data for this project were taken on two observing runs, both using the same equipment in Hawaii, in August 1995 and August 1997. On the 1995 run, seven objects were studied, all radio galaxies featuring in the 3C catalogue. The seven radio galaxies, at redshifts $0.7<z<1.2$, were not selected according to any statistical criterion, but formed a representative sample of the different morphologies present in this redshift band. With the hindsight provided by Eales et al., one presumes that 3C galaxies, displaying the strongest alignment effect, are also likely to display the strongest polarizations.
The second run, August 1997, looked at three objects. One, 3C 441, had featured in the first run but had not yielded a conclusive polarization value. Another, MRC 0156$-$252, possessed the brightest known absolute $V$-band magnitude for a radio galaxy at $z \sim 2$, and had featured in Eales & Rawlings’s comparison of radio galaxies at redshifts $z
\sim 1$ and $z > 2$. Finally, LBDS 53W091 is a very red radio galaxy visible at a very high ($z=1.552$) redshift — especially interesting since comparisons of its spectrum with synthetic and real elliptical galaxies suggest that it must be at least 3.5 Gyr old [@Dunlop+96a; @Spinrad+97a] – which is only consistent with its high redshift in certain cosmologies.
The instruments and procedure followed are summarised below. Consideration is also given to possible contaminating polarization due to Galactic dust. Full details of the nine sources investigated and other data concerning them can be found together with our results in Chapter \[resulch\].
As Mark Neeser reminds us in the cautionary note to his recent thesis [@Neeser-96a], such small samples of powerful objects are not likely to be typical of the Universe at large. But [*this*]{} author hopes that the results and discussion presented here in the final chapters will add a little to our understanding of the powerful objects that are high redshift radio galaxies; and especially that the methodology presented in Chapter \[stoch\] will be a useful guide to those who follow this work and take polarizations of statistically meaningful samples of these objects in future.
Instrumentation
---------------
All the $K$-band polarised images taken for this project were obtained using the IRPOL2 instrument, designed by the University of Hertfordshire and installed at UKIRT (the United Kingdom InfraRed Telescope, Hawaii). Our August 1995 run was the first common-user project undertaken by this instrument after its commissioning run.
The IRPOL2 polarimeter consists of a rotatable half-wave plate and a Wollaston prism, working in conjunction with the IRCAM3 InSb array detector. We used IRCAM3 at the default pixel scale, 0.286 arcsec/pixel [@Aspin-94a], with a $K$-band filter. The IRPOL2 system has negligible instrumental polarization [@hpc; @Chrysostomou-96a]. The Wollaston prism – a two-channel polarimeter following the paradigm of §\[paradigm\] – causes each source in its field of view to appear as a pair of superimposed images with orthogonal polarizations, separated by $-0.93$ pixels in right ascension and $+69.08$ pixels in declination [@Aspin-95a]. A focal plane mask is available: if used, it divides the array plane into four horizontal strips imaging light of alternate orthogonal polarizations. If not used, one polarised image is displaced and superposed on its orthogonal complement.
The design of the instrument is such that when the waveplate is set to its 0 reference position, an object totally linearly polarised with its electric vector at 83 (i.e. celestially East of North) would appear only in the upper (Northern) image, and an object totally linearly polarised at -7 would only appear in the lower image. Hence the reference axis ‘R’ is oriented at $\eta_0 = 83\degr$. There are four standard offset positions for the waveplate: 0, 22.5, 45and 67.5.
Observing Procedure
-------------------
We used slightly different techniques on the two observing runs to build up our images. On the first run, the focal plane mask was not used: a slight error in pointing could have caused the extended target objects to lie partially behind the mask. The absence of the mask meant that more field objects (useful controls for instrumental and local Galactic polarization) would also be imaged. The array was shifted equally in right ascension and declination to build up final image. On the second run, to reduce the background noise around our target objects, the mask was used; and so the array was shifted principally in right ascension to build up a final image.
### First run procedure
For each target object, we took a ‘mosaic’ of nine images with the waveplate at the 0 offset. One image consisted of a 60 second exposure (the sum of six ten-second co-adds), and the mosaic was built up by taking one image with the target close to the centre, and eight images with the frame systematically offset from the first by $\pm 28$ pixels (8 arcsec) horizontally and/or vertically, as illustrated in Figure \[mospat1\]. Since IRCAM3 is a square array of 256 pixels each side, each individual image had side 73, and the final mosaics had side 92, with the greatest sensitivity being achieved in the central square of side 54.
(126,60)(-30,-30) (-12,-4)[1]{} (0,0) (0,0)[(1,0)[28]{}]{} (28,0)[(0,1)[28]{}]{} (28,28)[(-1,0)[28]{}]{} (0,28)[(-1,0)[28]{}]{} (-28,28)[(0,-1)[28]{}]{} (-28,0)[(0,-1)[28]{}]{} (-28,-28)[(1,0)[28]{}]{} (0,-28)[(1,0)[28]{}]{} (31,-30)[9]{} (60,-10)[(1,0)[28]{}]{} (60,-13)[(0,1)[6]{}]{} (88,-13)[(0,1)[6]{}]{} (70,-8)[8]{}
The same source was then similarly observed with the waveplate at the 22.5, 45 and 67.5offsets, completing one cycle of observations; hence one such cycle took 36 minutes of integration time. Between two and five observation cycles were performed over the three nights for each target; the total integration time for each target is given with the observational data in Table \[extab\]. Not all of the times quoted are exact multiples of 36 minutes, as in some cases, mosaics were corrupted by problems with the UKIRT windblind, and excluded from our analysis. An example of a mosaic, 3C 54 and its surrounding field, observed with the waveplate at 22.5, is shown in Figure \[54image\]. This image shows clearly the effect of using the Wollaston prism without the focal plane mask: note the double images of most of the objects, and the partnerless objects on the right hand side whose upper or lower channels fell outside the detector array.
North is at the top, East at the left. The image was composed by mosaicing a series of nine 60-second exposures taken with the waveplate offset at 22.5.) \[54image\]
### Second run procedure
For our 1997 run, making use of the instrument’s focal plane mask, we created mosaics of each source by combining seven 60-s exposures at horizontal spacings of $9 \arcsec$ and vertical spacings of $\pm 1
\arcsec$ – see Figure \[mospat2\]. This procedure creates rectangular strip images measuring $127\arcsec \times 20\arcsec$, of which the central $19\arcsec \times
16\arcsec$ displays the maximum sensitivity. The total exposure time, summed over all four waveplate settings, is again listed in Table \[extab\].
Data Reduction
--------------
### First run procedure {#firstrunDR}
The raw data from IRCAM3 were stored as [*Starlink*]{} [ndf]{} ([foo.sdf]{}) images. These were converted to [iraf]{} ([foo.imh]{}) format for subsequent analysis by one of two methods: the earliest data to be analysed was handled by conversion to intermediate [fits]{} files, which were then read into [iraf]{} by its [rfits]{} routine[^4]; later, when the [*Starlink*]{} [figaro]{} package was enhanced, its new one-step [ndf2iraf]{} routine was employed. \[Documentation for [iraf]{} and [*Starlink*]{} packages can be found at their respective websites, [iraf.noao.edu]{} and [star-www.rl.ac.uk]{}.\]
We reduced the data by marking bad pixels, subtracting dark frames, and flat-fielding. Dr Stephen Eales at Cardiff University (private communication) made available software containing a list of known bad pixels on IRCAM3, which was used to mark the hot and dead pixels; he also provided mosaicing software which ignored such pixels when combining shifted frames to produce a final composite mosaic. The dark frames used were the means of 10-s dark exposures made at the start, middle and end of the night on which the corresponding target images were taken. Flat-fielding frames were obtained by median-combining the nine images of each mosaic without aligning them, and normalizing the resulting image by its mean pixel value. In order to align each set of nine images, we chose a star present on each frame, and measured its position with the [apphot.center]{} routine in the [iraf]{} package. Using these positions, the nine images were melded into a single mosaic image.
Photometry was performed on the pairs of images of field stars and of radio galaxies in each mosaic using [apphot.phot]{} from the [iraf]{} package. For each target, we chose one mosaic arbitrarily, and tested this to determine the best photometry aperture as follows: Using the arbitrary mosaic, we performed photometry on the two images of the source at a series of radii increasing in unit pixel steps. The measured magnitude in each aperture decreases as the light included in the aperture increases; we noted the first pair $(r,r+1)$ of radii where the change in magnitude was less than half the measured error on the magnitude. We then earmarked the next radius, $(r+2)$, for use in determining the polarization. In this way, we hope to include most of the source light but as little as possible of the surrounding sky. Where the $(r+2)$ aperture sizes differed for the two channels’ images of the source, we earmarked the larger of the two. This chosen aperture was then used as the photometric aperture on all the mosaics containing the target (i.e. on every waveplate setting from every observing cycle), with [iraf]{} output yielding a set of flux counts, magnitudes and errors.
The [apphot.phot]{} routine corrects for the sky brightness by measuring the modal light intensity in an annulus around the target; the position of the annulus was chosen in each case such that the outer radius did not extend to the nearest neighbouring object, and the inner radius was normally set one pixel greater than the photometry aperture. (Where we attempted to perform photometry on a knot within a larger structure, the inner radius of the annulus was set sufficiently large to exclude all the knots comprising the object.) The output file of [apphot.phot]{} contains data on the sky brightness and consequent errors on each photometric measurement, as well as ‘magnitudes’ (relative to an arbitrary zeropoint) for each target. These values were extracted and analyzed on a PC spreadsheet package ([*Microsoft Works*]{}). The spreadsheet programming follows the scheme described in Chapter \[stoch\] and is documented in Appendix \[codeapp\].
The spreadsheet analysis calculated both absolute $(Q,U)$ and normalized $(q,u)$ Stokes parameters for the $\eta_0 = 83\degr$ reference frames, together with the best estimates of the errors in each case. Finally, the optimal normalised Stokes parameters were used as input to [fortran]{} routines (also documented in Appendix \[codeapp\]) designed to follow Steps \[findperr\] thru \[findangle\] of Chapter \[stoch\]’s data reduction scheme. The program’s resultant output values consist of point estimates and confidence intervals of the debiased polarization and orientation measures, and an estimate of the probability that the source is actually polarised.
### Second run procedure
Data reduction for the August 1997 run was performed using the standard UKIRT [ircamdr]{} software to flat field images, and [iraf]{} to align and combine the mosaics. Since spreadsheet analysis of the August 1995 data had shown no source of systematic error, it was not felt necessary to repeat the spreadsheet system of rigorous checks before producing algebraic optimal estimates of the normalised Stokes parameters. Rather, in this case, the images were then averaged to form a single image for each waveplate, as this is equivalent to the algebraic method which gives the best signal-to-noise in the final polarimetry [@Leyshon-98a; @Maronna+92a; @Clarke+83b]. The 3C 441 data was then combined, at each waveplate position, with our stacked 1995 images. Results are discussed in the next chapter, and summarised in Table \[restab\].
Our Samples {#GalPolims}
-----------
Our first observing run took place on the nights of 1995 August 25, 26, and 27, and covered a sample of seven high-redshift 3C radio galaxies at redshifts $0.7<z<1.2$. These objects were not selected to form any kind of statistically complete sample; rather, they form a representative sample of the different morphologies present in this redshift band. Since this run represented the first known attempt to obtain infrared polarimetry of high redshift radio galaxies, objects known to be bright in the $K$-band were preferred.
The seven objects surveyed included 3C 22 and 3C 41, which are bright and appear pointlike; 3C 114 and 3C 356, which display complex knotted morphologies with large scale alignments between the $K$-band structure and the radio axis; and 3C 54, 3C 65 and 3C 441, which are faint sources with some indication of $K$-band structure. Of these three faint sources, 3C 54 displays an alignment between the $K$-band morphology and the radio axis [@Dunlop+93a], 3C 65 shows no preferred direction in its $H$-band structure [@Rigler+94a], and 3C 441 has a broad-band optical polarization which is roughly perpendicular to its radio structure [@Tadhunter+92a]. Five of the sources (not 3C 22 or 3C 356) were early radio sources observed by Longair .
The second observing run, conducted on the nights of 1997 August 18 and 19, was awarded primarily to study the controversial radio galaxy LBDS 53W091. This galaxy (see §\[D53W091\]) appears to be at least 3.5 Gyr old [@Dunlop+96a; @Spinrad+97a] — which is only consistent with its $z=1.552$ redshift in certain cosmologies. The timing of the run also made it possible to re-observe 3C 441, for which our 1995 data were inconclusive, and to observe MRC 0156$-$252, which has the brightest known absolute $V$-band magnitude for a radio galaxy at $z \sim 2$.
Table \[extab\] lists all the sources observed together with their redshifts and the rest-frame wavelength of the observed light. As discussed below in §\[ULGalPol\], it is possible to estimate an upper limit on the interstellar polarization imposed on $K$-band light during its passage through our Galaxy. Hence the [ned]{} values for the Galactic extinction $A_B$ at each observed source and the corresponding upper limits on $p_{\rm K}$ are given in this Table, too.
----------------------------------------------------------------------------------------------------------
Source IAU form $\lambda_r$($\mu$m) $t_{\rm $A_B$ $p_K$
int}$(min)
---------------- ------------ ------- --------------------- -------------- ------- ------- ------- -------
3C 22 0048+509 0.936 1.14 72 122.9 -11.7 1.09 0.76
3C 41 0123+329 0.794 1.23 108 131.4 -29.1 0.17 0.12
3C 54 0152+435 0.827 1.20 135 135.0 -17.6 0.37 0.26
3C 65 0220+397 1.176 0.92 72 141.5 -19.5 0.16 0.11
3C 114 0417+177 0.815 1.21 189 177.3 -22.2 1.26 0.88
3C 356 1723+510 1.079 1.06 99 77.9 34.2 0.10 0.07
3C 441 (1995) 2203+292 0.707 1.29 $\dagger$144 84.9 -20.9 0.34 0.24
3C 441 (1997) 2203+292 0.707 1.29 $\dagger$112 84.9 -20.9 0.34 0.24
LBDS 53W091 1721+501 1.552 0.86 364 76.9 +34.5 0.08 0.06
MRC 0156$-$252 0156$-$252 2.016 0.73 196 208.6 -74.8 0.00 0.00
----------------------------------------------------------------------------------------------------------
Key: $z$: redshift; $\lambda_r$ ([$\umu$m]{}): rest-frame equivalent of observed-frame 2.2[$\umu$m]{}; $t_{\rm int}$ (min) : total integration time (min) summed over all waveplate settings ($\dagger$: 3C 441 data from the two runs was pooled giving a composite image of 256 minutes integration time in total) ; $l$ (): Galactic longitude ([ned]{}); $b$ (): Galactic latitude ([ned]{}); $A_B$: Blue-band extinction (mag), from [ned]{}, derived from Burstein & Heiles ; $p_K$: maximum Galactic interstellar contribution to $K$-band polarization (per cent).
Calibration
-----------
The purpose of the two observing runs made for this project was to obtain polarimetry of faint objects rather than to obtain absolute photometry. Two-channel polarimetry depends on measuring the difference in signal between two channels simultaneously; this eliminates systematic errors which might arise due to imperfect calibration if the channels were instead measured consecutively. Such measurements can be carried out in atmospheric conditions less stringent than those required for accurate photometry. Normalised Stokes parameters, by definition, do not require calibration as they depend on the ratio of two intensities measured by the same system. The source intensities and absolute Stokes Parameters calculated in the course of the data reduction are not reproduced in this thesis; they were expressed in IRCAM3 data number rates throughout the reduction process.
The [iraf apphot.phot]{} routine produces a set of output magnitudes for the target objects, relative to a calibrated zero point. We did not use the resultant magnitudes, but the raw data number counts from the photometry aperture, as explained in Appendix \[codeapp\] – so it was not necessary to calibrate the zero-point for this purpose. We did not, therefore, observe photometric standard stars as part of the observations for this project.
A [*polarimetric*]{} standard was measured as part of the August 1997 run to verify the accuracy of the hardware and software forming our polarization reduction chain. The standard star HD 215806, recorded by Whittet et al. to have polarization 0.55% $\pm$ 0.06% at 77, was measured twice by our data reduction chain. The first measurement yielded 0.36% $\pm$ 0.11% (debiased to 0.35%) at 74 $\pm$ 6: clearly consistent within a $2\sigma$ error box. The second measurement similarly yielded 0.30% $\pm$ 0.11% (debiased to 0.29%) at 73. If instrumental polarization were present, it would increase the measured value, and probably skew it to a different orientation; the agreement between the published and measured orientations confirms that instrumental polarization is negligible in the IRPOL2 chain. The fact that both measurements are lower than the published value is not statistically significant, but is interesting enough that future workers measuring HD 215806 might want to check for variability in its polarization.
In general, it should be noted that polarized standards must be carefully chosen as polarization angle can only be measured to 0.1 and varies with wavelength [@Dolan+86a] — but with such inaccurate measurements being returned for polarization orientations on our radio galaxies, this is hardly relevant in this particular case. Neither would attempting to refine the reference axis zero point $\eta_0$ using the standard star add any meaningful accuracy to the measured orientations, given their large error figures.
Galactic Interstellar Polarization {#ULGalPol}
----------------------------------
In the previous chapter, we studied at length the best way to recover a measurement of the true polarization of light from a noisy system. One step remains, however, before the figure obtained can be said to be that of the active galaxy: the interstellar medium of both our own galaxy and the active nucleus’s host galaxy may modify the linear polarization of light passing through it.
Ordinary stars in our own galaxy are not expected to emit intrinsically polarised light; and measured polarization of starlight is presumed to be due to transmission through dust grains. Gehrels was the first to note that the interstellar polarization varied over visible wavelengths, and subsequent work by Serkowski and colleagues [@Coyne+74a; @Serkowski+75a] found an empirical relationship applicable throughout the visible spectrum: $$\label{skemper}
p/p_{\rm max} = \exp [-K \ln^{2}({{\lambda}_{\rm{max}}}/\lambda)]$$ where ${{\lambda}_{\rm{max}}}$ is the wavelength at which the polarization peaks, usually around 0.5[$\umu$m]{}, and empirically in the range 0.3–0.8[$\umu$m]{} [@Serkowski+75a; @Wilking+80a].
The parameter $K$ was fitted as a constant by Serkowski, Mathewson & Ford , who found the best value to be $K = 1.15$. Wilking et al. , however, investigated whether Serkowski’s empirical formula remained valid at infrared wavelengths, and found that a better fit was obtained by taking $K$ to be linearly dependent on ${{\lambda}_{\rm{max}}}$; an adequate approximation for our purposes is $K = 1.7\,$[$\umu$m]{}$^{-1}\, {{\lambda}_{\rm{max}}}$. That the empirical formula, so modified, holds up to around 2[$\umu$m]{}, was confirmed by Martin & Whittet . The best value of the constant coefficient was refined slightly by Whittet et al. but the value 1.7 remains an adequate approximation for our purposes.
In general, $p_{\rm max}$ for a given set of galactic co-ordinates is not known. But suppose we take the ratio of polarizations in two wavebands, $V$ and $K$, and rearrange: $$\label{sratund}
p_{K} = p_{V} \exp \left\{ -3.4 {{\lambda}_{\rm{max}}} \left[ \ln
\left( \frac{{{\lambda}_{\rm{max}}}}{\sqrt{\lambda_{K}\lambda_{V}}}\right) \ln
\left(\frac{\lambda_{V}}{\lambda_{K}}\right) \right] \right\}.$$ Hence $p_{K} = c.p_{V}$ where $c$ depends on ${{\lambda}_{\rm{max}}}$ but for $0.3<{{\lambda}_{\rm{max}}}<0.8$ we find $0.15<c<0.30$.
An empirical upper limit for $p_{V}$ (expressed as a percentage) is given by Schmidt–Kaler as $p_{V} \leq
9E_{B-V}$ and typically, $p_{V} = 4.5E_{B-V}$. \[Clarke & Stewart point out that determinations of an empirical upper limiting polarization also tend to find an empirical lower limit, suggesting imperfect debiasing, and that the true empirical upper limit is in fact lower than the one determined.\] It is well established [@Savage+79a; @Koorneef-83a; @Rieke+85a] that the ratio of total to selective extinction is $A_{V}/E_{B-V} \sim 3$; and so we can use the extinctions $A_{B}$ [@Burstein+82a figures can be obtained from the [ned]{} database] to obtain $E_{B-V} = A_{B} - A_{V} = A_{B}/4$.
Taking the maximum values, $c=0.3$ and $p_{V} \leq 9E_{B-V}$, we find an upper limit for infrared polarization $p_{K} \leq 0.7A_{B}$. We have seen (Table \[extab\]) that none of the objects surveyed for this thesis lie beyond regions of our Galaxy with $A_B > 1.3$, and all but two have $A_B < 0.4$, so the Galactic medium cannot contribute more than 1.2% to the $K$-band polarization of the two high-extinction sources, or more than 0.3% to $K$-band polarization of any of the others. A further check can be made by measuring the polarizations of sources (presumably intrinsically unpolarized stars) which lie in the same fields as our target objects (this will be commented on during discussions of individual targets in the next chapter).
Like our own Galaxy, the host galaxies of the active nuclei may contain dust regions capable of polarizing light passing through them. Goodrich & Cohen argue that 3C 109 is polarised in this way; 3C 234 could be a similar example [@Tran+98a]. Such polarization effects affect the observed $K$-band light at its rest-frame wavelength and will be considered in the context of the discussion of individual sources – since the source of such polarization is, by definition, the host galaxy of an active galactic nucleus. The possibility of polarizing effects being produced at intermediate wavelengths by any intergalactic medium cannot be ruled out [*a priori*]{}, but we invoke Occam’s Razor to assume the absence of significant amounts of any intergalactic medium without evidence to the contrary (e.g. from the observed colours of the target galaxies).
General Observations and Individual Objects {#resulch}
===========================================
> Look up at the heavens and count the stars – if indeed you can count them.
>
> ------------------------------------------------------------------------
>
> — Genesis 15:5 (NIV).
The observational datasets obtained were reduced and analysed using the procedure described in the previous chapter. As is recommended for polarimetry, the normalised Stokes parameters $q$ and $u$ were obtained; the reference axis for IRPOL2 is $\eta_0 = 83\degr$, i.e. that $q>0, u=0$ corresponds to a polarization orientation of 83 E of N and that for $q=0, u>0$, the polarization orientation is 128. Table \[obstab\] gives the polarizations of all target objects and associated objects, but not of the field objects also analyzed.
Some target objects displayed extended structure: Reference is made in the text of this chapter to the ‘moment analysis’ of Dunlop & Peacock , who devised an automated routine to evaluate a position angle for the extended structure they saw in $K$-band sources. For those sources in which there are clear distinct components to the structure, the identification of these components is given on the labelled images which follow. In most of our images, the target object is labelled [T]{}; other objects have been labelled following earlier maps in the literature, where available. Composite images are shown in each case with the images from all waveplate positions stacked together; in some cases, edited images are also provided where one set of images from the Wollaston prism’s ‘double image’ have been removed using the image patching facilities of the [starlink gaia]{} package.
Source $r$() $q (\sigma_q) $(%) $u (\sigma_u)$(%) $P \pm \sigma_P$(%) $2\sigma$.UL Prob $\theta$()
---------------------------- ------- -------------------- ------------------- ---------------------------- -------------- ------ ------------ ----- ----- -- -- -- --
3C 22 2.6 -1.3 (1.4) -3.2 (1.4) 3.3 $\pm$ 1.4 - 0.95 27 -71 +17
3C 41 2.3 +0.8 (1.1) -3.1 (1.1) 3.1 $\pm$ 1.1 - 0.98 45 -14 +79
3C 54 4.0 -1.4 (2.5) -6.0 (2.5) 5.9 $\pm$ 2.6 - 0.94 32 -79 +17
3C 65 2.9 -4.3 (4.2) -1.2 (4.0) 2.2 $\pm$ [${^{L}}$]{}4.5 10 0.42 1
3C 114 (Whole) 3.6 +11 (3) -4.1 (2.7) 11 $\pm$ 3 - 0.99 73 -21 +34
3C 114 (Knee) 1.7 +3.0 (1.6) +4.3 (1.8) 5 $\pm$ 1.7 - 0.99 111 -63 +14
3C 356 (Whole) 4.0 -10 (5) +3.5 (6) 9 $\pm$ 5 16 0.85 172
3C 356 $a$ (North) 2.3 -13 (8) +4.6 (8) 13 $\pm$ 8 41 0.62 164
3C 356 $b$ (SE) 2.6 -10 (9) -19 (17) 19 $\pm$ 15 24 0.78 24 -54 +33
3C 441 [**a**]{} 3.1 +4 (5) -0.7 (5) 1 $\pm$ [${^{L}}$]{}6 10 0.46 78
3C 441 B$\ddagger$ 3.1 +0.5 (5) -1.7 (5) 0.1 $\pm$ [${^{L}}$]{}2.8 7 0.14 47
3C 441 [**c**]{}$\ddagger$ 3.1 +1.3 (12) +10 (11) 3.5 $\pm$ [${^{L}}$]{}16 24 0.54 124
3C 441 E 2.6 +5 (12) +19 (13) 18 $^{+9}_{-8}$ 36 0.91 120
3C 441 F 2.0 +6 (19) +16 (22) 6 $\pm$ [${^{L}}$]{}28 43 0.48 118
3C 441 G 2.6 -0.1 (13) -6 (13) 0.3 $\pm$ [${^{L}}$]{}10 21 0.18 38
3C 441 H 2.3 -12 (14) -9 (15) 7 $\pm$ [${^{L}}$]{}22 33 0.67 11
LBDS 53W091 $\natural$ 1.1 0 (17) -7.5 (22) 0.4 $\pm$ [${^{L}}$]{}8 31 0.11 38
LBDS 53W091 $\flat$ 1.1 +0.6 (16) -3.4 (18) 0.17 [${^{L,U}}$]{} 22 0.20 43
Object 3a $\natural$ 1.1 -12 (18) +17 (18) 16 $\pm$ [${^{L}}$]{}14 43 0.70 146
Object 3a $\flat$ 1.1 -4.5 (21) +21 (20) 10 $\pm$ [${^{L}}$]{}21 46 0.67 134
MRC 0156$-$252 2.3 -2.5 (7) +1 (7) 0.14 $\pm$ [${^{L}}$]{}4.3 10.5 0.14 161
MRC 0156$-$252 2.7 -0.5 (8) -0.5 (8) 0.04 [${^{L,U}}$]{} 4 0.01 60
MRC 0156$-$252 3.4 +0.4 (8) +0.8 (9) 0.05 [${^{L,U}}$]{} 5.5 0.01 114
\[polvals\] \[restab\]
Have We Detected Polarization? {#havewe}
------------------------------
Equation \[propol\] allows us to quantify the probability that a given object is polarised. The probabilities of each object being polarised are listed in Table \[polvals\]. Three of our nine sources have a 95% or better probability of being polarised; and of these, 3C 22 and 3C 41 are polarised at the 3 per cent level, and 3C 114 at the 12 per cent level.
The number of prominent starlike objects (in addition to the target) featuring on the 1995 set of 3C object mosaics varies between one and seven, depending on the target. (We will refer to these as ‘stars’ but have no spectroscopic evidence to confirm their identity as such.) Where possible, we have performed polarimetry on these stars; out of the 21 stars so observed, only one has a greater than 95% probability of being polarised. This object was a bright starlike object on the mosaic containing 3C 114, but is only polarised at the 0.7 per cent level, which is explicable by the interstellar medium (see Table \[extab\]). Even without such special pleading, it would not be surprising for random noise to cause one star out of 21 to appear to be polarised at such a level.
Within the bin of sources having a probability 80-95% of being polarized, fall three further stars; of these, one is extremely faint, and another appears to be polarised at only the 0.3 per cent level. The third falls on the same mosaic as 3C 54, and appears to be polarised at the 5.6 $\pm$ 2.6 per cent level, with a 94% chance of the polarization being genuine. This star, however, straddles the edge of three of the nine component frames of the mosaics, so the validity of the result is called into question. Two of our sources also fall in the 80-95% probability bin: 3C 54 itself, polarised at the 6 per cent level, and 3C 356, at the 9 per cent level. (Object E of the 3C 441 complex may also fall in this bin, on the basis of the pooled 1995 and 1997 observations.)
For the 1995 observations, given that 17 out of 21 stars, but only 2 out of 7 sources, have a probability of less than 80% of being polarised at all, we feel confident of having detected polarization in three 3C sources, and possibly in a further two. The three targets for the 1997 run were faint objects in comparison to most of those observed in the earlier run: one component of the 3C 441 complex displays marginal evidence for polarization, but there is no strong evidence for polarization in 53W091 or MRC 0156$-$252.
Individual Objects
------------------
In the following object-by-object discussion, we will examine each target object in the context of other observational data about the same object from the literature. Evidence for parallel or perpendicular alignments will be noted, but discussion of the implications of our data for the properties of the central engines and host galaxies will be deferred to the next chapter. In this context, the [*Hubble Space Telescope*]{} is abbreviated [*HST*]{}. Certain papers will be cited very frequently, and will be abbreviated in this chapter: Dunlop & Peacock will be denoted D&P, Leyshon & Eales is abbreviated L&E, and a series of papers by Best, Longair & Röttgering will be denoted BLR-I , BLR-II , and BLR-III .
The polarizations of other objects on the target frames will be noted here, using normalised Stokes parameters of the form $\tilde{q} \pm
\sigma_{\tilde{q}} , \tilde{u} \pm \sigma_{\tilde{u}}$, as obtained through Data Reduction Step \[hereNSPs\]; estimates of $P$ (debiased) and orientation will also be quoted. Since the probability that these objects are polarized is low, formal errors on the nominal degree and angle of polarization are not quoted; these can easily be calculated from the normalised Stokes parameters and their errors if required. These field objects serve as useful controls which would immediately indicate regions of high $K$-band Galactic polarization – though they cannot, by themselves, rule out the presence of polarizing material beyond their locations.
### 3C 22 {#D3C22}
#### Structure {#structure .unnumbered}
Radio galaxy 3C 22 appears close to three other starlike objects which we have designated A, B and C (see Figure \[22figs\]); our star A is object C in the notation of Riley, Longair & Gunn . UKIRT $K$-band imaging shows a red companion about 4 to the south-west. This is placed at bearing 237 from the core by D&P’s moment analysis algorithm; they note that it was not apparent in the optical image of McCarthy . Our $K$-band image shows no evidence for extended structure, though this companion is clearly visible in the UKIRT $K$-band image of BLR-II, and also (clearly resolved as a separate object) in their [*HST*]{} image. At the resolution of the [*HST*]{}, the true bearing of this companion is seen to be 208 $\pm$ 1.
Raw (above) and annotated negative (below) $K$-band structure of 3C 22 – lower channel objects have been edited out of the negative image. \[22figs\]
The most recent review of the radio, visible and infrared properties of 3C 22 (BLR-II §3, and references therein) finds two slight extensions in [HST]{} images of 3C 22 itself. They find that the nucleus can be fitted as a combination of a point source and a de Vaucouleurs law, with the point source contributing 37% of the total $K$-band intensity (BLR-III). The $J-K$ colour of 3C 22 is typical of a radio galaxy at its redshift, but the $R-K$ colour is one of the reddest of the 3CR subsample of D&P. The $\sim 0.5\arcsec$ extensions are interpreted (BLR-II) as a possible close companion, marginally redder than the core, due south (bearing $\sim 180\degr$), and as a blue extension slightly south of west (bearing $\sim 250\degr$).
The radio position angle of 3C 22 is given as 102 (equivalently 282) by Schilizzi, Kapahi & Neff and Jenkins, Pooley & Riley ; the 8.4GHz VLA radio map of BLR-II confirms this to $\pm 1\degr$.
#### Polarimetry {#polarimetry .unnumbered}
Polarimetry results for 3C 22, designated T (for Target), and the three field objects, are reported in Table \[3C22objtab\]. All three companions produce normalised Stokes parameters around $1\sigma$. 3C 22 itself has a 95 per cent chance of truly being polarised, and its debiased polarization is $3.3\pm1.4$%. We expect that no more than 0.8% is due to the interstellar medium; most of the polarization is therefore intrinsic to the source. In the optimal 2.6 radius aperture, the two extended features observed by the [*HST*]{} will be included, but light from the red SW companion at 4 will not.
-------- ------- ------------ ------- ------------ ------ -------- -----------------
Source $q$ $\sigma_q$ $u$ $\sigma_u$ Prob $P$(%) $\theta(\degr)$
A -3.05 3.92 -3.84 4.03 53.2 4.22 18.8
B -3.13 2.23 2.62 2.23 81.3 3.77 153.0
C -0.61 0.60 -0.72 0.58 72.0 0.84 17.9
T -1.27 1.38 -3.16 1.38 95.3 3.26 27.0
-------- ------- ------------ ------- ------------ ------ -------- -----------------
: Normalised Stokes Parameters: Objects in 3C 22 field.[]{data-label="3C22objtab"}
The measured orientation of the [$E$]{}-vector is ${+27}^{+17}_{-71} \degr$ East of North. The error is large, but at a nominal 27$(= 207\degr)$ our measurement suggests that the true direction is more likely to be perpendicular to the radio axis, than parallel to it. Comparing the polarization orientation with extended structure, we find that the blue western optical extension is not remarkably close to being perpendicular or parallel to the polarization orientation; the red southern extension/companion might be in parallel alignment for a plausible error in the orientation angle. The red companion to the SW at bearing 208 effectively lies on the nominal polarization axis.
Jannuzi [@Elston+97a] has performed imaging polarimetry on 3C22 at shorter optical wavelengths, and reports $3\sigma$ upper limits in $V$ and $H$ of 5 per cent and 3 per cent respectively.
### 3C 41 {#D3C41}
#### Structure {#structure-1 .unnumbered}
3C 41 appears with a field object labelled B (see Figure \[41figs\]) following the notation of Riley, Longair & Gunn . The radio position angle of 3C 41 is 147 [@Longair-75a], confirmed to $\pm 2\degr$ by the 8.4GHz VLA radio map of BLR-II.
Raw (above) and annotated negative (below) images of 3C 41 – lower channel objects have been edited out of the negative image. \[41figs\]
Our $K$-band image shows no evidence for extended structure, but BLR-II detect two distinct companions in their [*HST*]{} image which can also be discerned as ‘extensions’ in their UKIRT $K$-band image; the WSW extension can also be distinguished in the $K$-band contour map of Eisenhardt & Chokshi , who note that $K$-band emission from the source extends for at least 12. Both companions are more than a magnitude bluer than the core: positioned ESE and WSW of the core they lie on a line oriented at a position angle of $127\degr \pm 2\degr$. They are hence misaligned with the radio axis by about 20 (BLR-II).
#### Polarimetry {#polarimetry-1 .unnumbered}
Normalised Stokes parameter measurements for 3C 41 (Target T) and field object B are given in Table \[3C41objtab\]. 3C 41 has a 98 per cent chance of having a nonzero underlying polarization, which we measure to be 3.1 $\pm$ 1.1 per cent. Our upper limit for extinction-induced polarization is only 0.1 per cent, so we are confident of having detected intrinsic polarization in this object. The orientation of the [$E$]{}-vector is ${+45}^{+79}_{-14} \degr$ East of North.
-------- ------- ------------ ------- ------------ ------ -------- -----------------
Source $q$ $\sigma_q$ $u$ $\sigma_u$ Prob $P$(%) $\theta(\degr)$
B -0.27 0.16 0.18 0.16 86.7 0.30 156.6
T 0.82 1.11 -3.08 1.11 98.4 3.08 45.4
-------- ------- ------------ ------- ------------ ------ -------- -----------------
: Normalised Stokes Parameters: Objects in 3C 41 field.[]{data-label="3C41objtab"}
Jannuzi [@Elston+97a] have firm $V$ and $H$ band polarizations for this source: at $V$, the polarization is 9.3 $\pm$ 2.3% at 58$\pm$ 7; at $H$, the polarization is 6.6 $\pm$ 1.6% at 57$\pm$ 7. The [$E$]{}-vector orientations in the three wavebands are consistent with one another. We therefore find a very good perpendicular alignment between the radio structure and optical polarization axes; the small errors on the $V$ and $H$ polarizations show that their alignment is perpendicular to the radio structure rather than the [*HST*]{} visible structure. Our $K$-band polarization orientation error is large enough to permit it to be perpendicular to the optical structure rather than the radio; but Occam’s razor invites us to assume that the true orientation in $K$ should correspond to that in $V$ and $H.$
### 3C 54 {#D3C54}
#### Structure {#structure-2 .unnumbered}
No wide-aperture image of 3C 54 and its surrounding field could be found in the literature; the target was identified on the grounds that it lay close to the nominal position at the centre of our UKIRT images and displayed a slight southern extension corresponding to that seen in D&P’s $K$-band contour map. This extension is known to be more prominent in $K$ than in $J$ [@Eisenhardt+90a], and is also known as structure $b$ at a bearing of 200 in the $R$-band [@McCarthy+87a]. Bright field objects have been designated A to D as indicated in Figure \[54figs\]; the target object itself may be seen more clearly in Figure \[54image\].
The position angle of 3C 54’s radio structure is 24[@Longair-75a]. A moment analysis gives the $K$-band structure’s major axis orientation as 27 (D&P), essentially parallel to the radio axis; the visible continuum structure is reported as very similar to the infrared, while the \[O[ii]{}\] emission is less similar, although elongated in the same sense [@McCarthy+87a; @McCarthy-88a]. The $J-K$ colour is typical of similar objects at the same redshift [@Eisenhardt+90a].
Raw (above) and annotated negative (below) images of 3C 54 – lower channel objects have been edited out of the negative image. \[54figs\]
#### Polarimetry {#polarimetry-2 .unnumbered}
The target and those field objects (A, B) bright enough to be analysed have their normalised Stokes parameters reported in Table \[3C54objtab\]. The probability that 3C 54 is polarised is 94 per cent; our measured value of polarization is 5.9 $\pm$ 2.6% at ${+32}^{+17}_{-79} \degr$ East of North. Dust is not expected to contribute more than 0.3%. We therefore appear to have a genuine polarization oriented parallel to both the radio and extended optical structures of this source.
-------- ------- ------------ ------- ------------ ------ -------- -----------------
Source $q$ $\sigma_q$ $u$ $\sigma_u$ Prob $P$(%) $\theta(\degr)$
A 1.82 2.75 2.41 3.17 39.3 1.08 109.5
B 5.82 2.50 -0.35 2.66 93.4 5.54 81.3
T -1.36 2.52 -6.01 2.55 94.6 5.88 31.6
-------- ------- ------------ ------- ------------ ------ -------- -----------------
: Normalised Stokes Parameters: Objects in 3C 54 field.[]{data-label="3C54objtab"}
### 3C 65 {#D3C65}
#### Structure {#structure-3 .unnumbered}
Source 3C 65 was identified using the chart provided by Gunn et al., and its field objects have been designated A–G as indicated in Figure \[65figs\].
Raw (above) and annotated negative (below) images of 3C 65 – lower channel objects have been edited out of the negative image. \[65figs\]
BLR-I describe 3C 65 as a fairly round central object, and one of the reddest in the 3CR sample with $V-K\sim 6$. A 4000Å break in the off-nuclear spectrum [@Lacy+95a; @Stockton+95a] indicates the presence of an old stellar population, $\sim 3-4$ Gyr. Their [*HST*]{} visible image shows that 3C 65 is slightly elongated NE–SW. Both their [*HST*]{} and UKIRT images show a blue companion galaxy 3 to the west, lying approximately on the radio axis, but this was too faint to be distinguished from the noise on our $K$-band images. Their 8.4GHz VLA radio map shows a radio structure at position angle $100\degr \pm 3\degr$.
Lacy et al. have claimed evidence for an infrared point source, possibly an obscured quasar nucleus, in the infrared core of 3C 65; Rigler & Lilly , however, found that the infrared profile could be satisfactorily fitted by a de Vaucouleurs law. BLR-II&III’s [*HST*]{} image of 3C 65 also yields an adequate fit for a cD galaxy (de Vaucouleurs law plus faint halo) model.
#### Polarimetry {#polarimetry-3 .unnumbered}
Field object D proved too faint for accurate photometry; normalized Stokes parameters for 3C 65 (target T) and the other field objects are reported in Table \[3C65objtab\]. Our polarimetry indicates a 57% probability that 3C 65 is an [*unpolarized*]{} source; our nominal polarization orientation angle (from Table \[obstab\]) is perpendicular to the radio axis, but the statistical significance of our $K$-band polarization measurement is at best dubious.
-------- ------- ------------ ------- ------------ ------ -------- -----------------
Source $q$ $\sigma_q$ $u$ $\sigma_u$ Prob $P$(%) $\theta(\degr)$
A -0.99 4.00 -1.51 6.58 4.6 0.09 21.3
B 4.21 15.12 1.97 14.73 4.6 0.23 95.5
C 2.12 4.32 2.95 4.20 30.6 1.27 110.2
E -0.07 2.29 2.41 2.21 44.9 1.87 128.8
F -0.71 5.16 -2.81 3.07 33.0 1.00 30.9
G -0.05 0.41 0.14 0.44 5.4 0.01 137.2
T -4.33 4.25 -1.17 4.03 42.9 2.12 0.6
-------- ------- ------------ ------- ------------ ------ -------- -----------------
: Normalised Stokes Parameters: Objects in 3C 65 field.[]{data-label="3C65objtab"}
### 3C 114 {#D3C114}
#### Structure {#structure-4 .unnumbered}
Raw (above) and annotated negative (below) images of 3C 114 – lower channel objects have been edited out of the negative image. \[114figs\]
The field containing 3C 114 is illustrated in Figure \[114figs\]; a close-up image detailing the structure of knots is shown as Figure \[114image\]. Its distinctive shape (D&P) makes it easily identifiable; it consists of at least four knots, where the brightest knot forms the knee of a $\Gamma$-shaped structure. The radio position angle of 3C 114 is 44 [@Strom+90a], and the large size of the radio structure (54) is noteworthy.
(North is at the top, East is on the left.) \[114image\]
D&P’s $K$-band contour map reveals more detail in the structure of 3C 114 with additional knots of lower intensity close to the four obvious ones in our image; the three major knots forming the NE–SW line seem to be joined by an underlying luminous structure. This structure dominates a moment analysis of the whole source, yielding a $K$-band optical structure angle of 52, close to the 44 for radio structure.
#### Polarimetry {#polarimetry-4 .unnumbered}
Only the knee knot proved bright enough to analyze on its own; Table \[polvals\] includes results for both that knot and the structure as whole. There is a probability in excess of 99 per cent that there is genuine polarization in both the knee knot and the overall structure.
-------- ------- ------------ ------- ------------ ------- --------- -----------------
Source $q$ $\sigma_q$ $u$ $\sigma_u$ Prob $P$(%) $\theta(\degr)$
A 0.77 0.23 0.12 0.29 99.7 0.77 87.3
B 5.56 3.20 0.78 3.21 78.5 5.14 87.0
C 5.55 3.20 0.67 3.21 78.3 5.12 86.5
D 0.12 2.86 0.43 2.85 1.2 $\ddag$ $\ddag$
E -0.80 1.12 -1.39 1.13 63.4 1.41 23.1
F 1.90 2.81 3.29 2.74 61.2 3.32 113.0
G 5.28 3.31 -0.18 3.14 72.0 4.75 82.0
T0 11.19 3.07 -4.14 2.68 100.0 11.73 72.9
T2 3.00 1.63 4.31 1.77 99.0 5.10 110.6
-------- ------- ------------ ------- ------------ ------- --------- -----------------
: Normalised Stokes Parameters: Objects in 3C 114 field.[]{data-label="3C114objtab"}
We measured polarization in the knee of 5.1 $\pm$ 1.7 per cent, at ${+111}^{+14}_{-63} \degr$ East of North. Overall, the whole object has a polarization of 11.7 $\pm$ 3.0 per cent at ${+73}^{+34}_{-20} \degr$. The extinction contribution could be as high as 0.9 per cent, but our detections of polarization are much higher than this, so the polarization appears to be intrinsic. With the radio and optical structure axes at $\sim 48\degr$, there is no clear alignment of polarization either parallel or perpendicular to the structure.
### 3C 356 {#D3C356}
#### Structure {#structure-5 .unnumbered}
Source 3C 356 has provoked much discussion in the literature. The radio structure is a large (72) double with position angle 161 [@Leahy+89a]. D&P’s $K$-band contour map displays three knots: two brighter knots lying NW–SE along the radio axis, and a much fainter knot off-axis to the south-west. Both bright knots are associated with radio cores [@Fernini+93a], and it is not clear which hosts the radio source; both lie at $z=1.079$ (BLR-II). In the convention established by LeFèvre, Hammer & Jones , and used by others [@Eales+90a; @Eisenhardt+90a; @Lacy+94a; @Cimatti+97a], the NW component is denoted $a$ and the SE component, $b$. Fainter components are labelled following BLR-II.
The SE component, $b$, is elongated roughly perpendicularly to the radio source [@Rigler+92a]. Lacy & Rawlings note how this SE radio core has a flatter spectral index and may be the host to the radio source, with galaxy $a$ interacting with the jet; $b$’s spectral index is $\alpha \approx 0.1$ between 8.4GHz and 5GHz (BLR-II), which is typical of a compact core in an extended radio source. Eales & Rawlings also favoured $b$ as the radio core due to its colour, magnitude and shape being typical of radio galaxies.
The more recent [*HST*]{} observations of BLR-I&II, however, reveal the NW component $(a)$ to have the same dumbbell morphology they observe in other radio galaxy hosts, while the SE object $(b)$ seems much more diffuse than their other 3CR sources. Component $a$ is also favoured as the radio core by Eisenhardt & Chokshi and McCarthy : it has bluer infrared–optical colours and dominates the visible continuum and \[O[ii]{}\] images. D&P dispute LeFèvre et al.’s claim that $b$ has the bluer colours; [*HST*]{} observations show that $b$ is redder, but some of the diffuse emission is as blue as $a$ [@Eisenhardt+90a BLR-II]. Component $a$’s spectral index is that of a compact steep spectrum source, $\alpha \approx 1.1$, and the 8.4GHz radio flux is only a quarter of that of component $b$ (BLR-II).
BLR-III find that 3C 356 (presumably meaning the NW component, measured in a 5 aperture), can be well modelled purely by a de Vaucouleurs profile. A 4000Å spectral break has been detected in both $a$ and $b$ [@Lacy+94a], indicating that both components contain stars and are aged at least 10 Myr. D&P’s $K$-band image moment analysis algorithm gives a position angle of 159 for the overall structure, but Cimatti et al. state that the two $K$-band knots taken together as a single structure lie at a position angle of 145, with the two dumbbell components of $a$ separated along a line at 152.
Our $K$-band image of 3C 356 (Figure \[356figs\]) reveals the three knots indicated: $b$ at the south-east, $a$ at the north-west, and the very faint south-west component denoted $d$ in the [*HST*]{} image of BLR-II. A prominent field star, object C of Riley, Longair & Gunn , is also labelled.
Raw (above) and annotated negative (below) images of 3C 356 – lower channel objects have been edited out of the negative image. \[356figs\]
#### Polarimetry {#polarimetry-5 .unnumbered}
Cimatti et al. have obtained Keck I spectropolarimetry of 3C 356’s two main components for light emitted between 200nm and 420nm. They find polarization which rises towards the ultraviolet: source $b$’s polarization was low, reaching $4.0 \pm
1.2 \%$ at 200nm, while $a$’s polarization rose from 3% at 420nm to about 15% at 200nm. A distinct Mg[ii]{}$\lambda$2800 line was also detected in component $a$’s light, polarized to the same degree as the continuum, and with the same orientation of 64.
The perpendicular axis to the polarization vector lies at 154, and is therefore within two degrees of the dumbbell separation observed in object $a$, nine degrees anticlockwise of the radio structure, and seven degrees clockwise of the $a$-$b$ axis.
Our $K$-band images show the galaxy in light emitted in a band centred on 1060nm. The SW knot (object $d$) was not bright enough to permit polarimetry. Normalised Stokes parameters for the two major components ($a$, $b$), the 3C 356 complex as a whole (T0) and field object C are recorded in Table \[3C356objtab\]. We find no strong evidence for $K$-band polarization in 3C 356; the probability of non-zero $K$-band polarization being present in $a$ and $b$ is 62% and 79% respectively, and the polarization orientations derived from our noisy measurements do not include 64 in their $\pm 1\sigma$ error boxes.
-------- -------- ------------ -------- ------------ ------ -------- -----------------
Source $q$ $\sigma_q$ $u$ $\sigma_u$ Prob $P$(%) $\theta(\degr)$
C 0.15 0.15 0.27 0.21 71.0 0.27 113.4
T0 -9.64 4.94 0.35 5.63 85.1 8.97 172.0
$a$ -13.32 7.95 4.60 8.38 62.4 18.76 23.9
$b$ -10.10 9.39 -18.88 16.62 78.9 12.90 163.5
-------- -------- ------------ -------- ------------ ------ -------- -----------------
: Normalised Stokes Parameters: Objects in 3C 356 field.[]{data-label="3C356objtab"}
### 3C 441 {#D3C441}
#### Structure {#structure-6 .unnumbered}
3C 441 appears in a rich field (Figure \[441figs\]) with five neighbours; identification of the radio core is based on the observations of Riley, Longair & Gunn and is apparently confirmed by the work of McCarthy and of Eisenhardt & Chokshi . Figure \[441figs\] is based on our August 1995 data (L&E) which was taken without a focal plane mask. The objects labelled [**a**]{} and [**c**]{}, and the position of unseen object [**d**]{}, follow the notation of Lacy et al. ; the star B is labelled as in Riley et al. , and the remaining objects are labelled E thru H. An unedited image with scale bar is given later as Figure \[Cpic\]. Object [**a**]{} itself is shown to have 0.5 mag bluer extension protruding to the south-west in its [*HST*]{} image (BLR-II). Object [**c**]{} is more compact in the infrared than in $R$-band imaging, while object [**a**]{} appears more extended east-west in the infrared than in $R$ [@Eisenhardt+90a BLR-II].
Raw (above) and annotated negative (below) images of 3C 441 – lower channel objects have been edited out of the negative image. North is at the top and East to the left; the scalebar is given in Figure \[Cpic\]. \[441figs\]
Fabry-Perot imaging by Neeser shows that none of the other objects in the field lie within a velocity range $(-1460,
+1180)$ kms$^{-1}$ of 3C 441 itself, but since the $J-K$ colours of most of these neighbouring objects lie between 1.6 and 1.85 (BLR-II), 3C 441 could be part of a cluster. Neeser questions whether the identification of 3C 441 is correct – arguing that it may in fact be our object F. Recent imaging by Lacy et al. suggests that 3C 441’s jet is impacting object [**c**]{} to its north-west – this object has the same infrared colour as object [**a**]{} and is the only other area of strong \[O[ii]{}\] emission in the field [@Eisenhardt+90a BLR-II].
The 5GHz radio map of 3C 441 [@Longair-75a] shows a double radio source with the separation between sources running East-West, and extended structure trailing off to the South-East. Lacy et al. show that the North-West jet (at 8GHz) is deflected to the south where it would otherwise have encompassed object [**c**]{}.
Total integration time 4 hours 16 minutes. \[1 hour 52 minutes (1997) plus 2 hours 24 minutes (1995).\] North is at the top, East at the left. The star B is labelled as in Riley, Longair & Gunn . \[Cpic\]
[nb]{} Any future worker planning to observe 3C441 should note that there is a $z=4.4$ quasar which falls in the same field [@McCarthy+88a]. Judicious planning could enable studies of this object to be conducted in parallel with the radio galaxy.
#### Polarimetry {#objE .unnumbered}
It is known that 3C 441 has a broad-band optical polarization of 1.5 $\pm$ 0.7% at 70 $\pm$ 13 in a 2 diameter aperture about the core: this orientation is roughly perpendicular to the radio structure [@Tadhunter+92a]. Our 1995 and 1997 images were stacked together to obtain polarization measurements of objects [**a**]{} (the putative core of 3C 441) and E thru H; the 1995 data alone was used to obtain polarization data on B and [ **c**]{}. Given the uncertainty posited by Neeser over the identification of 3C 441, and the interest in object [**c**]{} of Lacy et al. , we performed polarimetry on all the objects on the field, with the full results presented in Table \[obstab\].
The only object with a strong indication (90 per cent chance genuine) of polarization is E. The orientation is 120, which would be roughly parallel with the radio jet — but the position angle which E makes with the presumed core [**a**]{} is close to 0, which means that a model of E scattering light from [**a**]{} is possible. It would not be necessary for light from [**a**]{} to be beamed into E; if E subtends only a small solid angle of the light emitted by [**a**]{}, any light from [**a**]{} scattered by E would be quasi-unidirectional.
There is a weak indication that object [**c**]{} might be polarised. If so, the best estimate is 3.5% polarization at 124 – an orientation roughly parallel to any jet from [**a**]{} which might be scattered into our line of sight. Similarly, object H might possibly be polarised at 6.7%, 11, which is roughly perpendicular to its line of sight with [**a**]{}, and parallel to that with B. We have, however, no redshift data on any source other than [**a**]{}, and therefore cannot eliminate chance alignments should any of these sources be located at other redshifts.
For the presumed radio galaxy at [**a**]{}, the best estimate polarization is 1% at 78— consistent with both the magnitude and orientation of Tadhunter et al.’s broad-band visible measurement — but there is a 54 per cent chance that [**a**]{} is unpolarised with this result being merely an artifact of the noise. Even our measurement for E has a ten percent chance of being a noise-induced spurious result.
### LBDS 53W091 {#D53W091}
#### Structure {#structure-7 .unnumbered}
The galaxy LBDS 53W091 has aroused great excitement in recent years. First investigated by Dunlop et al. as an extremely red radio source, it was found to be a very red radio galaxy visible at a very high ($z=1.552$) redshift. Its spectrum exhibits late-type absorption features, and no prominent emission features. Comparisons of its spectrum with synthetic and real elliptical galaxies suggest that it must be at least 3.5 Gyr old [@Dunlop+96a; @Spinrad+97a] – which is only consistent with its high redshift in certain cosmologies, requiring a low density Universe $(\Omega \sim 0.2)$, or else an unacceptably low Hubble constant $(H_0 \la
50\,$kms$^{-1}$Mpc$^{-1})$ in an $\Omega=1$ cosmology. \[See Leyshon, Dunlop & Eales for further discussion of 53W091’s age and internal chemistry, considering issues which do not affect the interpretation of its polarization.\]
The lack of emission features suggests that the active nucleus responsible for its $\sim 25$ mJy 1.4GHz radio emission contributes very little light to the optical/ultraviolet; Dunlop et al. argue that the galaxy is unlikely to be an obscured quasar. Yet unpublished evidence (Chambers, private communication) suggests that 53W091 has a high infrared polarization — of order 40% — which would be extremely difficult to account for in an object with such a weak active nucleus.
Spinrad et al. note that radio galaxies with weak active nuclei ($S_{\rm 1.4GHz}<$ 50 mJy) generally are not expected to be dominated by optical nuclear emission, and do not display the alignment effect. Their 4.86GHz radio map of 53W091 reveals a double-lobed FR-II steep spectrum radio source, where the radio lobes are separated by approximately 43 at position angle 131.
Our data of 53W091 were stacked together with earlier observations made by Dr James Dunlop (private communication) in July 1997, and the total image is seen in Figure \[Wpic\]. The companion object to the south-east of 53W091 is known to be at the same redshift, and is labelled ‘3a’ in accordance with the labelling of Spinrad et al. . The position of their third component at the same redshift is also marked (labelled ‘4’) although there is not a distinct source on our image.
Total integration time 5 hours 56 minutes. \[4 hours 4 minutes (August) plus 1 hour 52 minutes (July).\] (North is at the top, East at the left. The image has been overlaid with contours fitted for 8%, 16% and 24% of the peak intensity present.) \[Wpic\]
Two stars on our image (the star on the top right of Figure \[Wpic\] and a brighter one in the lower slot of the focal plane mask, not shown) were identified with stars whose B1950 co-ordinates were obtained from the [*Digitized Sky Survey*]{} [@Lasker+90a]. Offsetting from these stars, the B1950 co-ordinates of the [[$K$]{}]{}-band sources were obtained and are given in Table \[positab\].
-------- -------------------------- -----------------------------------------------
Source $\alpha$ $\delta$
53W091 17h 21m 17898 $\pm$ 0057 $+50\degr 08\arcmin 48\farcs34 \pm 0\farcs29$
3a 17h 21m 18156 $\pm$ 0029 $+50\degr 08\arcmin 46\farcs34 \pm 0\farcs57$
-------- -------------------------- -----------------------------------------------
: B1950 co-ordinates of [[$K$]{}]{}-band sources in 53W091 field.[]{data-label="positab"}
Key: $\alpha$: B1950 Right Ascension; $\delta$: B1950 Declination.
Spinrad et al. ask whether 3a and 53W091 together might form a system displaying the alignment effect, but note that both sources’ colours suggest they are composed of old stars, for which there is no plausible alignment mechanism. (They allow that some interaction of jets from the active nucleus with the material surrounding the galaxy may cause some appearance of filamentary structure.) The axis connecting the two objects is at a position angle of 126, comparable to the radio axis at 131.
It is noteworthy that the diagonal distance between 53W091 and 3a is 4, the same distance as between the radio lobes in the 4.86GHz map of Spinrad et al. . The radio positions suggest that the south-east radio lobe lies due south of 53W091 (which is the north-west partner of the $K$-band pair): we cannot definitively claim that the radio and infrared pairs are congruent, nor that the infrared source of 53W091 lies between the two radio lobes. Systematic error in registering the astrometry between the two wavebands might allow either eventuality.
#### Polarimetry {#53Wgeom .unnumbered}
Polarimetry was performed on both 53W091 and on Object 3a; results for both are given in Table \[restab\]. Since the objects were very faint and close together, the photometry aperture was not chosen according to the method in §\[firstrunDR\], but was set to a radius of 4 pixels ($1\farcs1$). We also attempted to prune the frames with the greatest noise from our data[^5] and repeated the polarimetric analysis. Results for both the natural ($\natural$) and ‘despiked’ ($\flat$) data are given in Table \[restab\].
There is a weak indication that Object 3a is polarised, with a 70 per cent chance of the polarisation being genuine. If there truly is polarisation at a level of 10–15%, then 30–45 per cent of the light from object 3a could be scattered, and the source could consist entirely of scattered light within the error bars. (Dust scattering and non-perpendicular electron scattering will not result in total linear polarization of the scattered light.)
Is it possible that a beam from 53W091 is being scattered by a cloud at the position of 3a? The geometry suggests that this cannot be the case, since the polarization orientation is around 140, which is nearly parallel to the line connecting 3a to 53W091. If 3a were a hotspot induced by a beam emerging from 53W091, a polarization orientation nearer 30 would have been expected.
The core of 53W091 itself provides no evidence for polarization, and it would be difficult to obtain results as high as the 40% which Chambers (private communication) has suggested; nevertheless, short integration times on faint objects are subject to large errors in their polarimetry, so such a result is not impossible. \[James Dunlop (private communication) reports that Chambers’ integration time was not greater than three hours in total, compared to our six hours.\] Our polarization orientation, interestingly, is 38, nearly perpendicular to the radio axis and line to object 3a; but this is unlikely to be significant with errors of $\pm60\degr$ on our formal measure of the polarization angle.
### MRC 0156$-$252 {#D0156-252}
#### Structure {#structure-8 .unnumbered}
Eales & Rawlings have compared radio galaxies at redshifts $z \sim 1$ and $z > 2$, and find that those radio galaxies at $z>2$ have brighter absolute $V$-band magnitudes, very low Ly$\alpha/$H$\alpha$ ratios, and may be subject to strong reddening by dust. Such results might be attributed to evolution in radio galaxies, or to a selection bias for more powerful active nuclei at high redshift.
MRC 0156$-$252 has the brightest known absolute $V$-band magnitude for a radio galaxy at $z \sim 2$. The cause of its high luminosity is uncertain: it may be being viewed during an epoch of star formation, or Eales & Rawlings have suggested that the source is actually a quasar obscured by dust. Broad H$\alpha$ lines suggest that some of its light is originating in an active nucleus. McCarthy et al. earlier classified it as a radio galaxy and suggested [@McCarthy-93a], on the basis of its red spectral energy distribution, that it was a galaxy at an advanced stage of evolution. The galaxy appears unresolved in our $K$-band image (Figure \[Mpic\]), verifying the findings of McCarthy, Persson & West , who did, however, find extended structure in their visible-band images.
Total integration time 3 hours 44 minutes – including one cycle of observations which was not used for our subsequent polarimetry. North is at the top, East at the left. \[Mpic\]
#### Polarimetry {#polarimetry-6 .unnumbered}
The criterion used by L&E to select the aperture for polarimetry did not yield a unique result for this object, so photometry is given in Table \[restab\] for apertures of radius 8, 10 and 12 pixels. In all cases the best point estimate of the polarization is less than 0.15 per cent; and for the 10 and 12 pixel apertures, the formal $1\sigma$ confidence interval indicates that the source is totally unpolarised.
Discussion
==========
> $\star$ There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.$\star$ There is another which states that this had already happened.
>
> ------------------------------------------------------------------------
>
> — Douglas Adams, [*The Restaurant at the End of the Universe*]{}.
The nine objects studied for this thesis project were selected for their diverse properties, and do not form any kind of statistically complete sample. No one single approach, therefore, can interpret all the new data presented in this thesis. In some cases, the published data available in the literature complements our polarimetry and enables a more detailed analysis to be made of the properties of certain targets.
One factor which can be calculated for all radio galaxies is some constraint on the contribution of the active nucleus to the total light intensity observed in the $K$-band. This is performed using some basic assumptions about the undiluted polarization of the active nucleus. Best, Longair & Röttgering have fitted radial profiles to UKIRT $K$-band images of some of our targets, and in each of these cases their figure can be used in place of our assumptions, allowing an estimate of the undiluted polarization to be made.
Spatial and spectral modelling can be performed for some interesting sources. 3C 22 and 3C 41 have had polarizations determined at other wavebands, and taken with our data the properties of the scattering medium can be modelled. The complicated morphology of sources like 3C 114 and 3C 356 invites consideration of what spatial scattering processes may be at work, and this is modelled for a simple axisymmetric scattering geometry.
We have already noted how Cimatti et al. reviewed the properties of high-redshift radio galaxies whose optical polarizations had been measured by 1993. The results of this thesis, and of the recent literature (Table \[litpol\]) can be used to extend the parameter space in which Cimatti et al. looked for trends, and this will form the final part of our discussion.
As in the previous chapter, a series of papers by Best, Longair & Röttgering will be denoted BLR-I , BLR-II , and BLR-III . The dust scattering model paper of Manzini & di Serego Alighieri is abbreviated MdSA.
The Fractional Contribution of Quasar Light
-------------------------------------------
As we have reviewed in Chapter \[reviewRG\], radio galaxies are thought to have quasar nuclei at their cores, but to be oriented such that no direct radiation from the core can reach us. Nearby radio galaxies are known to have the morphology of giant ellipticals; spectral modelling of sources at higher redshifts suggests that more distant radio galaxies, too, are dominated by old, red stars. If our target objects are typical radio galaxies, the total light received from our sources will be a combination of starlight, light from the active nucleus scattered into our line of sight, and nebular continuum emission. Direct optical power-law emission from the active nuclei of radio galaxies is normally considered to be totally obscured, but in this analysis we will also need to consider the potential contribution of such light: in some radio galaxies the obscuring material may be less efficient owing to its orientation or optical depth.
### The dilution law for polarization
Let us denote by $\Phi_{\mathrm W}$, the fraction of the total flux density, ${{F}_{\rm{W}}}$, in a given waveband, W, which originates in the active nucleus and is scattered into our line of sight. We expect that in the visible wavebands, $\Phi_{U,B,V}$ will be a significant fraction of unity. From our observations, we wish to determine whether $\Phi_{K}$ is small, or whether a significant component of the stellar-dominated infrared also arises in the active nucleus. If we denote the flux density scattered into our line of sight from the quasar core by ${{F}_{\rm{Q,W}}}$, then $\Phi_{\mathrm W} = {{F}_{\rm{Q,W}}}/{{F}_{\rm{W}}}$ — at this stage we make the assumption that there is no contribution by direct radiation from the active nucleus.
Following MdSA, we assume that only the scattered component of the light from radio galaxies is polarised. Recalling from Equation \[ppol\] that the degree of linear polarization is defined by $P = I_p/(I_p + I_u)$, we define the unpolarized component to be $I_c$ for the scattered core light only, and $I_c + I_h$ for the scattered core light and host galaxy together. Hence defining ${{P}_{\rm{Q,W}}}$ as the intrinsic polarization produced by the scattering process, and ${{P}_{\rm{W}}}$ as the observed polarization after dilution, it follows that $$\label{fracflux}
\frac{{{P}_{\rm{W}}}}{{{P}_{\rm{Q,W}}}} = \frac{I_p/(I_p + I_c + I_h)}{I_p/(I_p + I_c)} =
\frac{I_p + I_c}{I_p + I_c + I_h} = \frac{{{F}_{\rm{Q,W}}}}{{{F}_{\rm{W}}}} =
\Phi_{\mathrm W}.$$ Therefore the fraction of the total light which is the scattered nuclear component, is equivalent to the ratio of the diluted and undiluted polarizations.
### The dilution law in a Unification Model scattering geometry
If we know the restrictions on possible values of the intrinsic polarization ${{P}_{\rm{Q,W}}}$, we can use our corresponding measurements of ${{P}_{\rm{W}}}$ to put limits on $\Phi_{\mathrm W}$ for the measured sources.
The appropriate restrictions depend on the nature of the scattering centres. If the scattering centres are electrons [@Fabian-89a], then Thomson scattering will take place: the effects of the geometry and of the wavelength can be treated independently. The spectral energy distribution of the light scattered in a given direction is independent of the scattering angle: ${{P}_{\rm{Q,W}}}$ will be the same constant ${{P}_{\rm{Q}}}$ at all wavelengths. The degree of polarization of the scattered light is given simply by $$\label{elecpol}
{{P}_{\rm{Q}}} = \frac{1-\cos^2\chi}{1+\cos^2\chi},$$ where $\chi$ is the scattering angle. For an AGN observed as a radio galaxy, we assume [@Barthel-89a] an orientation 45 $\leq \chi
\leq$ 90, whence $1/3 \leq {{P}_{\rm{Q}}} \leq 1\ \forall\ $W.
The case where the scattering centres are dust grains has been modelled recently (MdSA); the fraction of the light scattered by the dust, $f_{\rm
W}$, and the polarization of the scattered light, ${{P}_{\rm{Q,W}}}$, both depend strongly on wavelength. The exact relationship depends critically on the size distribution of the dust grains, and the amount of extinction they introduce; MdSA provide a series of graphs for the variation of $f_{\rm W}$ and ${{P}_{\rm{Q,W}}}$ with rest-frame wavelength 0.1[$\umu$m]{} $< \lambda_r <$ 1.0[$\umu$m]{}, corresponding to many different dust grain compositions and size distributions. At the redshifts of the objects in our sample, light observed in the $H$ and $J$ bands originates at rest-frame wavelengths below 1.0[$\umu$m]{}, but $K$-band light originates in the region 1.0[$\umu$m]{} $< \lambda_r <$ 1.15[$\umu$m]{}. To accommodate the $K$-band light within our models, we linearly extrapolated MdSA’s curves to $\lambda_r = 1.15$[$\umu$m]{}.
In MdSA models where the smallest dust grains have a radius not less than 40nm, ${{P}_{\rm{Q,W}}}$ approaches zero twice: once at a (rest frame) wavelength around 0.2[$\umu$m]{}, and again at some wavelength between 0.1[$\umu$m]{} and 0.7[$\umu$m]{} which depends critically on the dust grain size distribution. But in all cases, ${{P}_{\rm{Q,W}}}$ extrapolated into the 1.0[$\umu$m]{} $< \lambda_r < 1.15$[$\umu$m]{} region gives 0.3 $< {{P}_{{\rm{Q,}}K}} <$ 0.5; the MdSA graphs show that the intrinsic polarizations ${{P}_{{\rm{Q,}}V}}$ and ${{P}_{{\rm{Q,}}H}}$ should be lower than ${{P}_{{\rm{Q,}}K}}$ for the objects where we have $V$- and $H$-band polarimetry.
### Constraints on the nuclear component intensity of our sources {#constraincalc}
For the sources in which we have evidence of $K$-band polarization (here we will consider those with a $\geq 80\%$ probability of genuine polarization), we can hence estimate $\Phi_{K}$ under both electron and dust models. The values and upper limits are given in Table \[quasfrac\]. For the dust models, we take $ 1/{{P}_{\rm{Q,W}}} = 2.5 \pm 0.5$; the error takes into account all dust models, and the different redshift corrections for the different galaxies, but assumes that the scattering angle is 90. If the scattering angle is less, we assume that less polarization occurs (see MdSA, Figure 20), and hence $\Phi_{K}$ will be greater than our estimate. For the electron models, we multiply the observed polarization by $1/{{P}_{\rm{Q,W}}} = 2 \pm 1$; this takes into account all possible $\chi \geq 45\degr$ orientation effects.
Similarly, in all sources we can at least estimate an upper limit for the nuclear contribution contingent on our assumption of a quasar core and a $\chi \ge 45\degr$ scattering geometry. In sources which we evaluate as having a less than 80% probability of genuine polarization, we will multiply the $2\sigma$ upper limit on their measured polarizations by the $1\sigma$ upper limit on the reciprocal of the modelled intrinsic polarizations — which for both dust and electrons under the above assumptions requires a multiplication by three. These limiting values, too, are listed in Table \[quasfrac\].
\[quasfrac\]
Source ${{P}_{\rm{K}}}$ ${\Phi_{K}}_{\it e}$ ${\Phi_{K}}_{\it d}$ ${\Phi_{K}}_{\it r}$ ${\Phi_{K}}_{\it s}$
--------------------- --------------------- ---------------------- ---------------------- ---------------------- ----------------------
3C 22 3.3 [$\pm\ $]{}1.4 7 [$\pm\ $]{}4 8 [$\pm\ $]{}4 37 –
3C 41 3.1 [$\pm\ $]{}1.1 6 [$\pm\ $]{}4 8 [$\pm\ $]{}3 24 –
3C 54 5.9 [$\pm\ $]{}2.6 12 [$\pm\ $]{}8 15 [$\pm\ $]{}7 – –
3C 65 $<9.7$ $<8$ 1
3C 114 (T0) 11.7 [$\pm\ $]{}3.0 23 [$\pm\ $]{}13 29 [$\pm\ $]{}10 – –
3C 114 (T2) 5.1 [$\pm\ $]{}1.7 10 [$\pm\ $]{}6 13 [$\pm\ $]{}5 – –
3C 356 (T0) 9 [$\pm\ $]{}5 18 [$\pm\ $]{}13 23 [$\pm\ $]{}13 $<14$ 16
3C 356 $a$ $<41$ $<14$ –
3C 356 $b$ $<24$ $<14$ –
3C 441 [**a**]{} $<10$ $<5$ 4
3C 441 [**c**]{} $<24$ – –
3C 441 E 18 [$\pm\ $]{}9 35 [$\pm\ $]{}26 44 [$\pm\ $]{}25 – –
LBDS 53W091 $\flat$ $<22$ – –
53W091-3a $\flat$ $<46$ – –
MRC 0156$-$252 (23) $<10.4$ – –
: Percentage of $K$-band light estimated to be arising from the postulated active nucleus in our sample of radio galaxies.
The effects of multiple scattering have been ignored for both models; multiple scattering tends to depolarise light, and so the true value of $\Phi_{K}$ under multiple scattering will again be greater than our estimate. The only physical mechanism which could cause the true $\Phi_{K}$ to be [*lower*]{} than our estimate, is polarization of light in transit by selective extinction; and as we have already seen (Table \[extab\]), any such contribution to the polarization of our targets will be small.
### Constraints on the nuclear polarization based on BLR-III data {#blr3constr}
Another approach to estimating the possible contribution of a quasar component is to fit the radial intensity profile of the radio galaxy with a combination of a de Vaucouleurs law and a point source. This has been done with [*HST*]{} and UKIRT imaging for five of our 3C sources (BLR-III), and the fitted values or upper limits (${\Phi_{K}}_{\it r}$) are also reproduced in Table \[quasfrac\]. The same paper made a further estimate of the fraction of nuclear light present by fitting the spectrum as a combination of a (nuclear) flat spectrum and an old stellar population with the spectral energy distribution of Bruzual & Charlot ; again, these fractions (${\Phi_{K}}_{\it s}$) are reproduced in Table \[quasfrac\]. All of the estimates of $\Phi_{K}$ are consistent with one another; the significance in individual objects will be considered below.
It should be noted, however, that BLR-III’s ‘nuclear light’ need not include only scattered light – their simple spectral model distinguishes light from an evolved stellar population but lumps everything else (scattered central engine light, the spectral profile of newly formed stars, and nebular emission) into the ‘nucleus’. Similarly the radial profile fit distinguishes only the light sources which contribute to the de Vaucouleurs structure of a normal galaxy. Our earlier definition makes $\Phi_K$ the fraction of the total light which is scattered nuclear light. In particular, if the active nucleus is not perfectly shielded, the nuclear light fractions derived from BLR-III’s analysis will include direct nuclear light and may therefore be higher than the $\Phi_K$ values derived from our $K$-band data.
In two cases (3C 65 and 3C 441 [**a**]{}), the BLR-III radial fitting yields upper limits which are in fact much lower than (but obviously consistent with) the upper limits estimated on the basis of our $K$-band polarimetry. Dividing the nominal $K$-band polarization by ${\Phi_{K}}_{{\it r\!,}{\mathrm max}}$ yields a nominal lower limit to the intrinsic polarization of the nucleus, namely 27% for 3C 65 and 21% for 3C 441 [**a**]{}; since our $K$-band polarizations are consistent with zero within their error bars, however, these nominal nuclear polarizations are of limited usefulness.
Similarly, the spectral fitting approach yielded definite values of ${\Phi_{K}}_{{\it s}}$ in these two objects; dividing the measured polarization by this light-fraction gave nominal nuclear polarizations of 215% [*(sic)*]{} in 3C 65 and 26% in 3C 441 [**a**]{}. But the errors on the measured polarizations make any intrinsic polarization between zero and 100% possible. If the spectral fitting figure ${\Phi_{K}}_{{\it s}} = 1\%$ is accurate for 3C 65, then our nominal 2.15% diluted polarization measurement for this object is clearly too high, assuming the polarization occurs only in the nuclear component of the light.
In 3C 356 (T0), we find a better-constrained case: given a spectral fit light-fraction ${\Phi_{K}}_{{\it s}} = 16$%, the intrinsic nuclear polarization is ${{P}_{{\rm{Q,}}K}} = 57 \pm 31 \%$ (or for radial fitting, the [*lower limit*]{} nuclear polarization is 65 $\pm$ 35%). Again, the large error on our polarization measurement gives us a constraint of limited usefulness, but we might cautiously conclude (with 1$\sigma$ confidence) that the nuclear source in 3C 356 is at least 25% polarized.
The recent finding [@Eales+97a] that 3C galaxies at $z \sim 1$ are 1.7 times as bright as the radio-weaker 6C/B2 galaxies in a similar sample requires that $\Phi_{K} \ga 40\% (= 7/17)$ for 3C galaxies, if the scattering of nuclear nonstellar light is responsible for the brighter $K$-band magnitudes of 3C galaxies. Most of the results presented in Table \[quasfrac\], whether based on the polarimetry of this thesis or the BLR-III data, produce $\Phi_{K}$ values which are somewhat lower. This may be indicative of some correlation between the strength of the active nucleus and the number of passively evolving stars in the galaxy, allowing the polarizations and spectral and spatial fits to produce lower $\Phi_{K}$ values, while still producing the enhanced $K$-band brightnesses measured in the most powerful (3C) radio galaxies.
Optically Compact Sources
-------------------------
The most obvious division which can be made in our sample of nine sources is between those whose $K$-band image is dominated by a clear source object, and those where there is a complex structure of knots or components of comparable brightness. First we consider as a group the ‘optically compact sources’, dominated by one bright object: 3C 22, 3C 41, 3C 54, 3C 65 and MRC 0156$-$252.
Among the optically compact sources, the radio galaxies 3C 22 and 3C 41 are particularly noteworthy for the compact rounded morphologies in their observed $K$-band structure. They are also prominent for having the brightest $K$-band excess over the mean locus of the $K$-$z$ Hubble plot for 3C galaxies. When BLR-III fitted radial profiles to [*HST*]{} visible and UKIRT $K$-band images of eight high redshift radio galaxies, six could be modelled by a simple elliptical galaxy de Vaucouleurs profile; but in these two galaxies alone, an additional point source was required to give a good fit in the central region. These two sources also have polarization figures available in visible wavebands, data which enable simple spectral modelling to be performed for these two sources.
### Determination of the scattering angle from the BLR-III light-fraction
The BLR-III radial profile fitting suggests that in 3C 22, the point source contributes 37% of the total $K$-band intensity, and in 3C 41, 24%; but their fitting method is known to be biased low for sources with a high point component. By simulating the effects of an additional point source, a revised estimate could be made, suggesting that 3C 22 actually had a $50^{+20}_{-10}\%$ nuclear contribution, and 3C 41’s light included $31^{+10}_{-8}\%$ from the nucleus. Flat-spectrum component fitting was not attempted in these cases.
Clearly these very high nuclear contributions are not easily reconciled with the low polarizations of order 3% which we have measured in the $K$-band, provided our model assumptions are correct. If we assume the nuclear source is obscured from direct view and has no diluting effect on the scattered light, then the intrinsic nuclear polarizations implied by our polarization and BLR-III’s fits are of the order of 10%. Such a value is too low to be consistent with a $\chi > 45\degr$ scattering geometry.
If we assume a shallower scattering angle, we should also allow for the diluting effect of nuclear light which may have become visible along a direct line of sight to the nucleus. We can put a lower limit on the scattering angle by neglecting this component; if electron scattering is taking place, we can identify the limiting scattering angle. Substituting Equation \[elecpol\] into Equation \[fracflux\] and rearranging yields $$\label{getchi}
\cos^{2}(\chi) = \frac{\Phi_{K}-P_{K}}{\Phi_{K}+P_{K}}.$$
We find that a 3.3% polarised 3C 22 with 50% of its $K$-band emission originating in a nuclear source could be scattering light at 21 $\pm$ 5 if the scattering medium were electrons, or could contain dust scattering light at a slightly higher angle. Similarly the figures for 3C 41 imply electron scattering at 25$\pm$ 6, or a correspondingly higher angle for dust.
These models suggest that these two objects are not oriented at the $\chi \geq 45\degr$ positions of radio galaxies but are being viewed ‘down the jet’. Either the direct contribution of nuclear light is weak owing to some other factor (e.g. dust obscuration) and these angles are correct; or the direct contribution is stronger, in which case the true scattering angles are somewhat higher.
Both objects are also atypical in other properties: 3C 22 appears to emit a broad line component [@Rawlings+95a; @Economou+95a], and both 3C 22 [@Fernini+93a] and 3C 41 (BLR-II) exhibit radio jets, a feature rare in radio galaxies at high redshift. For these reasons, independently of the evidence of the $K$-band light fraction and polarization, it has been already suggested that these two objects may be oriented close to the threshold between radio galaxy and radio quasar properties. This is entirely consistent with the polarization which is suggesting that these objects are oriented at $\chi \sim 30\degr$ with some direct nuclear light contribution.
A further line of enquiry is open to us, since polarization measurements in other optical wavebands are available in the literature or by private communication for these two sources. These additional wavebands give a broad baseline against which models of the scattering medium can be tested.
### Modelling the scattering medium given multiwaveband polarimetry
Where the magnitude, W, has been measured in a given waveband for which the zero-magnitude flux density is ${{F}_{\rm{W0}}}$, the total flux density can be calculated: $$\label{totint}
{{F}_{\rm{W}}}={{F}_{\rm{W0}}}.10^{-0.4{\rm W}}.$$
The optical flux density of quasars can be modelled well by a power law of the form $C.\nu^{-\alpha}$ where $\alpha$, the ‘spectral index’, is of order unity [@Peterson-97a §1.3]. Since the efficiency with which a given species of scattering centre scatters light may depend on wavelength, we denote that efficiency by $f_{\rm W}$, and the scattered quasar component can be expressed $$\label{scatcomp}
{{F}_{\rm{Q,W}}}= C.f_{\rm W}.\nu^{-\alpha}.$$
It will be computationally convenient to define an ‘unscaled model light ratio’, ${{\phi}_{\rm{W}}}$, as $$\label{phiunsc}
{{\phi}_{\rm{W}}} =
\frac{f_{\rm W}.\nu^{-\alpha}}{{{F}_{\rm{0W}}}.10^{-0.4{\rm W}}};$$ then the actual model light ratio will be ${{\Phi}_{{{\mathrm W}}}}=C.{{\phi}_{\rm{W}}}$. There is clearly an upper limit set on $C$ by the fact that ${{\Phi}_{{{\mathrm W}}}}$ may not exceed unity in any waveband; hence $C \leq 1/{{\phi}_{\rm{W}}}$. Allowing for errors in the observed magnitudes, $\Delta$W, the maximum permissible value of $C$ in a model can be constrained by inspecting every relevant waveband, since the inequality must hold in all bands: $$\label{conmax}
C_{\rm max} = {\rm min} \left[ \frac{{{F}_{\rm{0W}}}.10^{-0.4({\rm
W}-\Delta{\rm W})}}{f_{\rm W}.\nu^{-\alpha}}\right], \forall\ {\rm W}.$$ Hence, for a given choice of scattering model, which determines the value for $\alpha$ and the form of $f_{\rm W}$, $C_{\rm max}$ is the minimum value obtained by substituting W by each wavelength observed, in turn.
Given a measurement of the magnitude of a radio galaxy, we can predict its polarization as a function of wavelength, up to a multiplicative constant. Rearranging Equation \[fracflux\], and employing our ‘unscaled model light ratio’, we first define an ‘unscaled model polarization’ ${{\Pi}_{\rm{W}}} = {{P}_{\rm{Q,W}}}.{{\phi}_{\rm{W}}}$, and so express our modelled polarization as: $$\label{polmod}
{{P}_{\rm{W,modelled}}} = {{P}_{\rm{Q,W}}}.{{\Phi}_{{W}}} =
C.{{P}_{\rm{Q,{\mathrm W}}}}.{{\phi}_{\rm{W}}} = C.{{\Pi}_{\rm{W}}}.$$
To fit a dust scattering model, we can calculate ${{\Pi}_{\rm{W}}}$ by obtaining $f_{\rm W}$ and ${{P}_{\rm{Q,W}}}$ from suitable curves in MdSA. For electron models, the wavelength-independent term $f_{\rm W}$ can be considered to have been absorbed into the multiplicative constant, $C$, while ${{P}_{\rm{Q}}}$, also wavelength-independent, can be assumed to be its minimum value, 1/3. We cannot separately identify $C$ and ${{P}_{\rm{Q}}}$; the physical constraints on these constants are $1/3 \leq {{P}_{\rm{Q}}} \leq 1$ and $0 \leq C
\leq C_{\rm max}$. If ${{P}_{\rm{Q}}}$ is greater than the assumed 1/3, then $C$ will be correspondingly smaller.
Given a set of $N$ polarization measurements ${{P}_{\rm{W}}} \pm {{\sigma}_{\rm{W}}}$, and a corresponding set of unscaled model polarizations, ${{\Pi}_{\rm{W}}} \pm {{\epsilon}_{\rm{W}}}$, based on measured magnitudes, we can calculate the deviation of the fit: $$\label{fitdev}
\delta = \sqrt{\frac{1}{N}.\sum_{{\mathrm W}_{1}\ldots
{\mathrm W}_{N}}{\frac{({{P}_{\rm{W}}}-C.{{\Pi}_{\rm{W}}})^2}{{{\sigma}_{\rm{W}}}^2+(C.{{\epsilon}_{\rm{W}}})^2}}}.$$ The best fit is that with the value of $C$ which minimizes $\delta$, subject to the physical constraint $0\leq C\leq C_{\rm max}$.
### A model for 3C 41
The source for which we had the most data was 3C 41, with polarimetry in 3 bands: $P_V = 9.3$ [$\pm\ $]{}2.3%; $P_H = 6.6$ [$\pm\ $]{}1.6%; $P_K=3.1$ [$\pm\ $]{}1.1 %. We attempted to fit two models; an electron model and a typical dust model with a minimum grain radius of 80nm. To fit the models to the observed polarizations, we tested a discrete series of possible spectral indices, $-0.5 \leq \alpha \leq 2$, with a step size of $1/3$. For each value of $\alpha$ we calculated the ‘unscaled polarizations’ $\Pi_V \pm \epsilon_V$, $\Pi_H \pm \epsilon_H$, $\Pi_K \pm \epsilon_K$. We then iteratively determined the best fit value of $C$ for each $\alpha$, and took as our overall best fit that combination of $\alpha$ and $C$ which gave the lowest $\delta$.
Lilly & Longair’s data shows that 3C 41, at $K = 15.95$ [$\pm\ $]{}0.10, is significantly brighter than the mean $K$-$z$ relationship, by about 0.6 mag. Magnitudes for 3C 41 were available in 5 bands: $J$, $H$ and $K$ [@Lilly+84a] and the narrow filters g and r$_S$ [@dpc]. The $H$ and $K$ values yielded direct estimates of the corresponding ‘unscaled polarizations’ $\Pi_H$ and $\Pi_K$; $\Pi_V$ was estimated by linear interpolation between ${{\Pi}_{\rm{g}}}$ and ${{\Pi}_{\rm{{r_{\it S}}}}}$.
Solid line: dust model, $\alpha = 1.167$; dashed line: electron model, $\alpha = 1.733$. \[pmodel41\]
Figure \[pmodel41\] shows the measured polarizations for 3C 41 as triangles () and the magnitude-based polarization estimates, after best-fit scaling, as stars (). The lines give the error envelope on the modelled polarization (based only on the errors on the magnitudes). The solid lines correspond to the dust model, and the dashed lines to the electron model.
As can be seen from Figure \[pmodel41\], the model curves lie below the data point at $H$, but above those at $K$ and $V$. The shape of the curve depends more strongly on the measured magnitudes (and on $f_{\rm
W}$ for dust) than on the spectral index, and the models for all reasonable values of $\alpha$ will have a broadly similar shape; the best fit will necessarily pass below the measured point at $H$, and above that at $V$.
Consider the electron model. Figure \[pmodel41\] shows us that the theoretical polarization curve for electron scattering is concave with respect to the origin, whereas a curve through the three data points would be convex; clearly it will not be possible to obtain a close fit for the central ($H$-band) point. Fitting the electron model curve, we found that the best fit occurred for $\alpha = 1.733$, with a deviation $\delta = 0.430$. From Table \[quasfrac\], we have $\Phi_{K} = 6 \pm
4$ per cent for 3C 41. Multiplying the polarizations observed at $H$ and $V$ by $1/{{P}_{\rm{Q,W}}} = 2 \pm 1$, we predict ${{\Phi}_{{H}}}=13 \pm 7$ per cent, and ${{\Phi}_{{V}}}=19 \pm 10$ per cent in these bands.
The dust model chosen as typical from MdSA was that for a cloud of spherical dust grains, with radii 250nm $> a >$ 80nm, with the number density per unit dust mass following an $a^{-3.5}$ law. This model produced a curve which fitted the data points very well. The best fit indicated that the optimum spectral index was $\alpha = 1.167$, for which $\delta = 0.177$.
This dust model was also used to estimate the proportion of scattered light at shorter wavelengths: $\Phi_{K} = 8 \pm 4$%, $\Phi_{H}
= 24 \pm 6$%, and $\Phi_{V} = 155 \pm 38$% [*(sic)*]{}. MdSA’s curve for polarization as a function of rest-frame wavelength predicts a 6 per cent polarization in the observed $V$-band, lower than the 9 per cent [*after dilution*]{} measured by Jannuzi [@Elston+97a]. This is still consistent, within error bars, as long as the true value of $\Phi_{V}$ for 3C 41 is less than, but very close to, unity; the observed $V$-band corresponds to the near-ultraviolet in the rest frame of 3C 41, and it is reasonable (MdSA) to suppose that the scattered quasar light in that band could form in excess of eighty per cent of the total light.
We noted earlier that the shape of the dust model polarization curve between 0.2[$\umu$m]{} and 0.7[$\umu$m]{} (rest frame) is very sensitive to the choice of dust grain distribution. The particular dust model chosen approaches zero polarization at a wavelength corresponding to the r$_S$ band when redshifted into our frame. This causes the ‘well’ visible in the model polarization curve (Figure \[pmodel41\]), whose presence is essential for the dust model curve to fit the data points closely.
We also considered other dust models from the selection given by MdSA. Of those which differed significantly from the ‘typical’ one considered so far, many of them will not predict polarizations in the observed $V$-band which are sufficiently high to be reconciled with the observed 9 per cent; and those which do, do not possess the deep well needed to fit the polarimetry across the spectrum. We conclude, therefore, that the best model for 3C 41 is that of an obscured quasar core with $\alpha
\sim 1.2$ beaming its optical radiation into a dust cloud, although an electron model cannot be ruled out within the error bars.
### A model for 3C 22
As we have already seen, 3C 22 is suspected of being an obscured quasar [@Dunlop+93a; @Rawlings+95a]; according to data in Lilly & Longair , its $K$-band magnitude ($15.67 \pm 0.10$) is brighter than the mean $K$-$z$ relationship by about 0.9 mag.
For 3C 22, we have one firm polarimetry point (this thesis) and two upper limits in $V$ and $H$ [@jpc] [@Elston+97a]; magnitudes were available in 4 bands including $J$, $H$ and $K$ [@Lilly+84a], and a crude eye estimate in r [@Riley+80a]. The measurements and models are shown in Figure \[pmodel22\], with the same symbols as Figure \[pmodel41\]; open triangles represent upper limits. We have taken $\alpha = 1$, and normalised the theoretical curves to the $K$-band data point.
Solid line: dust model; dashed line: electron model. \[pmodel22\]
Here the models are inconclusive. In the observed near-infrared, both models can easily fit, with some slight scaling, within the $K$-band measurement error bars; both models suggest that about 8 per cent of the $K$-band light arises in the active nucleus (Table \[quasfrac\]). The error on the observed r-band magnitude is so large that both models are consistent with the observed $V$-band upper limit of polarization.
### Other compact sources
Manzini & di Serego Alighieri (MdSA) comment that given the range of possible dust models, ‘the wavelength dependence of polarization is not necessarily a discriminant between electron and dust scattering.’ The data available to us are insufficient to indicate whether the scattering centres in these objects are electrons or dust; it is not possible to give an unambiguous fit of the polarization curves with so few data points, although 3C 41 does seem to fit a model (MdSA) with a minimum dust radius of 80nm particularly well, and we suggest that it does indeed consist of a quasar obscured by dust.
Since our sample of sources was selected for the variety rather than homogeneity of sources, it is difficult to come to any general conclusions on the properties of radio galaxies as a whole. There is no obvious trend of alignments, either with elongated structure of the compact optical source or the faint optical companions. $K$-band polarization may be present at any level from at least 6% down to zero. Table \[compalign\] indicates the different alignments (with respect to the radio jets) observed in the faint companions, optical cores, and polarizations of these objects; alignments are identified as parallel or perpendicular rather than skew if the $2\sigma$ error bars allow an aligned interpretation.
Source Companions Optical Core Polarization
---------------- ------------- -------------- --------------
3C 22 $\perp$ $\angle$ $\perp$
3C 41 $\angle$ $\odot$ $\perp$
3C 54 $\cdot$ $\parallel$ $\parallel$
3C 65 $\parallel$ $\perp$ $\perp$
MRC 0156$-$252 $\cdot$ $\parallel$ $\circ$
: Alignments in compact objects.[]{data-label="compalign"}
Two of these compact sources, MRC 0156$-$252 and 3C 54, appear to be truly isolated. Both of these are extremely faint sources in the exposures available to us (Figures \[54image\] and \[Mpic\]) but there is no indication of close companions of comparable brightness. BLR-II’s [*HST*]{} images of the other three reveal much fainter objects in close proximity (within 5), but there is no clear trend of alignment with the radio structure. 3C 22’s one companion is offset almost perpendicular to the radio jet; 3C 41 has two companions on opposite sides along an axis offset 20 from the radio structure position angle; and 3C 65 has one companion lying between the optical source and its northwestern radio lobe.
There is some evidence for structure within the source galaxies. 3C 65 is slightly elongated NE-SW (BLR-II), roughly perpendicular to its radio structure; 3C 54 is extended along its radio axis [@Dunlop+93a]; MRC 0156$-$252 is extended over 8 with three knots in the $r$-band visible (rest frame ultraviolet) corresponding to its radio core and lobes, though it appears compact in $J$, $H$ and $K$ (rest frame visible bands) [@McCarthy+92a]. Two tiny (subarcsecond) extensions are known south and west of 3C 22’s core, neither clearly related to the radio jet orientation (BLR-II).
There is no clear trend, therefore, for alignment in host galaxy structure or the positioning of companions. One skew companion is most likely a chance association of objects; two skew companions on the same axis, as seen in 3C 41, might be evidence for precession of the radio jets from the skew axis to their present position. Parallel core structure or companions suggest that the radio jet may be responsible in some way for the formation or excitation of structure along its path. Perpendicular structure could be indicative that the radio jet is orthogonal to the plane of rotation of the galaxy and its satellites. Any comprehensive model of radio galaxies clearly needs to allow, therefore, for the possibility of core structure and companions, both aligned with and orthogonal to, the large-scale radio structure.
Evidence that 3C 22 and 3C 41 may be close to the $\chi = 45\degr$ boundary between radio galaxy and quasar profiles encourages us to consider the other sources in this light. The most recent radio maps of MRC 0156$-$252 [@Carilli+97a] do not display a marked head-tail asymmetry in radio lobe intensity at 4710MHz or 8210MHz, although the eastern lobe has a much greater intensity in linearly polarized 4710MHz radiation than the core or western lobe. The formal $1\sigma$ confidence interval indicates that the source is totally unpolarised in $K$. At most, if we assume that it has radio jets perpendicular to our line of sight, 16 per cent of its [[$K$]{}]{}-band light arises in the active nucleus.
Nevertheless, it is possible that MRC 0156$-$252 is an obscured quasar [@Eales+96a], and McCarthy, Persson & West note that its properties are comparable to the red quasars observed by Walsh et al. . If this is the case, we must be looking close to ‘straight down the jet’ with a shallower scattering angle for infrared light; and hence our upper limit for $\Phi_{K}$ would be weaker. The fact that McCarthy, Persson & West detect $r$-band structure on the same 8 scale as the radio structure could support this, since at radio galaxy orientation, optical structure is usually less extended than radio structure, especially in the most powerful radio galaxies (BLR-I). Alternatively, if this galaxy’s high luminosity is due to star formation or direct emission from an active nucleus which is not well shielded, any scattered component might easily be diluted below a detectable level.
The five optically compact objects in our sample have been selected for their radio strength and should be presumed to share the radio emission mechanism common to all radio galaxies, unless proven otherwise. Their compact appearance gives no indication of a history of merging or recent star formation, unless the rest-frame ultraviolet elongated structure of MRC 0156$-$252 is interpreted as such. The detection of $K$-band polarization requires both the presence of a sufficiently dense scattering medium (probably situated within the host galaxy itself), and a well-shielded nucleus strong enough to yield sufficient scattered infrared light to be detectable despite dilution by the host galaxy.
The perpendicular polarizations of 3C 22, 3C 41 and (if genuine) the marginal 3C 65 are all, therefore, consistent with the Unification Hypothesis for radio galaxies. Failure to detect polarization in MRC 0156$-$252 is consistent with Unification given dilution, the absence of a suitable scattering medium, or an obscured quasar scenario. Finally, the parallel-polarized 3C 54 is the most difficult candidate to reconcile with the Unification model, requiring a special scattering geometry or possibly the presence of aligned dust grains causing polarization by transmission in the outer structure of what seems to be a quite diffuse source.
Spatial Modelling of Knotted Sources
------------------------------------
Our four remaining sources clearly display several distinct or joined knots in their $K$-band structure: 3C 114, 3C 356, 3C 441 and 53W091. The scattering models of the Alignment Effect invite us to investigate scenarios where one knot contains an active nucleus and other knots are regions of scattering material which have intercepted a particle jet or radiation cone emerging from the central engine.
### Polarized companions? {#polcom}
We have already discussed (§\[objE\]) companion E to 3C 441 as a possible case of nuclear light scattered and polarized by a cloud illuminated by the central engine. Other cases giving evidence for polarization by scattering have been published recently, as the new generation of telescopes and instruments begins to make possible high-resolution imaging polarimetry of high redshift radio galaxies.
Tran et al. used the Keck I to obtain extended imaging polarimetry of 3C 265, 3C 277.2 and 3C 324. In all three cases, the polarization maps displayed bipolar fans of polarization vectors centred on the nucleus, perpendicular to the optical structure and misaligned by tens of degrees with the radio axis. Earlier structural information on one radio galaxy was obtained by di Serego Alighieri, Cimatti & Fosbury ; their $V$-band polarimetry of the $z=0.567$ object $1336+020$ showed perpendicular polarization in a northern knot and in extended emission, higher than in the core.
Contour maps of the three Tran et al. sources are provided, at levels relative to the peak intensity of the central knot, and all three sources include companion objects. Little or no polarization is seen in the bright ($\sim 20$ per cent of peak) companions of 3C 277.2 and 3C 324. But in 3C 265, a faint companion object also exhibits the polarization seen in the fan – the object is a knot less than 8 per cent of peak intensity and lies beyond the extension of the $V$-band optical structure, in the same direction but unconnected with the optical core in contours down to 2 per cent of peak. Such a faint polarized knot could readily be identified with light redirected by a cloud of scattering particles.
Assuming an $\Omega_0 = 1.0, \Lambda=0$ cosmology with $H_0 = h_0$ kms$^{-1}$Mpc$^{-1}$, $h_0 = 100$, the knot in 3C 265 which lies about 9 from the core, is separated from the core by about 36$h_0$ kpc. The extended structure of $1336+020$ of about 3 corresponds to 12$h_0$ kpc. In comparison the 53W091 to $3a$ separation and the distance between 3C 441 [**a**]{} and E both correspond to approximately 16$h_0$ kpc. So the structure of these companion objects is of comparable scale to those in the literature.
### Alignments in the knotted sources: an overview
The most striking feature of our small sample of four knotted sources (Table \[knotalign\]) is that in all four cases, the core and at least one other knot lie along the line of the radio structure axis. In 3C 114, three prominent knots lie on this axis and the fourth is offset perpendicular; in 3C 356, the displacement vector between the major components $a$ and $b$ lies within a few degrees of the radio axis; 3C 441 has component [**c**]{} closely associated with the NW radio lobe and component F fairly close to the radio axis; and 53W091 is displaced from companion 3a, again within a few degrees of the radio axis. Only 3C 441 has relatively bright companions in skew positions; and since this source seems to be in a rich field, these companions can easily be accounted for as cluster members rather than effects of the active nucleus.
Source Companions Optical Core Polarization
-------- ------------------- -------------- ---------------------------------------
3C 114 $\parallel,\perp$ $\odot$ $\perp$ (T2), $\parallel$ (T0)
3C 356 $\parallel$ $\parallel$ $\parallel\ (a); \angle\ (b)$
3C 441 $\parallel,\perp$ $\parallel$ $\parallel$ (E); $\perp$ ([ **a**]{})
53W091 $\parallel$ $\angle$ $\parallel\ (3a); \perp$ (core)
: Alignments in knotted objects.[]{data-label="knotalign"}
### A conical sector model
We shall create a ‘toy model’ to help us investigate the properties of light scattered from a cloud of dust or electrons. Brown & McLean have modelled the case of axisymmetric Thomson scattering in a stellar envelope, and we can easily adapt this model to the case of scattering by an axisymmetric electron cloud in the conical region illuminated by a quasar nucleus embedded in an obscuring torus. We take an $(r,\theta,\phi)$ spherical co-ordinate system with the polar axis as the axis of the obscuring torus, and define $\mu = \cos(\theta)$. For convenience of integration we shall consider the electron cloud to have a constant number density, $n_0$, and to have the shape of a conical sector with boundaries $R_1 \le r \le R_2$ and $0 \le \theta \le
\Theta_1$ \[hence $1 \ge \mu \ge \mu_1 = \cos(\Theta_1)$\].
Brown & McLean’s treatment of the problem gives expressions for the light intensity (their Equation 5) and polarization (their Equation 16) of light scattered from an axisymmetric cloud of any number density $n(r,\mu)$ integrated over $r: 0 \rightarrow +\infty$ and $\mu: -1 \rightarrow +1$. They define the axial inclination, $i$, as the angle between the equatorial plane of the scattering cloud (in our case, the plane of the obscuring torus) and the sky plane (the perpendicular to the line of sight from the source to Earth). In this convention, our scattering angle, $\chi$, is such that $\chi = 180\degr - i$.
Substituting our special case of a constant density cloud with conical sector boundaries, we obtain: $$\label{scatintens}
I_1 = \frac{3 I_0 \sigma_T n_0}{16} (R_2 - R_1) \left[ (1-\mu_1)(2+\sin^2
i) + \frac{(1-{\mu_1}^3)(2-3\sin^2 i)}{3} \right],$$ and $$\label{scatpol}
\frac{1}{P} = 1 + 2 ({\mathrm cosec}^2 i) \left[
\frac{3(1-\mu_1)+(1-{\mu_1}^3)}{3(1-\mu_1)-3(1-{\mu_1}^3)} \right];$$ if $I_0$ is the intensity (power radiated into unit solid angle) of the central source, then $I_1$ is the intensity of the scattered radiation; $P$ is the polarization of this radiation, and $\sigma_T$ is the wavelength-independent cross-section for Thomson scattering. In this formalism, a negative $P$ corresponds to polarization perpendicular to the symmetry axis.
The most striking feature of Equation \[scatpol\] is that it is independent of the radial boundaries $R_1,R_2$. It follows that for small opening angles, where a spherical cap can be approximated as a disc of constant (Cartesian) $z$, the (undiluted) polarization of [**any**]{} axisymmetric distribution of electrons $n(r)$ depends only on the opening angle of the illuminating aperture; we can model any axisymmetric distribution of dust $n(r)$ as the sum of scattered light from infinitessimal slices of constant density.
We will now use our toy model to investigate two scenarios: scattering from a conical region of dust stretching from the origin $(R_1=0)$ to some finite distance $R_2$; and scattering from comparable clouds at different distances from the origin.
### A patchy cloud model
In order to use our toy model to investigate galaxy structures like that of 3C 114, we will consider what happens when similar small clouds are placed at different distances from a point source. We will consider all clouds to have the same volume, $V$, the same diameter, $w$, and the same fixed particle density, $n$. Further, for ease of integration, we will consider all clouds to be placed on the axis, and bounded by radial lines forming conical surfaces, and by spherical caps.
Let us consider a cloud which subtends some half-angle $\Theta_1$ and extends from $R_1$ to $R_1 + \Delta R$. Now the width of this cloud we can take as the linear distance subtended by the spherical cap at $R_1
+ \Delta R/2$, [*viz.*]{} $$w = 2 \sin (\Theta_1).(R_1 + \Delta R/2);$$ the volume is obtained by the trivial integration of $r^2 \sin \theta
\,dr\,d\theta\,d\phi$ over $(R_1,R_1+\Delta R)$ in $r$, 0 to $\Theta_1$ in $\theta$, and 0 to $2\pi$ in $\phi$, whence $$V = (2\pi/3).(1-\cos \Theta_1).[(R_1+\Delta R)^3 - {R_1}^3].$$ For given $w$, $V$ and distance from the illuminating source, $R_1$, the appropriate opening angle $\Theta_1$ and radial thickness $\Delta
R$ can easily be calculated. Trial and error revealed that with $V=6.6$ and $w=2$, the calculated $\Delta R$ values were also of order 2 at various radii $R_1$, yielding a model cloud about as broad as it is deep. Such a quasi-symmetric cloud seems the most appropriate for a toy model mimicking fairly symmetrical knots in galaxies. The different dimensions of this conical sector cloud at differing radii are given in Table \[cloudsize\].
\[cloudsize\]
$R_1$ $\Delta R$ $\Theta_1$ $\mu_1$
------- ------------ ------------ ---------
1 1.81 31.68 0.851
2 1.97 19.58 0.942
3 2.02 14.43 0.968
4 2.05 11.48 0.980
5 2.07 9.54 0.986
6 2.08 8.17 0.990
7 2.08 7.14 0.992
: Dimensions of a conical sector of fixed volume and width.
By substituting the boundaries for such a cloud into Equations \[scatintens\] and \[scatpol\], we can immediately obtain the polarization and relative intensity of the light scattered by similar clouds at different densities. The [*undiluted*]{} polarization is, in fact, independent of both the cloud density and the intensity of the illuminating source. The polarizations and intensities (relative to the $R=1$ case) are given in Table \[patchpol\]. It is also useful to calculate the surface brightness, $B=I_1/A$: approximating the side-on profile of the cloud as a sector subtending half-angle $\Theta_1$, the area bounded between $R_1$ and $R_1 + \Delta R$ is $A = 2\Theta_1 [(R_1 +
\Delta R)^2 - {R_1}^2]$.
\[patchpol\]
$R_1$ $P$(%) $I_r$(%) $B_r$(%) $P_r$(%)
------- -------- ---------- ---------- ----------
1 73.6 100.0 100.0 100.0
2 89.0 40.6 38.5 55.7
3 93.9 22.5 20.9 34.0
4 96.0 14.4 13.2 22.6
5 97.3 10.0 9.1 16.1
6 98.0 7.3 6.7 12.0
7 98.5 5.6 5.1 9.3
: Polarization and intensity of light scattered by a small cloud at various distances from an illuminating nucleus.
It is evident from Table \[patchpol\] that clouds of a given size lying further from their source of illumination are more strongly polarized, but the intensity of the polarized light is diminished. Obviously at greater distances, the angle (and hence fraction of the source emission) subtended by the cloud is smaller, but the beam entering the cloud is more collimated and hence there is less cancellation of polarization from light being scattered in opposite senses.
The toy model does not require that the illuminating source be shielded, only that it be pointlike; so a scattering cloud illuminated by an external small non-AGN source (e.g. a star cluster) would also produce polarised light in the manner of this model. If the scattering cloud’s only source of light is the external illumination, it is clearly not possible to obtain a structure of knots of similar brightness at different distances from the nucleus, since the relative intensity (and surface intensity) drop off so rapidly with distance. Where the intensity of the intrinsic emission of the knot, $I_k$, is much greater than that of the scattered light, the polarization of the diluted light can be approximated as $P_d = P.I_1/I_k$, and the values of $P_d$ relative to that at $R_1=1$ (since the absolute value depends on the knot’s intrinsic brightness) are also tabulated in Table \[patchpol\].
Is a scattering model adequate, therefore, to account for the knotted structures seen in many high-redshift radio galaxies? The central engine, by definition, must be many times brighter than any scattering clouds shining purely by scattering some small portion of its output; but if the nucleus is well-shielded then the residual starlight of the host galaxy might be of comparable brightness to a knot of scattered light. If there are multiple knots, those at greater distances from the nucleus should be much fainter, following Table \[patchpol\], unless they contain denser clouds of scattering material than the nearer clouds. Knots offset more than 45 from the radio axis ought to be shielded from the central engine by its obscuring torus; the light of such knots cannot easily be attributed to scattering.
### A continuous cloud model
Another useful model to examine is that of a scattering cloud bounded by $0 \le r \le R_1$ and $0 \le \theta \le \Theta_1$. This case can model an AGN whose conical region of illumination is filled with scattering particles close to the active nucleus. In a canonical model with $\Theta_1 = 45\degr$ where the particles fill the opening angle of the central engine, then $P = 53.3\%$. The undiluted polarizations owing to cones filled at other opening angles are given in Table \[conepol\]. Again, the undiluted polarizations are perpendicular to the conical axis and independent of the radial extent of the scattering region. One important consequence of this model is that any measurement implying an undiluted polarization much greater than 50% is indicative of a scattering medium subtending an angle smaller than the whole 45 illumination zone.
\[conepol\]
$\Theta_1$ 5 10 15 20 25 30 35 40 45 50 55
------------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
$P$ 99.2 97.0 93.4 88.5 82.6 75.9 68.7 61.1 53.3 45.6 38.2
: Polarization of scattering cones filled to various opening angles.
### 3C 114 {#M3C114}
We turn first to 3C 114, with three knots lying along the radio jet and one offset perpendicularly (Figure \[114figs\]). In this source, the ‘knee’ knot (T2) is presumably part of the parallel aligned structure, and its polarization is roughly perpendicular to the radio jet, as expected if some scattering process is at work inside the knot, perhaps a region of dust in the scattering cone as discussed immediately above. The rest of the structure, however, is difficult to interpret in terms of a scattering model. We have no polarimetry on the knots T3 or T4, but if T2 contains the central engine, T4 is brighter than T3 though it lies beyond T3. Clearly the light from T4 cannot primarily be scattered light and some other mechanism must be at work (e.g. jet induced star formation) to produce the Alignment Effect in this object.
When polarimetry is performed over the whole aperture, including the perpendicular knot T1, the overall polarization seems [*parallel*]{} to the radio jet — i.e. perpendicular to the axis connecting this anomalous knot to the main structure. Is this indicative of some scattering mechanism perpendicular to the main radio structure? It is difficult to conceive of a mechanism which allows a brilliant central engine to emit light in two perpendicular directions without also leaking such light towards Earth; yet the morphology (and indeed the high polarization which implies that any dilution is limited) do not suggest that we are seeing a central engine directly in T2. Could T1 be a chance alignment or a satellite galaxy? This would explain its presence but not the overall parallel polarization. The brightness of the knots in 3C 114 make it a prime candidate for spectroscopy or spectropolarimetry, which would shed more light on the chemistry and light emission mechanisms at work in each of the knots.
### 3C 356 {#M3C356}
We have already reviewed the structure of 3C 356 (§\[D3C356\]) and noted how the SE radio core $b$ has been proposed as the central engine with component $a$ interacting with its jet [@Lacy+94a; @Eales+90a]. More recent imaging (BLR-III) and spectropolarimetry [@Cimatti+97a], however, now point towards $a$ as being the more likely host of the active nucleus. Broad lines are clearly visible in the polarised spectrum of $a$, which itself is now known to have two components (BLR-II); two components are aligned along the $a-b$ axis with the component nearer $b$ providing 60% of the total emission from $a$. The Keck spectropolarimetry unequivocally demonstrates the presence of a polarized component strong in the near-ultraviolet and declining into the green; our $K$-band polarimetry cannot identify this polarized component in the near-infrared, suggesting that nebular emission and starlight are dominating this waveband.
The weight of evidence in the more recent literature, though not conclusive, indicates $a$ as the more likely host of the 3C radio source, though in this case $b$ seems to be a flat-spectrum radio galaxy in its own right. Comparison of the spectra of $a$ and $b$ presented by Cimatti et al. does not suggest that $b$’s main source of light is scattering from $a$’s emission, nor is the polarization in $b$ high; they also show that the polarization in $a$ can be modelled both by dust and by electron scattering. The high $(\sim 15\%)$ polarization present in the ultraviolet shows that scattered light must form a large proportion of the light from $a$ at these frequencies, and Cimatti et al. estimate that at 280nm, 50 [$\pm\ $]{}15% of the total flux from $a$ is scattered light, and the intrinsic undiluted polarization is 21 [$\pm\ $]{}7%. Our $K$-band data can only add to this the knowledge that the diluting component is much stronger in the infrared.
### 3C 441 {#M3C441}
Analysis of 3C 441 is complicated by the richness of the field in which it lies; without knowledge of the redshifts of all the objects, and hence of which could be true neighbours to the radio core, any interpretation must be tentative. Object E is known to be much fainter than the presumed active nucleus in $K$ (this thesis) and in $I$ and \[O[ii]{}\] [@Neeser-96a]. This is consistent with a scattered light hypothesis and it is highly plausible that companion E to 3C 441 is an illuminated object scattering light in the manner of the extended structure seen in 3C 265 and $1336+020$, pending confirmation of its redshift — though it must be noted that the result for E does have a ten percent chance of being a noise-induced spurious result.
### LBDS 53W091 {#M53W091}
Unlike the other objects in this thesis, LBDS 53W091 was chosen on the basis of the [*weakness*]{} of its radio emission. Its apparent age and redshift are hard to reconcile and the reported 40% polarization did not help to clarify the picture. We have ruled out a $K$-band polarization of that order; most of the discussion of this galaxy’s peculiar properties can be found in §\[D53W091\] and in more detail[^6] in Leyshon, Dunlop & Eales .
We have seen how radio-weak galaxies only display the Alignment Effect over small scales, up to about 15kpc [@Lacy+98b]; the separation between 53W091 and companion 3a is of this order (16$h_0$kpc) and so its good alignment is in keeping with what is known for 7C galaxies. We have seen (§\[53Wgeom\]), however, that the polarization in 3a cannot be reconciled with a scattering model; and the result for 3a is too tentative to warrant developing alternative models.
Radio Galaxy Trends: The Big Picture
------------------------------------
The data presented in this thesis represents the first $K$-band polarimetry of high redshift radio galaxies; that is, the measurements are the first on record for the rest frame near-infrared (0.7–1.3[$\umu$m]{}) polarizations of radio galaxies significantly less evolved than the local universe. The results can be compared with the infrared properties of nearby radio galaxies and with the visible-light properties of high redshift objects (§\[polevi\]) — always bearing in mind that our sample of nine diverse objects is of limited statistical significance.
Some nearby radio galaxies (Cen A, IC 5063, 3C 234) have been observed in the $K$-band (§\[polevi\]) and found to have polarizations of order 10%, oriented perpendicular to the radio jet. In these cases, it seems that the standard scattering hypothesis is the best explanation. 3C 233.1 is 5% polarized in $K$ but less than 0.5% in the visible, suggesting that dust extinction may reduce the contribution of visible scattered light. Our sources are being observed at rest-frame wavelengths somewhat shorter than the 2.2[$\umu$m]{} $K$-band, so if 3C 233.1 were used as a benchmark, we might expect perpendicular polarizations somewhat lower than 5%. In fact our measurements range between zero and 20%, all in the ‘ballpark’ defined by these earlier $K$-band observations.
We have noted (§\[poltrend\]) how Cimatti et al. analyzed the properties of 42 radio galaxies at $z\geq0.1$ and searched for trends with both the observed polarization and their estimate of the underlying nuclear polarization. All their conclusions were based on light emitted at rest frame wavelengths between 0.2[$\umu$m]{} and 0.7[$\umu$m]{}, so the observations of this thesis do not overlap in $\lambda_r$ with theirs. Nevertheless, it is valuable to try to interpret our results in the context of their trends analysis.
Cimatti et al. found that a good rule of thumb was that radio galaxies at $z>0.6$ were polarized above 8% and those at lower redshift, less than 7%. All of our sources lie at $z>0.6$, but some are certainly polarized below 7% in $K$-band, the firmest results being the unpolarized MRC 0156$-$252 and the 3% polarized 3C 22 and 3C 41. Jannuzi reports that 3C 22 is no more than 5% polarized in the $V$-band; 3C 41 fulfills the rule of thumb in $V$ but not in $H$. The rule, however, is a rule for radio galaxies, so if these objects are actually obscured quasars the rule is not applicable to them. Spectropolarimetry for 3C 356 $a$ [@Cimatti+97a] shows the rule satisfied in this source at 0.2[$\umu$m]{} but not at 0.4[$\umu$m]{}, while our results indicate that it probably obeys the rule at 1.1[$\umu$m]{}. The handy rule of thumb, therefore, must be used with the caveat that it applies only to light emitted at visible/ultraviolet rest wavelengths.
There is also a rule of thumb based on general results about alignments. Radio galaxies with $z>0.6$ and $P>5\%$ were always found to have perpendicular polarizations. Again, this is not found to carry into the $K$-band, since 3C 54’s polarization is parallel, and 3C 114 is ambiguous depending on which knots are included. Neither can we sustain the rule that if a radio galaxy polarization is in parallel alignment, the polarization is always lower than 5%: the same two objects and 3C 356 may all have higher parallel polarizations.
Except for 53W091, all the objects in our sample are very radio loud; all lie at high redshift. They would therefore be expected to display a clear Alignment Effect in the visible (§\[aligneff\]) and at least a marginal effect in the infrared. In fact the knotted sources (Table \[knotalign\]) do display parallel aligned structures, though some (most strikingly 3C 114) also have knots in perpendicular alignment — a feature not unknown in nearby radio galaxies [@Crane+97a]. Discerning alignments present in compact objects (Table \[compalign\]) is, by definition, more difficult, and no clear trend is apparent in our five compact sources. Clearly bright knots, where present, dominate any analysis of alignments and structure, whereas the slight extensions or faint companions of radio galaxies may not be related to the mechanism which sometimes causes aligned or perpendicular knots.
Conclusions
===========
> It seemed to me that in one of my innumerable essays, here and elsewhere, I had expressed a certain gladness at living in a century in which we finally got the basis of the Universe straight.
>
> ------------------------------------------------------------------------
>
> — Isaac Asimov, [*The Relativity of Wrong*]{}.
The work contained in this thesis has pushed forward scientific knowledge on two fronts: the practice of astronomical polarimetry, and our knowledge of the $K$-band polarization properties of radio galaxies. As always in science, new results reveal to us how little we know about the Universe at large and suggest future avenues of exploration. It is clear that there is great room for improvement in the polarimetric analysis software available to the astronomical community, and some recommendations are made here on functions which should be included in any comprehensive analysis package of the future.
Our sample of targets was selected for its diversity and provides a snapshot of some interesting objects; even so, some of these provide marginal results over the maximum realistic integration time on a world-class infrared telescope. Today, we can only dream of taking a sample large enough to yield good statistics: the number of objects and the integration times which would be required conspire to place such a project in the realms of spaceborne infrared telescopes, Keck-size telescopes, or weeks of dedicated observing time. In the meantime, individual $K$-band objects will surely be subjected to polarimetric analysis, and it is possible to give some pointers for properties to look out for.
Requirements for a Comprehensive Polarimetric Software Package
--------------------------------------------------------------
The software currently available for polarimetric analysis ([aaopol]{} from the Anglo-Australian Observatory and [polpack]{} from [*Starlink*]{}) concentrates on the generation of ‘vector maps’ illustrating the polarization of different parts of an image by means of arrows of appropriate length and orientation. Such software is fine for imaging polarimetry where there is a high-quality signal, but is inadequate to deal with pioneering research where the signal-to-noise ratio is low. The experience gained in performing the analysis for this thesis suggests that a future comprehensive polarimetry package should include the following features for two-channel Wollaston prism systems with a waveplate rotating in 22.5 steps.
### Generation of the Stokes Parameters
An imaging polarimetry package would normally function as an accessory to an imaging photometry system. It would be necessary to define one or more polarimetry apertures (a whole object, a series of knots, etc.) and define the binning resolution (the whole aperture, individual pixels, or some intermediate level). It should be possible to define a list of input images (possibly several for each waveplate) and tag each of them with the orientation of the waveplate used for that exposure; the orientation, $\eta_0$, of the reference axis of the waveplate system should also be noted. It would be desirable to provide automatic and manual facilities for registering the waveplate images rather than assuming perfect alignment.
Having defined the sampling apertures and resolution, the software should be capable of calling the photometry package, accepting the returned photometry data, and converting the results to absolute Stokes Parameters (with errors) relative to the waveplate reference axis. The reference angle and Stokes parameters for each pixel bin of each source on each image should be stored in a file for further analysis. This part of the software should implement Steps \[smallshot\] to \[noiseOK\] of Chapter \[stoch\].
### Stokes Parameter analysis routines
The nucleus of a polarimetry package should be a versatile system for performing analysis on sets of Stokes Parameters, absolute or normalized, derived from the photometry procedure detailed above, or directly entered from the literature. Among the analysis routines available should be the following:
1. \[optestfunc\] Derive the optimal estimates of the normalized Stokes Parameters from the absolute Stokes Parameters, following (i) Steps \[hereNSPs\] to \[nearnormal\], or (ii) the iterative method of §\[zpos\].
2. Estimate the probability that the true polarization of a bin is zero/non-zero using any of three methods: (i) the absolute Stokes Parameter confidence interval test (Step \[findconfr\]), (ii) the polarization debiasing test (Step \[estpolun\]) and (iii) the residual method (§\[zpos\]).
3. Convert normalized Stokes parameters to percentage polarization (Steps \[findperr\] to \[getint\]) and orientation (Steps \[propphi\] and \[findangle\]) format, providing point estimates and/or confidence intervals.
4. Convert data in the form of percentage polarization and orientation into Stokes Parameter form, both for ‘debiased’ polarizations and non-debiased crude estimates of the percentage polarization. (Such a function allows rapid conversion of data from the literature into a form comparable with other data.)
5. Test two sets of Stokes parameters for consistency with one another (for studying temporal variability etc.).
6. Convert Stokes Parameters from a given reference frame to a standard frame where the axis points North. (Working in the instrumental frame is best as the errors on the two channels are independent of one another; but standard orientation may be needed if data from two telescopes with different reference axes are to be combined.)
7. Combine two sets of normalized Stokes parameters. (Combining two sets of absolute Stokes Parameters is a trivial extension of function (\[optestfunc\]) unless the two reference axes are misaligned.)
A software package capable of performing all these analyses would be a powerful tool enabling the rigorous analysis of new polarimetric data and efficient comparison with the existing literature. Ideally the software should be able to output the debiased data in a form compatible with existing software for displaying polarization vector maps. Limited by the shot noise inherent in photometry of faint sources, and providing optimal estimates, such software would yield the most accurate estimates of true polarizations theoretically possible, and these recommendations are commended to astronomical programmers for their consideration.
A Summary of the Properties of Our Sample
-----------------------------------------
We have taken $K$-band polarimetry for seven 3CR radio galaxies, and found a diverse range of results. Out of our seven sources, two (3C 65 and 3C 441) display no evidence for polarization (though a companion to 3C 441 may be polarised). For the sources which do display some evidence of polarization, we have estimated the fraction of observed $K$-band light which originated in the active nucleus (Table \[quasfrac\]). Most of our findings are lower than the $\Phi_{K} \ga
40\%$ suggested by the recent findings of Eales et al. but are consistent with the hypothesis that radio galaxies consist of quasar nuclei embedded in giant elliptical galaxies.
All of the galaxies which appear to be polarised have large errors on the orientation of their [$E$]{}-vectors; hence any apparent alignment effects are suggestive rather than definitive. But with this caveat, we note that two sources (3C 54 and 3C 114) have high polarizations oriented in roughly parallel alignment with the radio axis and extension of the optical structure — i.e. in the opposite sense to the perpendicular alignment expected under a simple scattering model.
The compact galaxies 3C 22 and 3C 41 display significant polarizations of around 3% with a polarization alignment perpendicular to their radio axes; both appear in the $K$-band as pointlike objects. We suggest, therefore, that in these objects, infrared light from a quasar core is being scattered into our line of sight, and forms a significant part of the total $K$-band flux received from these sources; both objects may be inclined close to the $\chi \sim 45\degr$ ‘boundary’ between quasar and radio galaxy properties.
In the case of MRC 0156$-$252, which lies beyond a virtually dust-free part of our own Galaxy, we can be reasonably certain that this radio galaxy is not polarised, and the $K$-band light has not been scattered before reaching us. If some of the $K$-band light has originated in the active nucleus, its contribution should be smaller than at visible wavelengths [@Manzini+96a]; this being the case, subtraction of our image or a synthetic symmetrical galaxy could well reveal the structure of the active component at visible wavelengths, given the visible structure observed by McCarthy et al. . It is possible that this galaxy, like 3C 22 and 3C 41, is an obscured quasar at an intermediate orientation.
In LBDS 53W091, we can rule out the contribution of an active nucleus to providing more than $\sim 25$ per cent of the observed light. The majority of its $K$-band light, therefore, must be presumed to be due to its stellar population, and its $R-K$ colour remains consistent with an age in the range $2.5-3$ Gyr. The nature of its companion object $3a$, possibly polarised and of unclear physical relationship with 53W091, warrants further investigation.
In 3C 441, the polarization from object E may indicate that E is scattering light from [**a**]{} (whose identification as the central engine would thus be vindicated); the orientation of E’s polarization would not be consistent with the source being located within E or F and emitting jets at $\sim 145\degr$. Therefore, we favour the traditional identification of the central engine with [**a**]{}. Although this object was observed at two epochs, the observational errors cannot rule in or out temporal variability in polarization over a two year period.
It is noteworthy that when radial profile or spectral fitting estimates [@Best+98a] are combined with our $K$-band data (§\[blr3constr\]), there are hints that the true $K$-band nuclear polarizations of several sources (3C 65, 3C 356, 3C 441 [**a**]{}) are of the order of 25%. The measurement errors and uncertainties in the derivation mean that these figures are no more than indicative; but following Enrico Fermi’s rule of thumb that in a sufficiently complicated problem, uncertain contributions tend to cancel out each other, this can be taken as a very tentative indication for $K$-band nuclear polarizations of order 25% in radio galaxies.
Polarimetry of faint objects requires long integration times. The observing time available has permitted us to rule out the existence of very high polarizations in many of the objects studied, at least for light emitted along the line of sight to Earth. It would still be possible for light emitted in other directions from these objects to be polarised. ‘Polarization’ mentioned in these conclusions should be understood in the restricted sense of light leaving the source in the direction of Earth. Under the Unification Model, radio galaxies (a class of AGN assumed to be oriented with their jets perpendicular to that line of sight) would be more likely to display polarization originating in scattering or synchrotron radiation in the light travelling Earthwards than in directions closer to the jet.
Recommendations for Future $K$-band Radio Galaxy Polarimetry
------------------------------------------------------------
Interpreting polarization measurements is intimately linked with understanding the morphology of the knots or extensions which accompany a radio source. It is a distinct advantage to study sources in which the redshifts of the companion objects are known, so that chance alignments can be ruled out, and true near-neighbours can be identified as such. Where redshifts are not available and the source lies in a rich field, multifibre spectroscopy of the sources should be scheduled. Then, if an imaging polarization measurement suggests that a knot may be scattering nuclear light, the AGN spectrum could be scaled and subtracted from that of the knot.
The spectral and spatial profile fitting methods of Best, Longair & Röttgering greatly complement the data available from polarimetry, since stronger limits can be placed on the nuclear polarizations if the fraction of the light due to stars can be independently identified. There would be a distinct advantage in performing polarimetry on the other objects already analysed by them, or applying their analysis (de Vaucouleurs fitting could be done on the imaging polarimetry images in good seeing) to stacked polarimetric images.
Polarimetry is a cheap ‘overhead’ which could be seriously considered whenever imaging or photometry is being carried out on a radio galaxy whose linear extent is such that it can easily be viewed through a focal plane mask. Polarimetry complements mosaicing by spreading the image over the pixels available, and local effects are cancelled out when the waveplate is rotated through 45. Combining the two slits for imaging purposes can be integrated into the mosaicing process. The photon rate in each channel is 50% of that which would be achieved with no Wollaston prism (for both source and sky noise), so doubling the time per exposure would produce noise at the same level per integration as using the system without the Wollaston prism. In the same spirit, applying a Wollaston prism before a grating or multifibre spectroscopy system allows spectropolarimetry AND spectroscopy of a given quality to be done in just twice the time of spectroscopy alone.
Sources in which there is evidence for scattered light from a distinct knot – such as 3C 441 – are prime candidates for spectropolarimetry to be applied to their distinct knots. Tracing the nuclear spectrum in the polarized spectrum of the knot would confirm the scattered light hypothesis.
Spectropolarimetry should also be applied to sources in which there are indications of parallel polarization, such as 3C 114, 3C 54 and 53W091 3a. Only low parallel polarizations can be produced by flattened scattering discs, and untangling the spectrum of the parallel polarized component would give a better hint at what is taking place in such sources.
Sources in which their are indications of a skew Alignment Effect with infrared structure 10–20 out of line with visible structures (§\[misalign1\]) are also particularly interesting targets for further study, as are those where optical polarization is somewhat misaligned with radio structure (§\[misalign2\]). It would be a valuable exercise to perform a full literature review and observing campaign to compare the polarization orientation with the position angles determined for structure in the radio, near-infrared, visible and near-ultraviolet bands for every AGN with a published optical polarization. Understanding the wavelength-dependence of the skew alignment would probe the mechanism at work in these objects.
Finally, all polarimetric studies must be realistic about the time required to get a useful result. Low polarizations are harder to isolate than high ones; we spent more than an hour on 3C 22 and nearly two on 3C 41, our brightest sources; and the errors on these are still quite large. Six hours on the faint 53W091 has refuted any suggestion of 40% levels of polarization but cannot give a definitive answer on the presence or absence of lower levels. The author looks forward to the days when larger telescope mirrors, parallel use of Wollaston prisms and imaging spectroscopy allow the radio galaxy trends survey of Cimatti et al. to be extended to $K$-band observations, and the true contribution of scattered $K$-band light to the properties of high-redshift radio galaxies will be known.
> A man should keep his little brain-attic stocked with all the furniture that he is likely to use, and the rest he can put away in the lumber-room of his library, where he can get it if he wants it.
>
> ------------------------------------------------------------------------
>
> — Sherlock Holmes, [*The Five Orange Pips*]{}.
Aller L. H., 1987, PASP, 99, 1145
Angel J. R. P., Stockman H. S., 1980, ARAA, 18, 321
Antonucci R., 1982, [*Nature*]{}, 299, 605
Antonucci R., 1983, [*Nature*]{}, 303, 158
Antonucci R., 1984, ApJ, 278, 499
Antonucci R., 1993, ARAA, 31, 473
Antonucci R., Barvainis R., 1990, ApJ, 363, L17
Antonucci R., Hurt T., Kinney A., 1994, [*Nature*]{}, 371, 313
Antonucci R. R. J., Miller J. S., 1985, ApJ, 297, 621
Aspin C., 1994, IRCAM3 Operations Manual v1.07, Joint Astronomy Centre, Hilo, Hawaii; available on the World Wide Web at [url]{}\
`http://www.jach.hawaii.edu/~caa/i3/i3.html`
Aspin C., 1995, IRCAM3+IRPOL2 polarimetry data acquisition and reduction, Joint Astronomy Centre, Hilo, Hawaii; available on the World Wide Web at [ url]{}\
`http://www.jach.hawaii.edu/~caa/pol.doc`
Axon D. J., Ellis, 1976, MNRAS, 177, 499
Bailey J., Sparks W., Hough J., Axon D., 1986, [*Nature*]{}, 322, 150
Baron E., White S. D. M., 1987, ApJ, 322, 585
Barthel P. D., 1989, ApJ, 336, 606
Best P. N., Longair M. S., Röttgering H. J. A., 1996, MNRAS, 280, 9P (BLR-I)
Best P. N., Longair M. S., Röttgering H. J. A., 1997, MNRAS, 292, 758 {astro-ph/9707337} (BLR-II)
Best P. N., Longair M. S., Röttgering H. J. A., 1998, MNRAS, 295, 549 {astro-ph/9709195} (BLR-III)
Binette L., Robinson A., Courvoisier T. J.-L., 1988, A&A, 194, 65
Bithell M., Rees M. J., 1990, MNRAS, 242, 570
Blandford R. D., Königl A., 1979, ApJ, 232, 34
Boas, M. L. [ *Mathematical Methods in the Physical Sciences*]{}. John Wiley & Sons, New York, second edition, 1983.
Brindle C., Hough J. H., Bailey J. A., Axon D. J., Hyland A. R., 1986, MNRAS, 221, 739
Brown R. L., McLean I. S., 1977, A&A, 37, 141
Bruzual G., 1983, ApJ, 273, 105
Bruzual G., Charlot S., 1993, ApJ, 405, 538
Burstein D., Heiles C., 1982, AJ, 87, 1165
Carilli C. L., Röttgering H. J. A., van Ojik R., Miley G. K., van Breugel W. J. M., 1997, ApJS, 109, 1
Chambers K. C., Miley G. K., van Breugel W. J. M., 1987, [*Nature*]{}, 329, 604
Chambers K. C., Miley G. K., Joyce R. R., 1988, ApJ, 329, L75
Chambers K. C., Charlot S., 1990, ApJ, 348, L1
Chrysostomou A., 1996, Imaging Polarimetry with IRCAM3, Joint Astronomy Centre, Hilo, Hawaii; available on the World Wide Web at [url]{}\
`http://www.jach.hawaii.edu/UKIRT.new/instruments/irpol/IRCAM/ircampol.html`
Cimatti A., di Serego Alighieri S., 1995, MNRAS, 273, L7
Cimatti A., di Serego Alighieri S., Fosbury R. A. E., Salvati M., Taylor D. N., 1993, MNRAS, 264, 421
Cimatti A., di Serego Alighieri S., Field G. B., Fosbury R. A. E., 1994, ApJ, 422, 562
Cimatti A., Dey A., van Breugel W. J. M., Antonucci R., Spinrad H., 1996, ApJ, 465, 145
Cimatti A., Dey A., van Breugel W. J. M., Hurt T., Antonucci R., 1997, ApJ, 476, 677
Cimatti A., di Serego Alighieri S., Vernet J., Cohen M., Fosbury R. A. E., 1998, ApJ, 499, L21
Clarke D., Naghizadeh-Khouei J. 1994, AJ, 108, 687
Clarke D., Stewart, B. G., 1986, [*Vistas in Astronomy*]{}, 29, 27
Clarke D., Stewart B. G., Schwarz H. E., Brooks A., 1983, A&A, 126, 260 [^7]
Clarke D., Naghizadeh-Khouei J., Simmons J. F. L., Stewart B. G., 1993, A&A, 269, 617
Clarke, G. M. and Cooke, D. [*A Basic Course in Statistics*]{}. Edward Arnold, second edition, 1983.
Cohen M. H. et al. (9 co-authors), 1977, [*Nature*]{}, 268, 405
Cohen M. H., Vermeulen R. C., Ogle P. M., Tran H. D., Goodrich R. W., 1997, ApJ, 484, 193
Coyne G. V., Gehrels T., Serkowski K., 1974, AJ, 79, 581
Crane P., Vernet J., 1997, A&AS, 190, 391
Cristiani S., Vio R., 1990, A&A, 227, 385
Daly R. A., 1990, ApJ, 355, 416
Daly R. A., 1992, ApJ, 399, 426
Dey A., Cimatti A., van Breugel W. J. M., Antonucci R., Spinrad H., 1996, ApJ, 465, 157
Dey A., van Breugel W. J. M., Vacca W. D., Antonucci R., 1997, ApJ, 490, 698
Dickinson M., 1996, private communication
Dickson R., Tadhunter C., Shaw M., Clark N., Morganti R., 1995, MNRAS, 273, 29P
di Serego Alighieri S., Binette L., Courvoisier T., Fosbury R. A. E., Tadhunter C. N., 1988, [*Nature*]{}, 334, 591
di Serego Alighieri S., Fosbury R. A. E., Quinn P. J., Tadhunter C. N., 1989, [*Nature*]{}, 341, 307
di Serego Alighieri S., Cimatti A., Fosbury, R. A. E., 1993, ApJ, 404, 584 [^8]
di Serego Alighieri S., Cimatti A., Fosbury, R. A. E., 1994, ApJ, 431, 123
Dolan J. F., Tapia S., 1986, PASP, 98, 792
Dolan J. F. et al. (11 co-authors), 1994, ApJ, 432, 560
Dunlop J. S., Peacock J. A., 1993, MNRAS, 263, 936
Dunlop J., Peacock J., Spinrad H., Dey A., Jimenez R., Stern D., Windhorst R., 1996, [*Nature*]{}, 381, 581
Eales S. A., 1992, ApJ, 397, 49
Eales S. A., Rawlings S., 1990, MNRAS, 243, 1P
Eales S. A., Rawlings S., 1996, ApJ, 460, 68
Eales S. A., Rawlings S., Law-Green D., Cotter G., Lacy M., 1997, MNRAS, 291, 593
Economou F., Lawrence A., Ward M. J., Blanco P. R., 1995, MNRAS, 272, L5
Eggen O. J., Lynden-Bell D., Sandage A. R., 1962, ApJ, 136, 748
Eisenhardt P., Chokshi A., 1990, ApJ, 351, L9
Elston R., Jannuzi B., 1999, ApJL, in preparation
Fabian A. C., 1989, MNRAS, 238, 41P
Fanaroff B. L., Riley J. M., 1974, MNRAS, 167, 31P
Fernini I., Burns J. O., Bridle A. H., Perley R. A., 1993, AJ, 105, 1690
Fröberg C.-E., [*Numerical Mathematics*]{}. Addison-Wesley, 1985.
Fugmann W., Meisenheimer K., 1988, A&AS, 76, 145
Gehrels T., 1960, AJ, 65, 470
Gehrels T. (ed.), [*Planets, Stars and Nebulae studied with Photopolarimetry*]{}, chapter on Polarimetric Definitions (by D. Clarke), pages 45–53. The University of Arizona Press, 1974.
Ghisellini G., Padovani P., Celotti A., Maraschi L., 1993, ApJ, 407, 65
Goodrich R. W., Cohen M. H., 1992, ApJ, 391, 623
Goodrich R. W., Miller J. S., Martel A., Cohen M. H., Tran H. D., Ogle P. M., Vermeulen R. A., 1996, ApJ, 456, L9
Grandi P., Sambruna R. M., Maraschi L., Matt G., Urry C. M., Mushotzky R. F., 1997, ApJ, 487, 636
Gunn J. E., Hoessel J. G., Westphal J. A., Perryman M. A. C., Longair M. S., 1981, MNRAS, 194, 111
Hammer F., LeFèvre O. N., Angonin M. C., 1993, [*Nature*]{}, 362, 324
Hecht E., [*Optics*]{}. Addison-Wesley, second edition, 1987.
Hines D. C., Wills B. J., 1993, ApJ, 415, 82
Hoskin M. A., 1976, JHistA, 7, 169
Hough J., 1996, private communication
Hough J., Brindle C., Axon D., Bailey J., Sparks W., 1987, MNRAS, 224, 1013
Hsu J., Breger M., 1982, ApJ, 262, 732
Hurt T., Antonucci R., Cohen R., Kinney A., Krolik J., 1999, ApJ, 514, 579
Hutchings J. B., 1987, ApJ, 311, 526
Illingworth V. (ed.), [*Collins Dictionary of Astronomy*]{}, HarperCollins, Glasgow, 1994
Impey C. D., Malkan M. A., Tapia S., 1989, ApJ, 347, 96
Impey C. D., Lawrence C. R., Tapia S., 1991, ApJ, 375, 46
Jannuzi B., Elston R., 1991, ApJ, 366, L69
Jannuzi B., Green R., French H., 1993, ApJ, 404, 100
Jannuzi B., 1996, private communication
Jenkins C. J., Pooley G. G., Riley J. M., 1977, MemRAS, 84, 61
Killeen N. E. B., Bicknell G. V., Ekers R. D., 1986, ApJ, 302, 306
Kitchin C. R., [*Astrophysical Techniques*]{}. Hilger, Bristol, 1984.
Kollgaard R. I., Wardle J. F. C., Roberts D. H., Gabuzda D. C, 1992, AJ, 104, 1687
Koorneef J., 1983, A&A, 128, 84
Kormendy J., Richstone D., 1995, ARAA, 33, 581
Lacy M., Rawlings S., 1994, MNRAS, 270, 431
Lacy M., Rawlings S., Eales S. A., Dunlop J. S., 1995, MNRAS, 273, 821
Lacy M., Rawlings S., Blundell K. M., Ridgway S. K., 1998, MNRAS, 298, 966 {astro-ph/9803017}
Lacy M., Kaiser M. E., Hill G. J., Rawlings S., Leyshon G., 1999a, MNRAS, {astro-ph/9905357}
Lacy M., Ridgway S. E., Wold M., Lilje P. B., Rawlings S., 1999b, MNRAS, {astro-ph/9903314}
Laing R. A., Riley J. M., Longair M. S., 1983, MNRAS, 204, 151
Lasker B. M., Sturch C. R., McLean B. J., Russell J. L., Jenkner H., Shara M. M., 1990, AJ, 99, 2019
Lawrence A. et al. (16 co-authors), 1993, MNRAS, 260, 28
Leahy J. P., Muxlow T. W. B., Stephens P. W., 1989, MNRAS, 239, 401
LeFèvre O., Hammer F., Nottale L., Mazure A., Christian C., 1988a, ApJ, 324, L1
LeFèvre O., Hammer F., Jones J., 1988b, ApJ, 331, L73
Leyshon G., 1998, [*Experimental Astronomy*]{}, 1998, 2, 153 {astro-ph/9709164}
Leyshon G., Eales S. A., 1998, MNRAS, 295, 10 {astro-ph/9708085} (L&E)
Leyshon G., Dunlop J. S., Eales S. A., 1999, MNRAS {astro-ph/9905282}
Lilly S. J., Longair M. S., 1984, MNRAS, 211, 833
Longair M. S., 1975, MNRAS, 173, 309
Longair M. S., Best P. N., Röttgering H. J. A., 1995, MNRAS, 275, 47P
Manzini A., di Serego Alighieri S., 1996, A&A, 311, 79 (MdSA)
Maronna R., Feinstein C., Clocchiatti, A., 1992, A&A, 260, 525
Martin P. G., Whittet D. C. B., 1990, ApJ, 357, 113
Mathewson D., Ford V., 1970, MemRAS, 74, 139
McCarthy P. J., 1988, PhD thesis, University of California at Berkeley
McCarthy P. J., 1993, ARAA, 31, 639
McCarthy P. J., van Breugel W. J. M., Spinrad H., Djorgovski S., 1987a, ApJ, 321, L29
McCarthy P. J., Spinrad H., Djorgovski S., Strauss M. A., van Breugel W. J. M., Liebert J., 1987b, ApJ, 319, L39
McCarthy P. J., Dickinson M., Filippenko A. V., Spinrad H., van Breugel W. J. M., 1988, ApJ, 328, L29
McCarthy P. J., Kapahi V. K., van Breugel W. J. M., Subrahmanya C. R., 1990, AJ, 100, 1014
McCarthy P. J., van Breugel W. J. M., Kapahi V. K., 1991, ApJ, 371, 478
McCarthy P. J., Persson S. E., West S. C., 1992, ApJ, 386, 52
McLean I. S., [*Electronic Imaging in Astronomy*]{}, Wiley, Chichester, 1997
Miley G. K., 1980, ARAA, 18, 165
Miller J. S., Goodrich R. W., 1990, ApJ, 355, 456
Mood A. M., Graybill F. A., Boes D. C., [*Introduction to the Theory of Statistics*]{}. McGraw-Hill, third edition, 1974.
Moore R. L., Stockman H. S., 1984, ApJ, 279, 465
Naghizadeh–Khouei J., Clarke D., 1993, A&A, 274, 968
Naim A., Ratnatunga K. U., Griffiths R. E., 1997, ApJ, 476, 510
Neeser M. J., 1996, PhD thesis, Ruprecht-Karls-Universität, Heidelberg
NOAO. [*On-Line Documentation for IRAF:*]{} [apphot.phot.]{} On-line manual page for the [phot]{} command in the [digiphot.apphot]{} package of the IRAF data reduction system. Software support website: [iraf.noao.edu]{}.
Packham C., Hough J. H., Young S., Chrysostomou A., Bailey J. A., Axon D. J., Ward M. J., 1996, MNRAS, 278, 406
Packham C., Young S., Hough J. H., Axon D. J., Bailey J. A., 1997, MNRAS, 288, 375
Padovani P., Urry M. C., 1992, ApJ, 387, 449
Parsons R., [ *Statistical Analysis: A Decision-Making Approach*]{}. Harper & Row, New York, 1974.
Pentericci L., Röttgering H. J. A., Miley G. K., McCarthy P., Spinrad H., van Breugel W. J. M., Macchetto P., 1998, A&A {astro-ph/9809056v2}
Peterson B. M., [*An Introduction to Active Galactic Nuclei*]{}, Cambridge University Press, 1997
Prestage R. M., Peacock J. A., 1988, MNRAS, 230, 131
Rawlings S., Lacy, M., Eales S. A., 1991, MNRAS, 251, 17P
Rawlings S., Lacy, M., Sivia D. S., Eales S. A., 1995, MNRAS, 274, 428
Rees M. J., 1966, [*Nature*]{}, 211, 468
Ridgway S. K., Stockton A. N., 1997, AJ, 114, 511
Rieke G. H., Lebofsky M. J., 1985, ApJ, 288, 618
Rigler M. A., Lilly S. J., 1994, ApJ, 427, L79
Rigler M. A., Lilly S. J., Stockton A., Hammer F., LeFèvre O. N.,1992, ApJ, 385, 61
Riley J. M., Longair M. S., Gunn J. E., 1980, MNRAS 192, 233
Robson I., [*Active Galactic Nuclei*]{}, Wiley, Chichester, 1996
Roche N., Eales S. A., 1999, MNRAS [*in preparation*]{}
Rudy R. J., Schmidt G. D., Stockman H. S., Moore R. L., 1983, ApJ, 271, 59
Sánchez Almeida J., 1995, A&ASS, 109, 417
Sanders D. B., Soifer B. T., Elias J. H., Madore B. F., Matthews K., Neugebauer G., Scoville N. Z., 1988, ApJ, 325, 74
Sansom A. E. et al. (8 co-authors), 1987, MNRAS, 229, 15
Savage B. D., Mathis J. S., 1979, ARAA, 17, 73
Scarpa R., Falomo R., 1997, A&A, 325, 109
Scarrott S. M., Warren-Smith R. F., Pallister W. S., Axon D. J., Bingham R. G., MNRAS, 1983, 204, 1163
Scarrott S. M., Rolph C.D., Tadhunter C.D., 1990, MNRAS, 243, 5P
Scarrott S. M., Rolph C.D., Wolstencroft R. D., Walker H. J., Sekiguchi K., 1990, MNRAS, 245, 484
Schilizzi R., Kapahi V. K., Neff S., 1982, JAp&A, 3, 173
Schmidt–Kaler T., 1958, Zeitschrift für Astrofisik, 46, 145
Serjeant S., Rawlings S., Maddox S. J., Baker J. C., Clements D., Lacy M., Lilje P. B., 1998, MNRAS, 292, 494
Serkowski K., 1958, [*Acta Astronomica*]{}, 8, 135
Serkowski K., Mathewson D., Ford V., 1975, ApJ, 196, 261
Shaw M., Tadhunter C. N., Dickson R., Morganti R., 1995, MNRAS, 275, 703
Simmons J. F. L., Stewart B. G., 1985, A&A, 142, 100
Smith P. S., Balonek T. J., Heckart P. A., Elston R., 1986, ApJ, 305, 484
Spinrad H., 1982, PASP, 94, 397
Spinrad H., Djorgovski S. G., 1984a, ApJ, 280, L9
Spinrad H., Djorgovski S. G., 1984b, ApJ, 285, L49
Spinrad H., Dey A., Stern D., Dunlop J., Peacock J., Jimenez R., Windhorst R.,1997, ApJ, 484, 581
Sterken Chr., Manfroid J., [*Astronomical Photometry: A Guide*]{}. Kluwer Academic, 1992.
Stockman H. S., Angel J. R. P., Miley G. K., 1979, ApJ, 227, L55
Stockman H. S., Moore R. L., Angel J. R. P., 1984, ApJ, 279, 485
Stockton A., Kellogg M., Ridgway S. E., 1995, ApJ, 443, L69
Strom R. G., Riley J. M., Spinrad H., van Breugel, W. J. M., Djorgovski S., Liebert J., McCarthy P. J., 1990, A&A, 227, 19
Tadhunter C. N., Scarrott S. M., Draper P., Rolph, C., 1992, MNRAS, 256, 53P
Takalo L. O., Kidger M. R., de Diego J. A., Sillanpää A., 1992, A&A, 261, 415
Tinbergen J., [*Astronomical Polarimetry*]{}. Cambridge University Press, 1996.
Tran H. D., Cohen M. H., Goodrich R. W., 1995, AJ, 110, 2597
Tran H. D., Cohen M. H., Ogle P. M., Goodrich R. W., di Serego Alighieri S., 1998, ApJ, 500, 660
Urry M. C., Padovani P., 1995, PASP, 107, 803
Vinokur M., 1965, [*Annales d’Astrophysique*]{}, 28, 412
Walker G., [ *Astronomical Observations*]{}. Cambridge University Press, 1987.
Walsh D., Lebofsky M. J., Rieke G. H., Shone D., Elston R., 1985, MNRAS, 212, 631
Wardle J. F. C., Kronberg P. P., 1974, ApJ, 194, 249 [^9]
Webb W., Malkan M., Schmidt G., Impey C., 1993, ApJ, 419, 494
West M. J., 1992, MNRAS, 268, 79
Whittet D. C. B., Martin P. G., Hough J. H., Rouse M. F., Bailey J. A., Axon D. J., 1992, ApJ, 386, 562
Wieringa M. H., Katgert P, 1992, A&AS, 93, 399
Wilking B. A., Lebofsky M. J., Martin P. G., Rieke G. H., Kemp J. C., 1980, ApJ, 235, 905
Willott C. J., Rawlings S., Blundell K. M., Lacy M., 1998, MNRAS, 300, 625
Willott C. J., Rawlings S., Blundell K. M., Lacy M., 1999, MNRAS, {astro-ph/9905388}
Young S., Packham C., Hough J. H., Efstathiou A., 1996, MNRAS, 283, L1
Zirbel E. L., 1997, ApJ, 476, 489
Minimum Theoretical Errors in Stokes Parameters {#photapp}
===============================================
\[theorem\][Approximation]{}
The recent work of Sánchez Almeida , and of Maronna, Feinstein & Clocchiatti , considers the ideal case of polarimetry limited only by the shot noise intrinsic to quantized light. For completeness their results are presented here in the notation of this thesis, and extended slightly in the case of normalised Stokes Parameters. (See Chapter \[stoch\] for definitions of the Stokes Parameters.) This treatment also considers which estimators are optimal for estimating Stokes Parameters, and the consequences of binning the photon counts.
Light Intensity as a Poissonian Quantity
----------------------------------------
Consider a quasimonochromatic beam of light of intensity $I$, where the units of $I$ are photons per second. In a time interval $\tau$, the number of photons expected to arrive is $\lambda = I\tau$. As bosons, there will be some correlation between the arrival of individual photons, but this effect is negligible at optical wavelengths, and we can assume that the arrival of photons can be characterised by a Poisson distribution [@Walker-87a Ch. 2]. The number of photons actually arriving is hence a Poissonian random variable $X$, such that $$\label{poissdef}
P(X=x) = \frac{e^{-\lambda} \lambda^{x}}{x!}.$$ In the remainder of this appendix we will speak of such a Poisson distribution as having an intensity $I$, indicating that the mean of the distribution is $\lambda = I\tau$ for some arbitrary integration time $\tau$.
### Combining beams
Combining two beams of light of intensity $I_1$ and $I_2$ produces a Poissonian beam of intensity $I = I_1 + I_2$.
If the beams yield $\lambda_1\tau$ and $\lambda_2\tau$ photons respectively in the obvious notation, then $$\label{addbeam}
P(X=x) = \sum_{i=0}^{x} P(X_1=i).P(X_2=x-i) =
\sum_{i=0}^{x} \frac{e^{-\lambda_1}
{\lambda_1}^{i}}{i!}.\frac{e^{-\lambda_2} {\lambda_2}^{x-i}}{(x-i)!}$$ and factorizing out the exponential term, $$\label{addbeamb}
P(X=x) = e^{-(\lambda_1+\lambda_2)}
\sum_{i=0}^{x}\frac{{\lambda_1}^{i}{\lambda_2}^{x-i}}{i!(x-i)!}.$$
Anticipating the result, we substitute $\lambda=\lambda_1+\lambda_2$ into Equation \[poissdef\] and use the binomial expansion to obtain $$\label{poissex}
P(X=x) = \frac{e^{-(\lambda_1+\lambda_2)}}{x!} (\lambda_1+\lambda_2)^{x} =
\frac{e^{-(\lambda_1+\lambda_2)}}{x!}
\sum_{j=0}^{x} {\lambda_1}^{j}{\lambda_2}^{x-j} \frac{x!}{j!(x-j)!}.$$
Since the factors of $x!$ cancel and the indices $i$ and $j$ are summed over, then Equation \[addbeamb\] is identical to Equation \[poissex\] and the combined beams produce a Poisson distribution of mean intensity $I_1 + I_2$. [*QED.*]{}
[*It follows that when Poissonian light from two sources is combined – e.g. light from a pair of stars, or from a host galaxy and an active nucleus – the resultant beam is also a Poissonian.*]{}
### Attenuating beams
\[fracbeam\] Passing light of intensity $I$ through a filter which passes a fraction $f$ of the photons results in a Poissonian beam of intensity $fI$.
The number of photons arriving at the filter follows a Poisson distribution for intensity $I$, so $X$ photons arrive at the filter. There is a binomial distribution such that $W$ photons penetrate the filter given that $X$ arrive, where $$\label{binpen}
P(W=w\,|\,X) = \frac{X!}{w!(X-w)!} f^w (1-f)^{(X-w)}.$$ Overall the probability that $w$ arrive and penetrate is given by $$\label{binover}
P(W=w) = \sum_{x=w}^{\infty} P(X=x).\frac{x!}{w!(x-w)!} f^w (1-f)^{(x-w)}.$$
Making the substitution $k=x-w$ and expanding $P(X=x)$ from Equation \[poissdef\], we obtain $$\label{binoverb}
P(W=w) = \sum_{k=0}^{\infty}
\frac{e^{-\lambda}\lambda^{k+w}}{x!}.\frac{x!}{w!k!} f^w
(1-f)^k$$ and the $x!$ terms cancel. We take out the terms in $w$ and obtain $$\label{binoverc}
P(W=w) = \frac{(f\lambda)^w}{w!}.\left[e^{-\lambda} \sum_{k=0}^{\infty}
\frac{\lambda^k}{k!}(1-f)^k\right].$$
To complete the proof we must show that the term in square brackets is equivalent to $e^{-f\lambda}$. So, let the term in square brackets be denoted $G$. Expressing $e^{-\lambda}$ as a series, we have $$\label{binoverd}
G = \sum_{j=0}^{\infty} \frac{(-\lambda)^j}{j!} \sum_{k=0}^{\infty}
\frac{\lambda^k}{k!}(1-f)^k.$$ We can express the double series as a single series if we group together terms of the same power of $\lambda$: $$\label{binoverg}
G = \sum_{i=0}^{\infty} \lambda^i \left[
\sum_{j=0}^{i} \frac{(-1)^j}{j!}\frac{(1-f)^{i-j}}{(i-j)!}\right].$$ We now let $h=1-f$, and recast this as $$\label{binoverh}
G = \sum_{i=0}^{\infty} \lambda^i \left[
\sum_{j=0}^{i} \frac{(-1)^j}{j!}\frac{h^{i-j}}{(i-j)!}\right],$$ where we recognise the square bracket as $1/i!$ times the binomial expansion $$\label{binexh}
(h-1)^i = \sum_{j=0}^{i} \frac{(-1)^j h^{i-j} i!}{j!(i-j)!}.$$ Thus Equation \[binoverg\] can be simplified to $$\label{proof2}
G = \sum_{i=0}^{\infty} \frac{\lambda^i (h-1)^i}{i!} = \sum_{i=0}^{\infty}
\frac{\lambda^i (-f)^i}{i!} = \sum_{i=0}^{\infty} \frac{(-f\lambda)^i}{i!}.$$ Thus $G$ is shown to be the series expansion of $e^{-f\lambda}$ and so Equation \[binoverc\] is the Poisson probability for a distribution of mean $f\lambda$. This completes the proof.
[*Any optical filter which passes a fraction $f$ of the incident light in practice passes a fraction $f$ of the incident photons and removes the remainder. Theorem \[fracbeam\] shows that any filters employed in an astronomical experiment will not affect the Poissonian properties of a quasimonochromatic beam. (Naturally the wavelength dependence of the filter would change the spectrum of a polychromatic beam.) It also follows that a photon detector of quantum efficiency $f$ also produces a Poissonian output.*]{}
### Binning Poisson distributions {#binsec}
The work in the following sections is based on the ideal case of a detector which records photon counts limited only by the shot noise intrinsic to photons. In such a case, the arrival of detections is Poissonian. In practice, however, real astronomical detectors [@McLean-97a] first allow incoming photons to excite electrons which can be trapped, and then amplify and digitize the voltage due to these electrons.
Some incoming photons will fail to excite electrons, because the system will never have 100 per cent quantum efficiency; but Theorem \[fracbeam\] above shows that if the success or failure of a photon to do so is random (i.e. does not depend on the photon’s energy) then the photons which succeed in exciting electrons will also follow a Poisson distribution.
If we assume an idealized system where a fraction $f$ of the photons excite exactly one electron and the remainder go undetected, the next source of error is quantization error. The analogue-to-digital converter of the detector will measure the number of electrons with a conversion factor of $\Delta$ electrons per data number (DN). It can be shown [@Scarrott+83a; @McLean-97a] that quantization contributes a noise of $0.289\Delta$. More importantly, the output in DN no longer follows a Poisson distribution: the distribution has now been binned in units of width $\Delta$.
It is possible to give an exact formula for a DN distribution simply by adding up the Poisson probabilities for each number of photons which would yield a given quantized output: If $D$ is the random variable ‘count in DN units’ then (assuming $\Delta$ is an integer) we have $$\label{D_prob}
P(D=d) = \sum_{k=d\Delta}^{d\Delta+\Delta-1} e^{-\lambda}\lambda^{k}/k!,$$ and the mean of the distribution is hence $$\label{D_mean}
E(D) = \sum_{d=0}^{\infty} d.e^{-\lambda} \sum_{k=d\Delta}^{d\Delta+\Delta-1}
\lambda^{k}/k!.$$ There is no obvious analytic simplification of $E(D)$, but clearly binning the Poisson photon distribution in bins of width $\Delta$, since the photon distribution has $E(X) = \lambda,$ ${\mathrm{SD}}(X) =
\sqrt{\lambda}$, then the DN readout must have a mean of approximately $E(D) \simeq \lambda/\Delta$ and ${\mathrm{SD}}(D) \simeq (\sqrt{\lambda})/\Delta$.
The square root of $E(D)$ is hence $\sqrt{\lambda/\Delta}$, which is equivalent to $\sqrt{\Delta}$ times ${\mathrm{SD}}(D)$. Thus if a count, $d$, has been measured in DN units, it follows that ${\mathrm{SD}}(D) =
\sqrt{E(D)/\Delta}$ or equivalently ${{\mathrm{SD}}({D})}^2 =
E(D)/\Delta$. This approximation will be good for typical bin sizes ($\Delta \sim 6$) as long as the integration time is such that $\lambda > \Delta$. (This was verified empirically on a spreadsheet.)
We will not consider rigorously here the case of a detector where an incident photon is likely to excite more than one electron. Again, however, if the number of incident photons is significant over the integration time, it will be possible to define an overall gain $\Delta$ encompassing the photon-to-electron and analogue-to-digital conversions, and the crucial ${{\mathrm{SD}}(D)}^2 = E(D)/\Delta$ relationship will be retained to a first approximation.
Finally, in a realistic application (this thesis, §\[realnoise\]), the units of choice will often be ‘DN per unit time’. In this case, the system’s output will be a count rate of $O_D = E(D) / \tau$ with shot noise of $\sigma_{\mathsf{shot}} = {{\mathrm{SD}}(D)} / \tau$, whence the relationship between noise and signal becomes $$\label{shotSN}
\sigma_{\mathsf{shot}}^2 = [SD(D)]^2/\tau^2 = E(D)/\tau^2\Delta =
O_D/\tau \Delta.$$
Noise in a Generalised Polarimeter
----------------------------------
### Absolute Stokes Parameters
Sánchez Almeida considers the most general case of a polarimeter which splits the light from a source into $m$ different optical trains, which each produce a photo-count $n_i$. At least four distinct optical trains are needed to determine all four Stokes Parameters but this treatment also applies to systems with $m<4$ which can only determine $m$ Stokes Parameters.
We presume that this general polarimeter produces the Stokes Parameters $I, Q, U, V$ (or a subset if $m<4$) with their respective errors $\sigma_I, \sigma_Q, \sigma_U, \sigma_V$. (Sánchez Almeida takes these Stokes Parameters to be actual numbers of photons, but the treatment remains valid when normalized for unit time.) Using $S$ to denote any of $Q, U, V$, and where $\cal N$ is the total number of photons received summed over all $m$ optical trains, Sánchez Almeida proves the following (his Equation numbers denoted SA):
\[ithm\] The signal-to-noise on the intensity cannot be better than $\sqrt{\cal
N}$ : $$\label{sa_inoise}
\sigma_{I} \geq I / \sqrt{\cal N}.$$
It is possible to build ‘polarizers of minimum $I$ error’ [@Almeida-95a § 4.2] which have $\sigma_I =I / \sqrt{\cal N}$.
\[sthm\] The signal-to-noise on the other Stokes Parameters cannot be better than $\sqrt{\cal N}$: $$\label{sa_snoise}
\sigma_S \geq {{|S|}} / \sqrt{\cal N}.$$
[**For unpolarized light,**]{} the noise on the other Stokes Parameters is limited by the intensity, and cannot be better than $I/\sqrt{\cal N}$: $$\label{sa_unoise}
\forall S=0: \sigma_S \geq I / \sqrt{\cal N};$$ this does not necessarily hold true for polarised light.
[**For polarizers with $m=4$ and polarizers of minimum $I$ error,**]{} the errors on the Stokes Parameters are correlated such that $$\label{sa_anoise}
{\sigma_Q}^2 + {\sigma_U}^2 + {\sigma_V}^2 \geq {\sigma_I}^2 \geq I^2/
{\cal N}.$$
### The case of binned absolute Stokes Parameters
Sánchez Almeida’s logic can also be applied to Stokes Parameters expressed in DN units from a real detector, if the Poissonian substitution $n_i
\rightarrow {\sigma_i}^2$ is replaced by the binned substitution $n_i \rightarrow \Delta {\sigma_i}^2$.
Whence for Stokes Parameters $I_{D}$, $S_{D}$ and total count ${\cal N}_{D}$ expressed in DN units with $\Delta$ photons per DN:
\[DNithm\] The signal-to-noise on the intensity is restricted to: $$\label{DNsa_inoise}
\sigma_{I_{D}} \geq I_{D} / \sqrt{\Delta {\cal N}_{D}}.$$
\[DNsthm\] The signal-to-noise on the other Stokes Parameters cannot be better than $\sqrt{\cal N}$: $$\label{DNsa_snoise}
\sigma_{S_{D}} \geq {{|S_{D}|}} / \sqrt{\Delta{\cal N}_{D}}.$$
\[DNsa\_last\] [**For unpolarized light,**]{} the noise on the other Stokes Parameters is limited by the intensity: $$\label{DNsa_unoise}
\forall S_{D}=0: \sigma_{S_{D}} \geq I_{D} / \sqrt{\Delta {\cal N}_{D}};$$ this does not necessarily hold true for polarised light.
[*It follows that Theorems \[ithm\] to \[DNsa\_last\] allow us to estimate, [*a priori*]{}, the minimum errors obtainable when absolute Stokes Parameters are measured for an object of known magnitude, both for detectors registering raw photon counts and for the binned case.*]{}
### Extension to normalised Stokes Parameters
Sánchez Almeida’s method can be extended to give the minimum possible error on a normalised Stokes Parameter. He defines a calibration matrix $M_{ji}$ such that the measured Stokes Parameters (including $I=S_1$) are $$\label{meas_matx}
S_j = \sum_{i=1}^{m} M_{ji} n_i;$$ we could include division by the exposure time in the matrix $M_{ji}$ if we wish. We see that in the most general case the $j$th Stokes Parameter $S_j$ could depend on all $m$ optical trains, and hence the error on $S_j$ could depend on errors on all the $n_i$.
Now consider a normalized Stokes Parameter $s_j = S_j/I = S_j/S_1$. By the rule of adding errors in quadrature, the noise on $s_j$ must be given by $$\label{nsp_noise}
{\sigma_{s_j}}^2 = \sum_{k=1}^{m} \left( \frac{\partial s_j}{\partial n_k}
\right)^2 {\sigma_{n_k}}^2.$$ But because each photon-count is assumed to be affected by independent Poissonian noise, ${\sigma_{n_k}}^2 = n_k$ and so $$\label{nsp_noiser}
{\sigma_{s_j}}^2 = \sum_{k=1}^{m} \left( \frac{\partial s_j}{\partial n_k}
\right)^2 n_k.$$ Using $s_j = S_j/I$, this becomes $$\label{nsp_noises}
{\sigma_{s_j}}^2 = \sum_{k=1}^{m}
\left( \frac{\partial S_j}{\partial n_k} - s_j\frac{\partial S_1}{\partial
n_k} \right)^2 \frac{n_k}{I^2}.$$ Substituting the matrix form Equation \[meas\_matx\] into the partial derivatives yields $$\label{nsp_noisep}
{\sigma_{s_j}}^2 = \sum_{k=1}^{m}
(M_{jk} - s_j M_{1k})^2 {n_k}/{I^2}.$$ Expanding the brackets gives $$\label{nsp_noisee}
{\sigma_{s_j}}^2 = I^{-2} \sum_{k=1}^{m}
{M_{jk}}^2 n_k + {s_j}^2 n_k {M_{1k}}^2 - 2s_j n_k M_{1k} M_{jk}.$$ Sánchez Almeida shows (Equation SA 9a) that $$\label{sasum}
{\sigma_{S_j}}^2 = \sum_{i=1}^{m} {M_{ji}}^2 n_i$$ which allows us to substitute terms in Equation \[nsp\_noisee\] yielding $$\label{nsp_noisex}
{\sigma_{s_j}}^2 = I^{-2} \left[
{\sigma_{S_j}}^2 + {s_j}^2 {\sigma_{I}}^2 - 2s_j {\sigma_{\times}}^2
\right]$$ where we define the (not necessarily positive) quantity $$\label{xdef}
{\sigma_{\times}}^2 = \sum_{i=1}^{m}
n_i M_{1i} M_{ji}.$$
We already know the limits on $\sigma_I$ and $\sigma_{S_j}$ from Theorems \[ithm\] and \[sthm\]; to obtain a limit on $\sigma_{\times}$ we follow Sánchez Almeida’s use of the Cauchy-Schwarz inequality [@Froberg-85a §2.1] for series. Consider $$\label{cs_pow4}
{\sigma_{\times}}^4 =
\left[ \sum_{i=1}^{m} M_{ji}M_{1i}n_i \right]^2 \leq
\left[ \sum_{i=1}^{m} {M_{ji}}^2 n_i \right]
\left[ \sum_{k=1}^{m} {M_{1k}}^2 n_k \right]$$ by the Cauchy-Schwarz inequality. But the two bracketed terms on the right, by Equation \[sasum\], are errors on $I$ and $S_j$, whence $$\label{cs_ineq}
{\sigma_{\times}}^4 \leq {\sigma_{I}}^2 {\sigma_{S_j}}^2.$$ Taking square roots and not assuming the positive root, $$\label{cs_ineqr}
{{|{\sigma_{\times}}^2|}} \leq {\sigma_{I}} {\sigma_{S_j}}.$$ Returning to Equation \[nsp\_noisex\], we see that the first two terms in the square bracket must be positive, and the third term is in the range $\pm 2 s_j {\sigma_{I}} {\sigma_{S_j}}$. Recognising that both signs enable the bracket to be written as a square, and noting that $s_j$ is itself a signed quantity, we obtain:
\[gathm\] The error on the determination of a normalised Stokes Parameter $s_j$ satisfies: $$\label{cs_lims}
(\sigma_{S_j} - {{|s_j|}} \sigma_{I})^2 \leq I^2 {\sigma_{s_j}}^2
\leq (\sigma_{S_j} + {{|s_j|}} \sigma_{I})^2.$$
Now this theorem is not particularly useful for the general case; we can rewrite the lower bound as $$\label{cs_lowl} \sigma_{s_j} \geq \left| {\frac{\sigma_{S_j}}{I} - \frac{{{|s_j|}}
\sigma_{I}}{I}}\right| ,$$ and rearranging Theorems \[ithm\] and \[sthm\] (with ${{|s_j|}}
= {{|S_j|}}/I$) compare with $\sigma_I/I \geq 1/\sqrt{\cal N}$ and $\sigma_{S_j}/{{|s_j|}}I \geq
1/\sqrt{\cal N}$. It becomes apparent that both terms in the difference must be greater than or equal to $1/\sqrt{\cal N}$. This merely tells us that the error on the normalized Stokes Parameter must be non-negative; hardly a surprising result. (In the case of binned Stokes Parameters the same result is obtained, since the $\Delta$ terms cancel out by the time Equation \[cs\_lims\] is obtained.) But this result is presented here because Equation \[cs\_lims\] produces a useful result in the special case when $\sigma_I = \sigma_{S_j}$ (see Theorem \[gathmeq\] below).
Noise in a Two-Channel Polarimeter
----------------------------------
While Sánchez Almeida treated the general case of a polarimeter with an arbitrary number of optical trains, which could combine the data in every train to estimate $I$, Maronna, Feinstein & Clocchiatti consider the case of a two-channel polarimeter simultaneously measuring $I$ and [**one other**]{} Stokes Parameter $S$ to obtain $s=S/I$. They produce a number of theorems (denoted here by MFC) deduced by assuming that the polarised light arrives at the detector according to a Poissonian distribution.
### Optimal estimation {#optest}
As in Chapter \[stoch\], we must distinguish between the true values of the Stokes Parameters for a source, and the values which we measure in the presence of noise. We will use the subscript $0$ to denote the underlying values, and the subscript $i$ for individual measured values. We assume that $\nu_S$ individual sets of photon-count measurements have been made.
Consider a general normalized Stokes Parameter: $$s_i = \frac{S_i}{I_i} =
\frac{n_{1i} - n_{2i}}{n_{1i} + n_{2i}}.$$ Clarke et al. point out that the signal/noise ratio obtained by calculating $$\label{stilde}
\tilde{s} = \frac{\bar{S}}{\bar{I}} =
\frac{\sum_{i=1}^{\nu_S} S_i}{\sum_{i=1}^{\nu_S} I_i}$$ is much better than that obtained by simply taking the mean, $$\label{sbar}
\bar{s} = \frac{1}{\nu_s} \sum_{i=1}^{\nu_S} s_i
= \frac{1}{\nu_s} \sum_{i=1}^{\nu_S} \frac{S_i}{I_i},$$ since the Equation \[stilde\] involves the taking of only one ratio, where the two terms $\bar{S}$ and $\bar{I}$ have better signal/noise ratios than the individual $S_i$ and $I_i$ which are ratioed in Equation \[sbar\]. Maronna, Feinstein & Clocchiatti prove the following results:
\[MFC1\] $\tilde{s}$ is the maximum likelihood estimator of $s_0$;
Both $\tilde{s}$ and $\bar{s}$ are unbiased estimators of $s_0$.
### The maximum likelihood estimator of binned data
Consider again the case of taking polarimetric measurements using a device which produces 1 DN count for every $\Delta$ incoming photons, and where the population means for the number of photons arriving in the two channels of our detector are $\lambda_1$ and $\lambda_2$ respectively. Ultimately our interest is in estimating the normalised Stokes Parameter characteristing that population, $s_0 = (\lambda_1 -
\lambda_2)/(\lambda_1 + \lambda_2)$. The proof of Theorem \[MFC1\] hinges on the fact that the Maximum Likelihood Estimator (MLE) of a function is given by applying the function to the MLEs of its parameters (the so-called [*substitution principle*]{} of MLEs): since $\bar{S}$ and $\bar{I}$ are shown to be the MLEs of $S_0=
(\lambda_1 - \lambda_2)/\tau$ and $I_0=\lambda/\tau=(\lambda_1 + \lambda_2)/\tau$, the proof follows.
The probability $P(D=d)$ is given by Equation \[D\_prob\] as a function of $\lambda$. The MLE of $D$ is obtained by maximizing $P(D=d)$ with respect to $\lambda$, whence $$\label{D_MLEcalc}
0 = \frac{\mathrm{d}}{\mathrm{d}\lambda} P(D=d) =
\sum_{k=d\Delta}^{d\Delta+\Delta-1}
\frac{1}{k!} [k.e^{-\lambda}\lambda^{k-1} - e^{-\lambda}\lambda^{k}]
= e^{-\lambda} \sum_{k=d\Delta}^{d\Delta+\Delta-1} \frac{\lambda^{k-1}.k}{k!} - \frac{\lambda^{k}}{k!}.$$ Defining $T_k = \lambda^{k-1}.k/k!$, it follows that $T_{k+1} =
\lambda^{k}.(k+1)/(k+1)! = \lambda^{k}/k!$, allowing us to cast Equation \[D\_MLEcalc\] as
$$0 = \sum_{k=d\Delta}^{d\Delta+\Delta-1} T_k - T_{k+1}.$$
All the terms in the power series cancel out apart from the first and last, and substituting the limits of the sum gives $0= T_{d\Delta} -
T_{(d+1)\Delta}$. Using the definition of $T_k$ and rearranging terms yields $$\label{simD_MLE}
[(d_{\mathsf ML}+1)\Delta]!d_{\mathsf ML}\Delta = (d_{\mathsf
ML}\Delta)!\lambda^{\Delta}(d_{\mathsf
ML}+1)\Delta$$ and hence $$\label{D_MLE}
(d_{\mathsf ML}\Delta -1)!\lambda^{\Delta} = [(d_{\mathsf ML}+1)\Delta -1]!.$$
Now we would like Equation \[D\_MLE\] to provide $d_{\mathsf ML}$ as a function of $\lambda$ and $\Delta$ to see how the MLE, $d_{\mathsf ML}$, compares to the intuitive approximation $\lambda/\Delta$. The factorials allow no obvious analytic solution, but useful upper and lower limits may be obtained as follows: Equation \[D\_MLE\] can be recast as $$\lambda^{\Delta} =
(d_{\mathsf ML}\Delta)(d_{\mathsf ML}\Delta+1)(d_{\mathsf
ML}\Delta+2) \cdots (d_{\mathsf ML}\Delta+\{\Delta-1\}),$$ where the right hand side is a product of $\Delta$ distinct terms, none smaller than $d_{\mathsf
ML}\Delta$ and none larger than $d_{\mathsf ML}\Delta+(\Delta-1)$. The RHS $(=
\lambda^{\Delta} )$ is hence clearly larger than $(d_{\mathsf ML}\Delta)^\Delta$ and smaller than $[d_{\mathsf ML}\Delta+(\Delta-1)] ^\Delta$, whence $d_{\mathsf ML}\Delta <
\lambda <
d_{\mathsf ML}\Delta + (\Delta-1)$. Rearranging the inequalities yields $$\label{dlims_MLE}
\frac{\lambda+1}{\Delta} - 1< d_{\mathsf ML} < \frac{\lambda}{\Delta}.$$
The MLE of $D$ is hence slightly smaller than the simplistic $\lambda/\Delta$: this is not unexpected as a few photons failing to fill the highest bin will not be measured, and the binned measurement will be biased to slightly underestimate the photon count. But the MLE will never be lower than $1-\frac{1}{\Delta} < 1$ DN units (i.e. $<$ 1 DN unit) below the simplistic estimate.
### Normalized Stokes Parameters under binning {#optbin}
We noted above that the MLE of $s_0$ is obtained by substituting the MLEs of $\lambda_1,\lambda_2$ into $s =
(\lambda_1 - \lambda_2)/(\lambda_1 +
\lambda_2)$. Now we know that MLE$(D) = \frac{\lambda}{\Delta} -
\epsilon$, where $\epsilon =
\frac{1}{2} . \left( 1-\frac{1}{\Delta} \right) \pm \frac{1}{2} . \left( 1-\frac{1}{\Delta} \right)$. If we assume the Absolute Stokes Parameters $\bar{I_D}$ and $\bar{S_D}$ have been measured in DN units, then MLE$(\tau \bar{I_D}) = (\lambda/\Delta) - 2\epsilon$ and MLE$(\tau \bar{S_D}) = (\lambda_1-\lambda_2)/\Delta$. Taking their ratio, $$\label{MLEofs} \mathrm{MLE} (s) =
\frac{\bar{S_D}}{\bar{I_D}-2\epsilon} \simeq
\frac{\bar{S_D}}{\bar{I_D}} . \frac{1}{\left(1-
\frac{1-1\Delta}{\bar{I_D}}\right)}.$$
Equation \[MLEofs\] is not an exact formula for the MLE of $s$ since $\epsilon$ is an approximation half way between the known limits. But it is clear that use of the formula $\tilde{s} = \bar{S_D}/\bar{I_D}$ will give us within a factor $1/(1-2\epsilon/\bar{I_D})$ of the MLE, and this error factor may easily be calculated.
### Minimum errors on the normalized parameters
Returning to the case where $I$ and $S$ are measured in photons rather than DN, we note that errors on $\bar{S}$ and on $\bar{I}$ are not independent of one another. We can write: $$\label{getstilde} \tilde{s} = \frac{\bar{n}_{1} - \bar{n}_{2}}{\bar{n}_{1} + \bar{n}_{2}}.$$
If we propagate through the errors on the intensities, we find: $$\label{getsterr}
\sigma_{\tilde{s}} =
\frac{1}{\bar{n}_{1}+\bar{n}_{2}}.\sqrt{[(1-\tilde{s})\sigma_{\bar{n}_{1}}]^2
+ [(1+\tilde{s})\sigma_{\bar{n}_{2}}]^2}.$$
\[gathmeq\] The error on a normalised Stokes Parameter $s_j$ determined with a two-channel polarimeter cannot be better than $(1-{{|s_j|}})/{\sqrt{\cal N}}.$
We can put a lower limit on the error on $s$ using a special case of Theorem \[gathm\] defined above. In this case where the system is a two channel polarimeter taking a sum and difference of counts, then the errors on $S$ and $I$ are identical: the two channels have independent errors, and so $$\label{equiverr}
\sigma_{\bar{I}} = \sigma_{\bar{S}} = \sqrt{{\sigma_{\bar{n_{1}}}}^2 +
{\sigma_{\bar{n_{2}}}}^2 }.$$ Equation \[cs\_lowl\] hence simplifies to $$\label{cs_lowleq}
\sigma_{s_j} \geq \frac{\sigma_{I}}{I} {{|1-{{|s_j|}}|}}.$$ The lower limit for $\sigma_{I}/I$ may be substituted from Theorem \[ithm\] and hence Theorem \[gathmeq\] above is proven. Maronna, Feinstein & Clocchiatti follow an alternative treatment, as follows:
The errors associated with the two estimators of $s_0$ satisfy $$\label{best_err}
{\sigma_{\tilde{s}}}^2 = \frac{1}{\nu_S I_0}(1-{s_0}^2)\left[ 1
+\frac{1}{\nu_S I_0} + \frac{\tilde{b}}{(\nu_S I_0)^2} \right]$$ and $$\label{bar_err}
{\sigma_{\bar{s}}}^2 = \frac{1}{\nu_S I_0}(1-{s_0}^2)\left[ 1
+\frac{1}{I_0} + \frac{\bar{b}}{{I_0}^2} \right]$$ where $\tilde{b}$ and $\bar{b}$ are non-negative constants dependent on $\nu_S$ and $I_0$.
Equation \[best\_err\] should be consistent with the lower limit set by Theorem \[gathmeq\]: squaring the latter, we have a lower limit $$\label{sigmin}
{\sigma_{\mathsf{min}}}^2 = \frac{1}{\cal N} (1-{{|s_0|}})^2.$$ Substituting ${\cal N} = \nu_S I_0$ and $\iota = \left[ 1
+\frac{1}{\nu_S I_0} + \frac{\tilde{b}}{(\nu_S I_0)^2} \right]$, we can write Equation \[best\_err\] as $$\label{tilde_subbed}
{\sigma_{\tilde{s}}}^2 = \frac{1}{\cal N} (1-{{|s_0|}})(1+{{|s_0|}})(1+\iota)
= {\sigma_{\mathsf{min}}}^2 \frac{1+{{|s_0|}}}{1-{{|s_0|}}}(1+\iota).$$ Since ${{|s_0|}}$ lies between 0 and 1, and $\iota>0$, then clearly $\frac{1+{{|s_0|}}}{1-{{|s_0|}}}(1+\iota) > 1$ and $ {\sigma_{\tilde{s}}}$ will never be lower than $\sigma_{\mathsf{min}}$: [*QED*]{}.
### Estimating errors on the normalized parameters
We define a variance ${\sigma_0}^2$: $$\label{sig0def}
{\sigma_0}^2 = \frac{1-{s_0}^2}{\nu \tau I_0} = \frac{1-{s_0}^2}{\cal N}.$$ Equations \[best\_err\] and \[bar\_err\] show that both $\sigma_{\tilde{s}}$ and $\sigma_{\bar{s}}$ tend towards $\sigma_0$ for large $I$, as does $\sigma_{\tilde{s}}$ (but [ *not*]{} necessarily $\sigma_{\bar{s}}$) for large $\nu_S$.
Now define $\bar{\sigma}^2 = ({1-\bar{s}^2})/{\cal N}$ and $\tilde{\sigma}^2 =
({1-\tilde{s}^2})/{\cal N}$, in which case in can be shown [@Maronna+92a]:
\[MFC5\] $\tilde{\sigma}^2$ is the maximum likelihood estimator of $\sigma_0$;
\[MFC6\] $\tilde{\sigma}^2$ has the lowest variance of any possible estimator of the variance of $s$, and is hence the optimal error estimator.
The $\sigma_0$ variance can also be expressed ${\sigma_0}^2 =
\frac{1-s^2}{\nu_S \lambda}$, and substituting the MLE for $\lambda$ and (approximately) for $s$ in the case where $I_D$ is measured in binned units, we obtain the MLE variance in the binned case. $\lambda =
$MLE$(D)=\Delta [\tau.$MLE$(\bar{I_D}) + 2\epsilon]$, hence $$\label{bin5}
\mathrm{MLE}({\sigma_s}^2) = \frac{1-\tilde{s}^2}{\nu_S \Delta
(\tau\bar{I_D} +
2\epsilon)} \simeq \frac{1-\tilde{s}^2}{\nu_S \tau \Delta \bar{I_D}}.$$
N.B. The proof of Theorem \[MFC6\] depends on the proof that $\tilde{s}$ is an unbiased estimator for $s_0$. Since we have not given formal proof of an unbiased estimator of $s_0$ in the case of binned counts in DN units, we cannot extend this result to the binned case.
### The distribution of the normalized parameters
The Central Limit Theorem [@Boas-83a Ch. 16] suggests that even a normalised Stokes Parameter $s$ must be distributed approximately normally for a sufficiently large sample. Now if $I \rightarrow \infty$ then we have both
\[til\_tend\] $$\sigma(\tilde{s}) \rightarrow \sigma_0, \frac{\tilde{s}-s}{\sigma_0} \sim
N(0,1)$$
and
\[bar\_tend\] $$\sigma(\bar{s})
\rightarrow \sigma_0,
\frac{\bar{s}-s}{\sigma_0} \sim N(0,1).$$
Further, for many measurements of a low intensity source, $\nu_S
\rightarrow \infty$, and Theorem \[til\_tend\] still holds – but in this case, Theorem \[bar\_tend\] no longer holds. Furthermore, (MFC 5) both Theorems continue to hold under the same conditions if $\sigma_0$ is replaced by the optimal estimator $\tilde{\sigma}$.
### The effect of sky noise
The night sky is not totally dark, and contributes errors twice: from the sky superimposed on the target object, and from the measurement of adjacent sky used to make a sky correction. If the sky has a constant brightness per unit area, its intensity is subject to Poisson fluctuation like any other light source. But the sky brightness itself may vary from point to point, too.
Maronna, Feinstein & Clocchiatti consider the effects of sky noise, and show that (MFC 8 – for raw photon counts), subtracting the MLE of the sky noise from the total MLE of the light in the target aperture yields the MLE of the light from the source alone. Modern aperture photometry systems such as IRAF’s [apphot]{} automatically subtract the estimated sky noise from the total signal and provide the correct output for obtaining the MLE of the source. Maronna, Feinstein & Clocchiatti do not, however, evaluate whether any estimators of $s$ are biased by the presence of sky noise.
When counting photons in the presence of sky noise, the error on the resulting normalised Stokes Parameter is $\sigma_{\dag}$ such that, with $\nu_\dag$ measurements of the background sky and an expected photon count $2\phi$ from the sky, $$\label{MFC9}
{\sigma_{\dag}}^2 = \frac{1}{\nu_S \lambda} \left[ (1-s^2) +
\frac{2(1+s^2)\phi}{\lambda} +\frac{2s^2 \nu_S \phi}{\nu_{\dag} \lambda}
\right].$$
We can easily obtain MLE$(\sigma_{\dag})$ by substituting $\tau \bar{I}=$ MLE$(\lambda)$, $\tilde{s} =$ MLE$(s)$ and the sky half-intensity MLE$(\phi)$. Similarly, in a system binning counts we may substitute MLE$(\lambda)\simeq \tau \bar{I_D}\Delta$, MLE$(s) \simeq \tilde{s}$ and the sky half-intensity MLE$(\phi) \simeq \tau \bar{I}_{\mathsf
annulus}\Delta$. It is also shown (MFC 9) that $\tilde{s}$ calculated from noise-corrected values is normally distributed provided $\nu_S$ and $\nu_\dag$ both tend to infinity, and also if $\lambda$ tends to infinity with $\phi/\lambda$ bounded.
Mathematical glossary
---------------------
Since this Appendix uses a lot of mathematical terms common with Chapter \[stoch\], and a few which differ in definition, I have given both this Appendix and that Chapter a mathematical glossary defining the terms used. Latin symbols are listed in alphabetical order first, followed by Greek terms according to the Greek alphabet – except that terms of the form $\sigma_\aleph$ are listed under the entry for $\aleph$.
$D, d$
: The random variable $D$ and its particular value $d$ expressing the output of a photon detector in DN units.
$E(X)$
: The expected value (arithmetic mean) of the random variable $X$.
$f$
: The fraction of photons transmitted by an attenuating filter.
$I$
: The intensity of a beam of light in photons per second.
- The true value of $I$.
- The intensities of two component beams in a two-beam case.
- An estimate of $I_0$ such that $\bar{I} = \bar{X}/\tau$.
- The true SD of $I$.
- The measured SD of $I$.
- The standard error on $\bar{I}$.
- In a two-channel polarimeter, one of $\nu_S$ individual measurements of the light intensity: for photon counts $n_1, n_2$ in the two channels, $I_i = (n_{i1} + n_{2i})/\tau$.
- The intensity of a beam of light in DN per second.
$m$
: The number of independent optical trains in a general polarimeter.
MLE$(\bar{S})$
: The Maximum Likelihood Estimator of $\bar{S}$.
$n_\times$
: A photon count measured in one channel of a multi-channel polarimeter.
- The photon count measured in the $i$th optical train of a general polarimeter.
- The individual photon counts measured in the two channels of a two-channel polarimeter.
- The mean values of a series of $\nu_S$ photon counts measured in the two channels of a two-channel polarimeter.
- The noise (error) on an individual photon count measurement $n_i$ of the $i$th channel of a generalised polarimeter.
${\cal N}$
: The total photon count summed over the $m$ optical trains in a general polarimeter, ${\cal N} = \sum_{1}^{m} n_i$.
$P(X=x)$
: The probability that random variable $X$ is some given value $x$.
$Q, q$
: Absolute and normalised linear Stokes Parameters. See $S, s$.
$S$
: A generalised absolute Stokes Parameter illustrating the properties of $Q$ and $U$ (and, where applicable, $V$). It takes the same annotations as $I$.
- A generalised absolute Stokes Parameter in the sense that $S_1 \equiv I$, and $S_{2,3,4} \equiv Q,U,V$.
- In a two-channel polarimeter, one of $\nu_S$ individual measurements of the absolute Stokes Parameter $Q$ or $U$: for photon counts $n_1, n_2$ in the two channels, $S_i = (n_{1i} - n_{2i})/\tau$.
$s$
: A generalised normalised Stokes Parameter illustrating the properties of $q$ and $u$. It takes some of the same annotations as $I$.
- The true normalised Stokes Parameter of a source: $s_0 =
(\lambda_1 - \lambda_2)/(\lambda_1 + \lambda_2)$.
- A generalised absolute Stokes Parameter in the sense that $s_{2,3,4} \equiv q,u,v$, and $j \neq 1$.
- For a two-channel polarimeter, the ratio of individual absolute Stokes Parameters, $s_i = S_i / I_i$.
- For a two-channel polarimeter, the mean of the individual $s_i$, such that $\bar{s} = \sum_{1}^{\nu_S} s_i$.
- For a two-channel polarimeter, the ratio of the mean Stokes Parameters, $\tilde{s} = \bar{S} / \bar{I}$.
SD$(X)$
: The Standard Deviation of random variable $X$.
$U, u$
: Absolute and normalised linear Stokes Parameters. See $S, s$.
$V, v$
: Absolute and normalised circular Stokes Parameters. See $S$.
$W, w$
: The random variable $W$ and its particular value $w$ for photons counted from a beam of intensity $I$ attenuated by a factor $f$.
$X$
: A random variable: the number of photons which might arrive from a beam of intensity $I$ in time $\tau$.
- The $i$th measurement of a set of $\nu$ measurements of the random variable $X$.
- The arithmetic mean of a set of $X_i$, such that $\bar{X} = \sum_{1}^{\nu} X_i / \nu$.
- The standard deviation of a set of $X_i$, such that ${\sigma_{\bar{X}}}^2 = \left[ \sum_{1}^{\nu} {X_i}^2 / \nu \right]
- {\bar{X}}^2$.
- A possible value of the random variable $X$.
$\Delta$
: The [*integer*]{} number of photons which must be detected to give a count of 1 DN.
$\epsilon$
: The approximate amount $\pm \epsilon$ by which the intuitive $\lambda/\Delta$ overestimates the MLE of a binned measurement of $\lambda$.
$\lambda$
: The parameter characterising the Poisson distribution of the number of photons expected to be received in time interval $\tau$, such that $\lambda = I\tau$.
$\nu_S$
: The number of individual pairs of measurements made with a two-channel polarimeter in order to determine a set of $I_i$ and $S_i$.
$\nu_\dag$
: The number of individual pairs of measurements of empty sky made with a two-channel polarimeter in order to determine the sky noise.
$\sigma$
: A standard deviation. Terms of the form $\sigma_\aleph$ are listed under the entry for $\aleph$; note also:
- An idealised SD, such that ${\sigma_0}^2 =
(1-{s_0}^2)/{\cal N}$
- The SD corresponding to the MLE estimator of $s_0$, such that ${\tilde{\sigma}}^2 = (1-{\tilde{s}}^2)/{\cal N}$
- The SD corresponding to the mean estimator of $s_0$, such that ${\bar{\sigma}}^2 = (1-{\bar{s}}^2)/{\cal N}$
- The SD corresponding to the MLE estimator of $s_0$ in the presence of sky noise.
$\tau$
: The integration time for measuring light intensity.
$\phi$
: The parameter giving half the expected number of photons from the sky background which would be received in integration time $\tau$.
Computer Codes {#codeapp}
==============
The data reduction for this thesis was accomplished with the use of several home-made [fortran]{} routines and [*Microsoft Works*]{} spreadsheets. The programming of the most important routines and spreadsheets is recorded here for reference.
Spreadsheet Analysis of Two-Channel Photometry
----------------------------------------------
Any system for reducing and analyzing polarimetry begins with photometry. Polarized images were presented to [iraf]{}’s [apphot.phot]{} routine as described in Chapter \[obsch\]; photometry was performed on both images (i.e. channels) on a given mosaic. The [phot]{} output consisted of ASCII files rich in detail, including a calculation of the magnitude of the source in each specified aperture; the zeropoint of the magnitude scale was not calibrated, however, and this portion of the output was not used in the current data reduction scheme. The [phot]{} output also returned values, in data number count rate units, for the flux attributed to the source object (corrected for sky values using an annulus) and the error on this quantity. A [fortran]{} routine by the author (not recorded here) stripped these data fields from the output of [phot]{} into tabbed ASCII files which could be pasted into the analysis spreadsheet, [*Microsoft Works*]{}.
The spreadsheet was hand-coded with other data to accompany the aperture count rate and error: a normalization factor of either 1 or 10 was included because some images (those with initial reduction performed by Dr Stephen Eales) had been normalized for the 10 second exposure time of each co-added component. The number of components in the mosaic was also coded, normally 9 but less for those images where some frames of the mosaic had been corrupted and therefore rejected. The shot noise is calculated from the square root of the total flux, and the sky noise is obtained by subtracting the shot noise in quadrature from the overall error. The spreadsheet tests each individual count rate to ensure that the sky noise is much greater than the shot noise, and an error flag is set to indicate a warning if any of the individual count rates in the dataset fail this test, thus fulfilling Check \[smallshot\].
For each pair of photometry values, the spreadsheet next calculates the sum, difference, and common error on the sum and difference. The set of differences is summed, and if it exceeds three times the quadrature sum of the common errors, an error condition is flagged, thus implementing Check \[cbias\]. The sums, differences, and errors are repeated in the next group of columns, but rearranged to group together all the $Q_i$ values and then all the $U_i$ values. The normalised Stokes Parameters $q_i$ and $u_i$ are also calculated in these blocks, completing Step \[getthei\]. All normalised Stokes parameters throughout the reduction process were calculated in the instrumental $(\eta_0 = 83\degr)$ reference frame.
The standard errors ${{{\mathcal E}_{\mathsf{phot}}}}$ are calculated for the $Q_i$ and $U_i$, following Step \[getmean\]; the sample means and statistical errors ${{{\mathcal E}_{\mathsf{stat}}}}$ are also calculated. The difference between each individual measured error and ${{{\mathcal E}_{\mathsf{phot}}}}$ is calculated, and the spreadsheet extracts the maximum deviation. If this is more than 30% of ${{{\mathcal E}_{\mathsf{phot}}}}$, an error condition is flagged, satisfying Check \[maxbig\]. A normal distribution of sky noise (Step \[assumenorm\]) is automatically assumed. Check \[noiseOK\] is left to human inspection where the values of ${{{\mathcal E}_{\mathsf{phot}}}}$ and ${{{\mathcal E}_{\mathsf{stat}}}}$ are presented together.
The statistics for the $Q_i$ and $U_i$ samples are used to calculate the Student $t$ and normalized Gaussian $z$ statistics which can be used for hypothesis testing. The spreadsheet requires manual entry of the limits on $t$ or $z$ for a given confidence level in order to test the no-polarization hypothesis with that confidence (Step \[findconfr\]).
The errors ${{\varepsilon_{\mathsf{stat}}}}$ on the best-estimator Stokes Parameters are trivially calculated (Step \[hereNSPs\]) using the statistics obtained above; and having obtained the statistical value the spreadsheet can compare with the entire dataset to obtain the photometric errors ${{\varepsilon_{\mathsf{phot}}}}$. As before, the spreadsheet presents these errors for manual comparison, fulfilling Check \[stoeq\]. The more conservative (larger) errors are duly obtained and recorded (Step \[gotnorm\]).
Finally, the noise-normalised polarization $(m)$ and the nominal polarization orientation $(\phi)$ are calculated (Steps \[findperr\] and \[propphi\]) and returned in the results row, from which they can be manually passed on to the debiasing software documented below.
Debiasing Software
------------------
A number of [fortran]{} routines were developed in the course of the data analysis for this thesis. Most of these are not documented in detail here since they are trivial implementations of the formulae of Chapter \[stoch\], or else specific codes to convert [iraf]{} output into ASCII files suitable for [*Microsoft Works*]{}. The exceptions are the routines used for debiasing, since these require an iterative solution using Bessel functions. Double precision arithmetic is used, and the Bessel functions [dbesi0(y)]{} and [dbesi1(y)]{} are drawn from standard double-precision reference libraries.
### Program [debpol]{}
[Program [debpol]{}: takes in a noise-normalised measured polarization $m_0$ and returns output of the form $a_\ell , \hat{a}, a_u$ where $(a_\ell ,a_u)$ are the $1\sigma$ confidence limits and $\hat{a}$ the best point estimate of the true noise-normalised polarization. Note that the output must be multiplied by the normalizing error $\sigma$ used to obtain $m_0$ from $p$ in the first place, in order to obtain a meaningful value in percentage units.]{}
\[debpol\]
= = =\
`program debpol`\
\
implicit none\
\
double precision m0\
\
$*$ m0 is the measured normalized polarization P$/$sigmaP\
\
integer unit\_no\
\
character$*$80 my\_filename\
\
unit\_no = 1\
\
print $*$, “Welcome to DebPol: Debiaser for polarimetry”\
\
print $*$, “(c) Gareth Leyshon, 1997”\
\
print $*$, “ ”\
\
print $*$, “Output filename?”\
\
read $*$, my\_filename\
\
open (unit=unit\_no, file=(my\_filename), form=“formatted”)\
\
print $*$, “ ”\
\
print $*$, “N.B. input 0 for a blank output line, -1 to quit.”\
\
print $*$, “ ”\
\
145continue\
\
print $*$, “Enter the measured normed polarization:”\
\
read $*$, m0\
\
\
$*$ this version gets m0 from keyboard input\
\
$*$ can escape here to end of programme or output a blank and repeat\
\
\
if (m0.eq.-1.0d0) then\
go to 149\
else\
if (m0.eq.0.0d0) then\
write (unit\_no,$*$), “ ”\
go to 145\
end if\
endif\
\
call debcalc (unit\_no, m0)\
\
goto 145\
\
149 continue\
\
close(UNIT=unit\_no, STATUS=“KEEP”)\
\
print $*$, “ ”\
\
print $*$,“Routine concludes.”\
\
print $*$, “ ”\
\
end\
\
\
\
\
subroutine debcalc (unit\_no, m0)\
\
integer unit\_no\
\
double precision m0, aWK, aML, tol, ahat\
\
$*$ aWK and aML are the estimates of the Wardle & Kronberg & Maximum Likelihood\
\
double precision estML, estWK, mMLmax, mWKmin, den\
\
mMLmax = 1.5347d0\
\
mWKmin = 1.0982d0\
\
$*$ These are the fixed thresholds for applying different methods\
\
den = (mMLmax - mWKmin)\
\
tol = 1.0d-7\
\
$*$ allows the tolerance for convergence to be hard-wired into software\
\
aML = estML (m0,tol)\
\
$*$ makes a maximum likelihood estimate\
\
aWK = estWK (m0,tol)\
\
$*$ makes a Wardle and Kronberg estimate\
\
if (m0.lt.mWKmin) then\
\
ahat = aML\
\
else\
if (m0.gt.mMLmax) then\
\
ahat = aWK\
else\
\
ahat = (((m0-mWKmin)$*$aML$/$den) + ((mMLmax-m0)$*$aWK$/$den))\
\
end if\
\
end if\
\
$*$ has set the best estimator ahat according to most appropriate function\
\
\
write (unit\_no,$*$), m0,aML,aWK, ahat\
\
print $*$, “ ”\
\
print $*$, “Results for ”, m0\
\
print $*$, aML, ahat, aWK\
\
end\
\
\
$*$ outputs all the estimators (Maximum Likelihood, my best, Wardle & Kronberg)\
\
\
double precision function estML (m0,tol)\
\
double precision m0, y, yp, tol, dbesi0, dbesi1\
\
$*$ This works out the Maximum Likelihood Estimator of the true\
$*$ noise-normalised polarization\
\
$*$ y is the current value of the best estimate\
$*$ yp is its previous value during iteration\
\
$*$ m0 is the measured value\
$*$ tol the given tolerance\
\
$*$ debesi0 and dbesi1 are Bessel functions from double-precision libraries\
\
y = m0\
\
if (m0.lt.(1.4)) then\
\
y = 0.1d0\
\
else\
if (m0.lt.2.5d0) then\
y = m0$/$3\
end if\
\
end if\
\
$*$ the start value of y for iteration is a constant if m0 is small,\
$*$ otherwise a third of m0\
\
\
7827continue\
\
yp = y\
\
$*$ yp is the past value of y\
\
y = m0 $*$ m0 $*$ dbesi1(yp) $/$ dbesi0(yp)\
\
if ((y-yp).gt.tol) then\
\
go to 7827\
\
$*$ keep iterating until the change produced is lower than the given tolerance\
\
end if\
\
estML = y$/$m0\
\
end\
\
\
\
\
double precision function estWK (m0,tol)\
\
double precision m0, y, yp, tol, dbesi0, dbesi1\
\
$*$ This works out the Wardle & Kronberg estimate, notation as in ML case\
\
y = m0+0.8D0\
\
$*$ a fixed starting value is appropriate here\
\
\
7829continue\
\
yp = y\
\
y = ((m0 $*$ m0)-1.0d0) $*$ dbesi0(yp) $/$ dbesi1(yp)\
\
if ((y-yp).gt.tol) then\
\
go to 7829\
\
end if\
\
$*$ iterate until change is within specified tolerance\
\
estWK = y$/$m0\
\
end\
### Program [thcl]{}
[Program [thcl]{}: obtains the confidence limits for the phase-space angle $\theta$ (such that $\phi = \theta/2$ is the orientation of the polarization). It takes as input the best estimate of the true polarization, $\hat{a}$, and the point estimate of $\theta$ itself (in degrees). The output is in the form of the 67% and 95% confidence interval limits on $\theta$, also in degrees.]{}
\[thcl\]
= = = =\
program thcl\
\
implicit none\
\
double precision a0, t0\
\
integer unit\_no\
\
character$*$80 my\_filename\
\
unit\_no = 1\
\
print $*$, “Welcome to THCL: Theta Confidence Limits”\
\
print $*$, “(c) Gareth Leyshon, 1996”\
\
print $*$, “ ”\
\
my\_filename = “test.the”\
\
$*$ hard-wired filename for output - this could be changed\
\
open (unit=unit\_no, file=(my\_filename), form=“formatted”)\
\
print $*$, “ ”\
\
print $*$, “N.B. input 0 for a blank output line, -1 to quit.”\
\
print $*$, “ ”\
\
145continue\
\
print $*$, “Enter the best estimate of a:”\
\
read $*$, a0\
\
if (a0.eq.-1.0d0) then\
go to 149\
else\
if (a0.eq.0.0d0) then\
write (unit\_no,$*$), “ ”\
go to 145\
end if\
endif\
\
print $*$, “Enter the best estimate of theta:”\
\
read $*$, t0\
\
$*$ t\_0 is the best estimate\
\
call calculate (unit\_no, a0, t0)\
\
goto 145\
\
149 continue\
\
close(UNIT=unit\_no, STATUS=“KEEP”)\
\
print $*$, “ ”\
\
print $*$,“Routine concludes.”\
\
print $*$, “ ”\
\
end\
\
\
subroutine calculate (unit\_no, a0, t0)\
\
integer unit\_no\
\
double precision t0, Cp, a1, a2, a0, tx, getradians, degre\
\
real ao, tho, b1, b2, c1, c2\
\
\
tho = t0\
\
tx = getradians(t0)\
\
t0 = tx\
\
$*$ convert to radians\
\
print $*$, “ ”\
print $*$, “CALCULATE...”\
print $*$, “Best estimate of theta (deg):”, tho\
print $*$, “Best estimate of theta (rad):”, t0\
print $*$, “Best estimate of a:”, a0\
\
$*$ calculate for 67\
Cp = 0.67\
\
print $*$, “ ”\
print $*$, “Calling findconf for ”, Cp\
\
call findconf(a0,t0,Cp,a1,a2)\
\
b1 = degre(a1)\
\
b2 = degre(a2)\
\
$*$ store the results as b1, b2\
\
\
$*$ calculate for 95\
Cp = 0.95\
\
print $*$, “ ”\
print $*$, “Calling findconf for ”, Cp\
\
call findconf(a0,t0,Cp,a1,a2)\
\
c1=degre(a1)\
\
c2=degre(a2)\
\
$*$ store the results as c1, c2\
\
\
ao = a0\
\
write (unit\_no,$*$), ao,tho,b1,b2,c1,c2\
\
print $*$, “ ”\
\
print $*$, “Results for ”, ao,tho\
\
print $*$, b1,b2,c1,c2\
\
print $*$, “ ”\
\
end\
\
double precision function degre(alpha)\
\
double precision confac, alpha\
\
confac = 180/acos(-1.0d0)\
\
degre = alpha$*$confac\
\
end\
\
\
\
double precision function getradians(alpha)\
\
double precision confac, pi, alpha\
\
pi = acos(-1.0d0)\
\
confac = pi/(180.0)\
\
print $*$, “Degrees: ”, alpha\
\
print $*$, “confac: ”, confac\
\
getradians = alpha$*$confac\
\
print $*$, “Radians: ”, getradians\
\
end\
\
\
\
$*$ this is the main subroutine that finds the interval\
\
subroutine findconf(a0,t0,Cp,a1,a2)\
\
double precision Cp,a0,alist,blist,rlist,elist,theterr,\
$*$ epsabs,epsrel,result,t0,l,lstep,a1,a2,dstep,abserr\
\
logical flag\
\
integer ier,key,limit,neval,iord,last\
\
$*$ first set up constants for our integrating\
\
\
dstep=2.0d0\
key = 40\
limit = 20000000\
epsabs = 1.0d-20\
epsrel = 1.0d-20\
\
$*$ Cp is the confidence interval we want\
\
print $*$, “ ”\
\
print $*$, “Considering best est pol:”, a0\
\
print $*$, “Considering measured angle (rad):”, t0\
\
\
$*$ Now we are going to iterate for theterr.\
\
theterr = t0$*$(0.1d0)\
\
l = 1.0d-3\
\
lstep = 0.4d0\
\
556 continue\
\
a1 = t0-theterr\
\
a2 = t0+theterr\
\
\
print $*$, “ ”\
\
print $*$, “t0,err:”, t0, theterr\
\
call dqage(a0,t0,a1,a2,epsabs,epsrel,key,limit,result,\
$*$ abserr,neval,ier,alist,blist,rlist,elist,iord,last)\
\
$*$ dqage comes from a standard library for integrating under a curve\
\
print $*$, “Succeeded, area under curve is ”, result\
\
if ((abs(result-Cp)).gt.l) then\
\
$*$ here we iterate, decreasing our step size\
\
if (result.gt.Cp) then\
if (flag) then\
continue\
else\
lstep = lstep/dstep\
endif\
theterr = theterr - lstep\
flag=.true.\
else\
if (flag) then\
lstep = lstep/dstep\
end if\
theterr = theterr + lstep\
flag=.false.\
\
end if\
\
go to 556\
\
end if\
\
\
print $*$, “ ”\
\
print $*$, “Integration error?”, ier\
\
print $*$, “result = ”, result\
\
print $*$, “ ”\
\
print $*$, “We have found theterr = ”, theterr\
\
print $*$, “ ”\
\
$*$ depending on result/Cp, modify a and run it again.\
\
end\
\
\
$*$ HERE IS THE FUNCTION TO BE INTEGRATED, called by dqage\
\
double precision function f(a0,t0,t)\
\
double precision a0, t, t0, d, s, c, p, q, r, pi, half\
\
double precision sq, e, ix, ip\
\
half = (1.0d0)/(2.0d0)\
\
pi = acos(-1.0d0)\
\
ip = (1.0d0)/(pi$*$2.0d0)\
\
sq = sqrt(ip)\
\
d = t-t0\
\
s = a0$*$sin(d)\
\
c = a0$*$cos(d)\
\
e = ix(c)\
\
p = exp(-half$*$(s$*$$*$2))\
\
q = exp(-half$*$(c$*$$*$2))$*$ip\
\
r = (half+e)$*$c$*$sq\
\
f = p$*$(q+r)\
\
end\
\
\
\
double precision function ix(x)\
\
double precision x, y, derf, half\
\
$*$ derf is the double precision error function, erf(x)\
\
half = (1.0d0)/(2.0d0)\
\
y = derf(x$*$sqrt(half))\
\
ix = half $*$ y\
\
end\
[^1]: Note the definition in the frontmatter: the term ‘optical’ is used to encompass the near infrared, visible light and the near ultraviolet.
[^2]: This thesis will adopt the term ‘quasar’ for these objects regardless of radio intensity.
[^3]: Since the observational data recorded in this thesis concerns only imaging polarimetry, not polarized spectra, I will not review the finer features of raw and polarised AGN spectra here; the subject has been extensively treated in the literature.
[^4]: Doctoral thesis declaration: Thesis supervisor Dr Stephen Eales had already converted these images to [iraf]{} format before the author began his work; all subsequent data reduction was performed by the author.
[^5]: Doctoral thesis declaration: this despiking process and photometry of the despiked frames was carried out by Dr James Dunlop of the University of Edinburgh. Conversion of the photometry to polarimetry and subsequent debiasing was performed by the author. There is no qualitative difference between the natural and despiked results and so the details of the despiking process are not recorded here.
[^6]: Doctoral thesis declaration: that discussion is not reproduced in detail in this thesis as the major part of it was developed by Dr James Dunlop of the University of Edinburgh.
[^7]: In an attempt to use more consistent notation, my paper uses $\bar{s }$ for the arithmetic mean of a set of parameters, $\tilde{s}$ for a ratio of means, and $\hat{s}$ for the best (conservative) errors on certain quantities. Clarke et al., however, use $\bar{s}$ for the ratio of non-normalized mean Stokes Paramet ers, and $\tilde{s}(1)$ for the arithmetic mean.
[^8]: This thesis uses $\eta$ for the instrumental angle which di Serego Alighieri et al. call $\phi$.
[^9]: The Wardle & Kronberg paper reproduces Vinokur’s equation (my Equation \[thetadis\]) but omits the factor ‘${\mathrm sign}(x)$’ from Equation \[thetasup\] on the grounds (Wardle, private communication) that the probability of $x$ falling in the domain $x<0$ is negligibly small.
|
---
author:
- 'Yizhak Ben-Shabat'
- Stephen Gould
bibliography:
- 'references.bib'
title: 'DeepFit: 3D Surface Fitting via Neural Network Weighted Least Squares'
---
|
---
abstract: 'We define an *order polarity* to be a polarity $(X,Y,\operatorname{R})$ where $X$ and $Y$ are partially ordered, and we define an *extension polarity* to be a triple ${(e_X,e_Y,\operatorname{R})}$ such that $e_X:P\to X$ and $e_Y:P\to Y$ are poset extensions and $(X,Y,\operatorname{R})$ is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a pre-order structure on $X\cup Y$ such that the natural embeddings, $\iota_X$ and $\iota_Y$, of $X$ and $Y$, respectively, into $X\cup Y$ preserve the order structures of $X$ and $Y$ in increasingly strict ways. We define a Galois polarity to be an extension polarity where $e_X$ and $e_Y$ are meet- and join-extensions respectively, and we show that for such polarities there is a unique pre-order on $X\cup Y$ such that $\iota_X$ and $\iota_Y$ satisfy particularly strong preservation properties. We define morphisms for polarities, providing the class of Galois polarities with the structure of a category, and we define an adjunction between this category and the category of $\Delta_1$-completions and appropriate homomorphisms. We formalize the theory of extension polarities and prove a duality principle to the effect that if a statement is true for all extension polarities then so too must be its dual statement.'
author:
- Rob Egrot
title: Order polarities
---
Introduction
============
Background
----------
The concept of a *polarity*, i.e. a pair of sets $X$ and $Y$ and a relation $\operatorname{R}$ between them, was known to Birkhoff at least as far back as 1940 [@Bir40]. While, according to [@Bir95 p122], originally defined as a generalization of the dual isomorphism between polars in analytic geometry, the generality of the definition has lent itself to diverse applications in mathematics and computer science. For example, polarities under the name of *formal concepts* are fundamental in formal concept analysis [@GanWil99]. As another example, polarities appear bearing the name *classification* in the theory of information classification [@BarSel97 Lecture 4], where they are again a foundational concept.
For a more purely mathematical application, a particular kind of polarity, referred to as a *polarization*, was used in [@Tun74] to produce poset completions. The same paper also proves various results connecting properties of polarizations with properties of the resulting completion. More recently, this technique has been exploited to construct canonical extensions for bounded lattice expansions [@GehHar01], and also for posets [@DGP05], where they provide a tool for ‘completeness via canonicity’ results for substructural logics. Something similar also appears implicitly in [@GhiMel97], though neither polarizations nor polarities in general are mentioned explicitly.
The general idea behind these completeness results is, given a poset $P$ equipped with additional operations that are either order preserving or reversing in each coordinate, to show that there exists a completion of $P$ to which the additional operations can be extended. This roots of this technique appear in [@JonTar51], though not in the context of ‘completeness via canonicity’ results, as a generalization of Stone’s representation theorem to Boolean algebras with operators (BAOs). The approach there was to first (non-constructively) dualize to relational structures, then construct the canonical extension from these.
Early generalizations to distributive lattices used Priestly duality [@Pr70; @Pri72] in a similar way (see for example [@GehJon94; @S-S00a; @S-S00b]). More recent approaches using polarities bypass the dual construction, which is significantly more complicated outside of the distributive setting, and have the additional advantage of being constructive [@GehHar01; @DGP05]. Indeed, an innovation of [@DGP05] is to use the canonical extension of a poset to *construct* a dual, which can then play the same role in providing completeness results for substructural logics as the canonical frame does in the modal setting (see e.g. [@BRV01 Chapters 4 and 5]). For more on the development of the theory of canonical extensions see, for example, [@GehVos11] or the introduction to [@Gol18].
We note that for operations that are not *operators* in the sense of [@JonTar51], the canonical extension construction is ambiguous, as there are often several non-equivalent choices for the lifts of each operation, each of which may be ‘correct’ depending on the situation (see for example the epilogue of [@GehJon04] for a brief discussion of this). Moreover, for posets, what is meant by *the* canonical extension is even less clear than it is in the lattice case. This is a consequence of ambiguities surrounding the notions of ‘filters’ and ‘ideals’ in the more general setting. See [@Mor14] for a thorough investigation of this issue.
For canonical extensions in their various guises to play a role in ‘completeness via canonicity’ arguments, general results concerning the preservation of equations and inequalities are extremely useful. Some results of this sort can be found in [@Suz11a; @Suz11b], where arguments from [@GhiMel97] are extended to more general settings. One component of these arguments is the exploitation of the so called *intermediate structure*, an extension of the original poset intermediate between it and the canonical extension. The idea is that operations are, in a sense, lifted first to the intermediate structure, and then to the canonical extension.
More generally, the class of $\Delta_1$-completions [@GJP13] includes canonical extensions (however we define them), and also others such as the MacNeille (aka *Normal*) completion. Given a poset $P$, the $\Delta_1$-completions of $P$ are, modulo suitable concepts of isomorphism, in one-to-one correspondence with certain kinds of polarities constructed from the poset [@GJP13 Theorem 3.4]. Here also the intermediate structure appears. Indeed, a $\Delta_1$-completion is the MacNeille completion of its intermediate structure [@GJP13 Section 3].
What is done here
-----------------
In the existing literature, the intermediate structure emerges almost coincidentally from the construction of a completion. Given a polarity $(X, Y, \operatorname{R})$, first a complete lattice ${G(X,Y,\operatorname{R})}$ is constructed using the antitone Galois connection between $\wp(X)$ and $\wp(Y)$ induced by $\operatorname{R}$, as we explain in more detail in Section \[S:pol1\]. The intermediate structure is then found sitting inside it as a subposet. There are natural maps from $X$ and $Y$ into the intermediate structure, and, if these are injective, partial orderings are thus induced on $X$ and $Y$. When ${G(X,Y,\operatorname{R})}$ is an extension of a poset $P$, it will also follow that $X$ and $Y$ are extensions of $P$. It turns out that the pre-order on $X\cup Y$ induced by the intermediate structure agrees with $\operatorname{R}$ on $X\times Y$.
The broad goal of this paper is to take the idea of a polarity involving order extensions $e_X:P\to X$ and $e_Y:P\to Y$ as primitive, and develop a theory from this. More explicitly, we are interested in the interaction between the relation $\operatorname{R}$ and the orders on $X$ and $Y$, and, in particular, under what circumstances something corresponding to the ‘intermediate structure’ can be defined on $X\cup Y$. This issue raises several questions, depending on exactly what properties we think an ‘intermediate structure’ should have.
Based on our answers to these questions, we define a sequence of so-called *coherence conditions* for polarities. The bulk of this work is done in Section \[S:order\], where the main definitions are made, and in Section \[S:satisfaction\], where, among other things, we prove our defined conditions are strictly increasing in strength.
In Section \[S:Galois\] we define a *Galois polarity* to be a triple ${(e_X,e_Y,\operatorname{R})}$ satisfying the strongest of our coherence conditions, and with the additional property that $e_X$ is a meet-extension, and $e_Y$ is a join-extension. The ‘aptness’ of this definition is partly demonstrated by the fact that, if ${(e_X,e_Y,\operatorname{R})}$ is a Galois polarity, there is one and only one possible pre-order structure on $X\cup Y$ that agrees with the orders on $X$ and $Y$, agrees with $\operatorname{R}$ on $X\times Y$, and also preserves meets and joins from the base poset $P$ (see Theorem \[T:unique\] for a more precise statement).
Galois polarities are studied further in Section \[S:GaloiS1\]. First we justify the choice of terminology by demonstrating that, for Galois polarities, the unique pre-order structure described above can be defined in terms of a Galois connection between any join-preserving join-completion of $Y$ and any meet-preserving meet-completion of $X$. This requires some technical results on extending and restricting polarity relations, which we provide in Section \[S:ExtRes\]. Here we investigate the ‘simplest’ way we might hope to extend a relation between posets to a relation between meet- and join-extensions of these posets, and conversely the simplest way we might restrict a relation between extensions to a relation between the original posets. In particular we prove that it is rather common for coherence properties of a polarity to be preserved by extension and restriction as we define them.
By defining suitable morphisms, we can equip the class of Galois polarities with the structure of a category. This can be seen as a generalization of the concept of a $\delta$-homomorphism from [@GehPri08 Section 4]. We define an adjunction between this category and the category of $\Delta_1$-completions (see Theorem \[T:adj\]). This produces the correspondence between $\Delta_1$-completions of a poset and certain kinds of polarity from [@GJP13 Theorem 3.4] via the categorical equivalence between fixed subcategories.
Finally, in Section \[S:dual\] we characterize order polarities with various coherence levels as models of certain first- and second-order theories, and using this formulate a ‘duality principle’ for order polarities. This generalizes the familiar order duality for posets, and formalizes a labour-saving intuition to which we frequently appeal in proofs throughout the document.
In the long term we imagine handling lifting of operations, and the preservation of inequalities and so on, to ‘intermediate structures’ induced by polarities, and Galois polarities in particular. This is, of course, not an entirely new idea. Indeed, we have mentioned previously that lifting operations to canonical extensions is often done by first lifting to the intermediate structure. The hope is that, by shifting the focus a little from intermediate structures as they emerge in the construction of completions, to intermediate structures as algebraic objects of interest in their own right, some new insight might be gained. However, to control the length of this document, we leave the pursuit of this rather vague goal to future work.
Orders and completions {#S:comp}
======================
A note on notation
------------------
We use the following not entirely standard notations:
- Give a poset $P$ and $p\in P$, we define $$p^\uparrow = \{q\in P: q\geq p\} \text{ and }p^\downarrow= \{q\in P: q\leq p\}.$$
- Given a function $f:X\to Y$, and given $S\subseteq X$, we define $$f[S]=\{f(x):x\in S\}.$$
- With $f$ as above and with $y\in Y$ and $T\subseteq Y$ we define $$f^{-1}(y) = \{x\in X: f(x)=y\}$$ and $$f^{-1}(T)=\{x\in X:f(x)\in T\}.$$
- If $P$ is a poset then $P^\partial$ is the order dual of $P$.
- If $X$ and $Y$ are sets, then we may refer to a relation $\operatorname{R}\subseteq X\times Y$ as being a *relation on $X\times Y$*.
Extensions and completions
--------------------------
We assume familiarity with the basics of order theory. Textbook exposition can be found in [@DavPri02]. In this subsection we provide a brisk introduction to some more advanced order theory concepts. This serves primarily to establish the notation we will be using.
Let $P$ and $Q$ be posets. We say an order embedding $e:P\to Q$ is a **poset extension**, or just an *extension*. If $Q$ is also a complete lattice we say $e$ is a **completion**. If for all $q\in Q$ we have $q = {\bigwedge}e[e^{-1}(q^\uparrow)]$ then we say $e$ is a **meet-extension**, or a **meet-completion** if $Q$ is a complete lattice. Similarly, if $q = {\bigvee}e[e^{-1}(q^\downarrow)]$ for all $q\in Q$ then $e$ is a **join-extension**, or a **join-completion** when $Q$ is complete.
Note that it is common in the literature to refer to completions using the codomain of the function. For example, we might say “$Q$ is a completion of $P$" when talking about the completion $e:P\to Q$. This has the disadvantage of obfuscating the issue of what it means for two extensions to be isomorphic, as an isomorphism between codomains is not sufficient for extensions to be isomorphic in the sense used here. This rarely causes significant problems in practice, as it is usually clear from context what kind of isomorphism is required. However, we find the identification of extensions with maps to be more elegant, and will generally use this approach.
\[D:mapCat\] Given posets $P_1, P_2, Q_1, Q_2$, and monotone maps $f_1:P_1\to Q_1$ and $f_2:P_2\to Q_2$, a map, or morphism, from $f_1$ to $f_2$ is a pair of monotone maps $g_P:P_1\to P_2$ and $g_Q:Q_1\to Q_2$ such that the diagram in Figure \[F:mapHom\] commutes. If $g_p$ and $g_Q$ are both order isomorphisms then we say $f_1$ and $f_2$ are isomorphic.
If $f_1:P\to Q_1$ and $f_2:P\to Q_2$ are extensions of a poset $P$, then $f_1$ and $f_2$ are **isomorphic as extensions of $P$** if they are isomorphic in the sense described above and the map $g_P$ is the identity on $P$.
Definition \[D:mapCat\] equips the class of monotone maps between posets, and in particular the subclass of poset extensions, with the structure of a category. We will make frequent use of the idea of extensions being isomorphic, and we will return to the idea of a category of extensions in Section \[S:cat\].
$$\xymatrix{ P_1\ar[r]^{f_1}\ar[d]_{g_P} & Q_1\ar[d]^{g_Q} \\
P_2\ar[r]_{f_2} & Q_2
}$$
\[D:DM\] Given a poset $P$, the **MacNeille completion** of $P$ is a map $e:P\to {\mathcal N}(P)$ that is both a meet- and a join-completion.
The MacNeille completion was introduced in [@Mac37] as a generalization of Dedekind’s construction of ${\mathbb R}$ from ${\mathbb Q}$, it is unique up to isomorphism. The characterization used here is due to [@BanBru67]. See e.g. [@DavPri02 Section 7.38] for more information.
\[D:can\] The **canonical extension** of a lattice $L$ is a completion $e:L\to L^\delta$ such that:
1. $e[L]$ is *dense* in $L^\delta$. I.e. Every element of $L^\delta$ is expressible both as a join of meets, and as a meet of joins, of elements of $e[L]$.
2. $e$ is *compact*. I.e. for all $S, T\subseteq L$, if ${\bigwedge}e[S]\leq {\bigvee}e[T]$ then there are finite $S'\subseteq S$ and $T'\subseteq T$ with ${\bigwedge}S'\leq {\bigvee}T'$.
Canonical extensions are also unique up to isomorphism. This characterization, and the proof that such a completion exists for all $L$, is due to [@GehHar01]. It generalizes the definition of the canonical extension for Boolean algebras [@JonTar51], and distributive lattices [@GehJon94]. The construction used in [@GehHar01] can, as noted in Remark 2.8 of that paper, also be used for posets, and will again result in a dense completion. However, the kind of compactness obtained is weaker. This idea is expanded upon in [@DGP05]. The differences between the lattice and poset cases arise from the fact that definitions for filters and ideals which are equivalent for lattices are not so for posets. This issue is discussed in detail in [@Mor14]. One way to address this systematically is to talk about *the canonical extension of $P$ with respect to ${\mathcal F}$ and ${\mathcal I}$*, where ${\mathcal F}$ and ${\mathcal I}$ are sets of ‘filters’ and ‘ideals’ of $P$ respectively. By making the definitions of ‘filter’ and ‘ideal’ weak enough, this allows all notions of the canonical extension of a poset to be treated in a uniform fashion. This is the approach taken in [@MorVanA18], for example.
\[D:Del\] Given a poset $P$, a $\Delta_1$**-completion** of $P$ is a completion $e:P\to D$ such that $e[P]$ is dense in $D$.
$\Delta_1$-completions, introduced in [@GJP13], include both MacNeille completions and canonical extensions. As such they are not usually unique up to isomorphism, so it doesn’t make sense to talk about *the* $\Delta_1$-completion.
Let $P$ and $Q$ be posets. Then a **monotone Galois connection**, or just a *Galois connection*, between $P$ and $Q$ is a pair of monotone maps $\alpha:P\to Q$ and $\beta:Q\to P$ such that, for all $p\in P$ and $q\in Q$, we have $$\alpha(p)\leq q \iff p\leq \beta(q).$$ The map $\alpha$ is the **left adjoint**, and $\beta$ is the **right adjoint**.
An **antitone Galois connection** between $P$ and $Q$ is a Galois connection between $P$ and the order dual, $Q^\partial$, of $Q$.
A **pre-order** on a set is a binary relation that is reflexive and transitive. Every pre-order induces a **canonical partial order** by identifying pairs elements that break anti-symmetry.
Polarities for completions {#S:pol1}
--------------------------
Following [@Bir40], we define a **polarity** to be a triple ${(X,Y,\operatorname{R})}$, where $X$ and $Y$ are sets, and $\operatorname{R}\subseteq X\times Y$ is a binary relation. For convenience we will assume also that $X$ and $Y$ are disjoint. See the section on polarities in [@EKMS93] for several examples. Polarities have also been called *polarity frames* [@Suz14]. Given any polarity ${(X,Y,\operatorname{R})}$, there is an antitone Galois connection between $\wp(X)$ and $\wp(Y)$. This is given by the order reversing maps $(-)^R:\wp(X)\to\wp(Y)$ and ${}^R(-):\wp(Y)\to\wp(X)$ defined as follows: $$(S)^R = \{y\in Y: x\operatorname{R}y \text{ for all } x\in S\}.$$ $${}^R(T) = \{x\in X: x\operatorname{R}y \text{ for all } y\in T\}.$$
The set ${G(X,Y,\operatorname{R})}$ of subsets of $X$ that are fixed by the composite map ${}^R(-)\circ (-)^R$ is a complete lattice. Indeed, this is a closure operator on $\wp(X)$.
Polarities in the special case where $X$ and $Y$ are sets of subsets of some common set $Z$, where the relation $\operatorname{R}$ is that of non-empty intersection, and which also satisfy some additional conditions, have been referred to as *polarizations* in the literature [@Tun74; @MorVanA18]. Polarizations play an important role in the construction of canonical extensions.
There are maps $\Xi:X\to{G(X,Y,\operatorname{R})}$ and $\Upsilon:Y\to {G(X,Y,\operatorname{R})}$ defined by: $$\Xi(x) = {}^R(\{x\}^R)\text{ for $x\in X$, and}$$ $$\Upsilon(y) = {}^R\{y\} \text{ for $y\in Y$}.$$ $\Xi[X]$ and $\Upsilon[Y]$ join- and meet-generate ${G(X,Y,\operatorname{R})}$ respectively [@Geh06 Proposition 2.10]. Moreover, the (not usually disjoint) union $\Xi[X]\cup\Upsilon[Y]$ inherits an ordering from ${G(X,Y,\operatorname{R})}$. Thus the inclusion of the poset $\Xi[X]\cup\Upsilon[Y]$ into ${G(X,Y,\operatorname{R})}$ can be characterized as the MacNeille completion of $\Xi[X]\cup\Upsilon[Y]$. The order on $\Xi[X]\cup\Upsilon[Y]$ can be defined without first constructing ${G(X,Y,\operatorname{R})}$. We expand on this in Proposition \[P:order\] below.
\[P:order\] A pre-order on $\Xi[X]\cup\Upsilon[Y]$ is defined below. The partial ordering of $\Xi[X]\cup\Upsilon[Y]$ inherited from ${G(X,Y,\operatorname{R})}$ is the canonical partial order induced by this pre-ordering.
1. For $x_1,x_2\in X$ we have $\Xi(x_1)\leq \Xi(x_2)\iff (x_2 \operatorname{R}y\implies x_1 \operatorname{R}y$ for all $y\in Y)$.
2. For $y_1,y_2\in Y$ we have $\Upsilon(y_1)\leq \Upsilon(y_2)\iff (x\operatorname{R}y_1\implies x\operatorname{R}y_2$ for all $x\in X)$.
3. For $x\in X$ and $y\in Y$ we have $\Xi(x)\leq\Upsilon(y)\iff x\operatorname{R}y$.
4. For $x\in X$ and $y\in Y$ we have $$\Upsilon(y)\leq\Xi(x)\iff(x'\operatorname{R}y\text{ and }x\operatorname{R}y'\implies x'\operatorname{R}y'\text{, for all }x'\in X\text{ and }y'\in Y).$$
This is essentially [@Geh06 Proposition 2.7].
Proposition \[P:nat\] below provides another perspective on the conditions from Proposition \[P:order\].
\[P:nat\] Let ${(X,Y,\operatorname{R})}$ be a polarity. Then the following are equivalent:
1. $\preceq$ is the least pre-order definable on $\Xi[X]\cup \Upsilon[Y]$ such that:
1. $\Xi(x)\preceq\Upsilon(y)\iff x\operatorname{R}y$ for all $x\in X$ and $y\in Y$.
2. The restrictions of $\preceq$ to $\Xi[X]$ and $\Upsilon[Y]$ agree with the orders on these sets inherited from ${G(X,Y,\operatorname{R})}$.
2. $\preceq$ satisfies the conditions from Proposition \[P:order\]
Suppose $\preceq$ is any pre-order on $X\cup Y$ satisfying conditions 1(a) and 1(b). Then, by Proposition \[P:order\] we have $$\Xi(x_1)\preceq \Xi(x_2)\iff \Xi(x_1)\subseteq \Xi(x_2)\iff (x_2 \operatorname{R}y\implies x_1 \operatorname{R}y\text{ for all }y\in Y),$$ and thus \[P:order\](1) is satisfied. A similar argument works for \[P:order\](2), and \[P:order\](3) holds automatically. Finally, as $\preceq$ is transitive, we must have $$\begin{aligned}
\Upsilon(y) \preceq \Xi(x)\implies \Big(&\Xi(x')\preceq \Upsilon(y)\text{ and }\Xi(x)\preceq \Upsilon(y')\implies \Xi(x')\preceq \Upsilon(y')\\&\text{ for all }x'\in X\text{ and }y'\in Y\Big).\end{aligned}$$
Thus, by 1(a), any such pre-order $\preceq$ satisfies \[P:order\](1)-(3), and the ‘forward implication only’ version of \[P:order\](4).
To complete the proof it is sufficient to show that the ‘minimal’ $\preceq$ defined from $\operatorname{R}$ using conditions \[P:order\](1)-(4) defines a pre-order on $\Xi[X]\cup \Upsilon[Y]$ satisfying conditions 1(a) and 1(b). But this is what Proposition \[P:order\] tells us.
\[L:inj\] The following are equivalent:
1. The map $\Xi:X\to {G(X,Y,\operatorname{R})}$ is injective.
2. Whenever $x_1\neq x_2\in X$ there is $y\in Y$ such that either $(x_2,y)\in \operatorname{R}$ and $(x_1, y)\notin \operatorname{R}$, or vice versa.
3. Whenever $x_1\neq x_2\in X$ we have either $x_1\notin \Xi(x_2)$ and/or $x_2\notin \Xi(x_1)$.
The following are also equivalent:
1. The map $\Upsilon:Y\to{G(X,Y,\operatorname{R})}$ is injective.
2. Whenever $y_1\neq y_2\in Y$ there is $x\in X$ such that either $(x,y_2)\in \operatorname{R}$ and $(x,y_1)\notin \operatorname{R}$, or vice versa.
Observe that $\Xi(x) = \{z\in X: x\operatorname{R}y\implies z\operatorname{R}y\text{ for all }y\in Y\}$ for all $x\in X$. Let $x_1\neq x_2$ and suppose without loss of generality that there is $z\in \Xi(x_1)\setminus\Xi(x_2)$. Then $(z,y)\in\operatorname{R}$ for all $y\in Y$ with $(x_1,y)\in \operatorname{R}$, but there is $y'\in Y$ with $(x_2,y')\in\operatorname{R}$ and $(z,y')\notin \operatorname{R}$. For this $y'$ must have $(x_2,y')\in \operatorname{R}$ and $(x_1,y')\notin \operatorname{R}$. Thus $(1.a)\implies(1.b)$. That $(1.b)\implies(1.c)$ and $(1.c)\implies(1.a)$ is automatic. The proof for $\Upsilon$ is similar, but even more straightforward.
What if $X$ and $Y$ are not merely sets but also have a poset structure? We make the following definition.
A polarity ${(X,Y,\operatorname{R})}$ is an **order polarity** if $X$ and $Y$ are posets.
In this situation we might, for example, want the maps $\Xi$ and $\Upsilon$ to be order embeddings, which places constraints on $\operatorname{R}$. Building on lemma \[L:inj\] we have the following result.
\[P:ordEmb\] Let ${(X,Y,\operatorname{R})}$ be an order polarity. Then the map $\Xi:X\to {G(X,Y,\operatorname{R})}$ is an order embedding if and only if, for all $x_1,x_2\in X$, we have $$x_1\leq x_2\iff\text{ for all $y\in Y$ we have $x_2 \operatorname{R}y\implies x_1 \operatorname{R}y$}.$$
The map $\Upsilon:Y\to {G(X,Y,\operatorname{R})}$ is an order embedding if and only if, for all $y_1,y_2\in Y$, we have $$y_1\leq y_2\iff\text{ for all $x\in X$ we have $x\operatorname{R}y_1\implies x \operatorname{R}y_2$}.$$
We could appeal to proposition \[P:order\], but the direct argument is also extremely simple. Explicitly, $\Xi$ is an order embedding if and only if $x_1\leq x_2 \iff \Xi(x_1)\subseteq \Xi(x_2)$, and a little consideration reveals that $\Xi(x_1)\subseteq \Xi(x_2)$ if and only if $x_2 \operatorname{R}y\implies x_1 \operatorname{R}y$ for all $y\in Y$. Again, the argument for $\Upsilon$ is even more straightforward.
Propositions \[P:nat\] and \[P:ordEmb\], while essentially trivial in themselves, contain, in a sense, the seed of inspiration for the rest of the paper. In broad terms, we want to investigate the conditions for the existence of pre-orders on $X\cup Y$ such that similar results can be proved. This we do in the next section and onwards. First a little more notation.
Given disjoint sets $X$ and $Y$, we sometimes write ${X\cup_\preceq Y}$ to specify that we are talking about $X\cup Y$ ordered by a given pre-order $\preceq$.
Coherence conditions for order polarities {#S:order}
=========================================
The basic case
--------------
In the previous section we discussed polarities and order polarities from the perspective of ${G(X,Y,\operatorname{R})}$, and the inherited order structure on $\Xi[X]\cup\Upsilon[Y]$. In this situation the maps $\Xi$ and $\Upsilon$ may fail to be monotone, order reflecting, or even injective. In this section we forget about ${G(X,Y,\operatorname{R})}$, and ask instead, given an order polarity ${(X,Y,\operatorname{R})}$, under what circumstances can we define pre-orders on $X\cup Y$ that agree with $\operatorname{R}$ on $X\times Y$, and also extend the order structures of $X$ and $Y$? In other words, when are there pre-orders on $X\cup Y$ agreeing with $\operatorname{R}$ on $X\times Y$ such that the natural inclusions of $X$ and $Y$ into $X\cup Y$ are monotone? What about if we require the inclusions to be order embeddings, or to have stronger preservation properties? We will address these questions, but first some definitions.
Given a relation $\operatorname{R}\subseteq X\times Y$ we define the relation ${\operatorname{R}^d}\subseteq Y\times X$ by $$y{\operatorname{R}^d}x \iff (x'\operatorname{R}y\text{ and }x\operatorname{R}y'\implies x'\operatorname{R}y'\text{, for all }x'\in X\text{ and }y'\in Y).$$
Let ${(X,Y,\operatorname{R})}$ be an order polarity. Define ${\mathcal P}_{\operatorname{R}}$ to be the set of pre-orders on $X\cup Y$ agreeing with $\operatorname{R}$ on $X\times Y$, and extending the orders on $X$ and $Y$. I.e. $\preceq\in{\mathcal P}_{\operatorname{R}}$ if and only if:
(i) $\preceq|_{X\times Y} = \operatorname{R}$, and
(ii) the orders on $X$ and $Y$ are contained in $\preceq|_{X\times X}$ and $\preceq|_{Y\times Y}$ respectively.
\[T:ext\] Let ${(X,Y,\operatorname{R})}$ be an order polarity. Then ${\mathcal P}_{\operatorname{R}}$ is non-empty if and only if:
- For all $x_1,x_2\in X$ we have $x_1 \leq x_2 \implies (x_2 \operatorname{R}y\implies x_1 \operatorname{R}y$ for all $y\in Y)$.
- For all $y_1,y_2\in Y$ we have $y_1\leq y_2\implies (x\operatorname{R}y_1\implies x\operatorname{R}y_2$ for all $x\in X)$.
In addition, for all relations $\preceq$ on $(X\cup Y)^2$ we have $\preceq\in{\mathcal P}_{\operatorname{R}}$ only if the following conditions are satisfied:
- For $x_1,x_2\in X$ we have $x_1\leq x_2\implies x_1 \preceq x_2$.
- For $y_1,y_2\in Y$ we have $y_1\leq y_2\implies y_1 \preceq y_2$.
- For all $x\in X$ and $y\in Y$ we have $x\operatorname{R}y\iff x\preceq y$.
- ${\operatorname{R}^d}$ extends $\preceq$ on $Y\times X$.
Moreover, ${\mathcal P}_{\operatorname{R}}$ is closed under non-empty intersections, and, if it is non-empty, has a minimal element $\operatorname{\preceq_0}$, defined by:
- For all $x_1,x_2\in X$ we have $x_1 \operatorname{\preceq_0}x_2 \iff x_1\leq x_2$
- For all $y_1,y_2\in Y$ we have $y_1 \operatorname{\preceq_0}y_2\iff y_1\leq y_2$.
- For all $x\in X$ and $y\in Y$ we have $x\operatorname{\preceq_0}y\iff xR y$.
- There is no $x\in X$ and $y\in Y$ with $y \operatorname{\preceq_0}x$. I.e. $\operatorname{\preceq_0}|_{Y\times X}=\emptyset$.
Suppose first that $(A0)$ does not hold. Then there are $x_1\leq x_2\in X$, and $y\in Y$ with $x_2 \operatorname{R}y$ but not $x_1 \operatorname{R}y$. But this is impossible if there is a pre-order $\preceq$ on $X\cup Y$ agreeing with $\operatorname{R}$ and extending the order on $X$, as it would have to be transitive, and we would have $x_1 \preceq x_2$, and $x_2 \preceq y$, but not $x_1 \preceq y$. By a duality argument, which we discuss in Remark \[R:dual\] below, it follows that if either (A0) or (A1) fails then ${\mathcal P}_{\operatorname{R}}$ is empty.
Now suppose $\preceq$ is a relation on $(X\cup Y)^2$. Conditions (A2) and (A3) are just the statements that $\preceq$ extends the orders on $X$ and $Y$ respectively, and (A4) is just the statement that $\preceq$ agrees with $\operatorname{R}$ on $X\times Y$. Condition (A5) amounts to demanding a kind of transitivity: $$y\preceq x\implies y\operatorname{R}^d x \implies \Big((x'\operatorname{R}y \text{ and }x\operatorname{R}y')\implies x'\operatorname{R}y'\Big).$$ I.e. if $x'\preceq y$, $y\preceq x$, and $x\preceq y'$, then $x'\preceq y'$. Thus all these condition must certainly hold for $\preceq\in {\mathcal P}_{\operatorname{R}}$. It follows directly from this that any relation $\preceq\in {\mathcal P}_{\operatorname{R}}$ must contain $\operatorname{\preceq_0}$, so to complete the proof it remains only to show that, assuming (A0) and (A1), the relation $\operatorname{\preceq_0}$ is in ${\mathcal P}_{\operatorname{R}}$.
It follows from (A6) and (A7) that $\operatorname{\preceq_0}$ is reflexive, so it remains only to check transitivity. To do this we consider triples $(z_1,z_2,z_3) \in (X\cup Y)^3$, with $z_1 \operatorname{\preceq_0}z_2$, and $z_2\operatorname{\preceq_0}z_3$. A simple counting argument reveals there are eight cases, depending on the containment of each $z_i$ in $X$ or $Y$. The cases where the $z$ values are either all in $X$ or all in $Y$ follow from the fact that $\operatorname{\preceq_0}$ agrees with the orders on $X$ and $Y$. The cases that require $y\operatorname{\preceq_0}x$ are ruled out by (A9), so the only remaining cases are $(x_1, x_2, y)$, where $x_1,x_2\in X$ and $y\in Y$, and $(x, y_1, y_2)$ where $x\in X$ and $y_1,y_2\in Y$. These cases are covered by the assumption of (A0) and (A1), so we are done.
Finally, that ${\mathcal P}_{\operatorname{R}}$ is closed under non-empty intersections follows almost immediately from the definition of ${\mathcal P}_{\operatorname{R}}$ and the fact that intersections of pre-orders are also pre-orders.
\[R:dual\] In the proof of Theorem \[T:ext\] we appealed to a duality principle. This arises from the fact that (A0) and (A1) are, in a sense, dual to each other. Informally, it means something like “by switching some conditions to their (intuitively obvious) duals we could prove this using essentially the same argument", and this ad hoc approach usually suffices to reconstruct proofs as necessary. We formulate the concept precisely in Section \[S:dual\].
Note that the pre-order $\operatorname{\preceq_0}$ defined above is such that the inclusions of $X$ and $Y$ into $X\cup_{\operatorname{\preceq_0}} Y$ are not only monotone but order embeddings. Note also that conditions (A2)-(A5) are necessary but not sufficient for a relation on $(X\cup Y)^2$ to be in ${\mathcal P}_{\operatorname{R}}$. For example, a relation could satisfy these conditions but fail to be transitive when restricted to $X$.
\[D:Pe\] Let ${(X,Y,\operatorname{R})}$ be an order polarity. Define ${\mathcal P}^e_{\operatorname{R}}$ to be the set of pre-orders on $X\cup Y$ agreeing with $\operatorname{R}$ on $X\times Y$, and agreeing with the orders on $X$ and $Y$.
If ${(X,Y,\operatorname{R})}$ is an order polarity then ${\mathcal P}^e_{\operatorname{R}}$ is non-empty if and only if ${\mathcal P}_{\operatorname{R}}$ is non-empty. Moreover, ${\mathcal P}^e_{\operatorname{R}}$ is also closed under arbitrary non-empty intersections.
The first part follows directly from the definition of $\operatorname{\preceq_0}$ in Theorem \[T:ext\]. That ${\mathcal P}^e_{\operatorname{R}}$ is closed under arbitrary non-empty intersections is obvious.
In light of the discussion above we make the following definition.
\[D:0cons\] A polarity ${(X,Y,\operatorname{R})}$ is **0-coherent** if ${\mathcal P}_{\operatorname{R}}$ (or, equivalently, ${\mathcal P}_{\operatorname{R}}^e$) is non-empty. We may sometimes abuse notation slightly by referring to the relation $\operatorname{R}$ as being 0-coherent.
Given an order polarity ${(X,Y,\operatorname{R})}$ and $\preceq\in {\mathcal P}_{\operatorname{R}}$, we use ${X\uplus_\preceq Y}$ to denote the canonical partial order arising from ${X\cup_\preceq Y}$.
Extension polarities
--------------------
Suppose in addition that $X$ and $Y$ are both extensions of some poset $P$. In other words, that there are order embeddings $e_1:P\to X$ and $e_2:P\to Y$. What conditions must $\operatorname{R}$ satisfy in order for there to be $\preceq \in {\mathcal P}_{\operatorname{R}}$ such that the diagram in Figure \[F:fix1\] commutes? Note that in this figure $\iota_X$ and $\iota_Y$ stand for the compositions of the natural inclusion functions into ${X\cup_\preceq Y}$ with the canonical map from ${X\cup_\preceq Y}$ to ${X\uplus_\preceq Y}$.
$$\xymatrix{ P\ar[r]^{e_Y}\ar[d]_{e_X} & Y\ar[d]^{\iota_Y} \\
X\ar[r]_{\iota_X} & {X\uplus_\preceq Y}}$$
As this situation will be the focus of most of the rest of the document, we make the following definition.
An **extension polarity** is a triple $(e_X,e_Y,\operatorname{R})$, where $e_X:P \to X$ and $e_Y:P\to Y$ are order extensions of the same poset $P$, and $(X,Y,\operatorname{R})$ is an order polarity. When both $e_X$ and $e_Y$ are completions, we say $(e_X,e_Y,\operatorname{R})$ is a **completion polarity**. We sometimes say an extension polarity of form ${(e_X,e_Y,\operatorname{R})}$ **extends** $P$. The concept of 0-coherence from Definition \[D:0cons\] also applies, *mutatis mutandis*, to extension polarities.
Note that an order polarity is an extension polarity in the special case where $P$ is empty.
Let ${(e_X,e_Y,\operatorname{R})}$ be an extension polarity. Define ${\hat{\mathcal{P}}}_{\operatorname{R}}$ to be the subset of ${\mathcal P}_{\operatorname{R}}$ containing all $\preceq$ such that the diagram in Figure \[F:fix1\] commutes. We define ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ similarly (recalling Definition \[D:Pe\]).
We will use the following lemma.
\[L:conds\] Let ${(e_X,e_Y,\operatorname{R})}$ be an extension polarity. Suppose ${(e_X,e_Y,\operatorname{R})}$ satisfies $$\tag{$\dagger_0$} e_X(p) \operatorname{R}e_Y(p)\text{ for all }p\in P.$$ Then, if ${(e_X,e_Y,\operatorname{R})}$ satisfies $(A0)$ from Theorem \[T:ext\], it also satisfies $(\dagger_1)$ below. Similarly, if ${(e_X,e_Y,\operatorname{R})}$ satisfies $(A1)$ then it also satisfies $(\dagger_2)$.
- $x \leq e_X(p)\implies x\operatorname{R}e_Y(p)$ for all $p\in P$ and for all $x\in X$.
- $e_Y(p) \leq y \implies e_X(p) \operatorname{R}y$ for all $p\in P$ and for all $y\in Y$.
Moreover, if a polarity ${(e_X,e_Y,\operatorname{R})}$ satisfies either $(\dagger_1)$ or $(\dagger_2)$ then it also satisfies $(\dagger_0)$.
Suppose ${(e_X,e_Y,\operatorname{R})}$ satisfies (A0) and $(\dagger_0)$, and let $x\leq e_X(p)$ for some $x\in X$ and $p\in P$. Then $e_X(p) \operatorname{R}e_Y(p)$ by $(\dagger_0)$, and so $x \operatorname{R}e_Y(p)$ by (A0). Thus ${(e_X,e_Y,\operatorname{R})}$ satisfies ($\dagger_1$). The case where we assume (A1) and $(\dagger_0)$ to prove ($\dagger_2$) is dual. Suppose now that ${(e_X,e_Y,\operatorname{R})}$ satisfies ($\dagger_1$), and let $p\in P$. Then, as $e_X(p)\leq e_X(p)$, we have $e_X(p) \operatorname{R}e_Y(p)$ by ($\dagger_1$), and thus ${(e_X,e_Y,\operatorname{R})}$ satisfies $(\dagger_0)$. The case where we assume ($\dagger_2$) and prove $(\dagger_0)$ is again dual.
\[T:commute\] Let ${(e_X,e_Y,\operatorname{R})}$ be a 0-coherent extension polarity. Then ${\hat{\mathcal{P}}}_{\operatorname{R}}$ is non-empty if and only if:
- $e_X(p) \operatorname{R}e_Y(p)$ for all $p\in P$.
- $x \operatorname{R}e_Y(p)$ and $e_X(p) \operatorname{R}y\implies x\operatorname{R}y$ for all $p\in P$, for all $x\in X$ and for all $y\in Y$.
In addition, if ${(e_X,e_Y,\operatorname{R})}$ satisfies (B0) and (B1), then, given $\preceq\in {\mathcal P}_{\operatorname{R}}$ we have $\preceq\in {\hat{\mathcal{P}}}_{\operatorname{R}}$ if and only if it satisfies either (B2) or (B5), which are equivalent modulo these assumptions. In this case it also satisfies (B3) and (B4).
- $e_Y(p) \preceq e_X(p)$ for all $p\in P$.
- For all $x_1,x_2\in X$ and for all $p\in P$, if either
(i) $x_1\leq x_2$, or
(ii) $x_1 \operatorname{R}e_Y(p)$ and $e_X(p)\leq x_2$,
then $x_1\preceq x_2$.
- For all $y_1,y_2\in Y$ and for all $p\in P$, if either
(i) $y_1\leq y_2$, or
(ii) $y_1\leq e_Y(p)$ and $e_X(p) \operatorname{R}y_2$,
then $y_1\preceq y_2$.
- For all $x\in X$ and for all $y\in Y$, if there are $p,q\in P$ with $y\leq e_Y(p)$, with $e_X(p) \operatorname{R}e_Y(q)$, and with $e_X(q)\leq x$, then $y\preceq x$.
Moreover, if ${\hat{\mathcal{P}}}_{\operatorname{R}}$ is non-empty then it is closed under arbitrary non-empty intersections, and has a least element $\operatorname{\preceq_1}$ defined by the following conditions:
- For all $x_1,x_2\in X$ we have $x_1\operatorname{\preceq_1}x_2\iff$ either (i) $x_1\leq x_2$, or (ii) there is $p\in P$ with $x_1 \operatorname{R}e_Y(p)$ and $e_X(p)\leq x_2$.
- For all $y_1,y_2\in Y$ we have $y_1\operatorname{\preceq_1}y_2\iff$ either (i) $y_1\leq y_2$, or (ii) there is $p\in P$ with $y_1\leq e_Y(p)$ and $e_X(p) \operatorname{R}y_2$.
- For all $x\in X$ and $y\in Y$ we have $x\operatorname{\preceq_1}y\iff x\operatorname{R}y$.
- For all $x\in X$ and $y\in Y$ we have $y \operatorname{\preceq_1}x \iff$ there are $p,q\in P$ with $y\leq e_Y(p)$, with $e_X(p) \operatorname{R}e_Y(q)$, and with $e_X(q)\leq x$.
First of all, (B0) is clearly required if there is to be a pre-order agreeing with $\operatorname{R}$ on $X\times Y$ such that the diagram in Figure \[F:fix1\] commutes, and (B1) is implied by the transitivity of any $\preceq\in{\hat{\mathcal{P}}}_{\operatorname{R}}$.
Now, given $\preceq\in {\hat{\mathcal{P}}}_{\operatorname{R}}$, it is obviously necessary that (B2) hold, as otherwise the diagram will not commute. If $x_1\leq x_2\in X$, then the definition of ${\hat{\mathcal{P}}}_{\operatorname{R}}$ requires that $x_1\preceq x_2$. Suppose then that $x_1 \operatorname{R}e_Y(p)$ and $e_X(p)\leq x_2$ for some $x_1,x_2\in X$ and $p\in P$. Then $x_1\preceq e_Y(p)\preceq e_X(p)\preceq x_2$, and so we must have $x_1\preceq x_2$ by transitivity. It follows that $\preceq$ satisfies (B3), and the argument for (B4) is dual. Similarly, assuming (B2) holds and that $\preceq\in{\mathcal P}_{\operatorname{R}}$, if there are $p,q\in P$ with $y\leq e_Y(p)$, with $e_X(p) \operatorname{R}e_Y(q)$, and with $e_X(q)\leq x$, then $$y\preceq e_Y(p)\preceq e_X(p)\preceq e_Y(q)\preceq x,$$ and so $y\preceq x$ by transitivity, and thus (B2)$\implies$ (B5). Conversely, if we assume (B5) then setting $y = e_Y(p)$ and $x = e_X(p)$ produces (B2), and thus (B2) and (B5) are equivalent as claimed.
If $\preceq\in {\mathcal P}_{\operatorname{R}}$, and $\preceq$ satisfies (B2), then, assuming (B0), the diagram in Figure \[F:fix1\] clearly commutes, and so $\preceq\in {\hat{\mathcal{P}}}_{\operatorname{R}}$. Thus (B2) is a sufficient condition, as well as a necessary one.
We now show that, assuming (B0) and (B1) hold, $\operatorname{\preceq_1}$ as defined above is a pre-order such that the corresponding diagram commutes. That it is reflexive is automatic, so we show now that it is transitive. As in the proof of Theorem \[T:ext\], we consider the eight relevant cases of the triples $(z_1, z_2, z_3) \in (X\cup Y)^3$. Unfortunately we must proceed case by case, and each case may have several subcases.
- $(x_1,x_2,x_3)$: Here $x_1\operatorname{\preceq_1}x_2$, and $x_2\operatorname{\preceq_1}x_3$. This case breaks down into subcases, depending on the reason $\operatorname{\preceq_1}$ holds for each pair.
- If $x_1 \leq x_2$ and $x_2\leq x_3$ in $X$, then we have $x_1\leq x_3$, and thus $x_1\operatorname{\preceq_1}x_3$, by transitivity of $\leq$.
- Suppose instead that $x_1\leq x_2$, and that there is $p\in P$ with $e_X(p)\leq x_3$ and $x_2 \operatorname{R}e_Y(p)$. Then $x_1 \operatorname{R}e_Y(p)$ by (A0) of Theorem \[T:ext\], and so $x_1\operatorname{\preceq_1}x_3$ by (B6).
- Alternatively, if $x_1 \operatorname{R}e_Y(p)$, $e_X(p)\leq x_2$ and $x_2\leq x_3$, then $e_X(p)\leq x_3$, and so $x_1\operatorname{\preceq_1}x_3$ by (B6).
- Finally, suppose there are $p,q\in P$ with $x_1 \operatorname{R}e_Y(p)$, with $e_X(p)\leq x_2$, with $x_2 \operatorname{R}e_Y(q)$ and with $e_X(q)\leq x_3$. Then $e_X(p) \operatorname{R}e_Y(q)$ by (A0), and so $x_1 \operatorname{R}e_Y(q)$ by (B1), and thus $x_1\operatorname{\preceq_1}x_3$ by (B6).
- $(y_1,y_2,y_3)$: This case is dual to the previous one.
- $(x_1,x_2,y)$: Here we have $x_2 \operatorname{\preceq_1}y$, and thus $x_2 \operatorname{R}y$. We also have $x_1\operatorname{\preceq_1}x_2$, which breaks down into two cases.
- First suppose $x_1\leq x_2$. Then $x_1 \operatorname{R}y$ by (A0), and so $x_1\operatorname{\preceq_1}y$ as required.
- Suppose instead that there is $p\in P$ with $x_1 \operatorname{R}e_Y(p)$ and $e_X(p) \leq x_2$. Then $e_X(p) \operatorname{R}y$ by (A0), and so $x_1 \operatorname{R}y$ by (B1), and thus $x_1 \operatorname{\preceq_1}y$ as required.
- $(y,x_1,x_2)$: Here we have $y\operatorname{\preceq_1}x$, and so there are $p,q\in P$ with $y\leq e_Y(p)$, with $e_X(p) \operatorname{R}e_Y(q)$, and with $e_X(q)\leq x_1$, and $x_1\operatorname{\preceq_1}x_2$. There are two subcases.
- Suppose first that $x_1\leq x_2$. Then $e_X(q)\leq x_2$ and the result is an immediate application of (B9).
- Suppose instead that there is $r\in P$ with $x_1 \operatorname{R}e_Y(r)$ and $e_X(r) \leq x_2$. Then an application of (A0) produces $e_X(q) \operatorname{R}e_Y(r)$, and using this with (B1) provides $e_X(p) \operatorname{R}e_Y(r)$. Thus we get $y\operatorname{\preceq_1}x_2$ from (B9).
- $(x_1,y,x_2)$: We have $x_1 \operatorname{R}y$, and, by (B9), there are $p,q\in P$ with $y\leq e_Y(p)$, with $e_X(p) \operatorname{R}e_Y(q)$, and with $e_X(q)\leq x_2$. Then (A1) gives us $x_1 \operatorname{R}e_Y(p)$, and consequently (B1) produces $x_1 \operatorname{R}e_Y(q)$. Thus $x_1\operatorname{\preceq_1}x_2$ by (B6).
- $(y_1, x,y_2)$: Dual to the previous case.
- $(x,y_1,y_2)$: Dual to the $(x_1,x_2,y)$ case.
- $(y_1,y_2,x)$: Dual to the $(y, x_1,x_2)$ case.
From the above argument we conclude that $\operatorname{\preceq_1}$ is transitive, and thus defines a pre-order. To complete the argument that $\operatorname{\preceq_1}\in {\hat{\mathcal{P}}}_{\operatorname{R}}$, note first that $\operatorname{\preceq_1}$ obviously extends the orders on $X$ and $Y$, and so $\operatorname{\preceq_1}\in{\mathcal P}_{\operatorname{R}}$. Finally, that $\operatorname{\preceq_1}$ satisfies (B2) follows easily from (B9) and the fact that $e_Y(p)\leq e_Y(p)$, $e_X(p)\operatorname{R}e_Y(p)$ and $e_X(p)\leq e_X(p)$. Thus $\operatorname{\preceq_1}\in {\hat{\mathcal{P}}}_{\operatorname{R}}$ by a part of this theorem proved previously.
It follows from the fact that $\preceq\in {\hat{\mathcal{P}}}_{\operatorname{R}}$ must satisfy conditions (B3)-(B5) that $\operatorname{\preceq_1}$ is the smallest element of ${\hat{\mathcal{P}}}_{\operatorname{R}}$ when ${\hat{\mathcal{P}}}_{\operatorname{R}}$ is non-empty. That ${\hat{\mathcal{P}}}_{\operatorname{R}}$ is closed under non-empty meets is again essentially obvious.
\[D:1cons\] An extension polarity ${(e_X,e_Y,\operatorname{R})}$ is **1-coherent** if it is 0-coherent and also satisfies conditions (B0)-(B1) of Theorem \[T:commute\]. I.e. if ${\hat{\mathcal{P}}}_{\operatorname{R}}$ is non-empty.
\[C:emb\] Let ${(e_X,e_Y,\operatorname{R})}$ be a 1-coherent extension polarity. Then ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ is non-empty if and only if the following conditions are both satisfied:
- For all $x_1,x_2\in X$ and for all $p\in P$, if $x_1 \operatorname{R}e_Y(p)$ and $e_X(p)\leq x_2$, then $x_1\leq x_2$.
- For all $y_1,y_2\in Y$ and for all $p\in P$, if $y_1\leq e_Y(p)$ and $e_X(p) \operatorname{R}y_2$, then $y_1\leq y_2$.
In this case conditions (B3) and (B4) of Theorem \[T:commute\] are equivalent, respectively, to:
- For all $x_1,x_2\in X$, if $x_1\leq x_2$ then $x_1\preceq x_2$.
- For all $y_1,y_2\in Y$, if $y_1\leq y_2$ then $y_1\preceq y_2$.
Moreover, if ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ is non-empty then it is closed under non-empty intersections, and its least element is $\operatorname{\preceq_1}$ as in Theorem \[T:commute\].
Let $x_1,x_2\in X$, and let $p\in P$ with $x_1 \operatorname{R}e_Y(p)$ and $e_X(p)\leq x_2$. Suppose $\preceq\in{\hat{\mathcal{P}}}^e_{\operatorname{R}}$. Then from condition (B3) of Theorem \[T:commute\] we see that $x_1\preceq x_2$. Thus, since we are assuming the map induced by the inclusion of $X$ into ${X\cup_\preceq Y}$ is an order embedding we must have $x_1\leq x_2$. So (C0) is indeed a necessary condition. A similar argument shows the necessity of (C1).
Moreover, if (C0) holds then (B3) is clearly equivalent to (B3$'$), as the disjunction of $(i)$ and $(ii)$ from (B3) is then equivalent to $x_1\leq x_2$, and similarly if (C1) holds then the same is true for (B4) and (B4$'$). It is easily seen that $\operatorname{\preceq_1}$ is indeed an element of ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ whenever ${(e_X,e_Y,\operatorname{R})}$ satisfies (C0) and (C1), and so must be minimal, as it is minimal in ${\hat{\mathcal{P}}}_{\operatorname{R}}$. That ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ is closed under non-empty intersections is obvious.
\[D:2cons\] An extension polarity ${(e_X,e_Y,\operatorname{R})}$ is **2-coherent** if it is 1-coherent and also satisfies conditions (C0) and (C1) of Corollary \[C:emb\]. I.e. if ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ is non-empty.
As mentioned in the proof of Corollary \[C:emb\], in the case of 2-coherent extension polarities, the conditions (B3), (B4), (B6) and (B7) of Theorem \[T:commute\] simplify, as $(i)\vee(ii)$ and $(i)$ are equivalent in all cases.
We will provide examples showing that the strengths of the coherence conditions defined so far are strictly increasing, but we defer this till section \[S:strict\].
\[D:Pg\] Let ${(e_X,e_Y,\operatorname{R})}$ be an extension polarity. Define ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ to be the subset of ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ such that for all $\preceq\in{\hat{\mathcal{P}}}^g_{\operatorname{R}}$ the following both hold:
1. The induced map $\iota_X:X\to {X\uplus_\preceq Y}$ has the property that, for all $S\subseteq P$, if ${\bigwedge}e_X[S]$ is defined in $X$ then $\iota_X({\bigwedge}e_X[S]) = {\bigwedge}\iota_X\circ e_X[S]$.
2. The induced map $\iota_Y:Y\to {X\uplus_\preceq Y}$ has the property that, for all $T\subseteq P$, if ${\bigvee}e_Y[T]$ is defined in $Y$ then $\iota_Y({\bigvee}e_Y[T]) = {\bigvee}\iota_Y\circ e_Y[T]$.
\[T:commute1\] Let ${(e_X,e_Y,\operatorname{R})}$ be a 2-coherent extension polarity. Then ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is non-empty if and only if ${(e_X,e_Y,\operatorname{R})}$ satisfies the following conditions:
- For all $x\in X$ and $y_1,y_2\in Y$, and for all $S\subseteq P$ with ${\bigwedge}e_X[S]= x$, if $x \operatorname{R}y_2$ and $y_1\leq e_Y(p)$ for all $p\in S$, then $y_1\leq y_2$.
- For all $x_1,x_2\in X$ and $y\in Y$, and for all $T\subseteq P$ with $y = {\bigvee}e_Y[T]$, if $x_1 \operatorname{R}y$ and $e_X(q)\leq x_2$ for all $q\in T$, then $x_1\leq x_2$.
Moreover, if ${(e_X,e_Y,\operatorname{R})}$ satisfies (D0) and (D1), then, given $\preceq\in{\hat{\mathcal{P}}}^e_{\operatorname{R}}$ we have $\preceq \in {\hat{\mathcal{P}}}^g_{\operatorname{R}}$ if and only if it satisfies conditions (D2) and (D3) below.
- For all $y\in Y$, and for all $S\subseteq P$ with ${\bigwedge}e_X[S]$ defined in $X$, we have $\Big(y\leq e_Y(p)$ for all $p\in S\Big)\implies y\preceq {\bigwedge}e_X[S]$.
- For all $x\in X$, and for all $T\subseteq P$ with ${\bigvee}e_Y[T]$ defined in $Y$, we have $\Big(e_X(q)\leq x$ for all $q\in T\Big)\implies {\bigvee}e_Y[T]\preceq x$.
${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is closed under non-empty intersections, and, when non-empty, has a least element $\operatorname{\preceq_3}$ defined by:
- For all $x_1,x_2\in X$ we have $x_1\operatorname{\preceq_3}x_2\iff x_1\leq x_2$.
- For all $y_1,y_2\in Y$ we have $y_1\operatorname{\preceq_3}y_2\iff y_1\leq y_2$.
- For all $x\in X$ and $y\in Y$ we have $x\operatorname{\preceq_3}y\iff x\operatorname{R}y$.
- For all $x\in X$ and for all $y\in Y$ we have $y \operatorname{\preceq_3}x \iff$ either
1. there is $S\subseteq P$ with ${\bigwedge}e_X[S]$ defined in $X$, ${\bigwedge}e_X[S] \leq x$ and $y\leq e_Y(p)$ for all $p\in S$, or
2. there is $T\subseteq P$ with ${\bigvee}e_Y[T]$ defined in $Y$, ${\bigvee}e_Y[T] \geq y$ and $e_X(q)\leq x$ for all $q\in T$.
If (D1) does not hold for some $x_1,x_2\in X$, then for any $\preceq \in {\hat{\mathcal{P}}}^g_{\operatorname{R}}$ we would have $x_1\preceq x_2$, but not $x_1\leq x_2$, which would contradict the definition of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$. Thus (D1) is indeed a necessary condition for ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ to be non-empty. (D0) is necessary by duality.
Suppose now that ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is not empty, and let $\preceq\in {\hat{\mathcal{P}}}^g_{\operatorname{R}}$. To prove the necessity of (D2), let $S\subseteq P$ and suppose ${\bigwedge}e_X[S]$ exists. Suppose that $y\leq e_Y(p)$ for all $p\in S$. Then, by the assumption that $\preceq\in{\hat{\mathcal{P}}}^e_{\operatorname{R}}$ it follows that $y\preceq e_X(p)$ for all $p\in S$, and thus that $y$ is a lower bound for $e_X[S]$. By Definition \[D:Pg\](1), the map $\iota_X:X\to {X\uplus_\preceq Y}$ preserves meets, so we must have $y\preceq {\bigwedge}e_X[S]$ as claimed. Duality proves the necessity of (D3).
Conversely, suppose ${(e_X,e_Y,\operatorname{R})}$ satisfies (D0) and (D1), let $\preceq \in {\hat{\mathcal{P}}}^e_{\operatorname{R}}$ and suppose $\preceq$ satisfies (D2) and (D3). Let $S\subseteq P$ and suppose ${\bigwedge}e_X[S]$ exists in $X$. If $x\in X$ and $\iota_X(x)\leq \iota_X\circ e_X(p)$ for all $p\in S$, then $x \leq {\bigwedge}e_X[S]$, as $\iota_X$ is an order embedding (because $\preceq \in {\hat{\mathcal{P}}}^e_{\operatorname{R}}$), and so $\iota_X(x)\leq \iota_X({\bigwedge}e_X[S])$. Moreover, if $y\in Y$ and $\iota_Y(y)\leq \iota_Y\circ e_Y(p)$ for all $p\in S$, then $y\leq e_Y(p)$ for all $p\in S$, and so $\iota_Y(y)\leq\iota_X({\bigwedge}e_X[S])$, by (D2). From this and the dual result we see that $\preceq\in {\hat{\mathcal{P}}}^g_{\operatorname{R}}$ as claimed.
We must now show that $\operatorname{\preceq_3}$ as defined here induces a pre-order ${X\cup_\preceq Y}$ such that the diagram in Figure \[F:fix1\] commutes, and the maps $\iota_X$ and $\iota_Y$ are order embeddings with the required preservation properties. First of all, it’s obvious that $\operatorname{\preceq_3}$ is reflexive and that the $\iota$ maps are order embeddings. That the diagram commutes follows by using (D6) to get $e_X(p)\operatorname{\preceq_3}e_Y(p)$, and using (D7) with $S= \{p\}$ and $x = e_X(p)$ to give $e_Y(p) \operatorname{\preceq_3}e_X(p)$. Moreover, that $\operatorname{\preceq_3}$ satisfies (D2) and (D3) is automatic from the definition.
The main work now is showing that $\operatorname{\preceq_3}$ is transitive. Again this breaks down into eight cases of form $(z_1,z_2,z_3)$. The cases where a $y$ value does not appear before an $x$ value are covered by the proof of Theorem \[T:commute\] (noting the result of Corollary \[C:emb\]), so the proofs need not be repeated. There are four remaining cases.
- $(y,x_1,x_2)$: We have $x_1\leq x_2$, and two subcases.
- Suppose there is $S\subseteq P$ with ${\bigwedge}e_X[S] \leq x_1$ and $y\leq e_Y(p)$ for all $p\in S$. Then, since $x_1\leq x_2$ we have $y\operatorname{\preceq_3}x_2$ by (D7)(1).
- Suppose instead that there is $T\subseteq P$ with ${\bigvee}e_Y[T] \geq y$ and $e_X(q)\leq x_1$ for all $q\in T$. Then, as $x_1\leq x_2$ we have $e_Y(q) \leq x_2$ for all $t\in T$, and so we have $y\operatorname{\preceq_3}x_2$ by (D7)(2).
- $(y_1,y_2,x)$: Dual to the previous case.
- $(x_1,y,x_2)$: We have $x_1 \operatorname{R}y$ and two subcases.
- Suppose there is $S\subseteq P$ with ${\bigwedge}e_X[S] \leq x_2$ and $y\leq e_Y(p)$ for all $p\in S$. Then, given $p\in S$ we have $x_1 \operatorname{R}e_Y(p)$ by (A1) of Theorem \[T:ext\]. It then follows from (C0) of Corollary \[C:emb\] that $x_1\leq e_X(p)$, and so $x_1\leq {\bigwedge}e_X[S] \leq x_2$ as required.
- Suppose instead that there is $T\subseteq P$ with $y\leq {\bigvee}e_Y[T] $ and $e_X(q)\leq x_2$ for all $q\in T$. Then we have $x_1\operatorname{R}{\bigvee}e_Y[T]$ by (A1), and so $x_1\leq x_2$ by (D1).
- Dual to the previous case.
Thus $\operatorname{\preceq_3}$ is indeed a pre-order, and so is in ${\hat{\mathcal{P}}}_{\operatorname{R}}^g$ as claimed. Finally, that ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is closed under non-empty intersections is again obvious.
\[D:3cons\] An extension polarity ${(X,Y,\operatorname{R})}$ is **3-coherent** if it is 2-coherent and also satisfies conditions (D0) and (D1) of Theorem \[T:commute1\]. I.e. if ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is non-empty.
Note that, when ${(e_X,e_Y,\operatorname{R})}$ is 2-coherent, given $x\in X$ and $y\in Y$, and given $p,q\in P$ such that
1. $y\leq e_Y(p)$,
2. $e_X(p)\operatorname{R}e_Y(q)$, and
3. $e_X(q)\leq x$,
by setting $S = \{q\}$ we have ${\bigwedge}e_X[S]\leq x$, and also $y\leq e_Y(q)$ by (C1) from Corollary \[C:emb\]. It follows that (D2) is at least as strong as (B5) from Theorem \[T:commute\] as a constraint on pre-orders, and the same is true for (D3) by a dual argument. Example \[E:Psep\], later, demonstrates that they are strictly stronger, as, even when ${(e_X,e_Y,\operatorname{R})}$ is 3-coherent, a pre-order may satisfy (B5), but neither (D2) nor (D3).
Galois polarities {#S:Galois}
=================
Entanglement
------------
In applications of polarities to completion theory, the orders on the sets $X$ and $Y$ of an order polarity ${(X,Y,\operatorname{R})}$ are related to $\operatorname{R}$ via a property we present here as Definition \[D:ent\].
\[D:ent\] If ${(X,Y,\operatorname{R})}$ is an order polarity, we say $X$ and $Y$ are **entangled**, if:
- for all $x_1\not\leq x_2\in X$ there is $y\in Y$ with $(x_2,y)\in \operatorname{R}$ and $(x_1,y)\notin \operatorname{R}$, and
- for all $y_1\not\leq y_2\in Y$ there is $x\in X$ with $(x,y_1)\in \operatorname{R}$ and $(x,y_2)\notin \operatorname{R}$.
In this situation we also say that ${(X,Y,\operatorname{R})}$ is an **entangled polarity**. A similar definition applies to extension polarities.
For entangled polarities we can refine Theorem \[T:ext\] using the following lemma.
\[L:ent\] Let ${(X,Y,\operatorname{R})}$ be an entangled order polarity. Then ${(X,Y,\operatorname{R})}$ is 0-coherent if and only if:
- For all $x_1,x_2\in X$ we have $x_1\leq x_2 \iff (x_2\operatorname{R}y\implies x_1\operatorname{R}y$ for all $y\in Y)$.
- For all $y_1,y_2\in Y$ we have $y_1\leq y_2\iff (x\operatorname{R}y_1\implies x\operatorname{R}y_2$ for all $x\in X)$.
We claim that (A0$'$) and (A1$'$) here are equivalent, respectively, to (A0) and (A1) of Theorem \[T:ext\] when ${(X,Y,\operatorname{R})}$ is entangled. This is essentially immediate from the definitions.
In the case of entangled polarities, using (A0$'$) and (A1$'$) we could, if we were so inclined, restate various conditions from Theorems \[T:ext\], \[T:commute\] and \[T:commute1\] to avoid explicit reference to the orders on $X$ and $Y$. Lemma \[L:ent\] also has the following useful corollary.
\[C:ent\] Let ${(X,Y,\operatorname{R})}$ be an entangled order polarity. Then ${\mathcal P}^e_{\operatorname{R}}={\mathcal P}_{\operatorname{R}}$. Similarly, if ${(e_X,e_Y,\operatorname{R})}$ is an entangled extension polarity then ${\hat{\mathcal{P}}}^e_{\operatorname{R}} = {\hat{\mathcal{P}}}_{\operatorname{R}}$.
First note that ${\mathcal P}^e_{\operatorname{R}}\subseteq {\mathcal P}_{\operatorname{R}}$, so if ${\mathcal P}_{\operatorname{R}}$ is empty then so is ${\mathcal P}^e_{\operatorname{R}}$. Thus the case of interest is when ${\mathcal P}_{\operatorname{R}}$ is non-empty. So, appealing to Lemma \[L:ent\] we assume that (A0$'$) and (A1$'$) both hold. Let $\preceq\in {\mathcal P}_{\operatorname{R}}$, and let $x_1\not\leq x_2\in X$. Then, by entanglement, there is $y\in Y$ with $(x_2,y)\in \operatorname{R}$ and $(x_1,y)\notin \operatorname{R}$. So we cannot have $x_1 \preceq x_2$, as otherwise transitivity would produce $x_1 \preceq y$, and consequently $x_1 \operatorname{R}y$. So $\preceq\in {\mathcal P}^e_{\operatorname{R}}$, and thus ${\mathcal P}^e_{\operatorname{R}}={\mathcal P}_{\operatorname{R}}$ as required. That ${\hat{\mathcal{P}}}^e_{\operatorname{R}} = {\hat{\mathcal{P}}}_{\operatorname{R}}$ also follows from this argument.
Defining Galois polarities
--------------------------
A **Galois polarity** is a 3-coherent extension polarity ${(e_X,e_Y,\operatorname{R})}$ such that $e_X:P\to X$ is a meet-extension, and $e_Y:P\to Y$ is a join-extension.
The motivation for the name *Galois polarity* will become clear in section \[S:GC\]. Galois polarities have several strong properties, as we shall see.
\[L:Gent\] Galois polarities are entangled.
Let ${(e_X,e_Y,\operatorname{R})}$ be a Galois polarity, and let $x_1\not\leq x_2\in X$. Then, as $e_X$ is a meet-extension there is $p\in P$ with $x_1\not\leq e_X(p)$, and $x_2\leq e_X(p)$. Thus $x_2 \operatorname{R}e_Y(p)$ by $(\dagger_1)$ of Lemma \[L:conds\]. Moreover, if $x_1 \operatorname{R}e_Y(p)$ then $x_1 \leq e_X(p)$ by (C0) from Corollary \[C:emb\], which contradicts the choice of $p$. We conclude that (E0) holds. A dual argument works for (E1).
If ${(e_X,e_Y,\operatorname{R})}$ is a Galois polarity then ${\hat{\mathcal{P}}}^e_{\operatorname{R}} = {\hat{\mathcal{P}}}_{\operatorname{R}}$.
This follows immediately from Lemma \[L:Gent\] and Corollary \[C:ent\].
For Galois polarities, the structure of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is trivial, as we show in Theorem \[T:unique\]. First, the following technical lemma will be useful.
\[L:GalSimp\] If ${(e_X,e_Y,\operatorname{R})}$ is 3-coherent then a) and b) below both imply c) for all $x\in X$ and for all $y\in Y$. Moreover, if ${(e_X,e_Y,\operatorname{R})}$ is Galois then a), b) and c) are all equivalent for $x$ and $y$.
a) There is $S\subseteq P$ with ${\bigwedge}e_X[S] \leq x$ and $y\leq e_Y(p)$ for all $p\in S$.
b) There is $T\subseteq P$ with ${\bigvee}e_Y[T] \geq y$ and $e_X(q)\leq x$ for all $q\in T$.
c) For all $p,q\in P$, if $e_Y(p)\leq y$ and $x\leq e_X(q)$, then $p\leq q$.
As ${(e_X,e_Y,\operatorname{R})}$ is 3-coherent, ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$ is non-empty, so let $\preceq\in{\hat{\mathcal{P}}}^e_{\operatorname{R}}$. Suppose first that a) holds for $x$ and $y$, and let $p,q\in P$ with $e_Y(p)\leq y$ and $x\leq e_X(q)$. Then we have $$e_Y(p)\preceq y \preceq {\bigwedge}e_X[S] \preceq x \preceq e_X(q)$$ for some $S\subseteq P$, by appealing to (D2) from Theorem \[T:commute1\]. By commutativity of the diagram in Figure \[F:fix1\] we must therefore have $e_X(p)\preceq e_X(q)$, and so $p\leq q$. This shows a)$\implies$ c), and a dual argument shows b)$\implies$ c).
Suppose now that ${(e_X,e_Y,\operatorname{R})}$ is Galois and that c) holds for $x$ and $y$. As ${(e_X,e_Y,\operatorname{R})}$ is Galois we have $x = {\bigwedge}e_X[S]$ for $S = e_X^{-1}(x^\uparrow)$, and $y = {\bigvee}e_Y[T]$ for $T=e_Y^{-1}(y^\downarrow)$. By c) we have $q\leq p$ for all $q\in T$ and $p\in S$. Given $\preceq\in {\hat{\mathcal{P}}}^g_{\operatorname{R}}$ we thus have $e_X(q) \preceq e_Y(p)$, and as $\iota_X$ and $\iota_Y$ are meet- and join-preserving respectively, we must have $y = {\bigvee}e_Y[T] \preceq {\bigwedge}e_X[S] = x$, and thus $y\leq e_Y(p)$ for all $p\in S$, and $e_X(q)\leq x$ for all $q\in T$. It follows that c) implies both a) and b), and so we have the claimed equivalence.
\[T:unique\] If ${(e_X,e_Y,\operatorname{R})}$ is a Galois polarity then ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ contains only the element $\operatorname{\preceq_3}$ defined in Theorem \[T:commute1\], and the maps $\iota_X:X\to {X\uplus_{\operatorname{\preceq_3}} Y}$ and $\iota_Y:Y\to {X\uplus_{\operatorname{\preceq_3}} Y}$ are completely meet- and join-preserving respectively. Moreover, in this case $\operatorname{\preceq_3}$ can be defined as follows:
- For all $x_1,x_2\in X$ we have $x_1\operatorname{\preceq_3}x_2\iff x_1\leq x_2$ in $X$.
- For all $y_1,y_2\in Y$ we have $y_1\operatorname{\preceq_3}y_2\iff y_1\leq y_2$ in $Y$.
- For all $x\in X$ and $y\in Y$ we have $x\operatorname{\preceq_3}y\iff x\operatorname{R}y$.
- For all $x\in X$ and $y\in Y$ we have $y\operatorname{\preceq_3}x\iff p\leq q$ for all $p\in e_Y^{-1}(y^\downarrow)$ and $q\in e_X^{-1}(x^\uparrow)$.
We will start by proving that the alternative definition of $\operatorname{\preceq_3}$ is correct. Since (G0), (G1) and (G2) are identical to (D4), (D5) and (D6) respectively, and that (G3) is equivalent to (D7) follows immediately from Lemma \[L:GalSimp\].
That $\iota_X$ and $\iota_Y$ are completely meet- and join-preserving is a simple consequence of the definition of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ (Definition \[D:Pg\]) and the fact that $x = {\bigwedge}e_X[e_X^{-1}(x^\uparrow)]$ and $y = {\bigvee}e_Y[e_Y^{-1}(y^\downarrow)]$ for all $x\in X$ and $y\in Y$.
To see that $\operatorname{\preceq_3}$ is the only element of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ note first that it must be the smallest element, by definition. Moreover, if $\preceq\in{\hat{\mathcal{P}}}^g_{\operatorname{R}}$, then $\preceq$ is determined either by the orders on $X$ and $Y$, or by $\operatorname{R}$, everywhere except on $Y\times X$. So $\preceq\neq\operatorname{\preceq_3}$ if and only if there is $x\in X$ and $y\in Y$ with $y\preceq x$ and $y\cancel{\operatorname{\preceq_3}} x$. But this is impossible, as for any $p,q\in P$ with $e_Y(p)\leq y$ and $x\leq e_X(q)$ we are forced to have $p\leq q$ by the transitivity of $\preceq$ and the commutativity of the diagram in Figure \[F:fix1\].
Given a 0-coherent extension polarity $E={(e_X,e_Y,\operatorname{R})}$ where $e_X$ and $e_Y$ are meet- and join-extensions respectively, there is a simple necessary and sufficient condition for $E$ to be Galois.
\[P:GalSimp\] Let ${(e_X,e_Y,\operatorname{R})}$ be 0-coherent and let $e_X$ and $e_Y$ be, respectively, meet- and join-extensions of $P$. Then ${(e_X,e_Y,\operatorname{R})}$ is Galois if and only if the following both hold:
1. For all $p\in P$ and for all $x\in X$ we have $x\leq e_X(p)\iff x \operatorname{R}e_Y(p)$.
2. For all $p\in P$ and for all $y\in Y$ we have $e_Y(p)\leq y\iff e_X(p) R y$.
Suppose first that ${(e_X,e_Y,\operatorname{R})}$ is Galois, and let $p\in P$ and $x\in X$. Suppose $x\leq e_X(p)$. Then $x \operatorname{R}e_Y(p)$ by Lemma \[L:conds\]. Conversely, if $x\operatorname{R}e_Y(p)$ then $x\leq e_X(p)$ by (C0) of Corollary \[C:emb\]. Thus (S0) holds, and (S1) holds by a dual argument.
Suppose now that ${(e_X,e_Y,\operatorname{R})}$ is 0-coherent and satisfies (S0) and (S1), and also that $e_X$ and $e_Y$ are meet- and join-extensions respectively. We will show that the necessary conditions from Theorems \[T:commute\] and \[T:commute1\], and Corollary \[C:emb\], are satisfied.
- This is trivial.
- This follows from (S0) and (A0) from Theorem \[T:ext\].
- Let $x_1\operatorname{R}e_Y(p)$. Then $x_1\leq e_X(p)$ by (S0), and so if $e_X(p)\leq x_2$ then $x_1\leq x_2$ by transitivity of $\leq$.
- This is dual to (C0).
- Let ${\bigwedge}e_X[S] = x$, let $x\operatorname{R}y_2$, and suppose $y_1\leq e_Y(p)$ for all $p\in S$. Let $q\in P$ and suppose $e_Y(q)\leq y_1$. Then $q\leq p$ for all $p\in S$, and so $e_X(q)\leq x$. Thus $e_X(q) \operatorname{R}y_2$ by (A0), and so $e_Y(q)\leq y_2$ by (S1). As $e_Y$ is a join-extension it follows that $y_1\leq y_2$ as required.
- This is dual to (D0).
It follows from Proposition \[P:GalSimp\] that what we call a Galois polarity corresponds to what [@GJP13 Section 4] calls a $\Delta_1$-polarity. See also [@GJP13 Proposition 4.1], which tells us that the pre-order $\operatorname{\preceq_3}$ described in Theorem \[T:unique\] is the one arising naturally from ${G(X,Y,\operatorname{R})}$. Theorem \[T:unique\] says that this is in fact the only way we can pre-order $X\cup Y$ if we want the properties defining ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ to hold. Note of course that if ${(e_X,e_Y,\operatorname{R})}$ is not Galois then $\operatorname{\preceq_3}$ may not be a pre-order.
Since for Galois polarities ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ has only the one member, to lighten the notation we will from now on write e.g. $X\uplus Y$ in place of ${X\uplus_{\operatorname{\preceq_3}} Y}$ when working with Galois polarities.
The satisfaction and separation of the coherence conditions {#S:satisfaction}
===========================================================
Sets of coherent relations {#S:satisfaction1}
--------------------------
If $X$ and $Y$ are posets, it’s easy to see that the set of relations on $X\times Y$ such that the induced order polarity is 0-coherent is closed under arbitrary unions and intersections, and has $\emptyset$ and $X\times Y$ as least and greatest elements respectively. The situation for extension polarities and more restrictive forms of coherence is a little more delicate, as illustrated by Proposition \[P:relations\] below. First we introduce another definition.
Let $e_X:P\to X$ and $e_Y:P\to Y$ be poset extensions. For $\ast \in\{0,1,2,3\}$, define $\mathcal R^{(e_X,e_Y)}_\ast $ to be the set of relations on $X\times Y$ such that $\operatorname{R}\in \mathcal R^{(e_X,e_Y)}_\ast \iff{(e_X,e_Y,\operatorname{R})}$ is $\ast $-coherent.
\[P:relations\] Let $e_X:P\to X$ and $e_Y:P\to Y$ be poset extensions. Then $\mathcal R^{(e_X,e_Y)}_\ast $ is closed under arbitrary non-empty intersections for all $\ast \in\{0,1,2,3\}$.
Moreover, define the relation $\operatorname{R}_l$ by $$x \operatorname{R}_l y \iff e_X^{-1}(x^\uparrow)\cap e_Y^{-1}(y^\downarrow)\neq\emptyset.$$ Then $\operatorname{R}_l$ is the minimal element of $\mathcal R^{(e_X,e_Y)}_\ast $ for $\ast \in\{0,1,2\}$. If $e_X$ and $e_Y$ are meet- and join-extensions respectively, then the same is true for $\ast = 3$.
We dealt with the case where $\ast = 0$ in the preamble to this section. So, let $\ast \in\{1,2,3\}$, let $I$ be an indexing set, and, for all $i\in I$, let $\operatorname{R}_i$ be a relation on $X\times Y$ such that $(e_X, e_Y, \operatorname{R}_i)$ is $\ast $-coherent. To show that $(e_X, e_Y, \bigcap_I\operatorname{R}_i)$ is also $\ast $-coherent involves only a routine check of the relevant conditions from Theorems \[T:commute\] and \[T:commute1\], and Corollary \[C:emb\].
If $\operatorname{R}$ is a relation such that ${(e_X,e_Y,\operatorname{R})}$ is 1-coherent, then $\operatorname{R}$ must satisfy (A0), (A1) and (B0), and it follows that $\operatorname{R}_l\subseteq \operatorname{R}$. We should show that $\operatorname{R}_l$ does indeed produce a 2-coherent polarity $(e_X, e_Y, \operatorname{R}_l)$ for every choice of $e_X$ and $e_Y$, but this again is a routine check of the relevant conditions, so we omit the details.
Finally, suppose that $e_X$ is a meet-extension and $e_Y$ is a join-extension. We will check that $\operatorname{R}_l$ also satisfies (D0). Let $S\subseteq P$, and let $x = {\bigwedge}e_X[S]$ in $X$. Let $y_1,y_2\in Y$ and suppose that $y_1 \leq e_Y(p)$ for all $p\in S$, and that $x \operatorname{R}_l y_2$. Let $q\in P$ and suppose $e_Y(q)\leq y_1$. Then $e_Y(q)\leq e_Y(p)$, and thus $q\leq p$, for all $p\in S$. It follows that $e_X(q)\leq x$. Also, by definition of $\operatorname{R}_l$, there is $q'\in P$ with $x\leq e_X(q')$ and $e_Y(q')\leq y_2$. But then $q\leq q'$, and consequently $e_Y(q)\leq y_2$. This is true for all $q\in e_Y^{-1}(y_1^\downarrow)$, and so $y_1 \leq y_2$ as $e_Y$ is a join-extension. $\operatorname{R}_l$ also satisfies (D1) by duality, and so the proof is complete.
Note that when $e_Y$ is not a join-extension $\operatorname{R}_l$ may not satisfy (D0), as Example \[E:D0fail\] demonstrates. In this case $\mathcal R^{(e_X,e_Y)}_3$ is empty, as every $\operatorname{R}$ such that ${(e_X,e_Y,\operatorname{R})}$ is 3-coherent must contain $\operatorname{R}_l$, by the proof of Proposition \[P:relations\]. By duality, when $e_X$ is not a meet-extension $\operatorname{R}_l$ may not satisfy (D1), and in this case too $\mathcal R^{(e_X,e_Y)}_3$ will be empty, for the same reason.
\[E:D0fail\] Let $P$ be the poset in Figure \[F:D0failP\], and let $e_X$ and $e_Y$ be the extensions defined in Figures \[F:D0failX\] and \[F:D0failY\] respectively. Note that $e_X$ is a meet-extension, but $e_Y$ is not a join-extension. Let $S= \{p,q\}$. Then $x = {\bigwedge}e_X[S]$, and $x\operatorname{R}_l e_Y(r)$. But we also have $y\leq e_Y(p)$ and $y\leq e_Y(q)$, but $y\not\leq e_Y(r)$. So (D0) does not hold for $\operatorname{R}_l$.
$$\xymatrix{ \bullet_p & \bullet_q & \bullet_r
}$$
$$\xymatrix{ \bullet_p\ar@{-}[dr] & \bullet_q\ar@{-}[d] & \bullet_r\ar@{-}[dl] \\
& \circ_x &
}$$
$$\xymatrix{\bullet_p\ar@{-}[d] & \bullet_q\ar@{-}[dl] & \bullet_r \\
\circ_{y}
}$$
A strict hierarchy for coherence {#S:strict}
--------------------------------
Example \[E:D0fail\], taken with Proposition \[P:relations\], also demonstrates that it is possible for an order polarity to be 2-coherent but not 3-coherent (take $(e_X, e_Y, \operatorname{R}_l)$ from this example). Thus 3-coherence is a strictly stronger condition than 2-coherence. However, this example only applies when either $e_Y$ fails to be a join-extension, or, by duality, when $e_X$ fails to be a meet-extension. Example \[E:D0fail2\] below demonstrates that, even when $e_X$ and $e_Y$ *are* meet- and join-extensions respectively, there may be choices of $\operatorname{R}$ for which ${(e_X,e_Y,\operatorname{R})}$ is 2-coherent but not 3-coherent.
\[E:D0fail2\] Let $e_X$ and $e_Y$ be as in Figures \[F:D0failX2\] and \[F:D0failY2\] respectively, where the embedded images of elements of $P$ are represented using $\bullet$, and the extra elements of $X$ and $Y$ using $\circ$. Then it’s easy to see that $e_X$ and $e_Y$ are meet- and join-extensions respectively. Moreover, if we define $\operatorname{R}= \operatorname{R}_l\cup \{(x, y_2)\}$ then ${(e_X,e_Y,\operatorname{R})}$ is 2-coherent, as can be observed by noting the pre-order on $X\cup Y$ defined in Figure \[F:2cons\]. However, ${(e_X,e_Y,\operatorname{R})}$ is not 3-coherent, as any $\preceq \in {\hat{\mathcal{P}}}^e_{\operatorname{R}}$ preserving the meet that defines $x$ would necessarily have $y_1\preceq y_2$, which would contradict the definition of ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$.
$$\xymatrix{ \bullet\ar@{-}[dr] & \bullet\ar@{-}[d] \\
& \circ_x\\
\bullet\ar@{-}[ur] & \bullet\ar@{-}[u] & \bullet & \bullet
}$$
$$\xymatrix{\bullet\ar@{-}[d] & \bullet\ar@{-}[dl] & \circ_{y_2} \\
\circ_{y_1} & \\
\bullet\ar@{-}[u] & \bullet\ar@{-}[ul] & \bullet\ar@{-}[uu] & \bullet\ar@{-}[uul]
}$$
$$\xymatrix{\bullet\ar@{-}[dr] & \bullet\ar@{-}[d] & \circ_{y_2} \\
\circ_{y_1}\ar@{-}[ur]\ar@{-}[u] & \circ_{x}\ar@{-}[ur] & \\
\bullet\ar@{-}[ur]\ar@{-}[u] & \bullet\ar@{-}[u]\ar@{-}[ul] & \bullet\ar@{-}[uu] & \bullet\ar@{-}[uul]
}$$
2-coherence is also a strictly stronger condition than 1-coherence, as witnessed by Example \[E:1not2\] below.
\[E:1not2\] Let $P$ be the two element antichain $\{p,q\}$. Define $X\cong Y\cong P$, and let $e_X$ and $e_Y$ be isomorphisms. Define $\operatorname{R}= \operatorname{R}_l\cup \{(e_X(p), e_Y(q))\}$. Then ${(e_X,e_Y,\operatorname{R})}$ is 1-coherent, but not 2-coherent.
Separating the classes of pre-orders
------------------------------------
We have seen that the classes of extension polarities defined by the coherence conditions are strictly separated. It is also true that, even for a Galois polarity ${(e_X,e_Y,\operatorname{R})}$ we may have ${\hat{\mathcal{P}}}^g_{\operatorname{R}}\subset {\hat{\mathcal{P}}}^e_{\operatorname{R}}$, and for a 3-coherent ${(e_X,e_Y,\operatorname{R})}$ we may have ${\hat{\mathcal{P}}}^g_{\operatorname{R}}\subset {\hat{\mathcal{P}}}^e_{\operatorname{R}}\subset {\hat{\mathcal{P}}}_{\operatorname{R}}$ (from Corollary \[C:ent\] we know this is not true for Galois polarities). This is demonstrated in Examples \[E:Psep\] and \[E:Psep2\] respectively.
\[E:Psep\] Let $e_X$ and $e_Y$ be as in Figures \[F:enotgX\] and \[F:enotgY\] respectively. Let $\operatorname{R}=\operatorname{R}_l$. Then ${(e_X,e_Y,\operatorname{R})}$ is Galois, by Proposition \[P:relations\], but the pre-order represented in Figure \[F:enotgC\] is in ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$, but is not in ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$.
$$\xymatrix{\bullet & \bullet \\
& \circ_{x}\ar@{-}[u]\ar@{-}[ul]\\
\bullet\ar@{-}[ur] & \bullet\ar@{-}[u]
}$$
$$\xymatrix{
\bullet\ar@{-}[d] & \bullet\ar@{-}[dl] \\
\circ_{y} \\
\bullet\ar@{-}[u] & \bullet\ar@{-}[ul]
}$$
$$\xymatrix{\bullet & \bullet \\
\circ_{y}\ar@{-}[ur]\ar@{-}[u] & \circ_{x}\ar@{-}[u]\ar@{-}[ul] \\
\bullet\ar@{-}[ur]\ar@{-}[u] & \bullet\ar@{-}[u]\ar@{-}[ul]
}$$
\[E:Psep2\] Let $P$ and $e_X$ be as in Example \[E:Psep\], and let $e_Y$ be defined by the diagram in Figure \[F:noteY\]. Again let $\operatorname{R}=\operatorname{R}_l$. Then ${(e_X,e_Y,\operatorname{R})}$ is 3-coherent by Proposition \[P:relations\]. However, there is a pre-order in ${\hat{\mathcal{P}}}^e_{\operatorname{R}}\setminus {\hat{\mathcal{P}}}^g_{\operatorname{R}}$ based on that in Figure \[F:enotgC\], and a pre-order in ${\hat{\mathcal{P}}}_{\operatorname{R}}\setminus {\hat{\mathcal{P}}}^e_{\operatorname{R}}$ obtained by additionally setting $z_2\preceq z_1$.
$$\xymatrix{
\bullet\ar@{-}[d] & \bullet\ar@{-}[dl] \\
\circ_{y} & & \circ_{z_2}\\
\bullet\ar@{-}[u] & \bullet\ar@{-}[ul] & \circ_{z_1}\ar@{-}[u]
}$$
Extending and restricting polarity relations {#S:ExtRes}
============================================
Extension
---------
If $e:P\to Q$ is an order extension, then given another order extension $e':Q\to Q'$, the composition $e'\circ e$ is also an order extension. It is natural to ask whether an extension polarity ${(e_X,e_Y,\operatorname{R})}$ can be extended to something like $(e'_X\circ e_X, e'_Y\circ e_Y, \operatorname{R}')$, and under what circumstances the level of coherence of $\operatorname{R}$ transfers to $\operatorname{R}'$. This is of particular interest, for example, if we wish to extend $e_X$ and $e_Y$ to completions, as we shall do in Section \[S:GC\]. The next theorem provides some answers.
\[T:Galois\] Let ${(e_X,e_Y,\operatorname{R})}$ be an extension polarity, let $i_X:X\to{\overline{X}}$ and $i_Y:Y\to{\overline{Y}}$ be order extensions. Let $\operatorname{\overline{R}}$ be the relation on ${\overline{X}}\times{\overline{Y}}$ defined by $$x' \operatorname{\overline{R}}y' \iff \text{ there is } x\in X \text{ and } y\in Y \text{ with } x'\leq i_X(x)\text{, } i_Y(y)\leq y'\text{, and } x \operatorname{R}y.$$ Then:
1. $({\overline{X}}, {\overline{Y}}, \operatorname{\overline{R}})$ is 0-coherent.
2. For all $x\in X$ and for all $y\in Y$ we have $x \operatorname{R}y\implies i_X(x) \operatorname{\overline{R}}i_Y(y)$, and the converse is true if and only if $(X, Y, \operatorname{R})$ is 0-coherent.
3. If ${(e_X,e_Y,\operatorname{R})}$ is $\ast$-coherent then $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is $\ast$-coherent, for $\ast \in\{1, 2\}$.
4. If ${(e_X,e_Y,\operatorname{R})}$ is Galois, and if $i_X:X\to {\overline{X}}$ and $i_Y:Y\to{\overline{Y}}$ are meet- and join-extensions respectively, then $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is also Galois.
5. Let $\operatorname{S}\subseteq{\overline{X}}\times{\overline{Y}}$ satisfy (A0) and (A1), and suppose $x\operatorname{R}y\implies i_X(x)\operatorname{S}i_Y(y)$. Then $\operatorname{\overline{R}}\subseteq \operatorname{S}$.
6. If $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is not $\ast$-coherent, then there is no $\operatorname{S}\subseteq {\overline{X}}\times {\overline{Y}}$ satisfying the conditions from part (5) such that $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is $\ast$-coherent, for $\ast\in\{2,3\}$.
<!-- -->
1. We check that $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is 0-coherent using Theorem \[T:ext\]. We need only check (A0) as (A1) is dual. Let $x'_1\leq x_2'\in {\overline{X}}$, let $y'\in {\overline{Y}}$, and suppose $x_2' \operatorname{\overline{R}}y'$. Then there is $x\in X$ and $y\in Y$ with $x'_2\leq i_X(x)$, with $i_Y(y)\leq y'$, and with $x \operatorname{R}y$. But then $x_1' \operatorname{\overline{R}}y'$, by definition of $\operatorname{\overline{R}}$, so (A0) holds.
2. If $x \operatorname{R}y$ then that $i_X(x) \operatorname{\overline{R}}i_Y(y)$ follows directly from the definition. Conversely, suppose ${(e_X,e_Y,\operatorname{R})}$ is 0-coherent and $i_X(x_1) \operatorname{\overline{R}}i_Y(y_1)$. Then there is $x_2\in X$ and $y_2\in Y$ with $x_1\leq x_2$, with $x_2 \operatorname{R}y_2$, and with $y_2\leq y_1$. It follows from 0-coherence of ${(e_X,e_Y,\operatorname{R})}$ that $x_1\operatorname{R}y_1$ as required. Moreover, $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is always 0-coherent by (1), so, if the converse holds ${(e_X,e_Y,\operatorname{R})}$ inherits 0-coherence from $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$.
3. Now suppose ${(e_X,e_Y,\operatorname{R})}$ is 1-coherent. Appealing to Theorem \[T:commute\], we check that (B0) and (B1) hold for $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$.
- Let $p\in P$. Then $e_X(p) \operatorname{R}e_Y(p)$ as ${(e_X,e_Y,\operatorname{R})}$ is 1-coherent, and it follows easily that $i_X\circ e_X(p) \operatorname{\overline{R}}i_Y\circ e_Y(p)$. Thus (B0) holds for $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ as required.
- Let $x'\in {\overline{X}}$, let $y'\in {\overline{Y}}$, and let $p\in P$. Suppose $x' \operatorname{\overline{R}}(i_Y\circ e_Y(p))$ and $(i_X\circ e_X(p)) \operatorname{\overline{R}}y'$. Then there are $x_1\in X$ and $y_1\in Y$, with $x'\leq i_X(x_1)$, with $x_1 \operatorname{R}y_1$, and with $i_Y(y_1)\leq i_Y\circ e_Y(p)$, and also $x_2\in X$ and $y_2\in Y$ with $i_X\circ e_X(p)\leq i_X(x_2)$, with $x_2 \operatorname{R}y_2$, and with $i_Y(y_2)\leq y'$. As $i_X$ and $i_Y$ are order embeddings we have $y_1\leq e_Y(p)$ and $e_X(p)\leq x_2$. As ${(e_X,e_Y,\operatorname{R})}$ is 1-coherent we have $\operatorname{\preceq_1}\in{\hat{\mathcal{P}}}^e_{\operatorname{R}}$, and thus $$x_1\operatorname{\preceq_1}y_1 \operatorname{\preceq_1}e_Y(p)\operatorname{\preceq_1}e_X(p)\operatorname{\preceq_1}x_2\operatorname{\preceq_1}y_2.$$ So $x_1 \operatorname{R}y_2$ by transitivity of $\operatorname{\preceq_1}$ and the fact that it agrees with $\operatorname{R}$ on $X\times Y$. It follows that $x' \operatorname{\overline{R}}y'$, by the definition of $\operatorname{\overline{R}}$, and so (B1) holds.
Thus $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is 1-coherent. Suppose now that ${(e_X,e_Y,\operatorname{R})}$ is 2-coherent. Appealing to Corollary \[C:emb\], we check that (C0) holds for $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$. Let $x_1',x_2'\in {\overline{X}}$, and let $p\in P$. Suppose $x'_1 \operatorname{\overline{R}}(i_Y\circ e_Y(p))$, and $i_X\circ e_X(p)\leq x'_2$. Then there are $x\in X$ and $y\in Y$ with $x'_1\leq i_X(x)$, with $i_Y(y)\leq i_Y\circ e_Y(p)$, and with $x\operatorname{R}y$. As ${(e_X,e_Y,\operatorname{R})}$ is 2-coherent we know $\operatorname{\preceq_1}\in{\hat{\mathcal{P}}}^e_{\operatorname{R}}$, by Corollary \[C:emb\], and we have $$x \operatorname{\preceq_1}y\operatorname{\preceq_1}e_Y(p) \operatorname{\preceq_1}e_X(p).$$ So $x\leq e_X(p)$, by definition of ${\hat{\mathcal{P}}}^e_{\operatorname{R}}$, and consequently $$x_1'\leq i_X(x)\leq i_X\circ e_X(p).$$ Thus $x_1'\leq x_2'$, as $i_X\circ e_X(p)\leq x'_2$, and so (C0) holds. By duality (C1) also holds, and so $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is 2-coherent as claimed.
4. Suppose now that ${(e_X,e_Y,\operatorname{R})}$ is 3-coherent, and that the $i_X$ and $i_Y$ are meet- and join-extensions respectively. First, that $i_X\circ e_X$ and $i_Y\circ e_Y$ are meet- and join-extensions respectively follows from the corresponding properties of $i_X$, $e_X$, $i_Y$ and $e_Y$. It remains only to check that (D0) and (D1) hold for $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ and appeal to Theorem \[T:commute1\].
Let $x'\in {\overline{X}}$, let $y_1',y_2'\in {\overline{Y}}$, and let $S\subseteq P$. Suppose ${\bigwedge}(i_X\circ e_X[S]) = x'$. Suppose also that $x' \operatorname{\overline{R}}y_2'$, and that $y_1'\leq i_Y\circ e_Y(p)$ for all $p\in S$. Then there are $x\in X$ and $y\in Y$ with $x'\leq i_X(x)$ and $i_Y(y)\leq y_2'$, and with $x \operatorname{R}y$. We aim to prove that $y_1'\leq y_2'$.
Let $y_0\in Y$ be such that $i_Y(y_0)\leq y_1'$, and let $q\in e_Y^{-1}(y_0^\downarrow)$. Then $$e_Y(q) \leq y_0 \leq e_Y(p) \text{ for all } p\in S,$$ and so $i_X\circ e_X(q)\leq x'\leq i_X(x)$, and consequently $e_X(q)\leq x$. Since ${(e_X,e_Y,\operatorname{R})}$ is Galois, ${\hat{\mathcal{P}}}_{\operatorname{R}}^g$ contains $\operatorname{\preceq_3}$, and we have $y_0\operatorname{\preceq_3}x$ as the map $\iota_Y: Y\to X\uplus_{\operatorname{\preceq_3}} Y$ preserves joins of sets in $e_Y[P]$ and $y_0 = {\bigvee}e_Y[y_Y^{-1}(y_0^\downarrow)]$. So we have $$y_0 \operatorname{\preceq_3}x \operatorname{\preceq_3}y,$$ and thus $y_0\leq y$ for all $y_0$ with $i_Y(y_0)\leq y'_1$. But, as $i_Y$ is a join-extension, we have $$y_1' = {\bigvee}i_Y[i_Y^{-1}(y_1'^\downarrow)],$$ and so $y_1'\leq i_Y(y)\leq y'_2$, which is what we are trying to prove. It follows that (D0) holds for $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$, and thus by duality (D1) also holds.
5. Suppose $x'\operatorname{\overline{R}}y'$. Then there is $x\in X$ and $y\in Y$ with $x'\leq i_X(x)$, $x\operatorname{R}y$, and $i_Y(y) \operatorname{R}y'$. Let $\operatorname{S}\subseteq {\overline{X}}\times {\overline{Y}}$ satisfy the conditions from (5). Then $i_X(x) \operatorname{S}i_Y(y)$, and the result follows from (A0) and (A1).
6. From (5) we know that any relation on ${\overline{X}}\times{\overline{Y}}$ that ‘extends $\operatorname{R}$’ must contain $\operatorname{\overline{R}}$. Examination of the conditions (C0), (C1), (D0) and (D1) reveals that if they fail for $\operatorname{\overline{R}}$ they will also fail for any relation containing $\operatorname{\overline{R}}$.
Theorem \[T:Galois\], specifically parts (5) and (6), tells us that if we want to find a 2- or 3-coherent polarity extending ${(e_X,e_Y,\operatorname{R})}$, then it suffices to look at $\operatorname{\overline{R}}$, as if this does not produce the desired result then nothing will. Note that this does not apply for 1-coherence. To see this let $P=\{p\}\cong X\cong Y\cong {\overline{X}}\cong {\overline{Y}}$, and let $\operatorname{R}=\emptyset$. Then (B0) fails for $\operatorname{\overline{R}}$, but if $\operatorname{S}=\{(i_X\circ e_X(p)), i_Y\circ e_Y(p)\}$ then $(i_X\circ e_X,i_Y\circ e_Y,\operatorname{S})$ is obviously 1-coherent.
For 0-coherent polarities we can add converses to some of the statements in Theorem \[T:Galois\], but we will leave this till Corollary \[C:converses\]. Note that for $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ to be 3-coherent it is not sufficient for ${(e_X,e_Y,\operatorname{R})}$ to be 3-coherent, or even Galois. The additional restrictions on the extensions $i_X$ and $i_Y$ from Theorem \[T:Galois\](4) are necessary, as Example \[E:notSuf\] demonstrates below.
\[E:notSuf\] Let $P$ be the three element antichain from Figure \[F:D0failP\], and let $X \cong Y \cong P$. Let ${\overline{X}}$ and ${\overline{Y}}$ be the poset extensions illustrated in Figures \[F:D0failX\] and \[F:D0failY\] respectively. Define $\operatorname{R}$ on $X\times Y$ by $x\operatorname{R}y\iff$ there is $p\in P$ with $x=e_X(p)$ and $y= e_Y(p)$.
We can put a poset structure on $X\cup Y$ just by identifying copies elements of $P$ appropriately, in which case we end up with something isomorphic to $P$. Clearly the natural maps $\iota_X$ and $\iota_Y$ are meet- and join-preserving order embeddings, and so ${(e_X,e_Y,\operatorname{R})}$ is Galois. However, $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is not 3-coherent. Indeed, it follows from Example \[E:D0fail\] that there is no relation $\operatorname{S}$ such that $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is 3-coherent.
The following lemma says, roughly, that the extension of the ‘minimal’ polarity relation $\operatorname{R}_l$ is again the minimal polarity relation.
\[L:minExt\] Let $(e_X,e_Y,\operatorname{R}_l)$ be an extension polarity, where $\operatorname{R}_l$ is as in Proposition \[P:relations\], and let $i_X:X\to{\overline{X}}$ and $i_Y:Y\to{\overline{Y}}$ be order extensions. Then $\overline{\operatorname{R}_l} = \operatorname{S}_l$, where $\operatorname{S}_l\subseteq {\overline{X}}\times {\overline{Y}}$ is defined analogously to $\operatorname{R}_l$.
Let $x'\in {\overline{X}}$ and let $y'\in {\overline{X}}$. Then $$\begin{aligned}
x'\overline{\operatorname{R}_l} y' &\iff x'\leq i_X(x)\text{, }x\operatorname{R}_l y\text{ and }i_Y(y)\leq y'\text{ for some } x\in X \text{ and }y\in Y \\
&\iff x'\leq i_X(x)\text{, }i_Y(y)\leq y'\text{ and } e_X^{-1}(x^\uparrow)\cap e_Y^{-1}(y^\downarrow)\neq\emptyset \text{ for } x\in X \text{, }y\in Y\\
&\iff (i_X\circ e_X)^{-1}(x'^\uparrow)\cap (i_Y\circ e_Y)^{-1}(y'^\downarrow)\neq\emptyset \\
&\iff x' \operatorname{S}_l y'. \end{aligned}$$
Restriction
-----------
If $i_X:X\to {\overline{X}}$ and $i_Y:Y\to{\overline{Y}}$ are order extensions, then a polarity $({\overline{X}}, {\overline{Y}}, \operatorname{S})$ can be restricted in a natural way to a polarity $(X, Y, \operatorname{\underline{S}})$. The following theorem makes this precise.
\[T:rest\] Let $X$ and $Y$ be posets, and let $i_X:X\to {\overline{X}}$ and $i_Y:Y\to{\overline{Y}}$ be order extensions. Let $\operatorname{S}$ be a relation on ${\overline{X}}\times {\overline{Y}}$. Then there is a relation $\operatorname{\underline{S}}$ on $X\times Y$ defined by $$x \operatorname{\underline{S}}y \iff i_X(x) \operatorname{S}i_Y(y)$$ such that the following hold:
1. $(X,Y,\operatorname{\underline{S}})$ is an order polarity.
2. If $({\overline{X}}, {\overline{Y}}, \operatorname{S})$ is 0-coherent then so is $(X,Y,\operatorname{\underline{S}})$.
Moreover, if $P$ is a poset, and if $e_X:P\to X$ and $e_Y:P\to Y$ are order extensions, then both $(e_X,e_Y,\operatorname{\underline{S}})$ and $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ are extension polarities, and:
1. If $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is $\ast$-coherent then so is $(e_X,e_Y,\operatorname{\underline{S}})$ for $\ast \in\{1,2\}$.
2. Suppose $i_X$ preserve meets in $X$ of subsets of $e_X[P]$ whenever they exist, and let $i_Y$ likewise preserve joins in $Y$ of subsets of $e_Y[P]$. Then, if $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is $3$-coherent, so is $(e_X,e_Y,\operatorname{\underline{S}})$, and the same is true if we replace ‘3-coherent’ with ‘Galois’.
First of all, $(X,Y,\operatorname{\underline{S}})$ is obviously an order polarity as the definition requires only that $X$ and $Y$ are posets and $\operatorname{\underline{S}}$ is a relation between $X$ and $Y$.
Now, let $\preceq$ be a pre-order on ${\overline{X}}\cup {\overline{Y}}$, and consider the diagram in Figure \[F:restrict\]. Here $X\uplus_\preceq Y$ is the poset structure induced on $X\cup Y$ by the maps $\iota_X\circ i_X$ and $\iota_Y\circ i_Y$, and $\phi$ is the associated order embedding (which we can think of as an inclusion). It is easy to see that if $\preceq$ is in ${\mathcal P}_{\operatorname{S}}$, ${\hat{\mathcal{P}}}_{\operatorname{S}}$ or ${\hat{\mathcal{P}}}^e_{\operatorname{S}}$ then the restriction of $\preceq$ to $X\cup Y$ will be in ${\mathcal P}_{\operatorname{\underline{S}}}$, ${\hat{\mathcal{P}}}_{\operatorname{\underline{S}}}$ or ${\hat{\mathcal{P}}}^e_{\operatorname{\underline{S}}}$ appropriately. This proves (1), (2), and (3).
For (4) we can take the same approach to prove 3-coherence. Because of the restriction on $i_X$, meets in $X$ of subsets of $e_X[P]$ correspond to meets in ${\overline{X}}$ of subsets of $i_X\circ e_X[P]$, and similarly joins in $Y$ of subsets of $e_Y[P]$ correspond to joins in ${\overline{Y}}$, so the meet- and join-preservation properties of $\iota_{{\overline{X}}}$ and $\iota_{{\overline{Y}}}$, respectively, also apply to $\iota_X$ and $\iota_Y$.
Finally, suppose $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is Galois, and let $x_1'\not\leq x_2' \in {\overline{X}}$. Then, as $i_X\circ e_X$ is a meet-extension, there is $p\in P$ with $x_2' \leq i_X\circ e_X(p)$ and $x_1'\not\leq i_X\circ e_X(p)$. By writing $i_X\circ e_X(p)$ as $i_X(e_X(p))$ we see immediately that $i_X$ is a meet-extension, and $i_Y$ is a join-extension by duality. Similarly, let $x_1\not\leq x_2\in X$. Then $i_X(x_1)\not\leq i_X(x_2)$, so there is $q\in P$ with $i_X(x_2) \leq i_X\circ e_X(q)$ and $i_X(x_1)\not\leq i_X\circ e_X(q)$, and thus $x_2 \leq e_X(q)$ and $x_1\not\leq e_X(q)$. So $e_X$ is also a meet-extension, and $e_Y$ is a join-extension by duality. The result then follows, as we have already proved that $(e_X,e_Y,\operatorname{\underline{S}})$ will be 3-coherent.
$$\xymatrix{ P\ar[r]^{e_Y}\ar[d]_{e_X} & Y\ar[d]^{\iota_Y}\ar[r]^{i_Y} & {\overline{Y}}\ar[dd]^{\iota_{{\overline{Y}}}} \\
X\ar[r]_{\iota_X}\ar[d]_{i_X} & X\uplus_{\preceq}Y\ar[dr]^{\phi} \\
{\overline{X}}\ar[rr]_{\iota_{{\overline{X}}}} & & {\overline{X}}\uplus_{\preceq} {\overline{Y}}}$$
Unlike the situation in Theorem \[T:Galois\], partial converses for the implications in Theorem \[T:rest\] do not hold, as Example \[E:notConv\] demonstrates.
\[E:notConv\] Let $P$ be the poset represented by the $\bullet$ elements in Figure \[F:notConv\], let $P\cong X\cong Y\cong{\overline{X}}$, and let ${\overline{Y}}$ be represented by Figure \[F:notConv\]. Then the implicit maps $i_X$ and $i_Y$ are obviously meet- and join-extensions respectively, and are also, respectively, trivially completely meet- and join-preserving. Let $\operatorname{S}= \operatorname{R}_l\cup \{(p, y)\}$, where $\operatorname{R}_l\subseteq {\overline{X}}\times{\overline{Y}}$ is as in Proposition \[P:relations\]. Then $\operatorname{\underline{S}}= \operatorname{R}'_l$, where $\operatorname{R}_l'\subseteq X\times Y$ is defined analogously, and $(e_X, e_Y, \operatorname{\underline{S}})$ is Galois by Proposition \[P:relations\]. However, $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is not even 0-coherent, as we have $(p, y) \in S$ but $(q, y)\notin S$, so (A0) from Theorem \[T:ext\] fails.
$$\xymatrix{ \circ_y \\
\bullet\ar@{-}[u] & \bullet\ar@{-}[ul] & \bullet_p\\
& & \bullet_q\ar@{-}[u]
}$$
Using the notation of Theorems \[T:Galois\] and \[T:rest\] we can define a map $\overline{(-)}$ from the complete lattice of relations on $X\times Y$ to the complete lattice of relations on ${\overline{X}}\times{\overline{Y}}$, by taking $\operatorname{R}$ to $\operatorname{\overline{R}}$. Similarly, we can define a map $\underline{(-)}$ going back the other way by taking $\operatorname{S}$ to $\operatorname{\underline{S}}$. These maps are obviously monotone. We also have the following result.
\[L:Gcon\] Let $X$ and $Y$ be posets, let $i_X:X\to{\overline{X}}$ be an extension of $X$, and let $i_Y:Y\to{\overline{Y}}$ be an extension of $Y$. Then:
1. Let $\operatorname{R}$ be a relation on $X\times Y$. Then $\operatorname{R}\subseteq \underline{(\operatorname{\overline{R}})}$. Moreover, if $(X, Y, \operatorname{R})$ is 0-coherent then $\operatorname{R}= \underline{(\operatorname{\overline{R}})}$.
2. Let $\operatorname{S}$ be a relation on ${\overline{X}}\times {\overline{Y}}$. If $(i_X, i_Y, \operatorname{S})$ is 0-coherent then $\overline{(\underline{\operatorname{S}})}\subseteq \operatorname{S}$. Moreover, the opposite inclusion may fail, even when $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is Galois.
We start with (1). Let $x\in X$, let $y\in Y$ and suppose $x \operatorname{R}y$. Then $i_X(x) \operatorname{\overline{R}}i_Y(y)$ by definition of $\operatorname{\overline{R}}$, and so $x \underline{(\operatorname{\overline{R}})} y$ by definition of $\underline{(\operatorname{\overline{R}})}$. Suppose now that $(X, Y, \operatorname{R})$ is 0-coherent and let $x \underline{(\operatorname{\overline{R}})} y$. Then $i_X(x) \operatorname{\overline{R}}i_Y(y)$ by definition of $\underline{(\operatorname{\overline{R}})}$, and thus $x \operatorname{R}y$ by Theorem \[T:Galois\](2).
For (2), suppose first that $(i_X, i_Y, \operatorname{S})$ is 0-coherent, and let $x'\in {\overline{X}}$ and $y'\in {\overline{Y}}$ with $x' \overline{(\underline{\operatorname{S}})} y'$. Then, by definition of $\overline{(\underline{\operatorname{S}})}$ there are $x\in X$ and $y\in Y$ with $x'\leq i_X(x)$, with $i_Y(y)\leq y'$, and with $x \underline{\operatorname{S}} y$. But then $i_X(x) \operatorname{S}i_Y(y)$ by definition of $\underline{\operatorname{S}}$, and so $x'\leq i_X(x) \operatorname{S}i_Y(y)\leq y'$, and thus $x' \operatorname{S}y'$ by 0-coherence of $(i_X, i_Y, \operatorname{S})$. To see that the opposite inclusion may fail, see Example \[E:notEq\] below.
Note that the polarity $(i_X, i_Y, \operatorname{S})$ from Example \[E:notConv\] is not 0-coherent, but, appealing to Lemma \[L:minExt\], we have $\overline{(\underline{\operatorname{S}})}\subseteq \operatorname{S}$. Thus $(i_X, i_Y, \operatorname{S})$ being 0-coherent is strictly stronger than having $\overline{(\underline{\operatorname{S}})}\subseteq \operatorname{S}$.
Using the notation of Lemma \[L:Gcon\], let $L$ be the complete lattice of relations on $X\times Y$, and let $M$ be the complete lattice of 0-coherent relations on ${\overline{X}}\times{\overline{Y}}$. Then the maps $\overline{(-)}:L\to M$ and $\underline{(-)}:M\to L$ are, respectively, the left and right adjoints of a Galois connection.
First, recall the discussion at the start of Section \[S:satisfaction1\] for the lattice structure of $M$. Moreover, $\overline{(-)}:L\to M$ is well defined by Theorem \[T:Galois\](1). By Lemma \[L:Gcon\] we have $\operatorname{R}\subseteq \underline{(\operatorname{\overline{R}})}$ for all $R\in L$, and $\overline{(\underline{\operatorname{S}})}\subseteq \operatorname{S}$ for all $S\in M$, which is one of the equivalent conditions for two monotone maps to form a Galois connection (see e.g. [@DavPri02 Lemma 7.26]).
Using Theorem \[T:rest\] and Lemma \[L:Gcon\] we can get partial converses for Theorem \[T:Galois\].
\[C:converses\] With notation as in Theorem \[T:Galois\], suppose $(X,Y,\operatorname{R})$ is 0-coherent. Then:
1. If $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is $\ast$-coherent then so is ${(e_X,e_Y,\operatorname{R})}$ for $\ast \in\{1,2\}$.
2. Suppose $i_X$ preserves meets in $X$ of subsets of $e_X[P]$ whenever they exist, and let $i_Y$ likewise preserve joins in $Y$ of subsets of $e_Y[P]$. Then, whenever $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is $3$-coherent, so is ${(e_X,e_Y,\operatorname{R})}$, and this is also true if we replace ‘3-coherent’ with ‘Galois’.
For (1), if $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is $\ast$-coherent, then so is $(X,Y,\underline{(\operatorname{\overline{R}})})$, by Theorem \[T:rest\], and as $(X,Y,\operatorname{R})$ is 0-coherent we have $\operatorname{R}= \underline{(\operatorname{\overline{R}})}$, by Lemma \[L:Gcon\]. The proof of (2) is essentially the same.
\[E:notEq\] Let $P$ be the poset in Figure \[F:notEqP\], and let $X \cong Y \cong P$. Let ${\overline{X}}$ and ${\overline{Y}}$ be the posets in Figures \[F:notEqX\] and \[F:notEqY\] respectively, denoting embedded images of elements of $X$ and $Y$ with $\bullet$, and extension elements with $\circ$. Define $\operatorname{S}$ on ${\overline{X}}\times {\overline{Y}}$ so that $i_X\circ e_X(p) \operatorname{S}e_Y\circ i_Y(p)$ for all $p\in P$, and also $x' \operatorname{S}y'$. Then $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{S})$ is Galois, as can be seen by considering the poset in Figure \[F:notEqPos\], and defining $\iota_{{\overline{X}}}$ and $\iota_{{\overline{Y}}}$ in the obvious way. However, there is no $x\in X$ and $y\in Y$ with $x'\leq i_X(x)$, with $i_Y(y)\leq y'$, and with $x \underline{\operatorname{S}} y$. Thus $(x',y')\notin \overline{(\underline{\operatorname{S}})}$.
$$\xymatrix{ \bullet\ar@{-}[d] & \bullet\ar@{-}[d] \\
\bullet & \bullet
}$$
$$\xymatrix{ \bullet\ar@{-}[dr]\ar@{-}[d] & & \bullet\ar@{-}[dl]\ar@{-}[d] \\
\bullet & \circ_{x'} & \bullet
}$$
$$\xymatrix{ \bullet\ar@{-}[d] & \circ_{y'} & \bullet\ar@{-}[d] \\
\bullet\ar@{-}[ur] & & \bullet\ar@{-}[ul]
}$$
$$\xymatrix{ \bullet\ar@{-}[d]\ar@{-}[dr] & \circ_{y'} & \bullet\ar@{-}[d]\ar@{-}[dl] \\
\bullet\ar@{-}[ur] & \circ_{x'}\ar@{-}[u] & \bullet\ar@{-}[ul]
}$$
Galois polarities revisited {#S:GaloiS1}
===========================
Galois polarities via Galois connections {#S:GC}
----------------------------------------
Galois polarities are so named because the associated (unique) pre-order on $X\cup Y$ can be described in terms of a Galois connection. This idea is precisely articulated in Corollary \[C:Galois\] below.
When ${(e_X,e_Y,\operatorname{R})}$ is Galois, we know from Theorem \[T:unique\] that ${\hat{\mathcal{P}}}_{\operatorname{R}}^g$ contains only a single element, $\operatorname{\preceq_3}$. As mentioned previously, to lighten the notation we write e.g. $X\uplus Y$ in place of ${X\uplus_{\operatorname{\preceq_3}} Y}$ when working with Galois polarities. The following theorem collects together some useful facts.
\[T:commute2\] Let ${(e_X,e_Y,\operatorname{R})}$ be a Galois polarity, let $i_X:X\to {\overline{X}}$ be a completely meet-preserving meet-extension, and let $i_Y:Y\to{\overline{Y}}$ be a completely join-preserving join-extension. Then:
1. $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is Galois.
2. The map $\gamma:P\to X \uplus Y$ defined by $\gamma = \iota_X\circ e_X = \iota_Y\circ e_Y$ is an order embedding. Moreover, if $S,T\subseteq P$ and ${\bigwedge}S$ and ${\bigvee}T$ exist in $P$, then
1. $\gamma({\bigwedge}S) = {\bigwedge}\gamma[S] \iff e_X({\bigwedge}S)= {\bigwedge}e_X[S]$, and
2. $\gamma({\bigvee}T) = {\bigvee}\gamma[T] \iff e_Y({\bigvee}T)={\bigvee}e_Y[T]$.
3. $\gamma[P] = \iota_X[X]\cap \iota_Y[Y]$.
4. Define the map $$\phi: X\uplus Y\to {\overline{X}}\uplus {\overline{Y}}$$ by $$\phi(z) = \begin{cases} \iota_{{\overline{X}}}\circ i_X(z) \text{ if $z$ is (the equivalence class of) an element of $X$.} \\
\iota_{{\overline{Y}}}\circ i_Y(z) \text{ if $z$ is (the equivalence class of) an element of $Y$.}
\end{cases}$$ Then $\phi$ is a well defined order embedding, and the diagram in Figure \[F:Galois\] commutes.
<!-- -->
1. That $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is Galois is Theorem \[T:Galois\](4).
2. That $\gamma$ is well defined follows from 1-coherence of ${(e_X,e_Y,\operatorname{R})}$, and that $\gamma$ is an order embedding follows from 2-coherence of ${(e_X,e_Y,\operatorname{R})}$, as $\gamma$ is the composition of two order embeddings, $\iota_X\circ e_X$. That (a) and (b) hold follows from 3-coherence of ${(e_X,e_Y,\operatorname{R})}$, as, for example, $\gamma = \iota_X\circ e_X$ and $\iota_X$ preserves meets in $X$ of subsets of $e_X[P]$.
3. We obviously have $\gamma[P] \subseteq \iota_X[X]\cap \iota_Y[Y]$, so let $z\in \iota_X[X]\cap \iota_Y[Y]$. Then there are $x\in X$ and $y\in Y$ with $z=\iota_X(x)=\iota_Y(y)$. Thus, as $\iota_X(x)\leq\iota_Y(y)$ we have $e_X^{-1}(x^\uparrow)\cap e_Y^{-1}(y^\downarrow)\neq\emptyset$. Suppose $p\in e_X^{-1}(x^\uparrow)\cap e_Y^{-1}(y^\downarrow)$, and that $e_X(p)\not\leq x$. Then there is $q\in P$ with $x\leq e_X(q)$ and $e_X(p)\not\leq e_X(q)$. But this is a contradiction, as, since $\iota_Y(y)\leq \iota_X(x)$, (G3) of Theorem \[T:unique\] tells us that $p\leq q$. Thus $x=e_X(p)$, and so $z = \gamma(p)$. It follows that $\iota_X[X]\cap \iota_Y[Y]\subseteq \gamma[P]$ as claimed.
4. First note that since $(i_X\circ e_X, i_Y\circ e_Y, \operatorname{\overline{R}})$ is Galois, it makes sense to write ${\overline{X}}\uplus{\overline{Y}}$. Now, $\operatorname{R}= \underline{(\operatorname{\overline{R}})}$ by Lemma \[L:Gcon\], so the unique element of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ must be the same as the unique element of ${\hat{\mathcal{P}}}^g_{\underline{(\operatorname{\overline{R}})}}$, which is induced by the embeddings of $X$ and $Y$ into ${\overline{X}}\uplus {\overline{Y}}$ via $\iota_{{\overline{X}}}\circ i_X$ and $\iota_{{\overline{Y}}}\circ i_Y$ respectively, as discussed in the proof of Theorem \[T:rest\]. The result then follows from the commutativity of the diagram in Figure \[F:restrict\].
$$\xymatrix{ P\ar[r]^{e_Y}\ar[d]_{e_X}\ar[dr]^{\gamma} & Y\ar[d]^{\iota_Y}\ar[r]^{i_Y} & {\overline{Y}}\ar[dd]^{\iota_{{\overline{Y}}}} \\
X\ar[r]_{\iota_X}\ar[d]_{i_X} & X\uplus Y\ar[dr]^{\phi} \\
{\overline{X}}\ar[rr]_{\iota_{{\overline{X}}}} & & {\overline{X}}\uplus{\overline{Y}}}$$
The following fact will be useful.
\[P:GaloisExtension\] Let $P$ and $Q$ be posets, let $e_1: P\to J$ be a join-completion, and let $e_2:Q\to M$ be a meet-completion. Then any Galois connection $\alpha:P\leftrightarrow Q:\beta$ extends uniquely to a Galois connection $\alpha': J\leftrightarrow M:\beta'$.
This is [@Schm74 Corollary 2].
\[L:Galois\] Let $P$ be a poset, and let $e_X:P\to X$ and $e_Y:P\to Y$ be meet- and join-completions respectively. Then there is a unique Galois connection $\Gamma:Y\leftrightarrow X:\Delta$ such that $e_X = \Gamma\circ e_Y$ and $e_Y = \Delta\circ e_X$. The left and right adjoints of this Galois connection are defined, respectively, by $$\Gamma(y) = {\bigvee}e_X[e_Y^{-1}(y^\downarrow)],$$ $$\Delta(x) = {\bigwedge}e_Y[e_X^{-1}(x^\uparrow)].$$
$\Gamma$ and $\Delta$ are well defined as $X$ and $Y$ are complete. Using the fact that $e_X$ and $e_Y$ are, respectively, meet- and join-completions, we have $$\begin{aligned}
\Gamma(y) \leq x &\iff {\bigvee}e_X[e_Y^{-1}(y^\downarrow)] \leq x \\
&\iff x\leq e_X(q) \implies e_X(p)\leq e_X(q) \text{ for all } p\in e_Y^{-1}(y^\downarrow)\text{ and for all }q\in P \\
&\iff q\in e_X^{-1}(x^\uparrow) \text{ and } p\in e_Y^{-1}(y^\downarrow)\implies p\leq q\\
&\iff e_Y(p)\leq y \implies e_Y(p)\leq e_Y(q) \text{ for all } q\in e_X^{-1}(x^\uparrow)\text{ and for all }p\in P \\
&\iff y \leq {\bigwedge}e_Y[e_X^{-1}(x^\uparrow)] \\
&\iff y\leq \Delta(x). \end{aligned}$$
To see that this is the only such Galois connection between $X$ and $Y$ we apply Proposition \[P:GaloisExtension\] with $P=Q$ and the Galois connection produced by the identity function on $P$.
\[C:Galois\] Let ${(e_X,e_Y,\operatorname{R})}$ be a Galois polarity, let $i_X: X\to {\overline{X}}$ be a completely meet-preserving meet-completion of $X$, and let $i_Y:Y\to {\overline{Y}}$ be a completely join-preserving join-completion of $Y$. Let $\preceq$ be the unique element of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$. Let $\Gamma$ and $\Delta$ be as defined in Lemma \[L:Galois\], with respect to the maps $i_X\circ e_X$, and $i_Y\circ e_Y$. Then to define $\operatorname{\preceq_3}$ we can replace (G3) from Theorem \[T:unique\] by:
- For all $x\in X$ and $y\in Y$ we have $$y\preceq x\iff \Gamma(i_Y(y))\leq i_X(x)\iff i_Y(y)\leq \Delta(i_X(x)).$$
First note that the maps $\Gamma$ and $\Delta$ exist as $i_X\circ e_X:P\to {\overline{X}}$ and $i_Y\circ e_Y:P\to {\overline{Y}}$ are meet- and join-completions respectively. That (G3$'$) and (G3) are equivalent is an immediate consequence of the following equivalence:
$$\begin{aligned}
&\phantom{\iff i}\Gamma(i_Y(y))\leq i_X(x)\\ &\iff {\bigvee}i_X\circ e_X[(i_Y\circ e_Y)^{-1}(i_Y(y)^\downarrow)] \leq {\bigwedge}i_X\circ e_X[(i_X\circ e_X)^{-1}(i_X(x)^\uparrow)] \\
&\iff \Big(i_Y\circ e_Y(p)\leq i_Y(y) \text{ and } i_X(x) \leq i_X\circ e_X(q) \implies i_X\circ e_X(p)\leq i_X \circ e_X(q) \Big)\\
&\iff \Big(e_Y(p)\leq y \text{ and } x \leq e_X(q) \implies p\leq q \Big).\end{aligned}$$
Corollary \[C:Galois\] justifies the terminology ‘Galois polarity’, as the upshot of this result is that, for any Galois polarity ${(e_X,e_Y,\operatorname{R})}$, the unique element of ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ is directly defined by $\operatorname{R}$, the orders on $X$ and $Y$, and the Galois connection from Lemma \[L:Galois\]. Note that the choice of meet- and join-completions of $X$ and $Y$ here is constrained only by the requirement that they preserve meets and joins respectively. Indeed, we can weaken these conditions to just the preservation of meets and joins in $X$ and $Y$ respectively from $e_X[P]$ and $e_Y[P]$. So long as these requirements are met, the ordering induced by (G3$'$) will be the same as the one induced by (G3).
Polarity morphisms {#S:polHom}
------------------
\[D:compG\] A Galois polarity ${(e_X,e_Y,\operatorname{R})}$ is **complete** if $e_X$ and $e_Y$ are completions.
Noting Proposition \[P:GalSimp\], we see that [@GJP13 Theorem 3.4] establishes a one-to-one correspondence between what we call complete Galois polarities and $\Delta_1$-completions of a poset. Theorem \[T:delta\] below expands on the proof of this result, and in Section \[S:cat\] we reformulate it in terms of an adjunction between categories. First we need to define a concept of morphism between Galois polarities.
\[D:polHom\] Let $P$ and $P'$ be posets, let ${(e_X,e_Y,\operatorname{R})}$ be a Galois polarity extending $P$, and let $(e_{X'},e_{Y'},\operatorname{R}')$ be a Galois polarity extending $P'$. Then a **polarity morphism** between ${(e_X,e_Y,\operatorname{R})}$ and $(e_{X'},e_{Y'},\operatorname{R}')$ is a triple of monotone maps $(h_X:X\to X', h_P:P\to P', h_Y:Y\to Y')$ such that:
1. The diagram in Figure \[F:isom\] commutes.
2. For all $x\in X$ and $y\in Y$ we have $$\iota_Y(y)\leq \iota_X(x)\implies \iota_{Y'}\circ h_Y(y)\leq \iota_{X'}\circ h_X(x).$$
3. For all $x'\in X'$ and for all $y'\in Y'$, if $(x', y')\notin \operatorname{R}'$ then there is $x\in X$ and $y\in Y$ such that:
(i) $h^{-1}_X(x'^\uparrow)\subseteq x^\uparrow$.
(ii) $h_Y^{-1}(y'^\downarrow)\subseteq y^\downarrow$.
(iii) $h_X(a) \operatorname{R}' y' \implies a \operatorname{R}y$ for all $a\in X$.
(iv) $x' \operatorname{R}' h_Y(b) \implies x\operatorname{R}b$ for all $b\in Y$.
(v) $(x,y)\notin \operatorname{R}$.
If $h_X$, $h_P$ and $h_Y$ are all order embeddings, and also $h_X(x)\operatorname{R}' h_Y(y)\implies x\operatorname{R}y$ for all $x\in X$ and $y\in Y$, then $(h_X, h_P, h_Y)$ is a **polarity embedding**. If, in addition, all maps are actually order isomorphisms then $(h_X, h_P, h_Y)$ is a **polarity isomorphism**, and we say ${(e_X,e_Y,\operatorname{R})}$ and $(e_{X'},e_{Y'},\operatorname{R}')$ are isomorphic.
Sometimes we want to fix a poset $P$ and deal exclusively with isomorphism classes of Galois polarities extending $P$. In this case we say Galois polarities $E_1$ and $E_2$ are **isomorphic as Galois polarities extending $P$** if there is a polarity isomorphism $(h_X,h_P,h_Y):E_1\to E_2$ where $h_P$ is the identity on $P$.
$$\xymatrix{
X\ar[d]_{h_X} & \ar[l]_{e_X}P\ar[r]^{e_Y}\ar[d]^{h_P} & Y\ar[d]^{h_Y}\\
X' & \ar[l]^{e_{X'}}P'\ar[r]_{e_{Y'}} & Y'
}$$
Note that if $h_X$ and $h_Y$ are order embeddings then $h_P$ will be too, but this is not necessarily the case for order isomorphisms. Note also that Definition \[D:polHom\], while being similar in some respects, is largely distinct from the notion of a *bounded morphism between polarity frames* from [@Suz12]. It is also completely different to the frame morphisms of [@DGP05; @Geh06], which are duals to complete lattice homomorphisms, rather than ‘decomposed’ versions of certain maps $X\uplus Y\to X'\uplus Y'$. We will make this clear in Theorem \[T:polHom\] later.
\[L:xRy\] If $h= (h_X:X\to X', h_P:P\to P', h_Y:Y\to Y')$ is a polarity morphism, then for all $x\in X$ and for all $y\in Y$ we have $x\operatorname{R}y\implies h_X(x) \operatorname{R}' h_Y(y)$.
Suppose $(h_X(x),h_Y(y))\notin \operatorname{R}'$. Then, by \[D:polHom\](3) there are $x_0\in X$ and $y_0\in Y$ with $h_X^{-1}(h_X(x)^\uparrow)\subseteq x_0^\uparrow$, with $h_Y^{-1}(h_Y(y)^\downarrow)\subseteq y_0^\downarrow$ and with $(x_0,y_0)\notin \operatorname{R}$. From $h_X^{-1}(h_X(x)^\uparrow)\subseteq x_0^\uparrow$ it follows that $x_0\leq x$, and similarly we have $y\leq y_0$. Thus $(x,y)\notin \operatorname{R}$, as otherwise (A0) and (A1) of Theorem \[T:ext\] would force $x_0\operatorname{R}y_0$.
The following definition is due to Erné [@Ern91a]. This will be of interest to us as it precisely characterizes those maps between posets that lift (uniquely) to complete homomorphisms between their MacNeille completions [@Ern91a Theorem 3.1].
\[D:cut\] A monotone map $f:P\to Q$ is **cut-stable** if whenever $q_1\not\leq q_2\in Q$, there are $p_1\not\leq p_2\in P$ such that $f^{-1}(q_1^\uparrow)\subseteq p_1^\uparrow$ and $f^{-1}(q_2^\downarrow)\subseteq p_2^\downarrow$.
Condition (3) of Definition \[D:polHom\] is related to cut-stability, as we shall see in the proof of Theorem \[T:polHom\]. We can think of this as an adaptation of ideas from [@GehPri08 Section 4]. We extend from what, according to our terminology, is the special case of ${(e_X,e_Y,\operatorname{R})}$ where $e_X$ and $e_Y$ are the free directed meet- and join-completions respectively and $\operatorname{R}=\operatorname{R}_l$, to Galois polarities in general.
\[T:polHom\] Let $P$ and $P'$ be posets, let ${(e_X,e_Y,\operatorname{R})}$ be a Galois polarity extending $P$, and let $(e_{X'},e_{Y'},\operatorname{R}')$ be a Galois polarity extending $P'$. Let $\gamma:P\to X\uplus Y$ and $\gamma':P'\to X'\uplus Y'$ be the canonical maps as in Theorem \[T:commute2\]. Then, given a polarity morphism $(h_X, h_P,h_Y):{(e_X,e_Y,\operatorname{R})}\to(e_{X'},e_{Y'},\operatorname{R}')$, there is a unique, cut-stable monotone map $\psi:X\uplus Y \to X'\uplus Y'$ such that the diagram in Figure \[F:polHom\] commutes. Moreover, $\psi$ satisfies conditions (1)-(3) below.
1. $\psi\circ\gamma[P]\subseteq \gamma'[P']$,
2. $\psi\circ\iota_X[X]\subseteq \iota_{X'}[X']$, and
3. $\psi\circ\iota_Y[Y]\subseteq \iota_{Y'}[Y']$,
Conversely, given a cut-stable monotone map $\psi:X\uplus Y \to X'\uplus Y'$ satisfying (1)-(3), there is a unique polarity morphism $(h_X, h_P,h_Y)$ such that the diagram in Figure \[F:polHom\] commutes.
Finally, if $\psi$ and $(h_X,h_P,h_Y)$ are, respectively, a cut stable monotone map and a polarity morphism uniquely specifying each other according to the correspondence described above, then:
(a) $\psi$ is an order embedding if and only if:
1. $h_X$ and $h_Y$ are order embeddings, and
2. for all $x\in X$ and for all $y\in Y$ we have $h_X(x)\operatorname{R}' h_Y(y)\implies x\operatorname{R}y$.
I.e. if and only if $(h_X, h_P, h_Y)$ is a polarity embedding.
(b) If $h_X$ and $h_Y$ are both surjective then $\psi$ is surjective, but the converse does not hold in general.
Given $(h_X, h_P,h_Y)$, the commutativity of the diagram in Figure \[F:polHom\] demands that $\psi$ can only be defined by $$\psi(z) = \begin{cases}\iota_{X'}\circ h_X(z) \text{ when } z\in \iota_X[X] \\ \iota_{Y'}\circ h_Y(z) \text{ when } z\in \iota_Y[Y]\end{cases}$$
Abusing notation slightly, let $\preceq$ stand for the unique element of both ${\hat{\mathcal{P}}}^g_{\operatorname{R}}$ and ${\hat{\mathcal{P}}}^g_{\operatorname{R}'}$. If $x\in X$ and $y\in Y$, then, using Lemma \[L:xRy\], we have $$x\preceq y \iff x\operatorname{R}y\implies h_X(x)\operatorname{R}' h_Y(y)\iff h_X(x)\preceq h_Y(y).$$
If $y\preceq x$, then $\iota_Y(y)\leq \iota_X(x)$ by definition, and so $h_Y(y)\preceq h_X(x)$ by Definition \[D:polHom\](2). This shows $\psi$ is well defined, and along with the fact that $h_X$ and $h_Y$ are monotone proves $\psi$ is monotone.
To see that $\psi$ is cut-stable, let $z_1\not\leq z_2\in X'\uplus Y'$. Since $\iota_{X'}[X']$ and $\iota_{Y'}[Y']$ are, respectively, join- and meet-dense in $X'\uplus Y'$, there are $x'\in X'$ and $y'\in Y'$ with $\iota_{X'}(x')\leq z_1$, with $z_2\leq \iota_{Y'}(y')$, and with $\iota_{X'}(x')\not\leq \iota_{Y'}(y')$ (i.e. $(x',y')\notin \operatorname{R}'$). Thus by Definition \[D:polHom\](3) there are $x\in X$ and $y\in Y$ with the five properties described in that definition. We will satisfy the condition of Definition \[D:cut\] using the pair $\iota_X(x)\not\leq \iota_Y(y)$.
Let $z\in \psi^{-1}(z_1^\uparrow)$. We must show that $z\in \iota_X(x)^\uparrow$. We have $\psi(z)\geq z_1\geq\iota_{X'}(x')$. There are two cases. If $z= \iota_X(a)$ for some $a\in X$, then $\psi(z)= \iota_{X'}\circ h_X(a)$, and so $h_X(a)\geq x'$. Thus $a\in h_X^{-1}(x'^\uparrow)$, and so $a\in x^\uparrow$, by Definition \[D:polHom\](3.i). It follows that $z = \iota_X(a)\in \iota_X(x)^\uparrow$ as claimed. Alternatively, suppose $z=\iota_Y(b)$ for some $b\in Y$. Then $\psi(z)=\iota_{Y'}\circ h_Y(b)$, and so $\iota_{Y'}\circ h_Y(b)\geq \iota_{X'}(x')$, and consequently $x' \operatorname{R}' h_{Y}(b)$. It follows from Definition \[D:polHom\](3.iv) that $x\operatorname{R}b$, and thus that $\iota_X(x)\leq \iota_Y(b)=z$ as required. That $\psi^{-1}(z_2^\downarrow)\subseteq \iota_Y(y)^\downarrow$ follows by a dual argument, and so $\psi$ is cut-stable.
To see that condition (1) holds for $\psi$ note that $$\begin{aligned}
\psi\circ\gamma(p) &= \psi\circ\iota_Y\circ e_Y(p) \\
&= \iota_{Y'}\circ h_Y\circ e_Y(p) \\
&= \iota_{Y'}\circ e_{Y'}\circ h_P(p)\\
&= \gamma'(h_P(p)).\end{aligned}$$ That (2) and (3) hold is automatic.
Conversely, given monotone $\psi$ satisfying (2) and (3), if the diagram in Figure \[F:polHom\] is to commute, then $h_X$ and $h_Y$ must be $\iota_{X'}^{-1}\circ \psi\circ \iota_X$ and $\iota_{Y'}^{-1}\circ \psi\circ \iota_Y$ respectively. Here $\iota_{X'}^{-1}$ and $\iota_{Y'}^{-1}$ are the partial inverse maps, which are total on $\psi\circ \iota_X[X]$ and $\psi\circ \iota_Y[Y]$ by (2) and (3) respectively. The commutativity of this diagram also demands that, if $h_P$ exists, we have $$e_{X'}\circ h_P = h_X\circ e_X = \iota_{X'}^{-1}\circ \psi \circ \iota_X\circ e_X = \iota_{X'}^{-1}\circ \psi \circ\gamma,$$ and thus $h_P=e_{X'}^{-1}\circ \iota_{X'}^{-1}\circ \psi \circ\gamma$, if this is well defined. Consequently, assuming $\psi$ also satisfies (1), we can, and must, define $$h_P=\gamma'^{-1}\circ \psi\circ\gamma.$$ That $(h_X, h_P, h_Y)$ satisfies Definition \[D:polHom\](2) follows immediately from the definitions of $h_X$ and $h_Y$ and the fact that $\psi$ is monotone. If $\psi$ is also cut-stable, then to prove that $(h_X, h_P, h_Y)$ is a polarity morphism it remains only to check Definition\[D:polHom\](3).
So let $x'\in X'$, let $y'\in Y'$, and suppose $(x', y')\notin \operatorname{R}'$. Then $\iota_{X'}(x')\not\leq \iota_{Y'}(y')$, and thus by cut-stability there are $z_1\not\leq z_2\in X\uplus Y$ with $\psi^{-1}(\iota_{X'}(x')^\uparrow)\subseteq z_1^\uparrow$, and $\psi^{-1}(\iota_{Y'}(y')^\downarrow)\subseteq z_2^\downarrow$. As $\iota_X[X]$ and $\iota_Y[Y]$ are, respectively, join- and meet-dense in $X\uplus Y$, there are $x\in X$ and $y\in Y$ with $\iota_X(x)\leq z_1$, with $z_2\leq \iota_Y(y)$, and with $(x,y)\not\in \operatorname{R}$. It follows that $\psi^{-1}(\iota_{X'}(x')^\uparrow)\subseteq \iota_X(x)^\uparrow$ and $\psi^{-1}(\iota_{Y'}(y')^\downarrow)\subseteq \iota_Y(y)^\downarrow$. We will check the conditions required by Definition \[D:polHom\](3) are satisfied by the pair $(x,y)$:
(i) Let $a\in X$ and suppose $a\in h_X^{-1}(x'^\uparrow)$. Then $\iota_X(a)\in \psi^{-1}(\iota_{X'}(x'))\subseteq \iota_X(x)^\uparrow$, and thus $a\in x^\uparrow$ as required.
(ii) Dual to (i).
(iii) Let $a\in X$ and suppose $h_X(a)\operatorname{R}' y'$. Then $\iota_{X'}\circ h_X(a) \leq \iota_{Y'}(y')$, and thus $\psi \circ \iota_X(a) \leq \iota_{Y'}(y')$. It follows that $\iota_X(a)\in \psi^{-1}(\iota_{Y'}(y')^\downarrow)\subseteq \iota_Y(y)^\downarrow$, and so $a\operatorname{R}y$ as required.
(iv) Dual to (iii).
(v) By choice of $(x,y)$.
Finally, we check the claims (a) and (b). For (a), if $\psi$ is an order embedding then that $h_X$ and $h_Y$, and thus also $h_P$, are order embeddings follows directly from the commutativity of the diagram in Figure \[F:polHom\]. Moreover, ($\ddagger$) holds for the same reason. Conversely, suppose ($\dagger$) and ($\ddagger$) hold and consider the map $\psi$. Since we already know $\psi$ is monotone, suppose $z,z'\in X\uplus Y$ and that $\psi(z)\leq \psi(z')$. There are four cases.
If either $z,z'\in \iota_X[X]$, or $z,z'\in\iota_Y[Y]$, then that $z\leq z'$ follows again from the commutativity of the diagram in Figure \[F:polHom\]. In the case where $z = \iota_X(x)$ and $z'=\iota_Y(y)$ for some $x\in X$ and $y\in Y$, then $$\psi(z)\leq \psi(z')\iff h_X(x) \operatorname{R}' h_Y(y) \iff x\operatorname{R}y\iff \iota_X(x)\leq \iota_Y(y)\iff z\leq z'.$$ In the final case we have $z=\iota_Y(y)$ and $z'=\iota_X(x)$ for some $x\in X$ and $y\in Y$. Then $$\psi(z)\leq \psi(z')\iff \iota_{Y'}\circ h_Y(y) \leq \iota_{X'}\circ h_X(x).$$ If $p,q\in P$, and $e_Y(p)\leq y$ and $x\leq e_X(q)$, then $$\iota_{Y'}\circ h_Y\circ e_Y(p)\leq \iota_{Y'}\circ h_Y(y)\leq \iota_{X'}\circ h_X(x)\leq \iota_{X'}\circ h_X\circ e_X(q),$$ and thus $\iota_{Y'}\circ e_{Y'}(p)\leq \iota_{X'}\circ e_{X'}(q)$, by the commutativity of the diagram in Figure \[F:isom\], and it follows that $p\leq q$. Thus by the definition of $\preceq$ we have $y\preceq x$ as required.
For (b), if $h_X$ and $h_Y$ are onto then given $z'\in X'\uplus Y'$ we have either $z = \iota_{X'}(h_X(x))$ for some $x\in X$, or $z = \iota_{Y'}(h_Y(y))$ for some $y\in Y$. In either case it follows there is $z\in X\uplus Y$ with $\psi(z)= z$, and thus that $\psi$ is onto. To see that the converse may not hold see Example \[E:notSurj\].
$$\xymatrix{
& Y\ar[d]^{\iota_Y}\ar[r]^{h_Y} & Y'\ar[d]_{\iota_{Y'}}\\
P\ar[ur]^{e_Y}\ar[dr]_{e_X}\ar[r]^\gamma & X\uplus Y\ar[r]^\psi & X'\uplus Y' & \ar[dl]^{e_{X'}}\ar[ul]_{e_{Y'}}\ar[l]_{\gamma'}P' \\
& X\ar[u]_{\iota_X}\ar[r]_{h_X} & X'\ar[u]^{\iota_{X'}}
}$$
\[E:notSurj\] Let $P= X$ be a two element antichain, and let $Y$ be this two element antichain extended by adding a join for the two base elements. Let $P'= X'= Y' = Y$. Then the inclusion maps and the relation $\operatorname{R}_l$ define Galois polarities, and $X\uplus Y\cong Y\cong X'\uplus Y'$. Let $\psi:X\uplus Y\to X'\uplus Y'$ be map induced by the identity function on $X\cup Y$. Then $\psi$ is clearly monotone, surjective, cut-stable and satisfies (1)-(3) from Theorem \[T:polHom\]. However, the induced map $h_X$ cannot be surjective, as $2=|X|<|X'|=3$.
The class of Galois polarities and polarity morphisms forms a category.
Identity morphisms obviously exist, so we need only check composition. We will use Theorem \[T:polHom\]. It’s straightforward to show that the composition of maps satisfying conditions (1)-(3) of that theorem also satisfies these conditions, and compositions of monotone maps are obviously monotone. Moreover, cut-stability is preserved by composition [@Ern91a Corollary 2.10]. Thus it follows from Theorem \[T:polHom\] that polarity morphisms compose appropriately.
We will expand on this categorical viewpoint in Section \[S:cat\].
Galois polarities and $\Delta_1$-completions {#S:PolAndDel}
--------------------------------------------
Recall that given a poset $P$ we write, for example, $e:P\to {\mathcal{N}}(P)$ for the MacNeille completion of $P$ (see Definition \[D:DM\]).
\[L:polToDel\] If ${(e_X,e_Y,\operatorname{R})}$ is a Galois polarity, and if $e:X\uplus Y\to{\mathcal{N}}(X\uplus Y)$ is the MacNeille completion of $X\uplus Y$, then $e\circ\gamma:P\to {\mathcal{N}}(X\uplus Y)$ is a $\Delta_1$-completion (where $\gamma$ is as in Theorem \[T:commute2\]).
$X\uplus Y$ is join-generated by $\iota_X[X]$, and meet-generated by $\iota_Y[Y]$, and $X$ and $Y$ are meet- and join-generated by $e_X[P]$ and $e_Y[P]$ respectively. ${\mathcal{N}}(X\uplus Y)$ is both join- and meet-generated by $e[X\uplus Y]$. Thus every element of ${\mathcal{N}}(X\uplus Y)$ is both a join of meets, and a meet of joins, of elements of $e\circ\gamma[P]$ as required.
If ${(e_X,e_Y,\operatorname{R})}$ is a Galois polarity, define the $\Delta_1$-completion $e\circ\gamma$ constructed from ${(e_X,e_Y,\operatorname{R})}$ in Lemma \[L:polToDel\] to be the **$\Delta_1$-completion generated by ${(e_X,e_Y,\operatorname{R})}$**.
\[L:delToPol\] Let $d:P\to D$ be a $\Delta_1$-completion. Define $X_D$ and $Y_D$ to be (disjoint isomorphic copies of) the subsets of $D$ meet- and join-generated by $e[P]$ respectively. Define $e_{X_D}:P\to X_D$ and $e_{Y_D}:P\to Y_D$ by composing $d$ with the isomorphisms into $X_D$ and $Y_D$ respectively. Abusing notation by identifying $X_D$ and $Y_D$ with their images in $D$, define $\operatorname{R}_D$ on $X_D\times Y_D$ by $x \operatorname{R}_D y\iff x\leq y$ in $D$. Then $(e_{X_D}, e_{Y_D},\operatorname{R}_D)$ is a complete Galois polarity.
The inherited order from $D$ defines a pre-order on $X_D\cup Y_D$ that is a member of ${\hat{\mathcal{P}}}^g_{\operatorname{R}_D}$, so $(e_{X_D},e_{Y_D},\operatorname{R}_D)$ is a Galois polarity. Moreover, $X_D$ and $Y_D$ are complete because $D$ is.
If $d:P\to D$ is a $\Delta_1$-completion, define the complete Galois polarity $(e_{X_D}, e_{Y_D},\operatorname{R}_D)$ constructed from $d$ in Lemma \[L:delToPol\] to be the **Galois polarity generated by $d$**.
\[L:GalEmb\] Let ${(e_X,e_Y,\operatorname{R})}$ be a Galois polarity. Then there is a polarity embedding from ${(e_X,e_Y,\operatorname{R})}$ to $(e_{X_{{\mathcal{N}}}}, e_{Y_{{\mathcal{N}}}}, \operatorname{R}_{{\mathcal{N}}} )$, where the latter object is the Galois polarity generated by the $\Delta_1$-completion generated by ${(e_X,e_Y,\operatorname{R})}$. Moreover, if ${(e_X,e_Y,\operatorname{R})}$ is complete then this embedding is an isomorphism of polarities extending $P$.
Using Lemma \[L:polToDel\], $e\circ\gamma :P\to {\mathcal{N}}(X\uplus Y)$ is the $\Delta_1$-completion generated by ${(e_X,e_Y,\operatorname{R})}$, where $e:X\uplus Y\to {\mathcal{N}}(X\uplus Y)$ is the MacNeille completion. Recall that $\gamma = \iota_X\circ e_X = \iota_Y\circ e_Y$ by definition. To lighten the notation we write e.g. $X_{{\mathcal{N}}}$ for $X_{{\mathcal{N}}(X\uplus Y)}$. Define the map $h_X:X\to X_{{\mathcal{N}}}$ by $h_X = \mu_X\circ e\circ\iota_X$, where $\mu_X$ is the isomorphism used to define $X_{{\mathcal{N}}}$, as in Lemma \[L:delToPol\]. This is clearly an order embedding. Similarly define an order embedding $h_Y:Y\to Y_{{\mathcal{N}}}$ by $h_Y=\mu_Y\circ e\circ\iota_Y$. Define $h_P$ to be the identity map. Note that $e_{X_{{\mathcal{N}}}}$ is just $\mu_X\circ e\circ\gamma = \mu_X\circ e\circ\iota_X\circ e_X$, and similarly $e_{Y_{{\mathcal{N}}}}=\mu_Y\circ e\circ\iota_Y\circ e_Y$. Thus we trivially have the commutativity required by Definition \[D:polHom\](1).
To show that (2) is also satisfied, let $x\in X$, let $y\in Y$, and suppose $\iota_Y(y)\leq \iota_X(x)$. Then $e\circ \iota_Y(y)\leq e\circ \iota_X(x)$. The unique pre-order $\preceq = \operatorname{\preceq_3}$ on $X_{{\mathcal{N}}}\uplus Y_{{\mathcal{N}}}$ can only be the order inherited from ${\mathcal{N}}(X\uplus Y)$, so $\mu_X\circ e\circ\iota_X(x)\preceq \mu_Y\circ e\circ\iota_Y(y)$, and thus $\iota_{Y_{{\mathcal{N}}}}\circ h_Y(y) \leq \iota_{X_{{\mathcal{N}}}}\circ h_X(x)$ as required.
For (3), let $x'\in X_{{\mathcal{N}}}$, let $y'\in Y_{{\mathcal{N}}}$, and suppose $(x', y')\notin \operatorname{R}_{{\mathcal{N}}}$. Then $\mu^{-1}_X(x')\not\leq \mu^{-1}_Y(y')$. As $e\circ\iota_X[X]$ and $e\circ\iota_Y[Y]$ are, respectively, join- and meet-dense in ${\mathcal{N}}(X\uplus Y)$, there is $x\in X$ and $y\in Y$ with $e\circ\iota_X(x)\not\leq e\circ\iota_Y(y)$, with $e\circ\iota_X(x)\leq \mu^{-1}_X(x')$, and with $\mu^{-1}_Y(y')\leq e\circ\iota_Y(y)$. We check the necessary conditions are satisfied for this choice of $x$ and $y$:
(i) Let $a\in X$. Then $$\begin{aligned}
h_X(a)\geq x' &\iff \mu^{-1}_X\circ\mu_X\circ e\circ\iota_X(a)\geq \mu^{-1}_X(x')\\
&\implies e\circ\iota_X(a)\geq e\circ\iota_X(x)\\
&\iff a\geq x,\end{aligned}$$ and so $h_X^{-1}(x'^\uparrow)\subseteq x^\uparrow$ as required.
(ii) Dual to (i).
(iii) Let $a\in X$, let $y'\in Y_{{\mathcal{N}}}$, and suppose $h_X(a)\operatorname{R}_{{\mathcal{N}}} y'$. Then $$\mu^{-1}_X\circ\mu_X\circ e\circ\iota_X(a)\leq \mu_Y^{-1}(y')\leq e\circ\iota_Y(y),$$ and so $\iota_X(a)\leq \iota_Y(y)$, and thus $a\operatorname{R}y$ as required.
(iv) Dual to (iii).
(v) Since $e\circ\iota_X(x)\not\leq e\circ\iota_Y(y)$ we must have $(x, y)\notin \operatorname{R}$.
Now let $x\in X$, let $y\in Y$, and suppose $h_X(x)\operatorname{R}_{{\mathcal{N}}} h_Y(y)$. Then $$\mu_X^{-1}\circ\mu_X\circ e\circ\iota_X(x)\leq \mu_Y^{-1}\circ\mu_Y\circ e\circ\iota_Y(y),$$ and so $\iota_X(x)\leq \iota_Y(y)$, and thus $x\operatorname{R}y$, and we conclude that $(h_X, {\mathrm{id}}_P, h_Y)$ is a polarity embedding as claimed. Finally, when $X$ and $Y$ are complete, as taking MacNeille completions preserves all meets and joins, the maps $h_X$ and $h_Y$ will be surjective, and thus isomorphisms. As $h_P$ is the identity on $P$ the result follows.
\[L:delIsom\] Let $d:P\to D$ be a $\Delta_1$-completion. Then $d$ is isomorphic, as an extension of $P$, to the $\Delta_1$-completion generated by $(e_{X_D}, e_{Y_D},\operatorname{R}_D)$, where the latter object is the complete Galois polarity generated by $d$.
$X_D$ and $Y_D$ are (disjoint isomorphic copies of) the subsets of $D$ meet- and join-generated by $d[P]$ respectively. By definition, and abusing notation slightly, the inclusion of $X_D\cup Y_D$ into $D$ is a MacNeille completion. The unique pre-order $\preceq\in {\hat{\mathcal{P}}}^g_{\operatorname{R}_D}$ on $X_D\cup Y_D$ is just the one inherited from $D$. So composing the embedding $\gamma_D:P\to X_D\uplus Y_D$ with the MacNeille completion of $X_D\uplus Y_D$ we get something isomorphic to $d$ as an extension of $P$.
\[T:delta\] Let $P$ be a poset. There is a 1-1 correspondence between (isomorphism classes of) $\Delta_1$-completions and (isomorphism classes of) complete Galois polarities. Moreover, for a fixed poset $P$ this correspondence restricts to a 1-1 correspondence between (isomorphism classes of) $\Delta_1$-completions of $P$ and (isomorphism classes of) complete Galois polarities extending $P$.
Let, for example, $[d]$ stand for an isomorphism class of $\Delta_1$-completions, let $E$ stand for a complete Galois polarity, and let $\Theta$ be the map defined by $E\in \Theta([d])$ if and only if $E$ is isomorphic to a complete Galois polarity that generates a $\Delta_1$-completion isomorphic to $d$.
Now, if $E_1\in \Theta([d])$ and $E_1\cong E_2$, then $E_2\in \Theta([d])$ by definition of $\Theta$. Moreover, if $E_1,E_2\in \Theta([d])$ then there are complete Galois polarities $E_1'\cong E_1$ and $E_2'\cong E_2$ which generate $\Delta_1$-completions $d_1$ and $d_2$ respectively, and such that $d_1\cong d \cong d_2$. If $d_1$ and $d_2$ are isomorphic $\Delta_1$-completions, then it’s easy to construct a polarity isomorphism between the complete Galois polarities they generate, so we have $E_1\cong E_1'\cong E_2'\cong E_2$. Thus $\Theta([d])$ is an isomorphism class of complete Galois polarities, and this class does not depend on the choice of representative of $[d]$.
By Lemma \[L:delIsom\], every $\Delta_1$-completion is isomorphic to the $\Delta_1$-completion generated by the Galois polarity it generates, so $\Theta$ is a well defined map from the class of isomorphism classes of $\Delta_1$-completions to the class of isomorphism classes of complete Galois polarities.
By Lemma \[L:GalEmb\], if ${(e_X,e_Y,\operatorname{R})}$ is a complete Galois polarity then it is isomorphic to the complete Galois polarity generated by the $\Delta_1$-completion it generates, so $\Theta$ is surjective. It’s also easy to see that isomorphic Galois polarities generate isomorphic $\Delta_1$-completions, so $\Theta$ is injective.
Appealing to Proposition \[P:GalSimp\], the second claim is essentially [@GJP13 Theorem 3.4], and we can also obtain this result with the proof above by working modulo isomorphisms of extensions of $P$ and polarities extending $P$.
Before moving on we pause to consider a technical question regarding the polarity extensions discussed in Section \[S:ExtRes\]. Given a Galois polarity ${(e_X,e_Y,\operatorname{R})}$ which is not complete, by Lemma \[L:GalEmb\] there is a polarity embedding from ${(e_X,e_Y,\operatorname{R})}$ to $(e_{X_D}, e_{Y_D}, \operatorname{R}_D)$, where $e_{X_D}$ and $e_{Y_D}$ are meet- and join-completions respectively. By the definition of polarity embeddings, there are order embeddings $h_X:X\to X_D$ and $h_Y:Y\to Y_D$, and its easy to see these will be meet- and join-completions respectively.
Thus Theorem \[T:Galois\] applies and produces a Galois polarity $(e_{X_D}, e_{Y_D}, \operatorname{\overline{R}})$. We certainly have $\operatorname{\overline{R}}\subseteq \operatorname{R}_D$, by Theorem \[T:Galois\](5), but does the other inclusion also hold? The answer, in general, is no. To see this we borrow [@GJP13 Example 2.2], and lean heavily on the discussion at the start of Section 4 in that paper. The MacNeille completion of a poset $P$ can be constructed from the Galois polarity $(e_{{\mathcal F}_p}, e_{{\mathcal I}_p}, \operatorname{R}_l)$, where $e_{{\mathcal F}_p}:P\to {\mathcal F}_p$ and $e_{{\mathcal I}_p}:P\to {\mathcal I}_p$ are the natural embeddings into the sets of principal upsets and downsets of $P$ respectively.
Consider the poset $P= \omega \cup \omega^\partial$. I.e. $P$ is made up of a copy of $\omega$ below a disjoint copy of the dual $\omega^\partial$. Then ${\mathcal N}(P)=\omega\cup\{z\}\cup \omega^\partial$. I.e. $P$ with an additional element above $\omega$ and below $\omega^\partial$. Let $X = {\mathcal F}_p$ and let $Y={\mathcal I}_p$, so ${\mathcal N}(P)$ is generated by $(e_X, e_Y, \operatorname{R}_l)$. Let the complete Galois polarity arising from Theorem \[T:delta\] be $(e_{X_D}, e_{Y_D}, \operatorname{R}_D)$, where $X_D \cong X\cup \{z_X\}$, and $Y_D\cong Y\cup\{z_Y\}$. Now, to produce ${\mathcal N}(P)$ it is necessary that $z_X\operatorname{R}_D z_Y$, but $(z_X, z_Y)\notin \operatorname{\overline{R}}_l$, and thus $\operatorname{R}_D\neq \operatorname{\overline{R}}$ in this case.
It also follows from this that the $\Delta_1$-completion generated by ${(e_X,e_Y,\operatorname{R})}$ may not be isomorphic to the one generated by $(e_{X'}\circ e_X, e_{Y'}\circ e_Y, \operatorname{\overline{R}})$ from Theorem \[T:Galois\], even when $e_{X'}$ and $e_{Y'}$ are meet- and join-completions respectively.
A categorical perspective {#S:cat}
-------------------------
Here we assume some familiarity with the basic concepts of category theory. The standard reference is [@MacL98], and an accessible introduction can be found in [@Lein14].
We define a pair of categories, ${\mathrm{Pol}}$ and ${\mathrm{Del}}$ as follows:
- Let ${\mathrm{Pol}}$ be the category of Galois polarities and polarity morphisms (from Definition \[D:polHom\]).
- Let ${\mathrm{Del}}$ be the category whose objects are $\Delta_1$-completions, and whose maps are commuting squares as described in Definition \[D:mapCat\], with the additional property that $g_Q$ in that diagram is a complete lattice homomorphism.
\[T:adj\] Let $F:{\mathrm{Pol}}\to {\mathrm{Del}}$ and $G:{\mathrm{Del}}\to {\mathrm{Pol}}$ be defined as follows:
- Let $F:{\mathrm{Pol}}\to{\mathrm{Del}}$ be the map that takes a Galois polarity ${(e_X,e_Y,\operatorname{R})}$ to the $\Delta_1$-completion it generates (described in Lemma \[L:polToDel\]), and takes a polarity morphism $(h_X:X_1\to X_2, h_P:P_1\to P_2, h_Y:Y_1\to Y_2)$ to the map between $\Delta_1$-completions described in Figure \[F:Fmap\], where ${\mathcal{N}}(\psi)$ is the unique complete lattice homomorphism lift of the map $\psi:X_1\uplus Y_1\to X_2\uplus Y_2$ from Theorem \[T:polHom\] to the respective MacNeille completions, as described in [@Ern91a Theorem 3.1].
- Let $G:{\mathrm{Del}}\to {\mathrm{Pol}}$ be the map that takes a $\Delta_1$-completion $d:P\to D$ to the Galois polarity it generates (described in Lemma \[L:delToPol\]), and takes a $\Delta_1$-completion morphism as in Figure \[F:Dmap\] to the triple $(g|_{X_{D_1}}, f, g|_{Y_{D_1}})$, where, for example, $g|_{X_{D_1}}$ is (modulo isomorphism) the restriction of $g$ to $X_{D_1}$, where this is as defined in Lemma \[L:delToPol\].
Then $F$ and $G$ are functors and form an adjunction $F\dashv G$.
For ease of reading we will break the proof down into discrete statements which obviously add up to a proof of the claimed result.
- “*$F$ is well defined*". $F$ is certainly well defined on objects. For maps, as Theorem \[T:polHom\] says $\psi$ will be cut-stable, [@Ern91a Theorem 3.1] says that ${\mathcal{N}}(\psi)$ will be a complete lattice homomorphism. Moreover, the conjunction of these theorems also implies that the diagram in Figure \[F:Fmap\] commutes.
- “*$F$ is a functor*". To see that $F$ lifts identity maps to identity maps note that the identity morphism on ${(e_X,e_Y,\operatorname{R})}$ clearly lifts via Theorem \[T:polHom\] to the identity on $X\uplus Y$, and since taking MacNeille completions is functorial for cut-stable maps (see [@Ern91a Corollary 3.3]), that $F$ maps identity morphisms appropriately follows immediately.
Similarly, it follows from the uniqueness of the map $\psi$ in Theorem \[T:polHom\] that if $h_1=(h_{X_1}, h_{P_1}, h_{Y_1})$ induces the map $\psi_1$, and if $h_2=(h_{X_2}, h_{P_2}, h_{Y_2})$ induces the map $\psi_2$, then the composition $h_2\circ h_1$, if it exists, induces the map $\psi_2\circ\psi_1$. So $F$ respects composition as the MacNeille completion functor does.
- “*$G$ is well defined*". $G$ is also clearly well defined on objects. Consider now a map as in Figure \[F:Dmap\]. We must show that $g|_{X_{D_1}}:X_{D_1}\to X_{D_2}$, that $g|_{Y_{D1}}:Y_{D_1}\to Y_{D_2}$, and that $(g|_{X_{D_1}}, f, g|_{Y_{D1}})$ satisfies the conditions of Definition \[D:polHom\]. First, to lighten the notation define $g_X = g|_{X_{D_1}}$, and $g_Y = g|_{Y_{D1}}$.
Now, by commutativity of Figure \[F:Dmap\] we have $g\circ d_1[P_1]\subseteq d_2[P_2]$. Since $X_{D_1}$ and $X_{D_2}$ are (modulo isomorphism) the meet-closures of $d_1[P_1]$ and $d_2[P_2]$ respectively, and since $g$ is a complete lattice homomorphism, it follows that $g_X$ does indeed have codomain $X_{D_2}$, and $Y_{D_2}$ is the codomain of $g_Y$ by duality. We now check the conditions of Definition \[D:polHom\]:
1. This follows immediately from the definitions of $g_X$ and $g_Y$ and the commutativity of Figure \[F:Dmap\].
2. Abusing notation slightly, we can think of the $\iota$ maps as inclusion functions, and so the claim is just the statement that $y\leq x\implies g(y)\leq g(x)$, and thus is true as $g$ is monotone.
3. Abusing notation in the same way as before, let $x'\in X_{D_2}$, let $y'\in Y_{D_2}$, and suppose $x'\not\leq y' \in D_2$. Using the completeness of $D_1$, let $z_1 = {\bigwedge}g^{-1}(x'^\uparrow)$, and let $z_2 = {\bigvee}g^{-1}(y'^\downarrow)$. It follows easily from the fact that $g$ is a complete lattice homomorphism that $x'\leq g(z_1)$ and $g(z_2)\leq y'$, so if $z_1\leq z_2$ then $x'\leq g(z_1)\leq g(z_2) \leq y'$, as $g$ is monotone. Thus to avoid contradiction we must have $z_1\not\leq z_2$.
As $X_{D_1}$ and $Y_{D_1}$ are, respectively, join- and meet-dense in $D_1$, there is $x\in X_{D_1}\cap z_1^\downarrow$ and $ y\in Y_{D_1}\cap z_2^\uparrow$ with $x\not\leq y$. Now, as $x\leq z_1={\bigwedge}g^{-1}(x'^\uparrow)$ we have $g_X^{-1}(x'^\uparrow)\subseteq z_1^\uparrow\subseteq x^\uparrow$. Thus (i) holds for this choice of $x$, and (ii) holds for $y$ by a dual argument.
Moreover, suppose $a\in X_{D_1}$, and that $g(a)\leq y'$. Then $a\in g^{-1}(y'^\downarrow)$, and so $a\leq z_2$, by definition of $z_2$, and consequently $a\leq y$. Thus (iii) holds, and (iv) is dual. That (v) holds is automatic from the choice of $x$ and $y$.
- “*$G$ is a functor*". $G$ obviously sends identity maps to identity maps, and almost as obviously respects composition.
- “$F\dashv G$". The unit $\eta$ is defined so that its components are the embeddings of ${(e_X,e_Y,\operatorname{R})}$ into $GF{(e_X,e_Y,\operatorname{R})}$ described in Lemma \[L:GalEmb\]. We first show that $\eta$ is indeed a natural transformation. Let $A = (e_{X_1}, e_{Y_1}, \operatorname{R}_1)$ and $B = (e_{X_2}, e_{Y_2}, \operatorname{R}_2)$ be Galois polarities, and let $g =(g_X, g_P, g_Y)$ be a polarity morphism from $A$ to $B$. We aim to show that the diagram in Figure \[F:etaSquare\] commutes.
Consider the diagram in Figure \[F:etaNat\]. Here, for example, $h_{X_1}:X_1\to X_{{\mathcal{N}}_1}$ takes the role of $h_X$ from Lemma \[L:GalEmb\], embedding $X_1$ into $X_{{\mathcal{N}}_1}$. The map $g_X^+:X_{{\mathcal{N}}_1}\to X_{{\mathcal{N}}_2}$ is the $X$ component of $GFg$, which is, modulo isomorphism, the restriction of $Fg$ to $X_{{\mathcal{N}}_1}$, and so on. The inner squares commute by definition of $g$, and that the outer squares commute can be deduced from the commutativity of the diagram in Figure \[F:Xcom\], the commutativity of whose right square follows from the commutativity of the diagram in Figure \[F:polHom\].
Now, $GFg\circ \eta_A$ is the polarity morphism $(g^+_X\circ h_{X_1},g_P,g^+_Y\circ h_{Y_1})$, and $\eta_B\circ g$ is the polarity morphism $(h_{X_2}\circ g_X, g_P, h_{Y_2}\circ g_Y)$, and these are equal by the commutativity of the diagram in Figure \[F:etaNat\].
Now, let $E={(e_X,e_Y,\operatorname{R})}$ be a Galois polarity extending $P$. We will show that $\eta_E$ has the appropriate universal property (see e.g. [@Lein14 Theorem 2.3.6]). Let $d:Q\to D$ be a $\Delta_1$-completion, and let $h=(h_X, h_P, h_Y): E \to G(d)$ be a polarity morphism. We must find a map $g:F(E)\to d$ such that $Gg\circ\eta_E = h$, and show that $g$ is unique with this property.
Consider the diagram in Figure \[F:hCom\]. The upper triangle commutes as $\eta$ is a natural transformation (see Figure \[F:hCom2\]). The isomorphism between $G(d)$ and $GFG(d)$ is just $\eta_{G(d)}$, by Lemma \[L:GalEmb\] and the fact that $G(d)$ is a complete Galois polarity. Note that this is an isomorphism of Galois polarities extending $Q$, so is the identity map on $Q$. By Lemma \[L:delIsom\], there is an isomorphism, $\phi:FG(d)\to d$, of extensions of $Q$, and it follows that $\eta_{G(d)}^{-1} = G\phi$. Thus $\phi\circ Fh:F(E)\to d$ has the property that $$G(\phi\circ Fh)\circ \eta_E = G\phi\circ GFh\circ \eta_E = \eta_{Gd}^{-1}\circ GFh\circ\eta_E = h.$$ We must show that $\phi\circ Fh$ is unique with this property, so let $f:F(E)\to d$ be another ${\mathrm{Del}}$ morphism with $Gf\circ\eta_E = h$. Recall that $F(E) =e\circ\gamma :P\to {\mathcal{N}}(X\uplus Y)$. Then $f$ must agree with $\phi\circ Fh$ on $e[X\uplus Y]$, by definition of $G$. But $(\phi\circ Fh)|_{e[X\uplus Y]}$ is cut-stable, so extends uniquely to a complete lattice homomorphism (by [@Ern91a Theorem 3.1]). Thus $f = \phi\circ Fh$ as required, and so $F\dashv G$ as claimed.
$$\xymatrix{
P_1\ar[d]_{h_P}\ar[r]^{\gamma_1} & X_1\uplus Y_1\ar[r]^{e_1}\ar[d]^\psi & {\mathcal{N}}(X_1\uplus Y_1)\ar[d]^{{\mathcal{N}}(\psi)} \\
P_2\ar[r]\ar[r]_{\gamma_2} & X_2\uplus Y_2\ar[r]_{e_2} & {\mathcal{N}}(X_2\uplus Y_2)
}$$
$$\xymatrix{
P_1\ar[r]^{d_1}\ar[d]_{f} & D_1\ar[d]^g \\
P_2\ar[r]_{d_2} & D_2
}$$
$$\xymatrix{
A\ar[r]^{\eta_A}\ar[d]_g & GF(A)\ar[d]^{GFg} \\
B\ar[r]_{\eta_B} & GF(B)
}$$
$$\xymatrix{
X_{{\mathcal{N}}_1}\ar[d]_{g^+_X} & X_1\ar[d]\ar[l]_{h_{X_1}}\ar[d]^{g_X} & \ar[l]_{e_{X_1}}P_1\ar[r]^{e_{Y_1}}\ar[d]^{g_P} & Y_1\ar[r]^{h_{Y_1}}\ar[d]^{g_Y} & Y_{{\mathcal{N}}_1}\ar[d]^{g^+_Y}\\
X_{{\mathcal{N}}_2} & X_2\ar[l]^{h_{X_2}} & \ar[l]^{e_{X_2}}P_2\ar[r]_{e_{Y_2}} & Y_2\ar[r]_{h_{Y_2}} & Y_{{\mathcal{N}}_2}
}$$
$$\xymatrix{
{\mathcal{N}}(X_1\uplus Y_1)\ar[d]_{Fg} & \ar[l]_{e_1}X_1\uplus Y_1\ar[d]^\psi & \ar[l]_{\iota_{X_1}}X_1\ar[d]^{g_X} \\
{\mathcal{N}}(X_2\uplus Y_2) & \ar[l]^{e_2}X_2\uplus Y_2 & \ar[l]^{\iota_{X_2}}X_2
}$$
$$\xymatrix{
E\ar[d]_h\ar[rr]^{\eta_E} & & GF(E)\ar[d]^{GFh} \\
G(d)\ar[rr]_{\eta_{G(d)}} & & GFG(d)
}$$
$$\xymatrix{
E\ar[ddrr]_h\ar[drr]^{\eta_{G(d)}\circ h}\ar[rr]^{\eta_E} & & GF(E)\ar[d]^{GFh} \\
& & GFG(d)\ar@{<->}[d]^\cong \\
& & G(d)
}$$
The components of counit of the adjunction between $F$ and $G$ are provided by the isomorphisms produced in Lemma \[L:delIsom\]. Thus the subcategory, $\textrm{Fix}(FG)$, of ${\mathrm{Del}}$ is just ${\mathrm{Del}}$ itself. The canonical categorical equivalence between $\textrm{Fix}(GF)$ and $\textrm{Fix}(FG)$ produces a categorical version of the correspondence in Theorem \[T:delta\].
We end the section with a universal property for Galois polarities whose relation is the minimal $\operatorname{R}_l$.
\[P:univ\] Let $(e_X,e_Y,\operatorname{R}_l)$ be a Galois polarity extending $P$, let $Q$ be a poset, and let $f:X\to Q$ and $g:Y\to Q$ be monotone maps such that $f\circ e_X = g\circ e_Y$. Let $\preceq$ be the unique element of ${\hat{\mathcal{P}}}^g_{\operatorname{R}_l}$. Then the following are equivalent:
1. $y\preceq x \implies g(y)\leq f(x) $.
2. There is a unique monotone map $u:X\uplus Y\to Q$ such that the diagram in Figure \[F:univ\] commutes.
Suppose (1) holds. We define $u':X\cup Y \to Q$ by $$u(z) = \begin{cases} f(z) \text{ if } z\in X \\
g(z) \text{ if } z\in Y\end{cases}$$ We show that $u'$ is monotone with respect to the pre-order $\preceq$ and the order on $Q$. Let $z_1\preceq z_2\in X\cup_{\preceq} Y$. If $z_1$ and $z_2$ are both in $X$, or both in $Y$, then that $u'(z_1)\leq u'(z_2)$ follows immediately from the definition of $u$ and the fact that both $f$ and $g$ are monotone. If $z_1= x\in X$, and $z_2=y\in Y$ then there is $p\in e_X^{-1}(x^\uparrow)\cap e_Y^{-1}(y^\downarrow)$, and so $f(x)\leq g(y)$ by the assumption that $f\circ e_X = g\circ e_Y$. If $z_1= y\in Y$ and $z_2 = x\in X$ then that $g(y)\leq f(x)$ is true by (1), and so $u'(y)\leq u'(x)$ as required. Define $u$ by $u(\iota_X(x)) = f(x)$ and $u(\iota_Y(y))= g(y)$. Then $u$ is well defined and monotone by the monotonicity of $u'$, and that $u$ is unique with these properties is automatic from the required commutativity of the diagram.
Conversely, suppose (2) holds. Then $$y\preceq x \implies \iota_Y(y)\leq \iota_X(x)\implies u\circ \iota_Y(y)\leq u\circ \iota_X(x)\implies g(y)\leq f(x)$$ as required.
Proposition \[P:univ\] says that, if $e_X$ and $e_Y$ are fixed meet- and join-extensions respectively, the pair of maps $(\iota'_X, \iota'_Y)$ arising from $(e_X,e_Y,\operatorname{R}_l)$ is initial in the category whose objects are pairs of monotone maps $(f:X\to Q, g:Y\to Q)$ such that $f\circ e_X = g\circ e_Y$ and $y\preceq x \implies g(y)\leq f(x)$, and whose maps are commuting triangles as in Figure \[F:univMorph\] (here $h$ is monotone, and commutativity means $f_2 = h\circ f_1$ and $g_2 = h\circ g_1$). In particular this category contains all $(\iota_X,\iota_Y)$ arising from Galois polarities ${(e_X,e_Y,\operatorname{R})}$ based on $e_X$ and $e_Y$.
$$\xymatrix{
P\ar[r]^{e_Y}\ar[d]_{e_X} & Y\ar[d]^{\iota_Y}\ar@/^/[ddr]^g\\
X\ar[r]_{\iota_X}\ar@/_/[drr]_f & X\uplus Y\ar[dr]^u \\
& & Q
}$$
$$\xymatrix{
(X,Y)\ar[r]^{(f_1,g_1)}\ar[dr]_{(f_2,g_2)} & Q_1\ar[d]^h\\
& Q_2
}$$
A duality principle for order polarities {#S:dual}
========================================
The theory of order polarities
------------------------------
We want to think of order polarities in their various forms as the classes of models for certain theories. A similar approach is taken for ordinary (i.e. not ‘order’) polarities in [@Gol18 Section 5], but we must extend this system to deal with the additional features of extension polarities. For convenience we will use $\approx$ as a logical symbol representing equality.
Let ${\mathscr L}= \{{\mathcal P},{\mathcal X},{\mathcal Y},{\mathcal R},{\vartriangleleft}, e_{\mathcal X}, e_{\mathcal Y}\}$, where ${\mathcal P}$, ${\mathcal X}$, ${\mathcal Y}$ are unary predicates, and ${\mathcal R}$, ${\vartriangleleft}$, $e_{\mathcal X}$, $e_{\mathcal Y}$ are binary predicates. Then ${\mathscr L}$ is the **signature of extension polarities**.
\[D:axioms\] Let $E$ be an ${\mathscr L}$-structure. We can write down a first-order ${\mathscr L}$-sentence guaranteeing that:
1. For all $z\in E$ exactly one of ${\mathcal P}(z)$, ${\mathcal X}(z)$ and ${\mathcal Y}(z)$ holds.
2. ${\vartriangleleft}$ defines a partial ordering on $E$.
3. For all $z_1,z_2\in E$, if $z_1{\vartriangleleft}z_2$ then either ${\mathcal P}(z_1)$ and ${\mathcal P}(z_2)$, ${\mathcal X}(z_1)$ and ${\mathcal X}(z_2)$, or ${\mathcal Y}(z_1)$ and ${\mathcal Y}(z_2)$.
4. For all $z_1,z_2\in E$, if $z_1 {\mathcal R}z_2$ then ${\mathcal X}(z_1)$ and ${\mathcal Y}(z_2)$.
5. $e_{\mathcal X}$ corresponds to an order embedding from $\{z\in E:{\mathcal P}(z)\}$ to $\{z\in E: {\mathcal X}(z)\}$.
6. $e_{\mathcal Y}$ corresponds to an order embedding from $\{z\in E:{\mathcal P}(z)\}$ to $\{z\in E: {\mathcal Y}(z)\}$.
It will help us later to be explicit here, so define ${\mathscr L}$-sentences as follows:
1. $\forall z\Big(\big({\mathcal P}(z)\vee {\mathcal X}(z)\vee {\mathcal Y}(z)\big)\wedge \neg\big(({\mathcal P}(z)\wedge {\mathcal X}(z))\vee ({\mathcal P}(z)\wedge {\mathcal Y}(z)) \vee ({\mathcal X}(z)\wedge {\mathcal Y}(z))\big) \Big)$.
2. $$\begin{aligned}
\forall z_1z_2z_3\Big( (z_1{\vartriangleleft}z_1) &\wedge \big(((z_1{\vartriangleleft}z_2)\wedge (z_2{\vartriangleleft}z_1)){\rightarrow}z_1\approx z_2 \big)\\&\wedge \big(((z_1{\vartriangleleft}z_2)\wedge (z_2{\vartriangleleft}z_3)){\rightarrow}(z_1{\vartriangleleft}z_3)\big) \Big).\end{aligned}$$
3. $\forall z_1 z_2\Big( (z_1{\vartriangleleft}z_2){\rightarrow}\big( ({\mathcal P}(z_1)\wedge {\mathcal P}(z_2))\vee ({\mathcal X}(z_1)\wedge {\mathcal X}(z_2))\vee ({\mathcal Y}(z_1)\wedge {\mathcal Y}(z_2)) \big) \Big).$
4. $\forall z_1z_2\Big( {\mathcal R}(z_1, z_2){\rightarrow}\big({\mathcal X}(z_1)\wedge {\mathcal Y}(z_2)\big) \Big)$.
5. $$\begin{aligned}
\forall z_1\Big(&\big({\mathcal P}(z_1) {\rightarrow}\exists z_2({\mathcal X}(z_2) \wedge e_{\mathcal X}(z_1,z_2))\big)\\ &
\wedge \big(\exists z_2(e_{\mathcal X}(z_1,z_2)){\rightarrow}{\mathcal P}(z_1) \big)\\
& \wedge\forall z_2z_3 z_4 \big( ((z_1{\vartriangleleft}z_2)\wedge e_{\mathcal X}(z_1, z_3)\wedge e_{\mathcal X}(z_2,z_4)) {\rightarrow}( z_3{\vartriangleleft}z_4) \big) \\
& \wedge\forall z_2z_3 z_4 \big( ((z_3{\vartriangleleft}z_4)\wedge e_{\mathcal X}(z_1, z_3)\wedge e_{\mathcal X}(z_2,z_4)) {\rightarrow}( z_1{\vartriangleleft}z_2) \big) \Big).\end{aligned}$$
6. Like (5) but substituting ${\mathcal Y}$ for ${\mathcal X}$, and $e_{\mathcal Y}$ for $e_{\mathcal X}$.
We get a single sentence by taking the conjunction of all the sentences we have defined. Note that (5) and (6) requires (2) to guarantee that $e_{\mathcal X}$ and $e_{\mathcal Y}$ are well defined and injective. Denote the set of these axioms ${\mathrm{Tpol}}$.
\[P:EP\] If $E$ is an ${\mathscr L}$-structure and $E\models {\mathrm{Tpol}}$, then $$(e_X:\{z\in E : P(z)\}\to \{z\in E : X(z)\},e_Y:\{z\in E : P(z)\}\to \{z\in E : Y(z)\},\operatorname{R})$$ defines an extension polarity when $P$, $X$, $Y$, $\operatorname{R}$ are the interpretations of ${\mathcal P}$, ${\mathcal X}$, ${\mathcal Y}$, ${\mathcal R}$, and where $e_X$, $e_Y$, $\leq$ are defined using $e_{\mathcal X}$, $e_{\mathcal Y}$, ${\vartriangleleft}$ in the obvious way. Moreover, If ${(e_X,e_Y,\operatorname{R})}$ is an extension polarity then ${(e_X,e_Y,\operatorname{R})}$ can be naturally understood as an ${\mathscr L}$-structure, and ${(e_X,e_Y,\operatorname{R})}\models {\mathrm{Tpol}}$.
This is straightforward.
Define ${\mathrm{EP}}$ to be the class of ${\mathscr L}$-structures where ${\mathrm{Tpol}}$ holds.
Dual formulas and dual polarities
---------------------------------
Let $E$ be an ${\mathscr L}$-structure, and suppose the symbols of ${\mathscr L}$ are interpreted in $E$ as $P$, $X$, $Y$, $\leq$, $e_X$, $e_Y$, $\operatorname{R}$. Define $E^\partial$ to be the ${\mathscr L}$-structure whose underlying set is that of $E$, and whose interpretations of the symbols of ${\mathscr L}$ are as follows:
- ${\mathcal P}$ is $P$.
- ${\mathcal X}$ is $Y$.
- ${\mathcal Y}$ is $X$.
- ${\vartriangleleft}$ is $\leq^\partial$, which is defined using $z_1\leq^\partial z_2 \iff z_2\leq z_1$.
- ${\mathcal R}$ is $\operatorname{R}^\partial$, which is defined by $\operatorname{R}^\partial(z_1 ,z_2)\iff \operatorname{R}( z_2,z_1)$.
- $e_{\mathcal X}$ is $e_Y$.
- $e_{\mathcal Y}$ is $e_X$.
Let $\theta$ be a second-order ${\mathscr L}$-formula. We define the **dual**, $\theta^\partial$, recursively. As ${\mathscr L}$ is relational the only terms are variables. We define the dual for atomic ${\mathscr L}$-formulas by:
- $({\mathcal P}(z))^\partial = {\mathcal P}(z)$.
- $({\mathcal X}(z))^\partial = {\mathcal Y}(z)$.
- $({\mathcal Y}(z))^\partial = {\mathcal X}(z)$.
- $(e_{\mathcal X}(z_1,z_2))^\partial = e_{\mathcal Y}(z_1,z_2)$.
- $(e_{\mathcal Y}(z_1,z_2))^\partial = e_{\mathcal X}(z_1,z_2)$.
- $(z_1{\vartriangleleft}z_2)^\partial = z_2{\vartriangleleft}z_1$.
- $({\mathcal R}(z_1,z_2))^\partial = {\mathcal R}(z_2,z_1)$.
- $(z_1\approx z_2)^\partial = z_1\approx z_2$.
- If $Z$ is an $n$-ary predicate variable we define $Z(z_1,\ldots,z_n)^\partial = Z(z_1,\ldots,z_n)$.
We extend this to first-order ${\mathscr L}$-formulas by defining:
- $(\neg\theta)^\partial = \neg(\theta^\partial)$.
- $(\theta_1\wedge \theta_2)^\partial = \theta_1^\partial \wedge \theta_2^\partial$.
- $(\forall z \theta))^\partial = \forall z \theta^\partial$.
Finally, to extend to second-order formulas, suppose $Z$ is a predicate variable and define:
- $(\forall Z \theta)^\partial = \forall Z \theta^\partial$.
If $\Gamma=\{\theta_i:i\in I\}$ is a set of ${\mathscr L}$-formulas, then we define $\Gamma^\partial = \{\theta_i^\partial:i\in I\}$.
\[L:dual\] Let $E$ be an ${\mathscr L}$-structure, and let $\theta$ be a second-order ${\mathscr L}$-formula. Let $v$ be an assignment of variables for $E$, and note that $v$ also defines an assignment of variables for $E^\partial$. Then $E,v\models \theta \iff E^\partial,v\models \theta^\partial$.
We induct on formula construction. Again, as ${\mathrm{Tpol}}$ is a relational signature the only terms are variables. If $\theta$ is atomic there are nine cases:
- $\theta = z_1\approx z_2$: We have $v(z_1)= v(z_2)$ in both $E$ and $E^\partial$.
- $\theta = {\mathcal P}(z)$: We $P(v(z))$, and there is nothing to prove.
- $\theta = {\mathcal X}(z)$: We have $\theta^\partial = {\mathcal Y}(z)$, and ${\mathcal Y}$ is interpreted in $E^\partial$ as $X$. Since $X(v(z))$ there is nothing to do.
- $\theta = {\mathcal Y}(z)$: Similar to the preceding case.
- $\theta = e_{\mathcal X}(z_1,z_2)$: Here $\theta^\partial = e_{\mathcal Y}(z_1,z_2)$, and $e_{\mathcal Y}$ is interpreted in $E^\partial$ as $e_X$. Since $e_X(v(z_1),v(z_2))$ there is nothing to do.
- $\theta = e_{\mathcal Y}(z_1,z_2)$: Similar to the preceding case.
- $\theta = z_1{\vartriangleleft}z_2$: We have $$\begin{aligned}
E,v\models z_1{\vartriangleleft}z_2&\iff v(z_1)\leq v(z_2)\\&\iff v(z_2)\leq^\partial v(z_1)\\
&\iff E^\partial,v\models z_2{\vartriangleleft}z_1.\end{aligned}$$
- $\theta = {\mathcal R}(z_1, z_2)$: We have $\operatorname{R}(v(z_1), v(z_2)) \iff \operatorname{R}^\partial(v(z_2), v(z_1))$.
- $\theta = Z(z_1,\ldots,z_n)$ for some $n$-ary predicate variable $Z$: This is automatic.
Now for the inductive step we have four cases:
1. $\theta = \neg \psi$: In this case $$E,v\models \neg\psi \iff E, v\not\models \psi \iff E^\partial,v\not\models \psi^\partial \iff E^\partial,v\models \neg\psi^\partial.$$
2. $\theta = \psi_1\vee \psi_2$: We have either $E, v\models \psi_1$ or $E, v\models \psi_2$, and the result follows.
3. $\theta = \forall z \psi$: If $u$ is an assignment of variables for $E$ agreeing with $v$ everywhere except, possibly, at $z$, we have $E,u \models \psi$, and so $E^\partial, u \models \psi^\partial$, and thus $E^\partial, v \models \forall z \psi^\partial$. So $E,v\models\forall z \psi \implies E^\partial,v\models \forall z \psi^\partial$, and the argument for the converse is similar.
4. $\theta = \forall S \psi$: Essentially the same argument as in the preceding case.
\[L:axioms\] Consider the axioms ${\mathrm{Tpol}}$ from Definition \[D:axioms\]. (1)-(4) are self-dual, and (5) and (6) are dual to each other.
This is a routine check.
\[C:dual\] $E\in{\mathrm{EP}}\iff E^\partial\in{\mathrm{EP}}$.
This follows immediately from Lemmas \[L:dual\] and \[L:axioms\].
The conditions from Theorems \[T:ext\], \[T:commute\], \[T:commute1\], and Corollary \[C:emb\] all correspond to ${\mathscr L}$-sentences, as do the conditions that $e_X$ and $e_Y$ are meet- and join-extensions respectively. These conditions are all first-order, except (D0) and (D1) which require quantification over sets. Note that an order polarity $(X,Y,\operatorname{R})$ is an extension polarity ${(e_X,e_Y,\operatorname{R})}$ where $P$ is empty. I.e. which is a model of the self-dual ${\mathscr L}$-sentence $$\forall z (\neg {\mathcal P}(z)).$$
Bearing this in mind, and recalling Proposition \[P:EP\], we make the following definition.
For $\ast\in\{0,1,2,3\}$ define ${\mathrm{Tpol}}_\ast$ to be the finite set of ${\mathscr L}$-sentences defining $\ast$-coherence. Similarly, define ${\mathrm{Tpol}}_g$ to be the finite set of ${\mathscr L}$-sentences defining Galois polarities. Moreover, for $\ast\in\{0,1,2,3,g\}$ define ${\mathrm{EP}}_\ast$ to be the class of ${\mathscr L}$-structures satisfying ${\mathrm{Tpol}}_\ast$.
Note that ${\mathrm{Tpol}}_3$ uses some second-order axioms, as mentioned previously.
\[L:dualApp\] Let $\ast\in\{0,1,2,3,g\}$ and let $E$ be an ${\mathscr L}$-structure. Then $$E\in {\mathrm{EP}}_\ast\iff E^\partial\in{\mathrm{EP}}_\ast.$$
This follows from Lemma \[L:dual\] and the fact that the ${\mathscr L}$-sentences involved are all either self-dual or come in dual pairs. Lemma \[L:axioms\] proves this for ${\mathrm{EP}}$, and for the ${\mathrm{EP}}_\ast$ cases we just need to look at the conditions from the corresponding theorems. For example, both (B0) and (B1) are self-dual.
\[T:dual\] Let $\ast\in\{0,1,2,3,g\}$, and let $\theta$ be a second-order ${\mathscr L}$-sentence. Then ${\mathrm{Tpol}}_\ast\models \theta\iff {\mathrm{Tpol}}_\ast\models \theta^\partial$.
Suppose ${\mathrm{Tpol}}_\ast\models \theta$, and let $E\in{\mathrm{EP}}_\ast$. Then $E^\partial\in{\mathrm{EP}}_\ast$, by Lemma \[L:dualApp\]. So $E^\partial\models \theta$, and it follows from Lemma \[L:dual\] that $E^{\partial\partial} \models \theta^\partial$. But $E^{\partial\partial}= E$, so we have $E\models \theta^\partial$ as required. Thus ${\mathrm{Tpol}}_\ast\models \theta \implies {\mathrm{Tpol}}_\ast\models \theta^\partial$. The argument for the converse is similar.
Applying the duality principle
------------------------------
We have appealed to this duality principle several times during the course of the paper. We go through the details of a pair of representative examples here.
\[E:duality1\] Consider the proof of Lemma \[L:conds\]. First it is shown that $$(A0)\wedge (\dagger_0) \models (\dagger_1).$$ So, by Lemma \[L:dual\] we have $(A0)^\partial\wedge (\dagger_0)^\partial \models (\dagger_1)^\partial$, but $(A0)$ and $(\dagger_1)$ are dual to $(A1)$ and $(\dagger_2)$ respectively, and $(\dagger_0)$ is self-dual. So we have $$(A1)\wedge (\dagger_0) \models (\dagger_2)$$ as claimed.
In the next example we abuse notation slightly by writing, for example, $e_X(p)$ as a shorthand way to specify “the element $z$ such that $e_X(p,z)$".
\[E:duality2\] Consider the proof of Theorem \[T:commute\], specifically the proof of the transitivity of $\operatorname{\preceq_1}$, and, even more specifically, the $(x_1,y,x_2)$ case. Let $\theta_1$ be the ${\mathscr L}$-formula defined by $$\begin{aligned}
\theta_1 = &{\mathcal X}(x_1)\wedge {\mathcal X}(x_2)\wedge {\mathcal Y}(y)\wedge {\mathcal R}(x_1,y) \\
&\wedge \exists pq\big({\mathcal P}(p)\wedge {\mathcal P}(q)\wedge {\mathcal R}(e_{\mathcal X}(p), e_{\mathcal Y}(q)) \wedge (y{\vartriangleleft}e_Y(p)) \wedge (e_X(q){\vartriangleleft}x_2)\big),\end{aligned}$$ and define $\theta_2$ by $$\theta_2 = {\mathcal X}(x_1)\wedge {\mathcal X}(x_2) \wedge \exists p\big({\mathcal P}(p)\wedge {\mathcal R}(x_1,e_Y(p))\wedge (e_X(p){\vartriangleleft}x_2)\big).$$ Define $\theta$ by $$\theta = \forall x_1x_2y (\theta_1{\rightarrow}\theta_2).$$
Then the $(x_1,y,x_2)$ case of the transitivity proof is showing that ${\mathrm{Tpol}}_1\models \theta$, and so by the duality principle we have ${\mathrm{Tpol}}_1\models \theta^\partial = \forall x_1x_2y(\theta_1^\partial {\rightarrow}\theta_2^\partial)$.
But, if we substitute the variable symbols $y_2,y_1,x$ for $x_1,x_2,y$ respectively we get $$\begin{aligned}
\theta_1^\partial = &{\mathcal Y}(y_2)\wedge {\mathcal Y}(y_1)\wedge {\mathcal X}(x)\wedge {\mathcal R}(x,y_2) \\
&\wedge \exists pq\big({\mathcal P}(p)\wedge {\mathcal P}(q)\wedge {\mathcal R}(e_{\mathcal X}(q),e_{\mathcal Y}(p)) \wedge (e_X(p){\vartriangleleft}x) \wedge (y_1{\vartriangleleft}e_Y(q))\big),\end{aligned}$$ and $$\theta_2^\partial = {\mathcal Y}(y_2)\wedge {\mathcal Y}(y_1) \wedge \exists p\big({\mathcal P}(p)\wedge {\mathcal R}(e_X(p),y_2)\wedge (y_1{\vartriangleleft}e_Y(p))\big).$$ So $${\mathrm{Tpol}}_1\models \forall xy_1y_2 \theta^\partial,$$ which proves the $(y_1,x,y_2)$ case of transitivity.
As the reader has no doubt observed from these examples, formal application of the duality principle can involve some rather tedious bookkeeping. Fortunately, it is usually fairly easy to see where it applies, and the details can be safely suppressed.
[10]{}
B. Banaschewski and G. Bruns. Categorical characterization of the [M]{}ac[N]{}eille completion. , 18:369–377, 1967.
J. Barwise and J. Seligman. , volume 44 of [*Cambridge Tracts in Theoretical Computer Science*]{}. Cambridge University Press, Cambridge, 1997. The logic of distributed systems.
G. Birkhoff. . Am. Math. Soc., 1940.
G. Birkhoff. . Am. Math. Soc., 1995.
P. Blackburn, M. de Rijke, and Y. Venema. , volume 53 of [*Cambridge Tracts in Theoretical Computer Science*]{}. Cambridge University Press, Cambridge, 2001.
B. A. Davey and H. A. Priestley. . Cambridge University Press, New York, second edition, 2002.
J. Dunn, M. Gehrke, and A. Palmigiano. Canonical extensions and relational completeness of some substructural logics. , 70:713–740, 2005.
M. Erné. The [D]{}edekind-[M]{}ac[N]{}eille completion as a reflector. , 8(2):159–173, 1991.
M. Erné, J. Koslowski, A. Melton, and G. E. Strecker. A primer on [G]{}alois connections. In [*Papers on general topology and applications ([M]{}adison, [WI]{}, 1991)*]{}, volume 704 of [*Ann. New York Acad. Sci.*]{}, pages 103–125. New York Acad. Sci., New York, 1993.
B. Ganter and R. Wille. . Springer-Verlag, Berlin, 1999. Mathematical foundations, Translated from the 1996 German original by Cornelia Franzke.
M. Gehrke. Generalized [K]{}ripke frames. , 84(2):241–275, 2006.
M. Gehrke and J. Harding. Bounded lattice expansions. , 238(1):345–371, 2001.
M. Gehrke, R. Jansana, and A. Palmigiano. ${\Delta_1}$-completions of a poset. , 30(1):39–64, 2013.
M. Gehrke and B. Jónsson. Bounded distributive lattices with operators. , 40:207–215, 1994.
M. Gehrke and B. Jónsson. Bounded distributive lattice expansions. , 94(1):13–45, 2004.
M. Gehrke and H. Priestley. Canonical extensions and completions of posets and lattices. , 43:133–152, 2008.
M. Gehrke and J. Vosmaer. A view of canonical extension. In [*Logic, language, and computation*]{}, volume 6618 of [ *Lecture Notes in Comput. Sci.*]{}, pages 77–100. Springer, Heidelberg, 2011.
S. Ghilardi and G. Meloni. Constructive canonicity in non-classical logics. , 86(1):1–32, 1997.
R. Goldblatt. Canonical extensions and ultraproducts of polarities. , 79(4):Art. 80, 28, 2018.
B. Jónsson and A. Tarski. Boolean algebras with operators [I]{}. , 73:891–939, 1951.
T. Leinster. , volume 143 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 2014.
S. Mac Lane. , volume 5 of [ *Graduate Texts in Mathematics*]{}. Springer-Verlag, New York, second edition, 1998.
H. M. MacNeille. Partially ordered sets. , 42(3):416–460, 1937.
W. Morton. Canonical extensions of posets. , 72(2):167–200, 2014.
W. Morton and C. J. van Alten. Distributive and completely distributive lattice extensions of ordered sets. , 28(3):521–541, 2018.
H. A. Priestley. Representation of distributive lattices by means of ordered [S]{}tone spaces. , 2:186–190, 1970.
H. A. Priestley. Ordered topological spaces and the representation of distributive lattices. , 24:507–530, 1972.
J. Schmidt. Each join-completion of a partially ordered set is the solution of a universal problem. , 17:406–413, 1974.
V. Sofronie-Stokkermans. Duality and canonical extensions of bounded distributive lattices with operators, and applications to the semantics of non-classical logics. [I]{}. , 64(1):93–132, 2000.
V. Sofronie-Stokkermans. Duality and canonical extensions of bounded distributive lattices with operators, and applications to the semantics of non-classical logics. [II]{}. , 64(2):151–172, 2000.
T. Suzuki. Canonicity results of substructural and lattice-based logics. , 4(1):1–42, 2011.
T. Suzuki. On canonicity of poset expansions. , 66(3):243–276, 2011.
T. Suzuki. Morphisms on bi-approximation semantics. In [*Advances in modal logic. [V]{}ol. 9*]{}, pages 494–515. Coll. Publ., London, 2012.
T. Suzuki. On polarity frames: applications to substructural and lattice-based logics. In [*Advances in modal logic. [V]{}ol. 10*]{}, pages 533–552. Coll. Publ., London, 2014.
W. R. Tunnicliffe. The completion of a partially ordered set with respect to a polarization. , 28:13–27, 1974.
|
---
address: |
$^1$ Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, USA\
$^2$ Department of Brain and Cognitive Sciences, Seoul National University, Seoul, South Korea\
$^3$ School of Computing Science, Newcastle University, Claremont Tower, Newcastle upon Tyne NE1 7RU, UK\
$^4$ Institute of Neuroscience, Newcastle University, Framlington Place, Newcastle upon Tyne NE2 4HH, UK
author:
- 'Jinseop S. Kim$^{1,2}$ and Marcus Kaiser[^1]$^{3,4,2,*}$'
title: 'From [*Caenorhabditis elegans*]{} to the Human Connectome: A Specific Modular Organisation Increases Metabolic, Functional, and Developmental Efficiency'
---
\[firstpage\]
Introduction
============
In network representation, neural networks at different levels of organisation ranging from connections between individual neurons to connections between brain regions can be described coherently, if the individual neurons or brain regions are substituted by the nodes and the connection between them by the links. Also, the modular organisation found in different levels of neural networks can be exhibited by network modules, where a module is a subset of the nodes having many connections among them and few to the rest of the network [@kaiser_tutorial].
The first species to show neural networks are Coelenterates such as [*Cnidaria*]{} [@mackie; @arendt]. These animals show a diffuse two-dimensional nerve network called a lattice network. In such networks, neighbours are well connected but there are no long-distance connections. For functionally specialised circuits, however, a regular organisation is unsuitable. Starting with the formation of sensory organs and motor units, neurons segregate in modules; e.g. forming ganglia in the roundworm [*Caenorhabditis elegans*]{} [@white]. Forming such modules, ganglia can process one modality with little interference from neurons processing different kinds of information. At one point of growing complexity of organisms, having one module for one modality or function is not sufficient. An example is processing of visual information in primates where the visual module consists of two network components: nodes that form the dorsal pathway for processing object movement and nodes of the ventral pathway for processing object features such as colour and form. These networks where smaller sub-modules are nested within modules are a type of hierarchical network [@kaiser_dyn; @meunier; @krumnack].
The modularity $Q$ measures how modular a given network is [@newman]. The human brain network for the connections between brain regions or ROIs as well as the neuronal network for the connections between neurons show a high modularity compared to randomly connected networks [@meunier] and this modularity is preserved from at least 4 to 40 years [@Lim2014CerCor]. However, there are numerous ways of constructing modular networks with a given value of modularity. What are specific to the chosen biological organisations over alternative modular arrangements and what are the advantages of them? In this article, we address these questions on two different levels of organisation: the connections between individual neurons in [*C. elegans*]{}, the level of the micro-connectome [@seung], and the connections between different human brain regions, the level of the macro-connectome [@defelipe]. To investigate the connection specificity of these networks over alternative arrangements, we employ benchmark networks generated by a link swapping process which is controlled by the simulated annealing algorithm. Such rewired networks can serve as control groups, where the number of connections for each node and the modularity of networks are kept constant.
First, at both levels we find that the clustering coefficient, indicating how well information can be distributed locally, and the characteristic path length, indicating how difficult global integration is, are high compared to alternative networks of similar modularity. This shows a balance between the need for communication within local circuits (high neighbourhood connectivity within modules) and the reduction of interference between modules (fewer shortcuts linking different modules). Indeed, brain disorders such as schizophrenia [@sritharan] and epilepsy [@chavez] can be linked to changes in local and global efficiency. Second, the total wiring length is smaller compared to the alternative networks of similar modularity. The connectivity of the original network and alternative networks are compared through their network of modules, the coarse-grained network obtained when human brain areas are considered as new nodes instead of the ROIs. We find that the formation of fibre bundles, or the fasciculation, is correlated with the reduced total wiring length. A similar behaviour is observed from the network of neurons and the network of ganglia in [*C. elegans*]{}. To quantify this bundling behaviour, we introduce the novel measure of dispersion indicating how widely individual nodes are connected to different modules of the network. Third, both neural networks show lower algorithmic entropy than their alternative arrangements. As the algorithmic entropy quantifies the amount of information needed to construct an object, this suggests that fewer rules are needed to encode for the organisation of neural networks and the neural systems are efficiently organised from a developmental point of view [@roberts].
Materials and Methods
=====================
Data {#data .unnumbered}
----
The human brain network used in this article was from [@hagmann]. The connectivity was obtained from $5$ individual subjects using the diffusion spectrum imaging (DSI). The DSI is one of the protocols of diffusion magnetic resonance imaging (dMRI), which detects the diffusion pattern of water molecules in the brain to predict the trajectory of fibre tracts. In the DSI, first the brain was partitioned into anatomical areas called the Brodmann areas and then each of them was subdivided into a certain number of ROIs in such a way that each ROI has a similar surface area. The number of brain areas were chosen to be $R=66$ and the number of ROIs resulted in $N=998$. The ROIs were regarded as nodes and the brain areas as modules. Next, the tractography was constructed from the diffusion pattern and a link was assigned between two ROIs that are connected by the predicted fibre tract. The total number of links was $E=17,865$.
For [*C. elegans*]{}, a total of $N=279$ neurons and corresponding $E=2,990$ connections were used. These included $1,584$ unidirectional and $1,406$ bidirectional connections. Biologically, they represent $672$ gap junctions, $1,962$ chemical synapses and $376$ connections where both gap junctions and chemical synapses exist between the neuron pairs. As some network measures are defined only for undirected networks, all the unidirectional connections were replaced by bidirectional ones leading to a total of $2,287$ bidirectional links. Three-dimensional neuron coordinates were used as described in [@varier]. The information about the $R=10$ ganglia membership for modules was taken from [@ay].
The network of modules was defined as follows. The modules, corresponding to the anatomical areas for brain or the ganglia for [*C. elegans*]{}, were regarded as nodes in place of the ROIs or neurons. Correspondingly, two modules were assigned with a link between them only if there is at least one link between a pair of nodes each of which is contained by each module (see Supplementary Figure S1).
Network measures {#network-measures .unnumbered}
----------------
All the calculations, including measurement of modularity and simulated annealing procedure (see below), were performed by custom built codes in C programming language and MATLAB (routines are available at [http://www.biological-networks.org/]{}). The characteristic path length ($L$) was the average number of connections that have to be passed on the shortest paths between all pairs of network nodes. The clustering coefficient ($C$) was the proportion of actually present connections, out of all possible connections, among network nodes directly connected to a node. It was calculated as the average over all individual nodes of the network [@watts]. The small-world index was calculated as $\sigma_{\text{sw}}=(C/C_{\text{rand}}) / (L/L_{\text{rand}})$ or equivalently $\sigma_{\text{sw}}=(C/L) / (C_{\text{rand}}/L_{\text{rand}})$, where $C$ and $L$ defined as above were measured from the observed network and $C_{\text{rand}}$ and $L_{\text{rand}}$ were the average values from $100$ Erdős-Rényi (ER) random network [@humphries]. The generation rule for the ER network was as following. Initially $N$ nodes are given without any connection. At each time step, a link is added between a pair of nodes which are selected among the $N$ nodes at random, avoiding multiple times of selection. This step is repeated until the number of links becomes $E$. The small-worldness $\sigma_{\text{sw}}$ is larger than 1 for small-world networks, equal to 1 if the ratio between $C/C_{\text{rand}}$ and $L/L_{\text{rand}}$ is the same as for random networks (note that absolute $C$ and $L$ might still differ from those of random networks), and smaller than 1 when the clustering coefficient is smaller and/or the characteristic path length is larger than for random networks. The total wiring length ($W$) was the sum of the Euclidean distance between all connections of a network when the network nodes are provided with spatial locations.
Modularity and link swapping {#modularity-and-link-swapping .unnumbered}
----------------------------
For a network of $N$-nodes, $E$-links and $R$-modules, whose node index, $i$, runs from $1$ to $N$ and the module to which node $i$ belongs, $q_i$, can take value from $1$ to $R$, the modularity was defined as $$Q=\frac{1}{2E} \sum_{ij} \left[ A_{ij}-\frac{k_i k_j}{2E}\right]\delta(q_i,q_j),$$ where $A_{ij}$ is $(i,j)$ element of the adjacency matrix, $k_i$ is the number of connections, or the degree, of node $i$, and $\delta$ is Kronecker’s delta function [@newman]. It measures what fraction of the links connect two nodes within one module and its deviation from the case when the links are distributed at random. The modularity can be used for finding the modular structure of a given network when it is unknown. In such a setting, an optimal partitioning of the network nodes is searched, which maximises the modularity of the given network. Therefore, the assignments of nodes to modules are varied while the connections of the nodes are fixed. In this study, however, the predefined modules of respective networks, i.e. the anatomical areas of human brain and the ganglia of [*C. elegans*]{}, were regarded as fixed. Each node already has its intrinsic module membership. Instead, the connections between nodes were varied by link swapping controlled by simulated annealing.
The link swapping is a process which a pair of links are selected and then two nodes at an arbitrary end of each link are exchanged. Whereas the degree of each node, as well as its distribution for the entire network, is preserved before and after the manipulation, the modularity of the network can be increased, be decreased, or remain the same depending on the sort of the selected pair of links. It is increased if a pair of links are selected in such a way that at least two nodes at the ends of different links lie in one module and swapping is carried out to connect those two nodes. Likewise, the number of intra-module links determines the modularity of rewired network after the swapping (see Supplementary Figure S1).
To alter the modularity of the networks to have desired values, the selection of link pairs for swapping process was controlled by simulated annealing method as follows [@guimera]. At each step, the link swapping is attempted and the amount of change in modularity for the attempt, $\Delta Q$, is calculated. The attempt is accepted with probability $1$ if $\Delta Q \ge 0$ or with probability $e^{\Delta Q/T}$ if $\Delta Q<0$, where $T$ is the control parameter or temperature. Otherwise, the attempt is rejected and the swapping is reversed to recover the original connectivity. When $T \to 0$, link swappings are accepted only when the modularity increases and the simulated annealing becomes equivalent to the greedy algorithm for finding the maximum modularity. Originally, the simulated annealing was devised to avoid trapping into local extrema as the greedy algorithm often does, and $T$ is incrementally decreased from a finite value to infinitesimal so that the swapping happens a certain number of times at each $T$ value. The consequent maximum value during the entire time steps is expected to be the global maximum. In a similar manner, to obtain a network with the desired modularity $Q_d$, one can set the problem to minimise $\lvert Q-Q_d \rvert$.
However, in this study, we employed a simpler method since the minimisation procedure is computationally expensive and the networks from the two different methods are theoretically equivalent to each other. The alternative method took advantage of the fact that the modularity, unless small fluctuations, converges to a single value for a given temperature. After a sufficient number of link swappings are performed, the connection specificity of the original network is lost and the resulting network has desired modularity but otherwise maximally random. Any choice of network snapshot at this state is statistically identical to each other, and the entire set of such networks is the ensemble of networks with the given modularity. In practice, we first performed $800 \times E$ times of link swapping for a given temperature, and then sampled $100$ network snapshots during additional $200 \times E$ of steps.
Dispersion {#dispersion .unnumbered}
----------
We introduced the novel measure, dispersion $D$, of a network which shows how widely the connections are distributed across different modules. The dispersion of an individual node $i$ was defined as $D_i = R_i / R$ where $R_i$ is the number of different modules to which the node is connected to (brain areas for the human connectome or ganglia for the [*C. elegans*]{} connectome) and $R$ is the total number of modules ($66$ and $10$, respectively). The maximum dispersion of a node is 1 in the case where the node is connected to at least one node in all other modules of the network. The dispersion of a network is the average dispersion for all nodes: $D = \sum D_i / N$ where $N$ is the number of nodes ($998$ ROIs for the human connectome and $279$ neurons for the [*C. elegans*]{} connectome). Note that the modules in this study are anatomical units (brain areas or ganglia) and not the modules defined by network analysis module detection algorithms [@kaiser_tutorial]. However, alternative definitions for module can also be applied, and the dispersion could serve as a useful measure for future studies.
Algorithmic entropy {#algorithmic-entropy .unnumbered}
-------------------
Algorithmic entropy was used as a measure for the amount of information the networks bear. It was originally introduced as a conceptual measure for any kind of physical or abstract objects, and later a practical way to quantify it was devised [@li]. Assume an object saved in a computer storage device. If the object contains regularities, it can be described by a shorter message leading to less storage usage. A compression algorithm is a standard way to detect such regularities and reduce storage usage, and the compressed data size can give an estimate of the amount of information. To apply this to the neural networks, we saved the networks in the format of unweighted adjacency matrices into $N \times N$ [int8]{} arrays, whose $(i, j)$ element takes value $1$ if nodes $i$ and $j$ have a connection to each other and otherwise $0$. Any configuration of networks with the same number of nodes $N$ has $N^2$ bytes of data size. Then, minimum compression size for each of the adjacency matrix arrays for the original connectomes as well as the rewired network ensembles for different values of $Q$ was found by the simulated annealing method similar to above. The compression was performed by the [gzip]{} library which uses the Lempel-Ziv coding [@ziv]. The compression ratio, the ratio of the compressed data size to the original size of the array in bytes, was measured to indicate the relative amount of information in the networks. As the adjacency matrix of the networks are symmetric and sparse, more efficient data storing strategy could be devised. Although this can change the quantitative values of the compression ratio, it is unlikely that the qualitative trend of the result from the original and alternative networks would change.
For simulated annealing, the objective measure to minimise was the compression size $Z$ and the variable was node index assignment. Whereas the assignment of node index, i.e. which node becomes the node $i$, is arbitrary, the shape of the adjacency matrix depends on the index assignment and in turn the compression size depends on the shape. As the algorithmic entropy, by definition, aims to measure the upper bound of the amount of information, the node index assignment needs to minimise the size of compressed array. At each time step of the simulated annealing, the node indices were reassigned by exchanging the indices of two nodes $i$ and $j$, which is equivalent to exchanging the $i$-th and $j$-th column and row of the adjacency matrix. Then $\Delta Z$ was measured by comparing the $Z$ values before and after the reassignment, to determine such a reassignment should be kept or reverted with the probability of $1$ when $\Delta Z \le 0$ or with probability of $e^{-\Delta Z/T}$ when $\Delta Z>0$. $T$ was incrementally decreased from a finite value to infinitesimal so that the index reassignment happens a certain number of time steps at each $T$ value. The global minimum $Z$ during the entire time step was recorded.
Results
=======
To illustrate the connectivity of the neural networks, we calculated the network measures of the human and [*C. elegans*]{} connectome. Two relevant measures, $L$ and $C$, were compared to those of the ER random networks with the same number of nodes and links (Table \[table1\]). First, the characteristic path length $L$, related to the global efficiency of reaching other nodes at the global level, shows the average number of connections that need to be crossed to go from one network node to another. Second, the clustering coefficient $C$, related to the local efficiency of reaching nearby nodes, indicates how well neighbours of a node are connected, i.e. what proportion of potential links between neighbours actually exists. Third, the small-world index $\sigma_{\text{sw}}$ indicates to what extent the fraction of two small-world measures, $C/L$, of a network deviates from that of random networks. Finally, we observed the total wiring length that is the sum of the approximated metric lengths of all individual connections. Note that the Euclidean distance in three dimensions gives an estimate or lower bound of the length of a connection, as the curvature in actual wiring between nodes makes the real distance longer. More information on network measures can be found in [@kaiser_tutorial; @rubinov].
The human macro-connectome consists of $R=66$ brain areas (modules), $N=998$ ROIs (nodes), and $E=17,865$ connections (links) between ROIs in total. The average degree, $<k>$, is $35.80$. The characteristic path length, $L$, is $3.07$ and the clustering coefficient, $C$, is $0.47$. For comparison, the ER networks with the same number of nodes and links yield $L=2.22$ and $C=0.036$ (average over $100$ generated networks). The high small-world index $\sigma_{\text{sw}}$ value of $9.27$, as well as the high $C$ value compared to $C_{\text{rand}}$, suggests that the human brain connectome is a small-world network. On the other hand, it is interesting to note that $L$ is slightly larger than $L_{\text{rand}}$ which suggests the opposite. It is due to the fact that $L$ can be reduced drastically by only a few extremely long-range connections. While the ER networks can have such long-range connections, the connection range of human connectome is relatively limited. The total wiring length $W$ is $493.5$ m. The modularity $Q$ is $0.26$.
For the [*C. elegans*]{} micro-connectome of $N=279$ neurons and $E=2,287$ links, $L=2.43$ and $C=0.34$ whereas $L_{\text{rand}}=2.30$ and $C_{\text{rand}}=0.059$, which gives the small-world index $\sigma_{\text{sw}}=5.37$. Similar observations can be made as the case of human connectome: $C$ and $\sigma_{\text{sw}}$ indicate that the [*C. elegans*]{} connectome is a strongly small-world network, but its $L$ is sightly larger than $L_{\text{rand}}$ due to the lack of extremely long-range connections. The total wiring length $W$ is $588.2$ mm. The modularity $Q$ is $0.15$.
From these basic measures, the connection specificity of the networks can be roughly depicted. Both networks are small-world with few long-range connections and have modular organisation. Since the modularity values, $0.26$ for human and $0.15$ for [*C. elegans*]{}, are small compared to those of other networks known to have modular structure, the significance of the modular organisation could be questioned. However, these networks, though small, do have modularity indicated by the values when compared to the completely randomized, zero-modularity networks obtained by the link swapping as seen below.
$Q$ $D$ $L$ $L_{\text{rand}}$ $C$ $C_{\text{rand}}$ $\sigma_{\text{sw}}$
------------------ -------- -------- -------- ------------------- -------- ------------------- ----------------------
Human $0.26$ $0.12$ $3.07$ $2.231\pm0.001$ $0.47$ $0.036\pm0.002 $ $9.27$
[*C. elegans*]{} $0.15$ $0.46$ $2.43$ $2.300\pm0.002$ $0.34$ $0.059\pm0.001$ $5.37$
\[table1\]
To understand the connectivity in detail, next we compared the network measures of the connectome to their respective benchmark networks which were generated through the link-swapping process controlled by simulated annealing as described in Methods. Each node of the benchmark networks has one-to-one correspondence to a node of the original network, and has the same degree and membership to a module as the original node. By changing the control parameter $T$ of the simulated annealing process, the resulting benchmark networks with varying modularities were obtained. The relation between $T$ and resulting $Q$ is given in the Supplementary Figure S2 and Supplementary Table S1. Figure \[Figure1\] visualises the original neural networks and corresponding benchmark networks with different modularities.
{width="\textwidth"}
The network measures of the original and benchmark networks are shown in Figure \[Figure2\]. The quantities, $L$, $C$, $\sigma_{\text{sw}}$, and $W$, show strong positive or negative correlations to $Q$ for the benchmark networks, whereas the values from the original network deviate from the trends of the curves in all cases. In general, as the modularity grows, the number of local loops increases and the number of long-range connections decreases. Therefore, the increase in $L$ and $C$, as well as the decrease in $W$, with respect to growing $Q$ is easily understood. For all the network measures, the original neural networks show marked differences to alternative arrangements with the same modularity. In addition, some values for the original networks can only be reached for much higher modularity in alternative networks or cannot be reached at all ($L$, $C$, and $W$ for the human connectome). Note that the clustering coefficients of the original networks are higher than those of alternative networks of the same modularity, which suggests better local interaction efficiency. The high characteristic path length, on the other hand, suggests reduced global communication efficiency.
![ [**Small-worldness are different in the connectomes.**]{} The small-world measures, characteristic path length $L$, clustering coefficient $C$, small-world index $\sigma_{\text{sw}}$, and total wiring length $W$, of human () and [*C. elegans*]{} () connectomes with respect to modularity, $Q$, which is varied by link swapping. Note that $W$ is normalised with respect to the values of the original neural networks. Unobservable error bars lie within the symbols. The vertical dashed lines denote the values of the original networks. The original networks show more global segregation (higher $L$ suggests lower global efficiency) and more local integration (higher $C$ suggests higher local efficiency) at the same time.[]{data-label="Figure2"}](Figure2.pdf){width="50.00000%"}
What made the original neural networks deviate from the tendency of alternative benchmark networks, or what is specific to the connectivity of the original networks? The answer is that one module of the neural networks is connected only to a small number of other modules, and corollarily, a pairs of modules are connected to each other by a redundant number of links. A pair of modules are considered to be connected to each other if any member nodes of them are connected. To test such connectivity between modules, the network of modules for both connectomes and examples of their benchmark networks were visualised in Figure \[Figure3\]. A visual inspection immediately shows that Figure \[Figure3\]a for the human connectome is sparse and Figure \[Figure3\]b for a benchmark network of it is dense. This effect is also visible, though less apparent, from Figure \[Figure3\]d for the [*C. elegans*]{} connectome and Figure \[Figure3\]e for the benchmark network.
As discussed above, the number of links before and after the link swapping does not change. Therefore, the observed difference in link density must have come in during the process of coarse-graining the network of nodes into the network of modules. Note that the multiple number of links between a pair of modules converge into a single link on the network of modules. Accordingly, the number of links on the network of modules is determined by the number of other modules the modules are connected to. Sparse connectivity of the network of modules implies that each module is connected to only a small number of other modules on the network of nodes and that a pair of modules are connected to each other by a redundant number of links. This is observed as bundling of fibres towards relatively few target nodes in the brain connectome, and it is also found in [*C. elegans*]{} connectome, where neurons are able to follow early established pathways, e.g. in the ventral cord [@varier]. On the other hand, the benchmark networks lose such connection specificity during the link swapping. A part of the multiple links from a module to another in the original networks are redirected to multiple number of new modules during the link swapping process, making the number of modules to which they connect larger but the number of links between a given pair of modules smaller.
{width="\textwidth"}
As a way to measure this, we introduced a novel network property called the dispersion $D$. It measures the average proportion of modules to which a network node is connected. Note, that this is different from an existing measure, the participation coefficient, which is the proportion of a node’s connections that connects to other modules, as the dispersion also indicates to *how many* other modules a node is connected to. For the human connectome, the dispersion is $0.12$, indicating that each ROI is, on average, connected to $12\%$ of all anatomical brain areas (Figure \[Figure3\]c). For [*C. elegans*]{} connectome, with a dispersion of $0.46$, each neuron is, on average, connected to $46\%$ of all ganglia (Figure \[Figure3\]f). These values for the connectomes are much lower than those of the benchmark networks with similar modularity. Human benchmark networks with $Q=0.25$ have $D=0.31$ (larger than the value of human connectome by factor of $2.6$) and [*C. elegans*]{} benchmark networks with $Q=0.15$ have $D=0.60$ (factor of $1.3$). In addition, such low dispersion values can only be reached for much higher modularities in alternative networks of [*C. elegans*]{}, or cannot be achieved at all for alternatives of human connectome. Less distributed fibres also reduce the total wiring length, meaning that less energy is needed in connection establishment (myelination) and maintenance (recovery to the resting potential after transmitting an action potential) [@laughlin; @chklovskii; @karbowski].
These considerations on the costs of material and energy can be seen as related to the physical structure or ‘hardware’ of neural networks. However, costs of the neural ‘hardware’ are not the only potential evolutionary constraint [@Kaiser2011Network]. Complementary to the concept of ‘hardware’, the rules for changing the pattern of connections and connection weights can be considered as the ‘software’ of the brain. Connection weights can adapt through learning and connection can be rewired after a lesion or traumatic brain injury [@Johansen-Berg2007CurrBiol]. However, looking at changes during brain development, the early perinatal large-scale architecture seems to be remarkably stable. Eliminating activity propagation by blocking neurotransmitter release has little effect on the layer and cortico-cortical architecture [@verhage]. Such invariance in the organisation of neural systems could be considered as determined by genetics factors.
Hence, following question can be raised: how much genetic information is needed to encode the connectivity patterns in human and [*C. elegans*]{}? One estimate, based on earlier studies in metabolic networks [@nykter], is the algorithmic entropy or Kolmogorov complexity [@li]. The algorithmic entropy is the length of a “sentence” describing an object in a “language”. The upper bound of the amount of information embedded in any type of data, here the connectivity matrix, can be approximated by the size of compressed data compared to the size of original data. It can be simply calculated by saving the data in a standard format and then applying a data compression. The compression ratio is the size of the compressed data divided by the size of the original data in bytes. The compression ratio approaches $1$ when almost the same amount of information is needed to describe a network structure, whereas the ratio is close to $0$ when little information is needed to encode the connectivity. In biological terms, we can think of the compressed data as the genetic information, the decompression algorithm as the pattern formation mechanism that is guided through genetic factors, and the uncompressed connectivity matrix as the organisation of neural system that follows neural development.
As shown in Figure \[Figure4\], the amount of information in the benchmark networks decreases as modularity grows larger. The networks with locally constrained connections are easier to describe than those with many long-range connections, thus have less information. The original networks, however, largely deviate from the curve. The values are comparable to, or even smaller than, the case of maximum modularity. This is also a consequence of the abundant connections between modules. Even when there are a considerable number of connections that are not locally confined, they can be easily described if the connections direct towards similar destinations. The connection specificity of the human connectome, which is locally dense and has only a limited number of global connections between brain areas, requires less information in describing the topology. Similar observations and arguments are applied to the [*C. elegans*]{} connectome as well.
![ [**Compression ratio as a function of modularity.**]{} The compression ratio is defined as the size of the compressed network divided by the size of the original network in bytes when the networks are represented by the adjacency matrices. It is shown for the original (vertical dashed lines) and rewired networks of human (, left axis) and [*C. elegans*]{} (, right axis). []{data-label="Figure4"}](Figure4.pdf){width="50.00000%"}
Discussion
==========
Neural systems show a modular architecture at different hierarchical levels, ranging from the network of individual neurons to network of brain regions. Observing human and [*C. elegans*]{} neural networks, we showed that the original networks are markedly different from the alternative benchmark networks. From both of the connectomes, we found the evidences indicate that local information distribution is more efficient but global integration is less so by studying the clustering coefficient and the characteristic path length, respectively. We also found that metabolic costs for establishing neural connections are low, which is suggested by relatively small total wiring length. To explain these results with the connection specificity of the neural networks, we introduced the novel measure dispersion, the ratio of modules to which an individual node is connected on average. By quantifying the distribution of connections across the modules, we found that smaller dispersion is specific to the original neural networks. Third, both neural networks showed a low algorithmic entropy, which indicates less requirement for the rules to organise the architecture of neural networks.
Increased separation reduces spreading and interference {#increased-separation-reduces-spreading-and-interference .unnumbered}
-------------------------------------------------------
Characteristic path lengths of neural networks were high compared to benchmark networks with the same modularity. Relatively high path length makes rapid spreading of activity less likely, as for epileptic seizures [@kaiser_lsa]. Sparse connectivity between modules can become a bottleneck for information flow. On the other hand, higher connectivity between modules or merging of modules can enhance the likelihood of activity propagation—we previously described this bottleneck behaviour as [*topological inhibition*]{} [@kaiser_crit]. Recent studies of functional connectivity in epilepsy indeed found a reduced path length, measured through increased global efficiency, and more connections between modules [@chavez]. Therefore, a relatively large characteristic path length might be one of the features that support healthy cognitive functions [@kaiser_nonoptimal; @chen; @vertes]. At the same time, increased neighbourhood connectivity, as measured by high clustering coefficient, renders a strong local interaction possible within a functionally related brain area or ganglion. In the similar line of argument, a study of oscillatory dynamics on neural networks has shown that the modular structure enables strong synchronization within modules and weak between them [@zhao].
Reduced dispersion decreases total wiring length {#reduced-dispersion-decreases-total-wiring-length .unnumbered}
------------------------------------------------
Low total wiring length reduces metabolic costs for connection establishment and at the same time obstacles activity propagation in neural systems [@kaiser_nonoptimal; @chen; @cherniak]. For both the human and [*C. elegans*]{} connectome, we saw reduced dispersion of connections which is linked to the decreased total wiring length. As primate and nematode systems are close to the optimal arrangement for reducing wiring length [@kaiser_nonoptimal; @Markov2012CerCor], any re-arrangement of connections to spread more widely throughout the network will lead to the formation of longer connections in the system. A mechanism that can limit dispersion in fibre tract systems is fasciculation of axons. The fasciculation is a mechanism that a small number of pioneer neurons form pathways that guide the axons of the following neurons, resulting in a bundle of axon fibres. This might also be the case for [*C. elegans*]{} where some neurons in the ventral cord are formed early on [@varier] providing a pathway between anterior and posterior parts of the worm. The reliance on pioneer fibres might prevent more diverse connectivity to other areas located afar.
Given the relation between dispersion and other network properties that change in schizophrenia [@zalesky], autism [@courchesne], or epilepsy [@walker], a reduced coherence of fibre tracts might be an important component in the path towards developmental diseases. Moreover, the dispersion might be related to changes in diffusion imaging, since a more distributed pattern of connectivity would break apart the fascicular pattern of fibre tracts. Therefore, we would expect that higher values of dispersion are associated with lower values of fractional anisotropy (FA) and to a shift towards more regular networks with higher characteristic path length as well as clustering coefficient. For neural disorders, for example, a shift towards regular networks has been reported for epilepsy [@ponten] and lowered FA was reported for schizophrenia [@skudlarski; @heuvel]. Note, however, that lower FA might not only result from more diffuse fibre tracts within a voxel but also from reduced myelination.
Development of modular neural networks {#development-of-modular-neural-networks .unnumbered}
--------------------------------------
Both of the connectomes showed higher Kolmogorov complexity as measured through the compression ratio. This algorithmic entropy is different from the information theory inspired entropy, which has been applied to brain networks [@sporns]. Kolmogorov complexity shows how much code is needed to generate an object. The generation of neural networks is the process of neural development. It can be driven by several factors including genes, epigenetic factors, and self-organisation. Although we only begin to understand the relation between genes and connectomes [@kaufman; @fornito], it has been pointed out that gene expression patterns which mediate growth factors and guidance cues play an important role in determining the connectivity of neural systems [@ooyen]. However, gene expression and the inclusion of genes into the genome are costly endeavours that would be expected to be under evolutionary pressure. Indeed, neural systems try to reduce the amount of genetic encoding that is needed for neural networks. At early stages of development in [*C. elegans*]{}, most long-distance connections can be established when the neurons are nearby [@varier]. This can reduce the need to control axon growth over long distances. The lower dispersion, which we found in both connectomes, might be another mechanism to reduce the amount of code requirement. Altogether, this suggests that the neural system might be efficient not only for the metabolic ‘running costs’ [@laughlin] but also in terms of their developmental mechanisms.
Which developmental mechanism could influence the modular organisation of neural systems? Several potential biological mechanisms for generating hierarchical modular networks have been described. One way is to start with an existing network and generate copies of the network where the copies retain the same internal connectivity as the original network but also establish connections directly to the original network. Variations of this method can be used to generate hierarchical scale-free networks [@ravasz] and were also thought to lead to cortical connectivity-like networks [@krubitzer]. The timing of synaptogenesis and cell birth can also be crucial for development [@varier; @rakic]. For modular networks, time windows during development can lead to multiple modules where the module number, module size, and inter-module-connectivity is determined by the number, width and overlap of developmental time windows for synaptogenesis, respectively [@nisbach; @kaiser_growth].
Link swapping perturbs lattice structure {#link-swapping-perturbs-lattice-structure .unnumbered}
----------------------------------------
Neural systems can be seen as lattice networks, using two-dimensional sheets of tissue preferring to connect to nearby nodes [@kaiser_rule] with additional long-distance shortcuts to promote rapid processing and integration of information [@kaiser_nonoptimal]. The connections are established with geometrical constraints [@henderson]. Previous studies have shown that lattice networks show a low compression ratio compared to other topologies [@sun]. During the link swapping, however, such geometrical constraints becomes relaxed. A rewired link can establish a new connection with any node in the module (intra-module link) or any node in the entire network (inter-module link). As a result, the lattice structure of the original network is perturbed which leads to higher dispersion. High dispersion prohibits efficient compression and the Kolmogorov complexity of the perturbed networks becomes high. It has been claimed that other measures of the neural networks, such as characteristic path length, clustering coefficient, and modularity, can also be interpreted as those of regular networks [@henderson]. The current study rediscovers such findings by showing that perturbation in lattice structure makes those measures deviate from the original values.
Conclusions {#conclusions .unnumbered}
-----------
In summary, both [*C. elegans*]{} and the human connectome show reduced global efficiency (higher characteristic path length), increased local efficiency (higher clustering coefficient), and reduced metabolic cost (lower total wiring length) compared to random modular networks. A marked difference in the organisation of the connectomes that is relevant to those properties is their low dispersion. The specific modular organisation of the connectomes requires fewer rules to construct it (lower algorithmic entropy), or fewer genetic factors to develop such neural system. Together, these results show that neural systems across different levels, from the network of neurons to the network of brain regions, commonly have efficiencies in multiple aspects listed above. The hierarchical natures of the modular organisation of these connectomes and how they can be understood with respect to the multiple constraints given by various network measures [@meunier; @krumnack] remain a topic for future studies.
[99]{}
\[1\] Kaiser M. 2011 A tutorial in connectome analysis: Topological and spatial features of brain networks. [*Neuroimage*]{}, [**57**]{}, 892–907.
\[2\] Mackie GO. 1988 Nerve nets. [*Comparative neuroscience and neurobiology*]{}, 84–86. Springer, New York.
\[3\] Arendt D, Denes AS, Jekely G, Tessmar-Raible K. 2008 The evolution of nervous system centralization. [*Phil Trans R Soc Lond B Biol Sci*]{}, [**363**]{}, 1523–1528.
\[4\] White JG, Southgate E, Thomson JN, Brenner S. 1986 The structure of the nervous system of the nematode *[C]{}. elegans*. [*Phil Trans R Soc Lond B Biol Sci*]{}, [**314**]{}, 1–340.
\[5\] Kaiser M, Hilgetag CC, K[ö]{}tter R. 2010 Hierarchy and dynamics of neural networks. [*Front Neuroinformatics*]{}, [**4**]{}, 112.
\[6\] Meunier D, Lambiotte R, Bullmore ET. 2010 Modular and hierarchically modular organization of brain networks. [*Front Neurosci*]{}, [**4**]{}, 200.
\[7\] Krumnack A, Reid AT, Wanke E, Bezgin G, K[ö]{}tter R. 2010 Criteria for optimizing cortical hierarchies with continuous ranges. [*Front Neuroinfom*]{}, [**4**]{}, 7.
\[8\] Newman MEJ. 2004 Fast algorithm for detecting community structure in networks. [*Phys Rev E Stat Nonlin Soft Matter Phys*]{}, [**69**]{}, 066133.
\[9\] Lim S, Han C, Uhlhaas PJ, Kaiser M. 2013 Preferential detachment during human brain development: Age- and sex-specific structural connectivity in [D]{}iffusion [T]{}ensor [I]{}maging ([DTI]{}). [*Cereb Cortex*]{}, Advanced online.
\[10\] Seung HS. 2009 Reading the book of memory: sparse sampling versus dense mapping of connectomes. [*Neuron*]{}, [**62**]{}, 17–29.
\[11\] DeFelipe J. 2010 From the connectome to the synaptome: An epic love story. [*Science*]{}, [**330**]{}, 1198–1201.
\[12\] Sritharan S, Han CE, Rotarska-Jagiela A, Singer W, Deichmann R, Maurer K, Kaiser M, Uhlhaas PJ. Increased long-range connectivity in schizophrenia: A Graph theoretical analysis of Diffusion Tensor Image data. (submitted)
\[13\] Chavez M, Valencia M, Navarro V, Latora V, Martinerie J. 2010 Functional modularity of background activities in normal and epileptic brain networks. [*Phys Rev Lett*]{}, [**104**]{}, 118701.
\[14\] Roberts A, Conte D, Hull M, Merrison-Hort R, al Azad AK, Buhl E, Borisyuk R, Soffe SR. 2014 Can Simple Rules Control Development of a Pioneer Vertebrate Neuronal Network Generating Behavior? [*J Neurosci*]{}, [**34**]{}, 608–621.
\[15\] Hagmann P, Cammoun L, Gigandet X, Meuli R, Honey CJ, [*et al.*]{} 2008 Mapping the structural core of human cerebral cortex. [*PLoS Biol*]{}, [**6**]{}, e159.
\[16\] Varier S, Kaiser M. 2011 Neural development features: Spatio-temporal development of the *[C]{}. elegans* neuronal network. [*PLoS Comput Biol*]{}, [**7**]{}, e1001044.
\[17\] Achacoso TB, Yamamoto WS. 1992 [*AY’s Neuroanatomy of *[C]{}. elegans* for Computation*]{}. CRC Press, Boca Raton, FL.
\[18\] Watts DJ, Strogatz SH. 1998 Collective dynamics of ‘small-world’ networks. [*Nature*]{}, [**393**]{}, 440–442.
\[19\] Humphries M, Gurney K. 2008 Network ‘small-worldness’: a quantitative method for determining canonical network quivalence. [*PLoS ONE*]{}, [**3**]{}, e0002051.
\[20\] Guimera R, Amaral LA. 2005 Cartography of complex networks: modules and universal roles. [*J Stat Mech*]{}, 2005(P02001):nihpa35573.
\[21\] Li M, Vitanyi P. 1997 [*An Introduction to Kolmogorov Complexity and Its Applications*]{}. Springer, New York, second edition.
\[22\] Ziv J, Lempel A. 1977 Universal algorithm for sequential data compression. [*IEEE T Inform Theory*]{}, [**23**]{}, 337–343.
\[23\] Rubinov M, Sporns O. 2010 Complex network measures of brain connectivity: Uses and interpretations. [*Neuroimage*]{}, [**52**]{}, 1059–1069.
\[24\] Laughlin SB, de Ruyter Van Steveninck RR, Anderson JC. 1998 The metabolic cost of neural information. [*Nat Neurosci*]{}, [**1**]{}, 36–41.
\[25\] Chklovskii DB, Koulakov AA. 2004 Maps In The Brain: What Can We Learn from Them? [*Annu Rev Neurosci*]{}, [**27**]{}, 369–392.
\[26\] Karbowski J. 2014 Constancy and trade-offs in the neuroanatomical and metabolic design of the cerebral cortex. [*Front Neural Circuits*]{}, [**8**]{}, 9.
\[27\] Kaiser M, Varier S. 2011 Evolution and development of brain networks: From *[C]{}aenorhabditis elegans* to *[H]{}omo sapiens*. [*Network: Computation in Neural Systems*]{}, [**22**]{}, 143–147.
\[28\] Johansen-Berg H. 2007 Structural plasticity: rewiring the brain. [*Curr Biol*]{}, [**17**]{}, R141–4.
\[29\] Verhage M, Maia A, Plomp JJ, Brussaard AB, Heeroma JH, [*et al.*]{} 2000 Synaptic assembly of the brain in the absence of neurotransmitter secretion. [*Science*]{}, [**287**]{}, 864–869.
\[30\] Nykter M, Price ND, Larjo A, Aho T, Kauffman SA, [*et al.*]{} 2008 Critical networks exhibit maximal information diversity in structure-dynamics relationships. [*Phys Rev Lett*]{}, [**100**]{}, 058702.
\[31\] Kaiser M, Hilgetag CC. 2010 Optimal hierarchical modular topologies for producing limited sustained activation of neural networks. [*Front Neuroinformatics*]{}, [**4**]{}, 8.
\[32\] Kaiser M, G[ö]{}rner M, Hilgetag CC. 2007 Functional criticality in clustered networks without inhibition. [*New J Phys*]{}, [**9**]{}, 110.
\[33\] Kaiser M, Hilgetag CC. 2006 Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems. [*PLoS Comput Bio*]{}, [**2**]{}, e95.
\[34\] Chen Y, Wang S, Hilgetag CC, Zhou C. 2013 Trade-off between Multiple Constraints Enables Simultaneous Formation of Modules and Hubs in Neural Systems. [*PLoS Comput Biol*]{}, [**9**]{}, e1002937.
\[35\] V[é]{}rtes PE, Alexander-Bloch AF, Gogtay N, Giedd JN, Rapoport JL, Bullmore ET. 2012 Simple models of human brain functional networks. [*Proc Natl Acad Sci USA*]{}, [**109**]{}, 5868–5873.
\[36\] Zhao M, Zhou C, L[ü]{} J, Lai CH. 2011 Competition between intra-community and inter-community synchronization and relevance in brain cortical networks. [*Phy Rev E*]{}, [**84**]{}, 016109.
\[37\] Cherniak C. 1994 Component placement optimization in the brain. [*J Neurosci*]{}, [**14**]{}, 2418–2427.
\[38\] Markov NT, Ercsey-Ravasz MM, Ribeiro Gomes AR, Lamy C, Magrou L, Vezoli J, Misery P, Falchier A, Quilodran R, Gariel MA, Sallet J, Gamanut R, Huissoud C, Clavagnier S, Giroud P, Sappey-Marinier D, Barone P, Dehay C, Toroczkai Z, Knoblauch K, Van Essen DC, Kennedy H. 2014 A weighted and directed interareal connectivity matrix for macaque cerebral cortex. [*Cereb Cortex*]{}, [**24**]{}, 17–36.
\[39\] Zalesky A, Fornito A, Seal ML, Cocchi L, Westin CF, [*et al.*]{} 2011 Disrupted axonal fiber connectivity in schizophrenia. [*Biol Psychiatry*]{}, [**69**]{}, 80–89.
\[40\] Courchesne E, Pierce K. 2005 Why the frontal cortex in autism might be talking only to itself: local over-connectivity but long-distance disconnection. [*Curr Opin Neurobiol*]{}, [**15**]{}, 225–230.
\[41\] Walker MC, Daunizeau J, Lemieux L. 2011 Concepts of connectivity and human epileptic activity. [*Front Syst Neurosci*]{}, [**5**]{}, 12.
\[42\] Ponten S, Bartolomei F, Stam C. 2007 Small-world networks and epilepsy: Graph theoretical analysis of intracerebrally recorded mesial temporal lobe seizures. [*Clin Neurophysiol*]{}, [**118**]{}, 918–927.
\[43\] Skudlarski P, Jagannathan K, Anderson K, Stevens MC, Calhoun VD, [*et al.*]{} 2010 Brain connectivity is not only lower but different in schizophrenia: A combined anatomical and functional approach. [*Biol Psychiatry*]{}, [**68**]{}, 61–69.
\[44\] van den Heuvel MP, Mandl RCW, Stam CJ, Kahn RS, Pol HEH. 2010 Aberrant frontal and temporal complex network structure in schizophrenia: A graph theoretical analysis. [*J Neurosci*]{}, [**30**]{}, 15915–15926.
\[45\] Sporns O. 2006 Small-world connectivity, motif composition, and complexity of fractal neuronal connections. [*Biosystems*]{}, [**85**]{}, 55–64.
\[46\] Kaufman A, Dror G, Meilijson I, Ruppin E. 2006 Gene expression of *[C]{}. elegans* neurons carries information on their synaptic connectivity. [*PLoS Comput Biol*]{}, [**2**]{}, e167.
\[47\] Fornito A, Zalesky A, Bassett DS, Meunier D, Ellison-Wright I, [*et al.*]{} 2011 Genetic influences on cost-efficient organization of human cortical functional networks. [*J Neurosci*]{}, [**31**]{}, 3261–3270.
\[48\] van Ooyen A, Willshaw DJ. 2000 Development of nerve connections under the control of neurotrophic factors: parallels with consumer-resource systems in population biology. [*J Theor Biol*]{}, [**206**]{}, 195–210.
\[49\] Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barab[á]{}si AL. 2002 Hierarchical organization of modularity in metabolic networks. [*Science*]{}, [**297**]{}, 1551–1555.
\[50\] Krubitzer L, Kahn DM. 2003 Nature versus nurture revisited: An old idea with a new twist. [*Prog Neurobiol*]{}, [**70**]{}, 33–52.
\[51\] Rakic P. 2002 Neurogenesis in adult primate neocortex: An evaluation of the evidence. [*Nat Rev Neurosci*]{}, [**3**]{}, 65–71.
\[52\] Nisbach F, Kaiser M. 2007 Developmental time windows for spatial growth generate multiple-cluster small-world networks. [*Euro Phys J B*]{}, [**58**]{}, 185–191.
\[53\] Kaiser M, Hilgetag CC. 2007 Development of multi-cluster cortical networks by time windows for spatial growth. [*Neurocomput*]{}, [**70**]{}, 1829–1832.
\[54\] Kaiser M, Hilgetag CC, van Ooyen A. 2009, A simple rule for axon outgrowth and synaptic competition generates realistic connection lengths and filling fractions. [*Cereb Cortex*]{}, [**19**]{}, 3001–3010.
\[55\] Henderson JA, Robinson PA. 2011 Geometric Effects on Complex Network Structure in the Cortex. [*Phys Rev Lett*]{}, [**107**]{}, 018102.
\[56\] Sun J, Bollt EM, ben Avraham D. 2008 Graph compression—save information by exploiting redundancy. [*J Stat Mech Theor Exp*]{}, [**06**]{}, P06001.
[^1]: \* Author for correspondence ([M.Kaiser@ncl.ac.uk]{}).
|
---
abstract: 'By introducing local $Z_N$ symmetries with $N=11,13$ in two 3-3-1 models, it is possible to implement an automatic Peccei-Quinn symmetry, keeping the axion protected against gravitational effects at the same time. Both models have a $Z_2$ domain wall problem and the neutrinos are strictly Dirac particles.'
address:
- |
$^1$Instituto de Física, Universidade de São Paulo,\
05315-970 São Paulo, SP, Brazil
- |
$^2$Instituto de Física Teórica, Universidade Estadual Paulista,\
Rua Pamplona 145,\
01405-900 São Paulo, SP, Brazil
author:
- 'Alex G. Dias$^1$ and V. Pleitez$^2$'
title: ' Stabilizing the invisible axion in 3-3-1 models '
---
[2]{}
Introduction {#sec:intro}
============
Recently, observations of the core mass distribution in the cluster of Galaxies Abell 2029 using the NASA’s Chandra X-ray Observatory suggest the existence of cold dark matter (CDM) [@cdm]. On the other hand, the Wilkinson Microwave Anisotropy Probe (WMAP) measurements of the cosmic microwave background temperature anisotropy and polarization are also consistent with CDM and a positive cosmological constant [@wmap]. Although the exact nature of the CDM is not known yet, candidates for this sort of matter are elementary particles such as neutralinos or the invisible axion [@darkmatter]. However, early invisible axion models [@dine; @kim] are unstable against quantum gravitational effects [@gravity], which may generate a large axion mass and also spoil the value of the $\bar{\theta}_{\rm eff}$ parameter. One way to stabilize the axion is by considering large discrete gauge symmetries in the sense of Ref. [@kw] as was done in the multi-Higgs extension of the standard model [@axionsm], in the 3-3-1 model [@axion331] or in the supersymmetric model [@babu]. The search for dark matter is of course related to the search for new physics beyond the standard model which in turn is related to the existence of new fundamental energy scales. In the literature, the most easily recognized fundamental energy scales are those related to supersymmetry, the neutrino masses, grand unification, and superstring theory.
In this vein, it is worth recalling once more that it has been known for a long time that the measured value of the electroweak mixing angle $\sin^2\theta_W(M_Z)=0.23113\lesssim1/4$ appears to obey, at an energy scale $\mu$, an $SU(3)$ symmetry in such a way that $\sin^2\theta_W(\mu)=1/4$ [@sw72]. Hence, if the value of $\sin^2\theta_W$ is not an accident, it may be considered as an indication of a new fundamental energy scale of the order of a few TeVs. Notwithstanding, in models with $SU(3)$ electroweak symmetry there is trouble when we try to incorporate quarks. A solution to this issue is to introduce an extra $U(1)$ factor such as in 3-3-1 models [@331; @outros], to embed the model in a Pati-Salam-like model [@dimo], or even to embed it in theories of TeV gravity [@extra].
Independently of the axion or dark matter issues, 3-3-1 models are interesting possibilities, on their own, for physics at the TeV scale. At low energies they coincide with the standard model and some of them give at a least partial explanation of some fundamental questions that are accommodated but not explained by the standard model. For instance, [*i)*]{} in order to cancel the triangle anomalies the number of generations must be three or a multiple of three; [*ii)*]{} the model of Ref. [@331] predicts that $(g'/g)^2=\sin^2\theta_W/(1-4\sin^2\theta_W)$; thus there is a Landau pole at the energy scale $\mu$ at which $\sin^2\theta_W(\mu)=1/4$, and according to recent calculations $\mu\sim4$ TeV [@phf]; [*iii)*]{} the quantization of the electric charge [@pr] and the vectorial character of the electromagnetic interactions [@cp] do not depend on the nature of the neutrinos, i.e., whether they are Dirac or Majorana particles; and [*iv)*]{} the model possesses ${\cal N}=1$ supersymmetry naturally at the $\mu$ scale [@331s]. If right-handed neutrinos are considered to transform nontrivially, 3-3-1 models [@331; @outros] can be embedded in a model with 3-4-1 gauge symmetry in which leptons transform as $(\nu_l,l,\nu^c_l,\,l^c)_L\sim({\bf1},{\bf4},0)$ under each gauge factor [@su4].
Models with $SU(3)$ (or $SU(4)$) symmetry may have doubly charged vector bosons. These types of bileptons may be detected by measuring the left-right asymmetries in Møller scattering [@assi], for instance, at the E158 SLAC experiments (which use 48 GeV polarized electrons scattering off unpolarized electrons in a liquid hydrogen target [@e158]); or in future lepton-lepton accelerators. It is interesting that the weak interaction’s parity nonconservation has never been observed in lepton-lepton scattering. Those asymmetries may also be used for seeking a heavy neutral $Z^{\prime 0}$ vector boson, which is also a prediction of these models, in $e\mu$ collisions [@emu]. Singly and doubly charged vector bileptons may also be produced in $e^-\gamma$ [@egamma] or $\gamma\gamma$ [@gammagamma] or hadron [@dion] colliders. New heavy quarks are also part of the electroweak quark multiplets in the minimal model representation. They are singlets under the standard model $SU(2)_L\otimes U(1)_Y$ group symmetry. In some versions their electric charge is different from the usual one, so that it can be used to distinguish such a model from their viable competitors. In fact, the $p\overline{p}$ production and decay of these exotic quarks at the energies of the Tevatron have been studied in Ref. [@djm] where a lower bound of 250 GeV on their masses was found. This sort of models is also predictive with respect to neutrino masses [@numass]; the models can implement the large mixing angle MSW solution to the solar neutrino issue [@lma], and also the almost bimaximal mixing matrix in the lepton sector [@pmns].
Summarizing, from the present experimental data, say those from the CERN $e^{+}e^{-}$ collider LEP, 3-3-1 models are safe if the symmetry breaking from 3-3-1 to 3-2-1 occurs at the level of TeVs; however, they have rich phenomenological consequences as we mentioned above. It will be interesting to search for some of the new particles that are present in these models, as extra Higgs scalars, exotic quarks and vector bileptons, at the energies of the upgrade DESY $ep$ collider HERA and Tevatron [@dion; @mk]. The scalar sectors are equivalent to multi-Higgs-boson extensions of the standard model; for instance, under $SU(2)_L\otimes
U(1)_Y$ the model with three triplets has two doublets and several non-Hermitian singlets, while the model with a sextet has three doublets, a complex triplet, and several complex singlets. In particular the neutral singlet ($\chi^0$) is $Z$-phobic (its coupling with $Z^0$ vanishes when the scale of the $SU(3)_L$ symmetry goes to infinity) and for this reason it evades the LEP constraints. For a finite $SU(3)_L$ energy scale there are corrections that can be calculated by using the oblique S, T, and U radiative parameters which constrain the allowed masses for the leptoquarks and bileptons [@stu]. These masses are of the same order of magnitude, a few TeV, as those allowed by the running of the coupling constants. Through the condition $\sin^2\theta_W(\mu)=0.25$, the running is sensitive to a new degree of freedom. Hence, the masses of exotic scalars and bileptons run from hundreds of GeV to a few TeV [@running]. We will return to this point later.
Turning back to the axion, the interesting point is that a Peccei-Quinn (PQ) symmetry [@pq] is almost automatic in the classical Lagrangian of 3-3-1 models. It is only necessary to avoid a trilinear term in the scalar potential by introducing a $Z_2$ symmetry [@pal]. Unfortunately, even in this case the PQ symmetry is explicitly broken by gravity effects. In order to stabilize the axion, and at the same time automatically implement the PQ symmetry, we must introduce local discrete symmetries, $Z_N$. In fact, recently it was shown that in a version of the Tonasse and Pleitez 3-3-1 model [@outros] it is possible to implement both symmetries $Z_{13}$ and PQ automatically, thus the axion is naturally light and there is no domain wall problem [@axion331].
We will consider in this work two 3-3-1 models in which only the known leptons transform nontrivially under the gauge symmetry, as in Refs. [@331], but we add also right-handed neutrinos and exotic charged leptons transforming as singlets. In one model (model A) we consider a scalar sextet but it is possible to use only three scalar triplets (model B). Both models admit a large enough discrete $Z_N$ symmetry, implying a natural light invisible axion.
The axion in two 3-3-1 models {#sec:331m}
=============================
We will consider two versions of the 3-3-1 model of Ref. [@331]. In model A we use three scalar triplets and a sextet, while in model B we avoid the scalar sextet. In both models we introduce also a scalar singlet, $\phi\sim({\bf1},{\bf1},0)$, and lepton singlets.
The quark and lepton sectors have the same representation content in both models. We have quarks transforming, under $SU(3)_C\otimes SU(3)_L\otimes U(1)_N$, as follows: $Q_{mL}=(d_m,\, u_m,\, j_m)^T_L\sim ( {\bf3},
{\bf3}^{*},- 1/3),\;
m=1,2$; $Q_{3L}=(u_3,\, d_3,\,J)^T_L\sim ( {\bf3}, {\bf 3}, 2/3)$, and the corresponding right-handed components in singlets, $u_{\alpha R}\sim({\bf3},{\bf1},2/3)$, $d_{\alpha R}\sim({\bf3},{\bf1},-1/3)$, $\alpha=1,2,3$; $J_{R}\sim({\bf3},{\bf1},5/3)$; $j_{mR}\sim({\bf3},{\bf1},-4/3)$; the leptons are the known ones and transform as triplets $({\bf 3}_a,0)$, $\Psi_{aL}=(\nu_a,\,l_a,\,l^c_a)^T_L$; $a=e,\mu,\tau$, and we also add right-handed neutrinos and a charged lepton in the singlets $\nu_{aR}\sim({\bf1},{\bf1},0)$, $E_{L,R}\sim
({\bf1},{\bf1},-1)$ [@duong; @seesaw]. The scalar sector, in the minimal version, has only three triplets $\chi=(\chi^{-},\,\chi ^{--},\,
\chi^0)^T$, $\rho =(\rho^{+},\,\rho ^0,\,\rho ^{++})^T$, $\eta =(\eta^0,\,\eta
_1^{-},\, \eta _2^{+})^T$, transforming as $({\bf1}, {\bf 3},-1),( {\bf1}, {\bf
3},1)$ and $({\bf1},{\bf3},0)$, respectively, and a scalar singlet $\phi\sim({\bf1},{\bf1},0)$.
With the quark and scalar multiplets above we have the Yukawa interactions $$\begin{aligned}
-{\cal L}^q_Y&=&
\overline{Q}_{iL} ( F_{i\alpha }u_{\alpha
R}\rho ^{*}+\widetilde{F}_{i\alpha }d_{\alpha R}\eta ^{*})
+ \lambda _{im}\overline{Q}_{iL}j_{mR}\chi ^{*}\nonumber \\
&+& \overline{Q}_{3L} ( G_{1\alpha }
u_{\alpha R}\eta +
\widetilde{G}_{1\alpha }d_{\alpha R}\rho) +
\lambda _1\overline{Q}_{3L}J_{1R}^{\prime }\chi\nonumber \\
&+&H.c.,
\label{yu1} \end{aligned}$$ where repeated indices mean summation.
Model with a scalar sextet (Model A) {#subsec:modela}
------------------------------------
In this model we add a scalar sextet $S\sim({\bf1},{\bf6},0)$ with the following electric charge assignment: $$S=\left(\begin{array}{lll}
\sigma^0_1 & h^-_1 & h^+_2\\
h^-_1 & H^{--}_1 & \sigma^0_2 \\
h^+_2 & \sigma^0_2 & H^{++}_2
\end{array}\right),
\label{sextet}$$ and we will assume that only $\sigma^0_2$ gets a nonzero vacuum expectation value (VEV) in order to give the correct mass to the known charged leptons plus a mixing with the heavy leptons ($K_a$ and $K^\prime_a$ terms below). The Yukawa interactions in the lepton sector are given by $$\begin{aligned}
-{\cal L}^l_Y&=&H^\nu_{ab}\overline{\Psi}_{aL}
\nu_{bR}\,\eta +H^l_{ab}
\overline{\Psi}_{aL}S(\Psi_{bL})^c
+K_a\overline{\Psi}_{aL}E_R\rho \nonumber \\ &+&
K^\prime_a \chi^T \;\overline{E}_L\,(\Psi_{aL})^c+G_E\overline{E}_LE_R\,\phi
+H.c.
\label{yu2} %yu3 \end{aligned}$$ where $H^l_{ab}$ is a symmetric matrix in the generation space; we have omitted $SU(3)$ indices. Neutrinos are strictly Dirac particles since the total lepton number will also be an automatically conserved.
Next we impose a $Z_{13}$ discrete symmetry under which the fields transform as $Q_{iL}\to \omega^{-1}_2Q_{iL}$, $Q_{3L}\to\omega_0Q_{3L}$, $u_{\alpha R} \to \omega_3u_{\alpha R}$, $d_{\alpha R}\to \omega^{-1}_5 d_{\alpha R}$, $J_R\to \omega_4 J_R$, $j_{mR}\to\omega^{-1}_6 j_{mR}$, $\Psi_L\to \omega_6\Psi_L$, $E_L\to\omega_3E_L$, $\nu_R\to\omega^{-1}_4\nu_R$, $E_R\to\omega_1E_R$, $\eta\to \omega^{-1}_3\eta$, $\rho\to \omega_5\rho$, $\chi\to \omega^{-1}_4\chi$, $S\to \omega^{-1}_1S$, $\phi\to\omega_2\phi$, where $\omega_k=e^{2\pi
ik/13},\,k=0,\cdots,6$. Notice that if $N$ is a prime number the singlet $\phi$ can transform under this symmetry with any assignment (but the trivial one), otherwise we have to be careful with the way we choose the singlet $\phi$ to transform under the $Z_N$ symmetry. This symmetry implies that the lowest order effective operator that contributes to the axion mass is $\phi^{13}/M^9_{\rm Pl}$ which gives a mass of the order $(v_\phi)^{11}/M^9_{\rm Pl}$ and also keeps the $\bar{\theta}$ parameter small as discussed in Ref. [@axion331].
The most general scalar potential invariant under the gauge and $Z_{13}$ discrete symmetries is $$V^{(A)}_{\rm 331} = V_{\rm H} +
\left(\lambda_{\phi 1}\,\phi\,\epsilon^{ijk}\eta_i\rho_j\chi_k +
\lambda_{\phi 2} \chi^TS^\dagger\rho\phi^*+
\mbox{H. c.}\right),
\label{pea}$$ where $V_{\rm H}$ denotes the Hermitian terms of the potential. This scalar potential has the correct number of Goldstone bosons and an axion field.
After imposing the $Z_{13}$ symmetry defined above we have that both the total lepton number $L$ and the PQ symmetry are automatic. The PQ charge assignment is as follows: $$\begin{aligned}
u'_L= e^{-i\alpha X_u}u_L,\; d'_L= e^{-i\alpha X_d}d_L,\;\;
l'_L= e^{-i\alpha X_l}l_L,\nonumber \\
\nu'_L=e^{-i\alpha X_\nu}\nu_L,\;
j'_L=e^{-i\alpha X_j}j_L,\;
J'_L= e^{-i\alpha X_J}J_L,\; \nonumber \\
E'_L=e^{-i\alpha X_{E_L}}E_L,\;
E'_R=e^{-i\alpha X_{E_R}}E_R,
\label{pq1} \end{aligned}$$ and in the scalar sector we have the following PQ charges: $$\begin{aligned}
\eta^0:\; & &-2X_u = 2X_d =X_\nu-X_{\nu_R},\nonumber \\
\eta^-_1:\; & &-(X_u+X_d)=X_u+X_d =X_l-X_{\nu_R},\nonumber \\
\eta^+_2:\; & &-(X_J+X_u) = X_j+X_d = -(X_l+X_{\nu_R}),
\nonumber \\
\rho^0:\; & &2 X_u = -2 X_d =X_l-X_{E_R},\nonumber \\
\rho^+: \;& &-(X_u+X_d) = X_u+X_d
=X_\nu-X_{E_R} ,
\nonumber \\
\rho^{++}:\; & &- (X_J+X_d) = X_j+X_u = -(X_l+X_{E_R}),
\nonumber \\
\chi^{\prime-}:\; & &-(X_u+X_J) =X_d+X_j =X_\nu+X_{E_L},\nonumber \\
\chi^{--}:\; & &-(X_d+X_J) =X_u+X_j = X_{E_L}+X_l,\nonumber \\
\chi^0: \;& &-2X_J = 2X_j=X_{E_L}-X_l,\nonumber \\
\phi: \;& &-2 X_j, \nonumber \\
\sigma^0_1:\; & & X_d+3X_j=2X_\nu,\nonumber \\
h^-_1:\; & & 3X_j-X_d=X_l+X_\nu,\nonumber \\
h^+_2:\; & & 4X_j = -X_l+X_\nu,
\nonumber \\
\sigma^0_2:\; & & 4X_j -2X_d =0, \nonumber \\
H^{--}_1: \;& & 3(X_j-X_d)= 2X_l,
\nonumber \\
H^{++}_2:\; & & 5X_j-X_d =-2X_l.
\label{pq23} \end{aligned}$$ From Eqs. (\[pq23\]) we obtain the relations $X_j=-X_J=\frac{1}{2}X_d=-\frac{1}{2}X_u=-\frac{2}{3}X_l=
-\frac{2}{3}X_{\nu_R}
=\frac{2}{5}X_\nu=\frac{2}{5}X_{E_R}=2X_{E_L}=-\frac{1}{2}X_\phi$.
Notice that the mass scale related to the exotic charged lepton $E$, up to an arbitrary dimensionless constant $G_E\sim O(1)$, is related to the PQ energy scale since the requirement of the symmetries of the model imposes the Yukawa couplings in Eq. (\[yu2\]).
Notice also that, at the energy scale below the breakdown of the $SU(3)_L$ symmetry, this model has scalar multiplets transforming under $SU(3)_C\otimes
SU(2)_L\otimes U(1)_Y$ as follows: two doublets $(\rho^+,\,\rho^0)\sim({\bf1},{\bf2},+1)$, $(\eta^0,\,\eta^-)\sim({\bf1},{\bf2},-1)$ and a non-hermitian triplet $(H^{--},\,h^-_1,\sigma^0_1)\sim({\bf1},{\bf3},-2)$. With the lighter scalar multiplets, and the usual degrees of freedom, the energy scale at which $\sin^2\theta_W=0.25$ is 5.2 TeV. The doublets $(\chi^-,\,\chi^{--})\sim({\bf1},{\bf2},-2)$, $(h^+_2,h^0)\sim({\bf1},{\bf2},+1)$ and the extra vector bosons have masses proportional to $v_\chi$; the lepton singlet $E$ has a mass of the order of $v_\phi$. More details will be given elsewhere.
Model with three scalar triplets (Model B) {#subsec:modelb}
------------------------------------------
In this model we do not introduce the scalar sextet and the Yukawa interactions are $$\begin{aligned}
-{\cal L}^l_Y&=&H^\nu_{ab}\overline{(\Psi)}_{aL}
\nu_{bR}\,\eta +H^l_{ab}\epsilon_{ijk}
\overline{(\Psi)^c}_{iaL}\Psi_{jbL}\eta_k
\nonumber \\ &+&K_a\overline{\Psi}_{aL}E_R\rho +
K^\prime_a \chi^T \;\overline{E_L}\,(\Psi_{aL})^c+G_E\bar{E}_LE_R\,\phi^*
\nonumber \\ & +&H.c.,
\label{yu3} \end{aligned}$$ where $H^l_{ab}$ is now an antisymmetric matrix. In both Yukawa interactions above, a general mixing is allowed in each charge sector. As in the previous model, neutrinos are strictly Dirac particles. The charged leptons gain mass as in Ref. [@seesaw].
If we want to implement a given texture for the quark and lepton mass matrices we have to introduce more scalar triplets, and a larger $Z_N$ symmetry will be possible in the model.
Let us introduce a $Z_4$ symmetry with parameters denoted by $\tilde{\omega}_0$, $\tilde{\omega}_1$, $\tilde{\omega}^{-1}_1$, and $\tilde{\omega}_2\equiv \tilde{\omega}^{-1}_2$. $u_{\alpha R}$, $Q_{iL}$, and $\nu_{aR}$ transform with $\tilde{\omega}_1$; $d_{\alpha R}$, $Q_{3L}$, $\Psi_{aL}$, $E_R$, $\chi$, and $\phi$ transform with $\tilde{\omega}^{-1}_1$, $\eta$ transform with $\tilde{\omega}_2$, and all the other fields remain invariant, i.e., transform with $\tilde{\omega}_0$. After $Z_4$ is imposed, the total lepton number $L$ and the PQ and $Z_{11}$ symmetries are all automatically implemented in the Yukawa sector and in the scalar potential. The most general scalar potential is then $$V^{(B)}_{\rm 331} = V_{\rm H} +
\left(\lambda\,\phi\,\epsilon^{ijk}\eta_i\rho_j\chi_k +
\mbox{H. c.}\right).
\label{peb}$$
The following $Z_{11}$ symmetry is automatically implemented in both the Yukawa interactions and in the scalar potential: $Q_{iL}\to \omega_3Q_{iL}$, $Q_{3L}\to\omega_0Q_{3L}$, $u_{\alpha R} \to \omega_4u_{\alpha R}$, $d_{\alpha R}\to \omega^{-1}_1 d_{\alpha R}$, $J_R\to \omega^{-1}_5 J_R$, $j_{mR}\to\omega^{-1}_3 j_{mR}$, $\Psi_L\to \omega_2\Psi_L$, $E_L\to\omega_3E_L$, $\nu_R\to\omega^{-1}_5\nu_R$, $E_R\to\omega_1E_R$, $\eta\to \omega^{-1}_4\eta$, $\rho\to \omega_1\rho$, $\chi\to \omega_5\chi$, $\phi\to\omega^{-1}_2\phi$. It happens that, in addition to the $Z_{11}$ symmetry, the $U(1)_{\rm PQ}$ and the conservation of the total lepton number are also automatic i.e., a consequence of the gauge symmetry and renormalizability of the model, in the interactions in Eqs. (\[yu1\]), (\[yu3\]), and (\[peb\]). The PQ charge assignments for the fermions in the model are as in Eq. (\[pq1\]); and in the scalar sector we have constraints equations as in the previous subsection. In this case proceeding as in the model A, we obtain the relations $X_d=X_u=0$ and $X_l=X_{\nu_R}=X_\nu=X_{E_R}=-\frac{1}{2}X_j=
\frac{1}{2}X_J=-\frac{1}{3}X_{E_L}=\frac{1}{4}X_\phi$. Notice that for leptons the PQ transformations are not a chiral symmetry. As in Model A, we see from Eq.(\[yu3\]) that the mass scale related to the singlet charged lepton $E$ is related to $v_\phi$. Moreover, in this model we have that $\sin^2\theta_W=0.25$ at 4 TeV.
conclusions
===========
We have built two invisible axion models in which the axion is naturally light and protected against quantum gravity effects. In model A, the $Z_{13}$ symmetry has to be imposed but in model B the $Z_{11}$ symmetry is automatically implemented in the classical Lagrangian after imposing a $Z_4$ symmetry. With a $Z_{13}$ symmetry the axion is protected from gravitational effects even if $v_\phi\approx10^{12}$ GeV but, with a $Z_{11}$ symmetry, $v_\phi\lesssim10^{10}$ GeV. In both models the PQ symmetry is automatically implemented in the classical Lagrangian in the sense that it is not imposed on the Lagrangian but is just a consequence of the particle content of the model, its gauge invariance, renormalizability, and Lorentz invariance.
We would like to stress the strong constraint put on model building by the approach proposed in Refs. [@axionsm; @axion331]. Once the symmetry $Z_N$ is used, automatic or imposed, there is no choice for new interactions. In this vein, in both models neutrinos are strictly Dirac particles, and for this reason both models will be ruled out if the neutrinos turn to be Majorana particles, say by observation of the neutrinoless double beta decay.
In the PQ solution to the strong CP problem the quark contributions to $\bar\theta$ are such that $\bar\theta\to \bar\theta-2\alpha\sum_fX_f$, where $f$ denotes any quark. In both models we have $\bar\theta\to \bar\theta-2\alpha X_j$ and we have the domain wall problem related to $Z_2\subset U(1)_{\rm
PQ}$ for $X_j=2$.
Concerning the CDM, we would like to call attention to a new possible candidate: a light and stable scalar in nonsupersymmetric models [@cc; @ft]. Although the mass of this scalar is in the range $32\,{\rm GeV}\,\lesssim m<M_Z$, the model is still compatible with the LEP data since the lightest scalar is almost a singlet under $SU(2)_L\otimes U(1)_Y$, and it may be mistaken for a light, $m_\chi\lesssim 50$ GeV [@bottino], or for the usual, $m_\chi\stackrel{>}{\sim}50$ GeV, neutralino. We recall that the latter bound comes from LEP2 searches for the corresponding chargino $m_{\chi^\pm}\stackrel{>}{\sim} 100$ GeV.
This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and partially by Conselho Nacional de Ciência e Tecnologia (CNPq).
[99]{} A. D. Lewis, D. A. Buote, and J. T. Stocke, Astrophys. J. [**586**]{}, 135 (2003). C. L. Bennett [*et al.*]{}, Astrophys. J. Suppl. [**148**]{}, 1 (2003). M. Srednicki and N. J. C. Spooner, Phys. Rev. D [**66**]{}, 010001-173 (2002). M. Dine, W. Fischler and M. Srednicki, Phys. Lett. [**104B**]{}, 199 (1981). J. E. Kim, Phys. Rev. Lett. [**43**]{}, 103 (1979). R. Holman, T. W. Kephart and S-J. Rey, Phys. Rev. Lett. [**71**]{}, 320 (1993). L. M. Krauss and F. Wilczek, Phys. Rev. Lett. [**62**]{}, 1221 (1989). Alex G. Dias, V. Pleitez and M. D. Tonasse, hep-ph/0210172. Alex G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D [**67**]{}, 095008 (2003). K. S. Babu, I. Gogoladze and K. Wang, Phys. Lett. [**B560**]{}, 214 (2003). S. Weinberg, Phys. Rev. [**D5**]{}, 1962 (1972). F. Pisano and V. Pleitez, Phys. Rev. D [**46**]{}, 410 (1992); P. Frampton, Phys. Rev. Lett. [**69**]{}, 2889 (1992); R. Foot, O. F. Hernandez, F. Pisano and V. Pleitez, Phys. Rev. D [**47**]{}, 4158 (1993). V. Pleitez and M. D. Tonasse, Phys. Rev. D [**48**]{}, 2353 (1993); F. Pisano, J. C. Montero and V. Pleitez, Phys. Rev. D [**D47**]{}, 2918 (1993); R. Foot, H. N. Long, and T. A. Tran, Phys. Rev. D [**50**]{}, 34 (1994); W. A. Ponce, Y. Giraldo, and L. A. Sanchez, Phys. Rev. D [**67**]{}, 075001 (2003). S. Dimopoulos and D. E. Kaplan, Phys. Lett. [**B531**]{}, 127 (2002). S. Dimopoulos, D. E. Kaplan and N. Weiner, Phys. Lett. [**B534**]{}, 124 (2002); I. Gogoladze, Y. Mimura and S. Nandi, Phys. Lett. [**B554**]{}, 81 (2003). P. H. Frampton and T. W. Kephart, hep-ph/0306053; P. H. Frampton, hep-th/0306029. C. A. de S. Pires and O. Ravinez, Phys. Rev. D [**58**]{}, 035008 (1998). C. A. de S. Pires, Phys. Rev. D [**60**]{}, 075013 (1999). J. C. Montero, V. Pleitez and M. C. Rodriguez, Phys. Rev. D [**65**]{}, 035006 (2002). F. Pisano and V. Pleitez, Phys. Rev. D [**51**]{}, 3865 (1995). J. C. Montero, V. Pleitez, and M. C. Rodriguez, hep-ph/0204130, and references therein. See ` http://www.slac.stanford.edu/exp/e158`. J. C. Montero, V. Pleitez, and M. C. Rodriguez, Phys. Rev. D [**58**]{}, 097505 (1998). E. M. Gregores, A. Gusso, and S. F. Novaes, Phys. Rev. D [**67**]{}, 075001 (2002); D. V. Soa, T. Inami, and H. N. Long, hep-ph/0304300. G. Tavares-Velasco and J. J. Toscano, Europhys. Lett. [**53**]{}, 465 (2001); Phys. Lett. [**B472**]{}, 105 (2000). B. Dion, T. Grégoire, D. London, L. Marleau, and H. Nadeu, Phys. Rev. D [**59**]{}, 075006 (1999). P. Das, P. Jain and D. W. McKay, Phys. Rev. D [**59**]{}, 055011 (1999). T. Kitabayashi and M. Yasue, Phys. Rev. D [**63**]{}, 095006 (2001); T. Kitabayashi, Phys. Rev. D [**64**]{}, 057301 (2001). T. Kitabayashi and M. Yasue, Phys. Lett. [**B508**]{}, 85 (2001). J. C. Montero, C. A. de S. Pires, and V. Pleitez, Phys. Rev. D [**65**]{}, 095001 (2002). M. Kuze and Y. Sirois, Prog. Part. Nucl. Phys. [**50**]{}, 1 (2003). P. H. Frampton and M. Harada, Phys. Rev. D [**58**]{}, 095013 (1998); H. N. Long and T. Inami, Phys. Rev. D [**61**]{}, 075002 (2000). D. Ng, Phys. Rev. D [**49**]{}, 4805 (1994); P. Jain and S. D. Joglekar, Phys. Lett. B, [**407**]{}, 151 (1997). R. D. Peccei and H. Quinn, Phys. Rev. Lett. [**38**]{}, 1440 (1977); R. D. Peccei and H. Quinn, Phys. Rev. D [**16**]{}, 1791 (1977). P. B. Pal, Phys. Rev. D [**52**]{}, 1659 (1995). T. V. Duong and E. Ma, Phys. Lett. [**B306**]{}, 307 (1993). J. C. Montero, C. A. de S. Pires, and V. Pleitez, Phys. Rev. D [**65**]{}, 093017 (2002). Alex G. Dias, V. Pleitez and M. D. Tonasse, hep-ph/0204121. D. Fregolente and M. D. Tonasse, Phys. Lett. [**B555**]{}, 7 (2003); H. N. Long and N. Q. Lan, Europhys. Lett. [**64**]{}, 571 (2003). A. Bottino, F. Donato, N. Fornengo and S. Scopel, hep-ph/0307303.
|
---
abstract: 'The perovskite SrTiO$_3$-LaAlO$_3$ structure has advanced to a model system to investigate the rich electronic phenomena arising at polar interfaces. Using first principles calculations and transport measurements we demonstrate that an additional SrTiO$_3$ capping layer prevents atomic reconstruction at the LaAlO$_3$ surface and triggers the electronic reconstruction at a significantly lower LaAlO$_3$ film thickness than for the uncapped systems. Combined theoretical and experimental evidence (from magnetotransport and ultraviolet photoelectron spectroscopy) suggests two spatially separated sheets with electron and hole carriers, that are as close as 1 nm.'
author:
- 'R. Pentcheva'
- 'M. Huijben'
- 'K. Otte'
- 'W.E. Pickett'
- 'J.E. Kleibeuker'
- 'J. Huijben'
- 'H. Boschker'
- 'D. Kockmann'
- 'W. Siemons'
- 'G. Koster'
- 'H.J.W. Zandvliet'
- 'G. Rijnders'
- 'D.H.A. Blank'
- 'H. Hilgenkamp'
- 'A. Brinkman'
title: 'Parallel electron-hole bilayer conductivity from electronic interface reconstruction'
---
Polarity discontinuities at the interfaces between different crystalline materials are usually compensated by atomic reconstructions via defects or adsorbates as in conventional semiconductor interfaces. However, in complex oxides the mixed valence states provide an extra option for charge rearrangement by redistributing electrons at lower energy cost than redistributing ions. The remarkable electronic transport properties that occur at the interface between the band insulators SrTiO$_3$ (STO) and LaAlO$_3$ (LAO) [@ohtomo2004; @ohtomo2006; @nakagawa2006; @huijben2006; @thiel2006; @reyren2007; @brinkman2007; @caviglia2008; @basletic2008] have been attributed to this so called electronic reconstruction[@Sawatzky; @millis2004] but direct evidence has not yet been found.
The polarity of LAO arises from the LaO and AlO$_2$ layers being not charge neutral in the \[001\] direction, unlike the formally neutral TiO$_2$ and SrO layers of STO. In the ionic limit, LaO has a charge $q=$ +e and AlO$_2$ $q=-$e per unit cell. The screened dipole per unit cell is then $D=q \Delta z /\epsilon$, where the spacing $\Delta z = c/2$ ($c=3.9$ Å is the out of plane lattice parameter) and $\epsilon$ = 25 is the dielectric constant of LAO [@hayward2005]. Screening contributions come primarily from a strong lattice polarization of the LAO film (their contribution can be as high as $\sim 62\%$ [@pentcheva2009]), supplemented by electronic cloud deformation [@ishibashi2008]. For STO-LAO systems, the remaining screened dipole of 0.08 eÅ per cell is expected to give rise to an internal electric field of $2.4 \times 10^7$ V/cm, and a resulting build-up of electric potential of 0.9 V per LAO unit cell.
This potential shift explains quantitatively why, above a threshold of 3-4 unit cells, electrons are transferred from the surface, across the LAO slab, into the STO conduction band. The resulting insulator-to-metal transition has been observed experimentally for the $n$-type LaO/TiO$_2$-interface [@thiel2006]. However, the corresponding potential shifts across LAO have not been detected so far in experiments, which suggests that possibly non-electronic reconstructions occur during the growth, driven by the polar potential build-up. For a reconstructed STO-LAO interface, it should be noted that to avoid potential build-up also the LAO surface itself needs to reconstruct, either structurally, electronically, or chemically. After electronic charge transfer one would expect holes at the surface, which have also never been observed.
In this Letter we show that an additional STO capping layer circumvents structural and chemical reconstructions at the LAO surface. The O $2p$ band in the STO capping layer allows for hole doping, so that an electronic reconstruction mechanism comes into play. By means of the STO capping layer one enters a new regime in the field of electronically reconstructed oxide interfaces with two spatially separated 2D conducting sheets, one electron-like and the other hole-like, that can display new electronic behavior including the possibility of a 2D excitonic liquid phase.
The system consisting of a varying number of LAO monolayers (ML), $n=1-5$ ML, and of a STO capping layer, $m=0-2$ ML, stacked on an STO(001) substrate, was studied by DFT calculations in the generalized gradient approximation (GGA) (for details on the calculations see Refs. [@pentcheva2009; @EPAPS]). The calculated layer-resolved densities of states are presented in Fig. \[fig:bandshifts\]a for 2ML LAO with and without 1ML STO capping. The effect of the electric field within the LAO film is apparent from the shifts of bands, [[*e.g.*]{}]{} by $\sim$0.4 eV per LAO unit cell for the uncapped system [@pentcheva2009]. Note, that this potential shift is smaller than the mentioned 0.9 eV due to effects related to the well-known underestimation of band gaps by density functional theory (DFT).
![\[fig:bandshifts\] (a) Layer-resolved density of states (DOS) of STO(001)/2LAO (dotted line), and STO(001)/2LAO/1STO (black line, colored area) aligned at the bottom of the Ti $3d$ band at the interface. (b) Influence of the STO capping on the band structure of STO(001)/2LAO/$m$STO with increasing number of capping layers ($m=0-2$), showing the closing of the band gap due to overlap between surface O $2p$ states (black circles) and interface Ti $3d$ states (blue circles). (c) The electron density distribution in the TiO$_2$ layers of the $m=2$ sample shows at the interface electrons in the Ti $3d_{xy}$ orbitals (top) and holes in the O $2p_{\pi}$ bands at the surface (bottom). The electron density is integrated between -0.6 and 0.0 eV.](Fig1)
Adding a single unit-cell STO capping layer is found to have a dramatic impact on the calculated electronic structure: the band gap, being 1.2 eV for STO(001)/2LAO, is nearly closed for STO(001)/2LAO/1STO. While the ionic relaxation pattern [@pentcheva2009] changes significantly when a capping layer is added [@EPAPS], the net contribution of the buckled TiO$_2$ and SrO layers does not affect appreciably the total ionic dipole moment of the film (which scales with the number of LAO layers). Hence, the gap reduction has mainly an electronic origin.
The evolution of the band structure of STO(001)/2LAO/$m$STO with increasing number of capping layers ($m=0-2$) is depicted in Fig. \[fig:bandshifts\]b. The valence band maximum is defined by the O $2p$-states at the M($\pi$,$\pi$)-point in the surface layer, while Ti $3d$-states at $\Gamma$ at the $n$-type interface mark the bottom of the conduction band. In the capped systems a dispersive O $2p$ surface band extends 0.8 eV above the subsurface O $2p$ band and effectively reduces the band gap driving the insulator-to-metal transition at an LAO thickness of only 2ML compared to 4 ML in the uncapped case. This surface state is analogous to the one on the clean STO (001) surface [@kimura1995; @padilla1998]. Further STO layers ([[*e.g.*]{}]{} STO(001)/2LAO/2STO [@EPAPS]) increase the band overlap at the Fermi level, but have an overall weaker influence due to the lack of internal field in STO. The reduction of the band gap, and finally its closing, is thus due to three *electronic* effects: (i) the steady upward shift of the O $2p$ states as they approach the surface [@pentcheva2009] due to the internal polarity of LAO, (ii) the band discontinuity at the interface between LAO and the capping STO layer, and (iii) the dispersive O $2p$ surface band in the capped systems that extends 0.8 eV above the subsurface O $2p$ band.
![\[fig:transportdata\] (a) Sheet resistance as function of temperature for three different STO-LAO-STO samples: STO(001)/2LAO/1STO (red circles), STO(001)/2LAO/2STO (blue triangles), and STO(001)/2LAO/10STO (black squares). The sample of STO(001)/2LAO was found to be insulating. All samples are grown at $2 \times 10^{-3}$ mbar of oxygen. Inset: $dR/dT$ as function of temperature with different linear fits below and above 100 K. (b) Sheet resistance at room temperature of STO(001)/$n$LAO/10STO samples for varying $n$ LAO interlayer thickness and a fixed number of 10 unit cells of STO capping layer. Red squares indicate samples grown at relatively high oxygen pressure ($2 \times 10^{-3}$ mbar), blue circles indicate samples grown at lower oxygen pressure ($3 \times 10^{-5}$ mbar).](Fig2)
Experimentally, we confirm the crucial influence of a single monolayer of nonpolar material on the electronic interface reconstruction. STO(001)/$n$LAO/$m$STO samples were made by pulsed laser deposition of $n$ ML of LAO and $m$ ML of STO on TiO$_2$-terminated STO(001) substrates (for fabrication details, see Ref. ). While uncapped STO(001)/2LAO samples are found to be insulating (sheet resistance above 1 G$\Omega/\square$), samples with an additional single ML of STO are conducting (see Fig. \[fig:transportdata\]a). The conductivity is further enhanced in STO(001)/2LAO/2STO samples, but the influence of increasing the STO capping layer thickness weakens, as expected from the DFT results: the STO(001)/2LAO/10STO sample has almost the same conductivity as the STO(001)/2LAO/2STO sample. Samples with a single ML of LAO were found to be insulating except for the sample with a thick STO capping ($m\geq10$).
It is known that the sheet resistance in STO/LAO samples depends critically on the oxygen pressure during growth [@brinkman2007; @siemons2007; @herranz2007; @kalabukhov2007; @huijben2009] and can vary over many orders of magnitude [@brinkman2007]. Figure \[fig:transportdata\]b shows the sheet resistance for two different sets of STO/LAO/STO heterostructures with varying LAO interlayer thickness, grown at a relatively high oxygen pressure ($2 \times 10^{-3}$ mbar) and at lower oxygen pressure ($3 \times 10^{-5}$ mbar). For the coupled-interface samples, the influence of the oxygen pressure is now found to be much weaker. Apparently, the STO capping protects the underlying LAO surface against reconstruction via defects or adsorbates.
In order to obtain spectroscopic evidence for the *electronic* reconstruction, ultraviolet photoelectron spectroscopy (UPS) was performed *in situ* immediately after the growth of a STO(001)/2LAO/1STO sample. Figure \[fig:UPS\]a shows a gradual increase in intensity for the more surface sensitive measurements at lower detector angles. These states originate from the valence band of LAO as well as the valence band of the STO surface ML. Note, that the valence band states penetrate all the way to $E_F$, unlike studies on doped STO [@UPS], where only trapped states close to the conduction band are usually observed.
![\[fig:UPS\] (a) UPS spectra of a STO(001)/2LAO/1STO sample taken *in situ* after growth at 80 K under various detector angles (the inset shows how the angle towards the detector is defined). A gradual filling of the valence band towards the Fermi energy is shown for lower angles. Lower angle spectra have larger contributions from the surface layers. (b) STS conductance $(dI/dV)/(I/V)$ at 300 K for different tip-sample distances (current set-point respectively 1.5 nA, 2.0 nA, and 4.0 nA at a bias voltage of -1.5 V). The Fermi energy, $E_F$, at $V=0$ V (blue dashed line) is found to lie between the valence band at the M-point of the STO capping surface and the conduction band at the substrate-LAO interface. The right inset shows the current-voltage characteristics from which the conductance was derived and the left inset shows an STM topography image taken at 300 K with a bias voltage of -1.0 V, revealing the substrate steps.](Fig3)
To probe states around the Fermi level, scanning tunneling spectroscopy (STS) was performed in ultra high vacuum using a variable temperature cryostat. Figure \[fig:UPS\]b shows the local density of states, $(dI/dV)/(I/V)$, of a STO(001)/2LAO/1STO sample. At room temperature, the Fermi energy lies between the valence band of the STO surface and the conduction band of the substrate-LAO interface. The bandgap is (almost) closed, as predicted in Fig. \[fig:bandshifts\] by DFT (for more spectroscopic details, see [@EPAPS]). The observed density of states just below the Fermi energy, as measured by both UPS and STS, is consistent with the electronic reconstruction scenario and suggests the presence of holes in transport.
![\[fig:MRexp\] (a) Sheet resistance as function of magnetic field at different temperatures for a STO(001)/2LAO/1STO sample, exhibiting a positive magnetoresistance. (b) Hall coefficient ($R_H/H$) of the same sample as a function of magnetic field. (c) Sheet carrier densities and (d) mobilities as obtained from a two-band fit to the magnetoresistance and Hall data at each temperature.](Fig4)
To investigate the possibility of a parallel electron-hole bilayer and the sign of the charge carriers in capped systems, magnetoresistance and Hall data were analyzed. Because the intrinsic coupling between the layers would not allow to probe the transport properties of the layers individually, unless structures are realized on a sub-micron length scale, our measurements contain information on the layers in parallel. Fig. \[fig:MRexp\] displays a positive non-quadratic magnetoresistance and a Hall resistance whose slope increases for higher fields for all conducting STO/LAO/STO samples. Quantum oscillations can still be excluded because of the low mobility. A negative magnetoresistance contribution, observed for single-interface samples deposited at high oxygen pressure [@brinkman2007], only appears below 10 K. It is natural to interpret the observations in terms of multiband conductivity. Indeed, in the temperature range up to 100 K, the magnetoresistance as well as the Hall resistance can be fitted with a two band model [@EPAPS] (solid lines in Fig. \[fig:MRexp\] a and b). Two carrier concentrations and two mobilities could be obtained for the STO(001)/2LAO/1STO sample from fitting as a function of temperature (Fig. \[fig:MRexp\] c and d).
The positive carrier sign of one of the bands at low temperatures indicates hole-type conductivity, while the other band is of electron-type. We note that no fit to the data could be obtained for equal signs of the two carrier densities. Neither oxygen vacancy doping, nor doping by cation substitution, have ever been shown to give rise to hole conductivity in the STO/LAO system. The calculated electron density distribution in Fig. \[fig:bandshifts\]c displays electrons of Ti $3d_{xy}$ orbital character in the interface TiO$_2$ layer, while holes of O $2p_{\pi}$ type are present in the surface TiO$_2$ layer. Consequently, we attribute the hole band to the surface layer, while the electron band, with a lower carrier density but a much larger mobility, is naturally attributed to the Ti $3d_{xy}$ states at the interface to the substrate, consistent with the observations in Ref. [@salluzzo].
Note, that the hole density is about an order of magnitude larger than the electron density. However, the Hall effect is dominated by the electron band because of its large mobility (10$^{3}$ cm$^{2}$V$^{-1}$s$^{-1}$, similar to values obtained on single interface STO/LAO samples deposited at oxygen pressures $>10^{-3}$ mbar). The unequal number of electrons and holes illustrates that not all charge carriers are visible in transport measurements. While the effective electron and hole masses cannot be directly inferred from our data, the band structure calculations (neglecting strong correlation effects and defects) render 0.4 $m_e$ for the electrons (both in the capped and uncapped system) and a significantly higher effective mass of 1.2 $m_e$ for the holes. Above 100 K, neither magnetoresistance, nor a nonlinear Hall resistance were observed, because the mobilities become so low that no magnetoresistance effects are expected any longer ($\mu^2H^2\ll1$ in the two-band equations of [@EPAPS]). Therefore, no two-band fitting analysis can be performed in this case, and no statement can be done on the presence of electrons and holes.
In the STO(001)/2LAO/1STO sample evidence from different experimental techniques point to an electronic reconstruction mechanism. At the same time, we know that for uncapped thick LAO samples no potential build up has been observed [@segal2009], suggestive of non-electronic reconstruction scenarios. In order to verify whether it is only the capping that makes a difference, the magnetotransport was studied for a large number of samples with either thicker LAO or thicker capping STO [@EPAPS]. While for defect-free systems theory predicts an increase in the band overlap, and hence in the number of electrons and holes, no hole contribution was found experimentally beyond $n=2$ and $m=1$. Apparently, during growth structural reconstruction occurs whenever the potential build up exceeds a few eV. While conductivity arises also in this case, no evidence for pure electronic reconstruction exists any longer. The STO(001)/2LAO/1STO sample might be an example of a structure where the potential build-up during the growth of just 2 ML of LAO is just not yet large enough for a reconstruction. The subsequent capping ML protects the LAO surface from structural or chemical reconstruction, provides another potential increase (either during growth or cooldown), and can accomodate mobile holes, resulting in electronic reconstruction. This conclusion provides guidelines to enhancing electronic reconstruction effects in general.
As shown in Fig. \[fig:bandshifts\], the surface valence band has its maximum at the M=($\pi,\pi$) zone corner, whereas the substrate-LAO interface conduction band has its minimum at the zone center. This makes the band overlap distant not only in real space (across 12 Å or more, depending on capping layer thickness) but also indirect in momentum. As a practical consequence, an electron at the surface cannot move to the substrate without some mechanism to supply the momentum transfer. The obvious mechanism is via phonons, specifically M=($\pi,\pi$) phonons. These are zone boundary optical phonons, which typically lie at a few tens of meV energy. Equilibration of electrons and holes across the LAO slab will be slow at low temperature, but will occur rapidly as soon as optical phonons are excited.
In summary, the STO capping has enabled us to show that holes are present in electronically reconstructed oxide interface samples. Their mobility is low and it is expected that the holes can become localized or eliminated in uncapped STO/LAO systems more strongly ([[*e.g.*]{}]{} by absorbed molecules or by ionic surface reconstruction). This possibly explains the large sensitivity of uncapped samples to growth conditions and the possibility to manipulate the interface conductivity by an atomic force microscope tip [@cen2008; @cen2009].
A further consequence of this 2D electron-hole bilayer is that it provides the conditions necessary for formation of a 2D excitonic liquid [@keldysh1965; @cloizeaux1965] comprised of interacting indirect excitons. In this oxide nanostructure the separation of the 2D electron and hole gases can be varied by the choice of polar material as well as capping material. In analogy to other oxides, such as ZnO [@Kawasaki], it is expected that higher mobilities can be obtained by reducing the defect density. Furthermore, the carrier densities can be tuned by gating, allowing a substantial parameter range to be probed.
This work is financially supported by the Dutch Foundation for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO) through VIDI and VICI grants, NANONED, the Bavaria-California Technology Center (BaCaTeC), German Science Foundation (TRR80) and DOE’s Computational Materials Science Network, DOE Grant DE-FG02-04ER46111, and a grant for computational time at the supercomputer HLRBII at the Leibniz Rechenzentrum.
[29]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
.
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
|
---
abstract: 'The observed astrophysical phenomenon of dark matter has generated new interest in the problem of whether the principles underlying QFT are consistent with invisibility/inertness of energy-momentum carrying “stuff” as e.g. “unparticles”. We show that the 2-dim. model which has been used to illustrate the meaning of unparticles belong without exception to the class of former infraparticles. In d=1+3 infraparticle are identical to electrically charged particles which despite their nonlocality are our best particle physics “candles”. The “invisibility” in this case refers to the infinite infrared photon cloud with energies below the resolution of the measuring apparatus which can be made arbitrarily small by increasing the photon registering sensitivity but not eliminated. This is not quite the kind of invisibility which the unparticle community attributes to their invisible “stuff” and whose existence would probably contradict the asymptotic completeness property. The main aim of the present work is to show that knowledge about this part of QFT is still in its infancy and express the hope that the work on unparticles may rekindle a new interest in conceptually subtle old unsolved important problems instead of inventing new once which after some time increase the list of unsolved old ones.'
author:
- |
Bert Schroer\
CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil\
and Institut fuer Theoretische Physik der FU Berlin, Germany
date: October 2008
title: A note on Infraparticles and Unparticles
---
Previous incursions beyond the standard particle setting
========================================================
The quest for understanding the particle content of QFT beyond the standard mass gap hypothesis (one-particle states separated by a finite distance from the contiuum) has been an important topic for a long time. The study of these problems began after it became clear that interactions, which become sufficiently strong in the infrared regime, can and will change the conceptual basis of standard scattering theory. The oldest model to address this issue is the famous Bloch-Nordsiek model which, via the Yennie-Frautschi-Suura infrared treatment of scattering theory of charged particles [@YFS], led in the early 60s to the first ideas about *infraparticles* [@S1] i.e. charged particles permanently surrounded by an infinite cloud of soft photons below the visibility limit. The most recent attempt in this direction is Georgi's proposal of “unparticles” [@Geo] which are thought to lead to an “invisibility” of a certain kind of outgoing matter component, which shows its presence through the appearance of energy-momentum-carrying but otherwise undetectable “fractional particle stuff”.
Unlike the infraparticle concept this so formulated “unparticle”-problem[^1] is not imposed by any observational fact; it is at this stage a mere mind game, although dark matter is sometimes mentioned as a potential observational application. Clearly what is called “stuff” by those authors is outside the standard particle world and its conjectured appearance together with scattering of ordinary particles is part of the larger “asymptotic completeness” problem i.e. of the question whether it is possible that the particles which emerge asymptotically in a scattering process do not form a complete set of states but leave some stuff which dissipated into the vacuum in such a way that it cannot be accounted for in particle registering devices. So any set of physical assumption which leads to asymptotic completeness has a bearing on this problem; we will return to this point at the end of the paper.
It is our intention to test the consistency of this idea within the setting of local quantum fields and in particular to compare this unparticle proposal with the problem to detect infinite[^2] soft photon clouds around infraparticles*.* The study of infraparticles is a well researched subject which started in the 60s with the investigation of certain two-dimensional models [@S1] and reached a certain amount of conceptual maturity in the 80s, when it was shown that 4-dimenional (electrically) charged particles are infraparticles [@Fro][@Bu1][@Bu2].
Some of these old results will be reviewed; this is warranted because the new development has apparently been taking place without much awareness about the old achievements, This is of particular relevance since the task to explore the field-particle relation beyond the boundaries of standard particle physics (existence of discrete masses separated by gaps from the continuum) was already the aim of the infraparticle investigations. By comparing the presently still only half-baked unparticle idea with the more mature infraparticle physics one hops to learn if and in what sense the former represents a conceptually viable new trans-particle idea.
The unravelling of the relation between quantum fields and particles has been one of the most difficult tasks ever since field quantizations was discovered in the late 20s. The subtlety of this problem became first highlighted in the work of Furry and Oppenheimer [@F-O] when these authors found that in interacting QFTs every field, including the “fundamental” Lagrangian fields, never creates a pure one-particle state (and not even a one-particle state with a limited finite number of particle/antiparticle pairs), but its local creation is always in company with an infinite particle/antiparticle polarization “cloud”[^3] whose extension and shape depends on the kind of local interaction. This observation followed in the heels of Heisenberg's discovery of a more mild form of vacuum polarization associated the composites (Wick-products) of free fields while trying to define the “partial” charge localized in a compact spatial region which is associated with the spatial integral over bilinear current of a free complex scalar quantum field.
In a modern setting the F-O observation amounts to the nontriviality of the (connected) formfactors of any local operator $A$ in an interacting QFT, which by the crossing property[^4] are all related to a an analytic master function, usually identified with the vacuum polarization formfactor of $A$ $$^{out}\left\langle p_{1},...p_{n}\left\vert A\right\vert 0\right\rangle \neq0$$ These formfactors, as a result of their crossing analyticity, fulfill a kind of “Murphy's law”, stating that all channels, whose coupling is not forbidden by charge superselection rules, and their associated symmetries, are indeed coupled. This is very different from QM (even in its relativistic form, as the theory of direct particle interactions (DPI) [@interface]) where one can couple/decouple channels at will by manipulating the interaction potentials. Whereas this “no decoupling of channels” situation in QFT does not have the status of a theorem which can be found in the literature but is certainly consistent with the the nonexistence of any counterexample, there is the weaker statement, the Åks theorem [@Aks], showing that in $d\geq1+2$ a QFT must have on-shell particle creation if it has has any nontrivial elastic scattering at all. A breakdown of this kind of “benevolent Murphy’s law” in the mass gap setting is only possible in a setting which avoids the crossing property. This in turn can only happens in theories in which at least some generating massive fields are semiinfinite stringlike localized [^5]. The problem with this putative partial “on-shell blackening” is that no interacting demonstration model has yet been found.
The problems related to these first (Furry-Oppenheimer) observations were finally solved in the late 50s with a reasonably good first understanding of the field-particle relation. Contrary to the Fock space formulation of QM, interacting QFT connects fields with particles only through the asymptotic large time limits of scattering theory. The derivation of incoming/outgoing free fields (which lead to a Wigner-Fock space particle structure of standard QFT) from the short range nature of the connected part of correlation functions (a consequence of the mass-gap energy momentum spectrum and causal locality) has been one of the high points in the conceptual understanding of QFT, with profound experimental consequences.
The first necessity to go beyond this standard setting arose from the observation that electrically charged particles do not fit into this framework since the S-matrix, as represented by the on-shell restriction of time ordered correlations, has infrared divergencies which cannot be removed by renormalizing the parameters of QED. This divergencies are not a mere consequence of the violation of the gap hypothesis. As the Yukawa coupling between nucleons and massless pions which is infrared finite and fits perfectly into the standard particle/field framework shows, the infrared divergencies, which lead to a breakdown of the standard particle setting for electrically charged particles, are the result of an increase of interaction strength in the infrared of the coupling of photons to charged fields[^6], it does not happen in the mention Yukawa coupling since no matter how large the coupling paramter is, scalar or pseudoscalar couplings cannot reach the necessary infrared strength. The Bloch-Nordsiek method and its refinement in the work of Frautschi-Yennie-Suura [@YFS] shows that the infrared stable quantities, which replace the standard scattering amplitudes, are the inclusive cross sections in which the photons below a certain resolution (which varies with the sensitivity of the measuring hardware) escape undetected.
This led to a profound revision of the particle-field relation. The simplest way in which this new particle aspect revealed itself was through a change in the two-point functions: instead of the mass-shell delta function in the Kallen-Lehmann two-point function of a physical (gauge invariant) electrically charged field does has a “infraparticle” singularity which starts at $p^{2}=m^{2}$ in a inverse power-like fashion and extends into the multiparticle continuum. Unitarity (Hilbert space positivity) limits this interaction-dependent power to be milder than the mass shell singulariy. This in turn leads to a vanishing[^7] large time asymptotic LSZ limits, thus underlining the breakdown of standard scattering theory.
There has been steady progress in a nonperturbative structural understanding; the strongest result, a milestone in the conceptual conquest of infrared aspects of QFT, has been obtained by Buchholz. He showed that an appropriate formulation of the quantum Gauss law [@Bu2] is incompatible with the standard particle structure. The infraparticle structure, including the spontaneous breakdown of Lorentz invariance in electrically charged states, is a consequence of this observation.
The derivation of inclusive scattering formulas, which bypasses amplitudes and produces directly inclusive probabilities, remained however a still incompletely achieved goal. Although one believes to have all the relevant concepts in place [@Por], these attempts did not really lead to useful nonperturbative formulas for inclusive scattering probabilities which can match the formal elegance of the standard LSZ formulas.
A related problem, raised by the unparticle community is the question of the possible particle manifestation of conformal QFT. Here the LSZ limits of any field with noncanonical (anomalous) dimension again vanish, as in the case of infraparticles. But in addition one finds a stronger theorem stating that any conformal field with a canonical (free field) short distance behavior is necessarily itself a free field [@S2]. Every field has a scaling limit, but not all such limiting theories are conformal. On the other hand each conformal theory is believed to arise through a scaling limit of a standard theory. This raises questions about the possible particle physics aspects which can be extracted from a conformal theory.
Although according to the best of my knowledge there are no theorems, one believes that only highly inclusive cross sections can counteract the vanishing of the LSZ limits. Some people have pointed out that this may not be enough; in order to arrive at finite cross sections one should generalize the inclusiveness aspect as an averaging procedure over incoming configurations[^8], With other words, one expects such possibly “doubly inclusive cross sections” in a standard massive theory to remain finite in a scaling limit in which the inclusivness (resolution energy) parameters remain fixed. There seem to exist no proofs for these claims of relating inclusive cross sections with conformal QFT. Whereas the answer concerning the simple inclusive cross sections seems to be negative (infrared divergent), the feasibility of the double K-L-N appears still open.
Even though the informed reader will recognize an identity of motivation since both the infraparticle setting and the unparticle idea aim at conquering the ground beyond the standard particle theory, there is one difference with respect to the issue of “invisibility”. The infrared “stuff” which constantly oozes out from an infraparticle (the infinitely many soft photons below an energy resolution hover with nearly infinite extension around an electrically charge particle) is not invisible per se. The hard photons with energies above the resolution coming out of a scattering process involving charged infraparticles are very visible and the “invisibility” in the infrared is determined by outside resolution parameters on the registration side, although there is always an infinite cloud remaining which escapes detection. On the other hand, since the protagonists of unparticles do not seem to think in terms of resolving the hard component of their stuff, the only remaining possibility is that their unparticle stuff is intrinsically invisible, independent of the infrared strength of the interaction. But if this is the case, what role remains for the infrared properties of interactions as the cause of “invisibility”.
The next section contains some details on infraparticles which up to the present constitute the only known mechanism to transcend the limitation of standard particle physics. At the end of that section the reader is expected to understand why a critique of unparticles requires a good understanding of infraparticles.
A brief anthology of infraparticles
===================================
A conceptual understanding of infraparticles and their scattering in the realistic 3+1 dimensional case of charged particles in QED is much more difficult than the standard scattering theory of massive particles in the presence of a mass gap. Already in second order perturbation theory infrared divergencies arise in the scattering amplitudes. These divergencies cannot be absorbed into a renormalization of physical parameters. Although the suggestion that this indicates a *conceptual change in the notion of particles* is quite old, its concretization in the setting of QED turned out to be a long lasting scientific endeavor.
The infraparticle idea was first exemplified and tested in the “theoretical laboratory” of 2-dimensional models [@S1]. Those models which were known in the 60s[^9] had no genuine interactions in the sense of scattering, but they certainly contained interesting messages about a modified particle structure which could account for the infrared divergencies observed in collisions in which electrically charged particles participate [@YFS].
These models strongly suggested that it was the modification of the particle structure whose nonobservance led to the divergencies. But the change from particles to infraparticles also required to abandon the standard setting of scattering amplitudes and pass to inclusive scattering *probabilities*[^10]. It was found that the logarithmic infrared divergences in on-shell perturbative amplitudes of charged particles, calculated according to the standard rules, are compensated with divergencies in multi photon creation contributions to inclusive cross sections; the remaining finite terms summed up to an interaction dependent power law for small values in the inclusive resolution parameter.
This model observation suggested that the radical change of the particle concept amounts in momentum space to an amalgamation of the particle mass-shell with the continuum which, unlike the ubiquitous vacuum polarization contributions in any interacting QFT, cannot be gotten rid of by the large time asymptotics of scattering theory. But if the standard scattering theory breaks down, then the Hilbert space of the model does not have the form of a multiparticle Fock space. Indeed the “exponential massless Boson fields” which appear at first in [@S1] (and later in all the other 2-dim. infraparticle models), do carry a superselected charge and therefore cannot live in the bosonic Fock space defined by the creation/annihilation operators.of the well;defined current (derivative of the Boson field). Rather they generate a bigger algebra living in a bigger Hilbert space of which the chargeless algebra generated by the currents is a subalgebra and the original Fock space is a subspace of a space of charged states without a Fock space structure.
This change should reveal itself in the Källén-Lehmann representation as a modification of the mass-shell contribution and indeed this was precisely what one observed in the very first calculation [@S1] where instead of the mass-shell delta function one found a milder singularity in form of a cut starting at $p^{2}\geq m^{2},~p^{0}\geq m.$ In the setting of operators the basic field of this model (the derivative coupling of a massless scalar to a massive spinor) was described by a product of a free massive Dirac field with an exponential of a zero mass scalar Bose field i.e.$$\psi(x)=\psi_{0}(x):e^{ig\varphi(x)}: \label{field}$$ There is a subtle point concerning the precise meaning of this exponential since as mentioned before the massless scalar boson $\varphi$ itself is not to be considered as a bona fide pointlike object[^11] and the exponential is amalgamated with the free Dirac field in such a way that the Hilbert space of the model does not contain a $\psi
_{0}(x)$ Fock subspace.
There are different ways to make this point explicit. The simplest is to start from an exponential of a massive two-dimensional free field, which lives in the Fockspace of the free field[^12] and hence obeys the unrestricted Wick contraction rules, and to perform a zero mass limit within the vacuum expectation values. In the massless limit the exponential operators in the correlation function must be multiplied with a certain anomalous power in the mass which is chosen such that no correlation diverges, but not all vanish. The power needed is this is the same as given by a formal scaling argument.
Another method is to work in the Hilbert space of currents and define the desired exponentials as a limit of a bilocal line integral of the current with one end going to infinity; doing this inside the correlation functions leads again to the previous result.
The massive free field in 2 spacetime dimensions has no other physical representations than the usual charge-less vacuum representation. Its massless limit is however very special in that it leads to the only free field with continuously many charged representation formally generated by the exponential function, a fact which was noticed already by Jordan[^13]. However his attempt to sell this observation under the heading “neutrino theory of light” (believing erronously that this can be generalized to higher dimensions) was not very successful; it led to a very funny mocking song composed by his colleagues [@Pais].
A field which is a local function of free fields (and hence is local relative to free fields) has no interaction (no scattering), even though its correlation function (even its two-point function) look like anything but free. This also holds true for fields which result from a charge generating scaling limit procedure from free fields, as the above exponential.
What can and does happen through the use of such charge-carrying exponentials however, is that the Hilbert space obtained from the reconstruction using the limiting correlation functions is different from the Wigner-Fock space of the original particles. It is easy to see that the presence of the charge-carrying exponential modifies the mass-shell delta function into a fractional power (not a fractional number of particles as the unpartcle partisans claim) in terms of the Källén-Lehmann spectral variable $\kappa^{2}$, in short its defines an “infraparticle”[^14]. The field (\[field\]) originates from a Lagrangian which describes a conventional derivative coupling of a two-dimensional massless scalar with a massive spinor. All other soluble models (including those which have recently been used to explain the notion of unparticles [@Geo2]) see later) of the 60s and 70s, with the exception of the unmodified Schwinger model[^15], have these charge-carrying zero mass exponential factors. The local observables of these models always contain the current operator $\partial\varphi,$ whereas the exponentials are charge creators in the mentioned sense i.e. objects which interwine the different superselection sectors of the respective models.
The testfunction-smeared infraparticle operator applied the the vacuum yields a state which captures much more of the localized testfunction than just the mass shell restriction of its Fourier transform which the free field was able to extract. As a result the separation into a “particle” like contribution and the remaining “stuff” is not as well-defined as in standard particle theories. Note also that if one attributes to the word “stuff” the meaning of an uncountable substrate, it is not the emitted higher frequency photons (which enter the registring device), but rather the invisible uncountable long range part which deserved the predicate “stuff” and is at least partially (below the resolution on the observer side which can never be completely eliminated) “invisible”.
It is not uncommon that what is infrared-divergent in perturbations theory may sum up to be zero nonperturbatively. Indeed the LSZ limits of infraparticle fields as (\[field\]) are zero since Hilbert space positivity forces the mass-shell singularities to be milder than a delta function. This means that standard scattering theory is not applicable to infraparticles, but the objects beyond the standard setting are not necessarily “invisible”. A calculationally efficient formalism to compute inclusive cross section for infraparticles exists only in a rudimentary fashion [@Por]; the most efficient method is still the Yennie-Frautschi-Suura infrared regularization-based compensation method which in turn is a generalization of the Bloch-Nordsieck formalism.
The conjecture, based on the change of the mass-shell structure of the Kallen-Lehmann two-point function in those models, that the cause for the breakdown of the standard scattering theory was a rather radical change of notion of particles, was a bit audacious in the 60s. Merely viewing this change in the analytic setting of poles and singular cuts as a singular cut replacing the delta function in the K-L spectral function, would not reveal the full dynamical structure of infraparticles. What was needed was an understanding in terms of *spacetime localization properties*.
Partial results about the realistic case were found in [@Fro], and a more complete conceptual picture emerged in [@Bu1][@Bu2] (see also [@Por][@Haag]) where a theorem was proven according to which the infraparticle structure together with the spontaneous breakdown of the L-symmetry in charged sectors (related to the infinite cardinality of the infrared photon clouds) is a consequence of the appropriately formulated quantum version of the Gauss law. This limits the infraparticle nature to abelian gauge theories, but represents nevertheless (in my opinion) a high point for what can be achieved by rigorous structural arguments.
There remains a practical question namely how does a physical charge-carrying operator look like? From Buchholz’s theorem we can conclude that it must be extended up to infinity. The formal candidate which has the sharpest localization which is consistent with the Gauss law is of the Dirac-Jordan-Mandelstam form$$\begin{aligned}
\Psi(x,e) & =~"\psi(x)e^{\int_{0}^{\infty}ie_{el}A^{\mu}(x+\lambda
e)d\lambda}"\label{DJM}\\
\Psi(x,e) & \rightarrow D(\Lambda^{-1})\Psi(\Lambda x,\Lambda e)\nonumber\end{aligned}$$ where $e$ is a spacelike unit vector which characterizes the localization along the line $x+\mathbb{R}_{+}e$ and the electric charge is denoted by $e_{el}$. This expression fulfills all the formal requirements. It is gauge invariant and extends to infinity in accordance with Gauss law. It is a string-localized field which transforms covariantly (second line) but the L-invariance is spontaneously broken i.e. the implementing global unitaries of the algebraic automorphism do not exist [@Bu2]. It is an interesting and poorely understood question whether such formulas are a necessary structural consequence of the nature of the local observables. Combining an old idea of reconstructing charged fields from neutral currents by using a lightlike limit procedure which Langerholc and myself designed in the 60s [@La-Sc] Jacobs [@Jacobs] introduced the concept of gauge bridges and showed that at least in the abelian case and in the quasiclassical approximation of QED the above formula is canonically distinguished in the sense that it can be obtained in a natural way from local observables only. Unfortunately it is not clear whether this holds also for the quasiclassical approximation of the QCD model.
Another unexpected but related feature was that the different spacelike directions, which after smearing with directional functions $g(e)$ with small support become narrow spacelike cones, are defining superselection rules in addition to those of the electric charge (or the electric charge is the directional-independent part of a finer superselection structure). The physical mechanism behind is that these cones contain an infinite accumulation of soft photons which makes it impossible to pass from one cone direction to another by a local or at least quasilocal change [@Haag].
It takes tremendous computational stamina to proof that this formal expression (\[DJM\]) admits a renormalized version in every order of perturbation theory, but exactly this was accomplished by Steinmann[^16] [@St]. There are of course other noncovariant ways of organizing the localization in accordance with Gauss law as e.g. a Coulomb-like distribution which is rotationally invariant around x in a fixed reference frame, but the semiinfinite string localization which represents a singular limit of a spacelike cone is the best analogy to the point as the sharpest limit of a causally closed (i.e. double-cone shaped) compact region and in the sense of maintaining Poincaré covariance. Note that the line integral in the exponential corresponds to the zero mass scalar field in the 2-dim. setting in that its perturbative modifications in the exponential interaction strength also lead to momentum space logarithmic corrections (a power law modification after summation). This is in accord with the two-dim. exponential massless Boson calculation in [@S1] and [@Geo2], with the only difference that in the above case the gauge invariance prevents to attribute a separate Hilbert space meaning to the two factors in (\[DJM\]).
The spacetime analysis of infraparticle is not only more intricate than the study of the infraparticle structure of the two-point function near $p^{2}=m^{2},$ it is also much more revealing. For example it would be virtually impossible to conclude from the changed mass shell singularity structure of the two point function alone that the sharpest localization of infraparticles is semiinfinite spacelike and that Gauss’s Law is the cause of all these modifications.
Behind the esthetic flaw of having to do things “by hand” instead of getting them from the perturbative formalism as all the other expectations of pointlike fields, there exists a problem which becomes much more pressing in QCD, where *no consistent formula “by hand” for nonlocal gauge invariant operators which corresponds to (\[DJM\]) has been found*. This is of course related to the problem of gluon- and quark- confinement and possibly of dark matter (in the sense of matter which is largely inert with respect to standard matter but nevertheless appears to coexist in the same theory).
Such an “invisible” counterpart of the charged QED matter, if it exists, cannot be understood as part of the existing gauge theoretical formalism aiming at local observables which are identified as the gauge invariant part within an unphysical setting. Possibly nonlocal gauge invariants in QCD are a fortiori not part of the formalism but left to ingenious guesswork. Whereas in the above abelian case of physical charged fields this was still possible on a formal level (\[DJM\]) as well as under the more stringent conditions of renormalized perturbation , nothing is known about nonlocal operators in a physical Hilbert space in QCD-like models except those vague ideas associated for the last 4 decades with confinement of quarks and gluons[^17] which draw their main support come from placing quantum mechanical quarks into a vault created by the walls of a potential or from lattice gauge theory which is not even able to predict the simpler infraparticle properties of QCD. QFT does not dispose over such resources, contrary to the quantum mechanical vault mechanism its very restrictive causal locality principle only leave the infinite spacetime extension as the resource of “invisibility” of certain matter components. This resource was already used in a weak form by the undetected infrared photon component, but as mentioned, it is not a consequence of the presence of zero mass particles alone, one also needs an interaction which is sufficiently strong in the infrared; the $N$-$\pi$ interaction with massless $\pi$ does not have the strength to create infrared clouds.).
The problem starts when a zero mass gluon acts on itself. In a metaphoric picture interacting should inherit the partial invisibility of infrared photons, but on the other hand they are also required to behave like charged infraparticles whose “least nonlocal” localization is a semiinfinite string as (\[DJM\]) i.e. they have to be the source and that what it produces at the same time. How can these two tendencies be reconciled in a non-metaphoric way? It seems to me that the first step in this direction should be to look for a reformulation which loosens the shackles of gauge theory to local observables and get local and nonlocal observables (=“nonlocal gauge invariants”) under the same roof. But this can only be done mitibation on the gauge side i.e. by staying in a physical Hilbert space throughout the calculation.
Indeed some recent ideas about how to overcome this conceptual handicap go precisely into this deirction. In order to have also physical (alias gauge invariant) *nonlocal operators within a unified formalism*, one must leave the boundaries of gauge theory, because the latter by its very nature of being a quantized form of classical gauge theories is limited to local observables generated by pointlike fields.
There is a formulation for which free vectorpotentials are string-localized $A_{\mu}(x,e)$ where $e$ is a spacelike direction. This potential is covariant and fulfills the prerequisites of renormalization since its short distance dimension is sdd=1. It is transversal and fulfills the axial gauge condition $e^{\mu}A_{\mu}(x,e)=0$ in addition to transversality $\partial_{\mu}A^{\mu
}(x,e)$ [@MSY]$.$ One may call it the “axial gauge”, but one should be aware that strictly speaking *it is not a gauge* but a covariant string-localized field in the *physical* Hilbert space which naturally fluctuates in both variables $x$ and $e.$ With other words the direction $e$, unlike a gauge parameter, participates also in the L-transformations and is indistinguishable. from a point in 3-dim. de Sitter spacetime (space of spacelike directions) and finally should also be accountable for string localizations of charged fields (\[DJM\])
Different from the pointlike setting of free potentials in the BRST (or any other gauge fixing formalism), these covariant stringlike potentials share together with their field strength the same physical Hilbert space and, at least in the QED case, this continues in the presence of interactions. The difference in the covariant transformation law requires to keep the $e^{\prime}s$ (which participates in the Lorentz transformation) at generic values, unlike a fixed gauge parameter in the pointlike BRST approach. Since these stringlike potentials do not admit a Lagrangian description, one has to take recourse to the setting of “causal perturbation theory”. But this only exists for poinrlike fields; the occurrance of stringlike localization leads to a significant change in the perturbative iterative Epstein-Glaser formalism which makes the perturbation theory different from those of pointlike gauge fields in that counterterms may now also be string-localized. Ignoring this aspect and treating it as a gauge problem in the axial gauge one inevitably runs into the unmanageable infrared divergencies well-known to anybody who tried to lay his hand on this problem.
In the string-localized setting the origin of all these problems becomes obvious since the infrared problems are equivalent to short distance problems in a 3-dimensional de Sitter space, but unfortunately the problem does not factorize in Minkowski and de Sitter, so that it necessitates a nontrivial generalization of the Epstein-Glaser iteration step. This is presently beeing investigated [@M-S], but with only two people working on this problem, (one being well beyond retirement age and the other overburdened with teaching duties) this will take some time [@M-S]. One obvious observation should be mentioned, for correlations of alias gauge invariant fields the new setting leads to $e$-independence on the level of the same formal arguments as in the BRST gauge formulation.
The renormalization theory involving string-localized fields is much more demanding since the time-ordering does not only effect the starting points of the semiinfinite strings, but also involves the string line as a whole (which leads to the mentioned significant change in the Epstein Glaser iteration). It is very important to do the computation for generic values of the $e_{i}$ i.e. to treat them like independent points in de Sitter space and integrate, as one always does, over the inner $x_{i}$.
The question is then whether one should average over the internal $e_{i}$ (integration) i.e. as if the theory would be a QFT on de Sitter space, or whether one should smear all of them with the same testfuntion $g(e)$ supported around one point in de Sitter space which the above formula (\[DJM\]) would suggest. As long as one keeps the $g$ fixed on keeps the terrible infrared problems of the axial gauge at bay. Most of our numerical understanding about QCD comes from lattice analogs. But the use of lattice theory is not such a good idea for problems of a more structural kind. Lattic theory has not even been able to shed a light on the infraparticle problem, how can it reasonably be expected to solve such structural conundrums as invisibility in the sense of gluons, quarks or dark matter?
The advantage of the formulation in terms of string-localized potentials (instead of the standard formulation) is that the physical origin of the infrared problems of QCD is clearer. But the problem is anything but simple, and remembering how long ot took to get renormalization theory for pointlike fields into a manageable shape, it would be unrealistic to expect that its string analog can be worked out much faster than it took to elaborate renormalized perturbation theory for pointlike fields.
One would like to expect from the string reformulation of nonabelian Yang Mills theories some clarification of the following problems. The local degrees of freedom, which can be described by pointlike physical fields, do not account for all degree of freedom of the system. Wheras the fate of the remaining one’s in QED is well understood, in the QCD case this is terra incognita and expected to account for the “invisible” degrees of freedom which are carried by gluons and quark fields. Hence one would like to think that the issue of invisible, inert or dark matter is connected to a very strong indecomposable nonlocality beyond the well-known infraparticle properties [@S3][@S4] of undetected infrared photon clouds. The aim would be to show that certain infrared degrees of freedom cannot be registerd at all in counters which are at most quasilocal in their extension [@Haag].
Even at standing accused of being repetitious let me state again that in contrast to QM which can, by using appropriate potentials, keep matter “out of sight” by placing it into a confining vault potential (“confinement”), the only resource of QFT for creating its structural richness is causal locality; there are no confining vaults in the arsenal. of causal localization. There is of course no guaranty that properties as invisibility/darkness can be explained within QFT, but there can be no doubt that the only available resource is delocalization.
The analysis of irreducible representations has shown that there are two kinds of localization, pointlike or indecomposable stringlike. In its purest and strongest form the latter shows up in the stringlike fields associated with the Wigner infinite spin representations [@MSY]. For arguments in favor of their inertness relative to ordinary matter see [@S3][@S4]. The nonexistence of local operators which carry certain charge superselection rules as in (\[DJM\]) is only possible as a result of interactions.
A new string-localized formalism may also shed a new light on the Schwinger[^18]-Higgs formalism of charge screening and its nonabelian counterpart. Algebraically there is no difference between scalar QED (or its nonabelian counterpart) and the Higgs model, since the presence of a degree 4 term in a would be charged complex scalar field (needed for entering the “Mexican-hat” parameter region) is in any case *required by renormalization theor*y. In the presence of spin 1 fields, a charged Boson may *screen itself* in the presence of vectorpotentials and become a real field. The other degree of freedom in the complex field together (in agreement with a structural screening theorem by Swieca [@Bert]) with the two photon degrees of freedom combine to form a massive vectormeson.
At the end one has a fully pointlike local theory fitting into the standard framework of QFT and instead of the complex massive field obeying a charge selection rule the physical outcome consists in a real massive field without any rule which limits its copious production. Does this have an intrinsic meaning, can one experimentally tell that a model is the screened vversion of an originally charged one ? Hard to say. In any case this fully local theory is very different from the nonlocal electrically charged model, not to speak about the even more nonlocal invisible hypothetical gluon degrees of freedom in YM models.
The only conceptual recource which one has at one's disposal for the construction of interacting fields is Poincaré covariance and locality. In perturbation theory one also needs a minimality principle on the scaling degree, usually referred to as the renormalization principle. Although constructions based on operator algebraic methods had some recent success, the main source of qualitative and quantitative understanding is still the local Poincaré-invariant coupling between free fields of arbitrary spin and mass. Since all massive one-particle representations are pointlike generated, there is good reason to believe (apart from the remark at the end of the previous to last section) that, unless there are also zero mass representations entering the coupling, the resulting theory will be generated by pointlike fields. Adding global symmetries to these fields does not change the localization properties
Only the participation of higher helicity ($\geq1$) fields can do this. Power counting reqirements in limiting the dimension of the interaction to $sdd\leq4$ which excludes working with field strength and requires to use instead their covariant potentials which turn out to be semiinfinite string-localized. The problem is then to avoid that the whole theories is becomes string-localized; each QFT should at least have a subset of (generally composite) pointlike localized fields which generate the subalgebra of local observables. This would be the analog of the gauge invariant algebra in the approach built on local gauge invariance which the latter playing the role of separating a physical content from an unphysical embedding. The local gauge formulation not only misses out on “nonlocal gauge invariants” as the analogs of the string-localized generators, but it also creates the wrong impression that there is a mysterious gauge symmetry in analogy to an inner compact group symmetry, whereas it is really the existence requirement of a nontrivial local observables which restricts the interactions beween stringlike potentials and their coupling to massive matter. Nothing is known about such nonlocal degrees of freedom and their expected lsck of “visibility” beyond that related to the photon inclusivness in QED. It would be intereting if the present unparticle activities could be directed towards these gaping holes in our understanding.
An indication of non-intrinsicness comes from the computation where one starts from the sdd=2 massive vectormeson, uses BRST to lower the dimension to ssd=1 (which leads to renormalizability by power counting) and requires that the BRST cohomological formalism also works in higher orders. It turns out that this can only be achieved by introducing an additional physical degree of freedom whose simplest realization is a scalar massive particle (naturally with a vanishing one-point function which is the hallmark of the Higgs). But there is an unsatisfactory aspect in such a derivation since cohomological requirements in a an indefinite metric space are not representing physical principles.
The use of “ghostly crutches” could be avoided by using a string description of a massive vectormeson field which does the same job as BRST (namely reducing the sdd to 1 [@MSY]) without introducing indefinite metric and remaining in the physical Wigner-Fock space. In that case the only remaining principle for the presence of an additional scalar particle is that without its presence there would be a problem with locality i.e. the use of a string-localized vectormeson has made the interaction renormalizable in the sense of power counting and in order not to remain stuck with only semiinfinite strings we would need a locality restoring scalar particle. If the strings can be resoled in terms of pointlike fields without the presence of an additional scalar then we would have learned that there are fully local vectormeson theories without scalar companions. No matter what LHC will tell us, both outcomes would have fundamental consequences for the development of QFT. Arguments in favor of higher spin particles which only can interact among each other in the presence of lower spin particles would be much deeper than those based on higher symmetries as e.g. supersymmetry. Only in this way can one get away from the Mexican hat cooking recipe for the Schwinger-Higgs screening mechanism
What are unparticles and are they related to infraparticles ?
=============================================================
According to the existing literature unparticles [@Geo][@Geo2][@Grin] are hypothetical bursts of scale invariant invisible “stuff” which is formed in high energy collisions of ordinary particles and which dissipate without leaving direct traces (“invisibility”) through secondary interactions. Whereas the unobserved infrared photon “stuff” below the resolution leads to inclusive cross sections and changes the nature of the charged particles in a conceptually very radical way, the idea around unparticles is different namely after pealing off some unobserved long range “stuff” into the vacuum, the source particles remain the same ordinary particles as before the interaction. Such a process appears somewhat strange. since a short range source should not be able to give rise to “pealing off” long range stuff unless this stuff consists of massless particles but not as members of an infinite cloud. An example of such an reaction would be the before mentioned infrared-tame interaction of massive spinors with scalar massless (and quite visible) mesons.
In order to avoid getting lost in vagueness, we first look at the concrete and explicit 2-dim. illustration in [@Geo2] which consists of the modified Schwinger model i.e. $QED_{2}~$with a massive vectormeson instead of photon. As mentioned in the previous section all soluble models of the 60s [@Elcio], apart from the original Schwinger model itself, contain the subtle charge-carrying exponential Boson factor and other standard free fields; this is also the case for the modified Schwinger model. The authors chose for their illustration the chiral condensate operator [@L-S] which for the modified model has the form (here the details of how this operator comes about from a Lagrangian are irrelevant)$$\mathcal{O}(x)=:e^{i\alpha A(x)}::e^{i\beta\varphi(x)}:$$ where $A(x)$ is the gauge invariant massive field the exponential of which already appears in the gauge invariant solution of the Schwinger model. The second exponential is a charge carrying zero mass terms and $\alpha$ and $\beta$ are real parameter related to the ratio of the mass coming from the Schwinger-Higgs mechanism and the Lagrangian vectormeson mass $m_{0}.$ If it would not be for the second factor, the large distance limit would describe the Schwinger-Higgs chiral condensate with the massive one particle contribution being the next leading term in the expansion. The presence of the charge-carrying massless exponential undoes part of the screening and converts the leading term into a “infravacuum” wheras the next to leading term represents an infraparticle contribution in the previously explained sense[^19]. This charged “infravacuum”[^20] component of $\mathcal{O}\Omega~$is the only component which resembles separate scale-invariant “stuff” similar to what the authors envisage for unparticles; but try to have an interacting situation in which conformally invariant components coexist with massive ones in d=1+3 and watch yourself failing; to talk about a sector which is a little bit nonconformal is not much better than introducing the notion of a little bit pregnant in real life.
Two-dimensional models of the mentioned kind do not describe scattering.. Even though they are not free fields in the technical sense, they describe noninteracting charged “stuff”. So in order to utilize the infrared contributions to the two-point function in Feynman diagrams, the authors couple $\mathcal{O}$ to the square of another field [@L-S]. They use the fact that the infra/unparticle structure in d=1+1 allows for interaction-free illustrations (which only look like containing interactions) whereas according to our discussion in the previous section it is not possible in d=1+3 to separate kinematical from dynamical aspects.
Whereas d=1+1 infraparticles were introduced in order to understand the scattering of charge particles, the unparticles in the sense of representing scale invariant “stuff” which, unlike the soft photons clouds which never liberate themselves from the charged particles, are apparently not hooked on massive matter. Accepting for the sake of the argument the properties their protagonsists like to attribute to them it seems that they do not appear in the outgoing amplitudes and are not even accounted for in inclusive cross sections. It seems that the example of the extended Schwinger model (as all other examples with coupling to massless scalars) is not a good illustration for the creation of zero mass conformal stuff which, unlike that of infraparticles and its zero mass clouds, is supposed to separate itself from ordinary massive matter.
This model also points at two unsolved problems in QFT. The first one is: does it make sense to couple free fields with fields which are interacting from the start? Besides the question of practicality for a perturbative approach there looms an unsolved fundamental problem. One formulates interactions by coupling free fields not only for pragmatic computational reasons. One also believes that this insures the mentioned asymptotic completeness, which in the mass zero case amounts to a weaker form of completeness in terms of inclusive cross section. But it is doubtful that the coupling of anomalous dimensional conformal matter coupled to free fields stays in this setting.
The second difficulty which becomes particularly acute in d=1+3 is that coupling of massive to massless matter never leads to scale-invariant “sectors”; the only theory which does this is the tensor product of a conformal theory with a massive one. As much as it is meaningless to use expressions as “a little bit pregnant” in daily life, one cannot fight structural properties of QFT by notions of effective field theories or what is more specific to the situation at hand by Bank-Saks arguments which claim that it is possible to overrule such structural facts and make sense out of violating conformal invariance in a region of a theory. Ideas of effective actions may have their place of validity, but one should not try to use them for overturning structural properties.
As the example of the photon shows, its interpolating Heisenberg field is not scale covariant, only the registered outgoing free photons are. What remains however intact is the gapless zero mass energy-momentum spectrum and its ensuing long range character. It is also interesting to point out that even on a formal level the coupling of anomalous dimensional fields with ordinary matter does not improve the long range aspect; to the contrary, as the scale dimension increased, the infrared coupling becomes weaker. The strongest infrared couplings are those which involve string-localized potential associated with ($m=0,s\geq1$) representations for which the aforementioned vectorpotential $A_{\mu}(x,e)$ is the best studied case. The use of the string-localized description makes the long range which sets the infrared strength of the QED coupling manifest whereas (see previous section) in the gauge formulation this remains hidden and has to be brought out “by hand” through formulas as (\[DJM\]). Only they have a chance for accounting for the desired invisibility property.
Without wanting to lend support to the somewhat controversial physical interpretation of such couplings in the literature on unparticles, it may be interesting to mention that there are anomalous dimensional conformal fields which describe “stuff” in a more literal sense i.e. something which *certainly cannot be interpreted* as coming from a scaling limit from a standard theory and therefore cannot be associated with inclusive cross sections. These are the *conformal generalized free fields* as they e.g. arise from ordinary AdS free fields via the AdS-CFT correspondence. From a combinatorial point of view they behave as free fields[^21]; Duetsch and Rehren [@Du-Re] have investigated the suitability of the causality properties of such “stuff” for the formulation of a consistent perturbation theory and their results. The results are yet incomplete, but interesting and even somewhat encouraging. None of the unparticle lowest order calculations which only use unparticle two-point functions would change, if one uses these combinatorially much simpler fields.
Perhaps one should be careful with prematurely attaching physical attributes to unparticle calculations and rather study in more general terms what QFT has still in store once one goes beyond standard textbook particle physics. The most fruitful unexplored area seems to be the afore mentioned interacting massless higher helicity objects coupled to themselves and to standard massive matter. It is the generalization of the mentioned string-localized electromagnetic vector potential with scale dimension sdd=1 which has a good chance to lead to the kind of infrared singular interactions which one needs to get beyond the standard matter and create “stuff” which consists of infinitely many objects in a finite energy range. Zero mass is necessary but not sufficient; e.g. scalar zero mass couplings do not have the sufficient infrared strength. It is not the size of anomalous dimension but rather the algebraic form of the infrared coupling of higher helicity free string-localized potentials which increases with spin that increases this strength.
Among all at least partially studied models, the most promising are those which involve couplings among several string-localized potentials $A_{\mu
}^{(i)}(x,e).$ The experience with nonabelian gauge theories suggest that in order to find pointlike generated subalgebras the couplings must be related to each other in the way they are in gauge theories; if not one will get stuck with a string-localized theory which has no local subobservables at all. Assuming that the mentioned string theoretic generalization of renormalized perturbation theory works, one would have two kind of matter in such a setting: visible point-localized “glue-ball” matter and string-localized and presumably invisible gluon matter. I am convinced that without solving this problem one has no chance to understand the issue of invisibility versus asymptotic incompleteness. The understanding of the abelian counterpart QED where such string-localized fields represent the charged operators and where a rest of undetected soft photon “stuff” always remains outside observation is encouraging for a program of looking for stronger forms of invisibility. The unparticle project certainly shares this aim even if the proposals to implement it are quite different (apart from the shared low-dimensional illustrations).
The unparticle project tries to achieve invisibility of interaction generated “stuff” by using conformal matter with anomalous dimensions, in contrast the project favoured in this article is based on a generalization of gauge theory. Whereas the gauge formulation hides the nonlocality by introducing fake pointlike potentials together with BRST ghosts at the expense of the Hilbert space positivity and as a result tends to overlook (even in the abelian case (\[DJM\])) nonlocal operators in the *physical* Hilbert space, the string like description catches also those nonlocal field degrees of freedom which escape the pointlike description but nevertheless carry energy-momentum and hance react gravitationally. This still speculative project which generalizes the infraparticle idea is expected to explain the confinement of the gluon and quark degrees of freedom and to attribute physical reality to genuinly invisible dark matter. In the concluding section I will address this speculative issue in a more general context than un/infra particles.
Invisibility and lack of asymptotic completeness
================================================
Whereas in QM, which has no maximal velocity, fields are synonymous with particles and there is hardly any limit on the kind of interactions between them, QFT is more restrictive as a result that all of its properties at the end must be understood in terms of causal localization i.e. the localization in theories which have a maxial velocity [@interface]. The prize to pay for this is that its only measurable non-fleeting and genuinly intrinsic and stable objects, the particle states, are only appearing in the large time asymptotic limit of the fleeting field states respectively; an interacting theory with any interaction fulfilling the general principles will have no particles at any finite times! Since besides the stability (the existence of a lowest energy state) the realization of causal locality is the only handle at one’s disposal in order to control the asymptotic particle content, the study of admissible particle structure and their possible manifestations in the real world has remained the most subtle part of QFT.
Even the deeper understanding of the standard situation, in which one-particle states are separated from the continuum by a gap, has remained a 50 year challenge [@Bert]. It started with the (at that time surprising) observation that the number of phase space degrees of freedom in a finite phase space cell (which as everybody learns is finite in QM), is infinite in QFT; an infinity which originates from the realization of the causal localization principle. The hope was that the precise quantitative understanding of this infinity could explain why the Hilbert space of QFT apparently can be fully described (even beyond perturbation theory) as a Wigner-Fock Hilbert space; a fact which ceises to be valid in QED.
For free fields this set is compact and (as was shown later) even nuclear [@Haag] and there are good arguments that at least in physically reasonable theories (e.g. absence of Hagedorn temperature) it stays this way. Although many deep properties followed from this phase space structure, it is now agreed on that asymptotic particle completeness cannot be derived from phase space properties alone.
The infraparticle structure of electrically charged particles contained the important message that there are objects which cannot be generated by pointlike field and the formalism of gauge theories. Nonlocal objects in the physical Hilbert space as e.g. electrically charged particles are better constructed in a setting which permits stringlike localized potentials [^22].
It is interesting to note that as long as the mass gap hypothesis holds, the formulation and derivation of scattering theory between pointlike and semiinfinite stringlike fields [@Haag] is similar. The only significant difference is that the S-matrix and the formfactors of such models do not necessarily fulfill the important crossing property [@foun]; with other words there is no analytic master funcrion such that the different distributions of in and out particles are different boundary values of that master function. This has the intersting consequence that there may be a subterfuge to Aks theorem [@S3][@S4]. In that case it would be possible that certain channels cannot be produced in scattering processes despite the fact that there is no charge superselection rule which prevents them. this kind of partial inertness may have some potential interest in connection with dark matter. Massive strings can only exist in interacting matter, free massive fields are always pointlike generated. Massive strings once applied to the vacuum create states which are always generated in terms of pointlike states (which themselves cannot be obtained by applying pointlike fields from the interacting field algebra).
From the existing formulation of the unparticle project it is not clear if and how they fit into the balance of asymptotic completeness. For standard mass gap situation of believes that the coupling between free fields is not only popular because it is simple and one cannot think of anything else which is in agreement with the locality principle, but it also preempts the property of asymptotic completeness by having from the very beginning those fields and their Fock space which are as close as possible to the incoming/outgoing fields. In the case of electrically charged fields though for infraparticles one looses the Fock space structure, but since the scattering probabilities still add up (the resolution can be made arbitrarily small). In the case of QCD and for unparticles this is not so clear, but for different reasons. The difficulty in the latter case is that one starts already with anomalous dimensional fields which are not only very far removed from a Fock space structure, but for which it is not even clear whether they permit a doubly inclusive cross section setting which appears to be the least one needs to fulfill asymptotic completeness in the sense of probabilities.
Concluding remarks, resumé
==========================
The unparticle idea presents an opportunity to recall previous successful and less successful works which explored the region beyond the standard (mass gap) particle setting. The theory of infraparticles which aimed at the incorporation of the observed infrared aspects of electrically charged particles is an example of an successful attempt. Apart from the issue of “invisibility” which is the main motivation behind unparticles as a new kind of matter, the infraparticle models follow a similar construction recipe as those designed to illustrate unparticles, in fact *its two-dimensional illustrations are identical*.
On the other hand the popularity of unparticles may point our thinking again towards invisibility problems caused by noncompactly localized degrees of freedom within theories which we prematurely believed to have “solved” (e.g. by analogies with lattice theories). Some nice catch words as (gluon-, quark-) confinement provided us with a quiet conscience. But lattice theory lacks those strong principles which relate indecomposable positive energy representation of the Poincaré group for certain zero mass representation to semiinfinite string localization[^23], although it is quite efficient at emulating standard QFT containing only compactly localized representations. Interaction-free illustrations exist in the form of string-localized infinite spin positive energy representations in Wigner's list, but unfortunately they do not generate local subalgebras (they have no local conserved energy-stress tensor), at least not in this interaction-free form.
What one needs in order to have states “out of sight” coexisting with states generated by local observables is a situation in which the interaction is so strong in the infrared, that besides ordinary matter (described by a local subalgebra) there are degrees of freedom (not ghosts!) which have a localization as bad as that of the mentioned infinite spin representations.
The existence of ordinary matter in QCD-like theories has become part of the accepted folklore (dimensional transmutation) and phenomenological schemes have been designed to link such mechanisms with Lagrangian QCD. But the other side of the coin, namely the fate of the nonlocal (gluonic) degrees of freedom which escape the gauge theoretic formalism (which by its very nature is only focussed to the local ones) have been left in a conceptual limbo.
Looking at the literature, it seems that most people believe that they do not exist, i.e. that all the degrees of freedom went via dimensional transmutation into the local observables which in turn form an asymptotically complete system. But in view of the electrically charged particles whose sharpest possible localization is semiinfinite string-like, this is not very credible.
In order to see what is going on, we have started to investigate quadrilinear interactions between stringlike free fields with special attention to those for which the string-like nature is not enforced by imposing renormalizability but is naturally emerging from the Wigner representation struture [@MSY][@M][@M-S]. Since such fields have sdd=1 independent of spin, there is no problem with the power counting prerequiste for renormalizability. The really hard problem, as mentioned before, is the perturbative Epstein-Glaser iteration for semiinfinite strings instead of pointlike particles [@M-S].
There is no disagreement with the aims of unparticle physics since almost all the deep unsolved problems are in the infrared. Whatever the outcome will be, it is important to get particle physics away from those metaphoric inventions as e.g. supersymmetry and string theory back on track addressing the old unsolved problems with new ideas.
My attitude with respect to unparticles has been one of criticism (mainly connceted to the knowledge which was lost during the last 3 decades) but at the same time encouragement because unparticles could serve as a catalyzer for a return to particle theory’s most rich research area, which, unfortunately left too many unsolved problems on the wayside[^24]. One central problem which was first formulated in the early 60s [@Swieca] is: *when is a theory of quantized fields a theory of particles*, or more specifically. the problem of asymptotic completeness. The phase space structure of QFT (which is significantly different from that of QM) was identified as one local structural property which plays an important role in the understanding of the global particle structure. It was already clear at that time that there are field models which do not fulfill asymptotic completeness (generalized free fields, conformal models) and they should be excluded. But physically important structures as electrically charged particles and more general infraparticles are still inside an appropriately extended asymptotic completeness notion. Their history shows that structural problems of QFT cannot be solved by lowest order perturbation theory; the first order coupling between a conformal field theory and massive matter evaluated for the two point function, as used by unparticle followers, does not reveal anything.
It is far from being clear why, what was considered to be an important physical principle or at least a successful working hypothesis in those times, should be ignored now. After all there is presently not the slightest indication that nature does not like the old principles nor is there a theoretical guide outside of at least some form of asymptotic completeness in the sense of probabilty conservation when unparticle “stuff” fades away into the vacuum.
[99]{}
H. Georgi, Phys. Rev. **98**, (2007) 221601
B. Schroer, Fortschr. Phys. **11** (1963) 1
B. Grinstein, K. Intriligator and I. Rothstein, *Comments on Unparticles*, arXiv:0801.1140
D. Yenni, S. Frautschi and H. Suura, Ann. of Phys. **13**, (1961) 370
J. Froehlich, G. Morchio and F. Strocchi, Ann. Phys.119, (1979) 241
W. H. Furry and J. R. Oppenheimer, Phys. Rev. **45**, (1934) 245
B. Schroer, *Localization and the interface between quantum mechanics, quantum field theory and quantum gravity*, arXiv: 07114600
S. Aks, J. Math. Phys. **6**, (1965) 516
D. Buchholz and K. Fredenhagen, Commun. Math. Phys. **84**, (1982) 1
B. Schroer, String theory and the crisis in particle physics (or the ascent of metaphoric argument), arXiv:0805.1911
B. Schroer, Phys. Lett. B **494**, (2000) 124
J. Mund, B. Schroer and J. Yngvason, Commun. Math. Phys. **268**, (2006) 621
D. Buchholz, Commun. Math. Phys. **85**, (1982) 49
D. Buchholz, Phys. Lett. B **174** (1986) 331
J. Langerholc, and B. Schroer, *Can Current Operators Determine a Complete Theory?*, Commun. Math. Phys., **4**, (1967) 123
http://www-library.desy.de/preparch/desy/thesis/desy-thesis-09-009.pdf
P. Jordan, *Beitraege zur Neutrinotheorie des Lichts*, Zeitschr. fuer Physik **114**, (1937) 229 and earlier papers quoted therein
M. Porrmann, Commun. Math. Phys. **248**, (2004) 269 and many previous contributions cited therein
T. Kinoshita, J. Math. Phys. **3**, (1962) 650, T. D. Lee and M. Nauenberg, Phys. Rev. **133**, (1964) 1549
E. Abdalla, E.C. Abdalla, K. Rothe, *Non perturbative methods in two dimensional quantum field theory*, World. Scientific Publishing 1992
J. Lowenstein and J.A. Swieca, Ann. of Phys. **68**, (1971) 172
W. Kunhardt, *On infravacua and the Localization of sectors*, math-ph/980B. Schroer, *Localization and the interface between quantum mechanics, quantum field theory and quantum gravity*, arXiv: 071146006003
R. Haag, *Local Quantum Physics*, Springer Verlag 1996
A. Salam, Phys. Rev. **82**, (1951) 217
S. Weinberg, Phys. Rev. **118**, (1959) 838
O. Steinmann, Ann. Phys. (NY) **157**, (1984) 232
J. Mund and B. Schroer, work in progress
B. Schroer, *Indecomposable semi-infinite string-localized positive energy matter and “darkness”*, arXiv:0802.2098
B. Schroer, *Do confinement and darkness have the same conceptual roots?* arXiv:0802.3526
B. Schroer, *A critical look at 50 years particle theory from the perspective of the crossing property*, arXiv:0905.4006
M. Duetsch and K-H. Rehren, Ann. Henri Poincaré **4**, (2003) 613, arXiv:math-ph/0209025
B. Schroer, Particle Physics in the 60 and 70 and the legacy of contributions by J. A. Swieca, to be published in a special volume of the Brazilean Journal of Physics, arXiev:
H. Georgi and Y. Kats, *An Unparticle Example in 2D*, arXiev:0805.3953
J. Mund, *String-Localized Quantum Fields, Modular Locaization and Gauge theories*, manuscript based on a talk given at the Rio 2007 Math. Phys. Conference
B. Schroer, *particle physics in the 60s and 70s and the legacy of contributions by J. A. Swieca*, arXiv:0712.0371
[^1]: The terminology is taken from the papers and probably refers to their presumed invisibility. The name “stuff” is also used on several occasions in those papers.
[^2]: The presence of photon “clouds” i.e. an infinite number of soft photons requires their exact masslessness.
[^3]: In finite order of perturbation theory there is a finite admixture which increases with the order and becomes a “cloud” in the limit.
[^4]: In standard QFT (pointlike interpolating fields, mass shells with spectral gaps) the vacuum polarisation formfactor is related by crossing to the formfactors with different distributions of in and out particles in ket and bra states.
[^5]: Buchholz and Fredenhagen proved that semiinfinite string-localization is the most general possibility allowed by the mass gap hypothesis in conjunction with the assumption of the existence of a point-like generated neutral observable subalgebra [@Bu-Fr].
[^6]: If the authors of the statement “Observed, known particle physics is based on theories which have a mass gap and/or are free in the infrared” [@Grin] really mean what they write, they would have missed out on QED.
[^7]: It is not uncommon that an object which diverges in perturbation theory vanishes on structural grounds.
[^8]: Such doubly inclusive cross setions play a role in the Kinoshita-Lee-Nauenberg theorem [@K-L-N].
[^9]: Besides the mentioned model [@S1] list of those models consisted of the (massless) Thirring model, the Schwinger model and variations and combinations thereof.
[^10]: “Inclusive” meant that infrared photons below a certain sensitivity $\Delta$ of the registering apparatus were summed over.
[^11]: This “field operator” is really an operator-valued distribution whose testfunctions space is restricted to those Schwartz test functions whose total integral vanishes. Its Hilbert space is generated by polynomials in the well-defined associated current (its derivative, the field is a line integral in terms of this current).
[^12]: This property is lost in the zero mass limit when the free field diverges in the infrared (but its derivatives stay finite) and the Wick rules for exponentials suffer restrictions from the charge conservation.
[^13]: Unfortunately Jordan used his correct observation of what we nowadays would call Bosonization/Fermionization to base his pet idea of “the neutrio theory of light” on [@Jor]. The relation of the gauge invariant content of the Schwinger model in terms of a exponential massive free Boson which has no charge sectors to its short distance limit with infinitely many charge sectors is an impressive allegory for the transition from confinement to short distance charge liberation.
[^14]: The only restriction on its range comes from the requirement that the exponential acts in a Hilbert space which follows from the mentioned charge superselection rules.
[^15]: It was shown in [@L-S] the gauge invariant content of the Schwinger model is an exponential of a massive scalar field. As mentioned before, this exponential passes to a charge-carrying exponential zero mass operator. This illustrates the (gauge-invariant) “Schwinger-Higgs charge sceening” with the unscreened charges appearing in the “asymptotic freedom limit”. It also illustrates a peculiarity of 2-dim. free fields which already played a role in Jordan’s “neutrino theory of light” [@Jor].
[^16]: The computational effort necessary to assure the perturbative existence of these DJM formulas goes beyond what any standard renormalization formalism as [@Salam][@Weinberg] or any of the more recent refinements can achieve. Steinmann had to develop a technique especially for this problem.
[^17]: The putative link between asymptotic freedom and infrared slavery has the flaw that it does not account for all degrees of freedom which were initially there.
[^18]: Schwinger was thinking of a screened “phase” in spinor QED, where a perturbative implementation of screening is not possible. To make his point more convincing, he invented the Schwinger model.
[^19]: A footnote in [@Geo2] reveals that the authors are aware of this connection.
[^20]: The quotation marks are there in order to distinguish this situation from a more radical notion of infravacuum [@Ku] which cannot be viewed as the application of a charge carrying zero mass field to the standard vacuum. .
[^21]: They are not the standard anomalous dimensional fields which are “interacting” in some sense which can be made precise.
[^22]: From a point of view of positive energy irreducible representations of the Poincare group, the necessity to introduce stringlike generators only arises for the zero mass potentials of the pointlike helicity $\geq1$ field strength and in a much stronger form for the so-called infinite spin representations (which possess no pointlike generators). There is no reason which forces one to go generators on higher dimensional submanifolds as branes.
[^23]: The problems of lattice approximatio of QFT is similar to the approximatabilty of operator algebras by matrix algebras. Although for hyperfinite von Neumann algebras this is possible, there is no way of keeping track of the richness of infinite class of hyperfinite algebras by looking at finite matrix algebras. Most of the interesing physical mechanism are only accounted for in the infinite limit.
[^24]: Inasmuch I have lamented the loss of criticism in another context, I am of course also aware of the danger of calling an idea to account for conceptual-mathematical rigor in a too early stage; there are several important ideas in particle physics which started out on a wrong track.
|
---
abstract: 'A new type of Langevin equation exhibiting a non trivial phase transition associated with the presence of multiplicative noise is introduced. The equation is derived as a mesoscopic representation of the microscopic annealed Ising model (AIM) proposed by Thorpe and Beeman, and reproduces perfectly its basic phenomenology. The AIM exhibits a non-trivial behavior as the temperature is increased, in particular it presents a disorder-to-order phase transition at low temperatures, and a order-to-disorder transition at higher temperatures. This behavior resembles that of some Langevin equations with multiplicative noise, which exhibit also two analogous phase transitions as the noise-amplitude is increased. By comparing the standard models for noise-induced transitions with our new Langevin equation we elucidate that the mechanisms controlling the disorder-to-order transitions in both of them are essentially different, even though for both of them the presence of multiplicative noise is a key ingredient. PACS: 05.40.+j'
address:
- ' $^{1}$ Dipartimento di Fisica, Universitá di Roma “La Sapienza”, P.le A. Moro 2, I-00185 Roma, Italy'
- ' $^2$ Instituto Carlos I de F[í]{}sica Te[ó]{}rica y Computacional, Universidad de Granada, E-18071 Granada, Spain.'
author:
- 'Walter Genovese$^{1}$, Miguel A. Mu[ñ]{}oz $^{1,2}$, and P. L. Garrido $^{2}$'
title: 'Mesoscopic description of the annealed Ising model, and multiplicative noise'
---
Introduction
============
A great deal of attention has been recently devoted to the study of physical effects induced by the presence of noise, i.e. phenomena appearing in stochastic systems, which would be absent in the sole presence of the deterministic part of the corresponding Langevin equation [@HL]. By now it is clear that noise can generate quite unexpected and counterintuitive behaviors as, for example, the [*stochastic resonance*]{} [@sr], in which the output to input ratio of a bistable system subjected to the presence of an oscillating force is strongly enhanced by the presence of an additional stochastic term. Other examples are the resonant activation [@sa], and the noise induced spatial patterns [@n1]. Another type of phenomena the noise is at the base of, are the so called [*noise induced phase transitions*]{}. These came to light in an interesting paper by Van der Broek, Parrondo and Toral [@Raul] (see also [@HL; @BK; @Pik]). These authors pointed out the fact that some Langevin equations may exhibit a noise-induced ordering transition (NIOT), i.e. a phase transition that is not expected from the analysis of the deterministic part of such equation. The phenomenology is as follows:
i\) For low enough noise amplitudes the system is disordered (i.e. the order parameter takes a zero value).
ii\) At a certain critical value of the noise amplitude the system exhibits a NIOT and, in a range of noise intensities above it the system remains ordered.
iii\) Finally, for noise amplitudes larger than a second critical value, the noise operates in a more standard way, this is, disordering again the system. We refer to this second phase transition as noise induced disordering transition (NIDT).
A physical explanation of the NIOT was proposed in [@Raul]; The ordering of the system is the consequence of the interplay between the noise and the spatial coupling [@Kawai]. In particular, the noise generates a short time instability at every single site, and the presence of a spatial coupling renders stable the non trivial state generated in that way.
A minimal model capturing the essence of the NIOT has been recently proposed [@GMs]. It has been clarified that the essence of the NIOT is purely multiplicative, this is, in order to generate an ordering transition the noise has to appear multiplied by the field variable. In this way, it has been possible to recognize that the NIOT is characterized by a set of critical exponents other than those of well established universality classes (as for example, that of the Ising model) [@Noi; @GMs]. It has also been shown that due to the multiplicative origin of this transition it is possible to observe the phenomena in $d=1$, dimension at which is very unusual to observe phase transitions.
Other results concerning NIOTs and NIDTs can be found in the literature [@Sancho; @Muller; @Kim]. A common feature of all the previously referred models, is that they are defined by means of Langevin equations, this is, equations describing the physics at a mesoscopic, coarse-grained scale (in fact, the concept of noise is meaningful only at this level). In this context, it is an interesting task that of analyzing microscopic models that exhibit similar non-trivial behaviors; one of which is the anneal Ising model [@TB]. By studying the connection between a microscopic systems and their respective mesoscopic representation one could shed some light on the way in which microscopic mechanisms generate the very non-trivial effects described at a mesoscopic scale.
In what follows we introduce the time honored anneal Ising model. It was proposed and described more than twenty years ago by Thorpe and Beeman [@TB]. A more detailed description of it will be presented in the next section; here we summarize the main properties we are interested in. The system is an Ising model in which the interactions, $J$, among spins, are annealed (not quenched) random variables that change from bond to bond and are extracted from a fixed probability distribution, $P(J)$. Under certain conditions (this is, for some distributions $P(J)$ to be specified later), the system phenomenology is as follows:
i\) For low temperatures the system is disordered, i.e. the averaged magnetization is zero.
ii\) At a critical value of the temperature, $T_1$, the system exhibits a second order phase transition. As the temperature is further increased above $T_1$ the averaged magnetization keeps on growing until it reaches a maximum value and it starts decreasing if $T$ is increased further.
iii\) At a second critical temperature, $T_2$, the system exhibits another phase transition (analogous to the well known ferromagnetic-paramagnetic disordering transition of the standard pure Ising model). The system remains disordered for temperatures higher than $T_2$.
This phase diagram resembles very much the behavior of the previously described noise induced transitions in Langevin equations. It is our purpose here to analytically derive a coarse-grained, mesoscopic, representation, in terms of a Langevin equation, of the microscopic annealed Ising model (AIM) to further explore the eventual relations between both phenomena.
The annealed Ising model
========================
Let us consider a d-dimensional impure Ising model in the sense that the value of the coupling constant among spins, $J$, changes from bond to bond, being an annealed random variable with a fixed temperature-independent probability distribution, $P(J)$ (which is not quenched but annealed at every site). Following the strategy proposed by Thorpe and Beeman [@TB] the model can be exactly mapped into a standard pure Ising model with an effective parameter, $K =J/T$, that depends on $P(J)$ and $T$, and we write as $K_{eff}(T)$. In particular [@TB], $$\int dJ \frac{P(J)}{coth[ K_{eff}- J/T ]-\epsilon
(K_{eff})}=0
\label{eff}$$ where $\epsilon(K)$ is the correlation function of two nearest-neighbor spins in the pure Ising model. By solving the implicit equation Eq. (\[eff\]) one obtains $K_{eff}$ as a function of the temperature and the parameters characterizing $P(J)$. Note that, in particular, for the two-dimensional case, the Onsager’s solution [@Ons] provides an explicit value for $\epsilon(K)$ and therefore Eq. (\[eff\]) can be solved and, furthermore, the system magnetization can be expressed as a function of $T$. Let us suppose now that, in particular, the distribution $P(J)$ is centered at a positive value $J_0$ (favoring ferromagnetic ordering), and has a variable width (standard deviation), $\delta J$. The resulting magnetization for this particular type of distribution is qualitatively represented in figure 1 (see also [@TB]).
Observe that for narrow distributions of $J$ the magnetization curve is similar to its corresponding in the pure Ising model. Instead, as $\delta J$ is increased, a disordering tendency is observed at low temperatures, and in particular, for a large values of the width (as for example, $\delta J_4$ in Fig.1) the system is disordered at low temperatures, and exhibits a disorder-to-order phase transition at a certain temperature. The standard ferromagnetic-paramagnetic (order-to-disorder) transition is also present and occurs at a variable value of $T$ for different values of $\delta J$.
The physical mechanism leading to the previous behavior was argued in [@TB] to be the competition between ferromagnetic and antiferromagnetic types of interactions that emerges when sufficiently large values of $\delta J$ are considered. In particular, when $\delta J> J_0$ both positive and negative values of the coupling constant are accessible at each bond, and in that case , for low temperatures the system is in a [*frustrated state*]{} in which ferromagnetic and antiferromagnetic domains compete. That frustration makes the ferromagnetic order parameter to vanish. As the temperature is further raised the thermal noise activates annihilation of domain walls and the system is more likely to ordinate. As a result, the averaged magnetization grows with increasing temperature. At a given point this effect ceases, and the standard role of the temperature as a disorganizing source sets to work.
Continuous representation
=========================
Let us now follow a standard procedure [@Amit] to cast the previous AIM into a continuous Langevin equation. For that purpose we first consider the pure Ising model case, and write down its associated equilibrium partition function: $$Z=\sum_{\{s\}}exp\left( \sum_{ij}K_{ij}s_{i}s_{j}\right).
\label{amit1}$$ Introducing auxiliary Gaussian integrals in terms of continuous variables $\phi_i$ (with $i$ varying from 1 to the total number of spins, $N$, in the lattice), and performing the change of variables $\psi_i= K_{ij}^{-1}\phi_j$ we obtain [@Amit] $$Z\propto \int d{\psi}_{1}...d{\psi}_{N}exp\left[-\frac{1}{4}{\psi}_{i}
K_{ij}{\psi}_{j}+\sum_{i} \log \cosh(K_{ij}{\psi}_{j})\right].
\label{amit2}$$ Expanding the hyperbolic-cosine in power series, performing a transformation to Fourier space, considering only the leading dependence on the temperature, and performing the continuous limit we finally obtain [@Amit] $$\begin{aligned}
Z & \propto &\int d[\psi] \ e^{-H} \nonumber \\
H &=& \frac{1}{4} \int d^{d}x \
[K_{0}(1-2K_{0}){\psi}^{2}(x)+{\rho}(4K_{0}-1){(
\nabla \psi)}^{2} \nonumber \\
&+& \frac{1}{3}{K_{0}}^{4}
{\psi}^{4}(x)].
\label{amit2b}\end{aligned}$$ with $K_0 = \int d^d x K(x)$, and $\rho=1/2 \int d^d x K(x) x^2 $. In this way we have derived a Ginzsburg-Landau coarse grained Hamiltonian for the Ising model. This could have been guessed a priori by using heuristic arguments, but we have preferred to follow the previous procedure that permits to obtain explicit expressions for the coefficients as function of the microscopic parameters. In this way, observe, for example, that both the diffusion constant and the coefficient of the quadratic term depend on the coupling through $K_0$, therefore in order to simplify the notation we define the diffusion constant, $D=\rho (4 K_0 -1)$. Taking only the main relevant dependences on $D$ we can write $$H=\int d^{d}x \
\left[\frac{a D}{2}{\psi}^{2}+\frac{b}{4}{\psi}^{4}+\frac{D
}{2}({\nabla \psi})^{2}\right],
\label{amit3}$$ where $a$ is a tuning parameter proportional to the distance to the critical temperature, and $b$ is a positive parameter. Let us stress once more that we are neglecting higher order dependences of $b$ and $a$ on $D$, and we assume them to be unessential to reproduce the microscopic phenomenology of interest at mesoscopic level (this hypothesis will be verified afterwards). The simplest Langevin equation with a stationary distribution characterized by a Gibbsian distribution with the Hamiltonian in Eq. (\[amit3\]) is well known to be [@HH; @Gardiner] $${\partial}_{t} \psi =-(aD+b{\psi}^{2})\psi +D{\nabla}^{2}\psi+\eta(t)
\label{amit4}$$ where $\eta(t)$ is a Gaussian white noise with $\langle \eta(x,t) \rangle = 0$, and $\langle \eta(x,t) \eta (x',t') \rangle = \delta^d(x-x') \delta(t-t')$.
At this point we can analyze the effects of an annealed distribution of $J$ in the microscopic AIM at the level of Langevin equations. For that purpose let us observe that in order to mimic the variability of the coupling in the AIM we can just substitute $D$ at each site in Eq. (\[amit4\]) by a stochastic variable, namely: $ D \rightarrow D+{\xi(x,t)} $, with $\langle \xi(x,t) \rangle = 0$ and $\langle \xi(x,t) \xi (x',t') \rangle = \sigma_D^2 \delta^d(x-x')
\delta (t-t')$, where $D$ and ${\sigma}_{D}$ play the role of $J_0$ and $\delta J$ respectively in the microscopic model. In this way we obtain, $${\partial}_{t} \psi
=-[a(D+{\xi})+b{\psi}^{2}]\psi +D{\nabla}^{2}\psi+\nabla
({\xi}\nabla \psi)+\eta(t).
\label{final}$$ This equation (intended in the Ito interpretation [@Gardiner]) constitutes our continuous representation of the AIM. Let us underline that the differences with respect to the pure case, Eq. (\[amit4\], are two: the presence of a [*multiplicative noise*]{}, and an extra term that couples spatial fluctuations of $D$ with $\nabla \psi$. Changes of $a$, parameter which appears multiplying both the linear term and the multiplicative noise, correspond to temperature variations.
We have analyzed Eq. (\[final\]) in mean field approximation [@Kawai; @Max; @GMs], and by performing systematic numerical simulations in two dimensions. The mean field approximation is performed along the lines discussed in [@Kawai; @Max; @GMs]. For the numerical simulation we have employed the Euler method [@Max], in a 32\*32 lattice, with lattice spacing $\Delta a=1$, and considered a time mesh $\Delta t=0.001$. Without lost of generality the parameters $b$ and $D$ have been fixed to $1$ and $10$ respectively. Different noise amplitudes, $\sigma_D$, have been considered. The main results we have obtained are as follows: in both, mean field approximation and in the numerical simulation, we reproduce the qualitative behavior of the order parameter as a function of the temperature characteristic of the microscopic model (see Fig.2 and Fig.3 and compare them with Fig.1).
In mean field approximation the order-to-disorder critical point is located at $a=0$, and in numerical we obtain also a close to zero critical value which does not depend on $\sigma_D$. On the contrary, the location of the disorder-to-order transition, depends on $\sigma_D$, analogously as the location of $T_1$ depends on $\delta J$ in the AIM. Observe that this transition is not sharp in the lowermost curve of Fig.3 due to finite size effects. Curves in Fig.2 and Fig.3 change with increasing $\sigma_D$ in the same way they do in the AIM when increasing $\delta J$, i.e., the larger the noise the smaller the ordering.
Let us stress once more that in order to obtain the transition, we change both the coefficient of the linear term and of the multiplicative noise term. If one of these two coefficients was kept fixed while the other was changed the microscopic phenomenology would not be reproduced. [*The presence of the multiplicative noise term is essential to generate the disorder-to-order transition*]{}. We have performed a numerical study of Eq. (\[final\]) omitting the term proportional to $\nabla
({\xi}\nabla \psi)$ , and conclude that none of the previous conclusions is qualitatively affected by this suppression; by omitting this term the disorder-to-order critical point is shifted to a lower value of $a$, and consequently this term has only a disorganizing effect. We could consequently write down a minimal model just by dropping out this unnecessary term, in the same way we omitted other irrelevant higher order dependences on $D$ in the derivation of the Langevin equation. We conclude that [ *the proposed Langevin equation with multiplicative noise in the Ito representation reproduces qualitatively all the interesting properties of the anneal Ising model, and in particular the reentrant phase transition.*]{} Therefore, once more it is shown that the multiplicative noise is the key ingredient of highly non-trivial phenomena in stochastic systems at a mesoscopic level.
Let us finally remark that the phenomenon we have just described [*is not*]{} the usual noise-induced transition as reported in previous works [@Raul; @Sancho; @GMs]. First of all, in those works only the multiplicative noise amplitude has to be changed to obtain a NIOT, while in our case the transition is obtained by varying the parameter $a$ that multiplies both the multiplicative noise and the linear term. Consequently in our case the disorder-to-order transition is not purely noise-induced. Second, considering a Stratonovich representation of the Langevin equation with multiplicative noise is essential in those works to generate noise-induced ordering. In fact, standard Langevin equations as those described in [@Raul; @Sancho; @GMs] do not exhibit NIOTs when intended in the Ito representation ([@preprint]). On the other hand, in the model presented here, the Langevin equation is intended in the Ito sense, and due to its peculiar structure, namely the coupling between $a$ and $\xi(t)$, that we have justified from a microscopic point of view, it can exhibit a rather rich phenomenology. In particular the system shows an ordering and a disordering transition as the temperature is increased but it does not exhibit, for example, the short time instability characteristic of the phenomena discussed in [@Raul; @Kawai; @GMs].
It is a pleasure to acknowledge L. Pietronero, M. Scattoni and S. Pellegrini and Juan Ruiz-Lorenzo for useful comments and discussions, and J.M.Sancho for a critical reading of the first draft and interesting correspondence. This work has been partially supported by the European Union through a grant to M.A.M. ERBFMBICT960925, and by the TMR ’Fractals’ network, project number EMRXCT980183.
See W. Horsthemke e R. Lefever, [*Noise Induced Transitions*]{}, Springer Verlag, Berlin, 1984, and references therein.
R. Benzi, A. Sutera and A. Vulpiani, J. Phys. A [ **14**]{}, L453 (1981). K. Weisenfeld and F. Moss, Nature [**373**]{},33 (1995)
C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. [ **69**]{}, 2318 (1992). P. Pechukas and P. Hänggi, Phys. Rev. Lett. [ **73**]{}, 2772 (1994).
J. Garc[í]{}a-Ojalvo, A. Hernández-Machado and J. M. Sancho, Phys. Rev. Lett. [**71**]{}, 1542 (1993); J. M. Parrondo, C. Van den Broeck, J. Buceta and F. J. de la Rubia, Physica [**A 224**]{}, 153 (1996).
C. Van den Broeck, J. M. R. Parrondo and R. Toral, Phys. Rev. Lett. [**73**]{}, 3395 (1994).
A. Becker and L. Kramer, Phys. Rev. Lett. [**73**]{}, 955 (1994).
A. S. Pikovsky and J. Kurths, Phys Rev. [**E 49**]{}, 898 (1994).
C. Van den Broeck, J. M. R. Parrondo, R. Toral and R. Kawai, Phys. Rev. E [ **55**]{}, 4084 (1997).
W. Genovese, M.A. Mu[ñ]{}oz and J. M. Sancho, Phys. Rev. E [**57**]{}, R2495 (1998).
G. Grinstein, M.A. Mu[ñ]{}oz and Y. Tu, Phys. Rev. Lett. [**76**]{}, 4376 (1996). Y. Tu, G. Grinstein and M.A. Mu[ñ]{}oz, Phys. Rev. Lett. [**78**]{}, 274 (1997).
J. Garc[í]{}a-Ojalvo, J. M. R. Parrondo, J.M. Sancho and C. Van den Broeck, Phys. Rev. E [**54**]{}, 6918 (1996).
R. Müller, K. Lippert, A. Kühnel and U. Behn, Phys. Rev. [**E 56**]{}, 2658 (1997).
S. Kim, S. H. Park and C. S. Ryu, Phys. Rev. Lett. [**78**]{}, 1616 (1997).
M. F. Thorpe, D. Beeman, Phys. Rev. B [**14**]{},188 (1976)
L. Onsager, Phys. Rev. [**75**]{}, 117 (1944)
D. Amit, [*Field theory, the renormalization group, and critical phenomena*]{}, World Scientific, 1984.
P. C. Hohenberg and B. J. Halperin, Rev. Mod. Phys. [**49**]{}, 435, (1977).
C. W. Gardiner, [*Handbook of Stochastic Methods*]{}, Springer-Verlag, Berlin and Heidelberg, 1985.
M. San Miguel and R. Toral, [*Stochastic Effects in Physical Systems*]{}, to be published in [*Instabilities and Nonequilibrium Structures*]{}, VI, E. Tirapegui and W. Zeller, eds. Kluwer Academic Pub. (1997). (Cond-mat/9707147).
An extended discussion of this point will be presented elsewhere.
|
---
author:
- 'Rados[ł]{}aw Wojtak$^{1}$, Steen H. Hansen$^{1}$ & Jens Hjorth$^{1}$'
title: Gravitational redshift of galaxies in clusters as predicted by general relativity
---
Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark
**The theoretical framework of cosmology is mainly defined by gravity, of which general relativity is the current model. Recent tests of general relativity within the $\Lambda$ Cold Dark Matter (CDM) model have found a concordance between predictions and the observations of the growth rate and clustering of the cosmic web[@Rap10; @Rey10]. General relativity has not hitherto been tested on cosmological scales independent of the assumptions of the $\Lambda$CDM model. Here we report observation of the gravitational redshift of light coming from galaxies in clusters at the $99$ per cent confidence level, based upon archival data[@Aba09]. The measurement agrees with the predictions of general relativity and its modification created to explain cosmic acceleration without the need for dark energy ($f(R)$ theory[@Car04]), but is inconsistent with alternative models designed to avoid the presence of dark matter[@Mil83; @Bek04].**
According to the theory of general relativity[@Ein16], light emitted from galaxies moving in the gravitational potential well of galaxy clusters is expected to be redshifted proportionally to the difference in gravitational potential $\Phi$ between the clusters and an observer, i.e., $z_{\rm gr}=\Delta\Phi/c^{2}$, where $c$ is the velocity of light in vacuum. For typical cluster masses of $\sim 10^{14}M_{\odot}$, where $M_{\odot}$ is the Sun’s mass, the gravitational redshift is estimated to be[@Cap95; @Bro00; @Kim04] $cz_{\rm gr}\approx 10$ km s$^{-1}$ which is around two orders magnitude smaller than the Doppler shift due to the random motions of galaxies in clusters. The method of disentangling the kinematic Doppler effect from gravitational redshift relies on the fact that the former gives rise to a symmetric broadening of the observed velocity distribution, whereas the latter shifts its centroid. A critical factor in detecting such a velocity shift is the number of galaxies with spectroscopically measured velocities and the number of galaxy clusters. Both should be sufficiently high in order to reduce the error due to the Doppler width of the velocity distribution and eliminate the sensitivity to irregularities in cluster structure, e.g. substructures, asphericity.
The data are compiled from the SDSS[@Aba09] Data Release 7 and the associated Gaussian Mixture Brightest Cluster Galaxy catalogue[@Hao10] containing the positions and redshifts of galaxy clusters identified in the survey. The cluster sample is richness-limited with a threshold corresponding to a cluster mass of $10^{14}M_{\odot}$. The mean, 5- and 95-percentile values of the cluster richness[@Hao10] are $16$, $8$, and $86$ and correspond to cluster masses of around $2\times 10^{14}M_{\odot}$, $10^{14}M_{\odot}$ and $10^{15}M_{\odot}$. The typical number of spectroscopic redshifts per cluster (within a $6$ Mpc aperture and a $\pm 4000$ km s$^{-1}$ velocity range around the mean cluster velocities) varies from $10$ for low-richness clusters to $140$ for the richest ones.
Fig. 1 shows the histograms of galaxy velocities calculated in four bins of the projected cluster-centric distance centred at $0.6$, $1.6$, $3.3$ and $5.2$ Mpc. The cluster centres and redshifts were approximated by the coordinates and redshifts of the brightest cluster galaxies, hereafter BCGs. The observed velocity distributions consist of two clearly distinct parts: a quasi-flat distribution of galaxies not belonging to the clusters (observed due to projection effect) and a quasi-Gaussian component associated with galaxies gravitationally bound to the clusters[@Woj07]. The latter is expected to reveal the signature of gravitational redshift in terms of a systematic shift of its velocity centroid. Analysis of mock kinematic data generated from cosmological simulations shows that the number of redshifts and clusters is sufficient to reduce all expected sources of noise such as substructures, cluster asphericity, non-negligible off-set between BCGs and clusters centres[@Ski11] (both in the position on the sky and redshift space), and to allow for detection of gravitational redshift at nearly $3\sigma$ confidence level (see SI).
We search for gravitational redshift by measuring the mean velocity $\Delta$ of the quasi-Gaussian component of the observed velocity distribution. We carry out a Monte Carlo Markov Chain analysis of the data using a two-component model for the velocity distribution which includes both a contribution from the cluster and non-cluster galaxies (SI). Constraints on the mean velocity are obtained by marginalising the likelihood function over the set of nuisance parameters defining the shape of both components of the velocity distribution. The best fitting models of the velocity distributions are shown in Fig. 1 and the resulting measurements of the mean velocity as a function of the projected cluster-centric distance $R$ are presented in Fig. 2. The obtained mean velocity is negative at all radii with a clear tendency to decline with increasing radius. The negative values arise from the fact that the rest frames of the clusters are defined by the observed velocities of the central galaxies. This choice of the reference frame implies that the gravitational redshift manifests itself as a blueshift[@Kim04] (negative mean velocity) varying with the projected cluster-centric distance from $0$ at the cluster centre to $-|\Phi(0)|/c$ at large projected radii $R$.
The detection of gravitational redshift is significant at the $99$ per cent confidence level. The integrated signal within the $6$ Mpc aperture amounts to $\Delta=-7.7\pm3.0$ km s$^{-1}$ which is consistent with the gravitational potential depths of simulated galaxy clusters of[@Kim04] $\Delta=-(5-10)$ km s$^{-1}$. A more quantitative comparison with theoretical predictions requires explicit information about the mean gravitational potential profile and the distribution of cluster masses in the sample. We make use of the velocity dispersion profile of the composite cluster to constrain both functions. Then we calculate the gravitational redshift in terms of the mean velocity $\Delta$ by convolving the individual profiles of the clusters with their mass distribution (SI). The resulting profile (red profile in Fig. 2; see also discussion on the effect of the anisotropy of galaxy orbits in SI) is fully consistent with the gravitational redshift inferred from the velocity distributions. The fact that the same gravitational potential underlies galaxy motions and gravitational redshift of photons in clusters provides observational evidence of the equivalence principle on the scale of galaxy clusters.
We confront the obtained constraints on gravitational redshift with the predictions of alternative theories of gravity. We consider two popular models of gravity, the tensor-vector-scalar (hereafter TeVeS) theory[@Mil83; @Bek04] and the $f(R)$ model[@Car04], designed to alleviate the problem of dark matter or to recover the expansion history of the Universe, respectively. Theoretical profiles of gravitational redshift are calculated using the relations between the generalised gravitational potentials of these models and the Newtonian potential (SI). The Newtonian potential is inferred from the observed velocity dispersion profile of the composite cluster under the assumption of the most reliable anisotropic model of galaxy orbits (see SI for more details), and constitutes the reference basis for the calculations. For TeVeS we assume that the total masses of galaxy clusters make up $80$ per cent of those recovered under assumption of the Newtonian gravity. This factor lowers the ratio of dynamical-to-baryonic mass in galaxy clusters to the value resulting from fitting Modified Newtonian Dynamics[@Mil83] (to which TeVeS is a relativistic generalisation) to cluster data[@Poi05]. The resulting profile of gravitational redshift does not match the data, deviating from the observations at the $95$ percent confidence level (the blue dashed line in Fig. 3). This discrepancy increases with projected radius and is mostly caused by a logarithmic divergence of the scalar field in the regime of small accelerations, i.e., $g<a_{0}$ and $a_{0}\approx 10^{-10}$ m s$^{-2}$, which is responsible for a $1/r$ modification of the gravitational acceleration. This result points to a critical problem for TeVeS (or Modified Newtonian Dynamics) in recovering the true gravitational potential at large distances around the cluster centres. Considering the $f(R)$ model, we choose the least favourable set of free parameters maximising the departure from Newtonian gravitational acceleration[@Sch10]. Despite this choice, the resulting profile of gravitational redshift is consistent with the data (the blue solid line in Fig. 2).
The obtained constraints on gravity are consistent with recent tests verifying the concordance between gravity, cosmological model and observations of the large scale structure of the Universe[@Rap10; @Rey10]. An important advantage of using gravitational redshift effect is that this method does not depend on cosmology (see also SI) allowing to probe gravity in a direct way. In particular, this implies that the discrepancy between TeVeS theory and the observations[@Rey10] is unlikely to be a consequence of a specific choice of cosmological parameters, but indeed points to the inadequacy of this model to describe the Universe on very large scales.
Our results complement a series of experiments and observations aimed at confirmation of the predicted gravitational redshift on different scales of the Universe. Fig. 3 shows a summary of the detections in terms of the relative accuracy of the measurements as a function of the scale of the gravitational potential well. The positions of data points vary from $20$ m, for the first ground-based experiment[@Pou59; @Pou64], to the $1-10$ Mpc scale for galaxy clusters. On the scale of the solar radius we plot the measurement of gravitational redshift for the Sun[@Lop91], and on the $2$ orders of magnitude smaller scale constraints from the observations of Sirius B white dwarf[@Gre71; @Bar05] and space-borne hydrogen maser[@Ves80]. These results make gravitational redshift the only effect predicted by general relativity which has been confirmed on spatial scales spanning $22$ orders of magnitude. Studying this effect in more detail relies on the size of the redshift sample and therefore will be possible with the advent of the next generation redshifts surveys, e.g., the EUCLID satellite.
[99]{}
Rapetti, D., [*et al.*]{} The observed growth of massive galaxy clusters - III. Testing general relativity on cosmological scales. [*Mon. Not. R. Astron. Soc.*]{} [**406**]{}, 1796–1804 (2010).
Reyes, R., [*et al.*]{} Confirmation of general relativity on large scales from weak lensing and galaxy velocities. [*Nature*]{} [**464**]{}, 256–258 (2010).
Abazajian, K. N., [*et al.*]{} The Seventh Data Release of the Sloan Digital Sky Survey. [*Astrophys. J. Suppl. S.*]{} [**182**]{}, 543–558 (2009)
Carroll, S., [*et al.*]{}Is cosmic speed-up due to new gravitational physics? [*Phys. Rev. D*]{} [**70**]{}, 043528 (2004)
Milgrom, M. A modification of the Newtonian dynamics– Implications for galaxies. [*Astrophys. J.*]{} [**270**]{}, 371–389 (1983)
Bekenstein, J. D. Relativistic gravitation theory for the modified Newtonian dynamics paradigm. [*Phys. Rev. D*]{} [**70**]{}, 083509 (2004)
Einstein, A. Die Grundlage der allgemeinen Relativitätstheorie. [*Astron. Nachr.*]{} [**354**]{}, 769–822 (1916)
Cappi, A. Gravitational redshift in galaxy clusters. [*Astron. Astrophys.*]{} [**301**]{}, 6–10 (1995)
Broadhurst, T. & Scannapieco, E. Detecting the Gravitational Redshift of Cluster Gas. [*Astrophys. J.*]{} [**533**]{}, 93–97 (2000)
Kim, Y.-R. & Croft, R. A. C. Gravitational Redshifts in Simulated Galaxy Clusters. [*Astrophys. J.*]{} [**607**]{}, 164–174 (2004)
Hao, J., [*et al.*]{} A GMBCG Galaxy Cluster Catalog of 55,424 Rich Clusters from SDSS DR7. [*Astrophys. J. Suppl. S.*]{} [**191**]{}, 254–274 (2010)
Wojtak, R., [*et al.*]{} Interloper treatment in dynamical modelling of galaxy clusters. [*Astron. Astrophys.*]{} [**466**]{}, 437–449 (2007)
Skibba, R. A., [*et al.*]{} Are brightest halo galaxies central galaxies? [*Mon. Not. R. Astron. Soc.*]{} [**410**]{}, 417–431 (2011)
Pointecouteau, E. & Silk, J. New constraints on modified Newtonian dynamics from galaxy clusters. [*Mon. Not. R. Astron. Soc.*]{} [**364**]{}, 654–658 (2005)
Schmidt, F. Dynamical masses in modified gravity. [*Phys. Rev. D*]{} [**81**]{}, 103002 (2010)
Pound, R. V & Rebka, G. A. Gravitational Red-Shift in Nuclear Resonance. [*Phys. Rev. Lett.*]{} [**3**]{}, 439–441 (1959)
Pound, R. V. & Snider, J. L. Effect of Gravity on Nuclear Resonance. [*Phys. Rev. Lett.*]{} [**13**]{}, 539–540 (1964)
Lopresto, J. C., Schrader, C. & Pierce, A. K. Solar gravitational redshift from the infrared oxygen triplet. [*Astrophys. J.*]{} [**376**]{}, 757–760 (1991)
Greenstein, J. L., Oke, J. B. & Shipman, H. L. Effective Temperature, Radius, and Gravitational Redshift of Sirius B. [*Astrophys. J.*]{} [**169**]{}, 563–566 (1971)
Barstow, M. A., [*et al.*]{} Hubble Space Telescope spectroscopy of the Balmer lines in Sirius B. [*Mon. Not. R. Astron. Soc.*]{} [**362**]{}, 1134–1142 (2005)
Vessot, R. F. C., [*et al.*]{} Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser. [*Phys. Rev. Lett.*]{} [**45**]{}, 2081–2084 (1980)
Acknowledgments {#acknowledgments .unnumbered}
===============
The Dark Cosmology Centre is funded by the Danish National Research Foundation. R.W. wishes to thank D. Rapetti, G. Mamon and S. Gottlöber for fruitful discussions and suggestions. The mock catalogues of galaxy clusters have been obtained from a simulation performed at the Altix of the LRZ Garching.
Author contributions {#author-contributions .unnumbered}
====================
R.W., analysis of the velocity distributions and the velocity dispersion profile, predictions for the models of modified gravity, drafting the manuscript; S.H.H., comparison with the constraints on gravitational redshift on different scales, writing and commenting the paper; J.H., conceiving the idea of the measurement, writing and commenting the paper;
[**Figure 1** Velocity distributions of galaxies combined from $7,800$ SDSS galaxy clusters. The line-of-sight velocity ($v_{\rm los}$) distributions are plotted in four bins of the projected cluster-centric distances $R$. They are sorted from the top to bottom according to the order of radial bins indicated in the upper left corner and offset vertically by an arbitrary amount for presentation purposes. Red lines present the histograms of the observed galaxy velocities in the cluster rest frame and black solid lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not belong to the cluster and a quasi-Gaussian contribution from the cluster members (see SI for more details). The cluster rest frames and centres are defined by the redshifts and the positions of the brightest cluster galaxies. The error bars represent Poisson noise.]{}
[**Figure 2** Constraints on gravitational redshift in galaxy clusters. The effect manifests itself as a blueshift $\Delta$ of the velocity distributions of cluster galaxies in the rest frame of their BCGs. Velocity shifts were estimated as the mean velocity of a quasi-Gaussian component of the observed velocity distributions (see Fig. 1). The error bars represent the range of $\Delta$ parameter containing $68$ per cent of the marginal probability and the dispersion of the projected radii in a given bin. The blueshift (black points) varies with the projected radius $R$ and its value at large radii indicates the mean gravitational potential depth in galaxy clusters. The red profile represents theoretical predictions of general relativity calculated on the basis of the mean cluster gravitational potential inferred from fitting the velocity dispersion profile under the assumption of the most reliable anisotropic model of galaxy orbits (see SI for more details). Its width shows the range of $\Delta$ containing $68$ per cent of the marginal probability. The blue solid and dashed lines show the profiles corresponding to two modifications of standard gravity: $f(R)$ theory[@Car04] and the tensor-vector-scalar (TeVeS) model[@Mil83; @Bek04]. Both profiles were calculated on the basis of the corresponding modified gravitational potentials (see SI for more details). The prediction for $f(R)$ represents the case which maximises the deviation from the gravitational acceleration in standard gravity on the scales of galaxy clusters. Assuming isotropic orbits in fitting the velocity dispersion profile lowers the mean gravitational depth of the clusters by $20$ per cent. The resulting profiles of gravitational redshift for general relativity and $f(R)$ theory are still consistent with the data and the discrepancy between prediction of TeVeS and the measurements remains nearly the same. The arrows show characteristic scales related to the mean radius $r_{\rm v}$ of the virialized parts of the clusters.]{}
[**Figure 3** The measured-to-predicted ratio of the gravitational redshift. The figure shows the results of different observations or experiments as a function of the spatial scale of the gravitational potential well. Blue and red symbols refer to detections of gravitational redshift $z_{\rm g}$ in: ground-based experiment[@Pou59] (blue circle), observations of Sirius B white dwarf[@Gre71; @Bar05] (blue triangles), space-based experiment[@Ves80] (blue square), observation of the Sun[@Lop91] (blue star), analysis of the cluster data reported in this work (red circle). All measurements are compared with the predictions of general relativity (solid symbols). Results obtained for galaxy clusters are also compared with the predictions of $f(R)$ theory and TeVeS model (red empty symbols). As a measure of gravitational redshift in galaxy clusters we used the signal integrated within the aperture of $6$ Mpc. The green square and circle show the measurement of the rate of growth of cosmic structure[@Rap10] and the probe of gravity $E_{\rm g}$ combining the properties of galaxy-galaxy lensing, galaxy clustering and galaxy velocities[@Rey10]. Both results are compared with the prediction of general relativity with a standard $\Lambda$CDM cosmological model. All error bars represent standard deviations. The relative accuracy of the measurement from space-born experiment[@Ves80] is beyond the resolution of the plot and amounts to $10^{-4}$.]{}
[Measuring gravitational redshift of galaxies in clusters]{} {#measuring-gravitational-redshift-of-galaxies-in-clusters .unnumbered}
============================================================
The gravitational potential depth in typical galaxy clusters, expressed in terms of the velocity shift, is estimated at around[@Kim04] $-10$ km s$^{-1}$. This is two orders of magnitude smaller than the Doppler shift arising from random motions of galaxies in clusters. Bearing in mind that the signature of the gravitational redshift lies in the mean of the velocity distribution one can show that in order to reduce the error on the gravitational redshift to the level of the effect itself one needs at least $10^{4}$ velocities. This number should grow by a factor of $4$, if one requires a $2\sigma$ detection of the effect, and probably by another factor of $2$ in order to account for a non-negligible number of background galaxies which do not contribute to the effect, but give rise to the shape of the observed velocity distribution. Needles to say, the only means to collect as many as $10^{4}-10^{5}$ galaxy velocities in clusters is stacking redshift data of sufficiently large number of clusters ($>10^{3}$ clusters, assuming that current redshift surveys typically provide $10$ redshifts per cluster).
Combining redshift data from a number of clusters also allows to reduce the error resulting from local irregularities of velocity distributions in individual clusters. Such irregularities arise naturally from the presence of substructures or filaments along the line of sight, deviation from spherical symmetry, residual streaming motions etc. In order to address the impact of these factors on the error of the gravitational redshift estimate, one needs to refer to cosmological simulations. Such an analysis was carried out by Kim & Croft[@Kim04] who concluded that the minimum number of galaxy clusters required to confirm the gravitational redshift effect at the $2\sigma$ confidence level is $\sim 3000$. With this number of clusters the gravitational redshift may be traced up to $6$ Mpc, which is $3-4$ times larger than the size of the virialized part of clusters– a natural boundary condition for all methods of the mass measurement based on the assumption of virial equilibrium.
Another important source of inaccuracy in the measurement of gravitational redshift, which was partly taken into account by Kim & Croft[@Kim04], is the choice of a central cluster galaxy which ideally would be an object at rest at the bottom of the gravitational potential well. In our work, we approximate such galaxy by a brightest cluster galaxy, hereafter BCG. In general, such choice is not fully justified because the positions and velocities of BCGs exhibit some deviations from those defined by the cluster mass centres[@Lin07; @Ski11]. For example, the typical dispersion of the random velocities of BCGs may reach $30-40$ per cent of the total velocity dispersion in galaxy clusters[@Ski11]. Yet, among all cluster galaxies, BCGs are those whose positions and velocities coincide mostly with the location and velocities of the cluster centres. In order to reduce the error caused by the non-vanishing velocities of BCGs to the level required for detection of the gravitational redshift effect, one needs to combine the data from a sufficiently large number of clusters, e.g., around $2500$ clusters for a $2\sigma$ detection[@Kim04].
[Data]{} {#data .unnumbered}
========
In order to compile statistically uniform and possibly the largest sample of galaxy redshifts in clusters, we make use of the SDSS[@Aba09] Data Release 7 whose integral part is a flux-limited spectroscopic survey providing the redshifts of nearly million galaxies brighter than Petrosian $r$-magnitude $17.77$ over the area $7400$ deg$^2$. The positions and redshifts of galaxy clusters come from a Gaussian Mixture Brightest Cluster Galaxy cluster catalogue[@Hao10] which is the most up-to-date catalogue of galaxy clusters assembled on the basis of the SDSS DR7. The catalogue comprises $55,000$ galaxy clusters at redshifts $0.1<z<0.55$ detected by means of searching for red-sequence galaxies and BCGs. It provides positions and redshifts of BCGs residing in galaxy clusters selected from the SDSS above a certain richness limit. For the purpose of our analysis, we neglect all clusters whose BCGs do not have spectroscopic redshifts. This reduces the cluster sample by $63$ per cent without affecting the relative fractions of poor and rich clusters.
We approximate the coordinates and redshift of cluster centres by the positions and redshifts of BCGs. Then we search for all galaxies within a $6$ Mpc aperture around the cluster centres. The radius of this aperture is $\approx 3.5$ larger then the virial radius and corresponds to the turn-around radius[@Cup08] at which the expansion of the Universe starts to dominate over peculiar velocities of galaxies and the velocity cut-off separates all potential cluster galaxies from the galaxies of background or foreground. In order to separate potential cluster members from distant interlopers (galaxies of background or foreground), we select only those galaxies whose velocities $v_{\rm los}$ in the rest frame of a related BCG, i.e. $$v_{\rm los}=c\frac{z-z_{\rm BCG}}{1+z_{\rm BCG}},$$ where $z$ and $z_{\rm BCG}$ are the redshifts of a given galaxy and related BCG respectively, lie within the $\pm 4000$ km s$^{-1}$ range. This velocity cut-off is sufficiently wide to include all cluster members with no respect to the cluster mass (the minimum velocity cut-off corresponding to the most massive clusters at small radii is around $\pm 3000$ km s$^{-1}$; however, wider velocity range is required for precise modelling of the interloper contribution to the observed velocity distribution[@Rin03]).
Our final sample comprises $7,800$ clusters with the mean redshift of $0.24$ and on average $16$ galaxies with spectroscopic redshift per cluster. Those clusters with less than $5$ redshifts were not included into the sample. As the final step we combine redshift data of all clusters into one. The velocity diagram of the resulting composite cluster is shown in Supplementary Fig. 1.
[Analysis of the velocity distributions]{} {#analysis-of-the-velocity-distributions .unnumbered}
==========================================
To place constraints on the mean of the observed velocity distribution of cluster galaxies induced by the gravitational redshift effect we carry out a Monte Carlo Markov Chain (MCMC) analysis of the velocity distribution with the likelihood function defined by $$\label{like}
L=\prod_{i=1}^{N}f(v_{\rm los,i}|\Delta,{\mathbf a}),$$ where $f(v_{\rm los}|\Delta,{\mathbf a})$ is a model of the velocity distribution, $\Delta$ is the mean of the velocity distribution of cluster galaxies, ${\mathbf a}$ is a vector of nuisance parameters describing the shape of the velocity distribution and $N$ is the number of redshifts. In order to account for the presence of the interlopers (galaxies of foreground or background observed due to projection effect), we use the following two-component model of the velocity distribution[@Woj07] $$\label{pdf}
f(v_{\rm los})=(1-p_{\rm cl})f_{\rm b}(v_{\rm los}|{\mathbf a})+p_{\rm cl}f_{\rm cl}(v_{\rm los}|\Delta,{\mathbf a}),$$ where $f_{\rm cl}(v_{\rm los})$ and $f_{\rm b}(v_{\rm los})$ are the velocity distributions of the cluster members and the interlopers, respectively (both normalised to $1$), and $p_{\rm cl}$ is a free parameter describing the probability of a given galaxy to be a cluster member. The choice of the functional form of $f_{\rm b}(v)$ depends on the operational definition of cluster membership and may vary from a wide Gaussian distribution[@Mam10], if one regards all galaxies beyond the virial sphere as the background, to a uniform distribution[@Woj07], if only gravitationally unbound galaxies contribute to the background. For the purpose of our study a uniform background is appropriate, since all gravitationally bound galaxies, regardless of their positions with respect to the virial sphere, contribute to the expected signal of gravitational redshift. However, close inspection of the data reveals that a uniform model of the background must be generalised to account for a subtle asymmetry between the number of interlopers with negative and positive velocities. This asymmetry arises from the fact that a flux-limited limited survey tends to include slightly more galaxies which are closer and, therefore, have negative Hubble velocities in the cluster rest frame. We find that in order to account for this effect it is sufficient to assume that $f_{\rm b}(v_{\rm los})$ is linear in velocity $v_{\rm los}$. The slope of this velocity distribution is the second nuisance parameter of our model, after $p_{\rm cl}$. We also note that the velocity distribution of cluster members cannot exhibit a similar asymmetry because it is dominated by random motions of galaxies which are independent of the positions with respect to the cluster centre.
The observed velocity distributions of cluster galaxies are not Gaussian at all projected radii $R$ (see Fig. 2). This deviation from Gaussianity is expected and arises mostly from combining data from clusters of different masses[@Dia96], from the fact that the radial bins are wider than the scale of variation of the velocity dispersion profile and from an intrinsic non-Gaussianity of velocity distributions of individual clusters[@Woj09]. Modelling these effects is beyond the scope of this work and, for our purpose, it is sufficient to invoke a phenomenological model of $f_{\rm cl}(v_{\rm los})$ providing a satisfactory fit to the data. We find that approximating $f_{\rm cl}(v_{\rm los})$ by a sum of two Gaussians with the same mean velocity $\Delta$ satisfies this condition. This introduces three additional parameters into the model given by eq. (\[pdf\]): two velocity dispersions and the ratio of the relative weights of both Gaussian components. Performing a K-S test, we verified that the final fits of our model are fully consistent with the data ($p=0.99$).
We carry out the MCMC analysis of the velocity distributions in $4$ radial bins of the projected cluster-centric distance (see Fig. 2). The number of redshifts in subsequent bins varies from $15,000$ in the two innermost bins to $45,000$ for the remaining two. The choice of these numbers is motivated by finding a balance between bin spacing and the local number of cluster galaxies (proportional to $p_{\rm cl}$ which varies from $0.9$ at the cluster centre to $0.3$ at $R\gtrsim 3$ Mpc). BCGs were not included in the first bin, otherwise the estimate of the mean would be biased towards $0$. The number of clusters contributing to the subsequent bins varies from $\sim 1000$ in the innermost bin to $\sim 2000$ in the outermost one.
For the conversion between the angular and physical scales we adopted a flat $\Lambda$CDM cosmology with $\Omega_{\rm m}=0.3$ and the Hubble constant $H_{0}=70$ km s$^{-1}$ Mpc$^{-1}$. We note, however, that all galaxy clusters used in this work lie at low redshifts ($z\approx 0.2$) and, therefore, the impact of using a particular cosmological model on the final results is negligible.
[Gravitational redshift profile]{} {#gravitational-redshift-profile .unnumbered}
==================================
Assuming spherical symmetry, the gravitational redshift profile of a single galaxy cluster (in terms of velocity blueshift of the velocity distribution) can be calculated using the following formula[@Cap95] $$\label{vel-shift-proj}
\Delta_{\rm s} (R)=\frac{2}{c\Sigma(R)}\int_{R}^{\infty}[\Phi(r)-\Phi(0)]\frac{\rho(r)r\textrm{d}r}
{\sqrt{r^2-R^2}},$$ where $R$ is the projected cluster-centric distance, $\Phi(r)$ is the gravitational potential, $\rho(r)$ and $\Sigma(R)$ are the 3D and surface (2D) density profiles of galaxies. In order to estimate this effect for the data combined from a cluster sample, one needs to convolve this expression with the distribution of cluster masses in the sample. Then the resulting profile of the blueshift takes the following form $$\label{shift-stack}
\Delta (R)=\frac{\int\Delta_{\rm s}(R)\Sigma(R)(\textrm{dN}/\textrm{d}M_{\rm v})\textrm{d}M_{\rm v}}
{\int\Sigma(R)(\textrm{d}N/\textrm{d}M_{\rm v})\textrm{d}M_{\rm v}},$$ where $M_{\rm v}$ is the virial mass and $\textrm{d}N/\textrm{d}M_{\rm v}$ is the mass distribution. We note that $\Delta_{\rm s}$ depends implicitly on the virial mass and the shape of the gravitational potential. The virial mass and radius are defined in terms of the overdensity parameter $\delta_{c}=3M_{\rm v}/(4\pi r_{\rm v}^{3}\rho_{c})$, where $\rho_{c}$ is the present critical density. In our calculations we adopted $\delta_{c}=102$ (see e.g. [Ł]{}okas & Hoffman[@Lok01]).
The main unknown factor in equation (\[shift-stack\]) is the mass distribution. This may be estimated by means of dynamical modelling of the observed velocity dispersion profile. The left panel of Supplementary Fig. 2 shows the velocity dispersion profile estimated in radial bins (black points) by fitting eq. (\[pdf\]) with $f_{\rm cl}(v)$ approximated by a single Gaussian. The profile is truncated at $R=1.2$ Mpc which is the virial radius corresponding to the anticipated lower limit of all virial masses in the sample which is around $10^{14}M_{\odot}$. We note that the velocity dispersion profile is flatter than typical profiles observed in single galaxy clusters. This property arises naturally from the fact that the cluster sample is not uniform in terms of the cluster mass (more massive clusters give rise to growth of velocity dispersions at large radii). Here we use this effect to place constraints on the mass distribution in the cluster sample.
In analogy with equation (\[shift-stack\]), one can show that the velocity dispersion profile for kinematic data combined from a set of clusters can be expressed as $$\label{dispersion}
\sigma_{\rm los}(R)=\Big(\frac{\int\widehat{\sigma}_{\rm los}^{2}(R)\Sigma(R)
(\textrm{dN}/\textrm{d}M_{\rm v})\textrm{d}M_{\rm v}}
{\int\Sigma(R)(\textrm{dN}/\textrm{d}M_{\rm v})\textrm{d}M_{\rm v}}\Big)^{1/2},$$ where $\widehat{\sigma}_{\rm los}(r)$ refers to the velocity dispersion profile of a single cluster. We approximate the mass distribution by a power-law, i.e., $\textrm{d}N/\textrm{d}M_{\rm v}\propto M_{\rm v}^{-\alpha}$, where $\alpha$ is a free parameter. This parameterisation is mostly motivated by the fact that it resembles the observed cluster counts as a function of the virial mass[@Roz10]. In order to account for the richness threshold of the cluster catalogue and cosmological decay of the mass function at high masses we impose cut-offs on the mass distribution at low and high masses respectively. The cut-offs are fixed at $10^{14}M_{\odot}$ and $2\times10^{15}M_{\odot}$ which are the limits of the mass range spanned by the clusters of the maxBCG catalogue[@Koe07; @Joh07; @Roz10]– the predecessor of the Gaussian Mixture Brightest Cluster Galaxy catalogue[@Hao10] used in this work.
In order to calculate the velocity dispersion profile of a single cluster, $\widehat{\sigma}_{\rm los}(R)$ in eq. (\[dispersion\]), we make use of a model of the distribution function presented by Wojtak [*et al.*]{}[@Woj08]. The model is constructed under assumption of spherical symmetry, constant mass-to-light ratio (galaxies trace dark matter) and for a wide range of possible profiles of the orbital anisotropy. We approximate the dark matter density profile by the NFW formula[@Nav97], i.e. $\rho(r)\propto 1/[(r/r_{\rm v})(1+c_{\rm v}r_{\rm v})^{2}]$, where $c_{\rm v}$ is the concentration parameter. Since fitting the velocity dispersion profile does not allow to constrain the mass profile and the anisotropy of galaxy orbits at the same time (the problem known as the mass-anisotropy degeneracy), we fix all parameters related to the orbital anisotropy. In order not to loose generality of our analysis we consider two models of the orbital anisotropy: anisotropic– with the anisotropy parameter $\beta(r)=1-\sigma_{\theta}^{2}(r)/\sigma_{r}^{2}(r)$ varying from $0$ in the centre to $0.4$ at the virial radius, and isotropic with $\beta(r)=0$, where $\sigma_{r}$ and $\sigma_{\theta}$ are the radial and tangential velocity dispersions. We note that these two models are two limiting cases of a whole family of the anisotropy profiles found both in simulations[@Cue08; @Han10] and observations[@Biv04; @Woj10]
We evaluate the velocity dispersion numerically as the second moment of the projected phase-space density at fixed projected radii $R$. Then we correct all resulting dispersions for the effect of non-vanishing random velocities of BCGs[@Ski11]. Such correction relies on replacing all velocity dispersions $\sigma_{\rm los}(R)$ by $(\sigma_{\rm los}(R)^{2}+\sigma_{\rm BCG}^{2})^{1/2}$, where $\sigma_{\rm BCG}$ is a typical velocity dispersion for BCGs. For our analysis, we assume that $\sigma_{\rm BCG}$ equals $35$ per cent of the total velocity dispersion within the virial radius.
Fitting the velocity dispersion profile to the data, we obtain constraints on the concentration parameter and the slope of the mass distribution (the right panel of Supplementary Fig. 2). For simplicity, we assume that the concentration parameter does not vary with the virial mass. The obtained concentrations are smaller by $20-30$ per cent than those found for simulated dark matter halos[@Kly10]. This effective flattening of the mass profile most (smaller concentration parameters) arises from the fact that the gravitational potential of clusters is measured with respect to BCGs whose positions exhibit random off-sets from the true cluster centres[@Lin07].
We make use of the constraints on $c_{\rm v}$ and $\alpha$ to calculate the profile of gravitational redshift given by eq. (\[shift-stack\]), where the gravitational potential takes the NFW form, i.e. $$\label{psiNFW}
\Phi(r)=-(GM_{\rm v}/r_{\rm v})^{1/2}g(c_{\rm v})^{1/2}\frac{\ln(1+r/r_{\rm v})}{r/r_{\rm v}}$$ and $g(c_{\rm v})=1/[\ln(1+c_{\rm v})-c_{\rm v}/(1+c_{\rm v})]$, and the number density of galaxies is proportional to that of dark matter. The resulting $\Delta$ profiles are shown in Supplementary Fig. 3. The blue (red) profile corresponds to an isotropic (anisotropic) model of galaxy orbits and the widths of the profiles are the $1\sigma$ ranges obtained by marginalising over all free parameters of the model. Both profiles are fully consistent with the constraints on $\Delta$ inferred from the observed velocity distributions (black points). Since the systematic errors induced by a choice of the anisotropy model are much smaller than the random errors associated with the gravitational redshift measurement, in the final comparison we only consider the profile calculated for an anisotropic model of galaxy orbits– a more reliable model in recovering the kinematics of galaxy clusters[@Woj10]. We also note the gravitational potential resulting from the analysis with anisotropic galaxy orbits is the basis for the calculations of the gravitational redshift effect in alternative models of gravity (see the last section of SI).
The theoretical calculations presented in this section rely on the extrapolation of the NFW profile beyond the virial sphere which is still robust to $2r_{\rm v}$, but is probably less justified for larger radii[@Tav08]. In order to check the potential impact of this assumption on the estimation of $\Delta(R)$, we considered a set of simple modifications of the density profile at large radii. We found that varying the asymptotic slopes of the dark matter (or galaxy number) density profile by $\pm 0.5$ at $r>2r_{\rm v}$ does not change the $\Delta$ profiles by more than $4$ per cent. We also found that the mass limits imposed on the mass distribution have a negligible effect on the final $\Delta$ profile. We verified that changing both mass limits by $50$ per cent induces error which do not exceed the uncertainty due to the unknown anisotropy of galaxy orbits.
[Test on a mock data sample]{} {#test-on-a-mock-data-sample .unnumbered}
==============================
In order to test the statistical robustness of the gravitational redshift detection, we analyse mock kinematic data generated from cosmological $N$-body simulations of a standard $\Lambda$CDM cosmological model (for details of the simulations see Wojtak [*et al.*]{}[@Woj08]). Such test allows to check whether all effects related to the internal substructure of clusters and their perturbed velocity distributions are sufficiently reduced in the procedure of stacking kinematic data of a large number of galaxy clusters.
We generate mock redshift data by drawing randomly dark matter particles from $7,800$ cylinders of observation (the number of cylinders corresponds to the number of clusters in the real data sample compiled from the SDSS). The cylinders of observations are defined by the $\pm 4000$ km s$^{-1}$ velocity range and the $6$ Mpc aperture. Their viewing angles are chosen at random and geometrical centres are located at the centre of one of $80$ cluster-mass dark matter halos. In order to simulate the offset between the positions of the cluster central galaxies and the cluster mass centres, we introduce a random shift between the positions and velocities of the cylinders and the corresponding halo mass centres. The maximum value of this shift is $0.1$ $h^{-1}$ Mpc for the positions[@Lin07] and $\pm35$ per cent of the total velocity dispersion for velocities projected onto the line of sight[@Ski11]. The total number of velocities in a composite cluster is fixed at $10^{5}$ or $10^{6}$. The former corresponds to the number of redshifts in the current SDSS sample and the latter represents a forecast for the future.
In order to perform a test of detectability of the gravitational redshift effect, we consider two data samples. The first consists of velocities which are not corrected for the gravitational redshift and the second takes into account an additional velocity shift proportional to the local gravitational potential. In both cases particle positions remain the same. Both mock data samples are analysed in the same manner as the SDSS data. The obtained constraints on the mean velocity as a function of the projected radius are shown in Supplementary Fig. 4, where blue and red colour refer to the samples with $10^{5}$ and $10^{6}$ redshifts, respectively.
The mean velocity obtained for the sample without gravitational redshift (points with the dashed error bars) are consistent with $\Delta=0$ at all radii. This shows that the number of clusters used in our work is sufficient to reduce all local effects giving rise to the fluctuations of the mean velocity. Velocity shifts obtained for the second data sample (points with the solid error bars) clearly indicate the presence of the gravitational redshift effect. They also trace the true profile of this effect averaged over the halo sample used for generating mock kinematic data (black solid line). We find that the measured velocity shifts for $10^{5}$ data points (blue points with the solid error bars) deviate from $\Delta=0$ profile at nearly the same significance level as the results obtained for the SDSS data. We also note that an increase of the number of redshifts per cluster improves the errors of the measurement according to the rule of the inverse square root proportionality.
[Gravitational redshift in alternative models of gravity]{} {#gravitational-redshift-in-alternative-models-of-gravity .unnumbered}
===========================================================
The only difference in the calculation of the gravitational redshift for models of modified gravity relies on replacing the Newtonian gravitational potential in eq. (\[vel-shift-proj\]) by the potential emerging from a given gravity. For the $f(R)$ model, the gravitational potential is calculated using phenomenological relations obtained by Schmidt[@Sch10] who ran a series of cosmological simulations of this model and quantified its effect on cluster dynamics in terms of the effective enhancement of gravitational acceleration with respect to the standard gravity. In our work we consider the most critical case of his results (with $|\textrm{d}f/\textrm{d}R|\approx10^{-4}$) leading to a homogeneous amplification of gravitational acceleration at all radii by factor of $1.33$.
Gravitational potential in TeVeS theory is a sum of the Newtonian potential $\Phi_{\rm N}$ and the scalar field $\phi$. The former is calculated assuming $80$ per cent of the total mass inferred from the velocity dispersion profile in the framework of Newtonian gravity. This factor lowers the total-to-barynic mass ratio to the value estimated in galaxy clusters under assumption of the Modified Newtonian Dynamics[@Poi05]. Assuming spherical symmetry, the scalar field $\phi$ is a solution of the following equation[@Bek04] $$\label{scalar}
\mu(y)\nabla\phi=(k/4\pi)\nabla\Phi_{\rm N},$$ where $y=(\nabla\phi)^2kl^{2}$, $k<<1$ and $\sqrt{k}/(4\pi l)\approx a_{0}=10^{-10}$ m s$^{-2}$. In our calculations we adopted $\mu(y)=\sqrt{y}/(1+\sqrt{y})$, which is one of the commonly used interpolating function in Modified Newtonian Dynamics[@Mil83], and $k=0.01$.
In both schemes of the calculation, the reference gravitational potential of Newtonian gravity is given by the constraints from fitting the velocity dispersion profile with an anisotropic model of galaxy orbits. We note that using the results of an isotropic model does not change the main conclusion of the test summarised in Fig. 3: the $f(R)$ model is still consistent with the data, whereas TeVeS yields even more divergent profile of the gravitational redshift.
von der Linden, A., [*et al.*]{} How special are brightest group and cluster galaxies? [*Mon. Not. R. Astron. Soc.*]{} [**379**]{}, 867–893 (2007)
Cupani, G., Mezzetti M. & Mardirossian, F. Mass estimation in the outer non-equilibrium region of galaxy clusters. [*Mon. Not. R. Astron. Soc.*]{} [**390**]{}, 645–654 (2008)
Rines, K., [*et al.*]{} CAIRNS: The Cluster and Infall Region Nearby Survey. I. Redshifts and Mass Profiles. [*Astron. J.*]{} [**126**]{}, 2152–2170 (2003)
Mamon, G. A., Biviano, A. & Murante, G. The universal distribution of halo interlopers in projected phase space. Bias in galaxy cluster concentration and velocity anisotropy? [*Astron. Astrophys.*]{} [**520**]{}, 30 (2010)
Diaferio, A. & Geller, M. J. Galaxy Pairwise Velocity Distributions on Nonlinear Scales. [*Astrophys. J.*]{} [**467**]{}, 19–37 (1996)
Wojtak, R., [*et al.*]{} The mass and anisotropy profiles of galaxy clusters from the projected phase-space density: testing the method on simulated data. [*Mon. Not. R. Astron. Soc.*]{} [**399**]{}, 812–821 (2009)
okas, E. L. & Hoffman, Y. The Spherical Collapse Model in a Universe with Cosmological Constant. [*Proc. 3rd International Workshop, The Identification of Dark Matter. World Scientific*]{}, 121 (2001)
Rozo, E., [*et al.*]{} Cosmological Constraints from the Sloan Digital Sky Survey maxBCG Cluster Catalog. [*Astrophys. J.*]{} [**708**]{}, 645–660 (2010)
Koester, B. P., [*et al.*]{} A MaxBCG Catalog of 13,823 Galaxy Clusters from the Sloan Digital Sky Survey. [*Astrophys. J.*]{} [**660**]{}, 239–255 (2007)
Johnston, D. E., [*et al.*]{} Cross-correlation Weak Lensing of SDSS galaxy Clusters II: Cluster Density Profiles and the Mass–Richness Relation. [*arXiv0709.1159*]{} (2007)
Wojtak, R., [*et al.*]{} The distribution function of dark matter in massive haloes. [*Mon. Not. R. Astron. Soc.*]{} [**388**]{}, 815–828 (2008)
Navarro, J. F., Frenk, C. S. & White, S. D. M. A Universal Density Profile from Hierarchical Clustering. [*Astrophys. J.*]{} [**490**]{}, 493–508 (1997)
Cuesta, A., [*et al.*]{} The virialized mass of dark matter haloes. [*Mon. Not. R. Astron. Soc.*]{} [**389**]{}, 385-397 (2008)
Hansen, S. H., Juncher, D. & Sparre, M. An Attractor for Dark Matter Structures. [*Astrophys. J.*]{} [**718**]{}, 68–72 (2010)
Biviano, A. & Katgert, P. The ESO Nearby Abell Cluster Survey. XIII. The orbits of the different types of galaxies in rich clusters. [*Astron. Astrophys.*]{} [**424**]{}, 779–791 (2004)
Wojtak, R. & [Ł]{}okas, E. L. Mass profiles and galaxy orbits in nearby galaxy clusters from the analysis of the projected phase space. [*Mon. Not. R. Astron. Soc.*]{} [**408**]{}, 2442–2456 (2010)
Klypin, A., Trujillo-Gomez, S. & Primack, J. Halos and galaxies in the standard cosmological model: results from the Bolshoi simulation. [*arXiv1002.3660*]{} (2010)
Tavio, H., [*et al.*]{} The dark outside: the density profile of dark matter haloes beyond the virial radius. [*arXiv:0807.3027*]{} (2008)
[**Supplementary Figure 2** Velocity dispersion profile of the composite cluster (left panel) and constraints on the concentration parameter $c_{\rm v}$ and the logarithmic slope of the mass distribution $\alpha$ (right panel) from fitting the velocity dispersion profile with an isotropic (blue) or anisotropic (red) model of galaxy orbits. The solid lines in the left panel show the best-fitting profiles of the velocity dispersion profile. The contours in the right panel are the boundaries of the $1\sigma$ and $2\sigma$ confidence regions of the likelihood function. The error bars in the left panel represent the range containing $68$ per cent of the marginal probability.]{}
|
---
abstract: 'It is shown that QCD is able to predict very tiny features of multiplicity distributions at very high energies which demonstrate that the negative binomial distribution (and, more generally speaking, any infinitely divisible distribution) is inappropriate for precise description of experimental data. New precise fits of high energy multiplicity distributions can be derived.'
author:
- |
**I.M. DREMIN\
**
title: |
**MULTIPLICITY DISTRIBUTIONS IN QCD\
AT VERY HIGH ENERGIES\
**
---
———- X-Sun-Data-Type: default X-Sun-Data-Description: default X-Sun-Data-Name: talk X-Sun-Content-Lines: 133
In this report, I briefly review the results of several cited below papers in which it has been shown that it is possible to solve the strongly non-linear integro-differential equations of QCD for the generating function of multiplicity distributions. From the solutions obtained, it follows the prediction about quite peculiar behavior of cumulants of the distribution which was never found before. The accelerator data support the prediction. Thus, more precise fits can be proposed to use in the cosmic ray range of very high energies.
First, let us introduce some definitions related to the normalized multiplicity distribution $$P_n = \sigma _n /\sum _{n=0}^{\infty }\sigma _n$$ of probabilities $P_n$ for $n$-particle (prong) events. Its generating function $G(z)$ is defined as $$G(z)=\sum _{n=0}^{\infty }(1+z)^{n} P_n ,$$ factorial moments are $$F_q = \frac {\sum n(n-1)...(n-q+1)P_n}{(\sum nP_n )^q} = \frac {1}{\langle n
\rangle ^q} \frac {d^{q}G(z)}{dz^{q}}\vert _{z=0}$$ and cumulants are $$K_q = \frac {1}{\langle n \rangle ^{q}} \frac {d^{q}\log G(z)}{dz^{q}}\vert
_{z=0}.$$ We use also their ratio $$H_q = K_q /F_q.$$
For the sake of simplicity, we consider here only a single gluon jet with the total energy in its c.m.s. equal to $Q$ and introduce the evolution parameter as $y=\ln (Q^2 /Q_{0}^{2})$, where $Q_{0}^{2}$=const. The generating function satisfies the QCD equation \[1,2\] $$G'(y)=\int _{0}^{1}dx (\frac {1}{x}-\Phi _{r}(x))\gamma _{0}^{2} [G(y+\ln
x)G(y+\ln (1-x))-G(y)],$$ where $\Phi _{r}(x)=(1-x)(2-x(1-x))$ is the regular part of the Altarelli-Parisi kernel and $\gamma _{0}^{2}=6\alpha _{S}/\pi $ ($\alpha _{S}$ is QCD coupling constant). It is the integro-differential non-linear equation with shifted arguments in non-linear part and, therefore, it seems impossible to find its solution. However, in a series of papers \[3-7\] it was shown that such a solution exists in higher-order perturbative QCD for the running coupling constant. Moreover, in the case of the fixed coupling constant, one was able to get an exact solution of the equation \[8,9\] (both in gluodynamics and in QCD with quarks and gluons). The decisive role in that progress is played by the usage of formulae (3),(4) proposed in \[4\] and by notion of the well-known relation between factorial moments and cumulants: $$F_q = \sum _{m=0}^{q-1} C_{m}^{q-1}K_{q-m}F_m ,$$ where $C_{m}^{q-1}$ are the binomial coefficients. Usage of (3),(4),(6) enables one to get additional relation between $F_q$ and $K_q$ which together with (7) provides the knowledge of any function $F_q, K_q, H_q$ and, therefore, of multiplicity distribution $P_n$. I shall not delve into mathematical details leaving room for physics discussion. One can learn them from cited papers \[3-10\].
MAIN PHYSICS CONCLUSIONS ARE:
1\. KNO-scaling is valid at very high energies.
2\. The shape of the distribution is much narrower than in the lowest perturbative approximation (called DLA).
3\. The tail of the distribution falls off approximately as $\exp [-a(n/\langle
n \rangle )^{\mu }]$ with $\mu > 1$.
4\. The cumulants $K_q$ acquire negative values while factorial moments are always larger than 1.
5\. Factorial moments are steadily increasing with their rank $q$ while cumulants oscillate with increasing amplitude at very large $q$. (Integer values of $q$ are considered. The extension to non-integer values was proposed in \[11\].)
6\. The ratio $H_q$ also oscillates with first minimum positioned at $q=5$ that reveals a new expansion parameter of QCD.
7\. This parameter shows that one should be careful in application of perturbative QCD to multiparticle production.
8\. The property 6) has been confronted to experiment and has found very good support of it (see \[12\]).
9\. The properties 4), 6) demonstrate that the negative binomial distribution (so popular nowadays) is inappropriate for precise fits of multiplicity distributions because its cumulants are always positive.
10\. Moreover, all infinitely divisible distributions are inappropriate since they have positive cumulants.
11\. The cluster models with the Poissonian distribution of clusters are inappropriate too.
The implications of these conclusions are to be studied yet. However, it is remarkable that perturbative QCD becomes a powerful tool in describing the soft processes when properly treated with higher-order terms taken into account. Thus , the widely spread opinion that perturbative QCD is inapplicable to soft processes should be reconsidered. The only (however, important) shortcoming is the hadronization stage treated still either in the framework of the local parton-hadron duality hypothesis or in Monte-Carlo simulations with various assumptions. However, from above consideration it is clear that qualitative features are well reproduced (and even predicted!) by QCD at the partonic level. The demonstrated in \[3-10\] possibility of the proper treatment of the partonic stage provides some hope. In particular, the energy dependence of partonic multiplicities \[10\] and the ratio of the partonic multiplicities in gluon and quark jets \[8, 10\] give good chances for further studies of the hadronization stage when compared with experimental data.
What concerns cosmic ray studies, it is not clear yet how the above findings influence the fragmentation region multiplicities that is of upmost importance for these investigations. It should be analyzed in combination with the knowledge of rapidity spectra in hadron-hadron and hadron-nucleus collisions.
In conclusion, the non-linearity of QCD has important implications for soft hadronic processes and it can be accurately treated to provide new predictions for very high energy particle interactions.
This work has been supported by the Russian fund for fundamental research (grant 94-02-3815) and by Soros fund.
1\. I.V.Andreev, [*Quantum Chromodynamics and Hard Processes at High Energies*]{}, M., Nauka, 1981 (in Russian).
2\. Yu.L.Dokshitzer, V.A.Khoze, A.H.Mueller, S.I.Troyan, [*Basics of Perturbative QCD*]{}, Gif-sur-Yvette, Editions Frontieres, 1991.
3\. Yu.L.Dokshitzer, Phys.Lett. B305 (1993) 295.
4\. I.M.Dremin, Phys.Lett. B313 (1993) 209.
5\. I.M.Dremin, Mod.Phys.Lett. A8 (1993) 2747.
6\. I.M.Dremin, V.A.Nechitailo, JETP Lett. 58 (1993) 945.
7\. I.M.Dremin, B.B.Levtchenko, V.A.Nechitailo, J.Nucl.Phys. 57 (1994) 477.
8\. I.M.Dremin, R.C.Hwa, Phys.Lett. B324 (1994) 477.
9\. I.M.Dremin, R.C.Hwa, Phys.Rev. D49 (1994) 5805.
10\. I.M.Dremin, V.A.Nechitailo, Mod.Phys.Lett. A9 (1994) to be published.
11\. I.M.Dremin, JETP Lett. 59 (1994) 501.
12\. I.M.Dremin, V.Arena, G.Boca et al, Phys.Lett. B (1994) to be published.
|
---
abstract: 'There has been an increase in interest in experimental evaluations to estimate causal effects, partly because their internal validity tends to be high. At the same time, as part of the big data revolution, large, detailed, and representative, administrative data sets have become more widely available. However, the credibility of estimates of causal effects based on such data sets alone can be low. In this paper, we develop statistical methods for systematically combining experimental and observational data to obtain credible estimates of the causal effect of a binary treatment on a primary outcome that we only observe in the observational sample. Both the observational and experimental samples contain data about a treatment, observable individual characteristics, and a secondary (often short term) outcome. To estimate the effect of a treatment on the primary outcome while addressing the potential confounding in the observational sample, we propose a method that makes use of estimates of the relationship between the treatment and the secondary outcome from the experimental sample. If assignment to the treatment in the observational sample were unconfounded, we would expect the treatment effects on the secondary outcome in the two samples to be similar. We interpret differences in the estimated causal effects on the secondary outcome between the two samples as evidence of unobserved confounders in the observational sample, and develop control function methods for using those differences to adjust the estimates of the treatment effects on the primary outcome. We illustrate these ideas by combining data on class size and third grade test scores from the Project STAR experiment with observational data on class size and both third and eighth grade test scores from the New York school system.'
author:
- 'Susan Athey[^1]'
- 'Raj Chetty[^2]'
- 'Guido W. Imbens[^3]'
bibliography:
- 'references.bib'
date: 'First Version, August 2019; Current version JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember '
title: 'Combining Experimental and Observational Data to Estimate Treatment Effects on Long Term Outcomes [^4] '
---
**Keywords: Causality, Experiments, Observational Studies, Long Term Outcomes**
Introduction {#section:introduction}
============
There has been an influential movement in empirical studies in economics towards relying more on experimental as opposed to observational data to estimate causal effects ([*e.g.*]{}, @duflo2007using [@angrist2010credibility]). The internal validity of randomized experiments tends to be high, and their analyses are relatively straightforward (@athey2017econometrics). However, due to the practical challenges involved in running experiments ([*e.g.,* ]{}@glennerster2013running), experiments are often limited in their size, in the richness of the information collected, and in representatitiveness, raising concerns about external validity. At the same time, as part of the big data revolution, large, detailed, and by their nature representative, administrative data sets have become more widely available ([*e.g.*]{}, @chetty). However, it is challenging to use such datasets to estimate causal effects because observational studies often lack internal validity. In this paper, we develop statistical methods for systematically combining experimental and observational data in an attempt to leverage the strengths of both types of data. We focus on a canonical case where both experimental and observational data contain information about individual treatment assignments and a secondary (e.g. short term) outcome (where the datasets contain different individuals), but only the observational data contains information about the primary (often long term) outcome of interest.
We illustrate our methods combining data from the New York school system (the “observational sample”) and from Project STAR (the “experimental sample”, see @krueger2001effect for an earlier analysis). Our goal is to estimate the effect of class size on eighth grade test scores in New York (the “primary outcome”). However, these eighth grade test scores are not available in the Project STAR data; instead, the experimental sample includes test scores only through the third grade (the “secondary outcome”). See Table 1 for the average outcomes in each sample by treatment status.
[lccccc]{} & Project STAR &\
& 3rd Grade Score & 3rd Grade Score & 8th Grade Score\
\
Mean Controls (regular class size) & $0.011$^^ & $0.157$^^ & $0.155$^^\
& ($0.015$^^)^^ & ($0.001$^^)^^ & ($0.001$^^)^^\
\
Mean Treated (small class size) & $0.193$^^ & $0.070$^^ & $-0.028$^^\
& ($0.025$^^)^^ & ($0.001$^^)^^ & ($0.002$^^)^^\
\
Difference & $0.181$^^ & $-0.087$^^ & $-0.183$^^\
& ($0.029$^^)^^ & ($0.002$^^)^^ & ($0.002$^^)^^\
\
We find that (and this holds even after adjusting for observed pre-treatment variables) the estimated effects of class size on the third grade scores observed in both samples are very different in the experimental and observational samples. For the experimental sample from Project STAR we see that there is a substantial positive effect of the small class size, an increase of 0.181 in 3rd grade scores. On the other hand, in the observational sample from New York we see a substantial negative relationship between the treatment and test scores in both third and eighth grade, -0.087 for 3rd grade scores and -0.183 for 8th grade scores. We can interpret this difference in the 3rd grade tests results, 0.181 in Project STAR [*versus*]{} -0.087 in New York, in two ways. One interpretation is that the difference (even after adjusting for pre-treatment variables) is due to differences between the two populations, so that it reflects lack of external validity of the Project STAR sample. A second interpretation is that it reflects lack of internal validity or non-random selection into the treatment in New York, in other words, the presence of unobserved confounders in the New York sample. In this paper we focus on the latter explanation, and maintain the assumption that the experimental dataset has both internal and external validity; that is, we assume that after adjusting for pre-treatment variables, the underlying populations in the experimental and observational datasets are comparable, even if treatments are assigned differently. Under this maintained assumption, the negative relationship between the treatment and 3rd grade outcomes in New York must be due to unobserved confounding, for example, sorting of students who are likely to test poorly into schools with low class sizes.
The main question we address in this paper is how we can adjust the 8th grade results for New York in Table 1 in the light of the experimental Project STAR 3rd grade results and the New York 3rd grade results, under the assumption that the difference in 3rd grade results is due to endogenous selection into the treatment or lack of internal validity in New York. Our approaches uses makes use of both the observed relationships between the 3rd grade and 8th grade outcomes in New York, which allow us to estimate counterfactual 8th grade outcomes as a function of 3rd grade outcomes, and the observed differences in the distributions of 3rd grade outcomes between New York and Project STAR, which allow us to infer counterfactual outcomes for New York students from alternative class sizes.
Formally, we consider a set up with two datasets, the experimental sample and the observational sample. For each unit in the observational dataset, we observe pre-treatment variables, a binary treatment assignment, the primary outcome, and a (vector-valued) secondary outcome. For each unit in the experimental dataset, we observe the same variables as in the observational dataset, except that we do not observe the primary outcome. Table \[tabel1\] illustrates this observational scheme. This observation scheme is also studied in @rosenman2018propensity [@rosenman2020combining; @kallus2020role]. @rosenman2018propensity focuses on the problem where assignment is unconfounded in both samples. @kallus2020role considers the case where assignment in the combined sample is unconfounded, but not in each of the samples separately. @rosenman2020combining allow for unobserved confounders in the observational sample and consider shrinkage estimators. @kallus2018removing focus on a different case where the same variables are observed in the two samples, but as in our set up, unconfoundedness does not hold in the observational sample. This set up in this paper differs from the surrogate set up in @athey2019surrogate where in the observational study the treatment indicator $W_i$ is not observed. Typically, the primary outcome is a long-term outcome such as eventual educational attainment, long-term wages, or mortality, while the secondary outcome may be a multi-dimensional vector of shorter-term outcomes that are associated with the long-term outcome. The object of interest is a low-dimensional estimand, for example, the average causal effect of the treatment on the primary outcome. The role of the secondary outcome and the pretreatment variables is to aid in the effort of credibly estimating the average causal effect on the primary outcome.
\[tabel1\]
<span style="font-variant:small-caps;">Table 1. Observation Scheme: ${\checkmark}$ is observed,</span> [?]{} <span style="font-variant:small-caps;">is missing</span>
\[tabel\_sampling\]0.3cm
---------------------------------------------------- ------------- ----------- --------- ----------- --------------
Primary Secondary Pretreatment
Sample Treatment Outcome Outcome Variables
Units $G_{i}$ $W_{i}$ $Y_{i}$ $S_{i}$ $X_{i}$
1 to $N_{{{\rm E}}}$ ${{\rm E}}$ [?]{}
$N_{{{\rm E}}}+1$ to $N_{{{\rm E}}}+N_{{{\rm O}}}$ ${{\rm O}}$
---------------------------------------------------- ------------- ----------- --------- ----------- --------------
Our approach makes use of three maintained assumptions. First, the sample of units in the observational dataset is representative of the population of interest. This assumption is essentially a definition. However, the treatment is not randomly assigned in the observational data. Second, the treatment in the experimental study was randomly assigned, ensuring that the experimental study has internal validity. We can easily generalize this to the case where the maintained assumption is that treatment assignment in the second sample is unconfounded given a set of pretreatment variables (@rosenbaum1983central [@imbens2015causal]). In our application this assumption is satisfied by design. However, because the primary outcome is not observed in this data set, we cannot estimate the average effect of interest on the experimental sample alone. Third, we assume that the pretreatment variables capture the differences between the populations that the observational and experimental sample were drawn from, so that conditional on these pretreatment variables, estimates of the treatment effect in the experimental sample have external validity (@shadishcookcampbell [@hotz2005predicting]). However, these three maintained assumptions are not sufficient for identification of the average effect of the treatment on the primary outcome.
Our first contribution to this problem is to formulate a novel assumption, which we call “latent unconfoundedness.” In combination with the maintained assumptions latent unconfoundedness allows for point-identification of the average causal effect of the treatment on the primary outcome in the observational study. The critical assumption is that the the unobserved confounders that affect both treatment assignment and the secondary outcome in the observational study are the same unobserved confounders that affect both treatment assignment and the primary outcome. Formally the assumption links (without the need for functional form assumptions) the biases in treatment-control differences in the secondary outcome (which can be estimated given the presence of the experimental data) to the biases in treatment-control comparisons in the primary outcome (which the experimental data are silent about) using a control function approach (@heckman1979sample [@heckman1985alternative; @imbens2009identification; @wooldridge2010econometric]) that also bears some similarity to the Changes-In-Changes approach in @athey2006identification. For a unit in the observational sample the control function is essentially the rank of the secondary outcome in the distribution of secondary outcomes in the experimental sample with the same treatment. The method makes use of the fact that under our maintained assumptions, systematic differences in the estimated effect of the treatment between the experimental and observational sample must be due to violations of unconfoundedness in the observational data. In our second contribution, we propose three different approaches to estimation of the average treatment effect under the maintained assumptions in combinatio with latent uconfoundedness. The three approaches consist of $(i)$ imputating the missing primary outcome in the experimental sample, $(ii)$ weighting of the units in the observational sample to remove biases, and $(iii)$ control function methods. Our analyses show how the presence of the experimental data can be systematically exploited to relax the assumption of unconfoundedness that is common in observational studies. In our third contribution we apply the new methods to obtain estimates of the effect of small class sizes on 8th grade test scores in New York. The combination of the New York data with the experimental Project STAR data leads to a positive estimates of the effect of small classes, whereas an analysis using only the New York data and assuming unconfoundedness leads to implausible negative estimates.
Two Examples {#two_examples}
============
To lay out the conceptual issues at the heart of the current paper we considerin this section two simple examples in some detail. These two examples allow us to introduce the identifying assumptions and estimation strategies that are the main contribution of this paper.
Set Up
------
The basic set up is the same in both examples. Using the potential outcome set up developed for observational studies by @rubin1974estimating (see @imbens2015causal for a textbook discussion), let the pair of potential outcomes for this outcome for unit $i$ be denoted by $Y_{i}^{{\rm P}}(0)$ and $Y_{i}^{{\rm P}}(1)$, where the superscript “${{\rm P}}$” stands for “Primary”. In many applications this is a long term outcome. The treatment received by unit $i$ will be denoted by $W_{i}\in\{0,1\}$. There is also a secondary outcome, possibly a short term outcome, with the pair of potential outcomes for unit $i$ denoted by $Y_{i}^{{\rm S}}(0)$ and $Y_{i}^{{\rm S}}(1)$, where the superscript “${{\rm S}}$” stands for “Secondary”. In the two examples both the primary and secondary outcomes are scalars, but in applications the secondary outcome is likely to be vector-valued. The realized values for the primary and secondary outcomes are $Y^{{\rm P}}_i\equiv Y_i^{{\rm P}}(W_i)$ and $Y^{{\rm S}}_i\equiv Y_i^{{\rm S}}(W_i)$. We are interested in the average treatment effect on the primary outcome, $$\label{taup} \tau^{{\rm P}}\equiv{\mathbb{E}}\left[ Y^{{\rm P}}_i(1)-Y^{{\rm P}}_i(0)\right],$$ although other estimands such as the average effect on the treated can be accomodated in this set up. The average effect on the secondary outcome, $ \tau^{{\rm S}}\equiv {\mathbb{E}}\left[ Y^{{\rm S}}_i(1)-Y^{{\rm S}}_i(0)\right],$ is for the purpose of the current study not of intrinsic interest.
We have two samples to draw on for estimation of $\tau^{{\rm P}}$. In that sense the set up connects to the literature on combining data sets, [*e.g.,*]{} @hotz2005predicting [@pearl2014external; @ridder2007econometrics]. The first sample is from an observational study. It is a random sample from the population of interest. The concern is that the assignment mechanism may be confounded. For all units in this observational sample we observe the triple $(W_i,Y^{{\rm S}}_i,Y^{{\rm P}}_i)$, The second sample is a possibly selective sample from the same population, with the assignment completely random. For all units in this experimental sample we observe the pair $(W_i,Y^{{\rm S}}_i)$, but not the primary outcome. The motivation for considering this setting is that it is often expensive to conduct randomized experiments, and it may not be feasible to observe the primary outcome in the experiment.
Let $G_i\in\{{{\rm E}},{{\rm O}}\}$, be the indicator for the subpopulation or group a unit is drawn from. Then we can think of the combined sample as a random sample of size $N$ from an artificial super-population for which we observe the quadruple $(W_i,G_i,Y^{{\rm S}}_i,Y^{{\rm P}}_i{{\boldsymbol}{1}}_{G_i={{\rm O}}})$, where ${{\boldsymbol}{1}}_{G_i={{\rm O}}}$ is a binary indicator, equal to 1 if $G_i={{\rm O}}$ and equal to 0 if $G_i={{\rm E}}$.
A Binary Outcome Example
------------------------
For the purpose of the first example in this section, we assume both the secondary and primary outcome are binary, $Y^{{\rm P}}_i(w),Y^{{\rm S}}_i(w)\in\{0,1\}$ for $w\in\{0,1\}$. For expositional reasons we also assume in this section that there are no pretreatment variables.
Define for all outcome types $t\in\{{{\rm S}},{{\rm P}}\}$, all groups $g\in\{{{\rm E}},{{\rm O}}\}$, and all treatment levels $w\in\{0,1\}$ the sample averages and sample sizes $${\overline{Y}}^{t,g}_w\equiv \frac{1}{N^g_w}\sum_{i=1}^N Y^t_i {{\boldsymbol}{1}}_{G_i=g,W_i=w},
\hskip1cm {\rm and}\ \
N^g_w\equiv \sum_{i=1}^N {{\boldsymbol}{1}}_{G_i=g,W_i=w}.$$ Assuming that $N^{{\rm O}}_0$, $N^{{\rm O}}_1$, $N^{{\rm E}}_0$, and $N^{{\rm E}}_1$ are all positive, six of the eight average outcomes ${\overline{Y}}^{t,g}_w$ are well-defined and can be calculated from the data, ${\overline{Y}}^{{{\rm P}},{{\rm O}}}_0$, ${\overline{Y}}^{{{\rm P}},{{\rm O}}}_1$, ${\overline{Y}}^{{{\rm S}},{{\rm O}}}_0$, ${\overline{Y}}^{{{\rm S}},{{\rm O}}}_1$, ${\overline{Y}}^{{{\rm S}},{{\rm E}}}_0$, and ${\overline{Y}}^{{{\rm S}},{{\rm E}}}_1$. The remaining two, ${\overline{Y}}^{{{\rm P}},{{\rm E}}}_0$ and ${\overline{Y}}^{{{\rm P}},{{\rm E}}}_1$ are not well-defined because we do not observe the primary outcome in the experimental sample.
### Using the Two Samples Separately
Let us first consider estimation of $\tau^{{\rm P}}$ and $\tau^{{\rm S}}$ using one sample at a time. Using only the experimental sample there is no way to estimate the average treatment effect on the primary outcome, because this sample does not contain any information on the primary outcome. We can estimate the average effect on the secondary outcome, using the experimental sample, as $$\hat\tau^{{{\rm S}},{{\rm E}}}={\overline{Y}}^{{{\rm S}},{{\rm E}}}_1-{\overline{Y}}^{{{\rm S}},{{\rm E}}}_0.$$ This estimator $\hat\tau^{{{\rm S}},{{\rm E}}}$ would be unbiased for $\tau^{{\rm S}}$ if the experimental sample had external validity and could be considered a random sample from the population of interest (formally, if $G_i{\perp\!\!\!\perp}(Y_i(0),Y_i(1))$, what @hotz2005predicting call location unconfoundedness), but would not necessarily be so otherwise.
Using only the observational sample the natural estimator for the average causal effect on the primary and secondary outcomes would be $$\hat\tau^{{{\rm P}},{{\rm O}}}={\overline{Y}}^{{{\rm P}},{{\rm O}}}_1-{\overline{Y}}^{{{\rm P}},{{\rm O}}}_0,\hskip1cm {\rm and}\ \
\hat\tau^{{{\rm S}},{{\rm O}}}={\overline{Y}}^{{{\rm S}},{{\rm O}}}_1-{\overline{Y}}^{{{\rm S}},{{\rm O}}}_0
,$$ respectively. For these estimators to be consistent for the average causal effect of the treatment on the primary and secondary outcomes we would need something like unconfoundedness, which, in the absence of pretreatment variables, corresponds to: $$W_i\ {\perp\!\!\!\perp}\ \Bigl(Y^{{\rm S}}_i(0),Y^{{\rm S}}_i(0),Y^{{\rm P}}_i(0),Y^{{\rm P}}_i(0)\Bigr)\ \Bigr|\ G_i={{\rm O}}.$$ With only the observational sample, there is not much in terms of alternatives for obtaining point estimates of the average treatment effect on the primary and secondary outcomes.
### Combining the Two Samples
Now consider estimation of $\tau^{{\rm S}}$ in the presence of both experimental and observational samples. In this case we have two distinct estimators for $\tau^{{\rm S}}$, namely $\hat\tau^{{{\rm S}},{{\rm E}}}$ and $\hat\tau^{{{\rm S}},{{\rm O}}}$. If we find no difference between $\hat\tau^{{{\rm S}},{{\rm E}}}$ and $\hat\tau^{{{\rm S}},{{\rm O}}}$, or at least no statistically significant difference, then both would appear to be reasonable estimates. We might improve the precision of either estimator by combining them efficiently ([*e.g.,*]{} @rosenman2018propensity [@rosenman2020combining; @kallus2020role]). However, if we find a substantial and statistically significant difference between the two, we can infer that either the assignment in the observational sample is not random (unconfoundedness does not hold), or the experimental sample is not a random sample from the population of interest (no external validity). In that case there may still be efficiency gains in combining the data, as discussed in @rosenman2020combining. However, in large samples the two estimators will be converging to different limits. There is no information in the data to determine whether unconfoundedness in the observational sample, or external validity in the experimental sample, is violated. Note that these assumptions are of a very different nature. The researcher has to use [*a priori*]{} arguments to choose between $\hat\tau^{{{\rm S}},{{\rm E}}}$ and $\hat\tau^{{{\rm S}},{{\rm O}}}$ and the corresponding assumptions. Choosing for $\hat\tau^{{{\rm S}},{{\rm E}}}$ would imply being less concerned with the external validity of the experimental sample, wherease the choice for $\hat\tau^{{{\rm S}},{{\rm O}}}$ would imply that the internal validity of the observational study would be viewed as less of a concern. In many cases researchers have argued for the primacy of internal validity over external validity ([*e.g.*]{}, @shadishcookcampbell [@imbens2010]), though others have argued against that perspective ([*e.g.*]{}, @manski2013public [@deaton2010]). If one prefers $\hat\tau^{{{\rm S}},{{\rm O}}}$ (downplaying the concerns about internal validity of the observational sample), the natural estimator for $\tau^{{\rm P}}$ is $\hat\tau^{{{\rm P}},{{\rm O}}}$, and there is little use for the experimental sample. However, if the researcher prefers $\hat\tau^{{{\rm S}},{{\rm E}}}$, the question arises how to estimate $\tau^{{\rm P}}$.
The current paper is concerned with this question. We take the position that there is an [*a priori*]{} preference for $\hat\tau^{{{\rm S}},{{\rm E}}}$ over $\hat\tau^{{{\rm S}},{{\rm O}}}$, possibly after accounting for differences in covariate distributions to deal with some of the external validity concerns (@hotz2005predicting). Then we address the main question of how to adjust the estimator $\hat\tau^{{{\rm P}},{{\rm O}}}$ to take into account the difference between $\hat\tau^{{{\rm S}},{{\rm O}}}$ and $\hat\tau^{{{\rm S}},{{\rm E}}}$, Conceptually there are multiple natural ways of doing so. We discuss three of these ways. The first one is based on imputation of the missing primary outcomes in the experimental sample. The second one is based on weighting the units in the observational sample. The third one is based on a control function approach. In this simple nonparametric case with binary outcomes the three approaches lead to identical point estimates .
### Imputation
The first approach is to take a missing data perspective on the primary $Y_i^{{\rm P}}$ in the experimental sample and impute these missing values using the observational sample. Consider unit $i$ in the experimental sample with $W_i=w$ and $Y^{{\rm S}}_i=y^{{\rm S}}$. Assuming the missing data on $Y^{{\rm P}}_i$ are missing at random ([*e.g.,*]{} @rubin1976inference [@little2019statistical; @rubin2004multiple]) suggests using the distribution of $Y^{{\rm P}}_i$ among units in the observational sample with $W_i=w$ and $Y^{{\rm S}}_i=y^{{\rm S}}$ to impute the missing values. If we are interested in estimating the average effect, we can just use the average value of $Y^{{\rm P}}_i$ in this subsample as the imputed value. Denote this average value for all values of $y^{{\rm S}}$ and $w$ by: $${\overline{Y}}^{{{\rm P}},{{\rm O}}}_{w,y^{{\rm S}}}=
\sum_{i=1}^N {{\boldsymbol}{1}}_{G_i={{\rm O}}, W_i=w,Y^{{\rm S}}_i=y^{{\rm S}}}Y^{{\rm P}}_i
\Bigl/\sum_{i=1}^N {{\boldsymbol}{1}}_{G_i={{\rm O}}, W_i=w,Y^{{\rm S}}_i=y^{{\rm S}}}.$$ The imputed value for $Y^{{\rm P}}_i$ for unit $i$ in the experimental sample is then the average of $Y^{{\rm P}}_j$ in the observational sample over all units $j$ with the same treatment level, $W_j=W_i$, and the same value for the secondary outcome, $Y^{{\rm S}}_j=Y^{{\rm S}}_i$: $$\hat Y^{{\rm P}}_i={\overline{Y}}^{{{\rm P}},{{\rm O}}}_{W_i,Y^{{\rm S}}_i}.$$ Then the imputation estimator for $\tau^{{\rm P}}$ is the difference in average imputed values in the experimental sample by treatment status, leading to the first estimator for $\tau^{{\rm P}}$: $$\label{imp}\hat\tau^{{{\rm P}},{{\rm imp}}}\equiv \frac{1}{N^{{\rm E}}_1}
\sum_{i:G_i={{\rm E}}} W_i {\overline{Y}}^{{{\rm P}},{{\rm O}}}_{W_i,Y^{{\rm S}}_i}
-\frac{1}{N^{{\rm E}}_0}\sum_{i:G_i={{\rm E}}} (1-W_i) {\overline{Y}}^{{{\rm P}},{{\rm O}}}_{W_i,Y^{{\rm S}}_i}
.$$
### Weighting
The second approach to using the experimental secondary outcomes is to reweight the observational sample where we do observe the primary outcome. Consider the $N^{{\rm O}}_w$ units in the observational sample with $W_i=w$. The fraction of those treated units in the observational study with secondary outcome $Y^{{\rm S}}_i=1$ is $\hat p^{{{\rm S}},{{\rm O}}}_w=\sum_{i:G_i={{\rm O}},W_i=w} Y^{{\rm S}}_i/\sum_{i:G_i={{\rm O}},W_i=w} 1$. The experimental study tells us this fraction would have been approximately $\hat p^{{{\rm S}},{{\rm E}}}_w=\sum_{i:G_i={{\rm E}},W_i=w} Y^{{\rm S}}_i/\sum_{i:G_i={{\rm E}},W_i=w} 1$, had the treatment been randomly assigned. The comparison $\hat p^{{{\rm S}},{{\rm O}}}_w$ versus $\hat p^{{{\rm S}},{{\rm E}}}_w$ reflects on the possible violation of the unconfoundedness assumption in the observational sample. We can give these units a weight $\lambda_{w,1}=\hat p^{{{\rm S}},{{\rm E}}}_w/\hat p^{{{\rm S}},{{\rm O}}}_w$ to adjust for the bias stemming from such violations. Similarly, units with treatment $W_i=w$ and $Y^{{\rm S}}_i=0$ would be given a weight $\lambda_{w,0}=(1-\hat p^{{{\rm S}},{{\rm E}}}_w)/(1-\hat p^{{{\rm S}},{{\rm O}}}_w)$. We then use these to estimate the average effect on the primary outcome as the difference of weighted averages of the treated and control outcomes in the observational sample, leading to the second estimator: $$\hat\tau^{{{\rm P}},{{\rm weight}}}\equiv
\frac{\sum_{G_i={{\rm O}}} \lambda_{1,Y^{{\rm S}}_i} W_i Y^{{\rm P}}_i}{\sum_{G_i={{\rm O}}} \lambda_{1,Y^{{\rm S}}_i}W_i }
-\frac{\sum_{G_i={{\rm O}}} \lambda_{0,Y^{{\rm S}}_i}(1-W_i) Y^{{\rm P}}_i}{\sum_{G_i={{\rm O}}} \lambda_{0,Y^{{\rm S}}_i} (1-W_i)}
.$$ Simple algebra shows that the two estimators are identical, $\hat\tau^{{{\rm P}},{{\rm imp}}}=\hat\tau^{{{\rm P}},{{\rm weight}}}$. This algebraic result relies on the outcome model and the weights being fully nonparametric in this simple example with the secondary outcome taking on only two values. In settings with the secondary outcome continuous, the two approaches will generally give different answers in finite samples.
A Control Function Approach in a Linear Model Setting
-----------------------------------------------------
In our second example we consider a simple linear model that exhibits most clearly some of the key features of the approach in the current paper. Suppose we have a linear model for the secondary potential outcomes with a constant treatment effect: $$Y^{{{\rm S}}}_i(0)=X_i^\top\gamma^{{\rm S}}+\alpha_i^{{\rm S}},\hskip1cm Y^{{\rm S}}_i(1)=Y^{{{\rm S}}}_i(0)+\tau^{{\rm S}}.$$ This models holds for both the experimental and observational samples. The properties of the unobserved component $\alpha^{{\rm S}}_i$ are key, and they may differ in the two samples. In the experimental sample the randomization guarantees that we have the following conditional independence: $$W_i\ \perp\!\!\!\perp\ \alpha_i^{{\rm S}}\ \Bigl|\ X_i,G_i={{\rm E}}.$$ In fact the randomization implies even stronger conditions, but we do not need those here. In the observational study we do not in general have the same conditional independence: $$W_i\ \not\!\perp\!\!\!\perp\ \alpha_i^{{\rm S}}\ \Bigl|\ X_i,G_i={{\rm O}}.$$ This randomization in the experimental sample implies that we can estimate the parameters of the model for the secondary outcome, $\tau^{{\rm S}}$ and $\gamma^{{\rm S}}$, by least squares regression of $Y^{{\rm S}}_i$ on $W_i$ and $X_i$ using only the data from the experimental sample. In other words, the conditional mean of $Y^{{\rm S}}_i$ given $W_i$ and $X_i$ has a causal interpretation as a function of $W_i$ in the experimental sample, but not in the observational sample.
Now consider the primary outcome. We specify a similar linear model for the primary outcome, but allow the coefficients to be different from those of the model for the secondary outcome, $$Y^{{{\rm P}}}_i(0)=X_i^\top\gamma^{{\rm P}}+\alpha_i^{{\rm P}},\hskip1cm Y^{{{\rm P}}}_i(1)= Y^{{{\rm P}}}_i(0)+\tau^{{\rm P}}.$$ Again the concern is that in the observational sample the unobserved component might be correlated with the treatment: $$W_i\ \not\!\perp\!\!\!\perp\ \alpha_i^{{\rm P}}\ \Bigl|\ X_i,G_i={{\rm O}}.$$ Such a correlation would imply that a linear regression of $Y^{{\rm P}}_i$ on $W_i$ and $X_i$ using the data from the observational sample would not be consistent for the causal effect $\tau^{{\rm P}}$ because of endogeneity of $W_i$. Now a key assumption is that there is a relationship between the short term and long term unobserved components $\alpha^{{\rm P}}_i$ and $\alpha^{{\rm P}}_i$ that allows us to remove the endogeneity bias in the long term relationship using the difference between the short term results for the experimental and observational data using a control function approach (@heckman1985alternative [@imbens2009identification; @kline2019heckits]). The key assumption that links the endogeneity problems for the primary and secondary outcomes is $$\label{cf_linear}\alpha^{{\rm P}}_i=\delta\alpha^{{\rm S}}_i+\varepsilon_i^{{\rm P}},
\hskip1cm {\rm with}\ \
W_i\ \perp\!\!\!\perp\ \varepsilon_i^{{\rm P}}\ \Bigl|\ X_i,\alpha^{{\rm S}}_i,G_i={{\rm O}}.$$ Later we relax this assumption to remove the functional form dependence, but for the moment let us focus on this version with linearity and additivity. The key is that the residual for the primary outcome, $\alpha^{{\rm P}}_i$, is related to the residual for the secondary outcome, $\alpha^{{\rm S}}_i$, with the remainder, $\alpha_i^{{\rm P}}-{\mathbb{E}}[\alpha^{{\rm P}}_i|\alpha^{{\rm S}}_i]$ unrelated to the treatment.
Let us show in some detail how this assumptions aids in the identification of $\tau^{{\rm P}}$ in this linear example. First, we can estimate $\tau^{{\rm S}}$ and $\gamma^{{\rm S}}$ from the experimental sample by linear regression. Denote these least squares estimates by $\hat\tau^{{{\rm S}}}$ and $\hat\gamma^{{{\rm S}}}$. Then we can estimate the residual $\alpha_i^{{\rm S}}$ for the units in the observational sample as $$\label{alpha} \hat \alpha^{{\rm S}}_i=Y^{{\rm S}}_i-W_i\hat\tau^{{{\rm S}}}-X_i^\top\hat\gamma^{{{\rm S}}}.$$ If this model is correct, and if the assignment to treatment in the observational sample were random, and finally, if the observational and experimental samples were randomly drawn from the same population, the population value of these residuals $\alpha^{{\rm S}}_i$ would have mean zero and be uncorrelated with the treatment indicator in the observational sample. The presence of non-zero association of this residuals and the treatment is exploited to adjust the estimates of the treatment effect on the primary outcome. We can do so by including this residual as a control variable in the least squares regression with the long term outcome as the dependent variable, using the observational data. The key insight is that we can use the linear representation in (\[cf\_linear\]) to write the long te outcome as: $$\label{control} Y^{{{\rm P}}}_i=W_i\tau+X_i^\top\gamma+\delta\alpha^{{\rm S}}_i+\varepsilon^{{\rm P}}_i,\hskip1cm
{\rm with}\ \
W_i \perp\!\!\!\perp\ \varepsilon_i^{{\rm P}}\ \Bigl|\ X_i,\alpha^{{\rm S}}_i,G_i={{\rm O}}.$$ Therefore this regression will lead to a consistent estimator for $\tau^{{\rm P}}$ under the current assumptions.
To further develop intuition for the control function approach, consider the example where the primary and secondary outcomes are eight and third grade test scores, and the treatment is class size. Using the experimental sample we estimate the the average effect of the class size on third grade scores. We then calculate the residuals in the observational study. We may find that the residuals are larger on average for the treated individuals than for the control individuals. This suggests that the treatment assignment in the observational sample was correlated with the third grade potential outcomes, with individuals with high values for the potential outcomes more likely to be in the treatment group. We then use that information to compare eighth grade scores for individuals with the same residuals, so we adjust for the original non-random selection into treatment.
The Connection Between the Imputation and Control Function Approaches
---------------------------------------------------------------------
The control function approach to dealing with the endogeneity of the treatment in the observational study we used in the second example may appear at first sight to be conceptually quite different from the weighting and imputation approaches in the first example. In fact the two approaches are closely related. Consider the imputation of $Y^{{\rm P}}_i$ for a unit in the experimental sample given the linear model. Substituting for $\alpha^{{\rm S}}_i$ using Equation (\[alpha\]) into Equation (\[control\]) implies that we can write for the primary outcome in the observational sample: $$Y^{{{\rm P}}}_i=W_i\beta+X_i^\top\lambda+\delta Y^{{\rm S}}_i+\varepsilon_i,$$ where $$\beta=\tau^{{\rm P}}-\delta\tau^{{\rm S}},\hskip1cm{\rm and}\ \ \lambda=\gamma^{{\rm P}}-\delta\gamma^{{\rm S}}.$$ Hence the imputed value for $Y^{{\rm P}}_i$ in the experimental sample using the estimated parameters from the observational sample leads to (ignoring estimating error) $$\hat Y^{{{\rm P}}}_i=W_i\beta+X_i^\top\lambda+\delta Y^{{\rm S}}_i.$$ Using the omitted variable bias formula it is easy to see that regressing this imputed value $\hat Y^{{\rm P}}_i$ on $W_i$ and $X_i$ (but omitting $Y^{{\rm S}}_i$), in the experimental sample, leads to $$\hat Y^{{\rm P}}_i=W_i\bar \beta+X_i^\top\bar \lambda+\varepsilon_i,\hskip1cm {\rm
with}\ \
\bar\beta=\beta+\delta\beta^{{\rm S}}=\tau^{{\rm P}}.$$ Thus, the coefficient on $W_i$ in this regression of the imputed primary outcome on the treatment and the pretreatment variables is consistent for the causal effect of the treatment on the primary outcome.
The General Case
================
The two examples in the preceeding section convey much of the intuition for our approach:. The key assumption in the linear case is (\[cf\_linear\]), which connects the bias in causal estimates for the primary and secondary outcomesin the observational sample. Making such assumptions allows us improve upon estimates for $\tau^{{\rm P}}$ based on the observational sample alone. What we do in this section is generalize the first example to the case where $(i)$ the secondary and primary outcomes may be continuous and $(ii)$ the secondary outcome may be vector-valued, and $(iii)$ where pre-treatment variables are present. We also generalize the control function approach to $(i)$ the nonlinear case, and $(ii)$ the case with multiple secondary outcomes in order to allow the critical assumptions to be weakened. We present the formal assumptions that justify the weighting and imputation estimators, and present their general forms and how they relate to the control function approach.
The Set Up
----------
We are interested in causal estimands defined for the population of interest. At a general level such estimands include simple average treatment effects, but more generally also the average effect of a policy that assigns the treatment to individuals in this population on the basis of covariates ([*e.g.*]{}, @manski2004statistical [@dehejia2005program; @hirano2009asymptotics; @athey2017efficient; @zhou2018offline]).
Define $$\label{tau_general}\tau_{g}^{t}\equiv {\mathbb{E}}\left[\left.Y_{i}^{t}(1)-Y_{i}^{t}(0)\right|G_{i}=g\right],$$ is the average effect of the treatment on outcome $t\in\{{{\rm S}},{{\rm P}}\}$ for group $g\in\{{{\rm O}},{{\rm E}}\}$. The superscripts on the estimands denote the outcome, and subscripts denote the population. The primary estimand we focus on in this paper is the average effect of the treatment on the long term outcome in the observational study population: $$\label{tau}\tau\equiv \tau_{{{\rm O}}}^{{{\rm P}}}\equiv {\mathbb{E}}\left[\left.Y_{i}^{{{\rm P}}}(1)-Y_{i}^{{{\rm P}}}(0)\right|G_{i}={{\rm O}}\right],$$ where we drop the subscript and superscript to simplify the notation.
It will be useful to have notation for the following three conditional expectations that differ in their conditioning sets: $$\label{eq:mu} \mu(t,w,x,g)={\mathbb{E}}\left[ \left.Y_i^t(w)\right| W_i=w,X_i=x,G_i=g\right],$$ $$\psi(t,w,x,g)\equiv{\mathbb{E}}\left[\left.Y^t_i(w)\right|X_i=x,G_i=g\right],$$ and $$\label{eq:kappa}\kappa(w,x,g,y^{{\rm S}})={\mathbb{E}}\left[ \left.Y_i^{{\rm P}}(w)\right| W_i=w,X_i=x,Y^{{\rm S}}_i=y^{{\rm S}},G_i=g\right],$$ for $t\in\{{{\rm S}},{{\rm P}}\}$, $w\in\{0,1\}$, $x\in{\mathbb{X}}$, $g\in\{{{\rm O}},{{\rm E}}\}$, and $y^{{\rm S}}\in{\mathbb{Y}}^{{\rm S}}$. For the control function approach we also need the following cumulative distribution function: $$\eta(y^{{\rm S}},w,x)\equiv F_{Y^{{\rm S}}|W,X,G}(y^{{\rm S}}|w,x,{{\rm E}})\equiv {\rm Pr}\left(\left.Y_i^{{\rm S}}\leq y^{{\rm S}}\right|W_i=w,X_i=x,G_i={{\rm E}}\right).$$
Some of these expectations, and the cumulative distribution function are identified, given some regularity conditions, for some groups and outcomes from the joint distribution of the quintuple $(W_i,G_i,X_i,Y^{{\rm S}}_i,Y^{{\rm P}}_i{\bf 1}_{G_i={{\rm O}}})$. We state the following lemma without proof.
\[lemma\_kappa\][(Conditional Expectations)]{}\
$(i)$ $\mu({{\rm S}},w,x,g)$ is identified for all $w\in\{0,1\}$, $x$, and $g\in\{{{\rm E}},{{\rm O}}\}$, and $\mu({{\rm P}},w,x,{{\rm O}})$ is identified for all $w\in\{0,1\}$, $x$,\
$(ii)$, $\kappa(w,x,{{\rm O}},y^{{\rm S}})$ is identified for all $w\in\{0,1\}$, $x$, and $y^{{\rm S}}\in{\mathbb{Y}}^{{\rm S}}$,\
$(iii)$ $\eta(y^{{\rm S}},w,x)$ is identified for $y^{{\rm S}}\in{\mathbb{Y}}^{{\rm S}}$, $w\in\{0,1\}$, and $x\in{\mathbb{X}}$.
Three Maintained Assumptions
----------------------------
There are two key features of our set up. First, we are interested in the population that the units in the observational study were drawn from. That is, the observational study has external validity.
[(External Validity of the Observational Study)]{} The observational sample is a random sample of the population of interest. \[assumption:external\_observational\]
At some level this can be thought of as simply defining the estimand in terms of the population distribution underlying the observational sample. Second, we maintain throughout the paper the assumption that the treatment in the experimental sample is unconfounded.
\[assumption:random\][(Internal Validity of the Experimental Sample)]{} For $w=0,1$, $$\label{random} W_i\ \perp\!\!\!\perp\ \Bigl(Y_{i}^{{{\rm P}}}(w),Y_{i}^{{{\rm S}}}(w)\Bigr)\ \Bigr| \ X_i,G_i={{\rm E}}.$$ \[assumption:internal\_experimental\]
@kallus2020role make a different assumption here, $$\label{kallus}W_i\ \perp\!\!\!\perp\ \Bigl(Y_{i}^{{{\rm P}}}(w),Y_{i}^{{{\rm S}}}(w)\Bigr)\ \Bigr| \ X_i,$$ where unconfoundedness holds in the combined sample, rather than in the experimental sample. Assumption (\[kallus\]) does not imply our assumption (\[random\]), or the other way around. In our application, with assignment in the Project STAR experimental sample completely randomized, our assumption is satisfied by design, and in general (\[kallus\]) would not hold. In other settings, for example where the data are sampled from a single population rather than two separate populations, (\[kallus\]) may be more appropriate than our assumption.
However, the external validity of the experimental study is not guaranteed. Instead we assume that conditional on the pretreatment variables we have external validity (@hotz2005predicting):
\[assumption:conditional\][(Conditional External Validity)]{} The experimental study has conditional external validity if $$\label{eq:cond_ext} G_i\ \perp\!\!\!\perp\ \Bigl(Y_{i}^{{{\rm P}}}(0),Y_{i}^{{{\rm P}}}(1),
Y_{i}^{{{\rm S}}}(0),Y_{i}^{{{\rm S}}}(1)\Bigr)\ \Bigr|\ X_i.$$
This assumption implies that if we find systematic differences between in differences in average outcomes by treatment status conditional on covariates between the experimental and observational sample, these differences must arise from violations of unconfoundedness for the observational sample.
A direct implication of the conditional external validity assumption is that after adjusting for differences in pretreatment variables between the experimental and observational sample, the average effect on the primary outcome in the experimental sample is what we are interested in. Of course that does not help us directly, because we do not see the primary outcome in the experimental sample.
The first result is that these three maintained assumptions are in general not sufficient for point-identification of the average effect of interest. Of course this does not mean that these assumptions do not have any identifying power. They do in fact affect the identified sets in the spirit of the work by (@manski_bounds).
\[lemma1\] The combination of Assumptions \[assumption:external\_observational\]-\[assumption:conditional\] is not sufficient for point-identification of $\tau^{{\rm P}}$.
The proof for this result is given in the appendix.
Unconfoundedness for the Observational Sample
---------------------------------------------
Next, let us consider the assumption that assignment in the observational study is unconfounded.
\[assumption:unconfoundedness\] [(Unconfoundedness in the Observational Sample)]{}\
For $w=0,1$, $$W_{i}\ \perp\!\!\!\perp\ \Bigl(Y_{i}^{{{\rm S}}}(w),Y_{i}^{{{\rm P}}}(w)\Bigr)\ \Bigl|\ X_{i},G_i={{\rm O}},$$
This assumption is made, for example, in @rosenman2018propensity. This assumption is sufficient for identification of $\tau$, but it is stronger than necessary. Intuitively it implies that we do not need the experimental sample for identification because under unconfoundedness the observational sample is sufficient for identification of the average treatment effect. However, the experimental sample may still be useful for precision. The precise version of the unconfoundedness assumption here is slightly different from that in, say, @rosenbaum1983central where it is assumed that $W_i$ is independent of the full set of $(Y_i^{{\rm P}}(w),Y^{{\rm S}}_i(w))_{w\in\{0,1\}}$. It is what is referred to in @imbens2000 as “weak unconfoundedness.” This issue will come up later.
\[lemma2\] The set of Assumptions \[assumption:random\]-\[assumption:unconfoundedness\] has a testable implication: $$\label{eq:unconf2} G_i\ \perp\!\!\!\perp\ Y^{{\rm S}}_i\ \Bigl|\ X_i,W_i.$$
We can also use this result to assess whether a particular set of pre-treatment variables is sufficient for unconfoundedness. Here we are interested in finding a set of pretreatment variables $X_i$ such that $$G_i\ \perp Y^{{\rm S}}_i\ \Bigl|\ X_i=x,W_i,$$ holds.
Latent Unconfoundedness
-----------------------
Suppose that we reject the conditional independence in Lemma \[lemma2\], so that we know that the full set of maintained assumptions, \[assumption:random\]-\[assumption:unconfoundedness\], does not hold. If we maintain unconfoundedness in the experimental sample, it must be that either conditional external validity in the experimental study, or unconfoundedness in the observational study must be violated. If we interpret such a finding as evidence against conditional external validity, and are willing to maintain unconfoundedness of the treatment assignment in the observational study, we should simply put aside the experimental data set and focus on estimates based on solely on the observational study. In many cases, however, we may wish to maintain conditional external validity and interpret a finding that (\[eq:unconf2\]) does not hold as evidence that unconfoundedness does not hold for the observational study. Here we explore methods for using the difference between the estimates of the causal effects for the experimental study (which we know to be internally valid) and the estimates for the observational study (which need not be internally valid) to adjust long term estimates for the observational study.
The idea, although not the implementation, is somewhat similar to that in a Difference-In-Differences (@cardmariel [@cardkrueger1; @angrist2008mostly]) set up where the initial (pre-treatment) differences between a treatment and control group are used to adjust post-treatment differences between the treatment and control group. More specifically, it relates to the Changes-In-Changes approach in @athey2006identification where functional form assumptions are avoided. Here initial differences in treatment effects between an experimental and observational study are used to adjust subsequent treatment effects for the observational study.
They key additional assumption that links the biases, in the observational study, between adjusted comparisons for the primary and secondary outcomes, is the following.
\[assumption:new\][(Latent Unconfoundedness)]{}\
For $w\in\{0,1\}$, $$\label{new}
W_i\ \perp\!\!\!\perp\ Y_{i}^{{{\rm P}}}(w)\ \Bigl|\ X_{i},Y^{{\rm S}}_i(w),G_i={{\rm O}}.$$
This assumption is both novel as well as critical in the current discussion, so let us offer some remarks.
Compared to a regular unconfoundedness assumption, we add the variable $Y^{{\rm S}}_i(w)$ to the conditioning set. At first this may appear to be an innocuous addition. However, following the standard approach to exploiting unconfoundedness assumptions, we see that this is not the case. Typically we use such an assumption to create subpopulations defined by the conditioning variables, and then compare treated and control units. To be specific, suppose we wish to estimate ${\mathbb{E}}[Y^{{\rm P}}_i(1)|G_i={{\rm O}}]$. We would first estimate the conditional expectation ${\mathbb{E}}[Y^{{\rm P}}_i(1)|Y_i^{{\rm S}}(1)=y^{{\rm S}},W_i=1,X_i=x,G_i={{\rm O}}]$. Then, however, we would need to average this over the marginal distribution of $(Y_i^{{\rm S}}(1),X_i)$ in the observational sample, but in this observational sample we only see draws from the conditional distribution of $(Y_i^{{\rm S}}(1),X_i)$ given $W_i=1$, and this is not the same distribution because of the failure of unconfoundedness in the observational sample. To address this, we need to exploit the presence of the experimental sample.
To highlight the link to the control function literature (@heckman1979sample [@heckman1985alternative; @imbens2009identification; @wooldridge2010econometric; @athey2006identification; @kline2019heckits; @mogstad2018using; @mogstad2018identification; @wooldridge2015control]), let us model the primary and secondary outcomes as $$Y^{{\rm P}}_i(w)=h^{{\rm P}}(w,\nu_i,X_i),\hskip1cm {\rm and}\ \ Y^{{\rm S}}_i(w)=h^{{\rm S}}(w,\eta_i,X_i)
,$$ with $h^{{\rm S}}(w,\eta,x)$ strictly monotone in $\eta$. Now we can write the latent unconfoundedness assumption as $$W_i\ \perp\!\!\!\perp\ \nu_i\ \Bigl|\ X_{i},\eta_i,G_i={{\rm O}}.$$ Although it is not generally true that $W_i \perp\!\!\!\perp \nu_i| X_{i},G_i={{\rm O}}$, adding $\eta_i$ to the conditioning set restores the exogeneity of $W_i$ in the observational sample.
It is useful to contrast this with a control function in a non-parametric instrumental variables setting ([*e.g.,*]{} @imbens2009identification), where the two models are $$Y^{{\rm P}}_i(w)=h^{{\rm P}}(w,\nu_i,X_i),\hskip1cm {\rm and}\ \ W_i(z)=r(z,\eta_i,X_i)
,$$ with $r(z,\eta,x)$ strictly monotone in $\eta$. The key assumption here is that $$W_i\ \perp\!\!\!\perp\ \nu_i\ \Bigl|\ X_{i},\eta_i.$$ The model relating the outcome of interest and the endogenous regressor is essentially the same in the two settings, $Y^{{\rm P}}_i(w)=h^{{\rm P}}(w,\nu_i,X_i)$. In both cases we address the endogeneity by conditioning on an additional variable, the control variable $\eta_i$. This control variable is estimated using an auxiliary model. This auxilliary model differs between the set up in the current paper and the instrumental variables setting. In the instrumental variables setting we model the relation between the endogenous regressor and an additional variable, the instrument, and deriving the control variable from that relation. In the current setting we model the relation between the secondary outcome and the endogenous regressor and deriving the control variable from that relation. In both cases the auxiliary model has a strict monotonicity assumption. This shows some of the limitations of the approach: the unobserved confounder $\eta$ cannot have a dimension higher than that of the secondary outcome.
Formally, adding Assumption \[assumption:new\] (latent unconfoundedness) to Assumptions \[assumption:external\_observational\]-\[assumption:conditional\] allows us to point-identify the average effect of interest.
\[theorem\_main\] Suppose that Assumptions \[assumption:external\_observational\]-\[assumption:conditional\] and \[assumption:new\] hold, so that the experimental study is unconfounded and has conditional external validity, and the observational study has latent unconfoundedness. Then $\tau^{{\rm P}}_{{\rm O}}$, the average effect of the treatment on the primary outcome in the observational study is point-identified.
Missing At Random
-----------------
There is an interesting and close connection between Assumptions \[assumption:external\_observational\]-\[assumption:conditional\] and \[assumption:new\] and the Missing-At-Random (MAR) assumption in the missing data literature (@rubin1976inference [@little2019statistical; @rubin2004multiple]).
Suppose that Assumptions \[assumption:external\_observational\]-\[assumption:conditional\] and \[assumption:new\] hold. Then: $$\label{mar}G_i\ \perp\!\!\!\perp\ Y_{i}^{{{\rm P}}}\ \Bigl|\ W_{i},X_{i},Y_{i}^{{{\rm S}}}.$$
Because $G_i={{\rm E}}$ is equivalent to an indicator that $Y^{{\rm P}}_i$ missing, and because $W_i$, $X_i$, and $Y^{{\rm S}}_i$ are observed for all individuals in the sample, the conditional independence in (\[mar\]) is equivalent to a MAR assumption. The result does not go the other way around. The MAR assumption by itself has no testable implications, but the combination of Assumptions Assumptions \[assumption:external\_observational\]-\[assumption:conditional\] and \[assumption:new\] does imply some inequality restrictions on the joint distribution of the observed variables. @kallus2020role starts with a Missing-At-Random assumption, and uses that in combination with an unconfoundedness assumption on the full sample to identify the average effect of the treatment for the full sample.
Estimation and Inference
========================
In this section we extend the same three estimation strategies we discussed in the examples in Section \[two\_examples\], imputation, weighting, and control function methods, to the general case.
The Imputation Approach
-----------------------
First, consider the imputation approach. Estimate the conditional mean of the primary outcome given the secondary outcome, treatment and pre-treatment variables in the observational sample: $$\kappa(w,x,y,{{\rm O}})={\mathbb{E}}\left[ \left.Y_i^{{\rm P}}\right| W_i=w,X_i=x,Y^{{\rm S}}_i=y,G_i={{\rm O}}\right].$$ Then impute for the units in the experimental sample the primary outcome as $\hat Y^{{\rm P}}_i=\hat{\kappa}(W_i,X_i,Y^{{\rm S}}_i,{{\rm O}})$. Then use the standard program evaluation methods to adjust for differences in covariates if necessary. If in the experimental sample the treatment is completely random, we would estimate the average treatmet effect in the experimental sample as $$\hat\tau^{{\rm imp},{{\rm E}}}=\frac{1}{N^{{\rm E}}_1}\sum_{i:P_i={{\rm E}}} W_i \kappa(1,X_i,Y^{{\rm S}}_i,{{\rm O}})-
\frac{1}{N^{{\rm E}}_1}\sum_{i:P_i={{\rm E}}} (1-W_i)\kappa(0,X_i,Y^{{\rm S}}_i,{{\rm O}}).$$ However, we wish to estimate the average effect in the observational sample, which may have a different distribution of the pre-treatment variables. This requires one additional layer of adjustment that depends on the pre-treatment variables. Define $$r(x)={\rm pr}(G_i={{\rm O}}|X_i=x).$$ Then we weight the units by the ratio $r(X_i)/(1-r(X_i))$: $$\hat\tau^{{\rm imp}}=\frac{\sum_{i:P_i={{\rm E}}} W_i \kappa(1,X_i,Y^{{\rm S}}_i,{{\rm O}}) r(X_i)/(1-r(X_i)}{\sum_{i:P_i={{\rm E}}} W_i r(X_i)/(1-r(X_i)}$$ $$\hskip2cm -
\frac{\sum_{i:P_i={{\rm E}}} (1-W_i)\kappa(0,X_i,Y^{{\rm S}}_i,{{\rm O}})r(X_i)/(1-r(X_i))}{\sum_{i:P_i={{\rm E}}} (1-W_i)r(X_i)/(1-r(X_i))}.$$
The Weighting Approach
----------------------
Second, consider the weighting approach. Estimate the distribution of $(Y^{{\rm S}}_i,W_i)$ in the observational and experimental sample as $$f_{W,Y^{{\rm S}}|X,P}(w,y^{{\rm S}}|x,p),$$ for all $x\in{\mathbb{X}}$ and $p\in\{{{\rm E}},{{\rm O}}\}$. Then construct the weights for all units in the observational sample as a function of $(W_i,X_i,Y^{{\rm S}}_i)$: $$\lambda_i=\frac{f_{W,Y^{{\rm S}}|X,P}(W_i,Y_i^{{\rm S}}|X_i,{{\rm E}})}{f_{W,Y^{{\rm S}}|X,P}(W_i,Y_i^{{\rm S}}|X_i,{{\rm O}})}.$$ These weights adjust for the differences between the observational and experimental sample.
Assuming we have completely random assignment in the experimental sample, we estimate the average treatment effect as $$\hat\tau^{\rm weight}=\frac{\sum_{i:P_i={{\rm O}}} Y_i W_i \lambda_i}
{\sum_{i:P_i={{\rm O}}} (1-W_i) \lambda_i}-
\frac{\sum_{i:P_i={{\rm O}}} (1-W_i) \lambda_i}
{\sum_{i:P_i={{\rm O}}} W_i \lambda_i}.$$
If instead we have unconfounded treatment assignment in the experimental sample, we need the weights that adjust for the non-randomness in the experimental sample. By the maintained assumptions this requires only adjusting for the differences in pre-treatment variables. Let the propensity score be $$e(x,g)={\rm pr}(W_i=1|X_i=x,G_i=g).$$ This leads to $$\hat\tau^{\rm weight}=\frac{\sum_{i:P_i={{\rm O}}} Y_i W_i \lambda_i/e(X_i,{{\rm E}})}
{\sum_{i:P_i={{\rm O}}} (1-W_i) \lambda_i/e(X_i,{{\rm E}})}-
\frac{\sum_{i:P_i={{\rm O}}} (1-W_i) \lambda_i/(1-e(X_i,{{\rm E}}))}
{\sum_{i:P_i={{\rm O}}} W_i \lambda_i/(1-e(X_i,{{\rm E}}))}.$$
The Control Function Approach
-----------------------------
Finally, the control function approach. First estimate the conditional distribution of the secondary outcome given treatment and pre-treatment variables in both samples: $$F_{Y^{{\rm S}}|W,X,G}(y^{{\rm S}}|w,x,g).$$ Then calculate the control variable for each unit in the observational sample as $$\eta_i=F_{Y^{{\rm S}}|W,X,G}(Y_i^{{\rm S}}|W_i,X_i,{{\rm E}}).$$ Next, estimate the conditional mean of the primary outcome in the observational sample given treatment status, control variable, and pre-treatment variables: $$\gamma(w,h,x)= {\mathbb{E}}\left[\left. Y^{{\rm P}}_i\right|W_i=w,\eta_i=h,X_i=x,G_i={{\rm O}}\right].$$ Finally, estimate the average treatment effect $\tau$ as $$\hat\tau^{\rm cf}=\frac{1}{N^{{\rm E}}_1}\sum_{i:G_i={{\rm E}}} W_i \hat\gamma(1,\eta_i,X_i)-
\frac{1}{N^{{\rm E}}_0}\sum_{i:G_i={{\rm E}}} (1-W_i) \hat\gamma(1,\eta_i,X_i).$$
An Application
==============
To illustrate the ideas in this paper we analyze data on the effect of class size on educational outcomes. We use the data from the Project STAR experiment on class size, where we observe for all children whether they are in a regular or small class. As the short term outcome we use the third grade score. We also observe the pre-treatment variables gender, whether the student gets a free lunch, and ethnicity. For the observational data we use data from the New York school system. We observe the same variables, but also eighth grade scores.
In Table \[tab:\] we report the results from several ordinary least squares regressions. The first two columns show the results from a regression of the short term outcome on the treatment, separately for the two samples, and controlling for the pretreatment variables. In the experimental sample in Project STAR we find a positive effect of small class size of 0.157. In the observational sample the least squares estimate is negative, -0.048. This suggests that there are unmeasured confounders. If we regress the eighth grade scores on the treatment in the observational sample we still get a negative estimate, -0.074. Now we follow the control function approach and include in that linear regression the control function, that is, the estimated residual: $$\hat \alpha^{{\rm S}}_i=Y^{{\rm S}}_i-W_i\hat\beta^{{\rm S}}-X_i^\top\hat\gamma^{{\rm S}}_i.$$ Including this in the regression gives a coefficient of 0.640 with a standard error of 0.001. It changes the coefficient on the treatment to 0.061, now much more in line with what one would expect given the causal effect of a small class size on the third grade scores in the experimental sample. We can also do this through the imputation approach. First we use the regression of the long term outcome on short term outcome and covariates to predict the long term outcome for the observations in the experimental sample. Then we regress the predicted value on the treatment, leading to the same estimate of 0.061.
[lccccc]{} & Secondary & Secondary & Primary & Primary & Imputed Primary\
\
$W_i$ & $0.157$^^ & $-0.048$^^ & $-0.074$^^ & $0.061$^^ & $0.061$^^\
& ($0.028$^^)^^ & ($0.002$^^)^^ & ($0.003$^^)^^ & ($0.018$^^)^^ & ($0.018$^^)^^\
\
$\hat\alpha_i^S$ & & & & $0.640$^^ &\
& & & & ($0.001$^^)^^ &\
\
$N$ & $6,027$^^ & $1,131,339$^^ & $498,597$^^ & $498,597$^^ & $6,027$^^\
$R^2$ & $0.130$^^ & $0.060$^^ & $0.040$^^ & $0.420$^^ & $0.170$^^\
Sample & Experimental & Observational & Observational & Observational & Experimental\
Covariates & Yes & Yes & Yes & Yes & Yes\
![$\alpha^S_i$ in the experimental sample by treated and control[]{data-label="fig:figure1"}](alpha_s_experiment.pdf)
![$\alpha^S_i$ in the observational sample by treated and control[]{data-label="fig:figure2"}](alpha_s_observational.pdf)
We also investigate if the surrogacy assumption holds here. Estimating a regression of the primary outcome on the secondary outcome and the treatment indicator (and including pre-treatment variables), leads to a coefficient on the treatment indicator of -0.039, with a standard error of 0.002. Thus, it appears that third grade scores are not a valid surrogate for eighth grade scores. Thus, the effect of class size is not fully captured by third grade scores, but also arises through other channels. Thus, accounting for the latent confounder is important.
Conclusion
==========
In this paper, we develop new statistical methods for systematically combining experimental and observational data in an attempt to leverage the internal validity of the experimental studies and the external validity and high precision of the observational studies. We do so in a setting where the experimental sample contains information on a secondary outcome and the observational study contains information on both primary and secondary outcomes. We articular a new and critical assumption that allows us to link the biases in comparisons in the observational study between primary and secondary outcome exploing the bias-free information on the secondary outcome from the experimental data. We illustrate these new results by combining data from the Project STAR experiment with observational data from the New York school system. We find that the biases in the observational study are substantial, but that the adjustment procedure based on the experimental data leads to more plausible results.
@adams2006overweight [@d2006surrogate; @abadie2016matching; @alonso2006unifying]
Appendix: Proofs of Results
[**Proof of Lemma \[lemma1\]:**]{} To prove this result we show that we cannot infer from the joint distribution of $(W_i,X_i,G_i,Y^{{\rm S}}_i,Y^{{\rm P}}_i{\bf 1}_{G_i={{\rm O}}})$, in combination with the assumptions, the distribution of $Y^{{\rm P}}_i(1)$ conditional on $X_i$ and $G_i={{\rm E}}$. This distribution can be written as $$f_{Y^{{\rm P}}(1)|X,G={{\rm E}}}(y|x)= f_{Y^{{\rm P}}(1)|X,G={{\rm E}},W=1}(y|x) p(W=1|X=x,G={{\rm E}})$$ $$\hskip2cm +
f_{Y^{{\rm P}}(1)|X,G={{\rm E}},W=0}(y|x) p(W=0|X=x,G={{\rm E}}).$$ The data are not informative about the distribution of $Y^{{\rm P}}_i(1)$ given $W_i=0,$ $X_i$ and $G_i={{\rm E}}$. Assumption \[assumption:conditional\] implies that this distribution is the same as the distribution of $Y^{{\rm P}}_i(1)$ given $W_i=0,$ $X_i$ and $G_i={{\rm O}}$, but the data are not informative about that either. $\square$
[**Proof of Lemma \[lemma2\]:**]{} To prove the result we show that $$G_i\ \perp\!\!\!\perp\ Y^{{\rm S}}_i(1)\ \Bigl|\ X_i=x,W_i=1.$$ We can factor the conditional distribution of $(Y^{{\rm S}}_i(1),G_i)$ given $X_i$ and $W_i=1$ as $$f(Y^{{\rm S}}(1),G|X,W=1)=f(Y^{{\rm S}}(1)|G,X,W) f(G|X,W=1).$$ By the unconfoundedness assumptions, Assumptions \[assumption:random\] and \[assumption:unconfoundedness\] it follows that this is equal to $$f(Y^{{\rm S}}(1)|G,X) f(G|X,W=1).$$ By Conditional External Validity (Assumption \[assumption:conditional\]) this is equal to $$f(Y^{{\rm S}}(1)|X) f(G|X,W=1).$$ By Assumptions \[assumption:random\] and \[assumption:unconfoundedness\] this is equal to $$f(Y^{{\rm S}}(1)|X,W=1) f(G|X,W=1),$$ which implies the conditional independence we set out to prove. $\square$
[**Proof of Theorem \[theorem\_main\]:**]{}[^5] To be clear here, we index the expectations operator by the random variable that the expectation is taken over. By definition $$\tau^{{\rm P}}_{{\rm O}}=
{\mathbb{E}}_{Y_i^{{\rm P}}(1),Y_i^{{\rm P}}(0)}\left[ \left.Y_i^{{\rm P}}(1)-Y_i^{{\rm P}}(0)\right| G_i={{\rm O}}\right]=
{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[ \left.Y_i^{{\rm P}}(1)\right| G_i={{\rm O}}\right]
-{\mathbb{E}}_{Y_i^{{\rm P}}(0)}\left[ \left.Y_i^{{\rm P}}(0)\right| G_i={{\rm O}}\right].$$ We focus on identification of the first term, which by iterated expectations can be written as $$\label{een}{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[ \left.Y_i^{{\rm P}}(1)\right| G_i={{\rm O}}\right]= {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right|X_i,G_i={{\rm O}}\right]\right| G_i={{\rm O}}\right].$$ Identification of the second term follows by the same argument. By Conditional External Validity (Assumption \[assumption:conditional\]), we can write the inner expectation as $${\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right|X_i,G_i={{\rm O}}\right]=
{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right|X_i,G_i={{\rm E}}\right]
,$$ so that (\[een\]) is equal to $$\label{twee} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right|X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ By iterated expectations this is equal to $$\label{drie} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}(1)}\left[\left.{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right| Y_i^{{\rm S}}(1),X_i,G_i={{\rm E}}\right]\right|X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ By Conditional External Validity (Assumption \[assumption:conditional\]), this is equal to $$\label{vier} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}(1)}\left[\left.{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right| Y_i^{{\rm S}}(1),X_i,G_i={{\rm O}}\right]\right|X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ By Latent Unconfoundedness (Assumption \[assumption:new\]) this is equal to $$\label{vier} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}(1)}\left[\left.{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i(1)\right| Y_i^{{\rm S}}(1),W_i=1,X_i,G_i={{\rm O}}\right]\right|X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ By the definitions $Y^{{\rm P}}_i=Y^{{\rm P}}_i(W_i)$ and $Y^{{\rm S}}_i=Y^{{\rm S}}_i(W_i)$ this is equal to $$\label{vijf} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}(1)}\left[\left.{\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i\right| Y_i^{{\rm S}},W_i=1,X_i,G_i={{\rm O}}\right]\right|X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ Define $$h(y^{{\rm S}},x)\equiv {\mathbb{E}}_{Y_i^{{\rm P}}(1)}\left[\left.Y^{{\rm P}}_i\right| Y_i^{{\rm S}}=y^{{\rm S}},W_i=1,X_i=x,G_i={{\rm O}}\right],$$ so that (\[vijf\]) is $$\label{zes} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}(1)}\left[\left.
h(Y^{{\rm S}}_i(1),X_i)
\right|X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ Note that $h(y^{{\rm S}},x)$ is directly identified from the observational sample.
Because of the unconfoundedness in the experimental sample (Assumption \[assumption:random\]), (\[zes\]) is equal to $$\label{zeven} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}(1)}\left[\left.
h(Y^{{\rm S}}_i(1),X_i)
\right|W_i=1,X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ By the definition of $Y^{{\rm S}}_i=Y^{{\rm S}}_i(W_i)$, and because the conditional distribution of $Y^{{\rm S}}_i(1)$ conditional on $W_i=1,X_i,G_i={{\rm O}}$ is the same as the conditional distribution of $Y^{{\rm S}}_i$ conditional on $W_i=1,X_i,G_i={{\rm O}}$, we can change the random variable that the expectation is taken over and write this as $$\label{acht} {\mathbb{E}}_{X_i}\left[\left.{\mathbb{E}}_{Y_i^{{\rm S}}}\left[\left.
h(Y^{{\rm S}}_i,X_i)
\right|W_i=1,X_i,G_i={{\rm E}}\right]\right| G_i={{\rm O}}\right].$$ The inner expectation $$k(x)\equiv {\mathbb{E}}_{Y_i^{{\rm S}}}\left[\left.
h(Y^{{\rm S}}_i,X_i)
\right|W_i=1,X_i=x,G_i={{\rm E}}\right],$$ is identified from the experimental sample. The expectation $${\mathbb{E}}[k(X_i)|G_i={{\rm O}}],$$ is identified from the observational sample, which completes the proof. $\square$
[^1]: [Graduate School of Business, Stanford University, and NBER, athey@stanford.edu. ]{}
[^2]: [Department of Economics, Harvard University, and NBER, chetty@harvard.edu. ]{}
[^3]: [ Graduate School of Business, and Department of Economics, Stanford University, and NBER, imbens@stanford.edu. ]{}
[^4]: [We are grateful for discussions with Nathan Kallus, Xiaojie Mao, Dylan Small, and for comments from seminar participants at the University of Pennsylvania and Berkeley. We also want to acknowledge excellent research assistance from Kevin Chen. This research was funded through the Sloan Foundation, Schmidt Futures, and ONR grant N00014-17-1-2131.]{}
[^5]: We are grateful to Nathan Kallus and Xiaojie Mao for pointing out a mistake in an earlier version of the proof of this theorem.
|
---
abstract: 'We develop the theory of nonlinear localised modes ([*intrinsic localised modes*]{} or [*discrete breathers*]{}) in two-dimensional (2D) photonic crystal waveguides. We consider different geometries of the waveguides created by an array of nonlinear dielectric rods in an otherwise perfect linear 2D photonic crystal, and demonstrate that the effective interaction in such waveguides is [*nonlocal*]{}, being described by a new type of nonlinear lattice models with long-range coupling and nonlocal nonlinearity. We reveal the existence of different types of nonlinear guided modes which are also localised in the waveguide direction, and describe their unique properties including bistability.'
address:
- '$^1$ Optical Sciences Centre, Australian National University, Canberra ACT 0200, Australia'
- '$^2$ School of Mathematics and Statistics, Australian Defence Force Academy, Canberra ACT 2600, Australia'
author:
- 'Serge F. Mingaleev $^{1,*}$, Yuri S. Kivshar $^1$, and Rowland A. Sammut $^2$'
title: 'Long-range interaction and nonlinear localized modes in photonic crystal waveguides'
---
[2]{}
Introduction
============
In physics, the idea of localisation is generally associated with disorder that breaks translational invariance. However, research in recent years has demonstrated that localisation can occur in the absence of any disorder and solely due to nonlinearity, in the form of [*intrinsic localised modes*]{}, also called [*discrete breathers*]{}.[@review] A rigorous proof of the existence of time-periodic, spatially localised solutions describing such nonlinear modes has been presented for a broad class of Hamiltonian coupled-oscillator nonlinear lattices,[@mak] but approximate analytical solutions can also be found in many other cases, demonstrating a generality of the concept of [*nonlinear localised modes*]{}.
Nonlinear localised modes can be easily identified in numerical molecular-dynamics simulations in many different physical models (see, e.g., Ref. for a review), but only very recently the first experimental observations of spatially localised nonlinear modes have been reported in mixed-valence transition metal complexes,[@bishop] quasi-one-dimensional antiferromagnetic chains,[@sievers] and arrays of Josephson junctions.[@JJ] Importantly, very similar types of spatially localised nonlinear modes have been experimentally observed in [*macroscopic*]{} mechanical [@zolo] and guided-wave optical [@silb] systems.
Recent experimental observations of nonlinear localised modes, as well as numerous theoretical results, indicate that both effects, i.e. nonlinearity-induced localisation and spatially localised modes, can be expected in physical systems of very different nature. From the viewpoint of possible practical applications, self-localised states in optics seem to be the most promising ones; they can lead to different types of nonlinear all-optical switching devices where light manipulates and controls light itself, by varying the input intensity. As a result, the study of nonlinear localised modes in photonic structures is expected to bring a variety of realistic applications of intrinsic localised modes.
One of the promising fields where the concept of nonlinear localised modes may find practical applications is in the physics of [*photonic crystals*]{} \[or photonic band gap (PBG) materials\] — periodic dielectric structures that produce many of the same phenomena for photons as the crystalline atomic potential does for electrons.[@pbg] Three-dimensional (3D) photonic crystals for visible light have been successfully fabricated only within the past year or two, and presently many research groups are working on creating tunable band-gap switches and transistors operating entirely with light. The most recent idea is to employ nonlinear properties of band-gap materials, thus creating [*nonlinear photonic crystals*]{} that have 2D or 3D periodic nonlinear susceptibility.[@berger; @sukh]
Nonlinear photonic crystals or photonic crystals with embedded nonlinear impurities create an ideal environment for the generation and observation of nonlinear localised photonic modes. In particular, such modes can be excited at nonlinear interfaces with quadratic nonlinearity,[@sukh2] or along dielectric waveguide structures possessing a nonlinear Kerr-type response.[@mcgurn] In this paper, we analyse nonlinear localised modes in 2D photonic crystal waveguides. We consider the waveguides created by an array of dielectric rods in an otherwise perfect 2D photonic crystal. It is assumed that the dielectric constant of the waveguide rods depends on the field intensity (due to the Kerr effect), so that waveguides of different geometries can support a variety of nonlinear guided modes. We demonstrate here that localisation can occur in the propagation direction creating a 2D spatially localised mode (see Fig. \[fig:sol2d\] below). As follows from our results, the effective interaction in such nonlinear waveguides is nonlocal, and the nonlinear localised modes are described by a nontrivial generalisation of nonlinear lattice models with long-range coupling and nonlocal nonlinearity.
Model
=====
We consider a 2D photonic crystal created by a square lattice of parallel, infinitely long dielectric columns (or rods) in air. The system is characterized by the dielectric constant $\epsilon(\bbox{x})=\epsilon(x_1, x_2)$, and it is assumed that the rods are parallel to the $x_3$ axis. The evolution of the TM-polarised [@pbg] light \[with the electric field having the structure $\bbox{E}=(0,0,E)$\], propagating in the $(x_1, x_2)$-plane, is governed by the scalar wave equation $$\nabla^2 E(\bbox{x}, t) - \frac{1}{c^2} \, \partial_t^2
\left[ \epsilon(\bbox{x}) E \right] = 0 \; ,
\label{sys:eq-E-t}$$ where $\nabla^2 \equiv \partial_{x_1}^2 + \partial_{x_2}^2$. For monochromatic light, we consider the stationary solutions $$E(\bbox{x}, t) = e^{-i \omega t} \, E(\bbox{x} \,|\, \omega) \; ,
\label{sys:E-t-omega}$$ and the equation of motion (\[sys:eq-E-t\]) reduces to the simple eigenvalue problem $$\left[ \nabla^2 + \left( \frac{\omega}{c} \right)^2
\epsilon(\bbox{x}) \right]
E(\bbox{x} \,|\, \omega) = 0 \; .
\label{sys:eq-E-omega}$$ This eigenvalue problem can be easily solved (e.g., by the plane waves method [@Maradudin:1993:PBGL]) in the case of a perfect photonic crystal, for which the dielectric constant $\epsilon(\bbox{x}) \equiv \epsilon_{pc}(\bbox{x})$ is a periodic function $$\epsilon_{pc}(\bbox{x}+\bbox{s}_{ij}) =
\epsilon_{pc}(\bbox{x}) \; ,
\label{sys:eps-pc}$$ where $i$ and $j$ are arbitrary integer, and $$\bbox{s}_{ij} = i \, \bbox{a}_1 + j \, \bbox{a}_2
\label{sys:s-ij}$$ is a linear combination of the primitive lattice vectors $\bbox{a}_1$ and $\bbox{a}_2$ of the 2D photonic crystal.
For definiteness, we consider the 2D photonic crystal earlier analysed (in the linear limit) in Refs. , i.e. we assume that the rods are identical and cylindrical, with radius $r_0=0.18a$ and dielectric constant $\epsilon_0=11.56$. The rods form a perfect square lattice with the distance $a$ between two neighbouring rods, i.e. $\bbox{a}_1=a \bbox{x}_1$ and $\bbox{a}_2=a \bbox{x}_2$. The frequency band structure for this type of 2D photonic crystal, and for the selected polarisations of the electric field, is shown in Fig. \[fig:band-r0.18\]. As follows from the structure of the frequency spectrum, there exists a large (38%) band gap that extends from the lower cut-off frequency, $\omega=0.302 \times 2\pi c/a$, to the upper band-gap frequency, $\omega=0.443 \times 2\pi c/a$. Since the characteristics of a PBG material remain unchanged under rescaling, we can assume that this gap is created either in the infra-red or visible regions of the spectrum. For example, if we choose the lattice constant to be $a =0.58 \, \mu$m, the wavelength corresponding to the mid-gap frequency will be $1.55 \, \mu$m.
The TM-polarised light cannot propagate through the photonic crystal if its frequency falls inside the band gap. But one can excite guided modes inside the forbidden frequency gap by introducing some interfaces, waveguides, or defects. Here, we consider waveguides created by a row of identical defects with a Kerr-type nonlinear response. These defect-induced waveguides possess translational symmetry, and the corresponding guided modes can be characterized by the reciprocal space wave vector $k$ directed along the waveguide. Such a guided mode has a periodical profile inside the waveguide, and it decays exponentially outside it.
[*Linear photonic-crystal waveguides*]{} created by removing a row of dielectric rods have been recently investigated numerically [@Mekis:1996:PRL; @Mekis:1998:PRB] and experimentally.[@Lin:1998:SCI] In particular, highly efficient transmission of light, even in the case of a bent waveguide, has been demonstrated.
In the present paper, in contrast to Refs. where only linear waveguides were considered, we study the properties of [*nonlinear waveguides*]{} created by inserting an additional row of rods fabricated from a Kerr-type nonlinear material characterized by a third-order nonlinear susceptibility with the linear dielectric constant $\epsilon_d$. For definiteness, we assume that $\epsilon_d = \epsilon_0 = 11.56$. As we show below, changing the radius $r_d$ of these defect rods and their location within the crystal, we can create waveguides with quite different properties.
Effective discrete equations
=============================
Writing the dielectric constant $\epsilon(\bbox{x})$ as a sum of periodic and defect-induced terms, i.e. $$\epsilon(\bbox{x})=\epsilon_{pc}(\bbox{x})+\delta
\epsilon(\bbox{x} \,|\, E) \; ,
\label{sys:eps}$$ we can present Eq. (\[sys:eq-E-omega\]) as follows, $$\begin{aligned}
\left[ \nabla^2 + \left( \frac{\omega}{c} \right)^2
\epsilon_{pc}(\bbox{x}) \right] &&
E(\bbox{x} \,|\, \omega) \nonumber \\
= - && \left( \frac{\omega}{c} \right)^2
\delta \epsilon(\bbox{x} \,|\, E) \, E(\bbox{x} \,|\, \omega) \; .
\label{sys:eq-E-omega-delta}\end{aligned}$$ Equation (\[sys:eq-E-omega-delta\]) can also be written in the integral form $$E(\bbox{x} \,|\, \omega) = \left( \frac{\omega}{c} \right)^2
\!\!
\int d^2\bbox{y} \,\,\, G(\bbox{x}, \bbox{y} \,|\, \omega) \,
\delta \epsilon(\bbox{y} \,|\, E) \, E(\bbox{y} \,|\, \omega) \; ,
\label{sys:eq-green-int}$$ where $G(\bbox{x}, \bbox{y} \,|\, \omega)$ is the Green function which is defined, in a standard way, as a solution of the equation $$\left[ \nabla^2 + \left( \frac{\omega}{c} \right)^2
\epsilon_{pc}(\bbox{x}) \right]
G(\bbox{x}, \bbox{y} \,|\, \omega) = - \delta(\bbox{x}-\bbox{y}) \; ,
\label{sys:eq-green-omega}$$ with, accordingly to Eq. (\[sys:eps-pc\]), periodic coefficients. The properties of the Green function and the numerical methods for its calculation have been already described in the literature. [@Maradudin:1993:PBGL; @Ward:1998:PRB] Here, we notice that the Green function of a perfect 2D photonic crystal is [*symmetric*]{}, i.e. $$G(\bbox{x}, \bbox{y} \,|\, \omega) =
G(\bbox{y}, \bbox{x} \,|\, \omega)
\label{sys:green-symm}$$ and [*periodic*]{}, i.e. $$G(\bbox{x} + \bbox{s}_{ij}, \bbox{y} + \bbox{s}_{ij} \,|\, \omega) =
G(\bbox{x}, \bbox{y} \,|\, \omega) \; ,
\label{sys:green-period}$$ where $\bbox{s}_{ij}$ is defined by Eq. (\[sys:s-ij\]).
Let us consider a row of [*nonlinear defect rods*]{} embedded into the crystal along a selected direction. To describe such a row, we should define the rods positions along $\bbox{s}_{ij}$ with some specific values of $i$ and $j$. For example, let us first assume that the defect rods are located at the points $\bbox{x}_m = \bbox{x}_0 + m \, \bbox{s}_{ij}$. In this case, the correction to the dielectric constant is $$\begin{aligned}
\delta \epsilon(\bbox{x}) = \left\{\epsilon_{d} +
|E(\bbox{x} \,|\, \omega)|^2\right\}
\sum_m \theta (\bbox{x}-\bbox{x}_m) \; ,
\label{sys:delta-eps}\end{aligned}$$ where $$\theta (\bbox{x}) = \left\{
\begin{array}{c}
1 \; , \quad \mbox{for} \quad |\bbox{x}| \leq r_d \; , \\
0 \; , \quad \mbox{for} \quad |\bbox{x}| > r_d \; .
\end{array}
\right.$$ Assuming that the radius of the rods, $r_{d}$, is sufficiently small (so that the electric field $E(\bbox{x} \,|\, \omega)$ is almost constant inside the defect rods), we substitute Eq. (\[sys:delta-eps\]) into Eq. (\[sys:eq-green-int\]) and, averaging over of the cross-section of the rods, derive an approximate [*discrete nonlinear equation*]{} for the electric field $$\begin{aligned}
E_n = \sum_m J_{n-m}(\omega) (\epsilon_{d} + |E_m|^2)
E_m \; ,
\label{sys:eq-E-disc}\end{aligned}$$ where $$J_{n}(\omega) = \left( \frac{\omega}{c} \right)^2
\int\limits_{r_d} d^2 \bbox{y} \,\,\,
G(\bbox{x}_0, \bbox{x}_n + \bbox{y} \,|\, \omega ) \; .
\label{sys:Jn}$$ This type of discrete nonlinear equation for photonic crystals has been earlier introduced by McGurn [@mcgurn], for the special case of nonlinear impurities embedded in the linear rods. However, the analytical approach developed by McGurn for that model did not take into account the field distribution via the explicit dependence of the coupling coefficients $J_n(\omega)$ and, as a result, the equation (\[sys:eq-E-disc\]) was not solved exactly. Moreover, the analysis of Ref. was based on the nearest-neighbour approximation where the coupling coefficients are approximated as $J_n= J_0\delta_{n,0} + J_1 \delta_{n,\pm 1}$ with constant $J_0$ and $J_1$.
In a sharp contrast, in the present paper we provide a systematic numerical analysis of different types of nonlinear localised modes in the framework of a complete model. In particular, we reveal that the approximation of the nearest-neighbour interaction is very crude in many of the cases analysed. Since the effective coupling coefficients are defined by the Green function, this can be seen directly from Fig. \[fig:green-r0.18\] that shows a typical spatial profile of the Green function which, in general, characterises a long-range interaction, very typical for photonic crystal waveguides. As a consequence of that, the coupling coefficients $|J_n(\omega)|$ calculated from Eq. (\[sys:Jn\]) decrease exponentially with the site number $n$, and in the asymptotic region they can be presented as follows $$|J_n(\omega)| \approx \left\{
\begin{array}{lcc}
J_0(\omega) \; , & \mbox{for} & n=0 \; , \\
J_{*}(\omega) \, e^{-\alpha(\omega) |n|} \; ,
& \mbox{for} & |n| \geq 1 \; ,
\end{array}
\right.
\label{sys:Jn-exp}$$ where the characteristic decay rate $\alpha(\omega)$ can be as small as $0.85$, depending on the values of $\omega$, $\bbox{x}_0$, $\bbox{s}_{ij}$, and $r_d$, and it can be even smaller for other types of photonic crystals.
This result allows us to draw an analogy with a class of the nonlinear Schrödinger (NLS) equations that describe nonlinear excitations in quasi-one-dimensional molecular chains with long-range (e.g. dipole-dipole) interaction between the particles and local on-site nonlinearities.[@Johansson:1998:PRE] For such systems, it was shown that the effect of nonlocal interparticle interaction introduces some new features in the properties of existence and stability of nonlinear localised modes. Moreover, for our model the coupling coefficients $J_n(\omega)$ can be either non-staggered and monotonically decaying, i.e. $J_n(\omega)=|J_n(\omega)|$, or staggered and oscillating from site to site, i.e. $J_n(\omega)=(-1)^{n}|J_n(\omega)|$. We can therefore expect that effective nonlocality in both linear and nonlinear terms of Eq. (\[sys:eq-E-disc\]) will bring a number of new features in the properties of nonlinear localised modes.
Examples of nonlinear modes
===========================
As can be seen from the structure of the example Green function, presented in Fig. \[fig:green-r0.18\], the case of monotonically varying $J_n(\omega)$ can be obtained by locating the defect rods at the points $\bbox{x}_0=\bbox{a}_1/2$, along the straight line in the $\bbox{s}_{01}$ direction. In this case, the frequency of a linear guided mode, that can be excited in such a waveguide, takes the minimum value at $k=0$ (see Fig. \[fig:def-x2-0.10\]), and the corresponding nonlinear mode is expected to be non-staggered.
We have solved Eq. (\[sys:eq-E-disc\]) numerically and found that nonlinearity can lead to the existence of a new type of guided modes which are localised in both directions, i.e. in the direction perpendicular to the waveguide, due to the guiding properties of a channel created by defect rods, and in the direction of the waveguide, due to the self-trapping effect. Such nonlinear modes exist with frequencies below the frequency of the linear guided mode of the waveguide, i.e. below the frequency $\omega_A$ in Fig. \[fig:def-x2-0.10\], and are indeed non-staggered, with the bell-shaped profile along the waveguide direction shown in the left inset of Fig. \[fig:norm-x2-0.10\].
The 2D nonlinear modes localised in both dimensions can be characterized by the mode intensity which we define, by analogy with the NLS equation, as $$Q = \sum_n |E_n|^2.
\label{sys:norm}$$ This intensity is closely related to the energy of the electric field in the 2D photonic crystal accumulated in the nonlinear mode. In Fig. \[fig:norm-x2-0.10\] we plot the dependence of $Q$ on frequency, for the waveguide geometry shown in Fig. \[fig:def-x2-0.10\].
As can be seen from the example of the Green function shown in Fig. \[fig:green-r0.18\], the case of staggered coupling coefficients $J_n(\omega)$ can be obtained by locating the defect rods at the points $\bbox{x}_0=\bbox{a}_1/2$, along the straight line in the $\bbox{s}_{10}$ direction. In this case, the frequency dependence of the linear guided mode of the waveguide takes the minimum at $k=\pi/a$ (see Fig. \[fig:def-x1-0.10\]). Accordingly, the nonlinear guided mode localised along the direction of the waveguide is expected to exist with the frequency below the lowest frequency $\omega_A$ of the linear guided mode, with a staggered profile. The longitudinal profile of such a 2D nonlinear localised mode is shown in the left inset in Fig. \[fig:norm-x1-0.10\], together with the dependence of the mode intensity $Q$ on the frequency (solid curve), which in this case is again monotonic.
The results presented above are obtained for linear photonic crystals with nonlinear waveguides created by a row of defect rods. However, we have carried out the same analysis for the general case of [*a nonlinear photonic crystal*]{} that is created by rods of different size but made of the same nonlinear material. Importantly, we have found very small difference in all the results for relatively weak nonlinearities. In particular, for the photonic crystal waveguide shown in Fig. \[fig:def-x1-0.10\], the results for linear and nonlinear photonic crystals are very close. Indeed, for the mode intensity $Q$ the results corresponding to a nonlinear photonic crystal are shown in Fig. \[fig:norm-x1-0.10\] by a dashed curve, and for $Q<20$ this curve almost coincides with the solid curve corresponding to the case of a nonlinear waveguide embedded into a linear photonic crystal.
Let us now consider the waveguide created by a row of defects which are located at the points $\bbox{x}_0=(\bbox{a}_1+\bbox{a}_2)/2$, along a straight line in either the $\bbox{s}_{10}$ or $\bbox{s}_{01}$ directions. The results for this case are presented in Figs. \[fig:def-x12-0.10\]–\[fig:sol2d\]. The coupling coefficients $|J_n|$ are described by a slowly decaying function of the site number $n$, so that the effective interaction decays on scales much larger than those in the cases considered previously. Similar to the NLS models with long-range dispersive interactions [@Johansson:1998:PRE; @Gaididei:1997:PRE], for this type of nonlinear photonic crystal waveguide we find a non-monotonic behaviour of the mode intensity $Q(\omega)$ and, as a result, multi-valued dependence of the invariant $Q(\omega)$ for $\omega<0.347 \times 2\pi c/a$. Similar to the results earlier obtained for the nonlocal NLS models [@Johansson:1998:PRE], we can expect here that nonlinear localised modes corresponding, in our notations, to the positive slope of the derivative $dQ/d\omega$ are unstable and will eventually decay or transform into modes of higher or lower frequency. Such a phenomenon is known as [*bistability*]{}, and in this problem it occurs as a direct manifestation of the nonlocality of the effective (linear and nonlinear) interaction between the defect rod sites.
Conclusions
===========
Exploration of nonlinear properties of PBG materials is a new direction of research, and it may open up a new class of applications of photonic crystals for all-optical signal processing and switching, allowing an effective way to create tunable band-gap structures operating entirely with light. Nonlinear photonic crystals, and nonlinear waveguides embedded into photonic structures with periodically modulated dielectric constant, create an ideal environment for the generation and observation of nonlinear localised modes.
In the present paper, we have developed a consistent theory of nonlinear localised modes which can be excited in photonic crystal waveguides of different geometry. For several geometries of 2D waveguides, we have demonstrated that such modes are described by a new type of nonlinear lattice models that include long-range interaction and effectively nonlocal nonlinear response. It is expected that the general features of nonlinear guided modes described here will be preserved in other types of photonic crystal waveguides. Our approach and results can also be useful to develop the theory of nonlinear two-frequency parametric localised modes in the recently fabricated 2D photonic crystals with the second-order nonlinear susceptibility [@neal]. Additionally, similar types of nonlinear localised modes are expected in photonic crystal fibers [@russell] consisting of a periodic air-hole lattice that runs along the length of the fiber, provided the fiber core is made of a highly nonlinear material (see, e.g., Ref. ).
Acknowledgments {#acknowledgments .unnumbered}
===============
Yuri Kivshar is thankful to Costas Soukoulis for useful discussions and suggestions at the initial stage of this project. The work has been partially supported by the Large Grant Scheme of the Australian Research Council, the Australian Photonics Cooperative Research Centre, and the Planning and Performance Foundation grant of the Institute of Advanced Studies.
[99]{} On leave from the Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kiev 03143, Ukraine.
S. Flach and C.R. Willis, Phys. Rep. [**295**]{}, 181 (1998); see also Chap. 6 in O.M. Braun and Yu.S. Kivshar, Phys. Rep. [**306**]{}, 1 (1998).
R.S. MacKay and S. Aubry, Nonlinearity [**7**]{}, 1623 (1994); see also S. Aubry, Physica D [**103**]{}, 201 (1997).
B.I. Swanson, J.A. Brozik, S.P. Love, G.F. Strouse, A.P. Shreve, A.R. Bishop, W.Z. Wang, and M.I. Salkola, Phys. Rev. Lett. [**82**]{}, 3288 (1999).
U.T. Schwarz, L.Q. English, and A.J. Sievers, Phys. Rev. Lett. [**83**]{}, 223 (1999).
E. Trias, J.J. Mazo, and T.P. Orlando, Phys. Rev. Lett. [**84**]{}, 741 (2000); P. Binder, D. Abraimov, A.V. Ustinov, S. Flach, and Y. Zolotaryuk, Phys. Rev. Lett. [**84**]{}, 745 (2000).
F.M. Russel, Y. Zolotaryuk, and J.C. Eilbeck, Phys. Rev. B [**55**]{}, 6304 (1997).
H.S. Eisenberg, Y. Silberberg, R. Marandotti, A.R. Boyd, and J.S. Aitchison, Phys. Rev. Lett. [**81**]{}, 3383 (1998).
J.D. Joannopoulos, R.D. Meade, and J.N. Winn, [*Photonic Crystals: Molding the Flow of Light*]{} (Princeton University Press, Princeton, 1995).
V. Berger, Phys. Rev. Lett. [**81**]{}, 4136 (1998).
A.A. Sukhorukov, Yu.S. Kivshar, O. Bang, J. Martorell, J. Trull, and R. Vilaseca, Optics and Photonics News [**10**]{}, No. 12 (1999) pp. 34-35.
A.A. Sukhorukov, Yu.S. Kivshar, and O. Bang, Phys. Rev. E [**60**]{}, R41 (1999).
A. R. McGurn, Phys. Lett. A [**251**]{}, 322 (1999).
A. A. Maradudin and A. R. McGurn, in [*Photonic Band Gaps and Localization*]{}, [*NATO ASI Series B: Physics*]{}, Vol. 308, Ed. C. M. Soukoulis (Plenum Press, New York, 1993), pp. 247–268.
A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, Phys. Rev. Lett. [**77**]{}, 3787 (1996).
A. Mekis, S. Fan, and J. D. Joannopoulos, Phys. Rev. B [**58**]{}, 4809 (1998).
S.-Y. Lin, E. Chow, V. Hietala, P.R. Villeneuve, and J.D. Joannopoulos, Science [**282**]{}, 274 (1998).
A. J. Ward and J. B. Pendry, Phys. Rev. B [**58**]{}, 7252 (1998).
M. Johansson, Y. B. Gaididei, P. L. Christiansen, and K. [Ø]{}. Rasmussen, Phys. Rev. E [**57**]{}, 4739 (1998).
Y. B. Gaididei, S. F. Mingaleev, P. L. Christiansen, and K. [Ø]{}. Rasmussen, Phys. Rev. E [**55**]{}, 6141 (1997).
N.G.R. Broderick, G.W. Ross, H.L. Offerhaus, D.J. Richardson, and D.C. Hanna, e-print physics/9910036, Phys. Rev. Lett (2000) in press.
T.A. Birks, J.C. Knight, and P.St.J. Russell, Opt. Lett. [**22**]{}, 961 (1997).
B.J. Eggleton, P.S. Westbrook, R.S. Windeler, S. Spälter, and T.A. Strasser, Opt. Lett. [**24**]{}, 1460 (1999).
|
---
abstract: 'We consider a family of self-adjoint Ornstein–Uhlenbeck operators ${\mathcal L}_{\alpha} $ in an infinite dimensional Hilbert space $H$ having the same gaussian invariant measure $\mu$ for all $''\alpha \in [0,1]$. We study the Dirichlet problem for the equation $\lambda \varphi - {\mathcal L}_{\alpha} \varphi = f$ in a closed set $K$, with $f\in L^2(K, \mu)$. We first prove that the variational solution, trivially provided by the Lax—Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to $K$. Then we address two problems: 1) the regularity of the solution $\varphi$ (which is by definition in a Sobolev space $W^{1,2}_{\alpha}(K,\mu)$) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior $W^{2,2}_{\alpha}$ regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside $K$ belongs to $W^{1,2}_{\alpha}(H,\mu)$. In the second case we exploit the Malliavin’s theory of surface integrals which is recalled in the Appendix of the paper, then we are able to give a meaning to the trace of $\varphi$ at $\partial K$ and to show that it vanishes, as it is natural.'
address:
- |
Scuola Normale Superiore\
Piazza dei Cavalieri, 7\
56126 Pisa, Italy
- |
Dipartimento di Matematica\
Università di Parma\
Viale G.P. Usberti, 53/A\
43124 Parma, Italy
author:
- Giuseppe Da Prato
- Alessandra Lunardi
title: 'On the Dirichlet semigroup for Ornstein – Uhlenbeck operators in subsets of Hilbert spaces'
---
Introduction and setting of the problem
=======================================
In this paper we present some results on second order elliptic and parabolic equations with Dirichlet boundary conditions in a closed set of a separable real Hilbert space $H$ (norm $|\cdot|$, inner product $\langle \cdot,\cdot \rangle$).
A motivation for the study of Dirichlet problems in proper subsets of $H$ is to provide a natural development of the potential theory in infinite dimensions started in [@Gross]. Only a few results seem to be available in this field, see e.g. [@DPZ3] and the references therein.
The finite dimensional theory in spaces of continuous functions is hardly extendable to the infinite dimensional setting. While in finite dimensions smooth boundaries consist only of regular points in the sense of Wiener, in infinite dimensions this is not true: for instance, certain hyperplanes and the boundary of the unit ball contain dense subsets of irregular points for suitable Ornstein-Uhlenbeck operators ([@DPGZ]). This leads to the lack of regularity results up to the boundary.
Here we avoid a part of such difficulties working in suitable $L^2$ spaces.
To begin with, we consider a class of Ornstein–Uhlenbeck operators of the type $$\label{OU}
{\mathcal L}_{\alpha} \varphi(x)=\frac{1}{2}\;\mbox{\rm Tr}\;[Q^{1-\alpha}D^2\varphi(x)] -\frac{1}{2}\langle x,Q^{-\alpha}D \varphi(x) \rangle,$$ where $Q\in {\mathcal L}(H)$ is a symmetric positive operator with finite trace, and $0\leq \alpha \leq 1$.
The most popular among such operators are ${\mathcal L}_{0} $ and ${\mathcal L}_{1} $: $${\mathcal L}_{0} \varphi(x)=\frac12\;\mbox{\rm Tr}\;[Q D^2\varphi(x)] -\frac{1}{2}\langle x, D \varphi(x) \rangle,$$ is the operator that arises in the Malliavin calculus, while $${\mathcal L}_{1} \varphi(x)=\frac12\;\mbox{\rm Tr}\;[ D^2\varphi(x)] -\frac{1}{2}\langle x,AD \varphi(x) \rangle,$$ (with $A= Q^{-1}$) is the generator of the Ornstein-Uhlenbeck semigroup with the best smoothing properties. See e.g. [@DPZ3].
The operators ${\mathcal L}_{\alpha} $ exhibit an important common feature: the associated Ornstein-Uhlenbeck semigroups $T_{\alpha}(t) $ in $C_b(H)$ have the same invariant measure $\mu={\mathcal N}_{Q}$, the Gaussian measure of mean $0$ and covariance $Q $. In this paper we shall consider realizations of the operators ${\mathcal L}_{\alpha} $ in the space $L^2(K , \mu)$, where $K$ is a closed set in $H$ with non empty interior part $\oo{K}$.
A unique weak solution to the Dirichlet problem $$\label{e1.6a}
\left\{\begin{array}{l}
\lambda \varphi(x)-{\mathcal L}_{\alpha} \varphi(x)=f(x), \quad\mbox{\rm in}\; K,
\\
\\
\varphi(x)=0,\quad\mbox{\rm on}\; \partial K
\end{array}\right.$$ with $\lambda >0$ and $f\in L^2(K,\mu)$ is easily obtained via the Lax-Milgram Theorem, applied in a Hilbert space $\oo{W}^{1,2}_{\alpha} (K ,\mu)$ “naturally" associated to ${\mathcal L}_{\alpha} $ (see next section). This allows to define a dissipative self-adjoint operator $M_{\alpha}$ in $L^2(K,\mu)$ such that $\varphi = R(\lambda, M_{\alpha})f$. As all dissipative self-adjoint operators in Hilbert spaces, $M_{\alpha}$ is the infinitesimal generator of an analytic contraction semigroup.
We give an explicit expression of the semigroup generated by $M_{\alpha}$. Precisely, we identify it with the natural extension to $L^2(K,\mu)$ of the so-called [*stopped semigroup*]{} $T^{K}_{\alpha}(t)$. In analogy with the finite dimensional case (e.g., [@Friedman]), it is defined in $B_b(K)$ (the space of the bounded and Borel measurable functions defined in $K$) by $$\label{e1.8}
\begin{array}{lll}
T^{K}_{\alpha}(t) \varphi(x)&=&\ds{{\mathbb E}}[\varphi(X_{\alpha}(t,x)){1\!\!\!\;\mathrm{l}}_{\tau_x \ge t}]\\
\\
&=&\ds\int_{\{\tau_x \ge t\}}\varphi(X_{\alpha}(t,x))d{{\mathbb P}},\quad\forall\;x\in K,
\end{array}$$ where $\tau_x $ is the entrance time in the complement of $K$, $$\label{e1.7}
\tau_x :=\inf\{t\ge 0:\;X_{\alpha}(t,x)\in K^c\},\quad\forall\;x\in K,$$ and $X_{\alpha}(t,x)$ is the solution to $$\label{e1.6}
dX_{\alpha}(t,x)= -\frac{1}{2}A^{\alpha}X_{\alpha}(t,x)dt+A^{(\alpha-1)/2}dW(t),\quad X(0,x)=x.$$ Here $W(t)$ is a standard cylindrical Wiener process in $H$, defined in a filtered probability space $(\Omega, \mathcal F, ({ \mathcal F}_t)_{t\geq 0}, {{\mathbb P}})$.
The definition of $T^{K}_{\alpha}(t)$ is similar to the one in [@Tal], where the exit time from $ \oo{K}$, $\widetilde{\tau}_x :=\inf\{t\ge 0:\;X_{\alpha}(t,x)\in \oo{K} ^c\}$ was used instead of our $\tau_x$. In finite dimensions, if $K$ is the closure of a bounded open set with smooth boundary the two definitions are equivalent, and $T^{K}_{\alpha}(t)$ is the semigroup associated to the realization of ${\mathcal L}_{\alpha}$ with Dirichlet boundary condition ([@Friedman §6.5]). Therefore, a lot of regularity results, both interior and up to the boundary, are well known. In infinite dimensions, interior regularity results were given in [@Tal] for $\alpha >0$. We do not know regularity results up to the boundary, even in the case of very smooth bounded sets such as balls.
Here we prove that $\mu$ is a sub-invariant measure for $T^{K}_{\alpha}(t)$. Therefore, $T^{K}_{\alpha}(t)$ has a natural extension (still called $T^{K}_{\alpha}(t)$) to a contraction semigroup in $L^2(K,\mu)$. The domain of its generator $L^{K}_{\alpha}$ consists of the range of the resolvent operator, $$\label{e1.10a}
R(\lambda, L^{K}_{\alpha})f = \int_0^\infty e^{-\lambda t}T^{K}_{\alpha}(t) f dt,\quad f\in L^2(K,\mu),$$ which is well defined for $\lambda >0$ since $T^{K}_{\alpha}(t)$ is a contraction semigroup. We prove that for each $\lambda >0$ and $f\in L^2(K,\mu)$, the function $\varphi := R(\lambda, L^{K}_{\alpha})f$ belongs to the above mentioned space $\oo{W}^{1,2}_{\alpha} (K,\mu)$, and satisfies the weak formulation of . Therefore, $L^{K}_{\alpha} = M_{\alpha}$.
Our main tool in the proof is the approximating Feynman–Kac semigroup $$\label{e1.11}
P^{{\varepsilon}}_{\alpha}(t) \varphi(x)={{\mathbb E}}\left[\varphi(X_{\alpha}(t,x))e^{-\frac1{\varepsilon}\;\int_0^t V(X_{\alpha}(s,x))ds}\right],$$ where $V$ is a (fixed) bounded continuous function that vanishes in $K$ and has positive values in $K^c$. Its infinitesimal generator in $L^2(H,\mu)$ is the operator $M^{{\varepsilon}}_{\alpha} : D(M^{{\varepsilon}}_{\alpha}) = D(L_{\alpha})$ $\mapsto
L^2(H, \mu)$, $M^{{\varepsilon}}_{\alpha} \varphi = L_{\alpha}\varphi_{\varepsilon}- \frac1{\varepsilon}\,V\varphi$, and we prove that for each $\varphi \in L^2(K,\mu)$, $t>0$, $\lambda >0$ we have $$T^{K}_{\alpha}(t)\varphi = \lim_{{\varepsilon}\to 0} (P^{{\varepsilon}}_{\alpha}(t) \widetilde{\varphi})_{|K}, \quad R(\lambda, L^{K}_{\alpha})\varphi = \lim_{{\varepsilon}\to 0} (R(\lambda, M^{{\varepsilon}}_{\alpha})\widetilde{\varphi})_{|K}$$ in $L^2(K,\mu)$, where $\widetilde{\varphi}$ is the null extension of $\varphi$ to the whole $H$.
Problem is of interest for $\lambda =0$ too. Using the fact that $D(L^{K}_{\alpha})$ is compactly embedded in $L^2(K,\mu)$, in Sect. 3.3 we prove that $0\in \rho(L^{K}_{\alpha})$ and that a Poincaré estimate holds in $\oo{W}^{1,2}_{\alpha}(K, \mu)$, for $\alpha\in (0, 1]$. Therefore, the supremum of $\sigma (L^{K}_{\alpha})$ is negative.
These results are proved without additional assumptions on $K$. In particular, we do not require that $K$ is bounded, or that its boundary is smooth.
If the boundary of $K$ is suitably smooth, it is possible to define surface integrals and traces at the boundary of functions in the Sobolev spaces $W^{1,2}_{\alpha}(K, \mu)$. Then we prove that the traces of the functions in $\oo{W}^{1,2}_{\alpha} (K,\mu)$ vanish. Therefore, the Dirichlet boundary condition in is satisfied in the sense of the trace, and $T^{K}_{\alpha}(t)\varphi $ has null trace at the boundary for every $t>0$ and $\varphi\in L^2(K,\mu)$.
Surface integrals for gaussian measures in Hilbert spaces are not a straightforward extension of the finite dimensional theory. To our knowledge the best reference is [@Bo §6.10], where the Malliavin theory is presented. It deals with level surfaces of smooth functions $g$ in a more general context than ours, since Souslin spaces $X$ are considered instead of Hilbert spaces. A part of the theory may be simplified in our Hilbert setting, and moreover some of the smoothness assumptions on $g$ can be weakened. Therefore, we end the paper with an appendix describing surface measures for level surfaces of suitably regular functions $g:H\mapsto {{\mathbb R}}$.
Several related important problems remain open, even for bounded $K$ with smooth boundary. Among them:
- While in finite dimensions $\varphi = R(\lambda, L^{K}_{\alpha})f$ is a strong solution to and it belongs to $W^{2,2}(K,\mu)$ under reasonable assumptions on the boundary $\partial K$ ([@LMP]), in infinite dimensions we do not know whether $\varphi$ possesses second order derivatives in $L^2(K,\mu)$, even if $K$ is the closed unit ball. In fact, even in the case $\alpha =1$, the estimates found in [@DPGZ; @Tal] are very bad both near the boundary and near $t=0$, and it is not clear how to use them to get informations on the resolvent.
- We do not know whether $T^{K}_{\alpha}(t)$ is strong Feller in $K$ (i.e., it maps $B_b(K)$, the space of the bounded Borel functions in $K$, to $C_b(K)$). This problem is open even for $K = \{ x\in H:\; |x| \leq 1\}$.
- In finite dimensions, if $\partial K$ is regular enough there are several characterizations of the space $\oo{W}^{1,2}_{\alpha} (K,\mu)$, that coincides with $\oo{W}^{1,2}_{1}(K,\mu)$ for every $\alpha\in [0,1]$. The most obvious is the following: since $\mu$ is locally equivalent to the Lebesgue measure, $\oo{W}^{1,2}_{1} (K,\mu)$ coincides with the space of the functions $f\in W^{1,2}_{1}(K,\mu)$ whose trace at the boundary vanishes. We do not know whether a similar characterization holds in infinite dimensions.
Referring to problem (a), in the recent paper [@BDPT] a self-adjoint realization $L $ of $\mathcal L_1$ in $L^2(K,\mu)$ with Neumann boundary condition has been studied, in the case that $K$ is a convex set with regular boundary. By means of a different (and better) approximation procedure, it has been proved that the resolvent $R(\lambda, L )$ maps $L^2(K,\mu)$ into $W^{2,2}_{1}(K,\mu)$.
Here we prove interior $W^{2,2}_{\alpha}$ regularity, for those $\alpha$ such that Tr$[Q^{1-\alpha}]<\infty$. In this case we show that for every ball $B\subset K$ with positive distance from $\partial K$ and for every $\varphi \in D(L^{K}_{\alpha})$, the restriction $\varphi_{|B}$ belongs to $W^{2,2}_{\alpha}(B, \mu)$.
Notation and preliminaries
==========================
We denote by $\langle \cdot, \cdot\rangle $ and by $|\cdot|$ the scalar product and the norm in $H$. ${\mathcal L}(H)$ is the space of the linear bounded operators in $H$.
Let $Q$ be a symmetric (strictly) positive operator in ${\mathcal L}(H)$ with finite trace, and let $A:=Q^{-1}$. Accordingly, let $\{e_k\}$ be an orthonormal basis in $H$ consisting of eigenfunctions of $Q$, i.e. $$Qe_k=\lambda_k e_k,\;\;Ae_k = \frac{1}{\lambda_k}e_k, \quad \forall\;k\in {{\mathbb N}}.$$ We denote by $D_k$ the derivative in the direction of $e_k$ and by $D$ the gradient of any differentiable function. Moreover we set $x_k=\langle x,e_k \rangle$ for all $x\in H,\; k\in {{\mathbb N}}$.
Throughout the paper we consider the $\sigma$-algebra $\mathcal{B}(H)$ of the Borel subsets of $H$ and the Gaussian measure with center $0$ and covariance $Q$ in $\mathcal{B}(H)$, denoting it by $\mu$.
An orthonormal basis of $L^2(H, \mu)$ consists of the Hermite polynomials. More precisely, for each $n\in {{\mathbb N}}\cup\{0\}$ let $$H_n(\xi) := (-1)^n n!^{-1/2} e^{\xi^2/2} D^n(e^{-\xi^2/2}), \quad \xi \in {{\mathbb R}},$$ be the usual normalized $n$-th Hermite polynomial. We denote by $\Gamma $ the set of all $\gamma : {{\mathbb N}}\mapsto {{\mathbb N}}\cup \{0\}$ such that $\sum_{k=1}^\infty \gamma(k) <\infty$. For each $\gamma\in \Gamma$ let $$H_\gamma (x) := \prod_{k=1}^{\infty} H_{\gamma(k)}\bigg( \frac{x_k}{\sqrt{\lambda_k}} \bigg), \quad x\in H,$$ be the corresponding Hermite polynomial in $H$. Then, the linear span ${\mathcal H}$ of all the Hermite polynomials $H_{\gamma}$ is dense in $L^2(H, \mu)$, and the linear span $\Lambda_0$ of the functions $H_{\gamma} \otimes e_h$, with $\gamma \in \Gamma$ and $h\in {{\mathbb N}}$, is dense in the space $L^2(H,\mu; H)$ of all the (equivalence classes of) measurable functions $F:H\mapsto H$ such that $\int_{H}|F(x)|^2 \mu (dx)<\infty$.
Other important dense subspaces of $L^2(H, \mu)$ are the spaces ${\mathcal E}_{\alpha}(H)$, the linear spans of the real and imaginary parts of the functions $x\mapsto e^{i\langle x, h \rangle} $, with $h\in D(A^{\alpha})$, $0\leq \alpha \leq 1$.
Sobolev spaces over $H$
-----------------------
We have the following integration formula, $$\int_{H} D_k\varphi \, d\mu = \frac{1}{\lambda_k}\int_H x_k\varphi \,d\mu , \quad \varphi \in {\mathcal E}_{\alpha}(H) , \; k\in {{\mathbb N}}.
\label{intpartiE}$$ It may be extended to $$\int_{H} \langle D\varphi, G\rangle d\mu + \int_{H} \varphi \, \mbox{\rm div}\,G \,d\mu = \int_{H}\varphi \langle x, AG(x)\rangle d\mu, \quad \varphi \in C^1_b(H), \; G\in \Lambda_0,
\label{intpartiG}$$ where div$\,G (x)= \sum_{k=1}^{\infty}\langle DG(x), e_k\rangle$. The linear operator $Q^{(1-\alpha)/2}D$ is well defined from ${\mathcal E}_{\alpha}(H) \subset L^2(H, \mu)$ to $ L^2(H,\mu; H)$, by $$Q^{(1-\alpha)/2}D\varphi = \sum_{k=1}^{\infty} \lambda_{k}^{(1-\alpha)/2}D_k\varphi \,e_k.$$ Using formula it is easy to see that $ Q^{(1-\alpha)/2}D$ is closable. We still denote by $ Q^{(1-\alpha)/2}D$ its closure, and by $W^{1,2}_{\alpha}(H, \mu)$ the domain of the closure. (Note that for $\alpha=0$, $ Q^{1/2}D$ is nothing but the Malliavin derivative). $W^{1,2}_{\alpha}(H, \mu)$ is endowed with the inner product $$\label{prodscal1,2}
\begin{array}{lll}
\langle \varphi,\psi \rangle_{W_{\alpha}^{1,2}(H,\mu)}&=&\ds\int_H\varphi\psi \,d\mu
+\int_H\langle Q^{(1-\alpha)/2}D\varphi,Q^{(1-\alpha)/2}D\varphi \rangle d\mu\\
\\
&=&\ds \int_H\varphi\psi \,d\mu
+\sum_{k=1}^\infty\int_H\lambda^{1-\alpha }_k D_k\varphi D_k\psi \, d\mu.
\end{array}$$ So, $W^{1,2}_{\alpha}(H, \mu)$ is the completion of ${\mathcal E}_{\alpha}(H)$ in the norm associated to the scalar product . It is also possible to characterize it through the Hermite polynomials. We have $\varphi\in W_{ \alpha}^{1,2}(H,\mu)$ iff $$\sum_{\gamma\in \Gamma} \sum_{h=1}^{\infty} \gamma_h \lambda_h^{-\alpha}\varphi_{\gamma}^2 <\infty$$ in which case the above sum is equal to $\int_H |Q^{(1-\alpha)/2}D\varphi|^2d\mu$. Indeed, the proof in [@DPZ3 Sect. 9.2.3] for $\alpha =1$ works as well for any $\alpha \in [0,1)$.
From this characterization it is clear that $W^{1,2}_{\alpha}(H, \mu) \subset W^{1,2}_{0}(H, \mu)$ for every $\alpha \in (0, 1]$, with continuous embedding.
Similarly, $W^{2,2}_{\alpha}(H, \mu)$ is the completion of ${\mathcal E}_{\alpha}(H)$ in the norm associated to the scalar product $$\begin{array}{lll}
\langle \varphi,\psi \rangle_{W_{ \alpha}^{2,2}(H,\mu)}&=&\ds\langle \varphi,\psi \rangle_{W_{ \alpha}^{1,2}(H,\mu)}+\int_H \mbox{\rm Tr}\;[Q^{2-2\alpha}D^2\varphi D^2\psi] d\mu\\
\\
&=&\ds \langle \varphi,\psi \rangle_{W_{ \alpha}^{1,2}(H,\mu)}+ \sum_{h,k=1}^\infty\int_H\lambda^{1-\alpha}_h\lambda^{1-\alpha}_k D_{h,k}\varphi D_{h,k}\psi \, d\mu.
\end{array}$$
Next lemma is a consequence of [@Bo Lemma 5.1.12] or [@DPZ3 Lemma 9.2.7].
\[emb\] There is $C>0$ such that $$\int_H |x|^2 \varphi(x)^2d\mu \leq C\|\varphi\|^2_{W^{1,2}_{0}(H, \mu)} , \quad \varphi \in W^{1,2}_{0}(H, \mu).$$
Lemma \[emb\], together with , yields the integration by parts formula in $W^{1,2}_{0}(H, \mu)$ (and hence, in all spaces $W^{1,2}_{\alpha}(H, \mu)$), $$\int_{H} D_k\varphi \, \psi\, d\mu = - \int_{H} \varphi \, D_k\psi\, d\mu + \frac{1}{\lambda_k}\int_H x_k\varphi \,\psi \, d\mu , \quad \varphi, \;\psi \in W^{1,2}_{0}(H, \mu), \;k\in {{\mathbb N}}.
\label{intparti}$$
For $0\leq \alpha \leq 1$ let $ T_{\alpha}(t)$ be the Ornstein-Uhlenbeck semigroup $$\label{OUalpha}
T_{\alpha}(t)\varphi(x) := \int_H \varphi(y){ \mathcal N}_{e^{-tA^{\alpha}/2} x, Q_t}(dy), \quad t>0,$$ with $$Q_t := \int_0^t e^{-sA^{\alpha} }Q^{1-\alpha} ds = Q(I- e^{-tA^{\alpha} }).$$ $T_{\alpha}(t)$ is a Markov semigroup in $C_b(H)$, whose unique invariant measure is $\mu$. Its extension to $L^2(H, \mu)$ is a strongly continuous contraction semigroup, still denoted by $T_{\alpha}(t)$, whose infinitesimal generator $L_{\alpha}$ is the closure of ${\mathcal L}_{\alpha} :{\mathcal E}_{\alpha}(H) \mapsto L^2(H, \mu)$.
The domain of $L_{\alpha}$ is continuously embedded in $W_{ \alpha}^{2,2}(H,\mu)$. Moreover, for any $\varphi$, $\psi\in D(L_{\alpha})$ we have $$\label{e1.6aa}
\int_{H}L_{\alpha}\varphi\;\psi\,d\mu =-\frac12\;
\int_{H}\langle Q^{(1-\alpha)/2}D\varphi, Q^{(1-\alpha)/2}D\psi\rangle d\mu.$$
We refer to [@DPZ3 Ch. 9, 10] for the proofs of the above statements, and we add further properties of the spaces $W_{ \alpha}^{1,2}(H,\mu)$ that will be used later. For each $\varphi\in L^1(H, \mu)$ we denote by $ \overline{\varphi}$ the mean value of $\varphi$, $$\overline{\varphi} := \int_H \varphi\,d\mu .$$
\[proprieta’\] Let $0\leq \alpha \leq 1$. Then
- A Poincaré estimate holds in $W_{ \alpha}^{1,2}(H,\mu)$, and precisely $$\int_H (\varphi - \overline{\varphi})^2 d\mu \leq \lambda_1^{\alpha} \int_H |Q^{(1-\alpha)/2}D\varphi|^2 d\mu,
\label{Poincare}$$ where $\lambda_1$ is the maximum eigenvalue of $Q$.
- The space $W_{ \alpha}^{1,2}(H,\mu)$ is compactly embedded in $L^2(H, \mu)$ for $\alpha >0$.
A proof of statement (a) that follows the approach of Deuschel and Strook [@DS] is in [@DPZ3 Ch.10] for $\alpha =1$. The same procedure works for $\alpha\in [0,1)$, since the key points of the proof still hold. Precisely, we have
- $|Q^{(1-\alpha)/2}DT^{\alpha}(t)\varphi|^2 \leq e^{-t/\lambda_1^{\alpha}}T^{\alpha}(t)(|Q^{(1-\alpha)/2}D\varphi|^2), \; \varphi\in C^1_b(H), \;t>0;$
- $\displaystyle \int_H \varphi L_{\alpha}\varphi \,d\mu = -\frac{1}{2} \int_H |Q^{(1-\alpha)/2}D \varphi|^2d\mu , \quad \varphi \in D( L_{\alpha});$
- $\lim_{t\to \infty} T^{\alpha}(t)\varphi (x) = \overline{\varphi}, \quad \varphi\in {\mathcal E}_{\alpha}(H), \; x\in H.$
Once (i), (ii), (iii) are satisfied one can follow the proof of [@DPZ3 Prop. 10.5.2] step by step. (ii) and (iii) follow from [@DPZ3 Prop. 10.2.3, Prop. 10.1.1]. To check that (i) holds is easy and it is left to the reader.
Statement (b) should be well known, however we give here a simple proof following [@DP Thm. 10.16] that concerns the case $\alpha =1$. We write every element $\varphi $ of $L^2(H, \mu)$ as $\varphi = \sum_{\gamma \in \Gamma} \varphi_{\gamma}H_{\gamma}$, with $\varphi_{\gamma} = \langle \varphi, H_{\gamma}\rangle$. We already remarked that $\varphi\in W_{ \alpha}^{1,2}(H,\mu)$ iff $$\sum_{\gamma\in \Gamma} \sum_{h=1}^{\infty} \gamma_h \lambda_h^{-\alpha}\varphi_{\gamma}^2 <\infty .$$ If a sequence $(\varphi ^{(n)})$ is bounded in $W_{ \alpha}^{1,2}(H,\mu)$, say $\|\varphi ^{(n)}\|_{W_{ \alpha}^{1,2}(H,\mu)}\leq K$ for each $n\in {{\mathbb N}}$, a subsequence $(\varphi ^{(n_k)})$ converges weakly in $ W^{1,2}_{ \alpha}(H,\mu)$ to a limit $\varphi $, that still satisfies $\|\varphi \|_{W_{ \alpha}^{1,2}(H,\mu)}\leq K$. We shall show that $\lim_{k\to \infty } \| \varphi ^{(n_k)} -\varphi\|_{L^2(H, \mu)} =0$.
For each $N\in {{\mathbb N}}$, let $\Gamma_N =\{ \gamma \in \Gamma: \; \sum_{h=1}^{\infty} \gamma_h \lambda_h^{-\alpha} <\infty\}$. Then $$\begin{array}{l}
\displaystyle{ \int_H (\varphi ^{(n_k)} -\varphi)^2d\mu = \sum_{\gamma\in \Gamma_N} ( \varphi ^{(n_k)}_{\gamma} - \varphi_{\gamma})^2
+ \sum_{\gamma\in \Gamma_N^c} ( \varphi ^{(n_k)}_{\gamma} - \varphi_{\gamma})^2 }
\\
\\
\leq \displaystyle{ \sum_{\gamma\in \Gamma_N} ( \varphi ^{(n_k)}_{\gamma} - \varphi_{\gamma})^2 + \frac{1}{N}\sum_{\gamma\in \Gamma} \sum_{h=1}^{\infty} \gamma_h \lambda_h^{-\alpha}( \varphi ^{(n_k)}_{\gamma} - \varphi_{\gamma})^2 }
\\
\\
\leq \displaystyle{ \sum_{\gamma\in \Gamma_N} ( \varphi ^{(n_k)}_{\gamma} - \varphi_{\gamma})^2 + \frac{(2K)^2}{N}.}
\end{array}$$ For ${\varepsilon}>0$ fix $N\in {{\mathbb N}}$ such that $4K^2/N\leq {\varepsilon}$. Since $\alpha >0$, then $\lim_{h\to \infty} \lambda_h^{-\alpha} = +\infty$, so that the set $\Gamma_N$ has a finite number of elements. Since $ \varphi ^{(n_k)}$ converges weakly to $\varphi $ in $W_{ \alpha}^{1,2}(H,\mu)$, it converges weakly to $\varphi $ in $L^2(H, \mu)$; in particular $\lim_{h\to \infty} \varphi ^{(n_k)}_{\gamma} = \varphi_{\gamma} $ for each $\gamma \in \Gamma_N$. Therefore, for $k$ large enough we have $ \sum_{\gamma\in \Gamma_N} ( \varphi ^{(n_k)}_{\gamma} - \varphi_{\gamma})^2 \leq {\varepsilon}$, and the statement follows.
Sobolev spaces over $K$
-----------------------
Throughout the paper we assume that $K\subset H$ is a closed set with positive measure. To avoid trivialities, we assume that also $K^c$ has positive measure.
To treat the Dirichlet problem we introduce Sobolev spaces over $K$. We denote by $W^{1,2}_{\alpha} (K,\mu)$ the space of the functions $u: K \mapsto {{\mathbb R}}$ that have an extension belonging to $W^{1,2}_{\alpha} (H,\mu)$, endowed with the standard inf norm. Moreover we denote by $\oo{W}^{1,2}_{\alpha} (K,\mu)$ the subspace of $W^{1,2}_{\alpha} (K,\mu)$ consisting of the functions $u: K \mapsto {{\mathbb R}}$ whose null extension to the whole $H$ belongs to the Sobolev space $W^{1,2}_{\alpha}(H,\mu)$. Therefore, $$\|u\|^{2}_{W^{1,2}_{\alpha} (K,\mu)} = \int_K u^2d\mu + \int_K |Q^{(1-\alpha)/2}Du|^2d\mu, \quad u\in \oo{W}^{1,2}_{\alpha} (K,\mu),$$ so that the $W^{1,2}_{\alpha} (K,\mu)$-norm in $\oo{W}^{1,2}_{\alpha} (K,\mu)$ is associated to the inner product $$\label{prod.scal}
\langle u,v\rangle_{W^{1,2}_{\alpha}(K,\mu)} = \int_{ K} u\,v\,d\mu + \int_{ K} \langle Q^{(1-\alpha)/2}Du, Q^{(1-\alpha)/2}Dv \rangle \,d\mu .$$ From the results of the next section it will be clear that such a space is not trivial, since it coincides with the domain of $(I-L^{K}_{\alpha})^{1/2}$. Moreover, since $W^{1,2}_{\alpha}(H,\mu)$ is continuously embedded in $W^{1,2}_{0}(H,\mu)$, then $\oo{W}^{1,2}_{\alpha} (K,\mu)$ is continuously embedded in $\oo{W}^{1,2}_{0} (K,\mu)$, for every $\alpha \in (0, 1]$.
The weak solution to {#sect:Diri}
---------------------
The quadratic form $\mathcal {\mathcal Q}_{\alpha}$ associated to $\mathcal L_{\alpha}$, $$\label{Q}
{\mathcal Q}_{\alpha}(u,v) := \frac{1}{2}\int_{ K} \langle Q^{(1-\alpha)/2}Du, Q^{(1-\alpha)/2}Dv \rangle \,d\mu , \quad u,v\in \oo{W}^{1,2}_{\alpha} (K,\mu),$$ is continuous, nonnegative, and symmetric. Therefore, for every $\lambda >0$ and $f\in L^2( K,\mu)$ there exists a unique $\varphi \in \oo{W}^{1,2}_{\alpha} (K,\mu)$ such that $$\label{LM}
\lambda \int_{ K} \varphi \,v\,d\mu + \frac{1}{2}\int_{ K} \langle Q^{(1-\alpha)/2}D\varphi , Q^{(1-\alpha)/2}Dv \rangle \,d\mu = \int_{ K} f\,v\,d\mu, \quad \forall v\in \oo{W}^{1,2}_{\alpha} (K,\mu).$$ The function $\varphi$ may be considered a weak solution to . Moreover, there exists a dissipative self-adjoint operator $M_{\alpha}$ in $L^2( K,\mu)$ such that $\varphi = R(\lambda, M_{\alpha})f$. Like all dissipative self-adjoint operators in Hilbert spaces, $M_{\alpha}$ is the infinitesimal generator of an analytic contraction semigroup, and several properties of $M_{\alpha}$ follow. See e.g. [@Kato Ch. 6].
The Dirichlet semigroup
=======================
In this section we give an explicit representation formula for the semigroup generated by the operator $M_{\alpha}$ defined in section \[sect:Diri\], through the approximation procedure described in the introduction. Moreover we show some properties of the semigroup and of its generator.
The approximating semigroups
----------------------------
We fix once and for all a function $V\in C_b(H)$ such that $$V(x)=0, \;x\in K, \quad V(x)>0, \;x\in K^c .$$ For ${\varepsilon}>0$ let $P^{{\varepsilon}}_{\alpha}(t)$ be defined by .
\[p2.2\] For any $\varphi\in C_b(H)$ we have $$\label{e2.4}
\int_H(P^{{\varepsilon}}_{\alpha}(t)\varphi(x))^2\mu(dx)\le
\int_H\varphi^2(x)\mu(dx).$$ Consequently, $P^{{\varepsilon}}_{\alpha}(t)$ is uniquely extendable to a $C_0$-semigroup in $L^2(H,\mu)$ which we shall denote by the same symbol.
We have in fact, by the Hölder inequality $$(P^{{\varepsilon}}_{\alpha}(t)\varphi(x))^2\le{{\mathbb E}}\left(\varphi^2(X_{\alpha}(t,x))e^{-\frac2{\varepsilon}\;\int_0^tV(X_{\alpha}(s,x))ds}\right)\le T_{\alpha}(t)(\varphi^2)(x),$$ where $ T_{\alpha}(t)$ is the Ornstein-Uhlenbeck semigroup defined in . Since $\mu$ is invariant for $T_{\alpha}(t)$, then $$\begin{aligned}
\int_H (P^{{\varepsilon}}_{\alpha}(t)\varphi(x))^2\mu(dx)\le \int_H T_{\alpha}(t)(\varphi^2)(x)\mu(dx)= \int_H\varphi^2(x)\mu(dx). \end{aligned}$$
We denote by $M^{{\varepsilon}}_{\alpha}$ the infinitesimal generator of $P^{{\varepsilon}}_{\alpha}(t)$ in $L^2(H,\mu)$ and we want to show that $M^{{\varepsilon}}_{\alpha}=L_{\alpha} -\frac1{\varepsilon}\,V .$ To this aim, for $\lambda>0$ and $f\in L^2(H, \mu)$ we consider the resolvent equation $$\label{e2.3}
\lambda \varphi_{\varepsilon}-L_{\alpha}\varphi_{\varepsilon}+\frac1{\varepsilon}\;V\varphi_{\varepsilon}= f.$$
\[p2.4\] Let $\lambda>0$, ${\varepsilon}>0$, and $f\in L^2(H,\mu)$. Then equation has a unique solution $\varphi_{\varepsilon}\in D(L_{\alpha})$, and the following estimates hold. $$\label{e2.7}
\int_H\varphi^2_{\varepsilon}d\mu\le \frac1{\lambda^2}\;\int_Hf^2 d\mu,$$ $$\label{e2.8}
\int_H|Q^{(1-\alpha)/2}D\varphi _{\varepsilon}|^2 d\mu\le \frac2{\lambda}\;\int_Hf^2 d\mu,$$ $$\label{e2.9}
\int_{K^c} V \varphi^2_{\varepsilon}d\mu\le \frac{\varepsilon}{\lambda }\;\int_Hf^2 d\mu.$$
Fix $\lambda >0$ and ${\varepsilon}>0$. Since $L_{\alpha}$ is maximal dissipative and $\varphi\to\frac1{\varepsilon}\, V\varphi$ is bounded and monotone increasing in $L^2(H,\mu)$, it follows by standard arguments that the operator $$D(L_{\alpha})\mapsto L^2(H,\mu), \quad \varphi \mapsto L_{\alpha} \varphi -\frac1{\varepsilon}\;V\varphi,$$ is maximal dissipative. So, equation has a unique solution $\varphi_{\varepsilon}\in D(L_{\alpha})$, that satisfies .
Multiplying both sides of by $\varphi_{\varepsilon}$, integrating over $H$ and taking into account yields $$\label{e2.10a}
\lambda\int_H \varphi_{\varepsilon}^2d\mu+\frac12\;
\int_H|Q^{(1-\alpha)/2}D\varphi_{\varepsilon}|^2d\mu+\frac1{\varepsilon}\;\int_{K^c} V \varphi_{\varepsilon}^2d\mu =
\int_H f\varphi_{\varepsilon}d\mu.$$ The inequality $\lambda\int_H|\varphi_{\varepsilon}|^2d\mu \leq \int_H f\varphi_{\varepsilon}d\mu$ yields again . The inequality $$\frac12\;
\int_H|Q^{(1-\alpha)/2}D\varphi_{\varepsilon}|^2d\mu\le \int_H f\varphi_{\varepsilon}d\mu$$ implies , using the Hölder inequality in the right-hand side and then . The inequality $$\frac1{\varepsilon}\;\int_{K^c}V \varphi_{\varepsilon}^2d\mu \le
\int_H f\varphi_{\varepsilon}d\mu$$ implies , using again the Hölder inequality in the right-hand side and then .
\[p2.3\] Let $M^{{\varepsilon}}_{\alpha}$ be the infinitesimal generator of $P^{{\varepsilon}}_{\alpha}(t)$. Then $D(M^{{\varepsilon}}_{\alpha} ) = D(L_{\alpha})$ and $$\label{e2.5}
M^{{\varepsilon}}_{\alpha}\varphi=L_{\alpha}\varphi-\frac1{\varepsilon}\;V\varphi,\quad\forall\;\varphi\in D(L_{\alpha}).$$
Let us show that $D(L_{\alpha})\subset D(M^{{\varepsilon}}_{\alpha})$, and that holds.
First, let $\varphi\in D(L_{\alpha})\cap C_b(H)$. For $x\in H$, $h>0$ we have $$\label{numero}
P^{\varepsilon}_h\varphi(x)-\varphi(x)
= T_{\alpha}(h)\varphi(x)-\varphi(x) + {{\mathbb E}}\left[\left(e^{-\frac1{\varepsilon}\;\int_0^h V(X_{\alpha}(r,x))dr}-1\right)\varphi(X_{\alpha}(h,x))\right].$$ We recall that, since $A^{\alpha}$ is self-adjoint, $X_{\alpha}(\cdot,x)$ possesses a.s. continuous paths ([@K; @T]). Therefore the functions $r\mapsto \varphi(X_{\alpha}(r,x))$ and $r\mapsto V(X_{\alpha}(r,x))$ are continuous a.s. Dividing both sides of by $h$ and letting $h\to 0$, we obtain $\lim_{h\to 0}
(P^{\varepsilon}_h\varphi -\varphi )/h = L_{\alpha}\varphi - V\,\varphi/{\varepsilon}$ pointwise and (by dominated convergence) in $L^2(H,\mu)$, so that $\varphi\in D(M^{{\varepsilon}}_{\alpha})$ and holds.
Let now $\varphi\in D(L_{\alpha})$, and let $(\varphi_n)$ be a sequence of functions in $ \mathcal E_{\alpha}(H)$ that converges to $\varphi$ in $D(L_{\alpha})$. Then, $\varphi_n\to \varphi$ in $L^2(H, \mu)$, so that $\frac1{\varepsilon}\;V \varphi_n \to \frac1{\varepsilon}\; V \varphi$ in $L^2(H, \mu)$, moreover $L_{\alpha}\varphi_n\to L_{\alpha}\varphi $ in $L^2(H, \mu)$. It follows that $M^{{\varepsilon}}_{\alpha} \varphi_n\to M^{{\varepsilon}}_{\alpha} \varphi$ in $L^2(H, \mu)$, and since $M^{{\varepsilon}}_{\alpha}$ is closed, then $\varphi \in D(M^{{\varepsilon}}_{\alpha})$ and holds.
The other inclusion $D(M^{{\varepsilon}}_{\alpha})\subset D(L_{\alpha})$ is immediate. Indeed, for any $\varphi\in D(M^{{\varepsilon}}_{\alpha})$ set $f=\lambda \varphi-M^{{\varepsilon}}_{\alpha} \varphi$, and let $ \varphi_{\varepsilon}$ be the solution of . Then $ \varphi_{\varepsilon}\in D(L_{\alpha})\subset D(M^{{\varepsilon}}_{\alpha})$, so that $(\lambda-M^{{\varepsilon}}_{\alpha})^{-1}f=\varphi_{\varepsilon}=\varphi$ which implies that $\varphi\in D(L_{\alpha})$.
From the very beginning, one would be tempted to replace the continuous function $V$ by ${1\!\!\!\;\mathrm{l}}_{K^c}$ in the definition of $M_{{\varepsilon}}$. But with this choice the proof of Proposition \[p2.3\] does not work. Indeed, it is not obvious that $(P^{\varepsilon}_h\varphi -\varphi )/h$ converges as $h\to 0$ for any $\varphi \in C_b(H)\cap D(L_{\alpha})$, if $x\in \partial K$, because the function $r\mapsto {1\!\!\!\;\mathrm{l}}_{K^c}(X_{\alpha}(r,x))$ could be discontinuous at $r=0$. If $\mu(\partial K)=0$ this difficulty is not relevant, since we are interested in $L^2$ convergence rather than in pointwise convergence. However, we prefer to make no further assumptions on $\partial K$ in this first part of the paper.
Identification of $T^{K}_{\alpha}(t)$
-------------------------------------
Let $T^{K}_{\alpha}(t)$, $P^{{\varepsilon}}_{\alpha}(t)$ be defined by , respectively.
\[p2.1\] For any $\varphi\in B_b(H)$, $t>0$, and for any $x\in K$ we have $$\label{e2.1}
\lim_{{\varepsilon}\to 0}P^{{\varepsilon}}_{\alpha}(t)\varphi(x)=T^{K}_{\alpha}(t)\varphi_{|K}(x).$$ Moreover $T^{K}_{\alpha}(t)$ is a semigroup of linear bounded operators in $B_b(K)$.
Let $t>0$, $x\in K$. Then $$\{\tau_x^K\ge t\}=\{\omega\in\Omega:\;X_{\alpha}(s,x)\in K,\;\forall\;s\in[0,t)\}$$ and $$\{\tau_x^K<t\}=\{\omega\in\Omega:\;\exists\;s_0\in (0,t):\; X_{\alpha}(s_0,x)\in K^c\}$$ Then we have $$P^{{\varepsilon}}_{\alpha}(t)\varphi(x)=\int_{\{\tau_x^K\ge t\}}\varphi(X_{\alpha}(t,x))d{{\mathbb P}}+\int_{\{\tau_x^K<t\}}\varphi(X_{\alpha}(t,x))e^{-\frac1{\varepsilon}\;\int_0^t V(X_{\alpha}(s,x))ds}d{{\mathbb P}}$$ In view of the dominated convergence theorem, to prove the statement it is enough to show that $$\label{e2.2}
\lim_{{\varepsilon}\to 0} e^{-\frac1{\varepsilon}\;\int_0^t V(X_{\alpha}(s,x))ds}=0,$$ for a.a. $\omega$ such that $\tau_x^K(\omega)<t$.
We already mentioned that $X_{\alpha}(\cdot,x)$ possesses a.s. continuous paths. Let $\omega\in \Omega$ be such that $X_{\alpha}(\cdot,x)(\omega)$ is continuous. If $\tau_x^K(\omega)<t$, there exist $s_0 <t$, $\delta >0$ (depending on $\omega$) such that $$X_{\alpha}(s ,x) \in K^c,\quad\forall\; s\in [s_0-\delta ,s_0+\delta].$$ Since $V$ is continuous and it has positive values in $K^c$, then $$c:= \inf \{ V(X(s,x)):\; s\in [s_0-\delta ,s_0+\delta]\} >0.$$ It follows that $$e^{-\frac1{\varepsilon}\;\int_0^t V(X_{\alpha}(s,x))ds}\le e^{-\frac{2c}{{\varepsilon}}\;\delta }\to 0,\;\mbox{\rm as}\;{\varepsilon}\to 0.$$ So, holds. The last statement is straightforward.
In the next proposition we show that $\mu$ is sub-invariant for $T^{K}_{\alpha}(t)$. We use the following notation. For each $\varphi\in B_b(K)$ we set $$\widetilde{\varphi}(x)=\left\{\begin{array}{l}
\varphi(x),\quad\mbox{\rm if}\;x\in K,\\
0, \quad\mbox{\rm if}\;x\notin K.
\end{array}\right.$$
\[p3.1\] For any $\varphi\in B_b(K)$, $t>0$, we have $$\label{e3.2}
\int_K(T^{K}_{\alpha}(t)\varphi(x))^2\mu(dx)\le \int_K\varphi^2(x)\mu(dx).$$ Consequently, $T^{K}_{\alpha}(t)$ can be uniquely extended to a $C_0$ semigroup of contractions in $L^2(K,\mu)$.
By the Hölder inequality we have for all $x\in K$ $$(T^{K}_{\alpha}(t)\varphi(x))^2\le {{\mathbb E}}[\varphi^2(X_{\alpha}(t,x)){1\!\!\!\;\mathrm{l}}_{\tau_x^K\ge t}]
\le {{\mathbb E}}[\widetilde{\varphi}^2(X_{\alpha}(t,x)){1\!\!\!\;\mathrm{l}}_{\tau_x^K\ge t}]\le T_{\alpha}(t)(\widetilde{\varphi}^2)(x) .$$ Since $\mu$ is invariant for $T_{\alpha}(t)$, it follows that $$\begin{array}{l}
\ds \int_K(T^{K}_{\alpha}(t)\varphi(x))^2d\mu \le \int_KT_{\alpha}(t)(\widetilde{\varphi}^2)(x)d\mu \\
\\
\ds\le \int_HT_{\alpha}(t)(\widetilde{\varphi}^2)d\mu \le\int_HT_{\alpha}(t)(\widetilde{\varphi}^2)d\mu \le \int_H\widetilde{\varphi}^2d\mu =\int_K\varphi(x)^2d\mu .
\end{array}$$ The conclusion follows.
We shall denote by $L^{K}_{\alpha}$ the infinitesimal generator of $T^{K}_{\alpha}(t)$ in $L^2(K,\mu)$.
\[p3.2\] For any $f\in L^2(K,\mu)$ and $t>0$ we have $$\label{e3.4c}
\lim_{\epsilon\to 0} (P^{{\varepsilon}}_{\alpha}(t) \widetilde{ f})_{|K} = T^{K}_{\alpha}(t) f,\quad \mbox{\rm in}\;L^2(K,\mu)$$ and, for $\lambda >0$, $$\label{e3.4}
\lim_{\epsilon\to 0} (R(\lambda, M^{{\varepsilon}}_{\alpha} ) \widetilde{ f})_{|K} =(\lambda-L^{K}_{\alpha})^{-1}f,\quad\mbox{\rm in}\; L^2(K,\mu).$$
Let $f\in C_b(H)$. By Proposition \[p2.1\], $P^{{\varepsilon}}_{\alpha} f $ converges pointwise to $T^{K}_{\alpha}(t) f$ in $K$. Moreover, $| (P ^{{\varepsilon}}_{\alpha}(t)f)(x)| \leq \|f\|_{\infty}$, $|(T^{K}_{\alpha}(t)f)(x)| \leq \|f\|_{\infty}$ for each $x\in K$ and $t>0$. By dominated convergence, $\lim_{{\varepsilon}\to 0} \| P^{{\varepsilon}}_{\alpha}(t) f - T^{K}_{\alpha}(t) f\|_{L^2(K,\mu)} =0$.
Let now $f\in L^2(K,\mu)$. Since $C_b(H)$ is dense in $L^2(H,\mu)$, there is a sequence $(f_n)\subset C_b(H)$ such that $$\|\widetilde{f}-f_n\|_{L^2(H,\mu)}\le \frac1n,\quad\forall\;n\in {{\mathbb N}}.$$ Then we have $$\begin{array}{l}
\|T^{K}_{\alpha}(t) f-P^{{\varepsilon}}_{\alpha}(t)\widetilde{ f}\|_{L^2(K,\mu)} \le
\|T^{K}_{\alpha}(t) ( f - f_n)\|_{L^2(K,\mu)}\\
\\
+\|T^{K}_{\alpha}(t) f_n- P^{{\varepsilon}}_{\alpha}(t) f_n \|_{L^2(K,\mu)}+\| P^{{\varepsilon}}_{\alpha}(t) (f_n - \widetilde{f}) \|_{L^2(K,\mu)}
\\
\\
\le \ds \frac2n+\|T^{K}_{\alpha}(t) f_n- P^{{\varepsilon}}_{\alpha}(t) \widetilde{f}_n \|_{L^2(K,\mu)},\quad\forall\;n\in {{\mathbb N}},
\end{array}$$ and follows.
To prove , we use the identity (in $L^2(H,\mu)$) $$R(\lambda, M^{{\varepsilon}}_{\alpha} ) \widetilde{ f} = \int_{0}^{\infty} e^{-\lambda t} P^{{\varepsilon}}_{\alpha}(t)\widetilde{ f} \, dt .$$ Taking the restrictions to $K$ of both sides and using we obtain $$\lim_{\epsilon\to 0} (R(\lambda, M^{{\varepsilon}}_{\alpha} ) \widetilde{ f})_{|K} = \int_{0}^{\infty} e^{-\lambda t} T^{K}_{\alpha}(t) f\, dt ,$$ which coincides with .
\[Identificazione\] For every $\lambda >0$ and $f\in L^2( K,\mu)$, the function $\varphi:=R(\lambda, L^{K}_{\alpha})f$ belongs to $\oo{W}^{1,2}_{\alpha}(K,\mu)$ and satisfies . Therefore, $T^{K}_{\alpha}(t)$ is the semigroup generated by $M_{\alpha}$ in $L^2( K,\mu)$.
For ${\varepsilon}>0$ define $\varphi_{{\varepsilon}} := R(\lambda, M^{{\varepsilon}}_{\alpha} ) \widetilde{ f} $. By Proposition \[p2.3\], $\varphi_{{\varepsilon}}$ is the solution to , with $f$ replaced by $ \widetilde{ f} $. By Proposition \[p2.4\], the $W^{1,2}_{\alpha}(H, \mu)$-norm of $\varphi_{{\varepsilon}}$ is bounded by a constant independent of ${\varepsilon}$. Therefore, there is a sequence ${\varepsilon}_k\to 0$ such that $ \varphi_{{\varepsilon}_k} $ converges weakly in $W^{1,2}_{ \alpha}(H, \mu)$ to a function $\Phi$. Let us prove that $\Phi = \widetilde{\varphi}$.
For every $\psi \in L^2(K, \mu)$ we have $$\int_K \Phi \psi \,d\mu = \lim_{k\to \infty}\int_H \varphi_{{\varepsilon}_k}\widetilde{\psi}\,d\mu = \lim_{k\to \infty}\int_K \varphi_{{\varepsilon}_k}\psi\,d\mu
= \int_K \varphi \psi \,d\mu$$ since, by Proposition \[p3.2\], $\lim_{{\varepsilon}\to 0} \| \varphi_{{\varepsilon}|K} - \varphi\|_{L^2( K,\mu)} = 0$. Then, $\Phi_{|K} = \varphi$.
Moreover, $$\int_{K^c} \Phi^2V\,d\mu = \int_H \Phi\cdot \Phi V{1\!\!\!\;\mathrm{l}}_{K^c} d\mu = \lim_{k\to \infty}\int_H \varphi_{{\varepsilon}_k}\Phi V{1\!\!\!\;\mathrm{l}}_{K^c} d\mu ,$$ and by estimate and the Hölder inequality we have $$\bigg| \int_H \varphi_{{\varepsilon}_k}\Phi V{1\!\!\!\;\mathrm{l}}_{K^c} d\mu \bigg| \leq \bigg( \int_{K^c} \varphi_{{\varepsilon}_k}^2V\, d\mu \bigg)^{1/2}
\bigg( \int_{K^c} \Phi^2V\,d\mu \bigg)^{1/2} \to 0 \quad \mbox{\rm as}\;k\to \infty .$$ It follows that $\Phi_{|K^c} =0$. Therefore, $\Phi = \widetilde{\varphi}\in W^{1,2}_{\alpha}(H, \mu)$, that is $\varphi \in \oo{W}^{1,2}_{\alpha}(K,\mu)$.
For every $v\in \oo{W}^{1,2}_{\alpha}(K,\mu)$ and $k\in {{\mathbb N}}$ we have (since $\int_H V \varphi_{{\varepsilon}_k} \widetilde{v}\,d\mu = 0$) $$\lambda \int_H \varphi_{{\varepsilon}_k} \widetilde{v}\,d\mu + \frac{1}{2} \int_H \langle Q^{(1-\alpha)/2}D\varphi_{{\varepsilon}_k}, Q^{(1-\alpha)/2}D\widetilde{v} \rangle d\mu = \int_H fv \,d\mu ,$$ and letting $k\to \infty$ we obtain $$\lambda \int_H \widetilde{\varphi} \widetilde{v} \,d\mu + \frac{1}{2} \int_H \langle Q^{(1-\alpha)/2}D\widetilde{\varphi} , Q^{(1-\alpha)/2}D\widetilde{v} \rangle d\mu = \int_H f\widetilde{v} \,d\mu ,$$ so that $\varphi$ satisfies , and the statement follows.
Consequences
------------
We list here some consequences of the results of this section, that hold for every $\alpha \in [0,1]$.
- $T^{K}_{\alpha}(t)$ is an analytic semigroup in $L^p( K,\mu)$ for every $p\in (1, \infty)$.
- The space $ \oo{W}^{1,2}_{\alpha}(K,\mu)$ coincides with the domain of $(I-L^{K}_{\alpha})^{1/2}$.
- For each $f\in L^2( K,\mu)$ we have $$\int_K | Q^{(1-\alpha)/2}DT^{K}_{\alpha}(t)f|^2\mu(dx)\le\frac1{\sqrt t}\;\int_K f^2(x)\mu(dx),\quad t>0.$$
These statements follow in a standard way from the fact that the infinitesimal generator $L^K_{\alpha}$ of $T^{K}_{\alpha}(t)$ is the operator associated to the symmetric quadratic form $\mathcal Q_{\alpha}$ defined in , and that it is dissipative.
Less standard consequences are a Poincaré inequality in the space $\oo{W}^{1,2}_{\alpha}(K,\mu)$ and the invertibility of $L^K_{\alpha}$ for $\alpha >0$, proved in the next proposition.
\[Pr:poinc\] For $\alpha \in (0,1]$ the spaces $\oo{W}^{1,2}_{\alpha}(K,\mu)$ and $D(L^{K}_{\alpha})$ are compactly embedded in $ L^2(K,\mu)$. Moreover $0\in \rho (L^{K}_{\alpha})$, and a Poincaré inequality holds in $\oo{W}^{1,2}_{\alpha}(K,\mu)$, $$\|u\|_{L^2( K,\mu)} \leq C \int_K | Q^{(1-\alpha)/2}Du|^2\,d\mu, \quad u\in \oo{W}^{1,2}_{\alpha}(K,\mu).$$
Since the embedding $W^{1,2}_{\alpha}(H, \mu)\subset L^2(H, \mu)$ is compact by Proposition \[proprieta’\](b), the embedding $\oo{W}^{1,2}_{\alpha}(K,\mu) \subset L^2(K,\mu)$ is compact too. Indeed, a sequence $u_n$ is bounded in $\oo{W}^{1,2}_{\alpha}(K,\mu)$ iff the sequence $\widetilde{u}_n$ is bounded in $W^{1,2}_{\alpha}(H, \mu)$. In this case, there is a subsequence of $ \widetilde{u}_n $ that converges to a function $v\in L^2(H, \mu)$. Therefore, a subsequence of $u_n$ converges to the restriction $v_{|K}$, in $L^2(K,\mu)$.
Since the domain $D(L^{K}_{\alpha})$ is continuously embedded in $\oo{W}^{1,2}_{\alpha}(K,\mu)$, it is compactly embedded in $L^2( K,\mu)$. Therefore, the spectrum of $L^K_{\alpha}$ consists of (nonpositive) eigenvalues. Let us prove that $0$ is not an eigenvalue.
Let $u\in D(L^{K}_{\alpha})$ be such that $L^{K}_{\alpha}u=0$. Then $$0 = \int_K u L^{K}_{\alpha}u\,d\mu = -\frac{1}{2}\int_K |Q^{(1-\alpha)/2}Du|^2\,d\mu = -\frac{1}{2}\int_H |Q^{(1-\alpha)/2}D\widetilde{u}|^2d\mu ,$$ and by the Poincaré inequality in $W^{1,2}_{\alpha}(H,\mu)$ (Proposition \[proprieta’\](a)) we have $$\int_{H} (\widetilde{u} - \int_H\widetilde{u}\, d\mu)^2d\mu =0.$$ So, $\widetilde{u}$ is constant a.e. in $H$, but since it vanishes in $K^c$, whose measure is positive, then it vanishes a.e. in $H$. Therefore, $u=0$.
This implies that the seminorm $u\mapsto (\int_K |Q^{(1-\alpha)/2}Du|^2\,d\mu)^{1/2}$ is in fact an equivalent norm in $\oo{W}^{1,2}_{\alpha}(K,\mu)$, that is, a Poincaré inequality holds in $\oo{W}^{1,2}_{\alpha}(K,\mu)$. Indeed, since $-L^{K}_{\alpha}$ is invertible, also $(-L^{K}_{\alpha})^{1/2}$ is invertible, so that the seminorm $u\mapsto \|(-L^{K}_{\alpha})^{1/2}u\|_{L^2(K,\mu)} =
\frac{1}{2}\int_K |Q^{(1-\alpha)/2}Du|^2\,d\mu$ is an equivalent norm in $D((-L^{K}_{\alpha})^{1/2}) = \oo{W}^{1,2}_{\alpha}(K,\mu)$; in other words there is $C>0$ such that $\|u\|_{L^2(K,\mu)} \leq C \int_K |Q^{(1-\alpha)/2}Du|^2\,d\mu$ for every $u\in \oo{W}^{1,2}_{\alpha}(K,\mu)$.
Interior regularity
===================
In this section we prove an interior regularity result for the solution to for $ \alpha <1$. We use the following lemma.
\[prodotto\] For every $\varphi \in D(L_{\alpha})$ and for every $\beta\in {\mathcal E}_{\alpha}(H)$, the product $\varphi \beta$ belongs to the domain of $L_{\alpha}$, and $$L_{\alpha}(\varphi\beta) = \beta L_{\alpha}\varphi + \varphi L_{\alpha}\beta + \langle Q^{1-\alpha}D\varphi, D\beta \rangle.$$
Since ${\mathcal E}_{\alpha}(H)$ is dense in $D(L_{\alpha})$, there is a sequence $(\varphi_n)\subset {\mathcal E}_{\alpha}(H)$ that converges to $\varphi$ in $D(L_{\alpha})$. For every $n$, $\beta \varphi_n$ is still in ${\mathcal E}_{\alpha}(H)$, hence it belongs to $D(L_{\alpha})$ and the statement follows easily.
\[RegPalla\] Assume that $$\mbox{\rm Tr}\;Q^{1-\alpha} = \sum_{k=1}^{\infty}\lambda_k^{1-\alpha} <\infty.$$ Then for every $y\in \oo{K}$ and $r>0$ such that dist$(B(y, r),$ $ \partial K)>0$, the restriction to $B(y, r)$ of the solution $\varphi $ to belongs to $W^{2,2}_{\alpha}(B(y, r), \mu)$.
It is enough to prove that the statement holds for $y\in D(A^{\alpha/2})$. Indeed, since $D(A^{\alpha/2})$ is dense in $H$, for each $y\in \oo{K}$ and $r>0$ such that dist$(B(y, r),$ $ \partial K)>0$ there are $y_1\in \oo{K}\cap D(A^{\alpha/2})$ and $r_1>r$ such that $B(y, r)\subset B(y_1, r_1)$ and dist$(B(y_1, r),$ $ \partial K)>0$.
So, let $y\in D(A^{\alpha/2} )$ and let $r_1>r$ be such that the ball $B(y, r_1)$ is contained in $\oo{K}$. Let $\rho :{{\mathbb R}}\mapsto [0,1]$ be a $C^2$ function such that $$\rho (\xi)=1, \; \xi \leq r^2, \quad \rho(\xi )=0, \;\xi\geq r_1^2,$$ and define a cutoff function $\theta$ by $$\theta(x) := \rho( |x-y|^2), \quad x\in H.$$ Our aim is to show that the product $\widetilde{\varphi} \theta$ belongs to $ W^{2,2}_{\alpha}(H, \mu)$. Since the restriction to $B(y, r)$ of $\widetilde{\varphi} \theta$ coincides with the restriction to $B(y, r)$ of $\varphi$, the statement will follow.
The proof is in three steps. As a first step, we show that $\theta\in D(L_{\alpha})$. Then we show that $ \varphi_{{\varepsilon}} \theta $ belongs to $D(L_{\alpha})$ for every ${\varepsilon}>0$, where $ \varphi_{{\varepsilon}} = R(\lambda, M^{{\varepsilon}}_{\alpha} )\widetilde{f}$. Eventually, we prove that $\widetilde{\varphi} \theta \in W^{2,2}_{\alpha}(H, \mu)$.
[*First step: $\theta\in D(L_{\alpha})$.*]{} We approach each $x\in H$ by the sequence $x_n = \sum_{k=1}^{n}\langle x, e_k \rangle e_k$, and we consider the sequence of functions $$\theta_n(x):= \rho (|x_n-y_n|^2), \quad x\in H, \;n\in {{\mathbb N}}.$$ Each of them belongs to $D(L_{\alpha})$. This is because it depends only on the first $n$ coordinates, it is bounded and it has bounded first and second order derivatives, and in finite dimensions the inclusion $C^2_b(H) \subset D(L_{\alpha}) $ holds. Therefore, it is easy to see that there exists the limit $\lim_{t\to 0}( T_{\alpha}(t) \theta_n - \theta_n)/t = L_{\alpha} \theta_n$ in $L^2(H, \mu)$, where $$\label{exp}
\begin{array}{lll}
L_{\alpha} \theta_n (x) & =& \rho' (|x_n-y_n|^2) \displaystyle{\bigg( \sum_{k=1}^{n} \lambda_k^{1-\alpha} - \sum_{k=1}^{n}\lambda_k^{-\alpha} \langle x , e_k\rangle \langle x-y, e_k\rangle \bigg)}
\\
\\
& + & 2\rho '' (|x_n-y_n|^2)\langle Q^{1-\alpha}(x_n-y_n), x_n-y_n\rangle .
\end{array}$$
Letting $n\to \infty$, $ \rho' (|x_n-y_n|^2)$ and $\rho '' (|x_n-y_n|^2)\langle Q^{1-\alpha}(x_n-y_n), x_n-y_n\rangle $ converge in $L^2(H, \mu)$ to $\rho'(|x-y|^2)$ and to $\rho '' (|(x -y |^2)\langle Q^{1-\alpha}(x -y ), x -y \rangle $, respectively, by dominated convergence. The sum $ \sum_{k=1}^{n}\lambda_k^{-\alpha} \langle x , e_k\rangle \langle x-y, e_k\rangle $ converges too. Indeed, for $p<q\in {{\mathbb N}}$ we have $$\begin{array}{l}
\ds{ \| \sum_{k=p}^{q}\lambda_k^{-\alpha} \langle x , e_k\rangle \langle x-y, e_k\rangle \|_{L^2(H, \mu)}}
\\
\\
\ds{\leq \sum_{k=p}^{q}\| \lambda_k^{-\alpha/2} \langle x , e_k\rangle\|_{L^2(H, \mu)}\| \lambda_k^{-\alpha/2} \langle x-y, e_k\rangle \|_{L^2(H, \mu)} }
\\
\\
\ds{ = \sum_{k=p}^{q} \lambda_k^{(1-\alpha)/2} \lambda_k^{-\alpha/2} ( \lambda_k + | \langle y, e_k\rangle|^2)^{1/2} }
\\
\\
\ds{ \leq \sum_{k=p}^{q} \lambda_k^{(1-\alpha)/2}( \lambda_k^{(1-\alpha)/2} + \lambda_k^{-\alpha/2} | \langle y, e_k\rangle|)}
\\
\\
\ds{
\leq \sum_{k=p}^{q} \lambda_k^{1-\alpha } + \frac{1}{2} ( \lambda_k^{1-\alpha } + \lambda_k^{-\alpha} | \langle y, e_k\rangle| ^2),}
\end{array}$$ where $\sum_{k=1}^{\infty} \lambda_k^{1-\alpha } <\infty$ by assumption, and $\sum_{k=1}^{\infty} \lambda_k^{-\alpha} | \langle y, e_k\rangle|^2 <\infty $ because $y\in D(A^{\alpha/2})$ . Therefore, $$\exists L^2(H, \mu)-\lim_{n\to \infty} \sum_{k=1}^{n}\lambda_k^{-\alpha} \langle x , e_k\rangle \langle x-y, e_k\rangle := \langle x, A^{\alpha}(x-y)\rangle .$$ (Note that $\langle x, A^{\alpha}(x-y)\rangle$ is not defined pointwise). It follows that $ \rho' (|x_n-y_n|^2)\cdot$ $ \sum_{k=1}^{n}\lambda_k^{-\alpha} \langle x , e_k\rangle \langle x-y, e_k\rangle$ converges to $ \rho' (|(x -y |^2) \langle x, A^{\alpha}(x-y)\rangle$ in $L^2(H, \mu)$. Since $L_{\alpha}$ is closed, $\theta\in D( L_{\alpha})$.
[*Second step: $ \varphi_{{\varepsilon}} \theta $ belongs to $D(L_{\alpha})$.*]{}
Since $ \varphi_{{\varepsilon}} \in D(L_{\alpha})$ and ${\mathcal E}_{\alpha}(H)$ is a core of $L_{\alpha}$, there is a sequence of exponential functions $\beta_n$ that converges to $ \varphi_{{\varepsilon}}$ in $D(L_{\alpha})$. Since $\theta$ is bounded, $\beta_n\theta $ converges to $ \varphi_{{\varepsilon}} \theta $ in $L^2(H, \mu)$. By Lemma \[prodotto\], $\beta_n\theta$ belongs to $ D(L_{\alpha})$ for every $n$, and we have $$L_{\alpha}( \beta_n \theta) = \beta_n L_{\alpha}\theta + \theta L_{\alpha}\beta_n + \langle Q^{1-\alpha}D\beta_n, D\theta \rangle.$$ As $n \to \infty$, $\beta_n$ converges to $ \varphi_{{\varepsilon}} $, $L_{\alpha}\beta_n $ converges to $ L_{\alpha} \varphi_{{\varepsilon}} $, and $\langle Q^{1-\alpha}D\beta_n, D\theta \rangle = \langle Q^{(1-\alpha)/2}D\beta_n, Q^{(1-\alpha)/2} D\theta \rangle $ converges to $ \langle Q^{(1-\alpha)/2 }D
\varphi_{{\varepsilon}} , Q^{(1-\alpha)/2 }D\theta \rangle $ in $L^2(H, \mu)$ since $D(L_{\alpha})\subset W^{1,2}_{\alpha}(H, \mu)$ and $Q^{(1-\alpha)/2} D\theta$ is bounded. Therefore, $ L_{\alpha}( \beta_n \theta)$ converges in $L^2(H, \mu)$, and since $ L_{\alpha}$ is closed, $ \varphi_{{\varepsilon}} \theta $ belongs to $D(L_{\alpha})$ and $$L_{\alpha} ( \theta \varphi_{{\varepsilon}} ) = (L_{\alpha} \theta ) \varphi_{{\varepsilon}} + \langle Q^{(1-\alpha)/2}D\theta, Q^{(1-\alpha)/2}D \varphi_{{\varepsilon}}\rangle + \theta L_{\alpha} \varphi_{{\varepsilon}}.
\label{varphitheta}$$ [*Third step: $\widetilde{\varphi} \theta$ belongs to $W^{2,2}_{\alpha}(H, \mu)$.*]{} Using and we get $$\lambda \theta \varphi_{{\varepsilon}} - L_{\alpha} (\theta \varphi_{{\varepsilon}} ) = \theta \widetilde{f }-(L_{\alpha} \theta ) \varphi_{{\varepsilon}} - \langle Q^{(1-\alpha)/2}D\theta, Q^{(1-\alpha)/2}D \varphi_{{\varepsilon}}\rangle := f_{1, {\varepsilon}}.$$ The $L^2$ norm of the right hand side $f_{1, {\varepsilon}}$ is bounded by a constant independent of ${\varepsilon}$. Therefore, $\|\theta \varphi_{\varepsilon}\|_{D( L_{\alpha})}$ is bounded by a constant independent of ${\varepsilon}$, and since $D( L_{\alpha})$ is continuously embedded in $W^{2,2}_{\alpha}(H, \mu)$, also $\|\theta \varphi_{\varepsilon}\|_{W^{2,2}_{\alpha}(H, \mu)}$ is .
Let $\{{\varepsilon}_k\}$ be the sequence used in the proof of Proposition \[Identificazione\], so that $ \varphi_{{\varepsilon}_k}$ converges weakly in $W^{1,2}_{\alpha}(H, \mu)$ to $\widetilde{\varphi}$. Possibly taking a further subsequence, $(\theta \varphi_{{\varepsilon}_k})$ converges weakly in $W^{2,2}_{\alpha}(H, \mu)$ to a function $u$ that belongs to $W^{2,2}_{\alpha}(H, \mu)$. Then $u = \theta \widetilde{\varphi}$; indeed, for each $\psi \in L^2(H, \mu)$ we have $$\int_H u\,\psi\,d\mu = \lim_{k\to \infty}\int_H \theta \varphi_{{\varepsilon}_k}\psi\,d\mu = \lim_{k\to \infty}\int_H \theta \widetilde{\varphi}\psi\,d\mu .$$ So, $\theta \widetilde{\varphi} \in W^{2,2}_{\alpha}(H, \mu)$.
Domains with smooth boundaries
==============================
In this section we assume that $$K = \{ x\in H:\; g(x) \leq 1\}$$ where $g: H\mapsto {{\mathbb R}}$ is a $C^1$ function that belongs to $D(L_0)$ and satisfies . Moreover we assume that $\sup g >1$, so that $K$ is a proper subset of $H$, and $\inf g <1$, so that the interior part of $K$ is not empty and the surface measure $d\sigma$ is well defined in the boundary $\Sigma$ of $K$, $\Sigma= \{ x\in H:\; g(x) = 1\}$. See the Appendix, to which we refer for the definition and properties of surface measures.
The aim of this section is to give a reasonable definition of the trace at $\partial K$ of any function in $W^{1,2}_{\alpha}(H, \mu)$, and to show that the functions in $\oo{W}^{1,2}_{\alpha}(H, \mu)$ have null trace at $\partial K$. This implies that $R(\lambda, L^{K}_{\alpha})f$ satisfies the Dirichlet boundary condition in in the sense of the trace for every $f\in L^2(K, \mu)$, and that $T^{K}_{\alpha}(t)f$ has null trace at the boundary for every $t>0$ and $f \in L^2(K,\mu)$.
As a first step we prove integration formulas for functions in the core ${\mathcal E}_{0}(H)$.
\[p5.1\] Let $k\in {{\mathbb N}}$ be such that $D_kg/|Q^{1/2}Dg|\in W^{2,2}_{0}(H, \mu)$. Then for every $\varphi\in {\mathcal E}_{0}(H)$ we have $$\label{parti}
\int_{K} D_k\varphi \,d\mu = \frac{1}{\lambda_k} \int_{K} x_k\varphi \,d\mu + \int_{\Sigma} \frac{D_k g}{|Q^{1/2}Dg| }\varphi \,d\sigma.$$ If $ |Q^{1/2}Dg|\in W^{2,2}_{0}(H, \mu)$, then for every $\varphi\in {\mathcal E}_{0}(H)$ we have
$$\label{e5.1}
\ds{ \int_{\Sigma} \varphi^2 |Q^{1/2}Dg| \,d\sigma_1}
\left\{ \begin{array}{ll}
= \ds{ \int_K \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \,d\mu + \int_K L_0g \,\varphi^2 \,d\mu } & (a)
\\
\\
= - \ds{ \int_{K^c} \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \,d\mu - \int_{K^c} L_0g \,\varphi^2 \,d\mu } & (b)
\end{array}\right.$$
For small ${\varepsilon}>0$ define the pathwise linear function $\theta_{{\varepsilon}}$ by $$\theta_{{\varepsilon}}(\xi) :=\left\{ \begin{array}{ll}
2, & \xi\leq 1-{\varepsilon},
\\
\frac{1}{ {\varepsilon}}(1-\xi) +1, & 1-{\varepsilon}<\xi < 1+{\varepsilon},
\\
0, & \xi \geq 1+{\varepsilon}.
\end{array}\right.,$$ and set $$\rho_{{\varepsilon}}(x) := \theta_{{\varepsilon}}(g(x)), \quad x\in H.$$ Since $\theta_{{\varepsilon}}$ is Lipschitz continuous, then $\rho_{{\varepsilon}}\in W^{1,2}_{0}(H, \mu)$ ([@Bo Rem. 5.2.1]). Then the product $\rho_{{\varepsilon}}\varphi $ belongs to $W^{1,2}_{0}(H, \mu)$ and $D_k (\rho_{{\varepsilon}}\varphi )= \theta_{{\varepsilon}}'(g(x))D_k g(x)\varphi(x) + \rho_{{\varepsilon}}(x)D_k \varphi(x)$, so that $$\label{prima}
\int_H (D_k\varphi ) \rho_{{\varepsilon}} \,d\mu -\frac{1}{ {\varepsilon}} \int_{1-{\varepsilon}<g <1+{\varepsilon}} \varphi D_k g \,d\mu = \frac{1}{\lambda_k}
\int_H x_k\varphi \rho_{{\varepsilon}} \,d\mu, \quad k\in {{\mathbb N}}.$$ Let us prove . Letting ${\varepsilon}\to 0$, $\rho_{{\varepsilon}}$ converges pointwise to $2{1\!\!\!\;\mathrm{l}}_{K}$ in $H\setminus \Sigma$, whose measure is $1$. Since $\rho_{{\varepsilon}}\leq 2$, by dominated convergence we get $$\exists \lim_{{\varepsilon}\to 0} \frac{1}{2 {\varepsilon}} \int_{1-{\varepsilon}<g <1+{\varepsilon}} \varphi D_k g \,d\mu = \int_{K} D_k\varphi \,d\mu - \frac{1}{\lambda_k} \int_{K} x_k\varphi \,d\mu .$$ Let us identify the limit in the left hand side as a boundary integral. Since $\varphi D_kg |Q^{1/2}Dg|^{-1} $ $\in $ $W^{2,2}_{0}(H, \mu)$, by Remark \[rem:limite\] we have $$\lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{1-{\varepsilon}<g <1+{\varepsilon}} \varphi D_k g \,d\mu = \int_{\Sigma } \frac{D_kg}{ |Q^{1/2}Dg|} \varphi \,d\sigma$$ and follows.
Let us prove (a). For every ${\varepsilon}>0$ and $k\in {{\mathbb N}}$, the function $\rho_{{\varepsilon}}\varphi ^2D_kg$ still belongs to $W^{1,2}_{0}(H, \mu)$. Therefore we may replace $\varphi $ in by $\lambda_k \varphi^2D_kg$, and summing over $k$ (recall Lemma \[emb\]), we obtain $$\int_H 2\varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \rho_{{\varepsilon}} \,d\mu + \int_H 2L_0g \,\varphi^2 \rho_{{\varepsilon}} \,d\mu$$ $$= \frac{1}{ {\varepsilon}} \int_{1-{\varepsilon}<g <1+{\varepsilon}} \varphi^2 |Q^{1/2}Dg|^2 \,d\mu$$ Letting ${\varepsilon}\to 0$ as before, by dominated convergence we get $$\lim_{{\varepsilon}\to 0} \int_H \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \rho_{{\varepsilon}} \,d\mu =
\int_K \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \,d\mu,$$ $$\lim_{{\varepsilon}\to 0} \int_H L_0g \,\varphi^2 \rho_{{\varepsilon}} \,d\mu = \int_K L_0g \,\varphi^2 \,d\mu .$$ Therefore, there exists the limit $$\lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{1-{\varepsilon}<g <1+{\varepsilon}} \varphi^2 |Q^{1/2}Dg|^2 \,d\mu = \int_K \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \,d\mu + \int_K L_0g \,\varphi^2 \,d\mu$$ that we identify as a boundary integral. Indeed, since $\varphi^2 |Q^{1/2}Dg| \in W^{2,2}_{0}(H, \mu)$, by Remark \[rem:limite\] we have $$\lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{1-{\varepsilon}<g <1+{\varepsilon}} \varphi^2 |Q^{1/2}Dg|^2 \,d\mu = \int_{\Sigma} \varphi^2 |Q^{1/2}Dg| \,d\sigma.$$ So, (a) holds. To prove (b), we may follow the same procedure replacing $K$ by $K^c$ and $\theta_{{\varepsilon}}$ by $$\widetilde{\theta}_{{\varepsilon}}(\xi) :=\left\{ \begin{array}{ll}
0, & \xi\leq 1-{\varepsilon},
\\
\frac{1}{ {\varepsilon}}( \xi -1) +1, & 1-{\varepsilon}<\xi < 1+{\varepsilon},
\\
2, & \xi \geq 1+{\varepsilon},
\end{array}\right.$$ or else, we may use the equality $$\int_{K^c} \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \,d\mu + \int_{K^c} L_0g \,\varphi^2 \,d\mu$$ $$= - \int_K \varphi \langle Q^{1/2}D\varphi, Q^{1/2}Dg\rangle \,d\mu - \int_K L_0g \,\varphi^2 \,d\mu$$ that follows from $$\int_H L_0g \, \varphi^2 d\mu = - \frac{1}{2}\int_H \langle Q^{1/2}Dg, Q^{1/2}D(\varphi^2) \rangle d\mu
= - \int_H \langle Q^{1/2}Dg, Q^{1/2}D \varphi ) \rangle \varphi \, d\mu$$ (see formula ).
As a second step, with the aid of Proposition \[p5.1\] we prove an integration by parts formula in $W^{1,2}_{0}(H, \mu)$ and we define the [*trace*]{} $\varphi_{|\Sigma}$ at the boundary $\Sigma$ of any function in $W^{1,2}_{0}(H, \mu)$.
\[MaggTraccia\] Assume that $|Q^{1/2}Dg|\in W^{2,2}_{0}(H, \mu)$, and that $|Q^{1/2}Dg|$ is bounded and $L_0g$ has at most linear growth either on $K$ or on $K^c$. Then for every $\varphi \in W^{1,2}_{0}(H, \mu)$ there exists $\psi \in L^2(\Sigma, \sigma)$ with the following property: for each sequence $(\varphi_n)\in {\mathcal E}_{0}(H)$ such that $\lim_{n\to \infty} \|\varphi_n - \varphi \|_{W^{1,2}_{0}(H, \mu)} =0$, the sequence $(\varphi_n |Q^{1/2} Dg|^{1/2}_{|\Sigma})$ converges to $\psi$ in $L^2(\Sigma, \sigma)$.
It is sufficient to recall formula and Lemma \[emb\].
Note that the assumptions of Corollary \[MaggTraccia\] are satisfied by the functions $g$ in Example \[examples\] of the Appendix.
Under the assumptions of Corollary \[MaggTraccia\], for each $\varphi \in W^{1,2}_{0}(H, \mu)$ the trace of $\varphi$ at $\Sigma$ is defined by $$\varphi_{| \Sigma} = \frac{\psi}{|Q^{1/2} Dg|^{1/2}},$$ where $\psi$ is given by Corollary \[MaggTraccia\].
Note that in general $\varphi_{|\Sigma} $ does not belong to $L^2(\Sigma, \sigma)$, because $ |Q^{1/2} Dg|^{-1/2}$ may be unbounded in $\Sigma$. Of course, if $ |Q^{1/2} Dg|^{-1/2}$ is bounded in $\Sigma$ (that is, if $\inf_{\Sigma} |Q^{1/2} Dg| >0$), then $\varphi_{|\Sigma} \in L^2(\Sigma, \sigma)$ for every $\varphi \in W^{1,2}_{0}(H, \mu)$ and the mapping $W^{1,2}_{0}(H, \mu)\mapsto L^2(\Sigma, \sigma)$, $\varphi \mapsto \varphi_{|\Sigma}$ is continuous.
In general, we have the following lemma.
\[tracciaL1\] Under the assumptions of Corollary \[MaggTraccia\], for every $\varphi \in W^{1,2}_{0}(H, \mu)$, $\varphi_{|\Sigma} \in L^1(\Sigma, \sigma)$ and the mapping $W^{1,2}_{0}(H, \mu)$ $\mapsto $ $L^1(\Sigma, \sigma)$, $\varphi \mapsto \varphi_{|\Sigma}$ is continuous.
Since $\varphi_{|\Sigma} = \psi |Q^{1/2} Dg|^{-1/2}$ with $\psi \in L^2(\Sigma, \sigma)$, it is sufficient to prove that $|Q^{1/2} Dg|^{-1/2} \in L^2(\Sigma, \sigma)$. The assumptions $ \|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}/ |Q^{1/2}Dg |^2\in L^{2}(H, \mu)$ and $|Q^{1/2} Dg|^{-1 }\in L^4(H, \mu)$, that are contained in assumption , imply that the function $\widetilde{\varphi} := |Q^{1/2} Dg|^{-1 }$ belongs to $W^{1,2}_{0}(H, \mu)$. By Corollary \[MaggTraccia\], $\widetilde{\varphi} |Q^{1/2} Dg|^{1/2} $ $=$ $ |Q^{1/2} Dg|^{-1/2}$ has trace in $L^2(\Sigma, \sigma)$.
Let the assumptions of Corollary \[MaggTraccia\] be satisfied. The following statements hold for every $\alpha \in [0,1]$. \[PartiTraccia\]
- If $D_kg/|Q^{1/2}Dg| \in W^{2,2}_{0}(H, \mu)$, for every $\varphi \in W^{1,2}_{\alpha}(H, \mu)$ the integration by parts formula holds.
- If $\varphi \in \oo{W}^{1,2}_{\alpha}(K, \mu)$, its trace at $\Sigma_1$ vanishes.
Since $ W^{1,2}_{\alpha}(H, \mu)\subset W^{1,2}_{0}(H, \mu)$, and $ \oo{W}^{1,2}_{\alpha}(K, \mu)\subset \oo{W}^{1,2}_{0}(K, \mu)$, it is enough to prove that the statements hold for $\alpha =0$.
\(i) It is sufficient to approach every $\varphi \in W^{1,2}_{0}(H, \mu)$ by a sequence $(\varphi_n)\subset {\mathcal E}_{0}(H)$, and to recall Lemma \[tracciaL1\].
\(ii) If $\varphi \in \oo{W}^{1,2}_{0}(K, \mu)$, it vanishes a.e. in $K^c$, and formula (b) yields the statement.
Surface integrals
=================
We consider level surfaces of smooth functions $g$. We refer to [@Bo §6.10], where the functions $g$ under consideration belong to the space $W^{\infty}(H, \mu)$ defined by $$W^{\infty}(H, \mu):= \bigcap _{k\in {{\mathbb N}}, p>1}W^{k,p}(H, \mu)$$ and $W^{k,p}(H, \mu)$ is the completion of the smooth cylindrical functions$^($[^1]$^)$ in the norm $$\|f\|_{k,p} : = \|f\|_{L^p(H, \mu) } + \sum_{j=1}^{k} \bigg( \int_H \bigg[ \sum_{i_1, \ldots, i_j\geq 1} (\lambda_{i_1}\cdot \ldots \cdot \lambda_{i_k}D_{i_1}\ldots D_{i_k}f(x))^2 \bigg]^{p/2} \mu(dx) \bigg)^{1/p}$$ (In particular, the spaces $W^{k,2}(H, \mu)$ coincide with our $W^{k,2}_{0}(H, \mu)$ for $k=1, 2$).
Another assumption is $$|Q^{1/2}Dg|^{-1}\in \bigcap _{ p>1}L^{p}(H, \mu).$$ Our aim here is to give a simplified presentation of surface measures in the case of a Hilbert space setting, under less heavy (although less elegant) assumptions on $g$.
For any continuous $g:H\mapsto {{\mathbb R}}$ and $r$ in the range of $g$ let us define the level sets $$\Sigma_r:= \{x\in H:\; g(x)= r \}.$$ We shall define probability measures on the surfaces $\Sigma_r$ with $r$ in the interior part of the range of $g$. To this aim, a first step is the study of the image of $\mu$ on ${{\mathbb R}}$ under the mapping $g$, defined by $$(\mu \circ g^{-1})(I) := \mu(g^{-1}(I)), \quad I\in {{\mathcal B}}({{\mathbb R}}).$$ We shall give sufficient conditions for $\mu \circ g^{-1}$ have continuous (in fact, $W^{1,2}$) density $k$ with respect to the Lebesgue measure. Similarly, for $\rho \in L^1(H, \mu)$ we shall consider the signed measure $$(\rho \mu)(B) := \int_B \rho(x)\mu(dx), \quad B\in {{\mathcal B}}(H)$$ and its image under the mapping $g$, $$(\rho \mu \circ g^{-1})(I) := (\rho \mu)(g^{-1}(I)), \quad I\in {{\mathcal B}}({{\mathbb R}}),$$ and we shall give sufficient conditions for $\rho \mu \circ g^{-1}$ have continuous density $k_{\rho}$ with respect to the Lebesgue measure. A key role will be played by the function $\psi$ defined by $$\label{psi}
\psi := \frac{L_0g }{ |Q^{1/2}Dg|^2} - \frac{\langle Q^{1/2}D^2g \,Q^{1/2} \cdot Q^{1/2}Dg, Q^{1/2}Dg\rangle}{ |Q^{1/2}Dg|^4} ,$$ if $g\in D(L_0)$. We shall use the following lemma.
\[Pr:ek\] Let $g\in D( L_0)$ be such that $|Q^{1/2}Dg|^{-1}\in L^4(H, \mu)$. Then
- $\mu \circ g^{-1}$ is absolutely continuous with respect to the Lebesgue measure.
- If a function $\rho \in W^{1,1}_{0}(H, \mu)$ is such that $$\label{rho}
\psi \rho \in L^1(H, \mu), \quad \frac{|Q^{1/2}D\rho|}{|Q^{1/2}Dg|} \in L^1(H, \mu),$$ where $\psi$ is defined in , then $\rho \mu \circ g^{-1}$ is absolutely continuous with respect to the Lebesgue measure.
To prove statement (a) we shall show that there exists $C>0$ such that $$\label{condsuff}\bigg| \int_{{{\mathbb R}}}\varphi'(r) (\mu \circ g^{-1})(dr) \bigg| \leq C\| \varphi\|_{\infty}, \quad \varphi \in C_b^1({{\mathbb R}}).$$ For each $k\in {{\mathbb N}}$ we have $$\label{dercomp} D_k( \varphi \circ g )(x) = \varphi'(g(x))D_kg(x) , \quad x\in H,$$ so that $$\label{gradcomp} \langle D ( \varphi \circ g) (x) , QDg(x)\rangle = ( \varphi' \circ g)(x)|Q^{1/2}Dg(x) |^2, \quad x\in H,$$ i.e. $$\label{comp}( \varphi' \circ g)(x) = \frac{ \langle Q^{1/2}D ( \varphi \circ g) (x) , Q^{1/2}Dg(x)\rangle }{|Q^{1/2}Dg(x) |^2}, \quad a.e.\; x\in H.$$ Therefore, $$\int_{{{\mathbb R}}}\varphi'(r)( \mu \circ g^{-1})(dr) = \int_H \varphi' \circ g\,d\mu = \int_H \frac{ \sum_{k}\lambda_k D_k ( \varphi \circ g) (x) D_kg(x) }{|Q^{1/2}Dg(x) |^2} d\mu .$$ Integrating by parts and recalling that $$\label{derquoz} D_k \bigg( \frac{1}{ |Q^{1/2}Dg|^2} \bigg) = -2 \;\frac{\sum_{i} \lambda_iD_ig D_{ik}g}{ |Q^{1/2}Dg|^4}$$ we obtain $$\begin{array}{ll}
\ds {\int_H \varphi' \circ g\,d\mu } & = \ds{- \int_H \varphi \circ g \sum_{k} \lambda_k D_k\bigg( \frac{D_kg}{ |Q^{1/2}Dg|^2}\bigg) d\mu
+\int_H \varphi \circ g \sum_{k} \frac{x_kD_kg}{ |Q^{1/2}Dg|^2}\,d\mu}
\\
\\
& = \ds{- \int_H \varphi \circ g \sum_{k} \lambda_k \bigg( \frac{D_{kk}g }{ |Q^{1/2}Dg|^2} -2D_kg \frac{\sum_{i} \lambda_iD_ig D_{ik}g}{ |Q^{1/2}Dg|^4} \bigg)d\mu}
\\
\\
& + \ds{ \int_H \varphi \circ g \sum_{k} \frac{x_kD_kg}{ |Q^{1/2}Dg|^2}\,d\mu}
\\
\\
& = \ds{ - 2 \int_H (\varphi \circ g)(x)\psi(x)d\mu},
\end{array}$$ where the function $\psi$ is defined in . The first addendum in $\psi$, $ L_0g / |Q^{1/2}Dg|^2$, belongs to $L^1(H, \mu)$ since both $ L_0g $ and $1 / |Q^{1/2}Dg|^2$ are in $L^2(H, \mu)$. Concerning the second addendum we have $$\frac{|\langle Q^{1/2}D^2g\,Q^{1/2}\cdot Q^{1/2}Dg, Q^{1/2}Dg\rangle |}{ |Q^{1/2}Dg|^4} \leq \frac{ \| Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}}{ |Q^{1/2}Dg|^2} .$$ Recalling that there exists $C_0>0$ such that ([@Bo Thm. 5.7.1]) $$\|x\mapsto \| Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}\|_{L^2(H, \mu)} \leq C_0 \|g\|_{D(L_0)},$$ it follows that the second addendum in $\psi$ belongs to $L^1(H, \mu)$. Then formula follows, with $C= \|\psi\|_{L^1(H, \mu)} \leq $ const. $( \|g\|_{D(L_0)} + \| |Q^{1/2}Dg|^{-1}\|_{L^4(H. \mu)})$.
We prove statement (b) by the same procedure, replacing $\mu$ by $\rho \mu$. For every $\varphi \in C_b^1({{\mathbb R}})$ we have $$\begin{array}{lll}
\ds {\int_H ( \varphi' \circ g)\,\rho \, d\mu } & = &\ds{ \int_H \sum_{k}\lambda_k D_k ( \varphi \circ g) (x) D_kg(x)\frac{\rho(x) }{|Q^{1/2}Dg(x) |^2} d\mu }
\\
\\
&=& \ds{ \int_H \varphi \circ g \bigg( -2 \psi \rho - \frac{\langle Q^{1/2}Dg, Q^{1/2}D\rho\rangle}{|Q^{1/2}Dg(x) |^2} \bigg) d\mu }
\end{array}$$ where $\psi$ is the function defined in . Assumption implies that the functions $\psi \rho$ and $ \langle Q^{1/2}Dg, Q^{1/2}D\rho\rangle/|Q^{1/2}Dg(x) |^2$ belong to $L^1(H, \mu)$. Then, $$\bigg| \int_{{{\mathbb R}}}\varphi'(r) (\mu \circ g^{-1})(dr) \bigg| = \bigg| \int_H ( \varphi' \circ g)\,\rho \, d\mu \bigg|
\leq C\| \varphi\|_{\infty}, \quad \varphi \in C_b^1({{\mathbb R}})$$ with $C = 2\|\psi\rho\|_{L^1} + \| \frac{|Q^{1/2}D\rho|}{|Q^{1/2}Dg|}\|_{L^1}$. The statement follows.
\[Pr:contk\] Let the assumptions of Proposition \[Pr:ek\] hold. Then:
- If the function $\psi$ defined in belongs to $W^{1,2}_{0}(H, \mu)$, then the density $k$ of $\mu \circ g^{-1}$ belongs to $W^{1,1}({{\mathbb R}})$.
- If $\rho \in W^{1,1}_{0}(H, \mu)$ satisfies and moreover, setting $$\rho_1 := 2\psi \,\rho + \frac{\langle Q^{1/2}Dg, Q^{1/2}D\rho\rangle}{ |Q^{1/2}Dg |^2}$$ we have $\rho_1 \in W^{1,1}_{0}(H, \mu)$, $\psi \rho_1 \in L^1(H, \mu)$, $ \frac{|Q^{1/2}D\rho_1|}{|Q^{1/2}Dg|} \in L^1(H, \mu)$, then $k_{\rho}\in W^{1,1}({{\mathbb R}})$.
To prove statement (a) we shall show that there is $C_1>0$ such that $$\bigg| \int_{{{\mathbb R}}}\varphi ''(r) (\mu \circ g^{-1})(dr) \bigg| \leq C_1\| \varphi\|_{\infty}, \quad \varphi \in C_b^2({{\mathbb R}}).$$ Indeed, this implies that $k$ is weakly differentiable with $k ' \in L^1({{\mathbb R}})$.
Differentiating we get $$D_{kk}( \varphi \circ g )(x) = \varphi''(g(x))(D_kg(x))^2 + \varphi'(g(x))D_{kk}g(x), \quad x\in H,$$ and summing over $k$ $${\rm Tr}(QD^2(g\circ \varphi)) = \varphi''(g(x)) |Q^{1/2}Dg(x) |^2 + \varphi'(g(x)) {\rm Tr}(QD^2g(x))$$ so that $$\begin{array}{lll}
\varphi'' \circ g & = &\ds{ \frac{ {\rm Tr}(QD^2( \varphi \circ g)) }{ |Q^{1/2}Dg |^2} - (\varphi'\circ g) \frac{ {\rm Tr}(QD^2g) }{ |Q^{1/2}Dg |^2} }
\\
\\
& = & \ds{ \frac{ 2L_0( \varphi \circ g) +\langle x, D( \varphi \circ g)\rangle }{ |Q^{1/2}Dg |^2} - (\varphi'\circ g) \frac{ 2L_0g + \langle x, Dg \rangle}{ |Q^{1/2}Dg |^2} }
\\
\\
& = & \ds{ \frac{ 2L_0( \varphi \circ g) }{ |Q^{1/2}Dg |^2} - (\varphi'\circ g) \frac{ 2L_0g }{ |Q^{1/2}Dg |^2} .}
\end{array}$$ Using again we get $$\begin{array}{l}
\ds {\int_H ( \varphi'' \circ g) d\mu } =
\\
\\
= \ds{ \int_H \bigg( -\langle Q^{1/2}D( \varphi \circ g) , Q^{1/2}D( |Q^{1/2}Dg |^{-2})\rangle -2 (\varphi'\circ g) \frac{L_0g}{ |Q^{1/2}Dg |^2} \bigg) d\mu }
\\
\\
= \ds{ \int_H \varphi'\circ g \bigg( \langle Q^{1/2}Dg, 2 \frac{Q^{1/2}D^2g \,Q^{1/2}\cdot Q^{1/2}Dg}{ |Q^{1/2}Dg |^4}\rangle - 2 \frac{L_0g}{ |Q^{1/2}Dg |^2} \bigg) d\mu }
\\
\\
= \ds{ -2 \int_H (\varphi'\circ g ) \psi\,d\mu} ,
\end{array}$$ where $\psi$ is defined in . Then we may use Proposition \[Pr:ek\], with $\rho = \psi$. By assumption, $\psi\in W^{1,2}_{0}(H, \mu)\subset W^{1,1}_{0}(H, \mu)$, moreover $\psi^2 \in L^1(H, \mu)$ and $ \frac{|Q^{1/2}D\psi|}{|Q^{1/2}Dg|}\in L^1(H, \mu)$ since $|Q^{1/2}D\psi|\in L^2(H, \mu)$, $|Q^{1/2}Dg|^{-1}\in L^2(H. \mu)$. We get $| \int_H (\varphi'\circ g ) \psi\,d\mu| \leq C \|\psi\|_{W^{1,2}_{0}(H, \mu)} \|\varphi\|_{\infty}$, and statement (a) follows.
Concerning statement (b), the proof is similar, replacing $\mu$ by $\rho \mu$. For every $\varphi \in C_b^2({{\mathbb R}})$ we have $$\begin{array}{l}
\ds {\int_H ( \varphi'' \circ g) \rho d\mu = \int_H \bigg( \frac{ 2\rho L_0( \varphi \circ g) }{ |Q^{1/2}Dg |^2} - 2(\varphi'\circ g) \frac{ \rho L_0g }{ |Q^{1/2}Dg |^2}\bigg) d\mu }
\\
\\
= \ds{ \int_H \bigg( -\langle Q^{1/2}D( \varphi \circ g) , Q^{1/2}D\bigg( \frac{\rho}{ |Q^{1/2}Dg |^{2} }\bigg)\rangle -2 (\varphi'\circ g) \frac{\rho L_0g}{ |Q^{1/2}Dg |^2} \bigg) d\mu }
\end{array}$$ $$\begin{array}{l}
= \ds{ \int_H \varphi'\circ g \bigg( \langle Q^{1/2}Dg, 2 \frac{Q^{1/2}D^2g \,Q^{1/2}\cdot Q^{1/2}Dg}{ |Q^{1/2}Dg |^4}\rangle
- 2 \frac{L_0g}{ |Q^{1/2}Dg |^2} \bigg) \rho \,d\mu }
\\
\\
- \ds{ \int_H \varphi'\circ g \frac{\langle Q^{1/2}Dg, Q^{1/2}D\rho\rangle}{ |Q^{1/2}Dg |^2}\, d\mu }
\\
\\
= - \ds{ \int_H \varphi'\circ g \bigg( 2 \psi \,\rho + \frac{\langle Q^{1/2}Dg, Q^{1/2}D\rho\rangle}{ |Q^{1/2}Dg |^2} \bigg)\,d\mu = - \int_H \varphi'\circ g\,\rho_1 \,d\mu ,}
\end{array}$$ where the function $\rho_1 $ satisfies the assumptions of Proposition \[Pr:ek\](b). We obtain $| \int_H ( \varphi'\circ g )\rho_1d\mu|\leq C\|\varphi \|_{\infty}$ and the statement follows.
One can play with $\rho$ and $g$ in order that the assumptions of Proposition \[Pr:contk\](b) are satisfied. In the next proposition we give sufficient conditions that are useful for the sequel.
\[Rem1\] The assumptions of Proposition \[Pr:contk\](b) are satisfied by every $\rho \in W^{2,2}_{0}(H, \mu)$ provided $g\in D( L_0)$ is such that $$\label{cosaserve}
\left\{ \begin{array}{l}|Q^{1/2}Dg|^{-1}\in L^4(H, \mu), \quad \psi\in W^{1,4}_{0 }(H, \mu) ,
\\
\\ \frac{ \|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}}{|Q^{1/2}Dg |^2}\in L^{2}(H, \mu), \; \frac{ \|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}}{|Q^{1/2}Dg |^3}\in L^2(H, \mu).
\end{array}\right.$$ In this case there exists $C_2>0$, depending only on $g$, such that $$\bigg| \int_{{{\mathbb R}}}\varphi''(r) (\rho \mu \circ g^{-1})(dr) \bigg| \leq C_2\|\rho\|_{W^{2,2}_{0}(H, \mu)}
\| \varphi\|_{\infty}, \quad \varphi \in C_b^2({{\mathbb R}}).$$ Consequently, if $\rho_n \to \rho$ in $W^{2,2}_{0}(H, \mu)$ then $k_{\rho_n}\to k_{\rho}$ in $W^{1,1}({{\mathbb R}})$, hence $k_{\rho_n}\to k_{\rho}$ in $L^{\infty}({{\mathbb R}})$.
Since $\psi\in L^2$ and $|Q^{1/2}Dg |^{-1}\in L^4$, then $\rho_1\in L^1$. Computing $Q^{1/2}D\rho_1$ we obtain $$\begin{array}{l}
Q^{1/2}D\rho_1=
\\
\\
= \rho Q^{1/2}D\psi + \psi Q^{1/2}D\rho - \displaystyle{ \frac{Q^{1/2}D^2g\,Q^{1/2}\cdot Q^{1/2}D\rho + Q^{1/2}D^2\rho Q^{1/2} \cdot Q^{1/2}Dg}{|Q^{1/2}Dg|^2} }
\\
\\
+2 \langle Q^{1/2}Dg, Q^{1/2}D\rho \rangle \displaystyle{ \frac{Q^{1/2}D^2g\,Q^{1/2}\cdot Q^{1/2}Dg}{|Q^{1/2}Dg|^4}}.
\end{array}$$ Estimating each addendum we get
- $\rho |Q^{1/2}D\psi| \in L^1$, since $ |Q^{1/2}D\psi| \in L^2$;
- $\psi | |Q^{1/2}D\rho| \in L^1$, since $\psi\in L^2$;
- $\displaystyle{\frac{\|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}| Q^{1/2}D\rho | }{|Q^{1/2}Dg|^2} }\in L^1$, since $ \displaystyle{\frac{\|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)} }{|Q^{1/2}Dg|^2} }\in L^2$;
- $\displaystyle{ \frac{\|Q^{1/2}D^2\rho Q^{1/2}\|_{{\mathcal L}(H)}}{|Q^{1/2}Dg|} }\in L^1$, since $\displaystyle{ \frac{1 }{|Q^{1/2}Dg|} }\in L^2$;
- $|Q^{1/2}D\rho| \displaystyle{\frac{\|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)} }{|Q^{1/2}Dg|^2} }\in L^1$, as above.
Therefore $\rho_1\in L^1$, and $\|\rho_1\|_{L^1(H, \mu)} \leq c\|\rho\|_{W^{2,2}_{0}(H, \mu)}$.
The assumptions $\psi \in L^4$, $ \frac{1}{|Q^{1/2}Dg |}\in L^4$ imply that $\psi \rho_1\in L^1$.
To check that $ \frac{|Q^{1/2}D\rho_1|}{|Q^{1/2}Dg|} \in L^1$ we redo the estimates above, dividing each term by $|Q^{1/2}Dg|$. We get
- $\rho \displaystyle{\frac{ |Q^{1/2}D\psi| }{|Q^{1/2}Dg|}}
\in L^1$, since $ |Q^{1/2}D\psi| \in L^4$ and $\displaystyle{ \frac{1 }{|Q^{1/2}Dg|}} \in L^4$;
- $\psi \displaystyle{ \frac{ | |Q^{1/2}D\rho| }{|Q^{1/2}Dg|}}
\in L^1$, since $\psi\in L^4$ and $\displaystyle{ \frac{1 }{|Q^{1/2}Dg|}} \in L^4$;
- $\displaystyle{\frac{\|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)}| Q^{1/2}D\rho | }{|Q^{1/2}Dg|^3}} \in L^1$, since $\displaystyle{ \frac{\|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)} }{|Q^{1/2}Dg|^3}} \in L^2$;
- $\displaystyle{ \frac{\|Q^{1/2}D^2\rho Q^{1/2}_{{\mathcal L}(H)}}{|Q^{1/2}Dg|^2} } \in L^1$, since $\displaystyle{ \frac{1 }{|Q^{1/2}Dg|}} \in L^4$;
- $|Q^{1/2}D\rho| \displaystyle{\frac{\|Q^{1/2}D^2g\,Q^{1/2}\|_{{\mathcal L}(H)} }{|Q^{1/2}Dg|^3}} \in L^1$, as above.
Therefore, the norms $\|\psi \rho_1\|_{L^1}$ and $\| \frac{|Q^{1/2}D\rho_1|}{|Q^{1/2}Dg|}\|_{L^1}$ are bounded by $c\|\rho\|_{W^{2,2}_{0}(H, \mu)}$. Applying Proposition \[Pr:contk\](b) the statement follows.
\[examples\] Let us consider some simple examples.
- $g(x) = \langle b, x\rangle$, with $|b| =1$,
- $g(x) =\langle Tx, x\rangle$, with $T\in {\mathcal L}(H)$, $Te_k = t_ke_k$ for each $k\in {{\mathbb N}}$ and $t_k\neq 0$ for infinitely many $k$,
- $g(x) = \sum_{k=1}^{13}x_k^2$.
In all these cases $g$ satisfies the conditions of Proposition \[Rem1\].
In case (a) we have $Dg = b$, $D^2g =0$ so that $L_0 g = - \langle b, x\rangle /2= -g/2$ and $$\psi = -\frac{ \langle b, x\rangle }{2|Q^{1/2}b|^2}$$ which belongs to $W^{1,4}_{0} (H, \mu)$. The other conditions of Proposition \[Rem1\] are obviously satisfied.
In case (b) we have $Dg(x) =2Tx$, $D^2g(x) = 2T$ so that $L_0 g = {\rm Tr}[QT] - g$ and $$\label{psies}
\psi(x) = \frac {{\rm Tr}[QT] - \langle Tx, x\rangle}{2|Q^{1/2}Tx|^2 } - \frac {\langle Q^2T^3x , x\rangle }{|Q^{1/2}Tx|^2 } .$$ Since $t_k\neq 0$ for infinitely many $k$, then $x\mapsto |Q^{1/2}Dg(x)|^{-1}$ belongs to all spaces $L^p(H, \mu)$. Indeed, $ |Q^{1/2}Dg(x)|^2 \geq 4 \sum_{k=1}^{N} \lambda_i t_k^2x_k^2 $ where $N$ is so large that at least $[p]+1$ addenda do not vanish. The other assumptions of Remark \[Rem1\] are easily seen to be satisfied.
In case (c) we still have $g(x) = \langle Tx, x\rangle$ with $T\in {\mathcal L}(H)$, $Tx = \sum_{k=1}^{13} x_k e_k $, so that $t_k\neq 0$ only for $k=1, \ldots, 13$. However, $ |Q^{1/2}Dg(x)|^{-1} \leq c_0( \sum_{k=1}^{13} x_k^2)^{-1/2}$ with $c_0= 1/\min\{ \lambda_{k}^{1/2}:\; k=1, \ldots, 13\}$ so that $ |Q^{1/2}Dg |^{-1} \in L^p(H, \mu)$ for every $p<13$. The function $\psi$ is still given by on span$\{e_1, \ldots, e_{13}\}$ and it belongs to $ L^p(H, \mu)$ for every $p<13/3$, in particular it belongs to $ L^4(H, \mu)$, as well as $|Q^{1/2}D\psi|^{-1}$. The other conditions of Proposition \[Rem1\] are easily seen to be satisfied.
In cases (a) and (b) with $T=I$ it is possible to give a representation formula for $k$ that shows that $k\in C^{\infty}$, see [@Hertle]. In case (c) we have $ |Q^{1/2}Dg(x)|^{-1} \geq c_1 ( \sum_{k=1}^{13} x_k^2)^{-1/2}$ with $c_1= 1/\max\{\lambda_{k}^{1/2}:\; k=1, \ldots, 13\}$ so that $ |Q^{1/2}Dg |^{-1} \notin L^p(H, \mu)$ for $p\geq 13$.
The construction of the surface measures goes as follows. First, one constructs surface measures depending explicitly on $g$ by an approximation procedure.
One fixes once and for all a convex compact set $K$ which is symmetric with respect to the origin and has positive measure, say $\mu(K)>1/2$. Such a $K$ does exist. Indeed, it is well known that there are compact sets $\widetilde{K}$ with positive (arbitrarily close to $1$) measure (a simple proof is e.g. in [@DP Thm. 6.2]). The absolute convex hull $K$ of $\widetilde{K}$ is compact, symmetric with respect to the origin and contains $\widetilde{K}$, so that $\mu(K)\geq \mu(\widetilde{K})$.
Then we need a regular cutoff function. The proof of its existence follows closely [@Bo Prop. 5.4.12], with a few simplifications due to our Hilbert space setting.
Let $K\subset H$ be compact, convex, symmetric with respect to the origin, with $\mu(K) >1/2$. Then there exists a function $\theta\in W^{\infty}(H, \mu)$ such that $\theta \equiv 1$ on $K$, $\theta=0 $ a.e. outside $2K$ and $0\leq \theta (x) \leq 1$ for all $x\in H$.
By the $0-1$ law (e.g., [@Bo Thm. 2.5.5]), the vector space $E$ spanned by $K$ has measure $1$. Consequently, $\lim_{m\to \infty} \mu(mK) =1$. Fix $m\in {{\mathbb N}}$ such that $$\mu(mK) >\frac{8}{9}.$$ Let us consider the Minkowski functional defined by $p_K(x) : = \inf \{ \alpha >0:\;x\in \alpha K\}$ for $x\in E$, and the function $d(x) : = \inf \{p_K(x-y):\;y\in K\}$ if $x\in E$, $d(x)=1$ if $x\notin E$. We modify it setting $$\varphi (x) = 1- h(d(x)) , \quad x\in H,$$ where $h(t) = t$ for $ t\leq 1$ and $h(t)=1$ for $t\geq 1$. The function $\varphi $ is Borel measurable, has values between $0$ and $1$, $\varphi \equiv 1$ on $K$ and $\varphi \equiv 0$ outside $E$ and outside $2K $. We regularize it applying $T_{0}(t) $, where $t>0$ is chosen such that $$1-e^{-t/2} <\frac{1}{8}, \quad m\sqrt{1-e^{-t}} <\frac{1}{8}.$$ Since $\varphi \in {\mathcal B}_b(H)$, then $T_0(t)\varphi \in W^{\infty}(H, \mu)$ (e.g., [@Bo Prop. 5.4.8]).
Moreover, $$\label{eq:1}
T_0(t)\varphi(x) \geq \frac{2}{3} \;\forall x\in K, \quad T_0(t)\varphi(x) \leq \frac{3}{5} \;\forall x\in E\setminus 2K.$$ Indeed, let $x\in K$. Then $ e^{-t/2}x\in K$, and for each $y\in mK$ we have $\sqrt{1-e^{-t}}y\in K/8$. The sum $ e^{-t/2}x+\sqrt{1-e^{-t}}y $ belongs to $9K/8$, so that $d(e^{-t/2}x+\sqrt{1-e^{-t}}y)\leq 1/8$ and therefore $\varphi(e^{-t /2} x+ \sqrt{1-e^{-t}}y) \geq 7/8$. Since $\mu(H\setminus mK )\leq 1/9$, we get $T_0(t)\varphi(x)\geq 7/8 - 1/9 > 2/3$. Let now $x\in E\setminus 2K$. Since $e^{-t/2} >7/8$, $e^{-t/2}x \notin 7K/4 $ and consequently for every $y\in mK$ the sum $ e^{-t/2}x+\sqrt{1-e^{-t}}y $ does not belong to $7K/4 - K/8 = 13K/8$. Therefore, $d( e^{-t/2}x+\sqrt{1-e^{-t}}y) \geq 5/8$, so that $\varphi(e^{-t /2} x+ \sqrt{1-e^{-t}}y) \leq 3/8$. Again since $\mu(H\setminus mK )\leq 1/9$, we get $T_0(t)\varphi(x)\leq 3/8 +1/9 = 35/72 <3/5$, and is proved.
Now fix a function $\eta \in C^{\infty}({{\mathbb R}})$ such that $0\leq \eta \leq 1$, $\eta (t) = 0$ for $t\leq 3/5$, $\eta (t) =1$ for $t\geq 2/3$, and set $$\theta(x) = \eta ( T_0(t)\varphi (x)), \quad x\in H.$$ The function $\theta $ is what we were looking for. It has values between $0$ and $1$, it belongs to $W^{\infty}(H, \mu)$, $\theta(x) =1$ for $x\in K$ and $\theta(x) =0$ for $x\in E\setminus 2K$. Since $\mu(E)=1$, then $\theta(x) =0$ for almost all $x\in H\setminus 2K$. The statement follows.
Now we fix $\varphi_0 \in C^{\infty}_{c}({{\mathbb R}})$ with $0\leq \varphi_0\leq 1$, $\int_{{{\mathbb R}}} \varphi_0(t)dt =1$ and $\varphi_0\equiv 1$ in a neighborhood of $0$, $\varphi_0\equiv 0$ outside $(-1, 1)$. Then for each $r\in {{\mathbb R}}$ the sequence $\{ \varphi_0(j(t-r))dt/j\}$ converges weakly to the Dirac measure $\delta_r$.
For each $r$ in the interior part of $g(H)$ we set $$\theta_n(x) = \theta\bigg(\frac{x}{n}\bigg) ,\quad x\in H; \quad \quad \varphi_j(t) = \frac{\varphi_0(j(t-r))}{j}, \quad j\in {{\mathbb N}}, \;t\in {{\mathbb R}}.$$ The following proposition is proved in [@Bo]. Since in the Hilbert space case there are not simplifications with respect to the general setting of [@Bo], we refer to [@Bo Lemma 6.10.1, Thm. 6.10.2] for the proof.
\[costruzione\]
- For each $n\in {{\mathbb N}}$, the sequence of measures $$\nu_{n, j}(dx) = \theta_n(x)\frac{\varphi_j(g(x))}{k(g(x))}\mu(dx)$$ converges weakly to a measure $\nu_n$ concentrated on $\Sigma_r := g^{-1}(r)$. Moreover, for each continuous $f\in W^{2,2}_{0}(H)$ we have $$\label{e0}
\int_H f\, d\nu_n = \int_{ \Sigma_r} f\, d\nu_n = \frac{k_{f\theta_n}(r)}{k(r)}.$$
- In its turn, the sequence $\nu_n$ converges weakly to a probability measure $\sigma^{(g)}_r $ concentrated on $\Sigma_r $, such that for each continuous $f\in W^{2,2}_{0}(H)$ we have $$\label{e1}
\int_H f\, d\sigma^{(g)}_r = \int_{ \Sigma_r} f\, d\sigma^{(g)}_r = \frac{k_{f }(r)}{k(r)}.$$
\[MisSup\] For every Borel bounded function $\varphi :H\mapsto {{\mathbb R}}$ and for every $r$ in the interior part of $g(H)$ we set $$\int_{\Sigma_r} \varphi \,d\sigma_r : = k(r) \int_{\Sigma_r} \varphi|Q^{1/2}Dg| \,d\sigma^{(g)}_r .$$
\[rem:limite\] It is easy to see that for every $f:H\mapsto {{\mathbb R}}$ such that $f |Q^{1/2}Dg| \in W^{2,2}_{0}(H, \mu)\cap C(H)$ we have $$\int_{\Sigma_r}f \,d\sigma_r = \lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{r-{\varepsilon}\leq g(x)\leq r+{\varepsilon}} f |Q^{1/2}Dg| \,d\mu .$$ Indeed, applying Proposition \[Pr:contk\] we get $$\lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{r-{\varepsilon}\leq g(x)\leq r+{\varepsilon}} f |Q^{1/2}Dg| \,d\mu = \lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{r-{\varepsilon}}^{r+{\varepsilon}} d(f |Q^{1/2}Dg|\circ \mu)$$ $$= \lim_{{\varepsilon}\to 0} \frac{1}{2{\varepsilon}} \int_{r-{\varepsilon}}^{r+{\varepsilon}} k_{f |Q^{1/2}Dg|}(t)dt = k_{f |Q^{1/2}Dg|}(r).$$ On the other hand, by Proposition \[costruzione\](b) we have $$k_{f |Q^{1/2}Dg|}(r) = k(r)\int_{\Sigma_r} f |Q^{1/2}Dg| \,d\sigma^{(g)}_r$$ and the right hand side is just $\int_{\Sigma_r}f \,d\sigma_r$ by definition.
[999]{}
V. Barbu, G. Da Prato, L. Tubaro, *Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space*, Ann. Probab. [**37**]{} (2009), 1427–1458.
V.I. Bogachev, *Gaussian Measures*, American Mathematical Society, Providence, 1998.
G. Da Prato, *An introduction to Infinite-Dimensional Analysis*, Springer-Verlag, Berlin, 2006.
G. Da Prato, B. Goldys, J. Zabczyk, *Ornstein-Uhlenbeck semigroups in open sets of Hilbert spaces*, C. R. Acad. Sci. Paris, [**325**]{}, 433-438, 1997.
G. Da Prato, J. Zabczyk, *Second Order Partial Differential Equations in Hilbert Spaces*, London Mathematical Society, Lecture Notes, [**293**]{}, Cambridge University Press, Cambridge, 2002.
J. D. Deuschel, D. Stroock, *Large Deviations*, Academic Press, San Diego, 1984.
E. B. Dynkin, *Markov Processes*, Vol I, Springer-Verlag, Berlin, 1965.
A. Friedman, *Stochastic differential equations and applications. Vol.1*, Academic Press, New York, 1975.
L. Gross, [*P*otential Theory in Hilbert spaces]{} J. Funct. Analysis [**1**]{}, (1965), 123–189.
A. Hertle, *Gaussian surface measures and the Radon transform on separable Banach spaces,* Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979), 513–531, Lecture Notes in Math. [**794**]{}, Springer-Verlag, Berlin, 1980.
T. Kato, *Perturbation theory for linear operators*, Springer-Verlag, Berlin, 1966.
P. Kotelenez, *A submartingale type inequality with applications to stochastic evolution equations,* Stochastics, [**8**]{} (1982), 139–51.
A. Lunardi, G. Metafune, D. Pallara, *Dirichlet boundary conditions for elliptic operators with unbounded drift*, Proc. Amer. Math. Soc. [**133**]{} (2005), 2625–2635.
P. Malliavin, *Stochastic analysis*, Springer-Verlag, Berlin, 1997.
A. Talarczyk, *Dirichlet problem for parabolic equations on Hilbert spaces*, Studia Math. [**141**]{} (2000), 109–142.
L. Tubaro, *An estimate of Burkholder type for stochastic processes defined by the stochastic integral*, Stochastic Anal. Appl., [**2**]{} (1984), 187–192.
[^1]: that is, functions of the type $f(x) =\varphi( \langle x, x_1\rangle, \ldots , \langle x, x_n\rangle)$ with $x_1$, …, $x_n \in H$ and $\varphi \in C^{\infty}_{b}({{\mathbb R}}^n)$.
|
---
author:
- Riccardo Catena
- and Bodo Schwabe
title: Form factors for dark matter capture by the Sun in effective theories
---
Introduction
============
The quest for dark matter is at a turning point. Data from direct, indirect and collider searches for dark matter with unprecedented exposure, resolution and extension in energy will finally be available during the next 5-10 years [@Cakir:2014nba; @Catena:2013pka; @Ibarra:2012cc; @Bergstrom:2012vd; @Bringmann:2012ez; @Baudis:2012ig; @Bertone:2010at]. Efficient strategies to globally interpret these data in terms of dark matter particle mass and interaction properties are of prime importance in astroparticle physics [@Strege:2014ija; @Buchmueller:2013rsa; @Bechtle:2012zk].
Effective theory methods have proven to be a very powerful tool in the analysis of collider data [@Zhou:2013raa; @Rajaraman:2011wf; @Goodman:2010ku; @Goodman:2010yf], dark matter direct [@Gluscevic:2014vga; @Catena:2014uqa; @Catena:2014epa; @Catena:2014hla; @Panci:2014gga; @DelNobile:2013sia; @Fitzpatrick:2012ib; @Fornengo:2011sz; @DelNobile:2011uf] and indirect [@Chen:2013gya; @Rajaraman:2012fu; @Goodman:2010qn] detection experiments, and in combined studies of these different strategies [@Fedderke:2014wda; @Alves:2014yha; @Alves:2015pea]. The main advantage of the effective theory approach to dark matter is that it allows for a model independent interpretation of the different observations when all relevant interaction operators are simultaneously explored in multidimensional statistical analyses, as for instance in [@Catena:2014uqa; @Catena:2014epa; @Catena:2014hla]. In contrast, comparing a simplistic model for dark matter to observations, important physical properties can be missed, and spurious correlations among physical observables can be enforced by the inappropriately small number of model parameters.
In the context of effective theories for dark matter, the dark matter-nucleus interaction plays a special role, in that its exploration is complicated by non trivial properties related to the internal structure of the nuclei in analysis. The effective theory of dark matter-nucleon interactions [@Fitzpatrick:2012ix; @Fan:2010gt] predicts that 8 independent nuclear response functions - or form factors - can be generated in the dark matter scattering by nuclei. The interpretation of any dark matter experiment probing the dark matter-nucleus interaction is unavoidably affected by the uncertainties within which the 8 nuclear response functions are known. Experiments of this type are dark matter direct detection experiments, and neutrino telescopes searching for solar neutrinos from dark matter annihilations. In this work we concentrate on the latter ones.
Dark matter can be captured by the Sun while scattering in the solar medium. Dark matter particles accumulated in the Sun might annihilate producing a flux of potentially observable energetic neutrinos [@Edsjo:1997hp]. The solar neutrino flux from dark matter annihilations is strictly related to the rate of dark matter capture by the Sun (e.g. proportional to the latter, assuming equilibrium between capture and annihilation [@Edsjo:1997hp]). It is therefore a function of the cross-section for dark matter-nucleus scattering, which in turn depends on the nuclear response functions computed in this work. For constant spin-independent dark matter-nucleon interactions, nuclear response functions for the most abundant element in the Sun are approximately known [@Gondolo:2004sc]. For constant spin-dependent interactions, dark matter is assumed to scatter off Hydrogen only, and nuclei with a more complex structure are neglected. Finally, for momentum and velocity dependent dark matter-nucleon interactions, only simplified calculations have so far been performed in the literature. In Ref. [@Liang:2013dsa], for instance, the rate of dark matter capture by the Sun is calculated for 6 momentum/velocity dependent operators, considering dark matter scattering from Hydrogen only. Recently, momentum and velocity dependent dark matter-nucleon interactions have also been explored in the context of helioseismology [@Vincent:2013lua; @Lopes:2014aoa; @Vincent:2014jia].
Nuclear response functions for model independent analyses of dark matter direct detection experiments have been calculated in [@Vietze:2014vsa; @Fitzpatrick:2012ix] under the assumption of one-body dark matter-nucleon interactions. Two-body interactions have also been included in [@Klos:2013rwa; @Menendez:2012tm] in an investigation of spin-dependent dark matter-nucleus currents. In addition, two-body contributions to spin-independent dark matter-nucleon interactions have been claimed to be important in dark matter direct detection in [@Prezeau:2003sv; @Cirigliano:2013zta; @Cirigliano:2012pq]. The nuclear response functions for isotopes of Xe, I, Ge, Na, and F found in [@Fitzpatrick:2012ix] have been applied to complementary analyses of current direct detection experiments [@Catena:2014uqa; @Gresham:2014vja; @DelNobile:2013sia], and in studies of the prospects for dark matter direct detection [@Catena:2014epa; @Catena:2014hla].
In this paper we calculate the 8 nuclear response functions generated in the dark matter scattering by nuclei for the 16 most abundant elements in the Sun. We then use the novel response functions to calculate the rate of dark matter capture by the Sun within the general effective theory of isoscalar and isovector dark matter-nucleon interactions mediated by a heavy spin-1 or spin-0 particle. In the analysis, we comprehensively describe how the capture rate depends on specific dark matter-nucleon interaction operators, and on the elements in the Sun. This study constitutes the first step towards robust model independent analyses of dark matter induced neutrino signals from the Sun.
The paper is organized as follows. In Sec. \[sec:astro\] we provide the equations for computing the rate of dark matter capture by the Sun given an arbitrary dark matter-nucleon interaction. In Sec. \[sec:eft\] we review the effective theory of dark matter-nucleon interactions, while in Sec. \[sec:obdme\] we calculate the 8 nuclear response functions predicted by the theory for the most abundant elements in the Sun. We calculate the dark matter capture rate for all isoscalar and isovector dark matter-nucleon interactions in Sec. \[sec:rate\], and we conclude in Sec. \[sec:conc\]. The dark matter response functions and the single-particle matrix elements needed in the analysis are listed in the Appendixes \[sec:appDM\] and \[sec:appME\], respectively. Finally, in Appendix \[sec:appNuc\] we provide the nuclear response functions of this work in analytic form.
Dark matter capture by the Sun {#sec:astro}
==============================
Dark matter particles of the galactic halo with interactions at the electroweak scale can be gravitationally captured by the Sun. For a dark matter particle of mass $m_\chi$ at a distance $R$ from the center of the Sun, the rate of scattering from a velocity $w$ to a velocity less than the local escape velocity $v(R)$ is given by [@Gould:1987ir] $$\Omega_{v}^{-}(w)= \sum_i n_i w\,\Theta\left( \frac{\mu_i}{\mu^2_{+,i}} - \frac{u^2}{w^2} \right)\int_{E_k u^2/w^2}^{E_k \mu_i/\mu_{+,i}^2} {\rm d}E\,\frac{{\rm d}\sigma_{i}}{{\rm d}E}\left(w^2,q^2\right)\,.
\label{eq:omega}$$ In Eq. (\[eq:omega\]), $E_k=m_\chi w^2/2$, $d\sigma_i/dE$ is the differential cross-section for dark matter scattering by nuclei of mass $m_i$ and density $n_i(R)$ in the Sun, $q$ is the momentum transfer and $E=~q^2/(2m_i)$ the nuclear recoil energy. The sum in the scattering probability extends over the most abundant elements in the Sun, and the dimensionless parameters $\mu_i$ and $\mu_{\pm,i}$ are defined as follows $$\mu_i\equiv \frac{m_\chi}{m_i}\, \qquad\qquad \mu_{\pm,i}\equiv \frac{\mu_i\pm1}{2}\,.$$ The velocity $u$ in Eq. (\[eq:omega\]) is the velocity of the dark matter particle at $R\rightarrow \infty$, where the Sun’s gravitational potential is negligible. The relation between $u$ and $w$ is $w=\sqrt{u^2+v(R)^2}$, and therefore $\Omega_{v}^{-}(w)$ depends on $R$.
In Eq. (\[eq:omega\]), we consider the general case in which the differential scattering-cross section depends both on the momentum transfer $q$, and on the dark matter-nucleus relative velocity $w$. We therefore relax the assumption of constant total cross-section, commonly made in this context. This generalization is important in the study of arbitrary dark matter-nucleus interactions, as we will see in the next sections.
Consider now a population of halo dark matter particles with speed distribution at infinity given by $f(u)$. A fraction of them will be captured by the Sun, with a differential capture rate per unit volume given by [@Gould:1987ir] $$\frac{{\rm d} C}{{\rm d}V} = \int_{0}^{\infty} {\rm d}u\, \frac{f(u)}{u}\, w\Omega_{v}^{-}(w) \,.$$ The total capture rate takes the following form $$C = 4\pi \int_{0}^{R_{\odot}} {\rm d} R\, R^2\,\frac{{\rm d} C}{{\rm d}V} \left(R\right)\,,
\label{eq:rate}$$ where we integrate over a sphere of radius $R_{\rm \odot}$, corresponding to the volume of the Sun. The aim of this work is to evaluate Eq. (\[eq:rate\]) within the most general effective theory for dark matter-nucleon one-body interactions mediated by heavy spin-1 or spin-0 particles, using for each element in the Sun the appropriate nuclear response functions.
From Eq. (\[eq:rate\]), one can calculate the differential neutrino flux from dark matter annihilations in the Sun. It is given by [@Edsjo:1997hp] $$\frac{{\rm d \Phi_\nu}}{{\rm d E_\nu}} = \frac{\Gamma_A}{4\pi D^2} \sum_{f} B_{\chi}^{f} \, \frac{{\rm d N_\nu^f}}{{\rm d E_\nu}} \,
\label{eq:nuflux}$$ where $E_{\nu}$ is the neutrino energy, $\Gamma_A$ the total dark matter annihilation rate, $B_{\chi}^{f}$ the branching ratio for dark matter pair annihilation into the final state $f$, and $D$ the distance from the observer to the center of the Sun. ${\rm d} N_\nu^f/{\rm d} E_\nu$ is the energy spectrum of neutrinos produced by dark matter annihilation into the final state $f$. In general, $\Gamma_A = (C/2) \tanh^2(t/\tau)$, where $t$ is the time variable, and $\tau$ the characteristic time scale for the equilibration of dark matter capture and annihilation.
In our calculations we consider the most abundant elements in the Sun, and use the densities $n_i(R)$ and the velocity $v(R)$ as implemented in the [darksusy]{} code [@Gondolo:2004sc]. Accordingly, we include in the analysis the following 16 elements: H, $^{3}$He, $^{4}$He, $^{12}$C, $^{14}$N, $^{16}$O, $^{20}$Ne, $^{23}$Na, $^{24}$Mg, $^{27}$Al, $^{28}$Si, $^{32}$S, $^{40}$Ar, $^{40}$Ca, $^{56}$Fe, and $^{58}$Ni. Finally, we assume the so-called standard halo model [@Freese:2012xd], with a Maxwell-Boltzmann speed distribution for $f(u)$, and a local standard of rest velocity of 220 km s$^{-1}$. We leave an analysis of astrophysical uncertainties [@Bozorgnia:2013pua; @Catena:2011kv; @Catena:2009mf] in the evaluation of Eq. (\[eq:rate\]) for future work.
Dark matter-nucleus scattering {#sec:eft}
==============================
In this section we review the theory of dark matter scattering by nucleons and nuclei [@Fitzpatrick:2012ix].
Dark matter-nucleon effective interactions {#sec:dmnu}
------------------------------------------
The amplitude for dark matter-nucleon elastic scattering, $\mathcal{M}$, is restricted by energy and momentum conservation, and respects Galilean invariance, i.e. the invariance under constant shifts of the tridimensional particle velocities. These restrictions determine how $\mathcal{M}$ depends on the momenta of the incoming and outgoing particles. Let us denote by ${\bf p}$ (${\bf p'}$) and ${\bf k}$ (${\bf k'}$) the initial (final) dark matter and nucleon tridimensional momenta, respectively. Momentum conservation implies that only three out of these four momenta are independent in the scattering process. A possible choice of independent momenta is ${\bf p}$, ${\bf k}$, and ${\bf q}\equiv {\bf k}-{\bf k'}$, where ${\bf q}$ is the momentum transferred from the nucleon to the dark matter particle. Whereas ${\bf q}$ is Galilean invariant, ${\bf p}$ and ${\bf k}$ are not. Galilean invariance therefore implies that $\mathcal{M}$ must depend on the difference ${\bf v}\equiv {\bf p}/m_\chi - {\bf k}/m_N $, rather than on ${\bf p}$ and ${\bf k}$ separately. ${\bf v}$ is the initial relative velocity between a dark matter particle of mass $m_\chi$ and a nucleon of mass $m_N$. In addition to particle momenta, $\mathcal{M}$ can depend on the dark matter particle and nucleon spins, $j_\chi$ and $j_N$, respectively.
Any non-relativistic quantum mechanical Hamiltonian leading to a scattering amplitude obeying such restrictions can be expressed as a combination of the following five Hermitian operators $$\mathbb{1}_{\chi N} \qquad\quad i{\bf{\hat{q}}} \qquad\quad {\bf{\hat{v}}}^{\perp} \qquad\quad {\bf{\hat{S}}}_{\chi} \qquad\quad {\bf{\hat{S}}}_{N} \,.
\label{eq:5Op}$$ The five operators in Eq. (\[eq:5Op\]) act on the two-particle Hilbert space spanned by tensor products of dark matter and nucleon states, respectively $|{\bf p},j_\chi\rangle$ and $|{\bf k},j_N\rangle$. The operator $\mathbb{1}_{\chi N}$ is the identity in this space, whereas ${\bf\hat{S}}_{\chi}$ and ${\bf\hat{S}}_{N}$ denote the dark matter particle and nucleon spin operators. Finally, $i{\bf{\hat{q}}}$ is the Hermitian transfer momentum operator, and ${\bf{\hat{v}^{\perp}}}$ the relative transverse velocity operator. They are Galilean invariant, and characterized by the matrix elements $$\begin{aligned}
\label{eq:me1}
\langle {\bf p'},j_\chi; {\bf k'},j_N| \, i{\bf{\hat{q}}} \,|{\bf p},j_\chi; {\bf k},j_N \rangle &=& i {\bf q} \,e^{-i {\bf q} \cdot {\bf r}} \,(2\pi)^3\delta({\bf k'}+{\bf p'}-{\bf k}-{\bf p}) \\
\label{eq:me2}
\langle {\bf p'},j_\chi; {\bf k'},j_N| \, {\bf{\hat{v}^{\perp}}} \, |{\bf{p}},j_\chi; {\bf k},j_N \rangle &=& \left({\bf v}+\frac{{\bf q}}{2\mu_N}\right) e^{-i {\bf q} \cdot {\bf r}} \,(2\pi)^3\delta({\bf k'}+{\bf p'}-{\bf k}-{\bf p})\end{aligned}$$ where $\mu_N$ is the reduced mass of the dark matter-nucleon system, and ${\bf r}$ is the position vector from the nucleon to the dark matter particle. Notice that energy conservation implies ${\bf v}\cdot{\bf q}=-q^2/(2\mu_N)$, and hence $ {\bf{v}^{\perp}}\cdot {\bf q}=0$, with ${\bf{v}^{\perp}} \equiv {\bf v}+{\bf q}/(2\mu_N)$. This justifies the use of the notation $\bf{\hat{v}^{\perp}}$. In Eqs. (\[eq:me1\]) and (\[eq:me2\]) we adopt a non-relativistic normalization for single-particle states.
------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\hat{\mathcal{O}}_1 = \mathbb{1}_{\chi N}$ $\hat{\mathcal{O}}_9 = i{\bf{\hat{S}}}_\chi\cdot\left(\hat{{\bf{S}}}_N\times\frac{{\bf{\hat{q}}}}{m_N}\right)$
$\hat{\mathcal{O}}_3 = i\hat{{\bf{S}}}_N\cdot\left(\frac{{\bf{\hat{q}}}}{m_N}\times{\bf{\hat{v}}}^{\perp}\right)$ $\hat{\mathcal{O}}_{10} = i\hat{{\bf{S}}}_N\cdot\frac{{\bf{\hat{q}}}}{m_N}$
$\hat{\mathcal{O}}_4 = \hat{{\bf{S}}}_{\chi}\cdot \hat{{\bf{S}}}_{N}$ $\hat{\mathcal{O}}_{11} = i{\bf{\hat{S}}}_\chi\cdot\frac{{\bf{\hat{q}}}}{m_N}$
$\hat{\mathcal{O}}_5 = i{\bf{\hat{S}}}_\chi\cdot\left(\frac{{\bf{\hat{q}}}}{m_N}\times{\bf{\hat{v}}}^{\perp}\right)$ $\hat{\mathcal{O}}_{12} = \hat{{\bf{S}}}_{\chi}\cdot \left(\hat{{\bf{S}}}_{N} \times{\bf{\hat{v}}}^{\perp} \right)$
$\hat{\mathcal{O}}_6 = \left({\bf{\hat{S}}}_\chi\cdot\frac{{\bf{\hat{q}}}}{m_N}\right) \left(\hat{{\bf{S}}}_N\cdot\frac{\hat{{\bf{q}}}}{m_N}\right)$ $\hat{\mathcal{O}}_{13} =i \left(\hat{{\bf{S}}}_{\chi}\cdot {\bf{\hat{v}}}^{\perp}\right)\left(\hat{{\bf{S}}}_{N}\cdot \frac{{\bf{\hat{q}}}}{m_N}\right)$
$\hat{\mathcal{O}}_7 = \hat{{\bf{S}}}_{N}\cdot {\bf{\hat{v}}}^{\perp}$ $\hat{\mathcal{O}}_{14} = i\left(\hat{{\bf{S}}}_{\chi}\cdot \frac{{\bf{\hat{q}}}}{m_N}\right)\left(\hat{{\bf{S}}}_{N}\cdot {\bf{\hat{v}}}^{\perp}\right)$
$\hat{\mathcal{O}}_8 = \hat{{\bf{S}}}_{\chi}\cdot {\bf{\hat{v}}}^{\perp}$ $\hat{\mathcal{O}}_{15} = -\left(\hat{{\bf{S}}}_{\chi}\cdot \frac{{\bf{\hat{q}}}}{m_N}\right)\left[ \left(\hat{{\bf{S}}}_{N}\times {\bf{\hat{v}}}^{\perp} \right) \cdot \frac{{\bf{\hat{q}}}}{m_N}\right] $
------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Non-relativistic quantum mechanical operators constructed from Eq. (\[eq:5Op\]). Introducing the nucleon mass, $m_N$, in the equations all operators have the same mass dimension.[]{data-label="tab:operators"}
Only 14 linearly independent quantum mechanical operators can be constructed from (\[eq:5Op\]), if we demand that they are at most linear in $\hat{{\bf{S}}}_N$, $\hat{{\bf{S}}}_\chi$ and $\bf{\hat{v}^{\perp}}$. They are listed in Tab. \[tab:operators\], and labelled as in [@Anand:2013yka], where the operator $\hat{\mathcal{O}}_2={\bf{\hat{v}}}^{\perp}\cdot{\bf{\hat{v}}}^{\perp}$ was neglected since it cannot be a leading-order operator in effective theories. They are at most quadratic in the momentum transfer, with the exception of $\hat{\mathcal{O}}_{15}$, that is cubic in ${\bf{\hat{q}}}$. The restriction on the number of spin/transverse relative velocity operators is a constraint on the spin of the particle mediating the underlying relativistic interaction, that is assumed here to be less than or equal to 1. Tab. \[tab:operators\] also assumes that the mediating particle is heavy compared to the momentum transfer, i.e. long-range interactions are not included.
The most general Hamiltonian density for dark matter-nucleon interactions mediated by a heavy spin-0 or spin-1 particle is hence a linear combination of 14 non-relativistic quantum mechanical operators, $\hat{\mathcal{O}}_k$, and is given by $$\begin{aligned}
{\bf\hat{\mathcal{H}}}({\bf{r}}) &=& 2 \sum_{k=1}^{15} \left[ c_k^{p} \left( \frac{\mathbb{1}+ \tau_3}{2} \right) + c_k^{n} \left( \frac{\mathbb{1} - \tau_3}{2} \right) \right] \hat{\mathcal{O}}_{k}({\bf{r}}) \,.
\label{eq:H}\end{aligned}$$ In Eq. (\[eq:H\]), $c^{p}_k$ and $c^{n}_k$ are the coupling constants for protons and neutrons as implemented in [@Anand:2013yka], and have dimension mass$^{-2}$. By construction, $c_2^p=c_2^n=0$. $\tau_3$ is the third Pauli matrix, and $\mathbb{1}$ denotes the $2\times2$ identity in isospin space. The matrices $(\mathbb{1} \pm \tau_{3})/2$ project a nucleon state into states of well defined isospin, i.e. protons and neutrons. As for the building blocks in Eq. (\[eq:5Op\]), the operators $\hat{\mathcal{O}}_k$ act on the two-particle Hilbert space spanned by tensor products of dark matter and nucleon states, $|{\bf p},j_\chi\rangle$ and $|{\bf k},j_N\rangle$, respectively. In the calculation of nuclear matrix elements for dark matter scattering by nuclei with well defined isospin quantum numbers, it is convenient to rewrite Eq. (\[eq:H\]) in terms of isoscalar and isovector coupling constants: $$\begin{aligned}
{\bf\hat{\mathcal{H}}}({\bf{r}})&=& \sum_{\tau=0,1} \sum_{k=1}^{15} c_k^{\tau} \hat{\mathcal{O}}_{k}({\bf{r}}) \, t^{\tau} \,.
\label{eq:Hc0c1}\end{aligned}$$ In Eq. (\[eq:Hc0c1\]) $t^0=\mathbb{1}$, $t^1=\tau_3$, and the isoscalar and isovector coupling constants, respectively, $c^0_k$ and $c^{1}_k$, are related to $c^{p}_k=(c^{0}_k+c^{1}_k)/2$ and $c^{n}_k=(c^{0}_k-c^{1}_k)/2$.
Dark matter-nucleus effective interactions
------------------------------------------
We construct the dark matter-nucleus interaction Hamiltonian density, $\hat{\mathcal{H}}_{\rm T}({\bf{r}})$, from Eq. (\[eq:Hc0c1\]) under the assumption of one-body dark matter-nucleon interactions. Within this assumption, $\hat{\mathcal{H}}_{\rm T}({\bf{r}})$ is the sum of $A$ terms of type (\[eq:Hc0c1\]), one for each of the $A$ nucleons in the target nucleus $$\hat{\mathcal{H}}_{\rm T}({\bf{r}}) = \sum_{i=1}^{A} \sum_{\tau=0,1} \sum_{k=1}^{15} c_k^{\tau}\hat{\mathcal{O}}_{k}^{(i)}({\bf{r}}) \, t^{\tau}_{(i)} \,.
\label{eq:H_I}$$ The Hermitian and Galilean invariant quantum mechanical operators $\hat{\mathcal{O}}_{k}^{(i)}({\bf{r}})$, $k=1,\dots,15$, are listed in Tab. \[tab:operators\]. We use the index $i$ to identify the specific nucleon to which dark matter couples in the scattering. Distinct nucleons are characterized by different isospin matrices $t^{\tau}_{(i)}$.
Notice that within the effective theory approach reviewed here, the spin-dependent operators $\hat{\mathcal{O}}_{4}$ and $\hat{\mathcal{O}}_{6}$ are treated independently. In contrast, the non-relativistic limit of a contact axial-axial dark matter-nucleon interaction generates a linear combination of the two operators [@Engel:1992]. Standard spin-dependent form factors used to interpret direct detection experiments account for this linear combination. The situation is different in the context of dark matter searches with neutrino telescopes. Here the operator $\hat{\mathcal{O}}_{4}$ is considered separately, in that for spin-dependent interactions dark matter is assumed to only scatter off Hydrogen with a constant cross-section, and obviously with no nuclear form factor.
In the single-particle state of the $i$th-nucleon, it is convenient to separate the motion of the nucleus center of mass from the intrinsic motion (relative to the nucleus center of mass) of the nucleon itself. This separation induces the following coordinate space representations for ${\bf{\hat{q}}}$ and ${\bf{\hat{v}}}^{\perp}$: $$\begin{aligned}
\label{eq:qx}
{\bf{\hat{q}}} &=& - i \overleftarrow{\nabla}_{{\bf{x}}} \,\delta({\bf{x}}-{\bf{y}}+{\bf{r}}) - i\delta({\bf{x}}-{\bf{y}}+{\bf{r}}) \overrightarrow{\nabla}_{{\bf{x}}} \\
\nonumber\\
\label{eq:vtvn}
{\bf{\hat{v}}}^{\perp} &=& {\bf{\hat{v}}}^{\perp}_{T} + {\bf{\hat{v}^{\perp}}}_{N}\,,\end{aligned}$$ with $$\begin{aligned}
\label{eq:xrep0}
{\bf{\hat{v}^{\perp}}}_{T} &=& \delta({\bf{x}}-{\bf{y}}+{\bf{r}}) \left( i \frac{ \overrightarrow{\nabla}_{{\bf{x}}}}{m_T} - i \frac{\overrightarrow{\nabla}_{{\bf{y}}}}{m_\chi} \right) + \frac{1}{2\mu_T}{\bf{\hat{q}}}
\\
{\bf{\hat{v}^{\perp}}}_{N} &=& \frac{1}{2m_{N}}\left[i \overleftarrow{\nabla}_{{\bf{r}}} \,\delta({\bf{r}}-{\bf{r}}_i) - i\delta({\bf{r}}-{\bf{r}}_i) \overrightarrow{\nabla}_{{\bf{r}}} \right] \,.
\label{eq:xrep}\end{aligned}$$ The operator $\nabla_{\bf{x}}$ acts on the nucleus center of mass wave function at ${\bf{x}}$, whereas $\nabla_{\bf{y}}$ acts on the dark matter particle wave function at ${\bf{y}}$. In Eq. (\[eq:xrep0\]), $m_T$ is the target nucleus mass, and $\mu_T$ the dark matter-nucleus reduced mass. Finally, the operator $\nabla_{\bf{r}}$ in Eq. (\[eq:xrep\]) acts on the constituent nucleon wave function at ${\bf{r}}$, where ${\bf{r}}$ denotes the radial coordinate of the dark matter particle in a frame with origin at the nucleus center of mass (notice that in Sec. \[sec:dmnu\], [**[r]{}**]{} was the position vector from the single nucleon to the dark matter particle). Separating the center of mass motion from the intrinsic motion of the constituent nucleons, the only operator depending on the position of the nucleons relative to the nucleus centre of mass, ${\bf{r}}_i$, is ${\bf{\hat{v}^{\perp}}}_{N}({\bf{x}})$. Operators in Tab. \[tab:operators\] independent of ${\bf{\hat{v}^{\perp}}}_{N}({\bf{x}})$ act like the identity $\mathbb{1}_{i}$ on the $i$th-nucleon position ${\bf{r}}_i$. In the coordinate space representation $\mathbb{1}_{i}$ corresponds to $\delta({\bf{r}}-{\bf{r}}_i)$.
Combining Eqs. (\[eq:H\_I\]), (\[eq:vtvn\]) and (\[eq:xrep\]) with Tab. \[tab:operators\], we can finally write the most general Hamiltonian density for dark matter-nucleus interactions mediated by heavy spin-0 or spin-1 particles as a combination of (one-body) charge and nuclear currents [@Fitzpatrick:2012ix]: $$\begin{aligned}
\hat{\mathcal{H}}_{\rm T}({\bf{r}}) &=& \sum_{\tau=0,1} \Bigg\{
\sum_{i=1}^A \hat{l}_0^{\tau}~ \delta({\bf{r}}-{\bf{r}}_i)
+ \sum_{i=1}^A \hat{l}_{0A}^{\tau}~ \frac{1}{2m_N} \Bigg[i \overleftarrow{\nabla}_{\bf{r}} \cdot \vec{\sigma}(i)\delta({\bf{r}}-{\bf{r}}_i) -i \delta({\bf{r}}-{\bf{r}}_i) \
\vec{\sigma}(i) \cdot \overrightarrow{\nabla}_{\bf{r}} \Bigg] \nonumber \\
&+& \sum_{i=1}^A {\bf{\hat{l}}}_5^{\tau} \cdot \vec{\sigma}(i) \delta({\bf{r}}-{\bf{r}}_i) + \sum_{i=1}^A {\bf{\hat{l}}}_M^{\tau} \cdot \frac{1}{2 m_N} \Bigg[i \overleftarrow{\nabla}_{\bf{r}}\delta({\bf{r}}-{\bf{r}}_i) -i \delta({\bf{r}}-{\bf{r}}_i)\overrightarrow{\nabla}_{\bf{r}} \Bigg] \nonumber \\
&+& \sum_{i=1}^A {\bf{\hat{l}}}_E^{\tau} \cdot \frac{1}{2m_N} \Bigg[ \overleftarrow{\nabla}_{\bf{r}} \times \vec{\sigma}(i) \delta({\bf{r}}-{\bf{r}}_i) +\delta({\bf{r}}-{\bf{r}}_i)\
\vec{\sigma}(i) \times \overrightarrow{\nabla}_{\bf{r}} \Bigg] \Bigg\} t^{\tau}_{(i)}
\label{eq:Hx}\end{aligned}$$ where $\vec{\sigma}(i)$ denotes the set of three Pauli matrices representing the spin operator of the $i$th-nucleon in the target nucleus, and $$\begin{aligned}
\label{eq:ls}
\hat{l}_0^\tau &=& c_1^\tau + i \left( {{\bf{\hat{q}}} \over m_N} \times {\bf{\hat{v}}}_{T}^\perp \right) \cdot {\bf{\hat{S}}}_\chi ~c_5^\tau
+ {\bf{\hat{v}}}_{T}^\perp \cdot {\bf{\hat{S}}}_\chi ~c_8^\tau + i {{\bf{\hat{q}}} \over m_N} \cdot {\bf{\hat{S}}}_\chi ~c_{11}^\tau \nonumber \\
\hat{l}_{0A}^{\tau} &=& -{1 \over 2} \left[ c_7 ^\tau +i {{\bf{\hat{q}}} \over m_N} \cdot {\bf{\hat{S}}}_\chi~ c_{14}^\tau \right] \nonumber \\
{\bf{\hat{l}}}_5^{\tau}&=& {1 \over 2} \left[ i {{\bf{\hat{q}}} \over m_N} \times {\bf{\hat{v}}}_{T}^\perp~ c_3^\tau + {\bf{\hat{S}}}_\chi ~c_4^\tau
+ {{\bf{\hat{q}}} \over m_N}~{{\bf{\hat{q}}} \over m_N} \cdot {\bf{\hat{S}}}_\chi ~c_6^\tau
+ {\bf{\hat{v}}}_{T}^\perp ~c_7^\tau + i {{\bf{\hat{q}}} \over m_N} \times {\bf{\hat{S}}}_\chi ~c_9^\tau + i {{\bf{\hat{q}}} \over m_N}~c_{10}^\tau \right. \nonumber \\
&& \left. + {\bf{\hat{v}}}_{T}^\perp \times {\bf{\hat{S}}}_\chi ~c_{12}^\tau
+i {{\bf{\hat{q}}} \over m_N} {\bf{\hat{v}}}_{T}^\perp \cdot {\bf{\hat{S}}}_\chi ~c_{13}^\tau+i {\bf{\hat{v}}}_{T}^\perp {{\bf{\hat{q}}} \over m_N} \cdot {\bf{\hat{S}}}_\chi ~ c_{14}^\tau+{{\bf{\hat{q}}} \over\
m_N} \times {\bf{\hat{v}}}_{T}^\perp~ {{\bf{\hat{q}}} \over m_N} \cdot {\bf{\hat{S}}}_\chi ~ c_{15}^\tau \right]\nonumber \\
{\bf{\hat{l}}}_M^{\tau} &=& i {{\bf{\hat{q}}} \over m_N} \times {\bf{\hat{S}}}_\chi ~c_5^\tau - {\bf{\hat{S}}}_\chi ~c_8^\tau \nonumber \\
{\bf{\hat{l}}}_E^{\tau} &=& {1 \over 2} \left[ {{\bf{\hat{q}}} \over m_N} ~ c_3^\tau +i {\bf{\hat{S}}}_\chi~c_{12}^\tau - {{\bf{\hat{q}}} \over m_N} \times{\bf{\hat{S}}}_\chi ~c_{13}^\tau-i
{{\bf{\hat{q}}} \over m_N} {{\bf{\hat{q}}} \over m_N} \cdot {\bf{\hat{S}}}_\chi ~c_{15}^\tau \right] \,.\end{aligned}$$ Inspection of Eq. (\[eq:Hx\]) shows that dark matter couples to the constituent nucleons through the nuclear vector and axial charges (first line in Eq. (\[eq:Hx\])), the nuclear spin and convection currents (second line in Eq. (\[eq:Hx\])), and the nuclear spin-velocity current (last line in Eq. (\[eq:Hx\])). The 14 dark matter-nucleon interaction operators in Tab. \[tab:operators\] contribute to these couplings in different ways. For instance, the constant spin-independent operator $\mathcal{O}_1$ contributes to the vector charge coupling through the operator $\hat{l}_0^\tau$, whereas the constant spin-dependent operator $\mathcal{O}_4$ contributes to the nuclear spin current coupling through the operator ${\bf{\hat{l}}}_5^{\tau}$.
The interaction Hamiltonian relevant for nuclear matrix element calculations is finally obtained by integrating the Hamiltonian density (\[eq:H\_I\]) over space coordinates $$\begin{aligned}
H_{\rm T}= \int d^{3}{\bf{r}}\, \hat{\mathcal{H}}_{\rm T}({\bf{r}}) \,.
\label{eq:Hfinal}\end{aligned}$$ This latter integration eliminates the delta functions $\delta({\bf{r}}-{\bf{r}}_i)$ in Eq. (\[eq:Hx\]).
Dark matter-nucleus scattering cross-section
--------------------------------------------
From the dark matter-nucleus interaction Hamiltonian (\[eq:Hfinal\]), one can calculate the amplitude for transitions between initial, $|i\rangle$, and final, $|f\rangle$, scattering states. We denote initial nuclear states by $|{\bf{k}}_T,J,M_J,T,M_T\rangle$, where $J$ and $T$ are the nuclear spin and isospin, respectively, and $M_J$ and $M_T$ the associated magnetic quantum numbers. With this notation $|i\rangle~=~|{\bf{k}}_T,J,M_J,T,M_T\rangle \otimes |{\bf{p}},j_\chi,M_\chi\rangle$ ($M_\chi$ is the spin magnetic quantum number of the dark matter particle, omitted in previous equations for simplicity), and an analogous expression applies to $|f\rangle$. We can therefore write $$\langle f |H_{\rm T} |i\rangle = (2\pi)^3 \delta({\bf k}'_T+{\bf p}'-{\bf k}_T-{\bf p}) \,i \mathcal{M}_{NR}
\label{eq:M0}$$ with $$\begin{aligned}
i \mathcal{M}_{NR}&=&
\langle J,M_J, T, M_T | \sum_{\tau=0,1} \left[
\langle \hat{l}_0^\tau\rangle~ \sum_{i=1}^A ~e^{-i {\bf{q}} \cdot {\bf{r}}_i} \right. \nonumber \\
&+& \langle \hat{l}_{0A}^{\tau}\rangle~ \sum_{i=1}^A ~ {1 \over 2m_N} \left(i \overleftarrow{\nabla}_{{\bf{r}}_i} \cdot \vec{\sigma}(i)~ e^{-i {\bf{q}} \cdot {\bf{r}}_i}
-ie^{-i {\bf{q}} \cdot {\bf{r}}_i} \vec{\sigma}(i) \cdot \overrightarrow{\nabla}_{{\bf{r}}_i} \right) \nonumber \\
&+& \langle{\bf{\hat{l}}}_5^\tau\rangle \cdot \sum_{i=1}^A ~\vec{\sigma}(i)~e^{-i {\bf{q}} \cdot {\bf{r}}_i} \nonumber \\
&+& \langle{\bf{\hat{l}}}_M^\tau\rangle \cdot \sum_{i=1}^A ~ {1 \over 2m_N} \left(i \overleftarrow{\nabla}_{{\bf{r}}_i} e^{-i {\bf{q}} \cdot {\bf{r}}_i} -i e^{-i {\bf{q}} \cdot {\bf{r}}_i}
\overrightarrow{\nabla}_{{\bf{r}}_i} \right) \nonumber \\
&+& \langle{\bf{\hat{l}}}_E^\tau\rangle \left. \cdot \sum_{i=1}^A ~ {1 \over 2m_N} \left( \overleftarrow{\nabla}_{{\bf{r}}_i} \times \vec{\sigma}(i) e^{-i {\bf{q}} \cdot {\bf{r}}_i} +e^{-i {\bf{q}} \cdot {\bf{r}}_i} \vec{\sigma}(i) \times \overrightarrow{\nabla}_{{\bf{r}}_i} \right) \right] t^\tau_{(i)} ~ | J, M_J,T,M_T \rangle \,. \nonumber\\
\label{eq:M}\end{aligned}$$ In Eqs. (\[eq:M0\]) and (\[eq:M\]) we use the result $$\begin{aligned}
\langle {\bf{k}}'_T;{\bf{p}}',j_\chi,M_\chi |\,{\bf{\hat{l}}}[{\bf{\hat{q}}},{\bf{\hat{v}}}^{\perp}_T,{\bf{\hat{S}}}_\chi]\,| {\bf{k}}_T;{\bf{p}},j_\chi,M_\chi \rangle &=&e^{-i {\bf{q}} \cdot {\bf{r}}} (2\pi)^3\delta({\bf k}'_T+{\bf p}'-{\bf k}_T-{\bf p}) \nonumber\\ &\times& \langle j_\chi,M_\chi |\,{\bf{\hat{l}}}[{\bf{q}},{\bf{v}}^{\perp}_T,{\bf{\hat{S}}}_\chi]\,| j_\chi,M_\chi \rangle \,,
\label{eq:lme}\end{aligned}$$ and the notation $\langle {\bf{\hat{l}}} \rangle \equiv \langle j_\chi,M_\chi |\,{\bf{\hat{l}}}[{\bf{q}},{\bf{v}}^{\perp}_T,{\bf{\hat{S}}}_\chi]\,| j_\chi,M_\chi \rangle$, with ${\bf{\hat{l}}}$ equal to one of the operators $\hat{l}_0^\tau$, $\hat{l}_{0A}^{\tau}$, ${\bf{\hat{l}}}_5^\tau$, ${\bf{\hat{l}}}_M^\tau$, and ${\bf{\hat{l}}}_E^\tau$. Notice that on the right hand side of Eq. (\[eq:lme\]), ${\bf{q}}$ and ${\bf{v}}^{\perp}_T\equiv {\bf{v}} + {\bf{q}}/(2\mu_T)$ replace, respectively, ${\bf{\hat{q}}}$ and ${\bf{\hat{v}}}^{\perp}_T$, in agreement with Eqs. (\[eq:me1\]) and (\[eq:me2\]). From now on $|{\bf{v}}|=w$ denotes the dark matter-nucleus relative velocity in the Sun. Importantly, each line in the transition amplitude (\[eq:M\]) is equal to the product of a term containing information on the kinematics of the scattering and on the dark matter-nucleon coupling strength, i.e. $\langle {\bf{\hat{l}}}\rangle $, and a term given by a nuclear matrix element.
In order to evaluate the nuclear matrix elements in Eq. (\[eq:M\]), we perform a multipole expansion of the nuclear charges and currents using a spherical unit vector basis ${\bf {e}}_\lambda$ with z-axis along ${\bf{q}}$, and the identities $$\begin{aligned}
e^{i {\bf{q}} \cdot {\bf{r}}_i} &=& \sum_{L=0}^\infty \sqrt{4 \pi(2L+1)}~ i^L j_L(q r_i) Y_{L0}(\Omega_{{\bf{r}}_i}) \nonumber \\
e^{i {\bf{q}} \cdot {\bf{r}}_i} {\bf{e}}_{0} &=& \sum_{L=0}^\infty \sqrt{4 \pi(2L+1)}~ i^{L-1} {\overrightarrow{\nabla}_{{\bf{r}}_i} \over q\
} j_L(qr_i) Y_{L0}(\Omega_{{\bf{r}}_i})\nonumber\\
e^{i {\bf{q}} \cdot {\bf{r}}_i} {\bf{e}}_{\lambda} &=& \sum_{L = 1}^\infty \sqrt{2 \pi(2L+1)}~ i^{L-2} \left[
\lambda j_L(qr_i) {\bf Y}_{LL1}^\lambda(\Omega_{{\bf{r}}_i}) + { \overrightarrow{\nabla}_{{\bf{r}}_i} \over q} \times j_L(qr_i) {\bf Y}_{LL1}^\lambda
(\Omega_{{\bf{r}}_i}) \right], \lambda=\pm 1\nonumber\\
\label{eq:Y}\end{aligned}$$ together with $${\bf{A}}=\sum_{\lambda=0,\pm1} \left( {\bf{A}} \cdot {\bf{e}}_{\lambda} \right) {\bf{e}}^{\dagger}_{\lambda}\,,$$ that holds for any vector ${\bf{A}}$, given a spherical unit vector basis ${\bf {e}}_\lambda$. The vector spherical harmonics in Eq. (\[eq:Y\]) are defined in terms of Clebsch-Gordan coefficients and scalar spherical harmonics: $${\bf Y}^M_{LL'1}(\Omega_{{\bf{r}}_i}) = \sum_{m\lambda} \langle L'm1\lambda|L'1LM \rangle
Y_{L'm}(\Omega_{{\bf{r}}_i}) \, {\bf e}_\lambda \,.$$ They obey the identity ${\bf Y}^{\lambda\dagger}_{LL'1}=-(-1)^{\lambda}{\bf Y}^{-\lambda}_{LL'1}$. The multipole expansion of the nuclear spin current, for instance, leads to $$\begin{aligned}
\langle{\bf{\hat{l}}}_5^\tau\rangle \cdot \sum_{i=1}^A ~\vec{\sigma}(i)~e^{-i {\bf{q}} \cdot {\bf{r}}_i} &=& \sum_{L=0}^{\infty}
\sqrt{4\pi (2L +1)}(-i)^{L} i \Sigma''_{L0;\tau} (q) (\langle{\bf{\hat{l}}}_5^\tau\rangle \cdot {\bf{e}}_0) \nonumber \\
&-& \sum_{L=1}^{\infty}
\sqrt{2\pi (2L +1)}(-i)^{L} \sum_{\lambda=\pm1} \Big(\lambda \Sigma_{L-\lambda;\tau}(q)+i\Sigma'_{L-\lambda;\tau}(q)\Big) (\langle{\bf{\hat{l}}}_5^\tau\rangle \cdot {\bf{e}}_\lambda) \,, \nonumber\\
\label{eq:l5}\end{aligned}$$ with $$\begin{aligned}
\Sigma'_{LM;\tau}(q) &=& -i \sum_{i=1}^{A} \left[ \frac{1}{q} \overrightarrow{\nabla}_{{\bf{r}}_i} \times {\bf{M}}_{LL}^{M}(q {\bf{r}}_i) \right] \cdot \vec{\sigma}(i) t^{\tau}_{(i)}\nonumber\\
\Sigma''_{LM;\tau}(q) &=&\sum_{i=1}^{A} \left[ \frac{1}{q} \overrightarrow{\nabla}_{{\bf{r}}_i} M_{LM}(q {\bf{r}}_i) \right] \cdot \vec{\sigma}(i) t^{\tau}_{(i)}
\label{eq:S1S2}\end{aligned}$$ where ${\bf{M}}_{LL}^{M}(q {\bf{r}}_i)=j_{L}(q r_i){\bf Y}^M_{LL1}(\Omega_{{\bf{r}}_i})$ , and $M_{LM}(q {\bf{r}}_i)=j_{L}(q r_i)Y_{LM}(\Omega_{{\bf{r}}_i})$. Assuming that nuclear ground states are eigenstates of $P$ and $CP$, only multipoles that transform as even-even under $P$ and $CP$ contribute to the square modulus of the transition amplitude. With this assumption $\Sigma_{LM;\tau}(q)$ does not contribute at all, and is therefore not defined here. Expressions similar to Eq. (\[eq:l5\]) can be derived for the remaining charges and currents. Besides the two operators in Eq. (\[eq:S1S2\]), four additional nuclear response operators contribute to the transition probability, namely $$\begin{aligned}
M_{LM;\tau}(q) &=& \sum_{i=1}^{A} M_{LM}(q {\bf{r}}_i) t^{\tau}_{(i)}\nonumber\\
\Delta_{LM;\tau}(q) &=&\sum_{i=1}^{A} {\bf{M}}_{LL}^{M}(q {\bf{r}}_i) \cdot \frac{1}{q}\overrightarrow{\nabla}_{{\bf{r}}_i} t^{\tau}_{(i)} \nonumber\\
\tilde{\Phi}^{\prime}_{LM;\tau}(q) &=& \sum_{i=1}^A \left[ \left( {1 \over q} \overrightarrow{\nabla}_{{\bf{r}}_i} \times {\bf{M}}_{LL}^M(q {\bf{r}}_i) \right) \cdot \left(\vec{\sigma}(i) \times {1 \over q} \overrightarrow{\nabla}_{{\bf{r}}_i} \right) + {1 \over 2} {\bf{M}}_{LL}^M(q {\bf{r}}_i) \cdot \vec{\sigma}(i) \right]~t^\tau_{(i)} \nonumber \\
\Phi^{\prime \prime}_{LM;\tau}(q ) &=& i \sum_{i=1}^A\left( {1 \over q} \overrightarrow{\nabla}_{{\bf{r}}_i} M_{LM}(q {\bf{r}}_i) \right) \cdot \left(\vec{\sigma}(i) \times \
{1 \over q} \overrightarrow{\nabla}_{{\bf{r}}_i} \right)~t^\tau_{(i)} \,.
\label{eq:MDP}\end{aligned}$$ Squaring the amplitude (\[eq:M\]), summing (averaging) the result over final (initial) spin configurations, and demanding that only multipoles transforming as even-even under $P$ and $CP$ contribute, one finally obtains [@Fitzpatrick:2012ix] $$\begin{aligned}
P_{\rm tot}({w}^2,{q}^2)&\equiv&{1 \over 2j_\chi + 1} {1 \over 2J + 1} \sum_{\rm spins} |\mathcal{M}_{NR}|^2 \nonumber \\ &=& {4 \pi \over 2J + 1}
\sum_{ \tau=0,1} \sum_{\tau^\prime = 0,1} \Bigg\{ \Bigg[ R_{M}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right)~W_{M}^{\tau \tau^\prime}(y) \nonumber\\
&+& R_{\Sigma^{\prime \prime}}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~W_{\Sigma^{\prime \prime}}^{\tau \tau^\prime}(y)
+ R_{\Sigma^\prime}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~ W_{\Sigma^\prime}^{\tau \tau^\prime}(y) \Bigg] \nonumber\\
&+& {{q}^{2} \over m_N^2} ~\Bigg[R_{\Phi^{\prime \prime}}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~ W_{\Phi^{\prime \prime}}^{\tau \tau^\prime}(y) + R_{ \Phi^{\prime \prime}M}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~W_{ \Phi^{\prime \prime}M}^{\tau \tau^\prime}(y) \nonumber\\
&+& R_{\tilde{\Phi}^\prime}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~W_{\tilde{\Phi}^\prime}^{\tau \tau^\prime}(y)
+ R_{\Delta}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~ W_{\Delta}^{\tau \tau^\prime}(y) \nonumber\\
&+& R_{\Delta \Sigma^\prime}^{\tau \tau^\prime}\left({v}^{\perp 2}_{T}, {{q}^{2} \over m_N^2}\right) ~W_{\Delta \Sigma^\prime}^{\tau \tau^\prime}(y) \Bigg] \Bigg\} \,,
\label{eq:Ptot}\end{aligned}$$ where the dark matter response function $R_{M}^{\tau \tau^\prime}$, $R_{\Sigma^{\prime \prime}}^{\tau \tau^\prime}$, $R_{\Sigma^\prime}^{\tau \tau^\prime}$, $R_{\Phi^{\prime \prime}}^{\tau \tau^\prime}$, $R_{\Phi^{\prime\prime}M}^{\tau \tau^\prime}$, $R_{\tilde{\Phi}^\prime}^{\tau \tau^\prime}$, $R_{\Delta}^{\tau \tau^\prime}$ and $R_{\Delta \Sigma^\prime}^{\tau \tau^\prime}$ are quadratic combinations of the matrix elements $\langle {\bf{\hat{l}}} \rangle$ and are defined in Appendix \[sec:appDM\]. They depend on the momentum transfer, the dark matter-nucleus relative velocity, as well as on the dark matter-nucleon interaction strength.
The nuclear response functions in Eq. (\[eq:Ptot\]) are defined as follows $$W_{AB}^{\tau \tau^\prime}(y)= \sum_{L\in S_{AB}} \langle J,T,M_T ||~ A_{L;\tau} (q)~ || J,T,M_T \rangle \langle J,T,M_T ||~ B_{L;\tau^\prime} (q)~ || J,T,M_T \rangle \,
\label{eq:W}$$ where $A$ and $B$ correspond to pairs of operators in Eqs.(\[eq:S1S2\]) and (\[eq:MDP\]). When $A=B$, only one letter is used. $S_{AB}=\{0,2,\dots\}$ for the pairs of operators $A=B=M_{LM;\tau}$, $A=B=\Phi^{\prime\prime}_{LM;\tau}$ and $A=\Phi^{\prime\prime}_{LM;\tau}$, $B=M_{LM;\tau}$. $S_{AB}=\{1,3,\dots\}$ for the pairs of operators $A=B=\Sigma^{\prime}_{LM;\tau}$, $A=B=\Sigma^{\prime\prime}_{LM;\tau}$, $A=B=\Delta_{LM;\tau}$, and $A=\Delta_{LM;\tau}$, $B=\Sigma^{\prime}_{LM;\tau}$. Finally, $S_{AB}=\{2,4,\dots\}$ for the pair of operators $A=B=\tilde{\Phi}^{\prime}_{LM;\tau}$. The integer numbers in $S_{AB}$ select multipoles transforming as even-even under $P$ and $CP$. Notice that only two interference terms, i.e. $A\neq B$, can satisfy this requirement and thereby appear in Eq. (\[eq:Ptot\]).
The nuclear response functions in Eq. (\[eq:W\]) are expressed in terms of matrix elements reduced in the spin magnetic quantum number $M_J$. The reduction of a tensor operator $A_{LM;\tau}$ of rank $L$ is done by the Wigner-Eckart theorem $$\langle J,M_J |\,{A}_{LM;\tau}\,|J,M_J\rangle =(-1)^{J-M_J}\left(
\begin{array}{ccc} J&L&J\\
-M_J&M&M_J
\end{array}
\right)
\langle J ||\,{A}_{L;\tau}\,|| J \rangle \,,
\label{eq:red}$$ and it involves Wigner $3j$-symbols which cancel in Eq. (\[eq:W\]) after summing over spin configurations because of their orthonormality. In the next section, we will evaluate our nuclear response functions using the [Mathematica]{} package of Ref. [@Anand:2013yka], which assumes the harmonic oscillator basis with length parameter $b=\sqrt{41.467/(45 A^{-1/3}-25A^{-2/3})}$ fm for the single-particle states. In this case, the nuclear response functions in Eq. (\[eq:W\]) only depend on the dimensionless variable $y=(bq/2)^2$.
For the $i$th-element in the Sun, we can finally write the dark matter-nucleus differential cross-section as follows $$\frac{{\rm d} \sigma_i}{{\rm d} E}(w^2,q^2) = \frac{m_T}{2\pi w^2} \,P_{\rm tot}({w}^2,{q}^2) \,,
\label{eq:sigma}$$ which constitutes the particle physics input in the calculation of the rate of dark matter capture by the Sun.
Nuclear matrix element calculation {#sec:obdme}
==================================
In this section we calculate the reduced nuclear matrix elements that appear in Eq. (\[eq:W\]) for the most abundant elements in the Sun. We list analytic expressions for the associated nuclear response functions in Appendix \[sec:appNuc\]. These expressions can be used by the reader in analyses of dark matter induced neutrino signals from the Sun. We perform this calculation using the [Mathematica]{} package introduced in [@Anand:2013yka], which requires as an input the one-body density matrix elements (OBDME) for ground-state to ground-state transitions of the target nuclei in analysis. We compute these OBDME using the [Nushell@MSU]{} program [@NuShell], which allows for fast nuclear structure calculations based on the nuclear shell model.
In order to relate the nuclear matrix elements in Eq. (\[eq:W\]) to the underlying OBDME, we expand the nuclear operators in Eqs. (\[eq:S1S2\]) and (\[eq:MDP\]), here collectively denoted by $A_{LM;\tau}$, in a complete set of spherically symmetric single-particles states, $|\alpha\rangle$. Here we assume the nuclear harmonic oscillator model for the radial part of the wave functions associated with the states $|\alpha\rangle$. Within this assumption, single-particle states can be labelled by their principal, angular momentum and spin quantum numbers, respectively $n_\alpha$, $l_\alpha$ and $s_\alpha$, and by their total spin and isospin, respectively $j_{\alpha}$ and $t_{\alpha}$: $|\alpha\rangle=|n_\alpha(l_\alpha s_\alpha=1/2)j_\alpha m_{j_\alpha};t_{\alpha}=1/2,m_{t_\alpha}\rangle$. Here $m_{j_\alpha}$ and $m_{t_\alpha}$ denote the total spin and isospin magnetic quantum numbers, whereas $|\alpha|$ represents the set of all non magnetic quantum numbers, i.e. $|\alpha\rangle=||\alpha|,m_{j_\alpha};m_{t_{\alpha}}\rangle$. With this notation, the nuclear operators in Eqs. (\[eq:S1S2\]) and (\[eq:MDP\]) can be expanded as follows
$$\begin{aligned}
A_{LM;\tau} &=&\; \sum_{\alpha\beta}\langle \alpha|~A_{LM;\tau}~|\beta\rangle \,a_{\alpha}^{\dagger}a_{\beta} \nonumber\\
&=&\; \sum_{|\alpha||\beta|}\langle\left|\alpha\right|\vdots\vdots A_{L;\tau}\vdots\vdots \left|\beta\right|\rangle\frac{[a_{|\alpha|}^{\dagger}\otimes\tilde{a}_{|\beta|}]_{LM;\tau}}{\sqrt{(2L+1)(2\tau+1)}} \,,
\label{eq:keyeq}\end{aligned}$$
where $\tilde{a}_{|\beta|,m_{j_\beta},m_{t_{\beta}}}\equiv (-1)^{j_\beta-m_{j_\beta}+1/2-m_{t_\beta}}\,a_{|\beta|,-m_{j_\beta},-m_{t_{\beta}}}$, and $\vdots\vdots$ denotes a matrix element reduced in spin and isospin according to Eq. (\[eq:red\]). The creation and annihilation operators $a^{\dagger}_{\alpha}$ and $\tilde{a}_\beta$ transform as tensors under spin and isospin transformations, and their tensor product admits the following representation $$\begin{aligned}
[a_{|\alpha|}^{\dagger}\otimes\tilde{a}_{|\beta|}]_{LM;\tau} &=& \sqrt{(2L+1)(2\tau+1)}\sum_{m_{j_\alpha}m_{t_{\alpha}}m_{j_\beta}m_{t_{\beta}}}(-1)^{\,j_{\alpha}-m_{j_\alpha}+t_{j_\alpha}-m_{t_{\alpha}}}\nonumber\\
&\times&\begin{pmatrix}
j_{\alpha}&L&j_{\beta}\\
-m_{\alpha}&M&m_{\beta}
\end{pmatrix}\begin{pmatrix}
t_{\alpha}&\tau&t_{\beta}\\
-m_{t_{\alpha}}&0&m_{t_{\beta}}
\end{pmatrix}a_{\alpha}^{\dagger}a_{\beta}
\,.\end{aligned}$$ The reduced nuclear matrix elements in Eq. (\[eq:W\]) can be further reduced in nuclear isospin, and hence written as $$\begin{aligned}
\langle J, T, M_T ||~ A_{LM;\tau} ~ || J, T, M_T \rangle &=& (-1)^{T-M_T}
\begin{pmatrix}
T&\tau&T\\
-M_T&0&M_T
\end{pmatrix} \nonumber\\
&\times& \sum_{|\alpha||\beta|}\langle\left|\alpha\right|\vdots\vdots A_{L;\tau}\vdots\vdots \left|\beta\right|\rangle\frac{\langle J, T \vdots\vdots ~[a_{|\alpha|}^{\dagger}\otimes\tilde{a}_{|\beta|}]_{L;\tau}
~\vdots\vdots J, T\rangle}{\sqrt{(2L+1)(2\tau+1)}} \,.\nonumber\\\end{aligned}$$ Using the definition of ground-state to ground-state OBDME, namely, $$\psi^{L;\tau}_{|\alpha||\beta|} \equiv \frac{\langle J, T\vdots\vdots ~[a_{|\alpha|}^{\dagger}\otimes\tilde{a}_{|\beta|}]_{L;\tau}
~\vdots\vdots J, T\rangle}{\sqrt{(2L+1)(2\tau+1)}} \,,
\label{eq:OBDME}$$ we can finally write the reduced nuclear matrix elements in Eq. (\[eq:W\]) as follows $$\begin{aligned}
\langle J, T, M_T ||~ A_{LM;\tau} ~ || J, T, M_T \rangle
&=& (-1)^{T-M_T}
\begin{pmatrix}
T&\tau&T\\
-M_T&0&M_T
\end{pmatrix}
\sum_{|\alpha||\beta|}\psi^{L;\tau}_{|\alpha||\beta|} \,\langle \left|\alpha\right|\vdots\vdots A_{L;\tau}\vdots\vdots \left|\beta\right|\rangle \,, \nonumber\\
\label{eq:master}\end{aligned}$$ which is the master equation for nuclear matrix element calculations based on the assumption of one-body dark matter-nucleon interactions. Since the nuclear operators $A_{LM;\tau}$ depend on isospin through the matrices $t^{\tau}_{(i)}$ only, the doubly reduced matrix elements in Eq. (\[eq:master\]) can be further simplified as follows $$\langle \left|\alpha\right|\vdots\vdots A_{L;\tau}\vdots\vdots \left|\beta\right|\rangle = \sqrt{2(2\tau+1)} \,\langle n_{\alpha}(l_\alpha1/2)j_\alpha ||\,A_{L}\,|| n_{\beta}(l_\beta1/2)j_\beta \rangle\,,
\label{eq:math}$$ where $A_L$ is the part of the operator $A_{L;\tau}$ acting on nuclear spin and space coordinates. In Appendix \[sec:appME\], we provide explicit expressions for the reduced matrix elements on the right hand side of Eq. (\[eq:math\]), which in the case of the harmonic oscillator single-particle basis are known analytically, and depend on the momentum transfer through the variable $y$ defined above. The [Mathematica]{} package in Ref. [@Anand:2013yka] provides an efficient implementations of these expressions.
We now move on to the OBDME calculation. In this computation, the multipole number $L$ is bounded from above, i.e. $L\le 2J$, whereas $\tau=0,1$. In contrast, the indexes $|\alpha|$ and $|\beta|$ in principle span a complete set of infinite single-particle quantum numbers. The nuclear shell model provides a robust framework to restrict the set of relevant $|\alpha|$ and $|\beta|$ in the OBDME definition (\[eq:OBDME\]), and to consistently truncate the infinite sums in Eq. (\[eq:master\]).
Element $2J$ $2T$ P core-orbits valence-orbits Hamiltonian restrictions
------------- ------ ------ --- ------------- ---------------- --------------------------- ----------------------------
${}^{3}$He 1 1 + none s-p-sd-pf wbt [@Warburton:1992rh] none
${}^{4}$He 0 0 + none s-p-sd-pf wbt [@Warburton:1992rh] none
${}^{12}$C 0 0 + s p pewt [@Warburton:1992rh] none
${}^{14}$N 2 0 + s p pewt [@Warburton:1992rh] none
${}^{16}$O 0 0 + none s-p-sd-pf wbt [@Warburton:1992rh] $0d_{3/2}1s_{1/2}1p\,0f$
${}^{20}$Ne 0 0 + s-p sd w [@Wildenthal:1984mf] none
${}^{23}$Na 3 1 + s-p sd w [@Wildenthal:1984mf] none
${}^{24}$Mg 0 0 + s-p sd w [@Wildenthal:1984mf] none
${}^{27}$Al 5 1 + s-p sd w [@Wildenthal:1984mf] none
${}^{28}$Si 0 0 + s-p sd w [@Wildenthal:1984mf] none
${}^{32}$S 0 0 + s-p sd w [@Wildenthal:1984mf] none
${}^{40}$Ar 0 4 + s-p sd-pf sdpfnow [@Nummela:2001xh] $1p_{1/2}1p_{3/2}0f_{5/2}$
${}^{40}$Ca 0 0 + s-p sd-pf sdpfnow [@Nummela:2001xh] $1p_{1/2}1p_{3/2}0f_{5/2}$
${}^{56}$Fe 0 4 + s-p-sd pf gx1 [@Honma:2004xk] $1p_{1/2}0f_{5/2}$
${}^{58}$Ni 0 2 + s-p-sd pf gx1 [@Honma:2004xk] $1p_{1/2}0f_{5/2}$
: Summary of element specific input parameters needed for the calculation of the OBDME via the [Nushell@MSU]{} code. We use the notation of [@Brown:2001zz] in defining the major shells. For each element in the Sun, we use a model space comprising the core-orbits and valence-orbits reported in this table. In the “restrictions” column, we list the energetic orbits not included in the calculation in order to make the computation numerically feasible. The interaction Hamiltonians in the next to last column are described in the review [@Brown:2001zz], and in the corresponding references.[]{data-label="tab:inputs"}
In the nuclear shell model nucleons occupy single-particle states degenerate in the total spin magnetic quantum number called sub-shells, or orbits. Sub-shells are solutions of the Schrödinger equation for a given nuclear potential (e.g. harmonic oscillator potential, Woods-Saxon potential, etc…) and reflect a choice of single-particle basis. Orbits are labeled with conventions similar to those used for atomic orbitals, e.g. the orbit $0p_{1/2}$ has “principal quantum number” 0, orbital angular momentum 1 and total spin 1/2. Groups of energetically close sub-shells form major shells of progressively increasing energy. The set of fully occupied major shells forms the nuclear core. For instance, the orbits $0s_{1/2}$, $0p_{3/2}$ and $0p_{1/2}$ divide into the $s$ and $p$ major shells, and together form the core of, e.g, $^{20}$Ne, which contains 8 proton/neutron pairs. Analogously, the orbits $1s_{1/2}$, $0d_{3/2}$, and $0d_{5/2}$ form the $sd$ major shell, and the orbits $1p_{1/2}$, $1p_{3/2}$, $0f_{5/2}$, and $0f_{7/2}$ the $pf$ major shell. Nucleons that are not in the nuclear core are called valence nucleons. Not all orbits are accessible to valence nucleons since sizable energy gaps separate adjacent major shells. Restrictions on the number of nucleons allowed in the most energetic orbits are often imposed in order to reduce the computational effort. The set of orbits that are actually accessible in a calculation constitutes the so-called model space. Therefore, the original $A$-nucleon problem characterized by the bare nuclear interaction is simplified to a many-body problem restricted to the model space, and subject to an effective Hamiltonian. Effective Hamiltonians for nuclear shell model calculations can be computed microscopically from nuclear forces, or fitted empirically to observations. Within this framework, the sums in Eq. (\[eq:master\]) only extend over orbits in the assumed model space, since the remaining states do not contribute by construction. We refer to [@Brown:2001zz; @Caurier:2004gf] for a more extended introduction to the nuclear shell model.
![Dark matter capture rate $C$ as a function of the dark matter particle mass $m_\chi$ for $c_1^0\neq0$ (top left panel), $c_4^0\neq0$ (top right panel), $c_1^1\neq0$ (bottom left panel), and $c_4^1\neq0$ (bottom right panel). We report the total capture rate (thick black line), and partial capture rates specific to the 16 most abundant elements in the Sun. Conventions for colors and lines are those in the legends. []{data-label="fig:c1c4"}](./c10-eps-converted-to.pdf){width="\textwidth"}
![Dark matter capture rate $C$ as a function of the dark matter particle mass $m_\chi$ for $c_1^0\neq0$ (top left panel), $c_4^0\neq0$ (top right panel), $c_1^1\neq0$ (bottom left panel), and $c_4^1\neq0$ (bottom right panel). We report the total capture rate (thick black line), and partial capture rates specific to the 16 most abundant elements in the Sun. Conventions for colors and lines are those in the legends. []{data-label="fig:c1c4"}](./c40-eps-converted-to.pdf){width="\textwidth"}
![Dark matter capture rate $C$ as a function of the dark matter particle mass $m_\chi$ for $c_1^0\neq0$ (top left panel), $c_4^0\neq0$ (top right panel), $c_1^1\neq0$ (bottom left panel), and $c_4^1\neq0$ (bottom right panel). We report the total capture rate (thick black line), and partial capture rates specific to the 16 most abundant elements in the Sun. Conventions for colors and lines are those in the legends. []{data-label="fig:c1c4"}](./c11-eps-converted-to.pdf){width="\textwidth"}
![Dark matter capture rate $C$ as a function of the dark matter particle mass $m_\chi$ for $c_1^0\neq0$ (top left panel), $c_4^0\neq0$ (top right panel), $c_1^1\neq0$ (bottom left panel), and $c_4^1\neq0$ (bottom right panel). We report the total capture rate (thick black line), and partial capture rates specific to the 16 most abundant elements in the Sun. Conventions for colors and lines are those in the legends. []{data-label="fig:c1c4"}](./c41-eps-converted-to.pdf){width="\textwidth"}
![[*Left panel.*]{} Ratio of the capture rate of this work for $c_1^0\neq0$ to the capture rate computed with [darksusy]{} for spin-independent dark matter interactions. We report the ratio of total rates (thick black line), and the ratio of partial rates specific to the 16 most abundant elements in the Sun. The two total rates differ by at most 8%. [*Right panel.*]{} Same as in the left panel, but for $c_4^0\neq0$. In this case the comparison can be performed for the total rates and for H only, since elements heavier than H are not included in [darksusy]{} for dark matter spin-dependent interactions.[]{data-label="fig:comp"}](./c10ds-eps-converted-to.pdf){width="\textwidth"}
![[*Left panel.*]{} Ratio of the capture rate of this work for $c_1^0\neq0$ to the capture rate computed with [darksusy]{} for spin-independent dark matter interactions. We report the ratio of total rates (thick black line), and the ratio of partial rates specific to the 16 most abundant elements in the Sun. The two total rates differ by at most 8%. [*Right panel.*]{} Same as in the left panel, but for $c_4^0\neq0$. In this case the comparison can be performed for the total rates and for H only, since elements heavier than H are not included in [darksusy]{} for dark matter spin-dependent interactions.[]{data-label="fig:comp"}](./c40ds-eps-converted-to.pdf){width="\textwidth"}
The OBDME for orbits corresponding to the nuclear core can be analytically calculated. Only multipoles of nuclear response operators with $L=\tau=0$ contribute, since in a nuclear core all orbits are fully occupied. One finds [@Walecka1] $$\psi^{L;\tau}_{|\alpha| |\beta|}=\sqrt{2(2J+1)(2T+1)(2j_\alpha+1)}\, \delta_{|\alpha| |\beta|}\delta_{\tau0}\,\delta_{L0}\,.
\label{eq:core}$$ The calculation of the OBDME for the remaining orbits in the model space instead requires a numerical approach. We address this problem using the [Nushell@MSU]{} program [@NuShell; @Brown:2001zz]. This code mainly relies on three inputs: the target nucleus spin, isospin and parity; the Hamiltonian for valence nucleon interactions (several options are provided with the code); the model space, including restrictions on the number of nucleons in the most energetic orbits. The assumptions made in our calculations are listed in Tab. \[tab:inputs\], and closely follow the guidelines provided in Ref. [@Brown:2001zz], and references therein. Assigned these inputs, the code first calculates the many-body ground-state wave function of the valence nucleon system diagonalizing the assumed interaction Hamiltonian. Then it evaluates the overlap of this wave function with the single-particle states $|\alpha \rangle$ according to Eq. (\[eq:OBDME\]). The OBDME that we obtain for $^{23}$Na, $^{28}$Si and $^{19}$F using the [Nushell@MSU]{} w-interaction negligibly differ from those in the code [@Anand:2013yka] (here we use $^{19}$F for comparison only, but it does not enter in our calculation). The remaining interactions in Tab. \[tab:inputs\] were studied in [@Warburton:1992rh; @Nummela:2001xh; @Honma:2004xk]. For instance, in the full $pf$ model space the gx1 interaction was found to successfully describe binding energies, electro-magnetic transitions, and excitation spectra of Iron, and of various Nickel isotopes [@Honma:2004xk]. The major limitation of our numerical OBDME calculation hence resides in the use of model space restrictions. We adopt such restrictions because of limits in the available computing power: [Nushell@MSU]{} only runs on Windows machines, whereas our cluster for extensive calculations has a Unix architecture. For the gx1 interaction, the impact of restrictions on observable quantitates has been discussed in [@Honma:2004xk]. For the isotopes $^{56}$Fe and $^{58}$Ni, for example, a restriction of the model space where 5 or more nucleons are allowed to be excited from the $f_{7/2}$ orbit to higher orbits implies an underestimation of the binding energy of the order of a few percent.
Ultimately, the nuclear structure calculations performed here have to be considered explorative, due to the restrictions in Tab. \[tab:inputs\], and to the fact that more sophisticated interaction Hamiltonians could in principle be considered. At the same time, we are not aware of nuclear structure calculations of comparable complexity in the context of dark matter capture by the Sun.
Before concluding this section, we comment on the OBDME calculation for Hydrogen. In the present analysis, Hydrogen constitutes a special case, in that it consists of a single valence nucleon system with no-core. Its OBDME can be trivially calculated as follows [@Walecka1] $$\psi^{L;\tau}_{|\alpha| |\beta|}= \delta_{|\alpha| |\gamma|} \delta_{|\beta| |\gamma|}\,,$$ where $|\gamma|$ corresponds to the $0s_{1/2}$ orbit.
Using the OBDME resulting from the methods outlined above, we evaluate the reduced nuclear matrix elements in Eq. (\[eq:W\]), and hence the dark matter-nucleus scattering cross-section (\[eq:sigma\]) for the most abundant elements in the Sun. This cross-section will allow us to calculate the rate of dark matter capture by the Sun (\[eq:rate\]) for all interaction operators in Tab. \[tab:operators\], as we will see next. The nuclear response functions that we obtain in this analysis, i.e. Eq. (\[eq:W\]), are listed in Appendix \[sec:appNuc\], and can be used by the reader for other projects.
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_3$ and $\hat{\mathcal{O}}_5$.[]{data-label="fig:c3c5"}](./c30-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_3$ and $\hat{\mathcal{O}}_5$.[]{data-label="fig:c3c5"}](./c50-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_3$ and $\hat{\mathcal{O}}_5$.[]{data-label="fig:c3c5"}](./c31-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_3$ and $\hat{\mathcal{O}}_5$.[]{data-label="fig:c3c5"}](./c51-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_6$ and $\hat{\mathcal{O}}_7$.[]{data-label="fig:c6c7"}](./c60-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_6$ and $\hat{\mathcal{O}}_7$.[]{data-label="fig:c6c7"}](./c70-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_6$ and $\hat{\mathcal{O}}_7$.[]{data-label="fig:c6c7"}](./c61-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_6$ and $\hat{\mathcal{O}}_7$.[]{data-label="fig:c6c7"}](./c71-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_8$ and $\hat{\mathcal{O}}_9$.[]{data-label="fig:c8c9"}](./c80-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_8$ and $\hat{\mathcal{O}}_9$.[]{data-label="fig:c8c9"}](./c90-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_8$ and $\hat{\mathcal{O}}_9$.[]{data-label="fig:c8c9"}](./c81-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_8$ and $\hat{\mathcal{O}}_9$.[]{data-label="fig:c8c9"}](./c91-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{10}$ and $\hat{\mathcal{O}}_{11}$.[]{data-label="fig:c10c11"}](./c100-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{10}$ and $\hat{\mathcal{O}}_{11}$.[]{data-label="fig:c10c11"}](./c110-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{10}$ and $\hat{\mathcal{O}}_{11}$.[]{data-label="fig:c10c11"}](./c101-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{10}$ and $\hat{\mathcal{O}}_{11}$.[]{data-label="fig:c10c11"}](./c111-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{12}$ and $\hat{\mathcal{O}}_{13}$.[]{data-label="fig:c12c13"}](./c120-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{12}$ and $\hat{\mathcal{O}}_{13}$.[]{data-label="fig:c12c13"}](./c130-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{12}$ and $\hat{\mathcal{O}}_{13}$.[]{data-label="fig:c12c13"}](./c121-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{12}$ and $\hat{\mathcal{O}}_{13}$.[]{data-label="fig:c12c13"}](./c131-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{14}$ and $\hat{\mathcal{O}}_{15}$.[]{data-label="fig:c14c15"}](./c140-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{14}$ and $\hat{\mathcal{O}}_{15}$.[]{data-label="fig:c14c15"}](./c150-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{14}$ and $\hat{\mathcal{O}}_{15}$.[]{data-label="fig:c14c15"}](./c141-eps-converted-to.pdf){width="\textwidth"}
![Same as in Fig. \[fig:c1c4\], but for the interaction operators $\hat{\mathcal{O}}_{14}$ and $\hat{\mathcal{O}}_{15}$.[]{data-label="fig:c14c15"}](./c151-eps-converted-to.pdf){width="\textwidth"}
Numerical evaluation of the capture rate {#sec:rate}
========================================
In this section we numerically evaluate the dark matter capture rate by the Sun, Eq. (\[eq:rate\]), using the nuclear response functions derived in the previous section, and collected in analytic form in Appendix \[sec:appNuc\]. We study one operator at the time, and for each interaction operator in Tab. \[tab:operators\], we separately consider the corresponding isoscalar and isovector coupling constants. In the figures, we report the dark matter capture rate as a function of the dark matter particle mass, varying $m_\chi$ in the range 10 - 1000 GeV. When a coupling constant is different from zero, it takes the reference value of $10^{-3}\,m_v^{-2}$, with $m_v=246.2$ GeV. Using the same interaction strength in all panels allows for a straightforward comparison between capture rates associated with different operators. For definiteness, we assume a spin $j_\chi=1/2$ for the dark matter particle.
Constant spin-independent and spin-dependent interactions
---------------------------------------------------------
We start with the capture rate for the interaction operators $\hat{\mathcal{O}}_1$ and $\hat{\mathcal{O}}_4$, corresponding to constant, i.e. velocity and momentum independent, dark matter-nucleon interactions. Fig. \[fig:c1c4\] shows the capture rate $C$ as a function of $m_\chi$ for the two operators. The top panels refer to the couplings constants $c_1^0$ and $c_4^0$, whereas the bottom panels correspond to $c_1^1$ and $c_4^1$. In the plots we report the total capture rate (thick black line), and partial capture rates specific to the 16 most abundant elements in the Sun. Conventions for colors and lines are those in the legends.
In the case $c_1^0\neq0$ many elements contribute to $C$ in a comparable manner. The leading contributions come from $^{4}$He, $^{16}$O, and $^{56}$Fe, with an additional sizable contribution due to $^{20}$Ne for $m_\chi\gtrsim 400$ GeV. For $c_4^0\neq0$ the most effective isotopes in capturing dark matter are H and $^{14}$N, though the latter significantly contributes for $m_\chi\gtrsim100$ GeV only. Similarly, in the case $c_1^1\neq0$ the most important element is H, though also $^{56}$Fe gives a sizable contribution to $C$ for large values of $m_\chi$. Finally, for $c_4^1\neq0$ only H is relevant in the dark matter capture by the Sun.
Fig. \[fig:comp\] compares the isoscalar rates of Fig. \[fig:c1c4\] with the spin-independent and spin-dependent capture rates computed by [darksusy]{}. For constant spin-independent interactions, corresponding to the $\hat{\mathcal{O}}_1$ operator, [darksusy]{} uses a simplified version of Eq. (\[eq:omega\]), namely $$\Omega_{v}^{-}(w)= \sum_i n_i w \,\Theta\left( \frac{\mu_i}{\mu^2_{+,i}} - \frac{u^2}{w^2} \right)\frac{1}{E_k}\int_{E_k u^2/w^2}^{E_k \mu_i/\mu_{+,i}^2} {\rm d}E\,\sigma_i\frac{\mu_{+,i}^2}{\mu_i} \exp(-2y)\,,
\label{eq:omega2}$$ where $\sigma_i$ is the total dark matter-nucleus scattering cross-section in the limit of zero momentum transfer, $y=(bq/2)^2$, and $$b=\sqrt{\frac{2}{3}} \left[ 0.91 \left(\frac{m_i}{\rm GeV}\right)^{1/3} + 0.3 \right]~{\rm fm} \,,$$ which allows to analytically compute $\Omega_{v}^{-}(w)$. In the case of constant spin-dependent interactions, corresponding to the $\hat{\mathcal{O}}_4$ operator, [darksusy]{} calculates the capture rate for H only, and neglects other elements. Other interaction operators are not included in the program, and cannot be used for comparison.
The left panel of Fig. \[fig:comp\] shows the ratio of the capture rate of this work for $c_1^0\neq0$ to the capture rate computed with [darksusy]{} for spin-independent interactions. We report the ratio of total capture rates, and the ratio of partial rates specific to different elements in the Sun. Whereas for elements like $^{56}$Fe and $^{58}$Ni the two rates differ up to 25% for $m_\chi\simeq 1$ TeV, the total rate computed with our nuclear response functions and the one obtained from Eq. (\[eq:omega2\]) differ by at most 8%. We conclude that for constant spin-independent dark matter-nucleon interactions, the capture rate is only moderately affected by the use of refined nuclear response functions.
The capture rate for constant spin-dependent dark matter interactions computed with [darksusy]{} is systematically smaller than the capture rate of this work for $c_4^0\neq0$. This effect is however important for dark matter masses larger than $100$ GeV only. Neglecting elements heavier than H, and in particular $^{14}$N, induces an error on the total capture rate of about 25% for $m_\chi\simeq 1$ TeV, as shown in the right panel of Fig. \[fig:comp\].
In summary, the capture rate for the operators $\hat{\mathcal{O}}_1$ and $\hat{\mathcal{O}}_4$ found with the nuclear response functions of this work does not dramatically differ from that of previous studies. However, in the future errors at the 10-20% level on the capture rate induced by simplistic form factors might non negligibly alter the interpretation of a hypothetical signal in terms of dark matter particle mass and interaction properties.
Velocity and momentum dependent interactions
--------------------------------------------
We now move on to our results for the capture rate of the operators $\hat{\mathcal{O}}_i$, $i=3,5\dots,15$. We report these results in Figs. \[fig:c3c5\], \[fig:c6c7\], \[fig:c8c9\], \[fig:c10c11\], \[fig:c12c13\], and \[fig:c14c15\], which show total and partial capture rates as a function of the dark matter particle mass. In each panel the thick black line represents the total capture rate, whereas partial rates correspond to colored lines, as explained in the legends. Inspection of these figures shows that the most important element in the determination of $C$ significantly depends on the dark matter-nucleon interaction operator, on whether the coupling is of isoscalar or isovector type, and on the value of $m_\chi$. Elements that contribute the most to the capture rate for at least one interaction operator, and in a specific dark matter particle mass range are H, $^{4}$He, $^{14}$N, $^{16}$O, $^{27}$Al and $^{56}$Fe. The existence of a variegated sample of elements important in the dark matter capture process shows the significance of detailed nuclear structure calculations. This conclusion is in particular true for interaction operators that favor dark matter couplings to nuclei heavier, and more complex than H or $^{4}$He.
The properties of the 6 nuclear response operators in Eqs. (\[eq:S1S2\]) and (\[eq:MDP\]), and the solar nuclear abundances in Tab. \[tab:massfrac\] determine the most important isotopes for a given operator. In the small momentum transfer limit the response operator $M_{LM;\tau}$ measures the mass number $A$ of the nucleus, and is therefore larger for heavy elements, like for instance $^{56}$Fe. Operators coupling via $M_{LM;\tau}$ are $\hat{\mathcal{O}}_1$, $\hat{\mathcal{O}}_{5}$, $\hat{\mathcal{O}}_{8}$, and $\hat{\mathcal{O}}_{11}$, though with different velocity and momentum suppressions. For these operators a compromise between nuclear abundance and mass number determines the most relevant elements in the capture process. The response operators $\Sigma'_{LM;\tau}$ and $\Sigma''_{LM;\tau}$ measure the nucleon spin content of the nucleus, and favor nuclei with unpaired protons or neutrons, like H and $^{14}$N. These isotopes are important for operators like $\hat{\mathcal{O}}_4$, $\hat{\mathcal{O}}_{6}$, $\hat{\mathcal{O}}_7$, $\hat{\mathcal{O}}_{9}$, $\hat{\mathcal{O}}_{10}$, $\hat{\mathcal{O}}_{13}$, and $\hat{\mathcal{O}}_{14}$, that couple via $\Sigma'_{LM;\tau}$ and $\Sigma''_{LM;\tau}$. Similar interpretations exist for the remaining nuclear response operators. For instance, $\Delta_{LM;\tau}$ measures the nucleon angular momentum content of the nucleus, and $\Phi^{\prime \prime}_{LM;\tau}$ the content of nucleon spin-orbit coupling in the nucleus [@Fitzpatrick:2012ib]. It can be shown that $\Delta_{LM;\tau}$ favors nuclei with an unpaired nucleon in a non s-shell orbit, whereas $\Phi^{\prime \prime}_{LM;\tau}$ favors heavy elements with orbits of large angular momentum not fully occupied [@Fitzpatrick:2012ib]. For these reasons the element $^{56}$Fe is particularly important for interaction operators that generate the nuclear response operator $\Phi^{\prime \prime}_{LM;\tau}$, like $\hat{\mathcal{O}}_3$, $\hat{\mathcal{O}}_{12}$, and $\hat{\mathcal{O}}_{15}$.
For elements up to $^{16}$O, we assume the solar abundances reported in Ref. [@Bahcall:2004pz]. For heavier elements we consider the abundances of Ref. [@Grevesse:1998bj]. These are the abundances implemented in the [darksusy]{} program, which we use to calculate the average mass fractions in Tab. \[tab:massfrac\]. Capture rates linearly depend on the radial number densities $n_i$ (see Eq. (\[eq:omega\])), which are in turn proportional to the corresponding mass fractions. Assuming a different solar model, i.e. different mass fractions, would impact our results accordingly. Conservative relative uncertainties on the solar abundances are listed in Tab. 4 of [@Serenelli:2012zw], and range from 11.8% for $^{56}$Fe to 45.3% for $^{20}$Ne. Smaller uncertainties are quoted in [@Asplund:2009fu]. Elements not included in Tab. \[tab:massfrac\] (heavier or lighter than $^{58}$Ni) have abundances at least a factor of a few smaller than $^{58}$Ni, and are neglected in all present calculations. Whether the corresponding nuclear response functions can compensate for the small abundances of these isotopes, is an interesting question that we leave for future work.
Element Average mass fraction Element Average mass fraction
------------- ----------------------- ------------- -----------------------
H 0.684 ${}^{24}$Mg 7.30$\times10^{-4}$
${}^{4}$He 0.298 ${}^{27}$Al 6.38$\times10^{-5}$
${}^3$He 3.75$\times10^{-4}$ ${}^{28}$Si 7.95$\times10^{-4}$
${}^{12}$C 2.53$\times10^{-3}$ ${}^{32}$S 5.48$\times10^{-4}$
${}^{14}$N 1.56$\times10^{-3}$ ${}^{40}$Ar 8.04$\times10^{-5}$
${}^{16}$O 8.50$\times10^{-3}$ ${}^{40}$Ca 7.33$\times10^{-5}$
${}^{20}$Ne 1.92$\times10^{-3}$ ${}^{56}$Fe 1.42$\times10^{-3}$
${}^{23}$Na 3.94$\times10^{-5}$ ${}^{58}$Ni 8.40$\times10^{-5}$
: List of average mass fractions for the 16 most abundant elements in the Sun as implemented in the [darksusy]{} program [@Gondolo:2004sc]. The underlying solar model is introduced in [@Bahcall:2004pz].[]{data-label="tab:massfrac"}
Also the behavior of the capture rate as a function of the dark matter particle mass strongly depends on the nature of the dark matter-nucleon interaction. In the log-log planes of Figs. \[fig:c3c5\], \[fig:c6c7\], \[fig:c8c9\], \[fig:c10c11\], \[fig:c12c13\], and \[fig:c14c15\], we observe steeply decreasing lines, e.g. for $c_1^1\neq 0$, roughly flat lines, e.g. for $c_{11}^0\neq 0$, bumps, e.g. for $c_3^0\neq 0$, and even more complex behaviors, like in the case of $c_6^1\neq 0$. Different factors intervene in determining the exact dependence of the capture rate on $m_\chi$, including what element dominates the capture process, its nuclear structure and the resulting nuclear response functions, and the intrinsic momentum/relative velocity dependence of the operator in analysis.
Another important result of this work is to observe that the operators $\hat{\mathcal{O}}_1$ and $\hat{\mathcal{O}}_4$ do not necessarily dominate the dark matter capture process. We find that the operator $\hat{\mathcal{O}}_{11} = i{\bf{\hat{S}}}_\chi\cdot{\hat{\bf{q}}}/m_N$ generates a total dark matter capture rate larger than that associated with the operator $\hat{\mathcal{O}}_4$ for values of the dark matter particle mass larger than approximately 30 GeV. This result is clearly illustrated in Fig. \[fig:c1c4c11\], where we compare the total dark matter capture rate as a function of $m_\chi$ for the operators $\hat{\mathcal{O}}_1$, $\hat{\mathcal{O}}_4$ and $\hat{\mathcal{O}}_{11}$ assuming isoscalar interactions. As in the previous figures, we consider the same value of the coupling constant, i.e. $10^{-3}\,m_v^{-2}$, for the three operators. The relative strength of the three interactions in Fig. \[fig:c1c4c11\] is hence determined by the matrix elements of the nuclear response operators $M_{LM;\tau}(q)$, $\Sigma'_{LM;\tau}(q)$ and $\Sigma''_{LM;\tau}(q)$ when evaluated for the most abundant elements in the Sun, and by the intrinsic momentum/relative velocity dependence of the three operators. Notice that the response operator $M_{LM;\tau}(q)$ affects the cross-sections generated by $\hat{\mathcal{O}}_1$ and $\hat{\mathcal{O}}_{11}$, whereas a linear combination of $\Sigma'_{LM;\tau}(q)$ and $\Sigma''_{LM;\tau}(q)$ determines the cross-section associated with $\hat{\mathcal{O}}_{4}$.
![Total capture rate for the interaction operators $\hat{\mathcal{O}}_1$, $\hat{\mathcal{O}}_4$, and $\hat{\mathcal{O}}_{11}$. In the three cases we assume the same value for the isoscalar coupling constant, i.e. $10^{-3}\,m_v^{-2}$, with $m_v=246.2$ GeV (we set to zero the isovector coupling constant). The operator $\hat{\mathcal{O}}_{11}$, though never included in experimental analyses, generates a capture rate larger than that associated with the operator $\hat{\mathcal{O}}_4$ for $m_\chi\gtrsim30$ GeV.[]{data-label="fig:c1c4c11"}](./call-eps-converted-to.pdf){width="70.00000%"}
Conclusions {#sec:conc}
===========
We have calculated the 8 nuclear response functions generated in the dark matter scattering by nuclei, i.e. Eq. (\[eq:W\]), for the 16 most abundant elements in the Sun. We have carried out this calculation within the general effective theory of isoscalar and isovector dark matter-nucleon interactions mediated by a heavy spin-0 or spin-1 particle. This theory predicts 14 isoscalar and 14 isovector dark matter-nucleon interaction operators with a non trivial dependence on velocity and momentum transfer. In contrast, current experimental searches for dark matter focus on 2 [*constant*]{} spin-independent and spin-dependent interaction operators only.
We have used the nuclear response functions found in this work to calculate the rate of dark matter capture by the Sun for the 14 isoscalar and the 14 isovector dark matter-nucleon interactions separately. We find that different elements contribute to the dark matter capture rate in a significant manner. H, $^{4}$He, $^{14}$N, $^{16}$O, $^{27}$Al and $^{56}$Fe generate the leading contribution for at least one interaction operator, and in a specific dark matter particle mass range. Nuclear structure calculations, like those performed in this work, are hence crucial to accurately compute the rate of dark matter capture by the Sun, in particular for interaction operators that favor dark matter couplings to nuclei heavier, and more complex than H or $^{4}$He.
Another important result found in this work concerns the operator $\hat{\mathcal{O}}_{11} = i{\bf{\hat{S}}}_\chi\cdot{\hat{\bf{q}}}/m_N$, which couples to the nuclear vector charge operator. For $m_\chi\gtrsim30$ GeV, this operator generates a capture rate larger than the rate induced by the operator $\hat{\mathcal{O}}_{4} = {\bf{\hat{S}}}_\chi\cdot{\bf{\hat{S}}}_N $, i.e. the constant spin-dependent operator commonly considered in dark matter searches at neutrino telescopes. This result was not known previously, and should be kept in mind in the analysis of dark matter induced neutrino signals from the Sun. It is however not unexpected, in that $\hat{\mathcal{O}}_{11}$ is independent of the nucleon spin, i.e. $C\propto A^2$, and of the transverse relative velocity operator.
Our findings significantly extends previous investigations, where the dark matter capture rate was calculated for constant dark matter-nucleon interactions only (see however [@Liang:2013dsa] for an interesting exception), and using simplistic nuclear form factors. The nuclear response functions obtained in this work are listed in analytic form in Appendix \[sec:appNuc\], and can be used in model independent analyses of dark matter induced neutrino signals from the Sun.
This work has partially been funded through a start-up grant of the University of Göttingen. R.C. acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442).
Dark matter response functions {#sec:appDM}
==============================
Below, we list the dark matter response functions that appear in Eq. (\[eq:sigma\]). The notation is the same used in the body of the paper. $$\begin{aligned}
R_{M}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right) &=& c_1^\tau c_1^{\tau^\prime } + {j_\chi (j_\chi+1) \over 3} \left[ {q^2 \over m_N^2} v_T^{\perp 2} c_5^\tau c_5^{\tau^\prime }+v_T^{\perp 2}c_8^\tau c_8^{\tau^\prime }
+ {q^2 \over m_N^2} c_{11}^\tau c_{11}^{\tau^\prime } \right] \nonumber \\
R_{\Phi^{\prime \prime}}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right) &=& {q^2 \over 4 m_N^2} c_3^\tau c_3^{\tau^\prime } + {j_\chi (j_\chi+1) \over 12} \left( c_{12}^\tau-{q^2 \over m_N^2} c_{15}^\tau\right) \left( c_{12}^{\tau^\prime }-{q^2 \over m_N^2}c_{15}^{\tau^\prime} \right) \nonumber \\
R_{\Phi^{\prime \prime} M}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right) &=& c_3^\tau c_1^{\tau^\prime } + {j_\chi (j_\chi+1) \over 3} \left( c_{12}^\tau -{q^2 \over m_N^2} c_{15}^\tau \right) c_{11}^{\tau^\prime } \nonumber \\
R_{\tilde{\Phi}^\prime}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right) &=&{j_\chi (j_\chi+1) \over 12} \left[ c_{12}^\tau c_{12}^{\tau^\prime }+{q^2 \over m_N^2} c_{13}^\tau c_{13}^{\tau^\prime} \right] \nonumber \\
R_{\Sigma^{\prime \prime}}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right) &=&{q^2 \over 4 m_N^2} c_{10}^\tau c_{10}^{\tau^\prime } +
{j_\chi (j_\chi+1) \over 12} \left[ c_4^\tau c_4^{\tau^\prime} + \right. \nonumber \\
&& \left. {q^2 \over m_N^2} ( c_4^\tau c_6^{\tau^\prime }+c_6^\tau c_4^{\tau^\prime })+
{q^4 \over m_N^4} c_{6}^\tau c_{6}^{\tau^\prime } +v_T^{\perp 2} c_{12}^\tau c_{12}^{\tau^\prime }+{q^2 \over m_N^2} v_T^{\perp 2} c_{13}^\tau c_{13}^{\tau^\prime } \right] \nonumber \\
R_{\Sigma^\prime}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right) &=&{1 \over 8} \left[ {q^2 \over m_N^2} v_T^{\perp 2} c_{3}^\tau c_{3}^{\tau^\prime } + v_T^{\perp 2} c_{7}^\tau c_{7}^{\tau^\prime } \right]
+ {j_\chi (j_\chi+1) \over 12} \left[ c_4^\tau c_4^{\tau^\prime} + \right.\nonumber \\
&&\left. {q^2 \over m_N^2} c_9^\tau c_9^{\tau^\prime }+{v_T^{\perp 2} \over 2} \left(c_{12}^\tau-{q^2 \over m_N^2}c_{15}^\tau \right) \left( c_{12}^{\tau^\prime }-{q^2 \over m_N^2}c_{15}^{\tau \prime} \right) +{q^2 \over 2 m_N^2} v_T^{\perp 2} c_{14}^\tau c_{14}^{\tau^\prime } \right] \nonumber \\
R_{\Delta}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right)&=& {j_\chi (j_\chi+1) \over 3} \left[ {q^2 \over m_N^2} c_{5}^\tau c_{5}^{\tau^\prime }+ c_{8}^\tau c_{8}^{\tau^\prime } \right] \nonumber \\
R_{\Delta \Sigma^\prime}^{\tau \tau^\prime}\left(v_T^{\perp 2}, {q^2 \over m_N^2}\right)&=& {j_\chi (j_\chi+1) \over 3} \left[c_{5}^\tau c_{4}^{\tau^\prime }-c_8^\tau c_9^{\tau^\prime} \right].\end{aligned}$$
Single-particle matrix elements of nuclear response operators {#sec:appME}
=============================================================
Here we list the single-particle matrix elements of the nuclear response operators in Eqs. (\[eq:S1S2\]) and (\[eq:MDP\]). Eqs. (\[eq:m1\]), (\[eq:m2\]), (\[eq:m3\]), and (\[eq:m4\]) are implemented in the [Mathematica]{} package of Ref. [@Anand:2013yka].
Only 4 independent nuclear response operators are actually generated in the dark matter-nucleus scattering, and need to be considered in order to evaluate the dark matter-nucleus scattering cross-section. These are $M_{JM}(q{\bf{r}}_i)$, ${\bf{M}}_{JL}^M(q{\bf{r}}_i)\cdot{\vec{\sigma}(i)}$, ${\bf{M}}^M_{JL}(q{\bf{r}}_i)\cdot\frac{1}{q} \overrightarrow{\nabla}$, and ${\bf{M}}^M_{JJ+1}(q{\bf{r}}_i)\cdot\left({\vec{\sigma}(i)}\times\frac{1}{q} \overrightarrow{\nabla}\right)$. This result follows from the identities $$\begin{aligned}
\Sigma'_{JM;\tau}(q) &=& \sum_{i=1}^{A} \left[-\sqrt{\frac{J}{2J+1}}{\bf{M}}_{JJ+1}^{M}(q {\bf{r}}_i) + \sqrt{\frac{J+1}{2J+1}}{\bf{M}}_{JJ-1}^{M}(q {\bf{r}}_i) \right] \cdot \vec{\sigma}(i) t^{\tau}_{(i)}\nonumber\\
\Sigma''_{JM;\tau}(q) &=&\sum_{i=1}^{A} \left[\sqrt{\frac{J+1}{2J+1}} {\bf{M}}_{JJ+1}^{M}(q {\bf{r}}_i) + \sqrt{\frac{J}{2J+1}}{\bf{M}}_{JJ-1}^{M}(q {\bf{r}}_i) \right] \cdot \vec{\sigma}(i) t^{\tau}_{(i)} \,.
\label{eq:S1S2bis}\end{aligned}$$ The reduced single-particle matrix elements of the four independent nuclear response operators are given in the following, where to simplify the equations we use the notation $\langle \boldsymbol{\alpha}||\cdot|| \boldsymbol{\beta} \rangle \equiv \langle n_\alpha(l_\alpha1/2)j_\alpha||\cdot|| n_\beta(l_\beta1/2)j_\beta\rangle$, and $[\lambda]=\sqrt{2\lambda+1}$, for any index $\lambda$. They read as follows $$\begin{aligned}
\label{eq:m1}
\langle \boldsymbol{\alpha}||M_{J}(q{\bf{r}}_i)||\boldsymbol{\beta} \rangle &=&
\frac{1}{\sqrt{4\pi}}(-1)^{J+j_\beta+1/2}[l_\alpha][l_\beta][j_\alpha][j_\beta][J] \nonumber\\
&\times& \begin{Bmatrix}
l_\alpha&j_\alpha&\frac{1}{2}\\
j_\beta&l_\beta&J
\end{Bmatrix}
\begin{pmatrix}
l_\alpha&J&l_\beta\\
0&0&0
\end{pmatrix}
\Braket{n_\alpha l_\alpha j_\alpha |j_{J}(\rho)|n_\beta l_\beta j_\beta}\,\nonumber\\
$$ $$\begin{aligned}
\label{eq:m2}
\langle \boldsymbol{\alpha}||{\bf{M}}_{JL}(q{\bf{r}}_i)\cdot{\vec{\sigma}(i)}||\boldsymbol{\beta}\rangle &=&
\sqrt{\frac{3}{2\pi}}(-1)^{l_\alpha}[l_\alpha][l_\beta][j_\alpha][j_\beta][L][J] \nonumber\\
&\times&
\begin{Bmatrix}
l_\alpha&l_\beta&L\\
\frac{1}{2}&\frac{1}{2}&1\\
j_\alpha&j_\beta&J
\end{Bmatrix}
\begin{pmatrix}
l_\alpha&L&l_\beta\\
0&0&0
\end{pmatrix}\Braket{n_\alpha l_\alpha j_\alpha|j_{L}(\rho)|n_\beta l_\beta j_\beta} \nonumber \\
$$ $$\begin{aligned}
\label{eq:m3}
\langle \boldsymbol{\alpha}||{\bf{M}}_{JL}(q{\bf{r}}_i)\cdot\frac{1}{q} \overrightarrow{\nabla}||\boldsymbol{\beta}\rangle &=&
\frac{1}{\sqrt{4\pi}} (-1)^{L+j_\beta+1/2}[l_\alpha][j_\alpha][j_\beta][L][J]
\begin{Bmatrix}
l_\alpha&j_\alpha&\frac{1}{2}\\
j_\beta&l_\beta&J
\end{Bmatrix}\nonumber\\
&\times&\Bigg[-\sqrt{l_\beta+1}[l_\beta+1]
\begin{Bmatrix}
L&1&J\\
l_\beta&l_\alpha&l_\beta+1
\end{Bmatrix}
\begin{pmatrix}
l_\alpha&L&l_\beta+1\\
0&0&0
\end{pmatrix} \nonumber\\
&\times& \langle n_\alpha l_\alpha j_\alpha |j_{L}(\rho)\left(\frac{\text{d}}{\text{d}\rho}-\frac{l_\beta}{\rho}\right)| n_\beta l_\beta j_\beta\rangle + \nonumber\\
&+&\sqrt{l_\beta}[l_\beta-1]
\begin{Bmatrix}
L&1&J\\
l_\beta&l_\alpha&l_\beta-1
\end{Bmatrix}
\begin{pmatrix}
l_\alpha&L&l_\beta-1\\
0&0&0
\end{pmatrix}\nonumber\\
&\times&\langle n_\alpha l_\alpha j_\alpha |j_{L}(\rho)\left(\frac{\text{d}}{\text{d}\rho}+\frac{l_\beta+1}{\rho}\right)|n_\beta l_\beta j_\beta \rangle \Bigg] \nonumber\\\end{aligned}$$ $$\begin{aligned}
\label{eq:m4}
\langle \boldsymbol{\alpha}||{\bf{M}}_{JL}(q{\bf{r}}_i)&\cdot&\left({\vec{\sigma}(i)}\times\frac{1}{q} \overrightarrow{\nabla}\right)||\boldsymbol{\beta}\rangle = \nonumber\\
&=& (-1)^{l_\alpha}\frac{6i}{\sqrt{4\pi}}[l_\alpha][j_\alpha][j_\beta][J][L] \nonumber\\
&\times& \Bigg\{-[l_\beta+1]\sqrt{l_\beta+1}
\begin{pmatrix}
l_\alpha&L&l_\beta+1\\
0&0&0
\end{pmatrix}
\langle n_\alpha l_\alpha j_\alpha |j_{L}(\rho)\left(\frac{\text{d}}{\text{d}\rho}-\frac{l_\beta}{\rho}\right)|n_\beta l_\beta j_\beta \rangle \nonumber\\
&\times&\Bigg[ \sum_{L'} [L']^{2} (-1)^{J-L'}
\begin{Bmatrix}
L&1&L'\\
1&J&1
\end{Bmatrix}\begin{Bmatrix}
L&1&L'\\
l_\beta&l_\alpha&l_\beta+1
\end{Bmatrix}\begin{Bmatrix}
l_\alpha&l_\beta&L'\\
\frac{1}{2}&\frac{1}{2}&1\\
j_\alpha&j_\beta&J
\end{Bmatrix} \Bigg] \nonumber\\
&+& [l_\beta-1]\sqrt{l_\beta}
\begin{pmatrix}
l_\alpha&L&l_\beta-1\\
0&0&0
\end{pmatrix}
\langle n_\alpha l_\alpha j_\alpha|j_{L}(\rho)\left(\frac{\text{d}}{\text{d}\rho}+\frac{l_\beta+1}{\rho}\right)|n_\beta l_\beta j_\beta\rangle \nonumber \\
&\times& \Bigg[ \sum_{L'} [L']^{2} (-1)^{J-L'}
\begin{Bmatrix}
L&1&L'\\
1&J&1
\end{Bmatrix}\begin{Bmatrix}
L&1&L'\\
l_\beta&l_\alpha&l_\beta-1
\end{Bmatrix}\begin{Bmatrix}
l_\alpha&l_\beta&L'\\
\frac{1}{2}&\frac{1}{2}&1\\
j_\alpha&j_\beta&J
\end{Bmatrix}
\Bigg]\Bigg\} \,. \nonumber\\\end{aligned}$$ Eqs. (\[eq:m1\]), (\[eq:m2\]), and (\[eq:m3\]) also appear in the calculation of nuclear matrix elements for electroweak lepton-nucleus interactions. The latter expression is instead needed to evaluate the matrix elements of the nuclear response operators $\tilde{\Phi}'$ and $\Phi''$, specific to the dark matter-nucleus scattering. Different combinations of Wigner $3j$, $6j$ and $9j$ symbols appear in the equations above, which also depend on residual radial matrix elements of spherical Bessel functions and of their derivatives at $\rho=qr_i$. In the case of the harmonic oscillator single-particle basis, these radial matrix elements can be analytically evaluated as follows $$\begin{aligned}
&\langle n_\alpha l_\alpha j_\alpha | j_{L}(\rho)| n_\beta l_\beta j_\beta \rangle =\\ &\qquad=\frac{2^{L}}{(2L+1)!!}y^{L/2}e^{-y}\sqrt{(n_\beta-1)!(n_\alpha-1)!\Gamma(n_\alpha+l_\alpha+\frac{1}{2})\Gamma(n_\beta+l_\beta+\frac{1}{2})}\\
&\qquad\times\sum_{k=0}^{n_\beta-1}\sum_{k'=0}^{n_\alpha-1}\frac{(-1)^{k+k'}}{k!k'!}\frac{1}{(n_\beta-1-k)!(n_\alpha-1-k')!}\nonumber\\
&\qquad\times\frac{\Gamma[\frac{1}{2}(l_\beta+l_\alpha+L+2k+2k'+3)]}{\Gamma[l_\beta+k+\frac{3}{2}]\Gamma[l_\alpha+k'+\frac{3}{2}]}
\leftidx{_{1}}F_{1}[\frac{1}{2}(L-l_\beta-l_\alpha-2k-2k');L+\frac{3}{2};y],\end{aligned}$$ $$\begin{aligned}
&\langle n_\alpha l_\alpha j_\alpha |j_{L}(\rho)\left(\frac{\text{d}}{\text{d}r}-\frac{l_\beta}{r_i}\right)|n_\beta l_\beta j_\beta \rangle =\\ &\qquad=\frac{2^{L-1}}{(2L+1)!!}y^{(L-1)/2}e^{-y}\sqrt{(n_\beta-1)!(n_\alpha-1)!\Gamma(n_\alpha+l_\alpha+\frac{1}{2})\Gamma(n_\beta+l_\beta+\frac{1}{2})}\\
&\qquad\times\sum_{k=0}^{n_\beta-1}\sum_{k'=0}^{n_\alpha-1}\frac{(-1)^{k+k'}}{k!k'!}\frac{1}{(n_\beta-1-k)!(n_\alpha-1-k')!} \frac{\Gamma[\frac{1}{2}(l_\beta+l_\alpha+L+2k+2k'+2)]}{\Gamma[l_\beta+k+\frac{3}{2}]\Gamma[l_\alpha+k'+\frac{3}{2}]}\\
&\qquad\times\left\{-\frac{1}{2}(l_\beta+l_\alpha+L+2k+2k'+2)\leftidx{_{1}}F_{1}[\frac{1}{2}(L-l_\beta-l_\alpha-2k-2k'-1);L+\frac{3}{2};y]\right.\\
&\left.\qquad +(2k)\leftidx{_{1}}F_{1}[\frac{1}{2}(L-l_\beta-l_\alpha-2k-2k'+1);L+\frac{3}{2};y]\right\},\end{aligned}$$ $$\begin{aligned}
&\langle n_\alpha l_\alpha j_\alpha |j_{L}(\rho)\left(\frac{\text{d}}{\text{d}r_i}+\frac{l_\beta+1}{r_i}\right)| n_\beta l_\beta j_\beta \rangle =\\
&\qquad = \frac{2^{L-1}}{(2L+1)!!}y^{(L-1)/2}e^{-y}\sqrt{(n_\beta-1)!(n_\alpha-1)!\Gamma(n_\alpha+l_\alpha+\frac{1}{2})\Gamma(n_\beta+l_\beta+\frac{1}{2})}\\
&\qquad\times\sum_{k=0}^{n_\beta-1}\sum_{k'=0}^{n_\alpha-1}\frac{(-1)^{k+k'}}{k!k'!}\frac{1}{(n_\beta-1-k)!(n_\alpha-1-k')!}\frac{\Gamma[\frac{1}{2}(l_\beta+l_\alpha+L+2k+2k'+2)]}{\Gamma[l_\beta+k+\frac{3}{2}]\Gamma[l_\alpha+k'+\frac{3}{2}]}\\
&\qquad\times\left\{-\frac{1}{2}(l_\beta+l_\alpha+L+2k+2k'+2)\leftidx{_{1}}F_{1}[\frac{1}{2}(L-l_\beta-l_\alpha-2k-2k'-1);L+\frac{3}{2};y]\right.\\
&\left.\qquad +(2l_\beta+2k+1)\leftidx{_{1}}F_{1}[\frac{1}{2}(L-l_\beta-l_\alpha-2k-2k'+1);L+\frac{3}{2};y]\right\},\end{aligned}$$ where $y=(qb/2)^2$, and $\leftidx{_{1}}F_{1}$ is the confluent hypergeometric function.
Nuclear response functions {#sec:appNuc}
==========================
Below, we only list nuclear response functions different from zero.
Hydrogen (H) {#hydrogen-h .unnumbered}
------------
W\^[00]{}\_[M]{}(y)&= 0.0397887 & W\^[00]{}\_[\^]{}(y)&= 0.0397887 & W\^[00]{}\_[\^]{}(y)&= 0.0795775 &\
W\^[11]{}\_[M]{}(y)&= 0.0397887 & W\^[11]{}\_[\^]{}(y)&= 0.0397887 & W\^[11]{}\_[\^]{}(y)&= 0.0795775 &\
W\^[10]{}\_[M]{}(y)&= 0.0397887 & W\^[10]{}\_[\^]{}(y)&= 0.0397887 & W\^[10]{}\_[\^]{}(y)&= 0.0795775 &\
W\^[01]{}\_[M]{}(y)&= 0.0397887 & W\^[01]{}\_[\^]{}(y)&= 0.0397887 & W\^[01]{}\_[\^]{}(y)&= 0.0795775 &\
Helium ($^3$He) {#helium-3he .unnumbered}
---------------
W\^[00]{}\_[M]{}(y)&= 0.358099 e\^[-2y]{}& W\^[00]{}\_[\^]{}(y)&= 0.0397887 e\^[-2y]{} &W\^[00]{}\_[\^]{}(y)&= 0.0795775 e\^[-2y]{} &\
W\^[11]{}\_[M]{}(y)&= 0.0397887 e\^[-2y]{}& W\^[11]{}\_[\^]{}(y)&= 0.0397887 e\^[-2y]{} &W\^[11]{}\_[\^]{}(y)&= 0.0795775 e\^[-2y]{} &\
W\^[10]{}\_[M]{}(y)&= 0.119366 e\^[-2y]{} &W\^[10]{}\_[\^]{}(y)&= -0.0397887 e\^[-2y]{} &W\^[10]{}\_[\^]{}(y)&= -0.0795775 e\^[-2y]{} &\
W\^[01]{}\_[M]{}(y)&= 0.119366 e\^[-2y]{}& W\^[01]{}\_[\^]{}(y)&= -0.0397887e\^[-2y]{}& W\^[01]{}\_[\^]{}(y)&= -0.0795775 e\^[-2y]{} &\
Helium ($^4$He) {#helium-4he .unnumbered}
---------------
W\^[00]{}\_[M]{}(y)&= 0.31831 e\^[-2y]{}&\
Carbon ($^{12}$C) {#carbon-12c .unnumbered}
-----------------
W\^[00]{}\_[M]{}(y)&= 0.565882 e\^[-2y]{} (2.25 - y)\^2&\
W\^[00]{}\_[\^]{}(y)&= 0.0480805 e\^[-2y]{} &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.371134 + 0.164948 y) &\
Nitrogen ($^{14}$N) {#nitrogen-14n .unnumbered}
-------------------
W\^[00]{}\_[M]{}(y) &= e\^[-2y]{} (11.6979 - 11.1409 y + 2.67574 y\^2) &\
W\^[00]{}\_[\^]{}(y) &= 0.0230079 e\^[-2y]{} (1.20986 + y)\^2 &\
W\^[00]{}\_[\^]{}(y) &= 0.134532 e\^[-2y]{} (0.707578 - y)\^2\
W\^[00]{}\_[\^]{}(y)&= 0.0905048 e\^[-2y]{} &\
W\^[00]{}\_[\^]{}(y)&= 0.00126432 e\^[-2y]{} &\
W\^[00]{}\_(y)&= 0.0424075 e\^[-2y]{} &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-1.02414 + 0.483267 y)&\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0534451 - 0.0755325 y) &\
Oxygen ($^{16}$O) {#oxygen-16o .unnumbered}
-----------------
W\^[00]{}\_[M]{}(y)&= 0.000032628 e\^[-2y]{} (395.084 - 200.042 y + y\^2)\^2 &\
W\^[00]{}\_[\^]{}(y)&= 0.000032628 e\^[-2y]{} (3.66055 - y)\^2 &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.0471874 + 0.0367831 y - 0.00664641 y\^2 + 0.000032628 y\^3) &\
Neon ($^{20}$Ne) {#neon-20ne .unnumbered}
----------------
W\^[00]{}\_[M]{}(y)&= 0.0431723 e\^[-2y]{} (13.5766 - 9.05108 y + y\^2)\^2 &\
W\^[00]{}\_[\^]{}(y)&= 0.00348077 e\^[-2y]{} (2.50001 - y)\^2 &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.416077 + 0.443815 y - 0.1416 y\^2 + 0.0122586 y\^3)&\
Magnesium ($^{24}$Mg) {#magnesium-24mg .unnumbered}
---------------------
W\^[00]{}\_[M]{}(y)&= 0.123467 e\^[-2y]{} (9.63385 - 7.49299 y + y\^2)\^2 &\
W\^[00]{}\_[\^]{}(y)&= 0.0260816 e\^[-2y]{} (2.5 - y)\^2 &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-1.36673 + 1.6097 y - 0.567072 y\^2 + 0.056747 y\^3)&\
Sodium ($^{23}$Na) {#sodium-23na .unnumbered}
------------------
W\^[00]{}\_[M]{}(y)&= e\^[-2y]{} (42.0965 - 63.4498 y + 32.5913 y\^2 - 6.57878 y\^3 + 0.483166 y\^4) &\
W\^[11]{}\_[M]{}(y)&=e\^[-2y]{} (0.0795776 - 0.212207 y + 0.182941 y\^2 - 0.0543892 y\^3 + 0.00523012 y\^4) &\
W\^[10]{}\_[M]{}(y)&= e\^[-2y]{} (-1.83028 + 3.81972 y - 2.50445 y\^2 + 0.597822 y\^3 - 0.04545 y\^4) &\
W\^[01]{}\_[M]{}(y)&= e\^[-2y]{} (-1.83028 + 3.81972 y - 2.50445 y\^2 + 0.597822 y\^3 - 0.04545 y\^4) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0126672 - 0.0262533 y + 0.0401886 y\^2 - 0.010514 y\^3 + 0.00078605 y\^4)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.00917577 - 0.0167053 y + 0.0332751 y\^2 - 0.00765719 y\^3 + 0.000597676 y\^4)&\
W\^[10]{}\_[\^]{}(y)&=e\^[-2y]{} (0.0107811 - 0.020986 y + 0.0360971 y\^2 - 0.00876213 y\^3 + 0.000626718 y\^4) &\
W\^[01]{}\_[\^]{}(y)&=e\^[-2y]{} (0.0107811 - 0.020986 y + 0.0360971 y\^2 - 0.00876213 y\^3 + 0.000626718 y\^4) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0253345 - 0.0750847 y + 0.100235 y\^2 - 0.0384261 y\^3 + 0.00466396 y\^4)&\
W\^[11]{}\_[\^]{}(y)&=e\^[-2y]{} (0.0183515 - 0.0567009 y + 0.0887794 y\^2 - 0.0374699 y\^3 + 0.00477955 y\^4) &\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0215622 - 0.0652627 y + 0.0941439 y\^2 - 0.0379511 y\^3 + 0.00472138 y\^4)&\
W\^[01]{}\_[\^]{}(y)&=e\^[-2y]{} (0.0215622 - 0.0652627 y + 0.0941439 y\^2 - 0.0379511 y\^3 + 0.00472138 y\^4) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.612149 - 0.49308 y + 0.107832 y\^2)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.00940911 - 0.00747826 y + 0.00163204 y\^2)&\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.075893 + 0.060682 y - 0.0110124 y\^2)&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.075893 + 0.060682 y - 0.0110124 y\^2)&\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.000495589 - 0.00010394 y + 0.00000544981y\^2 )&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.00000616583 + 0.00008381 y + 0.0002848 y\^2 )&\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0000552785 - 0.000369894 y + 0.0000393968 y\^2 )&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0000552785 - 0.000369894 y + 0.0000393968 y\^2 )&\
W\^[00]{}\_(y)&=e\^[-2y]{} (0.0335711 - 0.0268568 y + 0.00656896 y\^2) &\
W\^[11]{}\_(y)&=e\^[-2y]{} (0.00772326 - 0.00617861 y + 0.0021619 y\^2) &\
W\^[10]{}\_(y)&=e\^[-2y]{} (0.0161021 - 0.0128817 y + 0.00362952 y\^2) &\
W\^[01]{}\_(y)&=e\^[-2y]{} (0.0161021 - 0.0128817 y + 0.00362952 y\^2) &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-5.07498 + 5.86765 y - 2.09908 y\^2 + 0.226345 y\^3)&\
W\^[11]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.0273574 + 0.0474719 y - 0.0213121 y\^2 + 0.00280825 y\^3)&\
W\^[10]{}\_[M\^]{}(y)&=e\^[-2y]{} (0.220651 - 0.382932 y + 0.17682 y\^2 - 0.0226015 y\^3) &\
W\^[01]{}\_[M\^]{}(y)&=e\^[-2y]{} (0.62922 - 0.727336 y + 0.243236 y\^2 - 0.0210943 y\^3) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0291634 + 0.0548817 y - 0.0305345 y\^2 + 0.00476387 y\^3)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0119052 + 0.0231539 y - 0.0164035 y\^2 + 0.00310235 y\^3) &\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.024821 + 0.0482732 y - 0.02884 y\^2 + 0.00481368 y\^3)&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.013988 + 0.0263236 y - 0.0171362 y\^2 + 0.00306717 y\^3)&\
Aluminium ($^{27}$Al) {#aluminium-27al .unnumbered}
---------------------
W\^[00]{}\_[M]{}(y)&= e\^[-2y]{} (87.0146 - 146.097 y + 83.5367 y\^2 - 18.5981 y\^3 + 1.43446 y\^4) &\
W\^[11]{}\_[M]{}(y)&= e\^[-2y]{} (0.119366 - 0.31831 y + 0.337291 y\^2 - 0.132526 y\^3 + 0.018155 y\^4) &\
W\^[10]{}\_[M]{}(y)&= e\^[-2y]{} (-3.22283 + 7.00266 y - 4.92756 y\^2 + 1.33587 y\^3 - 0.11524 y\^4) &\
W\^[01]{}\_[M]{}(y)&= e\^[-2y]{} (-3.22283 + 7.00266 y - 4.92756 y\^2 + 1.33587 y\^3 - 0.11524 y\^4) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0309465 - 0.0367242 y + 0.0265347 y\^2 - 0.00241606 y\^3 + 0.0110011 y\^4)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0218834 - 0.00944476 y + 0.011506 y\^2 + 0.000953537 y\^3 + 0.0104813 y\^4)&\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0260233 - 0.0210567 y + 0.0158643 y\^2 + 0.000606077 y\^3 + 0.0105713 y\^4)&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0260233 - 0.0210567 y + 0.0158643 y\^2 + 0.000606077 y\^3 + 0.0105713 y\^4)&\
W\^[00]{}\_[\^]{}(y)&=e\^[-2y]{} (0.0618929 - 0.210848 y + 0.244466 y\^2 - 0.0942682 y\^3 + 0.0243737 y\^4) &\
W\^[11]{}\_[\^]{}(y)&=e\^[-2y]{} (0.0437667 - 0.165622 y + 0.221193 y\^2 - 0.101991 y\^3 + 0.0277477 y\^4) &\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0520466 - 0.18713 y + 0.233007 y\^2 - 0.0985082 y\^3 + 0.0259327 y\^4)&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0520466 - 0.18713 y + 0.233007 y\^2 - 0.0985082 y\^3 + 0.0259327 y\^4)&\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (2.80498 - 2.24306 y + 0.455491 y\^2)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.021493 - 0.0156159 y + 0.00596886 y\^2)&\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.180417 + 0.137389 y - 0.0239615 y\^2)&\
W\^[01]{}\_[\^]{}(y)&=e\^[-2y]{} (-0.180417 + 0.137389 y - 0.0239615 y\^2) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0000680703 - 0.000376682 y + 0.00340251 y\^2 )&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.0149622 - 0.00563307 y + 0.00440385 y\^2) &\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0010092 + 0.00298228 y + 0.00281525 y\^2) &\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0010092 + 0.00298228 y + 0.00281525 y\^2) &\
W\^[00]{}\_(y)&= e\^[-2y]{} (0.126043 - 0.100835 y + 0.0237577 y\^2)&\
W\^[11]{}\_(y)&= e\^[-2y]{} (0.05736 - 0.045888 y + 0.012102 y\^2)&\
W\^[10]{}\_(y)&= e\^[-2y]{} (0.0850285 - 0.0680228 y + 0.016845 y\^2)&\
W\^[01]{}\_(y)&= e\^[-2y]{} (0.0850285 - 0.0680228 y + 0.016845 y\^2)&\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-15.6228 + 19.3589 y - 7.23234 y\^2 + 0.79705 y\^3) &\
W\^[11]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.0370794 + 0.0852545 y - 0.0449284 y\^2 + 0.00866992 y\^3)&\
W\^[10]{}\_[M\^]{}(y)&= e\^[-2y]{} (0.578632 - 1.00438 y + 0.491252 y\^2 - 0.0730693 y\^3)&\
W\^[01]{}\_[M\^]{}(y)&= e\^[-2y]{} (1.00112 - 1.15934 y + 0.40275 y\^2 - 0.0364952 y\^3)&\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0883243 + 0.185775 y - 0.104001 y\^2 + 0.0163635 y\^3)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0501045 + 0.114845 y - 0.0729898 y\^2 + 0.0131315 y\^3)&\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0742731 + 0.170242 y - 0.105744 y\^2 + 0.0188197 y\^3)&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0595834 + 0.125323 y - 0.0717204 y\^2 + 0.011398 y\^3)&\
Silicon ($^{28}$Si) {#silicon-28si .unnumbered}
-------------------
W\^[00]{}\_[M]{}(y)&= 0.281695 e\^[-2y]{} (7.44089 - 6.37784 y + y\^2)\^2 &\
W\^[00]{}\_[\^]{}(y)&=0.0739103 e\^[-2y]{} (2.5 - y)\^2 &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-2.68415 + 3.37434 y - 1.281 y\^2 + 0.144292 y\^3)&\
Sulfur ($^{32}$S) {#sulfur-32s .unnumbered}
-----------------
W\^[00]{}\_[M]{}(y)&= 0.580305 e\^[-2y]{} (5.92494 - 5.43118 y + y\^2)\^2&\
W\^[00]{}\_[\^]{}(y)&= 0.0765941 e\^[-2y]{} (2.5 - y)\^2&\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-3.12284 + 4.11173 y - 1.6721 y\^2 + 0.210827 y\^3)&\
Argon ($^{40}$Ar) {#argon-40ar .unnumbered}
-----------------
W\^[00]{}\_[M]{}(y)&= e\^[-2y]{} (31.8294- 65.9618 y + 48.5834 y\^2 - 15.194 y\^3 + 1.9036 y\^4 - 0.0595886 y\^5 &\
&+ 0.000544329 y\^6) &\
W\^[11]{}\_[M]{}(y)&= e\^[-2y]{} (0.318304 - 1.06524 y + 1.24846 y\^2 - 0.62249 y\^3 + 0.141618 y\^4 - 0.0138797 y\^5 &\
&+ 0.000480513 y\^6) &\
W\^[10]{}\_[M]{}(y)&= e\^[-2y]{} (-3.18299 + 8.62425 y - 8.02539 y\^2 + 3.19316 y\^3 - 0.554467 y\^4 + 0.0353797 y\^5 &\
&- 0.000511426 y\^6) &\
W\^[01]{}\_[M]{}(y)&= e\^[-2y]{} (-3.18299 + 8.62425 y - 8.02539 y\^2 + 3.19316 y\^3 - 0.554467 y\^4 + 0.0353797 y\^5 &\
&- 0.000511426 y\^6) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (0.299629 - 0.373798 y + 0.154895 y\^2 - 0.0238983 y\^3 + 0.00122474 y\^4)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.00414999 - 0.0181474 y + 0.0240755 y\^2 - 0.00926264 y\^3 + 0.00108115 y\^4)&\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0352627 + 0.0990955 y - 0.0683453 y\^2 + 0.0161561 y\^3 - 0.00115071 y\^4)&\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.0352627 + 0.0990955 y - 0.0683453 y\^2 + 0.0161561 y\^3 - 0.00115071 y\^4) &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-3.08821 + 5.12625 y - 2.89248 y\^2 + 0.653386 y\^3 - 0.0526576 y\^4 &\
&+ 0.000816493 y\^5)&\
W\^[11]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.036345 + 0.140282 y - 0.171917 y\^2 + 0.0770456 y\^3 - 0.0134973 y\^4 &\
&+ 0.000720769 y\^5)&\
W\^[10]{}\_[M\^]{}(y)&= e\^[-2y]{} (0.308826 - 0.709394 y + 0.515378 y\^2 - 0.153134 y\^3 + 0.0185641 y\^4 &\
& - 0.000767139 y\^5)&\
W\^[01]{}\_[M\^]{}(y)&= e\^[-2y]{} (0.363444 -1.17124 y + 1.09117 y\^2 - 0.373592 y\^3 + 0.0452762 y\^4 &\
& - 0.000767139 y\^5)&\
Calcium ($^{40}$Ca) {#calcium-40ca .unnumbered}
-------------------
W\^[00]{}\_[M]{}(y)&= 0.000016743 e\^[-2y]{} (1378.8 - 1387.54 y + 281.953 y\^2 - y\^3)\^2&\
W\^[00]{}\_[\^]{}(y)&= 0.0000376718 e\^[-2y]{} (13.117 - 8.74678 y + y\^2)\^2&\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.454214 + 0.759976 y - 0.432314 y\^2 + 0.0971138 y\^3 - 0.00730079 y\^4 &\
&+ 0.0000251146 y\^5)&\
Iron ($^{56}$Fe) {#iron-56fe .unnumbered}
----------------
W\^[00]{}\_[M]{}(y)&= e\^[-2y]{} (62.3888 - 160.428 y + 152.644 y\^2 - 67.2779 y\^3 + 14.478 y\^4 - 1.43665 y\^5 &\
&+ 0.0525291 y\^6) &\
W\^[11]{}\_[M]{}(y)&= e\^[-2y]{} (0.318309 - 1.27323 y + 1.99188 y\^2 - 1.54562 y\^3 + 0.622264 y\^4 - 0.122277 y\^5 &\
&+ 0.00921525 y\^6) &\
W\^[10]{}\_[M]{}(y)&= e\^[-2y]{} (-4.45633 + 14.6422 y - 18.2579 y\^2 + 10.8919 y\^3 - 3.2296 y\^4 + 0.446836 y\^5 &\
& - 0.0220016 y\^6) &\
W\^[01]{}\_[M]{}(y)&= e\^[-2y]{} (-4.45633 + 14.6422 y - 18.2579 y\^2 + 10.8919 y\^3 - 3.2296 y\^4 + 0.446836 y\^5 &\
&- 0.0220016 y\^6) &\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (4.22872 - 6.76595 y + 3.79067 y\^2 - 0.867433 y\^3 + 0.069506 y\^4)&\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.143378 - 0.229404 y + 0.144606 y\^2 - 0.0422756 y\^3 + 0.00486921 y\^4)&\
W\^[10]{}\_[\^]{}(y)&=e\^[-2y]{} (-0.778655 + 1.24585 y - 0.741661 y\^2 + 0.194658 y\^3 - 0.0183967 y\^4) &\
W\^[01]{}\_[\^]{}(y)&=e\^[-2y]{} (-0.778655 + 1.24585 y - 0.741661 y\^2 + 0.194658 y\^3 - 0.0183967 y\^4) &\
W\^[00]{}\_[M\^]{}(y)&= e\^[-2y]{} (-16.2427 + 33.8776 y - 25.2342 y\^2 + 8.30471 y\^3 - 1.20334 y\^4 + 0.0604243 y\^5)&\
W\^[11]{}\_[M\^]{}(y)&= e\^[-2y]{} (-0.213631 + 0.598168 y - 0.622338 y\^2 + 0.308014 y\^3 - 0.0735211 y\^4 &\
&+ 0.00669858 y\^5)&\
W\^[10]{}\_[M\^]{}(y)&=e\^[-2y]{} (1.16019 - 3.24853 y + 3.31473 y\^2 - 1.54264 y\^3 + 0.325833 y\^4- 0.0253084 y\^5) &\
W\^[01]{}\_[M\^]{}(y)&= e\^[-2y]{} (2.99085 - 6.23805 y + 4.81422 y\^2 - 1.74483 y\^3 + 0.288128 y\^4 - 0.015993 y\^5) &\
Nickel ($^{58}$Ni) {#nickel-58ni .unnumbered}
------------------
W\^[00]{}\_[M]{}(y)&= e\^[-2y]{} (66.9246 - 175.389 y + 169.877 y\^2 - 76.127 y\^3 + 16.6597 y\^4 - 1.6839 y\^5\
&+ 0.0628067 y\^6)\
W\^[11]{}\_[M]{}(y)&= e\^[-2y]{} (0.0795762 - 0.318305 y + 0.548985 y\^2 - 0.503018 y\^3 + 0.250492 y\^4\
& - 0.0603789 y\^5 + 0.00545169 y\^6)\
W\^[10]{}\_[M]{}(y)&= e\^[-2y]{} (-2.30773 + 7.63937 y - 10.3404 y\^2 + 6.95311 y\^3 - 2.30652 y\^4 + 0.350525 y\^5\
&- 0.0185041 y\^6)\
W\^[01]{}\_[M]{}(y)&= e\^[-2y]{} (-2.30773 + 7.63937 y - 10.3404 y\^2 + 6.95311 y\^3 - 2.30652 y\^4 + 0.350525 y\^5\
&- 0.0185041 y\^6)\
W\^[00]{}\_[\^]{}(y)&= e\^[-2y]{} (5.4697 - 8.75152 y + 4.88454 y\^2 - 1.10715 y\^3 + 0.0875404 y\^4)\
W\^[11]{}\_[\^]{}(y)&= e\^[-2y]{} (0.00977975 - 0.0156476 y + 0.0136707 y\^2 - 0.00592935 y\^3 + 0.00140426 y\^4)\
W\^[10]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.231284 + 0.370054 y - 0.264922 y\^2 + 0.0935201 y\^3 - 0.0110873 y\^4)\
W\^[01]{}\_[\^]{}(y)&= e\^[-2y]{} (-0.231284 + 0.370054 y - 0.264922 y\^2 + 0.0935201 y\^3 - 0.0110873 y\^4)\
W\^[00]{}\_[M\^]{}(y)&=e\^[-2y]{} (-19.1326 + 40.3764 y - 30.3339 y\^2 + 10.0435 y\^3 - 1.4629 y\^4 + 0.0741493 y\^5)\
W\^[11]{}\_[M\^]{}(y)&=e\^[-2y]{} (-0.0278969 + 0.0781112 y - 0.0956406 y\^2 + 0.0607914 y\^3 - 0.0211633 y\^4\
&+ 0.00276687 y\^5)\
W\^[10]{}\_[M\^]{}(y)&=e\^[-2y]{} (0.659741 - 1.84727 y + 2.0953 y\^2 - 1.10461 y\^3 + 0.25912 y\^4 - 0.0218459 y\^5)\
W\^[01]{}\_[M\^]{}(y)&= e\^[-2y]{} (0.809015 - 1.7073 y + 1.48687 y\^2 - 0.692274 y\^3 + 0.145722 y\^4\
&- 0.00939131 y\^5)
[10]{} url \#1[[\#1]{}]{}urlprefix\[2\]\[\][[\#2](#2)]{} Cakir A 2014 (*Preprint* )
Catena R and Covi L 2014 [*Eur.Phys.J.*]{} [**C74**]{} 2703 (*Preprint* )
Ibarra A and Wild S 2013 [*JCAP*]{} [**1302**]{} 021 (*Preprint* )
Bergstrom L, Bertone G, Conrad J, Farnier C and Weniger C 2012 [*JCAP*]{} [**1211**]{} 025 (*Preprint* )
Bringmann T and Weniger C 2012 [*Phys.Dark Univ.*]{} [**1**]{} 194–217 (*Preprint* )
Baudis L 2012 [*Phys.Dark Univ.*]{} [**1**]{} 94–108 (*Preprint* )
Bertone G 2010 [*Nature*]{} [**468**]{} 389–393 (*Preprint* )
Strege C, Bertone G, Besjes G, Caron S, Ruiz de Austri R [*et al.*]{} 2014 [*JHEP*]{} [**1409**]{} 081 (*Preprint* )
Buchmueller O, Cavanaugh R, De Roeck A, Dolan M, Ellis J [*et al.*]{} 2014 [*Eur.Phys.J.*]{} [**C74**]{} 2922 (*Preprint* )
Bechtle P, Bringmann T, Desch K, Dreiner H, Hamer M [*et al.*]{} 2012 [ *JHEP*]{} [**1206**]{} 098 (*Preprint* )
Zhou N, Berge D, Wang L, Whiteson D and Tait T 2013 (*Preprint* )
Rajaraman A, Shepherd W, Tait T M and Wijangco A M 2011 [*Phys.Rev.*]{} [ **D84**]{} 095013 (*Preprint* )
Goodman J, Ibe M, Rajaraman A, Shepherd W, Tait T M [*et al.*]{} 2010 [ *Phys.Rev.*]{} [**D82**]{} 116010 (*Preprint* )
Goodman J, Ibe M, Rajaraman A, Shepherd W, Tait T M [*et al.*]{} 2011 [ *Phys.Lett.*]{} [**B695**]{} 185–188 (*Preprint* )
Gluscevic V and Peter A H G 2014 [*JCAP*]{} [**1409**]{} 040 (*Preprint* )
Catena R and Gondolo P 2014 [*JCAP*]{} [**1409**]{} 045 (*Preprint* )
Catena R 2014 [*JCAP*]{} [**1407**]{} 055 (*Preprint* )
Catena R 2014 [*JCAP*]{} [**1409**]{} 049 (*Preprint* )
Panci P 2014 [*Adv.High Energy Phys.*]{} [**2014**]{} 681312 (*Preprint* )
Cirelli M, Del Nobile E and Panci P 2013 [*JCAP*]{} [**1310**]{} 019 (*Preprint* )
Fitzpatrick A L, Haxton W, Katz E, Lubbers N and Xu Y 2012 (*Preprint* )
Fornengo N, Panci P and Regis M 2011 [*Phys.Rev.*]{} [**D84**]{} 115002 (*Preprint* )
Del Nobile E and Sannino F 2012 [*Int.J.Mod.Phys.*]{} [**A27**]{} 1250065 (*Preprint* )
Chen J Y, Kolb E W and Wang L T 2013 [*Phys.Dark Univ.*]{} [**2**]{} 200–218 (*Preprint* )
Rajaraman A, Tait T M and Wijangco A M 2013 [*Phys.Dark Univ.*]{} [**2**]{} 17–21 (*Preprint* )
Goodman J, Ibe M, Rajaraman A, Shepherd W, Tait T M [*et al.*]{} 2011 [ *Nucl.Phys.*]{} [**B844**]{} 55–68 (*Preprint* )
Fedderke M A, Chen J Y, Kolb E W and Wang L T 2014 [*JHEP*]{} [**1408**]{} 122 (*Preprint* )
Alves A, Profumo S, Queiroz F S and Shepherd W 2014 [*Phys.Rev.*]{} [**D90**]{} 115003 (*Preprint* )
Alves A, Berlin A, Profumo S, and Queiroz F S (*Preprint* )
Fitzpatrick A L, Haxton W, Katz E, Lubbers N and Xu Y 2013 [*JCAP*]{} [ **1302**]{} 004 (*Preprint* )
Fan J, Reece M and Wang L T 2010 [*JCAP*]{} [**1011**]{} 042 (*Preprint* )
Edsjo J 1997 (*Preprint* )
Gondolo P, Edsjo J, Ullio P, Bergstrom L, Schelke M [*et al.*]{} 2004 [ *JCAP*]{} [**0407**]{} 008 (*Preprint* )
Liang Z L and Wu Y L 2014 [*Phys.Rev.*]{} [**D89**]{} 013010 (*Preprint* )
Vincent A C and Scott P 2014 [*JCAP*]{} [**01404**]{}, 019 (*Preprint* )
Lopes I, Panci P, and Silk J 2014 [*Astrophys.J.*]{} [**795**]{} 162 (*Preprint* )
Vincent A C, Scott P, and Serenelli A (*Preprint* )
Vietze L, Klos P, Menéndez J, Haxton W and Schwenk A 2014 (*Preprint* )
Klos P, Menéndez J, Gazit D and Schwenk A 2013 [*Phys.Rev.*]{} [**D88**]{} 083516 (*Preprint* )
Menendez J, Gazit D and Schwenk A 2012 [*Phys.Rev.*]{} [**D86**]{} 103511 (*Preprint* )
Prezeau G, Kurylov A, Kamionkowski M, and Vogel P 2003 [*Phys.Rev.Lett.*]{} [**91**]{} 231301 (*Preprint* )
Cirigliano V, Graesser M L, Ovanesyan G, and Shoemaker I M 2014 [*Phys.Lett.B*]{} [**739**]{} 293-301 (*Preprint* )
Cirigliano V, Graesser M L, and Ovanesyan G 2012 [*JHEP*]{} [**1210**]{} 025 (*Preprint* )
Gresham M I and Zurek K M 2014 [*Phys.Rev.*]{} [**D89**]{} 123521 (*Preprint* )
Gould A 1987 [*Astrophys.J.*]{} [**321**]{} 571
Freese K, Lisanti M and Savage C 2013 [*Rev.Mod.Phys.*]{} [**85**]{} 1561–1581 (*Preprint* )
Bozorgnia N, Catena R and Schwetz T 2013 [*JCAP*]{} [**1312**]{} 050 (*Preprint* )
Catena R and Ullio P 2012 [*JCAP*]{} [**1205**]{} 005 (*Preprint* )
Catena R and Ullio P 2010 [*JCAP*]{} [**1008**]{} 004 (*Preprint* )
Anand N, Fitzpatrick A L and Haxton W 2014 [*Phys.Rev.*]{} [**C89**]{} 065501 (*Preprint* )
Engel J, Pittel S and Vogel P 1992 [*Int.J.Mod.Phys.*]{} [**E1**]{} 1
Brown B A and Rae W D M 2007 [*MSU-NSCL report*]{}
Warburton E and Brown B A 1992 [*Phys.Rev.*]{} [**C46**]{} 923–944
Wildenthal B 1984 [*Prog.Part.Nucl.Phys.*]{} [**11**]{} 5–51
Nummela S, Baumann P, Caurier E, Dessagne P and Jokinen A [*et al.*]{} 2001 [ *Phys.Rev.*]{} [**C63**]{} 044316
Honma M, Otsuka T, Brown B A and Mizusaki T 2004 [*Phys.Rev.*]{} [**C69**]{} 034335 (*Preprint* )
Brown B A 2001 [*Prog.Part.Nucl.Phys.*]{} [**47**]{} 517–599
Caurier E, Martinez-Pinedo G, Nowacki F, Poves A and Zuker A P 2005 [*Rev.Mod.Phys.*]{} [**77**]{} 427 (*Preprint* )
Walecka J D 2005 [*Electron Scattering for Nuclear and Nucleon Structure*]{} (Cambridge University Press)
J. N. Bahcall, A. M. Serenelli, and S. Basu, [*Astrophys.J.*]{} [**621**]{} (2005) L85–L88, (*Preprint* )
N. Grevesse, and A. J. Sauval, [*Space Sci.Rev.*]{} [**85**]{} (1998) 161-174
A. Serenelli, C. Peña-Garay and W. C. Haxton, [*Phys. Rev.*]{} [**D87**]{} (2013) 043001, (*Preprint* )
M. Asplund, N. Grevesse, A. J. Sauval and P. Scott, [*Ann. Rev. Astron. Astrophys.*]{} [**47**]{} (2009) 481, (*Preprint* )
|
---
abstract: 'This document proves global boundedness and decay for axisymmetric perturbations of a known solution to the wave map problem from a slowly rotating $|a|\ll M$ Kerr spacetime to the hyperbolic plane. This problem is motivated by the general axisymmetric stability of Kerr conjecture and was first posed by Ionescu and Klainerman in [@IoKl]. Two particular developments in this paper, the treatment of terms near the axis of symmetry and the use of a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces, can be used for a variety of similar problems.'
author:
- John Stogin
bibliography:
- 'bib.bib'
title: 'Global Stability of the Nontrivial Solutions to the Wave Map Problem from Kerr $|a|\ll M$ to the Hyperbolic Plane under Axisymmetric Perturbations Preserving Angular Momentum'
---
Introduction
============
Estimates for the $\xi_a$ System {#xi_a_sec}
================================
Estimates for the $(\phi,\psi)$ System {#phi_psi_sec}
======================================
The Energy Estimates {#ee_sec}
====================
The Pointwise Estimates {#pointwise_sec}
=======================
The Structures of $\mathcal{N}_\phi$ and $\mathcal{N}_\psi$ {#nonlinear_sec}
===========================================================
Statement and Proof of the Main Theorem {#main_thm_sec}
=======================================
Regularity for Axisymmetric Functions {#regularity_sec}
=====================================
|
---
abstract: 'Gaussian and Chiral $\beta$-Ensembles, which generalise well known orthogonal ($\beta=1$), unitary ($\beta=2$), and symplectic ($\beta=4$) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like $\{\beta,N,n\}\Leftrightarrow \{4/\beta,n,N\}$ for all $\beta>0$, where $N$ and $n$ respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals of the Airy (Kontsevich) type are obtained. Consequences on the integral representation of the multiple orthogonal polynomials and the partition function of the formal one-matrix model are also discussed. Proofs rely on the theory of multivariate symmetric polynomials, especially Jack polynomials.'
address: 'Institut de Physique Théorique, CEA–Saclay, 91191 Gif-sur-Yvette cedex, France.'
author:
- Patrick Desrosiers
title: 'Duality in random matrix ensembles for all $\beta$'
---
[^1]
Introduction
============
It is the purpose of this article to obtain new dualities between different ensembles of random variables inspired by matrix models. Simple duality relations are given below, after a short review of the $\beta$-Ensembles.
$\beta$-Ensembles
-----------------
Let $x=(x_1,\ldots,x_N)$ denote a set of $N$ random variables. Moreover, let $\beta$ be a non-negative real number; it is called the Dyson index. The joint probability density function (p.d.f.) for the Gaussian $\beta$-Ensemble (${\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}$) is $$\label{gbe}
P_{{ {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}}(x)dx=\frac{1}{Z_{{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}}\prod_{1\leq i<j\leq N}|x_i-x_j|^\beta \prod_{i=1}^N e^{-x_i^2}dx_i.$$ The average of a function $f$ over the ensemble is given by $$\Big\langle f(x) \Big\rangle_{x\in {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}:= \int_{\mathbb{R}^N} f(x)P_{{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}(x)dx.$$ Note that $Z_{{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}$ is chosen such that the average of the identity equals one. Physically, the density $P_{{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}$ can be interpreted as the Boltzmann factor of a classical (two-dimensional) Coulomb gas at temperature $1/\beta$ [@DysonI]. There is also a quantum mechanical interpretation of the density: $\sqrt{P_{{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}}$ is the ground state of a $N$-body problem with pairwise potential interaction of the form $1/r^2$ [@Calogero]. The chiral Gaussian $\beta$-Ensemble (${\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}$) is defined similarly: $$\label{lbe}
P_{{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}(x)dx=\frac{1}{Z_{{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}}\prod_{1\leq i<j\leq N}|x_i-x_j|^\beta \prod_{i=1}^N x_i^\gamma e^{-x_i}dx_i,$$ where the variables $x_i$ are positive reals and the real part of $\gamma$ is greater then $-1$. Note that the Gaussian and chiral Gaussian ensembles are sometimes called Hermite and Laguerre (or even Wishart) ensembles respectively (cf. [@DF; @Dumitriu]).
In this article, the $\beta$-Ensembles are considered from a Random Matrix Theory [@ForresterBook; @Mehta] perspective. Let $\mathbf{X}=[X_{i,j}]$ be a random $N\times N$ Hermitian matrix whose entries are real ($\beta=1$), complex ($\beta=2$), or quaternion real ($\beta=4$) (see Appendix A for more detail). The trivial $\beta=0$ case corresponds to a real diagonal matrix. It is a classical result that Eq. provides the joint density for the eigenvalues of a Hermitian matrix $\mathbf{X}$ when the latter is drawn with probability $$\label{Mgbe}
e^{-{\ensuremath{\,\mathrm{tr}\,}}\mathbf{X}^2}(d\mathbf{X}), \quad$$ where $(d\mathbf{X})$ stands for the normalised product of all the real independent elements of $[dX_{i,j}]$. Similarly, if a positive definite Hermitian matrix $\mathbf{X}$ is distributed according to $$\label{Mlbe}
(\det \mathbf{X})^\gamma e^{-{\ensuremath{\,\mathrm{tr}\,}}\mathbf{X}}(d\mathbf{X}),$$ then the p.d.f. of its eigenvalues is given by Eq. . Moreover, set $\mathbf{X}=\mathbf{Y}^\dagger\mathbf{Y}$, where $\mathbf{Y}$ is a $N_1\times N_2$ rectangular matrix such that $N_1\geq N_2$. One can show that if $\mathbf{Y}$ has a Gaussian distribution, then the p.d.f. of $\mathbf{X}$ is given by with $\gamma=(\beta/2)(N_1-N_2+1-2/\beta)$ [@ForresterBook Chapter I].
It is worth mentioning that Eqs. and can also be realised, for all $\beta>0$, as the eigenvalue p.d.f. of tri-diagonal real symmetric matrices [@Dumitriu]. In the large $N$ limit, scaled versions of these tri-diagonal matrices can be seen as stochastic differential operators [@EdelmanSutton; @Ramirez].
Some results
------------
The aim of this paper is to prove the equivalence of some averages over $\beta$-ensembles and other averages over $4/\beta$-ensembles The averages contain a source matrix $\mathbf{S}$ of size $n$, and an external field matrix $\mathbf{F}$ of size $N$. These matrices are Hermitian with eigenvalues $s=(s_1,\ldots,s_N)$ and $f=(x_1,\ldots,x_N)$ respectively. Roughly speaking, the duality relations obtained here read $$\label{duality}
\left\lbrace
\begin{array}{c}
\,\,\,\beta\,\,\,\\
N\\
{\ensuremath{\mathbf{F}}}\\
n\\
{\ensuremath{\mathbf{S}}}\end{array}
\right\rbrace
\Longleftrightarrow
\left\lbrace
\begin{array}{c}
4/\beta\\
n\\
{\ensuremath{\mathbf{S}}}\\
N\\
{\ensuremath{\mathbf{F}}}\end{array}
\right\rbrace.$$
In order to formulate the results precisely, further notation has to be introduced. Let $\mathbf{1}$ be the identity matrix and let $$\left\langle F(\mathbf{X})\right\rangle_{\mathbf{X}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}}=\int (d\mathbf{X})e^{-{\ensuremath{\,\mathrm{tr}\,}}\mathbf{X}^2} F(\mathbf{X})\Big/ \int (d\mathbf{X})e^{-{\ensuremath{\,\mathrm{tr}\,}}\mathbf{X}^2},$$ which is the average of the complex-valued function $F$ over the Gaussian $\beta$-Ensemble of matrices with p.d.f. (not to be confused with the average over the eigenvalues). Special realisations of Eq. are given in the following propositions. Only expression relative to the Gaussian ensembles will be displayed for the moment; dualities for the Chiral ensembles will be given in Sections 3 and 4.
\[prop1\] Suppose that $\beta=1,2$ or $4$ and that $\beta'=4/\beta$. Then $$\begin{gathered}
\label{DualMat1}
e^{-\mathrm{tr}{\ensuremath{\mathbf{F}}}^2}\left\langle\prod_{j=1}^n\det\left(s_j{\ensuremath{\mathbf{1}}}\pm\mathrm{i}\sqrt{\frac{2}{\beta}}{\ensuremath{\mathbf{X}}}\right)e^{2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{F}}}}\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\\
e^{-\mathrm{tr}{\ensuremath{\mathbf{S}}}^2}\left\langle\prod_{j=1}^N\det\left({\ensuremath{\mathbf{Y}}}\pm\mathrm{i}\sqrt{\frac{2}{\beta}}f_j{\ensuremath{\mathbf{1}}}\right)e^{2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{S}}}}\right\rangle_{{\ensuremath{\mathbf{Y}}}\in {\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_n}.\end{gathered}$$
The previous result concerns the average of products of characteristic polynomials. Special cases of this duality previously appeared in the literature. For instance, the duality for moments of characteristic polynomials, which corresponds to $s_1=\ldots=s_n$ and ${\ensuremath{\mathbf{F}}}={\ensuremath{\mathbf{0}}}$, has been proved for $\beta=1,2,4$ and conjectured for all $\beta$ by Mehta and Normand [@Normand]. For ${\ensuremath{\mathbf{F}}}={\ensuremath{\mathbf{0}}}$ (i.e., no external field) and $\beta=2$, but for distinct $s$, this has been observed by Fyodorov and Strahov in [@FyodorovStrahaov]. Based on a previous work [@BrezinHikami00] and the supersymmetric method, Brézin and Hikami [@BrezinHikami01] have obtained dualities reproducing Eq. in the case ${\ensuremath{\mathbf{F}}}={\ensuremath{\mathbf{0}}}$ and $\beta=1,2,4$. The problem with a non-zero external field and $\beta=2$, has been recently solved by same authors [@BrezinHikami07].
A less common duality of the type , but not affecting $\beta$, concerns products of inverse characteristic polynomials.
\[prop2\]Suppose that $\beta=1,2$ or $4$ and that $\beta'=4/\beta$. Assume moreover that the variables $s$ and $f$ have non-zero imaginary parts. Then $$\begin{gathered}
e^{-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{F}}}^2}\left\langle\prod_{j=1}^n\det(s_j{\ensuremath{\mathbf{1}}}\pm {\ensuremath{\mathbf{X}}})^{-\beta/2}e^{2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{F}}}}\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\\
e^{-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2}\left\langle\prod_{j=1}^N\det({\ensuremath{\mathbf{Y}}}\pm f_j{\ensuremath{\mathbf{1}}})^{-\beta/2}e^{2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{S}}}}\right\rangle_{{\ensuremath{\mathbf{Y}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_n}.\end{gathered}$$
More general dualities will be exposed in Sections 3 and 4 after the introduction of multivariate hypergeometric functions in Section 2. To the best of the author’s knowledge, explicit dualities valid for $\beta$ general and $N$ finite, first appeared in a 1997 paper by Baker and Forrester [@Baker]. For instance, Eq. 5.31 in [@Baker] is equivalent to Eq. above with ${\ensuremath{\mathbf{F}}}={\ensuremath{\mathbf{0}}}$ and $s_1=\ldots=s_n=t$ say, and $\beta$ integer. The latter expression has been used in [@DF] for calculating asymptotic corrections to the global eigenvalue density when $\beta$ is even. The present work can be considered, to some extend, as a continuation of [@Baker]. Note also that Proposition 7, for $s_1=\ldots=s_n=t$ and $f=0$, provides a proof of the formula conjectured by Mehta and Normand for all $\beta>0$ [@Normand Eq. 3.29].
Closely related are the “particle-hole” dualities observed, when $\beta$ is rational, in the limit $N\rightarrow\infty$ of the correlation functions for the Sutherland model or equivalently, for the Circular $\beta$-Ensembles (see for instance [@Serban] and references therein). This can also be interpreted as a strong-weak coupling duality (see for example [@Jonke]).
Inter-relationships between orthogonal, unitary and symplectic ensembles have been also considered, at the level of the joint eigenvalue p.d.f. itself, for eigenvalues respecting interlacing inequalities [@ForresterRains]. Very recently, this work has been generalised by Forrester in [@Forrester07]. The duality found in the latter reference can be summarised as follows : setting $\beta'=2(r+1)=4/\beta$, the joint distribution of every $(r+1)$-st eigenvalue in a $\beta$-ensemble is equal to that of a $\beta'$-ensemble if the eigenvalues are properly ordered.
Before going further into the study of dualities, a last comment is in order. For $\beta=1,2$ and $4$, expectation values of products and ratios of characteristic polynomials can be expressed in terms of determinants or Pfaffians (see the extensive study by Borodin and Strahov [@BorodinStahov] and references therein). The size of the determinants depends on the number $n$ of characteristic polynomials but not on the size $N$ of the random matrix. Thus, one can exploit the determinant formulae for calculating the asymptotic behaviour of the correlation functions when $N\rightarrow\infty$. For general $\beta$, no such determinantal formulae are available. Then, solving a $\beta$-matrix model amounts to finding a reduced integral representation for the correlation functions. In other words, the aim is to obtain integral formulae whose dimension does not depend on the size of the matrix, thus providing a representation for the correlation functions that, in theory, allows to take the limit $N\rightarrow\infty$.
Preliminary definitions
=======================
This section furnishes a brief introduction to the theory of symmetric polynomials. More detail can be found in [@ForresterBook; @Kadell; @Mac; @Stan]
Partitions
----------
Let $\lambda=(\lambda_1,\lambda_2,\ldots,)$ denote a partition of $n$; that is, a sequence of non-negative integers such that $$\lambda_1\geq\lambda_2\geq \ldots\geq0,\qquad |\lambda|:=\sum_i\lambda_i=n.$$ One usually writes $(\lambda_1,\ldots,\lambda_\ell,0,\ldots,0)=(\lambda_1,\ldots,\lambda_\ell)$, where $\ell=\ell(\lambda)$ gives the number of non-zero parts in $\lambda$. A partition can also be expressed as follows: $\lambda=(\ldots,3^{n_3}, 2^{n_2},1^{n_1})$ where $n_k=n_k(\lambda)$ is the number of parts of $\lambda$ that are equal to $k$. The conjugate partition of $\lambda$, written $\lambda'=(\lambda'_1,\lambda'_2,\ldots)$, is such that $$\lambda'_k=\#\{\lambda_i \in \lambda : \lambda_i\geq k\}$$
To each partition, we associate a diagram by drawing $\lambda_1$ boxes on the first row, then $\lambda_2$ boxes under the first row, and so on, all boxes left justified. See Fig. \[fig1\]. The conjugation of the partition then corresponds to the transposition (as for matrices) of the diagram. The arm of a point $s=(i,j)$ in $\lambda$, written $a_\lambda(s)$, is the number of boxes to the right of $(i,j)$ in the $i$th row of the diagram $\lambda$. As illustrated in Fig. \[fig2\], similar definitions exist for the leg $l$, co-arm $a'$, and co-leg $l'$ of the point $s=(i,j)$.
(0,1)(9,6)
(0.5,4)[$\lambda\,\,=$]{}
(1.5,5)(2,5)(2.5,5)(3,5)(3.5,5) (1.5,4.5)(2,4.5)(2.5,4.5)(3,4.5) (1.5,4)(2,4)(2.5,4)(1.5,3.5)(2,3.5)(2.5,3.5) (1.5,3.0) (1.5,2.5)
(4.5,4)[$\lambda'\,\,=$]{}
(5.5,5)(6,5)(6.5,5)(7,5)(7.5,5)(8,5) (5.5,4.5)(6,4.5)(6.5,4.5)(7,4.5) (5.5,4)(6,4)(6.5,4)(7,4) (5.5,3.5)(6,3.5)(5.5,3.0)
The dominance ordering of partitions is defined as follows: $$\lambda\geq\mu \qquad\Longleftrightarrow\qquad \sum_{i=1}^k(\lambda_i-\mu_i)\geq0\qquad \forall \, k.$$ This ordering is partial. Note the obvious property: $$\lambda\geq\mu\qquad\Longleftrightarrow\qquad \lambda'\leq\mu'.$$
Four functions on partitions will be frequently used in the article. They are: the generalised factorial $$[u]^{(\alpha)}_\lambda:=\prod_{j\geq1}\frac{\Gamma\left(u-(j-1)/\alpha+\lambda_j\right)}{\Gamma\left(u-(j-1)/\alpha \right)};$$ the specialisation coefficient $$b^{(\alpha,N)}_\lambda=\prod_{s\in\lambda}\left(N-l'_\lambda(s)+\alpha a'_\lambda(s)\right);$$ and the “lower” and “upper” hook-lengths [@Stan] [^2] $$\label{eqHook}
h^\lambda_{(\alpha)}=\prod_{s\in\lambda}\left(l_\lambda(s)+1+\alpha a_\lambda(s)\right)\quad\mbox{and}\quad h_\lambda^{(\alpha)}=\prod_{s\in\lambda}\left(l_\lambda(s)+\alpha+\alpha\ a_\lambda(s)\right);$$ Note that $b^{(\alpha,N)}_\lambda=\alpha^{|\lambda|}[N/\alpha]^{(\alpha,N)}_\lambda$.
(-1,0)(5,4.5)
(0,0)(0,4)(6,4)(6,3)(5,3)(5,1)(3,1)(3,0.5)(1,0.5)(1,0)(0,0)
(0,2)(1.75,2) (1,2.25)[$a'_\lambda(s)$]{}
(2.25,2)(5,2) (3.75,2.25)[$a_\lambda(s)$]{}
(2,0.5)(2,1.75) (2.5,1.2)[$l_\lambda(s)$]{}
(2,2.25)(2,4) (2.5,3.0)[$l'_\lambda(s)$]{}
(2,2) (2,2)[$s$]{}
(-0.75,2)[$\lambda\;=$]{}
Jack polynomials
----------------
Let $x$ stand for the ordered set $(x_1,\ldots,x_N)$. A function $f$ is symmetric if it is invariant under permutations of the variables; that is, $f(x)=f(x_\sigma )$ for any element $\sigma$ of the symmetric group $S_N$, where $x_\sigma =(x_{\sigma(1)},\ldots,x_{\sigma(N)})$. The set of all symmetric polynomials in $N$ variables, whose coefficients are rational functions of $\alpha$, form an algebra over the ring $\mathbb{Q}(\alpha)$, witten $P^{S_N}$. A symmetric polynomial that is homogeneous of degree $n$ can be decomposed into the monomial basis $\{m_\lambda : |\lambda|=n\}$, where $$m_\lambda(x)=\frac{1}{n(\lambda)!}\sum_{\sigma\in S_N}x_\sigma^\lambda=\frac{1}{n(\lambda)!}\sum_{\sigma\in S_N}x_{\sigma(1)}^{\lambda_1}\cdots x_{\sigma(N)}^{\lambda_N}$$ and $
n(\lambda)!:=n_1(\lambda)! n_2(\lambda)!\cdots
$.
Another important basis for $P^{S_N}$ is provided by products of power sums $\{p_\lambda\}$, where $$p_\lambda(x):=p_{\lambda_1}(x)\cdots p_{\lambda_\ell}(x),\qquad p_k(x)=\sum_{i=1}^Nx_i^k.$$ The combinatorial (or Fock space) scalar product in the algebra of symmetric polynomials can be defined by $$\label{CombinatScalProd}
{\ensuremath{\langle\!\langle}}p_\lambda|p_\mu{\ensuremath{\rangle\!\rangle}}^{(\alpha)}=\alpha^{\ell(\lambda)}z_\lambda\delta_{\lambda,\mu},$$ where $z_\lambda=\prod_k k^{n_k}k!$. The parameter $\alpha$ is related to Dyson’s $\beta$ index as follows: $$\label{alphabeta}
\alpha=\frac{2}{\beta}.$$
The Macdonald automorphism will play an important role in the following paragraphs; it is given by $$\label{defHomo}
\omega_k p_n=(-1)^{n-1}k\,p_n$$ and satisfies $$\label{EqHomo}
{\ensuremath{\langle\!\langle}}\omega_k f|g{\ensuremath{\rangle\!\rangle}}^{(\alpha)}={\ensuremath{\langle\!\langle}}f|\omega_kg{\ensuremath{\rangle\!\rangle}}^{(\alpha)},\qquad
{\ensuremath{\langle\!\langle}}\omega_{1/\alpha} f|g{\ensuremath{\rangle\!\rangle}}^{(\alpha)}={\ensuremath{\langle\!\langle}}\omega_1 f|g{\ensuremath{\rangle\!\rangle}}^{(1)}.$$ for any symmetric polynomials $f$ and $g$.
The (monic) Jack polynomials, denoted by $P_\lambda(x)=P_\lambda(x;\alpha)$ or by $P^{(\alpha)}_\lambda(x)$, generalise many important symmetric polynomials: $$P^{(\alpha)}_\lambda=\begin{cases}e_{\lambda'},& \alpha=0\\
s_\lambda, &\alpha=1\\
Z_\lambda/h^\lambda_{(\alpha)}, &\alpha=2\\
m_\lambda&\alpha=\infty
\end{cases}$$ where $e_\lambda$, $s_\lambda$, and $Z_\lambda$ respectively stand for the elementary, Schur, and Zonal polynomials. $P^{(\alpha)}_\lambda$ is the unique symmetric polynomial with coefficients in $\mathbb{Q}(\alpha)$ satisfying $$\label{eqOrthoJack}
\begin{array}{lll}
(1)&\displaystyle P_{\lambda} = m_{\lambda} +\sum_{\mu < \lambda} c_{\lambda \mu}(\alpha) m_{\lambda}&\mbox{(triangularity)}\\
(2)&\displaystyle {\ensuremath{\langle\!\langle}}P_{\lambda}| P_{\mu} {\ensuremath{\rangle\!\rangle}}^{(\alpha)} =\Vert P_{\mu}^{(\alpha)}\Vert^2 \delta_{\lambda,\mu} &\mbox{(orthogonality)}
\end{array}$$ Alternatively, the Jack polynomials can be considered as the unique triangular symmetric polynomials that comply with $$D\,P_\lambda(x)=e_\lambda(\alpha) P_\lambda (x)\qquad \mbox{(eigenfunctions)},$$ where $$D=\sum_{i=1}^N\left(\alpha x_i^2\partial_{x_i}^2+2\sum_{j\neq i}\frac{x_ix_j}{x_i-x_j}\partial_{x_i}\right).$$ The eigenvalues are given by $e_\lambda(\alpha)=\sum_{i\geq1}\left(\alpha(i-1)\lambda'_i-(i-1)\lambda_i\right)$. The following specialisation and normalisation formulae will be used later: $$\label{eqSpecial}
P^{(\alpha)}_\lambda (1^N)=\frac{b_\lambda^{(\alpha,N)}}{h^\lambda_{(\alpha)}}\quad\mbox{and}\quad \Vert P^{(\alpha)}_\lambda\Vert^2=\frac{h_\lambda^{(\alpha)}}{h^\lambda_{(\alpha)}}.$$
The Jack polynomials possess several remarkable properties. Amongst them, it is worth mentioning their duality (see eq. ) $$\label{eqDualityJack}
\omega_\alpha \, P^{(\alpha)}_\lambda =\frac{1}{ \Vert P^{(\alpha')}_{\lambda'}\Vert^2} P_{\lambda'}^{(\alpha')}\qquad\mbox{with}\qquad \alpha':=\frac{1}{\alpha},$$ and the Cauchy type formula $$\label{eqCauchy}
\prod_{i=1}^N\prod_{j=1}^M\frac{1}{(1-x_iy_j)^{1/\alpha}}=\sum_{\lambda}\frac{1}{\Vert P^{(\alpha)}_\lambda\Vert^2}P^{(\alpha)}_\lambda (x)\,P^{(\alpha)}_\lambda (y).$$ By applying the former property to the latter equation, one gets $$\prod_{i=1}^N\prod_{j=1}^M(1+x_iy_j)=\sum_{\lambda}P^{(\alpha)}_\lambda (x)\,P^{(\alpha')}_{\lambda'} (y).$$
Multivariate hypergeometric functions
-------------------------------------
In multivariate analysis, the Jack polynomial $P_{\lambda}(x_1,x_2,\ldots)$ plays a role that is similar to that played by the monomial $x_1^{|\lambda|}$ for functions in one variable. As an example, the multivariate hypergeometric functions of two sets of variables are given by [@ForresterBook; @Yan]: $$\begin{gathered}
\label{eqHyper}
{\,_p{\mathcal{F}}_q}^{(\alpha)} (a_1,\ldots,a_p;b_1,\ldots,b_q;x_1,\ldots,x_N; y_1,\ldots,y_N)=\\
\sum_{\lambda}\frac{\alpha^{|\lambda|}}{h_\lambda^{(\alpha)}}\frac{[a_1]^{(\alpha)}_\lambda\cdots[a_p]^{(\alpha)}_\lambda}{[b_1]^{(\alpha)}_\lambda\cdots [b_q]^{(\alpha)}_\lambda}\frac{P_\lambda^{(\alpha)}(x) P_\lambda^{(\alpha)}(y)}{P^{(\alpha)}_\lambda(1^N)}.\end{gathered}$$ There is no simple “explicit forms” for the latter functions in general. However, one can prove that [@Kaneko] $$\label{eqHyperspecial}
{\,_0{\mathcal{F}}_0}^{(\alpha)}(x_1,\ldots,x_N;t,\ldots,t)=\prod_{i=1}^Ne^{tx_i}$$ $${\,_1{\mathcal{F}}_0}^{(\alpha)}(a;x_1,\ldots,x_N;t,\ldots,t)=\prod_{i=1}^N(1-tx_i)^{-a}.$$ Softwares are also available for computing “truncated” hypergeometric functions; they are based on the exact calculation of Jack polynomials (e.g., see [@Koev]).
As explained in [@Baker], the ${\,_p{\mathcal{F}}_q}^{(\alpha)}$’s provide generating functions for the multivariate Hermite and Laguerre polynomials, respectively written $\mathcal{H}_\lambda^{(\alpha)}$ and $\mathcal{L}_\lambda^{(\alpha,\gamma)}$. These polynomials were first defined by Lassalle in [@LassalleLaguerre; @LassalleHermite]. Explicitly, [^3] $$\label{eqHermite}
e^{-p_2(y)}{\,_0{\mathcal{F}}_0}^{(\alpha)}(2x;y)=\sum_\lambda \frac{1}{A_\lambda(\alpha,N)} \mathcal{H}_\lambda^{(\alpha)}(x){P}_\lambda^{(\alpha)}(y),$$ where $$A_\lambda(\alpha,N)=\frac{1}{(2\alpha)^{|\lambda|}}h_\lambda^{(\alpha)}{P}_\lambda^{(\alpha)}(1^N),$$ and $$\label{eqLaguerre}
e^{-p_1(y)}{\,_0{\mathcal{F}}_1}^{(\alpha)}(\gamma+q;x;y)=\sum_\lambda \frac{1}{B_\lambda(\alpha,\gamma,N)} \mathcal{L}_\lambda^{(\alpha,\gamma)}(x){P}_\lambda^{(\alpha)}(y),$$ where $$B_\lambda(\alpha,\gamma,N)=2^{|\lambda|}[\gamma+q]^{(\alpha)}_\lambda A_\lambda(\alpha,N), \qquad q=1+(N-1)/\alpha.$$ Multivariate orthogonal polynomials are deeply connected to the Dunkl operators and Calogero systems [@Baker; @vDiejen; @Rosler]. From these relations, it is possible to show, for instance, that the Hermite and Laguerre polynomials in many variables provide orthogonal bases for the algebra of symmetric polynomials: $$\label{eqOrthoH}
\Big\langle \mathcal{H}_\lambda^{(2/\beta)}(x) \mathcal{H}_\mu^{(2/\beta)}(x) \Big\rangle_{x\in {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=A_\lambda(2/\beta,N)\delta_{\lambda,\mu}$$ $$\label{eqOrthoL}
\Big\langle \mathcal{L}_\lambda^{(2/\beta,\gamma)}(x) \mathcal{L}_\mu^{(2/\beta,\gamma)}(x) \Big\rangle_{x\in {\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}=B_\lambda(2/\beta,\gamma,N)\delta_{\lambda,\mu}.$$
Expectation values of Jack polynomials
======================================
In this section, it is proved that the averages of Jack polynomials over the $\beta$-ensembles defined in and enjoy simple duality properties. Recall that the Jack polynomials form a basis for symmetric polynomials. Thus, in principle at least, dualities involving Jack polynomials provide general tools for reducing the dimension of the correlation functions.
Gaussian ensembles
------------------
The starting point is a conjecture by Goulden and Jackson [@Goulden] first proved by Okounkov in [@Okounkov]. Another proof is given below.
\[lemma1\] Let $\lambda$ be a partition of an even integer and let $\mu$ be the partition $(2^{|\lambda|/2})$. Also, let ${\ensuremath{\langle\!\langle}}f|g{\ensuremath{\rangle\!\rangle}}$ denote the combinatorial scalar product denied in Eq. . Then $$\label{EqProof1}
\left\langle {P^{(2/\beta)}_\lambda(x)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\left(\frac{\beta}{4}\right)^{\ell(\mu)}\frac{b_\lambda^{(2/\beta,N)}}{\ell(\mu)!}{\ensuremath{{\Big\langle}\!\!{\Big\langle}}}P^{(2/\beta)}_\Lambda\,\Big|\,p_{\mu}{\ensuremath{{\Big\rangle}\!\!{\Big\rangle}}}^{(2/\beta)}.$$ Alternatively, $$\label{EqProof1}
\left\langle {P^{(2/\beta)}_\lambda(x)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}={b_\lambda^{(2/\beta,N)}}\underset{p_\mu}{\mathrm{coeff}}
P^{(2/\beta)}_\Lambda,$$ where $\underset{x}{\mathrm{coeff}}f$ denote the coefficient of $x$ in $f$.
It is simpler to set $\alpha=2/\beta$. The orthogonality and Eq. immediately imply [@Baker Proposition 3.8] $$\left\langle{\,_0{\mathcal{F}}_0}^{(\alpha)}(2x;y){\,_0{\mathcal{F}}_0}^{(\alpha)}(2x;z) \right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}}=e^{p_2(y)+p_2(z)}{\,_0{\mathcal{F}}_0}^{(\alpha)}(2y;z).$$ Then, setting $z=0$ in the last equation and recalling that $P_\lambda(0)$ is equal to $1$ if $\lambda=(0)$ and 0 otherwise, one gets $$\left\langle{\,_0{\mathcal{F}}_0}^{(\alpha)}(2x;y) \right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}}=e^{p_2(y)}.$$ Now, the left-hand-side is developted in terms of the Jack polynomials and the combinatorial scalar product with respect to $P_\lambda^{(\alpha)}$ is taken (see Eq. ). This leads to $$\frac{(2\alpha)^{|\lambda|/2}}{h^\lambda_{(\alpha)}P^{(\alpha)}_\lambda(1^N)}\left\langle P^{(\alpha)}_\lambda(x)\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}}= \frac{1}{(|\lambda|/2)!}{\ensuremath{\langle\!\langle}}P^{(\alpha)}_\lambda| (p_2)^{|\lambda|/2}{\ensuremath{\rangle\!\rangle}}^{(\alpha)},$$ which, by virtue of Eq., is equivalent to the first equation of the lemma. The second equation simply follows from $${\ensuremath{\langle\!\langle}}p_\mu|p_\mu {\ensuremath{\rangle\!\rangle}}^{(\alpha)} = (2\alpha)^{|\lambda|/2}({|\lambda|}/{2})!\quad \mbox{if}\quad \mu=(2^{|\lambda|/2}).$$
\[theo1\] Set $x'=\mathrm{i}x\sqrt{2/{\beta}}$ and ${\beta'}=4/\beta>0$. Then, for every $N\geq\ell(\lambda)$ and $N'\geq \ell(\lambda')$, $$\label{eqTheo1}
\left\langle \frac{P^{(2/\beta)}_\lambda(x)}{P^{(2/\beta)}_\lambda(1^{N})}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\left\langle
\frac{P^{(2/\beta')}_{\lambda'}(x')}{P^{(2/\beta')}_{\lambda'}(1^{N'})}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_{N'}}.$$ Note that, as a consequence of the orthogonality , the expectation values are non-zero only if $|\lambda|$ is even.
Assume $N\geq \ell(\lambda)$ and $|\lambda|$ even. Following the notation used for the previous lemma, one has $$\label{eqProofTheo1}
\left\langle \frac{P^{(\alpha)}_\lambda(x)}{P^{(\alpha)}_\lambda(1^N)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\left(\frac{2}{\alpha}\right)^{\ell(\mu)}\frac{h^\lambda_{(\alpha)}}{\ell(\mu)!}\,{\ensuremath{\langle\!\langle}}P^{(\alpha)}_\lambda| p_\mu{\ensuremath{\rangle\!\rangle}}^{(\alpha)}.$$ The Macdonald automorphism is then exploited for getting $$\left\langle \frac{P^{(\alpha)}_\lambda(x)}{P^{(\alpha)}_\lambda(1^N)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=(-1)^{\ell(\mu)}{2}^{\ell(\mu)}\frac{h^\lambda_{(\alpha)}}{\ell(\mu)!}\,{\ensuremath{\langle\!\langle}}P^{(\alpha)}_\lambda| \omega_{1/\alpha}p_\mu{\ensuremath{\rangle\!\rangle}}^{(\alpha)}.$$ Moreover, the use of Eq. yields $$\begin{gathered}
\left\langle \frac{P^{(\alpha)}_\lambda(x)}{P^{(\alpha)}_\lambda(1^N)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=(-1)^{\ell(\mu)}{2}^{\ell(\mu)}\frac{h^\lambda_{(\alpha)}}{\ell(\mu)!}\,{\ensuremath{\langle\!\langle}}P^{(\alpha)}_\lambda| \omega_{1}p_\mu{\ensuremath{\rangle\!\rangle}}^{(1)}\\=(-1)^{\ell(\mu)}{2}^{\ell(\mu)}\frac{h^\lambda_{(\alpha)}}{\ell(\mu)!}{\ensuremath{\langle\!\langle}}\omega_{\alpha} P^{(\alpha)}_\lambda| p_\mu{\ensuremath{\rangle\!\rangle}}^{(1/\alpha)}.\end{gathered}$$ But according to the duality Eq. of the Jack polynomials, the last equation can be rewritten as $$\left\langle \frac{P^{(\alpha)}_\lambda(x)}{P^{(\alpha)}_\lambda(1^N)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=(-1)^{\ell(\mu)}{2}^{\ell(\mu)}\frac{h^\lambda_{(\alpha)}}{\ell(\mu)!} \frac{h^{\lambda'}_{(\alpha')}}{h_{\lambda'}^{(\alpha')}}\,{\ensuremath{\langle\!\langle}}P^{(\alpha')}_{\lambda'}| p_\mu{\ensuremath{\rangle\!\rangle}}^{(\alpha')},$$ where $\alpha'=1/\alpha$. Note that the last displayed equation is non-zero only if $N'\geq \ell(\lambda')$, where $N'$ denotes the number of variables in $P^{(\alpha')}_{\lambda'}$. Besides, from the definition of the lower and upper hook-lengths, one easily obtains $$h^\lambda_{(\alpha)}=\alpha^{|\lambda|}h_{\lambda'}^{(\alpha')}=\alpha^{2\ell(\mu)}h_{\lambda'}^{(\alpha')}$$ Hence, $$\left\langle \frac{P^{(\alpha)}_\lambda(x)}{P^{(\alpha)}_\lambda(1^N)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=(-1)^{\ell(\mu)}{2}^{\ell(\mu)} \alpha^{2\ell(\mu)}\frac{ h^{\lambda'}_{(\alpha')}}{\ell(\mu)!}\,{\ensuremath{\langle\!\langle}}P^{(\alpha')}_{\lambda'}| p_\mu{\ensuremath{\rangle\!\rangle}}^{(\alpha')}.$$ Finally, Lemma 1 or Eq. is used once again to obtain $$\left\langle \frac{P^{(\alpha)}_\lambda(x)}{P^{(\alpha)}_\lambda(1^N)}\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}={(-\alpha)^{\ell(\mu)}} \left\langle\frac{P^{(\alpha')}_{\lambda'}(y)}{P^{(\alpha')}_{\lambda'}(1^{N'})}\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_{N'}}.$$ which is the desired result.
The simplest example of such a duality is certainly the average of moments of the characteristic polynomial (cf. [@Normand]). Indeed, from $$(\det {\ensuremath{\mathbf{X}}})^n = (x_1\ldots x_N)^n = P^{(2/\beta)}_{(n^N)}(x_1,\ldots,x_N)$$ and the substitution of the the latter formula into Eq. , one concludes that $$\left\langle (\det {\ensuremath{\mathbf{X}}})^n\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=(-2/\beta)^{nN/2}\left\langle
(\det {\ensuremath{\mathbf{X}}})^N\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_{n}},$$ for $\beta=4/\beta'=1,2,4$. More generally, for all $\beta>0$, $$\left\langle (x_1\cdots x_N)^n\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=(-2/\beta)^{nN/2}\left\langle
(x_1\cdots x_n)^N\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_{n}}.$$
Another simple manifestation of Proposition \[theo1\] is related to the Hermite polynomials. It is well know that an orthogonal polynomial of degree $N$ has a $N\times N$ matrix integral representation. However, a classical orthogonal polynomial can also be realised as a single integral. For instance, the Hermite polynomial of degree $N$ can be written either as $$H_N(t)=2^N \Big\langle \prod_{i=1}^N(t-x_i)\Big\rangle_{x\in \mathrm{G2E}}= 2^N\Big\langle \det(t{\ensuremath{\mathbf{1}}}-{\ensuremath{\mathbf{X}}})\Big\rangle_{{\ensuremath{\mathbf{X}}}\in \mathrm{G2E}}$$ or as $$H_N(t)=\frac{2^N}{\sqrt{\pi}}\int_{-\infty}^\infty
dx\,e^{-x^2}(t\pm\mathrm{i}x)^N.$$ Proposition \[theo1\] allows to go directly from one representation to the other without referring to the orthogonality, determinantal representation, or differential equation. On the one hand, $$\prod_{i=1}^N(t-x_i)=\sum_{n=0}^{N}t^{N-n}(-1)^{n}e_n(x_1,\ldots,x_N)=\sum_{n=0}^{N}t^{N-n}(-1)^{n}P^{(\alpha)}_{(1^n)}(x_1,\ldots,x_N),$$ where $\alpha=2/\beta$, and $$P^{(\alpha)}_{(1^n)}(1^N)=\binom{N}{n},$$ so that $$\Big\langle \prod_{i=1}^N(t-x_i)\Big\rangle_{x\in {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\sum_{n=0}^{N}t^{N-n}(-1)^{n}\binom{N}{n}\left\langle \frac{P^{(\alpha)}_{(1^n)}(x)}{ P^{(\alpha)}_{(1^n)}(1^N)}\right\rangle_{{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}.$$ On the other hand, Proposition 1 and $(1^n)'=(n)$ immediately imply that the last equation is equal to $$\sum_{n=0}^{N}t^{N-n}(-1)^{n}\binom{N}{n}\left\langle \frac{P^{(1/\alpha)}_{(n)}(\mathrm{i}\sqrt{\alpha}x)}{ P^{(1/\alpha)}_{(n)}(1)}\right\rangle_{{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_1}= \sum_{n=0}^{N}t^{N-n}(-1)^{n}\binom{N}{n}\frac{1}{\sqrt{\pi}}\int_\mathbb{R}dxe^{-x^2}\left(\mathrm{i}\sqrt{\alpha}x\right)^n$$ Consequently, for all $\beta=2/\alpha>0$, $$\Big\langle \prod_{i=1}^N(t-x_i)\Big\rangle_{x\in {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^\infty
dx\,e^{-x^2}(t-\mathrm{i}\sqrt{\alpha}x)^N=\left(\frac{\sqrt{\alpha}}{2}\right)^NH_N(t/\sqrt{\alpha})$$
Chiral ensembles
----------------
Lemma \[lemma1\] and Proposition \[theo1\] can be easily adapted to the chiral case.
Let $\mu$ stand for the partition $(1^{|\lambda|})$, i.e., $\ell(\mu)=|\lambda|$. Then, for all partition $\lambda$, $\beta>0$, and $\gamma>-1$, $$\label{EqProof2}
\left\langle {P^{(2/\beta)}_\lambda(x)}\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}=[\gamma+q]^{(2/\beta)}_\lambda\left(\frac{\beta}{2}\right)^{|\lambda|}\frac{b_\lambda^{(2/\beta,N)}}{|\lambda|!}{\ensuremath{{\Big\langle}\!\!{\Big\langle}}}P^{(2/\beta)}_\Lambda\,\Big|\,p_{\mu}{\ensuremath{{\Big\rangle}\!\!{\Big\rangle}}}^{(2/\beta)},$$ where $q=1+\beta(N-1)/2$. Equivalently, $$\left\langle {P^{(2/\beta)}_\lambda(x)}\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}=[\gamma+q]^{(2/\beta)}_\lambda{b_\lambda^{(2/\beta,N)}}\underset{p_\mu}{\mathrm{coeff}}
P^{(2/\beta)}_\Lambda.$$
The orthogonality and Eq. give [@Baker Proposition 4.11] $$\label{eqProof2}
\left\langle{\,_0{\mathcal{F}}_1}^{(\alpha)}(\gamma+q;x;y){\,_0{\mathcal{F}}_1}^{(\alpha)}(\gamma+q;x;z) \right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}}=e^{p_1(y)+p_1(z)}{\,_0{\mathcal{F}}_1}^{(\alpha)}(\gamma+q;y;z)$$ for $\alpha=2/\beta$. Then, one sets $z=0$ and proceeds as in Lemma \[lemma1\].
Employing the same method as that exposed in the proof of Proposition \[theo1\], one shows that the last lemma and the duality for the Jack polynomials imply the following proposition.
\[theo2\] Set ${\beta'}=4/\beta>0$, $q=1+(N-1)\beta/2$, and $q'=1+(N'-1)2/\beta$. Moreover, assume that $N\geq\ell(\lambda)$, $N'\geq \ell(\lambda')$, $\gamma>-1$, and $\gamma'>-1$. Then, $$\label{eqProofTheo2}
\left\langle \frac{P^{(2/\beta)}_\lambda(x)}{P^{(2/\beta)}_\lambda(1^{N})}\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}=\,\frac{[\gamma+q]^{(2/\beta)}_\lambda}{[\gamma'+q']^{(2/\beta')}_{\lambda'}}\,
\left\langle
\frac{P^{(2/\beta')}_{\lambda'}(x)}{P^{(2/\beta')}_{\lambda'}(1^{N'})}\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta'\!\!\;\mathrm{E}}}^{\gamma'}_{N'}}.$$
As an example, choose $\lambda=(n^N)$ and $\gamma'=2(\gamma+1)/\beta-1$. Direct manipulations then give $$\frac{{[\gamma+q]^{(2/\beta)}_\lambda}}{{[\gamma'+q']^{(2/\beta')}_{\lambda'}}}=\left(\frac{\beta}{2}\right)^{nN}.$$ This implies that moments of the determinant also satisfy a simple duality in the Chiral $\beta$-Ensemble: $$\
\left\langle (x_1\cdots x_N)^n\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}=\left(\frac{\beta}{2}\right)^{nN}\left\langle (x_1\cdots x_n)^N\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta'\!\!\;\mathrm{E}}}^{\gamma'}_N}.$$
It is worth mentioning that the average in the Gaussian ensembles are limit cases of similar averages in the Chiral ensembles. This can be understood as follows. Set $y=\gamma+\sqrt{2\gamma}x$ with $\gamma>0$. Trivial manipulations give $$\gamma^{-1}e^\gamma y^\gamma e^{-y}=e^{-x^2}+\mathcal{O}(\gamma^{-1/2})$$ uniformly when $\gamma$ goes to $\infty$. Then, simple changes of variable allows one to conclude that $$\lim_{\gamma\rightarrow \infty} \left\langle F(-\sqrt{\gamma/2}+y/\sqrt{2\gamma})\right\rangle_{y\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N}=\left\langle F(x)\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}$$ for all multivariate polynomials $F$, positive integers $N$ and $\beta>0$. For instance, by setting $\gamma'=\gamma$ and taking the above limit in both sides of Proposition \[theo2\], one readily establishes Proposition \[theo1\].
Source – external field dualities
=================================
In this section, averages of (inverse) characteristic polynomials in $\beta$-Ensembles with external fields are studied. Central to the analysis is the use of Dunkl transforms (see for instance [@Rosler]).
Gaussian ensembles
------------------
The following formula is a generalisation of the Fourier transformation [@Baker; @Rosler]: $$\label{eqFourier}
e^{-p_2(y)} \left\langle{\,_0{\mathcal{F}}_0}^{(2/\beta)}(2x;y)F(x)\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}}=e^{\frac{1}{4}\Delta^{(2/\beta)}_y}F(y)$$ where $$\Delta^{(\alpha)}_x=\sum_{i=1}^N\left(\frac{\partial}{\partial {x_i}}+\frac{2}{\alpha}\sum_{j\neq i}\frac{1}{x_i-x_j}\right)\frac{\partial}{\partial {x_i}}.$$ The equation holds true for any analytic function $F$. It can be seen as a consequence of Eq. and the Lassalle formula $$\label{eqLassalle}
\mathcal{H}_\lambda^{(\alpha)}(x)=e^{-\frac{1}{4}\Delta_x^{(\alpha)}}P^{(\alpha)}_\lambda(x).$$
Now imagine that there exists a function $G(s;f)$ of two sets of variables, $s=(s_1,\ldots,s_n)$ and $f=(f_1,\ldots,f_N)$, satisfying for some $\alpha'$ $$\Delta_f^{(\alpha)}G(s;f)=\Delta_s^{(\alpha')}G(s;f).$$ Due to the commutativity of $\Delta_f^{(\alpha)}$ and $\Delta_s^{(\alpha')}$, this means that $$e^{\frac{1}{4}\Delta_f^{(\alpha)}}G(s;f)=e^{\frac{1}{4}\Delta_s^{(\alpha')}}G(s;f).$$ Therefore, as a direct consequence of the latter formula and Eq. , $$\begin{gathered}
e^{-p_2(f)} \left\langle{\,_0{\mathcal{F}}_0}^{(2/\beta)}(2x;f)G(s;x)\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}}= e^{-p_2(s)} \left\langle{\,_0{\mathcal{F}}_0}^{(2/\beta')}(2y;f)G(y;f)\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}}\end{gathered}$$ where it is understood that $\beta'=2/\alpha'$.
\[prop3\] Set $\beta'=4/\beta$. Then, for all positive integers $n$ and $N$, and all $\alpha>0$, $$\begin{gathered}
\label{eq1prop3}
e^{-p_2(f)} \left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j\pm \mathrm{i}\sqrt{\frac{2}{\beta}}x_k\right) \,_0{\mathcal{F}_0}^{(2/\beta)}(x,2f) \right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N} = \\
e^{-p_2(s)} \left\langle \prod_{j=1}^n\prod_{k=1}^N \left(y_j\pm \mathrm{i}\sqrt{\frac{2}{\beta}}f_k\right) \,_0{\mathcal{F}_0}^{(2/\beta')}(y,2s)\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_n}.\end{gathered}$$ If in addition the variables $s$ and $f$ are not real, then $$\begin{gathered}
\label{eq2prop3}
e^{-p_2(f)} \left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j\pm x_k\right)^{-\beta/2} \,_0{\mathcal{F}_0}^{(2/\beta)}(x,2f) \right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N} = \\
e^{-p_2(s)} \left\langle \prod_{j=1}^n\prod_{k=1}^N \left(y_j\pm f_k\right)^{-\beta/2} \,_0{\mathcal{F}_0}^{(2/\beta)}(y,2s)\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_n}.\end{gathered}$$ Remark that in the last equation, $\beta$ is not affected by the duality transformation.
Recall that $\alpha=2/\beta$ and let $$\prod(s;af)^b=\prod_{j=1}^n\prod_{k=1}^N\left(s_j-af_k\right)^b.$$ Following the above discussion on the Dunkl transform, it is sufficient prove that $$\Delta^{(\alpha)}_f \prod(s;af)^b= \Delta^{(\alpha')}_s \prod(s;af)^b$$ for appropriate values of $a$, $b$, and $\alpha'$. On the one hand, by using $$\frac{\partial}{\partial f_k} \prod(s;af)^b=-ab\sum_i\frac{1}{s_i-af_k} \prod(s;af)^b$$ and $$\sum_{i}\sum_{k\neq l}\frac{1}{s_i-af_k}\frac{1}{f_k-f_l}=\frac{a}{2}\sum_i\sum_{k\neq l}\frac{1}{s_i-af_k}\frac{1}{s_i-af_l},$$ it is simple to show that $$\begin{gathered}
\label{eqDeltaf}
\Delta^{(\alpha)}_f \prod(s;af)^b=\sum_{i,k}\frac{1}{s_i-af_k}\left( a^2b(b-1)\frac{1}{s_i-af_k}\phantom{\frac{a^2b}{\alpha}} \right. \\
\left.+a^2b^2\sum_{j\neq i}\frac{1}{s_j-af_k}
-\frac{a^2b}{\alpha}\sum_{l\neq k}\frac{1}{s_i-af_l}\right)\prod(s;af)^b\end{gathered}$$ On the other hand, exploiting $$\frac{\partial}{\partial s_k} \prod(s;af)^b= b\sum_k\frac{1}{s_i-af_k} \prod(s;af)^b$$ and $$\sum_{i\neq j}\sum_{k}\frac{1}{s_i-af_k}\frac{1}{s_i-s_j}=-\frac{1}{2}\sum_{i\neq j}\sum_{k}\frac{1}{s_i-af_k}\frac{1}{s_j-af_k},$$ one finds $$\begin{gathered}
\label{eqDeltas}
\Delta^{(\alpha')}_s \prod(s;af)^b=\sum_{i,k}\frac{1}{s_i-af_k}\left( b(b-1)\frac{1}{s_i-af_k}\phantom{\frac{a^2b}{\alpha}} \right. \\
\left.-\frac{b}{\alpha'}\sum_{j\neq i}\frac{1}{s_j-af_k}
+b^2\sum_{l\neq k}\frac{1}{s_i-af_l}\right)\prod(s;af)^b\end{gathered}$$ Note that $s$ and $f$ are kept generic. Thus, imposing the equality of Eqs. and requires $$a^2(b-1)=(b-1),\quad \alpha' a^2b=-1, \quad a^2=-b\alpha.$$ The only solutions to the latter system of equations are either $$a=\pm \mathrm{i}\sqrt{\alpha},\quad b=1,\quad \alpha'=1/\alpha.$$ or $$a=\pm1,\quad b=-1/\alpha,\quad \alpha'=\alpha.$$ This completes the proof of the proposition.
Consider for example the expectation value of a product of characteristic polynomials without external field, i.e., for $f=0$. According to Proposition \[prop3\], $$\left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j\pm \sqrt{\frac{2}{\beta}}x_k\right) \right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N} =
e^{p_2(s)} \left\langle \prod_{j=1}^n (\mathrm{i}y_j)^N \,_0{\mathcal{F}_0}^{(2/\beta')}(2y,-\mathrm{i}s)\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_n}.$$ The right-hand side can be written as $$e^{p_2(s)} \left\langle P^{(\beta/2)}_{(N^n)}(\mathrm{i}y)\,_0{\mathcal{F}_0}^{(2/\beta')}(2y,\mathrm{i}s)\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_n}.$$ Due to the Dunkl transform , the latter equation is also equal to $\exp\left(-\frac{1}{4}\Delta^{(\beta/2)}_s\right)P^{(\beta/2)}_{(N^n)}(s)$. Returning to the Lassalle formula , it becomes clear that averages of products of characteristic polynomials are multivariate Hermite polynomials: $$\label{eqHprod}
\mathcal{H}^{(\beta/2)}_{(N^n)}(s)= \left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j\pm\sqrt{\frac{2}{\beta}}x_k\right) \right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}.$$ This equivalence was first pointed out in [@Baker] by considering a limit of the multivariate Jacobi polynomial. Another proof consists in showing that the expectation value satisfies a Calogero-like eigenvalue problem.
Propositions \[prop1\] and \[prop2\] given in the Introduction are in fact special cases of Proposition \[prop3\]. The connection between the averages involving multivariate hypergeometric functions and the matrix integrals can be understood as follows. Let $\beta=1,2,$ or $4$, and let $({\ensuremath{\mathbf{U}}}^\dagger d{\ensuremath{\mathbf{U}}})$ stand for the normalised Haar measure for unitary matrices with real, complex or quaternion real elements respectively. It is a standard result [@ForresterBook; @Mehta] that $$(d{\ensuremath{\mathbf{X}}})\,F({\ensuremath{\mathbf{X}}})=\frac{1}{C} \prod_{i=1}^N dx\prod_{1\leq i<j\leq N}|x_i-x_j|^\beta ({\ensuremath{\mathbf{U}}}^\dagger d{\ensuremath{\mathbf{U}}})F({\ensuremath{\mathbf{U}}}x{\ensuremath{\mathbf{U}}}^\dagger).$$ for any Hermitian $\beta$-matrix ${\ensuremath{\mathbf{X}}}={\ensuremath{\mathbf{U}}}x {\ensuremath{\mathbf{U}}}^\dagger$, where $x=\mathrm{diag} (x_1,\ldots,x_N)$, and some constant $C$. When $F$ is not invariant, i.e., when $F({\ensuremath{\mathbf{U}}}x{\ensuremath{\mathbf{U}}}^\dagger)\neq F(x)$, the calculation of the expectation values containing $F$ requires the use of the theory of zonal polynomials (see Chapter VII in Macdonald’s classical book [@Mac]). One can show for instance that, if ${\ensuremath{\mathbf{X}}}$ and ${\ensuremath{\mathbf{Y}}}$ are $N\times N$ Hermitian $\beta$-matrices, $$\int ({\ensuremath{\mathbf{U}}}^\dagger d{\ensuremath{\mathbf{U}}}) P^{(2/\beta)}_\lambda ({\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{U}}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{U}}}^\dagger)=\frac{P^{(2/\beta)}_\lambda(x) P^{(2/\beta)}_\lambda(y)}{P^{(2/\beta)}_\lambda(1^N)}.$$ In the last equation, $P^{(2/\beta)}_\lambda ({\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{U}}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{U}}}^\dagger)=P^{(2/\beta)}_\lambda (z_1,\ldots,z_N)$, where $(z_1,\ldots,z_N)$ denotes the eigenvalues of ${\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{U}}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{U}}}^\dagger$. Now, by exploiting formula , one readily gets $$\label{angularint}
\int ({\ensuremath{\mathbf{U}}}^\dagger d{\ensuremath{\mathbf{U}}}) e^{{\ensuremath{\,\mathrm{tr}\,}}({\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{U}}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{U}}}^\dagger)}=\,_0{\mathcal{F}_0}^{(2/\beta)}(x;y),$$ and consequently, for any invariant function $g({\ensuremath{\mathbf{X}}})=g(x)$ and $\beta=1, 2$ or $4$, $$\begin{gathered}
\left\langle \,_0{\mathcal{F}_0}^{(2/\beta)}(x;f) g(x)\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}=\\
{\int (d{\ensuremath{\mathbf{X}}})e^{-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^2+{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{F}}}}g({\ensuremath{\mathbf{X}}})}\Big/{\int (d{\ensuremath{\mathbf{X}}})e^{-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^2}}= \left\langle e^{{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{F}}}} g({\ensuremath{\mathbf{X}}})\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}.\end{gathered}$$ Finally, the comparison of the latter formula with Eqs. and respectively establishes Propositions \[prop1\] and \[prop2\].
Chiral ensembles
----------------
There exists a Lassalle formula for the multivariate Laguerre polynomials. For $\alpha=2/\beta>0$ and $\gamma>-1$, it reads [@Baker] $$\label{eqLassalle2}
\mathcal{L}_\lambda^{(\alpha,\gamma)}(x)=e^{-\Delta_x^{(\alpha,\gamma)}}P^{(\alpha)}_\lambda(x),$$ where $$\label{Delta}
\Delta^{(\alpha,\gamma)}_x=\sum_{i=1}^N\left(x_i\frac{\partial}{\partial {x_i}}+\frac{2}{\alpha}\sum_{j\neq i}\frac{x_i}{x_i-x_j}+\gamma+1\right)\frac{\partial}{\partial {x_i}}.$$ From formula and Eq. , one can prove the following generalisation of the Hankel transform which is due to Dunkl [@Baker; @Rosler] : $$\label{eqHankel}
e^{p_1(y)} \left\langle {\,_0{\mathcal{F}}_1}^{(2/\beta)}(\gamma+q;x;-y)F(-x) \right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}_N^\gamma}=e^{-\Delta^{(2/\beta,\gamma)}_y}F(y)$$ with and $q=1+\beta(N-1)/2$. The strategy for getting dualities in the Chiral $\beta$-Ensemble is the same as in the Gaussian $\beta$-Ensemble; that is, find a function $G(s;f)$ such that $$\Delta_f^{(\alpha,\gamma)}G(s;f)=\Delta_s^{(\alpha',\gamma')}G(s;f),$$ and exploit the Dunkl transform in order to conclude that $$\begin{gathered}
e^{p_1(f)} \left\langle{\,_0{\mathcal{F}}_1}^{(2/\beta)}(\gamma+q;x;-f)G(s;-x)\right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}_N^\gamma}= \\
e^{p_1(s)} \left\langle{\,_0{\mathcal{F}}_1}^{(2/\beta')}(\gamma'+q';y;-s)G(-y;f)\right\rangle_{y\in{\ensuremath{\mathrm{ch}\:\!\!\beta'\!\!\;\mathrm{E}}}^{\gamma'}}\end{gathered}$$ for some $\beta'$, $\gamma'$, and $q'=1+\beta'(n-1)/2$.
\[prop4\] Set $\beta'=4/\beta$, $\gamma'=2(\gamma+1)/\beta-1$. Then, for all positive integers $n$ and $N$, all $\alpha>0$ and $\gamma>-1$, $$\begin{gathered}
\label{eq1prop4}
e^{p_1(f)} \left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j-{\frac{2}{\beta}}x_k\right) \,_0{\mathcal{F}_1}^{(2/\beta)}(\gamma+q;x,-f) \right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}_N^\gamma} = \\
(-1)^{nN}e^{p_1(s)} \left\langle \prod_{j=1}^n\prod_{k=1}^N \left(y_j-{\frac{2}{\beta}}f_k\right) \,_0{\mathcal{F}_1}^{(2/\beta')}(\gamma'+q';y,-s)\right\rangle_{y\in{\ensuremath{\mathrm{ch}\:\!\!\beta'\!\!\;\mathrm{E}}}_n^{\gamma'}}.\end{gathered}$$ Suppose moreover that $s$ and $f$ have non-zero imaginary parts, then $$\begin{gathered}
\label{eq2prop4}
e^{p_1(f)} \left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j+ x_k\right)^{-\beta/2} \,_0{\mathcal{F}_1}^{(2/\beta)}(\gamma+q;x,-f) \right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}_N^\gamma} = \\(-1)^{\beta nN/2}
e^{p_1(s)} \left\langle \prod_{j=1}^n\prod_{k=1}^N \left(y_j+ f_k\right)^{-\beta/2} \,_0{\mathcal{F}_1}^{(2/\beta)}(\gamma'+q';y,-s)\right\rangle_{y\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_n^{\gamma'}}.\end{gathered}$$ The latter equation is valid only if $\beta>2\gamma$, which ensures $\gamma'>-1$.
Define $$\prod(s;af)^b=\prod_{j=1}^n\prod_{k=1}^N\left(s_j-af_k\right)^b.$$ Direct calculations lead to the conclusion that $$\Delta^{(\alpha,\gamma)}_f \prod(s;af)^b= \Delta^{(\alpha',\gamma')}_s \prod(s;af)^b$$ for all $n$ and $N$, if and only if, $$ab\alpha'=-1,\quad a(b-1)=(b-1),\quad a(\gamma+1)=-(\gamma'+b),\quad a=-b\alpha.$$ The proposition follows from the fact that the unique solutions to these equations are $$a=-\alpha,\quad b=1,\quad \alpha'=1/\alpha, \quad \gamma'=\alpha(\gamma+1)-1$$ and $$a=1,\quad b=-1/\alpha,\quad \alpha'=\alpha, \quad \gamma'=1/\alpha -\gamma-1.$$
The comparison of Eqs. , , and leads to a simple generalisation of Eq. , that is, a random matrix representation of the multivariate Laguerre polynomial: $$\mathcal{L}^{(2/\beta',\gamma')}_{(N^n)}(s)= \left\langle \prod_{j=1}^n\prod_{k=1}^N\left(s_j-{\frac{2}{\beta}}x_k\right) \right\rangle_{x\in{\ensuremath{\mathrm{ch}\:\!\!\beta\!\!\;\mathrm{E}}}^\gamma_N},$$ where $\beta'=4/\beta$ and $\gamma'=2(\gamma+1)\beta-1$.
Further applications
====================
Multiple polynomials
--------------------
Ensembles of random Hermitian matrices with external field naturally lead to multiple polynomials when $\beta=2$. These polynomials satisfy orthogonality conditions for more than one scalar products; they can be interpreted as multi-parameter deformations of the usual orthogonal polynomials in one variable (see the review by van Assche in [@vAssche]).
It is known that the multiple Hermite polynomials of type II can be defined as follows: [^4] $$\label{mHermiteND}
\mathcal{H}_\mathbf{n}(z)=e^{-\frac{1}{4}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{F}}}^2}\left\langle \det(z{\ensuremath{\mathbf{1}}}-{\ensuremath{\mathbf{X}}})e^{{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{F}}}}\right\rangle_{{\ensuremath{\mathbf{X}}}\in \mathrm{GUE}_N}$$ Here the polynomial is chosen monic of degree $N$ while $\mathbf{n}=(n_1,\ldots,n_D)$ is the multiplicity vector for the eigenvalues of ${\ensuremath{\mathbf{F}}}$. In other words, ${\ensuremath{\mathbf{F}}}$ has $n_i$ eigenvalues equal to $g_i$ say, and $\sum_in_i=N$. One can write for instance $$f=g^\mathbf{n}=(g_1^{n_1},\ldots,g_D^{n_D}).$$ As a consequence of Proposition 1, it is clear that the multiple Hermite polynomial also has the following realisation: $$\label{mHermite1D}
\mathcal{H}_\mathbf{n}(z)=\frac{(-\mathrm{i})^N}{\sqrt{\pi}}\int_{\mathbb{R}} dy\,e^{-(y-\mathrm{i}z)^2}\prod_{j=1}^D(y-\mathrm{i}g_j/2)^{n_i}.$$ This formula has been first obtained by Bleher and Kuijlaars in [@Bleher] from the orthogonality relations satisfied by $P_{\mathbf{n}}$.
There is another interesting application of Proposition 1. Indeed, the comparison of Eqs. and indicates that the multiple Hermite polynomials can be represented as $\beta$-matrix integrals $$\mathcal{H}_\mathbf{n}(z)=e^{-\frac{1}{4}\frac{\beta}{2}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{F}}}^2}\left\langle \det\left(z{\ensuremath{\mathbf{1}}}-\sqrt{\frac{2}{\beta}}{\ensuremath{\mathbf{X}}}\right)e^{\sqrt{\frac{\beta}{2}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{F}}}}\right\rangle_{{\ensuremath{\mathbf{X}}}\in {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}$$ for $\beta=1,2,4$, or as $$\mathcal{H}_\mathbf{n}(z)=e^{-\frac{1}{4}\frac{\beta}{2}p_2(f)}\left\langle \prod_{j=1}^N\left(z-\sqrt{\frac{2}{\beta}}x_j\right)\,_0{\mathcal{F}_0}^{(2/\beta)}\left(x;\sqrt{\frac{\beta}{2}}f\right) \right\rangle_{x\in {\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}$$ for all $\beta>0$.
The multiple Hermite function of type I can also be written as a single integral [@Bleher]. $$Q_{\mathbf{n}}(z)=\lim_{\epsilon\rightarrow 0^+}\frac{1}{\pi}\Im \int_\mathbb{R} dy\,e^{-(y-z+\mathrm{i}\epsilon)}\prod_{j=1}^D(y-f_j/2)^{n_j} .$$ Note that taking the imaginary part is equivalent to closing the contour around the poles. Now suppose that $\beta/2$ is a positive integer and that $n_i=\beta m_i/2$ for all $i=1,\ldots,D$. In addition, construct a reduced sequence of variables (only the multiplicities are changed) $$\tilde{f}=g^\mathbf{m}=(g_1^{m_1},\ldots,g_D^{m_D}).$$ and set $M=\sum_j m_j=2N/\beta$. Then Proposition 2 furnishes new integral representations for the multiple Hermite functions of type II: $$Q_{\mathbf{n}}(z)=e^{-\frac{1}{4}p_2(\tilde{f})} \lim_{\epsilon\rightarrow 0^+}\frac{1}{\pi}\Im
\left\langle \prod_{j=1}^M(z-\mathrm{i}\epsilon-x_j)^{-\beta/2}\,_0{\mathcal{F}_0}^{(2/\beta)}(x;\tilde{f})\right\rangle_{x\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_M}.$$
Similar representations can be deduced for the multiple Laguerre polynomials of type I and II by exploiting the results given in Section 4.2.
Formal one-matrix models
------------------------
One point about matrix models encountered in theoretical physics and combinatorics deserves to be clarified.
Matrix integrals are often used for solving combinatorial problems related to 2D Quantum Gravity, such as counting graphs drawn on surfaces of a given genus (for a simple but precise introduction to the subject, see [@Zvonkin]; more advanced topics and recent references can be found in [@DiFrancesco]). Questions of convergence of the integrals are not an issue when enumerating objects; only formal power series are considered. One typically looks at functions involving traces of a random matrix averaged over the Gaussian ensemble of Hermitian matrices (usually $\beta=2$). Averages are obtained from the partition function of the one-matrix model: $$\label{FormalPart}
Z_{N,\beta}(t_1,t_2\ldots)=\int (d{\ensuremath{\mathbf{X}}}) \exp\left( -\sum_{k\geq 1} \frac{t_k}{k}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^k\right).$$
For $\beta=2$, Chekov and Makeenko [@Chekov] have shown, by exploiting Schwinger-Dyson equations, the equivalence of the one-matrix model to the Gaussian model with an external field. In fact, this remains true for $\beta=1$ and $4$. It can be proved quickly. First, set $$\label{eqt}
t_1=-2u+v{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^{-1},\quad t_2=2+v^2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^{-2},\quad\mbox{and}\quad t_k= v^k{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^{-k} \quad\mbox{for}\quad k\geq3,$$ where u is a formal parameter, $v=\pm\mathrm{i}\sqrt{2/\beta}$, and ${\ensuremath{\mathbf{S}}}$ is a $n \times n$ Hermitian or anti-Hermitian whose eigenvalues $s$ are non-zero (so, $t_k$ is essentially a $k$th power sum). Then, the formal development of $\sum_{k\geq 1} {t_k}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^k/k$ in terms of the eigenvalues of ${\ensuremath{\mathbf{S}}}$ yields $$\label{eqFormalProd}
\exp\left( -\sum_{k\geq 1} \frac{t_k}{k}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^k\right)=(\det {\ensuremath{\mathbf{S}}})^{-N}\prod_{j=1}^n\det(s_j{\ensuremath{\mathbf{1}}}-v{\ensuremath{\mathbf{X}}})\exp\left( -{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^2+2u{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}\right).$$ Finally, from the substitution of the last equation in Eq. and the comparison with Proposition 1, it holds that the formal one-matrix model is equivalent to a Gaussian model with an external field.
\[propFormal\] Let ${\ensuremath{\mathbf{X}}}$ be a $N\times N$ Hermitian $\beta$-matrix. Moreover, suppose that $\beta'=4/\beta$ and that ${\ensuremath{\mathbf{S}}}$ is a $n\times n$ Hermitian $4/\beta$-matrix satisfying Eq. . Then $$Z_{N,\beta}(t_1, t_2,\ldots)=
z_{N,\beta} \frac{e^{-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2+Nu^2}}{\det
{\ensuremath{\mathbf{S}}}^{N}}\left\langle\det\left({\ensuremath{\mathbf{Y}}}-\mathrm{i}\sqrt{\frac{\beta}{2}}u{\ensuremath{\mathbf{1}}}\right)^Ne^{2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{S}}}}\right\rangle_{{\ensuremath{\mathbf{Y}}}\in\mathrm{G}\beta'\mathrm{E}_n},$$ where $$z_{N,\beta}=\int (d{\ensuremath{\mathbf{X}}}) e^{-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^2}= \pi^{N/2}(\pi/2)^{\beta N(N-1)/4}.$$
The latter result remains true for all $\beta>0$ if one appropriately uses the generalised hypergeometric functions $\,_0\mathcal{F}_0(2y;s)$ in place of $e^{2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{S}}}}$ (cf. Section 4.1).
Matrices at the edge of the spectrum
------------------------------------
Consider a random matrix belonging to the ${\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N$ where $\beta=1,2$ or $4$. Suppose moreover that the size $N$ of the matrix goes to infinity. Then, a classical result [@ForresterEdge] says that that the eigenvalue correlation functions, when recentered and rescaled at the spectrum edge (i.e., $x_i\sim \sqrt{2N}$), can be expressed in terms of determinants (or Pfaffians) involving the Airy function and its derivative. Here it is proved that the average of products of characteristic polynomials can be expressed, at the edge of the spectrum, as a matrix Airy integral. For $\beta=2$, this integral has been first introduced by Kontsevich in [@Kontsevich] when studying the asymptotic behaviour of the one-matrix model’s partition function .
Let ${\ensuremath{\mathbf{W}}}$ and ${\ensuremath{\mathbf{F}}}$ be $N\times N$ Hermitian $\beta$-matrices. The matrix integral of the Kontsevich type is defined as follows: $$\label{matrixAiry}
\mathrm{Ai}^{(2/\beta)}({\ensuremath{\mathbf{F}}})=a^{(2/\beta)}_N\int (d{\ensuremath{\mathbf{W}}}) \exp \left( \frac{\mathrm{i}}{3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^3+\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}{\ensuremath{\mathbf{F}}}\right),$$ where $a^{(\alpha)}_N=({2\pi})^{-N- N(N-1)/\alpha}$. Obviously, the standard Airy function is recovered for $N=1$. Recall that the latter function satisfies $\mathrm{Ai}''(x)=x\mathrm{Ai}(x)$. From the derivation of Eq. and the use of integration by parts, a simple generalisation of the Airy differential equation is obtained: $$\Delta_{\ensuremath{\mathbf{F}}}\, \mathrm{Ai} ({\ensuremath{\mathbf{F}}})= {\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{F}}}\,\mathrm{Ai} ({\ensuremath{\mathbf{F}}}).$$ In the last equation, $\Delta_{\ensuremath{\mathbf{F}}}$ stands for the Laplacian of the matrix ${\ensuremath{\mathbf{F}}}$; that is, $$\Delta_{{\ensuremath{\mathbf{F}}}}=\sum_{i=1}^N \left(\frac{\partial}{\partial {F^0_{ii}}}\right)^2+\frac{1}{2}\sum_{a=0}^{\beta-1}\sum_{1\leq i<j\leq N}\left(\frac{\partial}{\partial {F^a_{ij}}}\right)^2.$$
In fact, the matrix integral is a special realisation of a more general multivariate Airy function (see Eq. ), $$\label{generalAiry}
\mathrm{Ai}^{(\alpha)}(f)=a^{(\alpha)}_N \int_{\mathbb{R}^N}e^{\frac{\mathrm{i}}{3}\sum_jw_j^3} {\,_0\mathcal{F}_0}^{(\alpha)}(w;\mathrm{i}f) \prod_{1\leq j<k\leq N} |w_j-w_k|^{2/\alpha} \, dw ,$$ which is valid for any $\alpha=2/\beta>0$. When $f_1=\ldots=f_N=x$ say, the latter function is equivalent to that previously introduced in [@DF] as the limiting eigenvalue density at the soft edge for the Chiral and Gaussian $\beta$-Ensembles ($\beta$ even). For general $f=(f_1,\ldots,f_N)$ and $\alpha>0$, one has $$\Delta^{(\alpha)}_f\, \mathrm{Ai}^{(\alpha)}(f)= p_1(f)\, \mathrm{Ai}^{(\alpha)}(f),$$ where $ \Delta^{(\alpha)}_f$ is the differential operator defined in Eq. and $p$ is a power sum (see Section 2.2). This multivariate Airy differential equation follows from the use of the Calogero-like equation [@Baker] $$\Delta^{(\alpha)}_x \,_0\mathcal{F}_0(x;y)=p_2(y)\, \,_0\mathcal{F}_0(x;y)$$ and simple manipulations in the integral of Eq. .
According to Eq. , a multivariate Hermite polynomial associated to a rectangular partition is equivalent to an average of products of characteristic polynomials. By virtue of Proposition \[prop1\], the latter quantity can be replaced by the expectation value of products of determinants in an ensemble of random matrices with an external field. Explicitly, $$\begin{gathered}
\label{eqHermiteMat}
\mathcal{H}^{(\beta/2)}_{(N^n)}(s)=\left\langle
\prod_{i=1}^n\det\left(s_j{\ensuremath{\mathbf{1}}}\pm\sqrt{\frac{2}{\beta}}{\ensuremath{\mathbf{X}}}\right)\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}\\
=e^{{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2}\left\langle (\det \mathrm{i}{\ensuremath{\mathbf{Y}}})^Ne^{-2\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{S}}}}\right\rangle_{{\ensuremath{\mathbf{Y}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta'\!\!\;\mathrm{E}}}_n}\end{gathered}$$ if $\beta'=4/\beta=1,2$ or $4$. The last integral representation suits perfectly for the large $N$ asymptotic analysis. The following result generalises the well known asymptotic expansion of the Hermite polynomial $H_N(x)$ in terms of the Airy function.
\[propEdge\] Let $G_{N,\beta}({\ensuremath{\mathbf{S}}})$ denote the first line of Eq. , where ${\ensuremath{\mathbf{S}}}$ is a Hermitian $4/\beta$-matrix whose eigenvalues are given by $s=(s_1,\ldots,s_n)$. Then, as $N\rightarrow\infty$, $$\label{eqPropEdge}
C^{-1}e^{-N^{1/3}\,{\ensuremath{\,\mathrm{tr}\,}}\, {\ensuremath{\mathbf{S}}}}\,G_{N,\beta}\left(\sqrt{2N}{\ensuremath{\mathbf{1}}}+\frac{1}{\sqrt{2N^{1/3}}}{\ensuremath{\mathbf{S}}}\right)\sim \mathrm{Ai}^{(\beta/2)}({\ensuremath{\mathbf{S}}})+\mathcal{O}\left(\frac{1}{N^{1/3}}\right).$$where $$\label{eqC}
C=2^{n(n-1)/\beta}\left(\frac{Ne}{2}\right)^{nN/2}(2\pi N^{1/3})^{n/2+n(n-1)/\beta}.$$
The correct proof of the proposition is given in Appendix B. It is purely technical and relies on the multidimensional steepest descent method. However, the asymptotic formula can be easily understood thanks to the following heuristic argument. First, by rescaling the matrix ${\ensuremath{\mathbf{Y}}}$ in Eq. , one finds $$\label{eqGMatrix}
G_{N,\beta}\left(\sqrt{2N}{\ensuremath{\mathbf{1}}}+\frac{1}{\sqrt{2N^{1/3}}}{\ensuremath{\mathbf{S}}}\right)=D\int (d{\ensuremath{\mathbf{Y}}}) \exp\left( Nf({\ensuremath{\mathbf{Y}}})+N^{1/3}g({\ensuremath{\mathbf{Y}}},{\ensuremath{\mathbf{S}}})\right)$$ where $$f({\ensuremath{\mathbf{Y}}})={\ensuremath{\,\mathrm{tr}\,}}\left(-2{\ensuremath{\mathbf{Y}}}^2-4\mathrm{i}{\ensuremath{\mathbf{Y}}}+\ln {\ensuremath{\mathbf{Y}}}\right),\quad g({\ensuremath{\mathbf{Y}}},{\ensuremath{\mathbf{S}}})=-2{\ensuremath{\,\mathrm{tr}\,}}\left({\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{S}}}+i{\ensuremath{\mathbf{S}}}\right)$$ and $$D=\frac{(-2N)^{nN/2}(2N)^{n/2+n(n-1)/\beta}}{\pi^{n/2}(\pi/2)^{n(n-1)/\beta}}e^{\frac{1}{2N^{1/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2}$$ Second, the following change of variables is made in order to eliminate the quadratic terms in $f({\ensuremath{\mathbf{Y}}})$: $$\label{changeW}
{\ensuremath{\mathbf{W}}}=2N^{1/3}\left({\ensuremath{\mathbf{Y}}}+\frac{\mathrm{i}}{2}{\ensuremath{\mathbf{1}}}\right).$$ Third, the function $f({\ensuremath{\mathbf{Y}}})$ is formally expanded in powers of $N^{-1/3}$. This yields $$\begin{gathered}
\label{EqHeuristic}
C^{-1} e^{-N^{1/3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}-\frac{1}{2N^{1/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2} G_{N,\beta}\left(\sqrt{2N}{\ensuremath{\mathbf{1}}}+\frac{1}{\sqrt{2N^{1/3}}}{\ensuremath{\mathbf{S}}}\right)=\\
\int (d{\ensuremath{\mathbf{W}}})\exp \left( \frac{\mathrm{i}}{3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^3+\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}{\ensuremath{\mathbf{S}}}\right)\exp\left(-\sum_{k\geq 4}\frac{\mathrm{i}^k}{kN^{(k-3)/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^k\right).\end{gathered}$$ Clearly, the last line is similar to the matrix Airy integral plus a correction of order less or equal to $N^{-1/3}$. However, the preceding approach is not rigorous: it doesn’t proves that Eq. furnishes the [*main contribution*]{} to the integral when $N$ is large; it doesn’t explain why ${\ensuremath{\mathbf{W}}}$ should be considered as a [*Hermitian*]{} $4/\beta$-matrix.
Proposition \[propEdge\] only provides the dominant term to the average of products of characteristic polynomials, or equivalently to multivariate Hermite polynomial, evaluated at the edge. One can nevertheless find an infinite (but not convergent) asymptotic series by exploiting Eq. . Note that the latter equation remains true when considering the steepest descent method used in Appendix B. Recall the well known identity (see e.g. [@Mac Chapter I]) $$\exp\left({\sum_{n\geq1}\frac{1}{n}v_n}\right)=\sum_\lambda \frac{1}{z_\lambda}v_\lambda,$$ where the sum is taken over all partitions $\lambda$, $ v_\lambda=v_{\lambda_1}\cdots v_{\lambda_\ell}$, and $z_\lambda$ is the quantity defined just above Eq. . The use of the latter formula in Eq. leads to $$\begin{gathered}
\label{EqHeuristic}
C^{-1} e^{-N^{1/3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}-\frac{1}{2N^{1/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2} G_{N,\beta}\left(\sqrt{2N}{\ensuremath{\mathbf{1}}}+\frac{1}{\sqrt{2N^{1/3}}}{\ensuremath{\mathbf{S}}}\right)=\\
\int (d{\ensuremath{\mathbf{W}}})\exp \left( \frac{\mathrm{i}}{3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^3+\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}{\ensuremath{\mathbf{F}}}\right)\sum_\lambda \frac{\mathrm{i}^{3\ell(\lambda)-|\lambda|}}{N^{|\lambda|/3}}\frac{1}{z_{\lambda_+}}p_{\lambda_+}({\ensuremath{\mathbf{W}}}).\end{gathered}$$ In the last equation, $\lambda_+=(\lambda_1+3,\ldots,\lambda_\ell+3)$ and $p_{\lambda}({\ensuremath{\mathbf{W}}})={\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^{\lambda_1}\cdots{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^{\lambda_\ell}$. Now, let $\mathbf{T}$ stand for an arbitrary $n\times n$ matrix with no symmetry property (i.e., $n^2$ independent elements). Then $$p_k(\partial_\mathbf{T})e^{\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}\mathbf{T}}:={\ensuremath{\,\mathrm{tr}\,}}(\partial_\mathbf{T})^k e^{\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}\mathbf{T}}= {\ensuremath{\,\mathrm{tr}\,}}(\mathrm{i}\mathbf{W})^ke^{\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}\mathbf{T}}$$ and the next proposition follows.
With the above notation, one has formally $$\begin{gathered}
C^{-1} e^{-N^{1/3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}-\frac{1}{2N^{1/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2}\,G_{N,\beta}\left(\sqrt{2N}{\ensuremath{\mathbf{1}}}+\frac{1}{\sqrt{2N^{1/3}}}{\ensuremath{\mathbf{S}}}\right)=\\
\left[\sum_{\lambda}\frac{(-1)^{|\lambda|}}{N^{|\lambda|/3}}\frac{1}{z_{\lambda_+}}\,p_{\lambda_+}(\partial_{\mathbf{T}})\,\mathrm{Ai}^{(\beta/2)}(\mathbf{T})\right]_{\mathbf{T}={\ensuremath{\mathbf{S}}}}.\end{gathered}$$
Conclusion
==========
New dualities between different ensembles of random Hermitian matrices have been obtained in the article. Their general form, valid for all $\beta>0$, has been given in Propositions \[theo1\], \[theo2\], \[prop3\], and \[prop4\].
In Section 5.2, Proposition \[prop3\] has been used for proving the equivalence of the formal one-matrix model and the Gaussian model with an external field. But the comparison of Eqs. and leads to another surprising conclusion: the partition function is a multivariate Hermite polynomial! Explicitly, $$\label{eqZHermite}
(\prod_{i\geq 1}s_i)^N\, Z_{n,\beta}(t_1,t_2,\ldots)=\mathcal{H}^{(\beta/2)}_{(N,N,\ldots)}(s_1,s_2,\ldots)$$ when $$t_k=2\delta_{k,2}+\left(\pm \frac{2}{\beta}\right)^{k/2} p_{k}(s_1^{-1},s_2^{-1},\ldots)$$ for all $k\geq1$. Similarly, the partition function, for $\beta$-matrices that are both Hermitian and positive (chiral ensembles), is equivalent to a multivariate Laguerre polynomial whose partition is rectangular. Since the partition function of a matrix model is related to the enumeration of maps, the above result suggests that there exists a combinatorial interpretation for the multivariate Hermite and Laguerre polynomials. This would generalise Viennot’s combinatorial work on classical polynomials in one variable [@Viennot].
Formal matrix models have been meanly studied when $\beta=2$. This case is simpler and many tools have been developed for calculating the “large N” or topological expansion of the partition functions. A new and very general approach, which seems relevant to the question of dualities, has been introduced recently by Eynard and Orantin [@EynardOrantin]. These authors have shown that the “free energy” (i.e., $-\ln Z$), for many matrix models or even algebraic curves, is invariant under a certain class of transformations. It would be interesting, on the one hand, to check if these transformations include precisely the source–external field exchange considered in Section 4 and, on the other hand, to determine if the invariance of the free energy can be generalised to the $\beta\neq2$ cases.
For $\beta=1,2,4$, the large $N$ limit of the partition function (equivalently, the expectation value of products of characteristic polynomials) has been evaluated in Section 5.3. It is proportional to a matrix Airy (or Kontsevich) integral. The proof given in Appendix B is valid only when one works with the matrices themselves, but not with their eigenvalues. However, the fact that the asymptotic formula has the same form for three distinct values of $\beta$ is non trivial. Thus, it is reasonable to surmise that formula Eq. remains the same for all $\beta>0$ if one uses Eq. , for the definition of the multivariate Airy function, instead of Eq. . The proof is still missing.
Finally, it is remarkable that the dualities concerning the product of [*inverse*]{} characteristic polynomials don’t involve a change in $\beta$ (for instance, compare Propositions \[prop1\] and \[prop2\]). This fact certainly indicates that averages involving ratios of characteristic polynomials are much more difficult to calculate. Recall that the knowledge of the latter quantities gives access to the eigenvalue correlation functions (or marginal densities) which, from a physical or probabilistic point of point, are of prime importance. Note that the average of ratios of characteristic polynomials is still an open problem for the Circular $\beta$-Ensembles [@Matsumoto] despite the fact that calculations in the latter ensembles are easier than in the Gaussian $\beta$-Ensembles. A simple exercise shows, however, that the average of ratios characteristic polynomials satisfies a Calogero equation related to a superalgebra, $gl(p|q)$ say, if $beta$ is rational. Superalgebraic systems of the circular (or trigonometric) type have been previously studied by Sergeev an Veselov [@VS; @VS2]. The solutions to these models are supersymmetric polynomials which belong to a special family of symmetric functions in two sets of variables (see examples 23 and 24 of Chapter I-3 in [@Mac]). Preliminary calculations indicate that solutions also exist for systems of the rational type (i.e., Calogero models connected to Gaussian $\beta$-Ensembles). The relation between the superalgebraic Calogero models and the eigenvalue correlation functions for the $\beta$-Ensembles will be the subject of a forthcoming paper.
The author is grateful to Peter J. Forrester for helpful discussions. Thanks also to Michel Bergère and Bertrand Eynard for stimulating discussions on related subjects. A small part of this work, supported by NSERC, was done while visiting the Centre de recherches mathématiques (CRM) de l’Université de Montréal and the Département de physique de l’Universié Laval; the author wishes to thank John Harnad and Pierre Mathieu for their hospitality.
$\beta$-Matrices
================
A $N\times N$ matrix whose entries are real ($\beta=1$), complex ($\beta=2$), or quaternionic real ($\beta=4$), can be represented as $$\mathbf{X}=\mathbf{X}^0+\sum_{k=1}^{\beta-1}\mathbf{X}^k\mathbf{e}_k,$$ where ${\ensuremath{\mathbf{X}}}^k$ ( for $0\leq k\leq \beta-1$) is a $N\times N$ matrix with real elements. The “imaginary numbers” $\mathbf{e}_k$ satisfy $\mathbf{e}_k^2=-1$, $\mathbf{e}_i\mathbf{e}_j=-\mathbf{e}_j\mathbf{e}_i$, and $\mathbf{e}_1\mathbf{e}_2=\mathbf{e}_3$. The conjugation is defined by $$\overline{\mathbf{X}}=\mathbf{X}^0-\sum_{k=1}^{\beta-1}\mathbf{X}^k\mathbf{e}_k.$$ $\mathbf{X}$ is Hermitian if $\mathbf{X}^\dagger:=(\overline{\mathbf{X}})^{\ensuremath{\mathrm{t}}}=\mathbf{X}$. Equivalently, ${\ensuremath{\mathbf{X}}}$ is Hermitian if $$(\mathbf{X^0})^{\ensuremath{\mathrm{t}}}=\mathbf{X^0},\quad ({\mathbf{X}^k})^{\ensuremath{\mathrm{t}}}=-\mathbf{X}^k \quad k\geq1.$$ The case $\beta=0$ is trivial: it is obtained when $\mathbf{X}=\mathbf{X}^0=\mathrm{diag}(x_1,\ldots,x_N)$.
Note that the case $\beta=4$ is special since quaternions are not commutative. Thus, in general, $\sum_{i,j} X_{i,j}Y_{j,i}\neq \sum_{i,j}Y_{i,j}X_{j,i}$ for $\beta=4$. The correct definition of the trace for quaternionic matrices is the following: $${\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}:=\frac{1}{2}\sum_{i=1}^N(X_{i,i}+\overline{X_{i,i}})\qquad (\beta=4).$$ This trace is real by definition and satisfies ${\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{Y}}}={\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}{\ensuremath{\mathbf{X}}}$ for all quaternionic matrices ${\ensuremath{\mathbf{X}}}$ and ${\ensuremath{\mathbf{Y}}}$. Alternatively, one can use the representation of quaternions in terms of the Pauli matrices: $${\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}=\frac{1}{2}{\ensuremath{\,\mathrm{tr}\,}}P({\ensuremath{\mathbf{X}}}) \qquad (\beta=4)$$ where $$P({\ensuremath{\mathbf{X}}})=\Big[P(X_{i,j})\Big]_{i,j=1}^N,\qquad P(X_{i,j})=\left[
\begin{array}{cc}
\phantom{-}X_{i,j}^0+\mathrm{i} X_{i,j}^1 & X_{i,j}^2+\mathrm{i} X_{i,j}^3 \\
-X_{i,j}^2+\mathrm{i} X_{i,j}^3 & X_{i,j}^0-\mathrm{i} X_{i,j}^1 \\
\end{array}
\right]$$ In other words, $P({\ensuremath{\mathbf{X}}})$ is $N\times N$ matrix with $2\times 2$ entries. For all $\beta$ and ${\ensuremath{\mathbf{X}}}$ Hermitian, $$\det \mathbf{X}=\exp ({\ensuremath{\,\mathrm{tr}\,}}\ln \mathbf{X})$$ is equal to the product of the eigenvalues of ${\ensuremath{\mathbf{X}}}$. Note however that, for $\beta=4$, ${\ensuremath{\,\mathrm{tr}\,}}P({\ensuremath{\mathbf{X}}})=2{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}$ and $\det\mathbf{X}=\sqrt{\det P(\mathbf{X})}$.
Finally, the measure on the space of Hermitian $\beta$-matrices is simply the product of the real independent elements of ${\ensuremath{\mathbf{X}}}$: $$(d\mathbf{X})=\prod_{i\leq j} d{X}^0_{i,j} \prod_{i< j}\prod_{k=1}^{\beta-1}dX^k_{i,j}.$$By using $\int_{\mathbb{R}} dx e^{-ax^2+kx}=\sqrt{\pi/a}e^{k^2/4a}$, one easily shows that $$\label{eqGauss}
\int (d{\ensuremath{\mathbf{X}}}) e^{-a{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}^2+{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{X}}}{\ensuremath{\mathbf{Y}}}}=\left(\frac{\pi}{a}\right)^{N/2}\left(\frac{\pi}{2a}\right)^{\beta N(N-1)/4}e^{\frac{1}{4a}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{Y}}}^2},$$ where both ${\ensuremath{\mathbf{X}}}$ and ${\ensuremath{\mathbf{Y}}}$ are Hermitian $\beta$-matrices of size $N$.
Proof of Proposition \[propEdge\]
=================================
Let ${\ensuremath{\mathbf{S}}}'$ be a fixed $n\times n$ Hermitian $\beta'=4/\beta$-matrix whose eigenvalues are close the spectrum edge of the ${\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N$: $$s'_j=\sqrt{2N}+\frac{1}{\sqrt{2N^{1/6}}}s_j .$$ The aim is to determine the asymptotic behaviour of $$G_{N,\beta}({\ensuremath{\mathbf{S}}}')= \mathcal{H}^{(\beta/2)}_{(N^n)}(s') =
\left\langle \prod_{i=1}^n\det\left( s'_j{\ensuremath{\mathbf{1}}}\pm\sqrt{\frac{2}{\beta}}{\ensuremath{\mathbf{X}}}\right)\right\rangle_{{\ensuremath{\mathbf{X}}}\in{\ensuremath{\mathrm{G}\:\!\!\beta\!\!\;\mathrm{E}}}_N}$$ The three cases (i.e., $\beta'=1,2,4$) must treated separately. Only the $\beta'=1$ case will be detailed below since the method is easily adapted for $\beta'=2$ and $\beta'=4$. Note that in the latter case, one has to use the appropriate definitions for the trace and the determinant (see Appendix A).
The starting point is the rescaled matrix integral . The integrand is an analytic function in $n+n(n-1)/2$ real variables $Y_{i,j}$. By using the Cauchy theorem, integrations along the real axis can be transformed into contour integrals in the complex plane. By convention, $-\pi\leq\mathrm{arg}(Y_{i,j})< \pi$ and the logarithm is defined on its principal branch. In order to ensure the convergence of the integrals, each variable $Y_{i,j}$ follows a path starting at $\infty e^{{\ensuremath{\mathrm{i}}}\phi_s}$ and ending at $\infty e^{{\ensuremath{\mathrm{i}}}\phi_e}$, where $$\label{cond1}
-\pi\leq\phi_s\leq-3\pi/4\quad \mbox{or}\quad 3\pi/4\leq\phi_s< \pi\quad\mbox{and}\quad \mbox -\pi/4\leq\phi_e\leq\pi/4$$ The function $f({\ensuremath{\mathbf{Y}}})$ in Eq. has double saddle points, noted $\eta_{i,j}$, if the following conditions are statisfied: $$\label{cond2}
\left.\frac{\partial}{\partial Y_{ij}}f({\ensuremath{\mathbf{Y}}})\right|_{Y=\eta}=0 ,\qquad \left.\frac{\partial^2}{\partial Y_{ij}\partial Y_{kl}}f({\ensuremath{\mathbf{Y}}})\right|_{Y=\eta}=0$$ together with some nonvanishing third order derivatives at the saddle point. Only constant solutions $\eta_{ij}$ (i.e., those that do not depend on the other $\eta_{kl}$) are suitable for the multidimensional steepest descent method. Direct calculations imply that the saddle point conditions are equivalent to following system of algebraic equations: $$-4\eta_{ii}-4{\ensuremath{\mathrm{i}}}+(\eta^{-1})_{ii}=0,\quad -4\eta_{ij}+(\eta^{-1})_{ij}=0,\quad -4-(\eta^{-1})_{ii}(\eta^{-1})_{ii}=0$$ together with $$-4-(\eta^{-1})_{ii}(\eta^{-1})_{jj}- (\eta^{-1})_{ij}(\eta^{-1})_{ij}=0, \quad (\eta^{-1})_{ik}(\eta^{-1})_{jk}=0$$ for all $1\leq i,j,k\leq n$ and $i<j$. Note that these equations, $\eta=(\eta_{ij})_{i,j}$ is interpreted as a matrix. By writting the inverse in terms of cofactors, which means $(\eta^{-1})_{ij}=(\det \eta )^{-1}\mathrm{cof} \eta_{ji}$ where $\det(\eta)=\sum_{j} \eta_{ij} \mathrm{cof} \eta_{ij}$ for some $i$, one shows that the unique set of solutions is: $$\eta_{jj}=\frac{1}{2{\ensuremath{\mathrm{i}}}} \quad\mbox{and}\quad \eta_{ij}=0$$ for all $1\leq i<j\leq n$.
Now, one has to ensure that the descent of $f({\ensuremath{\mathbf{Y}}})$ is maximum when the variables $Y_{ij}$ approach and leave $\eta_{ij}$. Let $\theta_{kl}$ denotes the argument of the complex variable $Y_{kl}=\eta_{kl}$ and let $$\phi_{jk}=\mathrm{arg}\left.\frac{\partial^3}{ \partial Y_{kl}^3} f({\ensuremath{\mathbf{Y}}})\right|_{\eta_{kl}}=-\frac{\pi}{2}$$ for all $1\leq k\leq l\leq n$. The contours of steepest descent must comply with $$e^{3{\ensuremath{\mathrm{i}}}\theta_{jk}+{\ensuremath{\mathrm{i}}}\phi_{jk}}=-1.$$ The three possible solutions are $\theta_{kl}=-5\pi/6,-\pi/6,$ and $\pi/2$. The two former angles are compatibles with conditions . Set $$W_{kl}=2N^{1/3}(Y_{kl}-\eta_{kl})$$ Each variable $W_{kl}$ follows the steepest descent path $\mathcal{D}$: it starts at $\infty e^{-5\pi{\ensuremath{\mathrm{i}}}/6}$, passes through the origin, and stops at $\infty e^{-\pi{\ensuremath{\mathrm{i}}}/6}$. In order to get a simple expression for $G_{N,\beta}({\ensuremath{\mathbf{S}}}')$, it is convenient to define the matrix ${\ensuremath{\mathbf{W}}}$ with elements $W_{kl}=W_{lk}$. Then, by collecting all the factors coming from $f({\ensuremath{\mathbf{Y}}})$ and $g({\ensuremath{\mathbf{Y}}},{\ensuremath{\mathbf{S}}})$ in Eq. , one shows that $G_{N,\beta}({\ensuremath{\mathbf{S}}}')$ is equal to [$$\label{limG1}
Ce^{N^{1/3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}+\frac{1}{2N^{1/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2}
\int_{\mathcal{D}} (d{\ensuremath{\mathbf{W}}})\exp \left(- \frac{\mathrm{i}}{3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^3-\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}{\ensuremath{\mathbf{S}}}\right)\exp\left(-\sum_{k\geq 4}\frac{1}{k\mathrm{i}^kN^{(k-3)/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^k\right).$$ ]{}where $\int_{\mathcal{D}} (d{\ensuremath{\mathbf{W}}})$ stands for the $n+n(n-1)/2$ contour integrals along the path $\mathcal{D}$, and $C$ is the constant given in Eq. . Therefore, the following limit holds : $$\label{limG2}
\lim_{N\rightarrow \infty} C^{-1}e^{-N^{1/3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}-\frac{1}{2N^{1/3}}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{S}}}^2}G_{N,\beta}({\ensuremath{\mathbf{S}}}') = \int_{\mathcal{D}} (d{\ensuremath{\mathbf{W}}})\exp \left( -\frac{\mathrm{i}}{3}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}^3-\mathrm{i}{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{W}}}{\ensuremath{\mathbf{S}}}\right).$$ Obviously, the first correction to the above limit is of order less or equal to $N^{-1/3}$. The complex functions involved in the integrals of the last equation are analytic, so Cauchy’s theorem can be applied once again. The integrals remains convergent if each variable goes from $\infty e^{i\theta_s}$ to $\infty e^{i\theta_s}$ where $$-\pi\leq\theta_s\leq-2\pi/3\quad \mbox{and}\quad \mbox -\pi/3\leq\theta_e\leq 0.$$ Contours along the real line are chosen so that ${\ensuremath{\mathbf{W}}}$ can be interpreted as a real symmetric matrix. The change ${\ensuremath{\mathbf{W}}}\mapsto-{\ensuremath{\mathbf{W}}}$ finishes the proof.
[99]{}
I. Andric and L. Jonke, [*Duality and quasiparticles in the Calogero-Sutherland model: Some exact results*]{}, Phys. Rev. [**A 65**]{} (2002) 34707, 4 pages.
W. van Assche, [*Multiple orthogonal polynomials*]{}, Chapter 23 in M. E. H. Ismail, [*Classical and Quantum Orthogonal Polynomials in One Variable*]{}, Cambridge University Press (2005), 724 pages.
T. H. Baker and P. J. Forrester, [*The Calogero-Sutherland Model and Generalized Classical Polynomials*]{}, Commun. Math. Phys. [**188**]{} (1997), 175–216.
P. M. Bleher and A. B. J. Kuijlaars, [*Integral representations for multiple Hermite and multiple Laguerre polynomials*]{}, Ann. Inst. Fourier [**55**]{} (2005), 2001–2014.
A. Borodin, E. Strahov, [*Averages of characteristic polynomials in random matrix theory*]{}, Commun. Pure and Appl. Math. [**59**]{} (2006), 161–253.
E. Brézin and S. Hikami, [*Characteristic Polynomials of Random Matrices*]{}, Commun. Math. Phys. [**214**]{} (2000), 111–135.
E. Brézin and S. Hikami, [*Characteristic Polynomials of Real Symmetric Random Matrices*]{}, Commun. Math. Phys. [**223**]{} (2001), 363–382.
E. Brézin and S. Hikami, [*Intersection theory from duality and replica*]{}, preprint (2007) <arXiv:0708.2210v1>, 16 pages.
F. Calogero, [*Ground State of a One-Dimensional N-Body System*]{}, J. Math. Phys. [**10**]{} (1969) 2197–2200; [*Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials*]{}, J. Math. Phys. [**12**]{} (1971) 419–436.
L. Chekov and Yu. Makeenko, [*A hint on the external field problem for matrix models*]{}, Phys. Lett. [**B 278**]{} (1992), 271–278.
P. Desrosiers and P. J. Forrester, [*Hermite and Laguerre $\beta$-ensembles: Asymptotic corrections to the eigenvalue density*]{}, Nucl. Phys. [**B 743**]{} (2006), 307-332.
P. Desrosiers and P. J. Forrester, [*Asymptotic correlations for Gaussian and Wishart matrices with external source*]{}, Int. Math. Res. Not. (2006), 27395, 43 pages.
J. F. van Diejen, [*Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement*]{}, Commun. Math. Phys. [**188**]{} (1997) 467–497.
P. Di Francesco, [*2D Quantum Gravity, Matrix Models and Graph Combinatorics*]{}, <arXiv:math-ph/0406013>, pages 33–88 in E. Brézin et al. (editors), [*Applications of Random Matrices in Physic*]{}, Springer (2006), 513 pages.
I. Dumitriu and A. Edelman, [*Matrix models for beta ensembles*]{}, J. Math. Phys. [**43**]{} (2002), 5830–5847.
I. Dumitriu and A. Edelman, [*Eigenvalues of Hermite and Laguerre ensembles : large beta asymptotics*]{}, Ann. I.H.P. Probab. stat [**41**]{} (2005), 1083–1099.
F. J. Dyson, [*Statistical Theory of the Energy Levels of Complex Systems. I*]{}, J. Math. Phys**3** (1962) 140–156.
A. Edelman and B. D. Sutton, [*From Random Matrices to Stochastic Operator*]{}, J. Stat. Phys**127** (2007), 1121–1165.
B. Eynard and N. Orantin, [*Invariants of algebraic curves and topological expansion*]{}, preprint (2007) <arXiv:math-ph/0702045>, 92 pages.
P. J. Forrester, [*The spectrum edge of random matrix ensembles*]{}, Nucl. Phys. [**B 402**]{} (1993), 709–728.
P. J. Forrester, [*Log-gases and RandomMatrices*]{}, book in preparartion. [www.ms.unimelb.edu.au/\~matpjf/matpjf.html](www.ms.unimelb.edu.au/~matpjf/matpjf.html).
P. J. Forrester and E. M. Rains, [*Inter-relationships between orthogonal, unitary and symplectic matrix ensembles*]{}, <arXiv:solv-int/9907008>, pages 171–208 in P.M. Bleher and A.R. Its (editors), [*Random matrix models and their applications*]{}, Cambridge University Press (2001), 448 pages.
P. J. Forrester, [*A random matrix decimation procedure relating $\beta = 2/(r+1)$ to $\beta = 2(r+1)$*]{}, preprint (2007), <arXiv:0711.1914>, 19 pages.
Y. V. Fyodorov, E. Strahov, [*An exact formula for general spectral correlation function of random Hermitian matrices*]{}, J. Phys. [**A 36**]{} (2003) 3203–3213.
I. P. Goulden and D. M. Jackson, [*Maps in locally orientable surfaces and integrals over real symmetric matrices*]{}, Canad. J. Math. [**49**]{} (1997), 865–882.
K. W. J. Kadell, [*The Selberg-Jack symetric functions*]{}, Adv.Math. [**130**]{} (1997), 33–102; first version accepted in 1987.
J. Kaneko, [*Selberg Integrals and Hypergeometric Functions Associated with Jack Polynomials*]{}, SIAM J. Math. Anal. [ **24**]{} (1993), 1086–1110.
P. Koev and A. Edelman, [*The efficient evaluation of the hypergeometric function of a matrix argument*]{}, Math. Comput. [**75**]{} (2006), 833–846.
M. Kontsevich, [*Intersection theory on the moduli space of curves and the matrix Airy function*]{}, Commun. Math. Phys. [**147**]{} (1992), 1–23.
M. Lassalle, [*Polynômes de Laguerre généralisés*]{}, C. R. Acad. Sci. Paris Sér. I Math. [**312**]{} (1991), 725–728.
M. Lassalle, [*Polynômes de Hermite généralisés*]{}, C. R. Acad. Sci. Paris Sér. I Math. (1991), 579–582.
, [Symmetric functions and [H]{}all polynomials]{}, 2nd ed., Clarendon Press, 1995.
S. Matsumoto, [*Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials*]{}, J. Phys. [ **A 40**]{} (2007), 13567-13586.
M. L. Mehta, [*Random Matrices*]{}, Third Edition, Academic Press (2004), 706 pages.
M. L. Mehta and J.-M. Normand, [*Moments of the characteristic polynomial in the three ensembles of random matrices*]{}, J. Phys. [**A 34**]{}, 4627–4639
A. Okounkov, [*Proof of a conjecture of Goulden and Jackson*]{}, Can. J. Math. [**49**]{} (1997), 883–886.
J. Ramirez, B. Rider, and B. Virag, [*Beta ensembles, stochastic Airy spectrum, and a diffusion*]{}, preprint (2006), <arXiv:math/0607331v3>.
M. Rösler, [*Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators*]{}, Commun. Math. Phys. [**192**]{} (1998) 519–542.
D. Serban, F. Lesage, V. Pasquier, [ *Single-particle Green function in the Calogero-Sutherland model for rational couplings $\beta=p/q$*]{}, Nucl. Phys., 499–512.
X. G. Viennot, [*Une théorie combinatoire des polynômes orthogonaux*]{}, Lecture Notes, Publication du LACIM, UQUAM (1984, 1991), 219 pages.
A.N. Sergeev and A. P. Veselov, *Deformed quantum Calogero-Moser problems and Lie superalgebras*, Commun. Math.Phys. [**245**]{} (2004), 249–278.
A.N. Sergeev and A. P. Veselov, [*Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials*]{}, Adv. Math. [**192**]{} (2005), 341–375.
R. P. Stanley, *Some combinatorial properties of Jack symmetric functions*, Adv. Math. [**77**]{} (1988), 76–115.
Z. Yan, [*A class of generalized hypergeometric functions in several variables*]{}, Canad. J. Math. [**44**]{} (1992) 1317–1338.
A. Zvonkin, [*Matrix Integrals and Map Enumeration: An Accessible Introduction*]{}, Math. Comput. Modelling [**26**]{} (1997) 281–304.
[^1]: IPhT–t08/013
[^2]: $h^\lambda_{(\alpha)}$ and $h_\lambda^{(\alpha)}$ are respectively noted $h_\lambda$ and $d'_\lambda$ in [@ForresterBook], while they are noted $c_\lambda$ and $c'_\lambda$ in [@Mac].
[^3]: All polynomials involved in Eqs. and are monic while those in [@Baker] are not. One goes from a convention to another via $\mathcal{H}_\lambda^{(\alpha)}(x)=2^{-|\lambda|}{P}_\lambda^{(\alpha)}(1^N) {H}_\lambda (x;\alpha)$ and $\mathcal{L}_\lambda^{(\alpha,\gamma)}(x)=(-1)^{|\lambda|}|\lambda|!{P}_\lambda^{(\alpha)}(1^N) {L}_\lambda^{\gamma}(x;\alpha)$. Additionally, ${P}_\lambda^{(\alpha)}(x)=\alpha^{-|\lambda|}h_\lambda^{(\alpha)} C^{(\alpha)}_\lambda (x)/|\lambda|!$.
[^4]: The factor $\exp(-{\ensuremath{\,\mathrm{tr}\,}}{\ensuremath{\mathbf{F}}}^2/4)$ comes from the absence of ${\ensuremath{\mathbf{F}}}$ in the definition of p.d.f. for GUE. See also Eq. .
|
---
author:
- |
[David Fifield[^1]]{}\
University of California, Berkeley
- |
[Lynn Tsai]{}\
University of California, Berkeley
- |
[Qi Zhong]{}\
University of California, Berkeley
bibliography:
- 'detecting-censor-detection.bib'
title: '**Detecting Censor Detection**'
---
Abstract {#abstract .unnumbered}
--------
Our goal is to empirically discover how censors react to the introduction of new proxy servers that can be used to circumvent their information controls. We examine a specific case, that of obfuscated Tor bridges, and conduct experiments designed to discover how long it takes censors to block them (if they do block at all). Through a year’s worth of active measurements from China, Iran, Kazakhstan, and other countries, we learn when bridges become blocked. In China we found the most interesting behavior, including long and varying delays before blocking, frequent failures during which blocked bridges became reachable, and an advancement in blocking technique midway through the experiment. Throughout, we observed surprising behavior by censors, not in accordance with what we would have predicted, calling into question our assumptions and suggesting potential untapped avenues for circumvention.
Introduction
============
Those who censor the Internet face a twofold challenge: not only do they have to block direct access to content, but they also must block access to proxy servers and other indirect means of circumventing their direct blocks. Because circumventors continually establish new proxy servers, effective censorship is therefore an ongoing task, requiring regular attention and upkeep. This aspect of the censorship problem—just how constrained censors are by limitations of resources, and how it limits their effectiveness—is not well understood, though circumvention would be improved by better knowledge of censors’ potential weaknesses. In this research, we seek to understand the ongoing behavior of censors as it relates to the specific question of the blocking of newly introduced Tor bridges. We do this through frequent active measurements, in multiple countries, that allow us to compute the “lag” between when a bridge is introduced and when it becomes blocked.
We limit our inquiry to what should be an easy case for the censor: the “default” bridges built into Tor Browser. These bridges use a traffic obfuscation protocol, their addresses are public, fairly static, and easily available to anyone who downloads the browser. Our usual assumptions about censors tell us that these bridges should be quickly blocked, and yet they remain unblocked almost everywhere in the world, even in places that are known to censor Tor. Even the famed Great Firewall of China (GFW), until recently, was delayed by days or weeks before blocking new bridges.
We ran measurements probes for a year, testing the reachability of default Tor bridges every 20 minutes from the U.S., China, Iran, and Kazakhstan. We recently began collaboration with established censorship measurement platforms to expand the tests to more countries. We found blocking of default bridges in China and Kazakhstan. In China, our measurements detected a change in behavior: around October 2016 the censor switched from blocking bridges only after release (after a delay of up to 35 days), to blocking bridges preemptively. The blocking in Kazakhstan is qualitatively different than the blocking in China, requiring different techniques to detect.
Our results demonstrate that a discrepancy exists between what we circumvention researchers assume about censors and what censors do in practice. In this work, we take only a few small steps toward explaining the discrepancy, performing targeted experiments to learn about how the censors in China extract bridge addresses from a software artifact. We hope to call attention to potential blind spots and weaknesses of censors that may be exploited for more effective circumvention. This is not a call for circumvention researchers to weaken their threat models; rather, we hope for richer and more precise threat models that take into account underappreciated vulnerabilities that may lead to more effective circumvention.
Related Work
============
There is not much research aimed at systematically measuring the reactions of censors to the advent of new or expanded forms of circumvention. We will list the works we are aware of that most closely resemble our goals and elaborate by contrast how our goals are different.
Dingledine [@five-ways-test-bridge-reachability] in 2011 enumerated ways of testing bridge reachability, among them our primary tool, direct scans. We attempt to shed light on some of the research questions he laid out, including knowing what bridges are blocked and where, and how quickly bridges become blocked. Dingledine considered the possibility that the very act of testing the reachability of a bridge could reveal the bridge’s existence to an alert censor. This consideration is less acute for us, because we limit ourselves to testing bridges whose addresses are already known to the public.
There is prior work on careful distribution strategies that seek to prevent a censor from discovering many secret proxy addresses; examples are Proximax [@McCoy2011a], rBridge [@Wang2013a], and Salmon [@Douglas2016a]. These limit the rate at which a censor can enumerate proxies and maintain a fraction of users whose proxies remain unblocked. We, on the other hand, start from a different assumption—that the proxy addresses are already public—and observe the nature of censors’ actual blocking reactions.
A rich body of research is devoted to innovating new protocols for disguising traffic, in order to make it harder to detect and censor. This present work relies on the properties of a specific protocol, obfs4 [@obfs4] (which is discussed further in Section \[sec:obfs4\]). Our purpose, however, is not directly to advance the state of the art of circumvention practice; but rather to better understand the interaction between censor and circumventor as it exists today.
Khattak et al. [@Khattak2013a] showed that understanding censors’ models is beneficial to facilitating evasion. By observing how the GFW processes certain packets, they were able to deduce some of the underlying weaknesses of the GFW and suggest ways to exploit them. Their work is similar to ours in spirit. While they study ways for a circumventor to defeat on-line detection, and we study how censors respond to the actions of circumventors, we have in common the desire for empirical measurement of actual censor capabilities,
The Open Observatory of Network Interference (OONI), a censorship measurement platform, began testing the reachability of Tor bridges [@ooni-bridge-reachability-study-and-hackfest], both by simple TCP probes and by attempts to bootstrap a complete Tor connection. The tests ran in a few specialized test locations from March 2014 until February 2015, then mostly lay dormant. The OONI tests did not specifically examine censors’ treatment of newly introduced bridges. They revived the test in December 2016 and began to test the same destinations we were testing. The results of working with the data of OONI and another measurement platform, ICLab, will appear in Section \[sec:ooni-iclab\].
In 2013, Zhu et al. [@Zhu2013a] looked into Chinese microblogging sites that implement internal censorship in order to follow the instructions of Chinese government restrictions; specifically, how long it takes for posts to be deleted. They discuss methods that detect censorship events within a few minutes of its occurrence. They discovered that deletions on microblogging sites occurred most frequently during the initial hour that it was posted, and 30% occurred in the first 30 minutes. They speculate that the censorship system they use contains a list of keywords that trigger different censorship behavior, and that if a post is deleted, most repost chains are deleted within five minutes of the original post’s. Their work is similar to ours because they, too, are concerned with time delays. However, they look at microblog deletion while we look at proxy blocking.
Nobori and Shinjo [@Nobori2014a] describe the experience of deploying a circumvention system, VPN Gate, and the Great Firewall’s reaction to it, over a period of two months in 2014. The GFW blocked VPN Gate’s centralized directory server only three days after initial deployment, and just one day later began to automatically harvest and block a list of mirror servers. The next day, the operators of VPN Gate discovered the IP address of the GFW’s automated scanner and blocked it; in response the GFW began scanning from multiple locations and cloud services. VPN Gate began poisoning its list of servers by mixing in unrelated IP addresses, but after another six days the GFW began verifying servers as belonging to VPN Gate before blocking them. During this whole process, the GFW suddenly and without explanation ceased blocking VPN Gate servers for about four days, then resumed again. The lesson of VPN Gate is that the GFW, at least, is capable of reacting quickly to a new circumvention system and build automation to block it.
In 2015, Ensafi et al. [@Ensafi2015b] did a detailed study on how the GFW uses active probing to quickly and dynamically discover a variety of types of proxy server. When the firewall detected a suspicious connection, it would send a followup request to the destination address to identify it. They noted that this type of probing for Tor already existed in early 2013. The discovery of active probing led to the development of probing-resistant protocols, obfs4 among them. That a national censor can be so sophisticated in some ways (active probing) and yet seem to lag in others (blocking of default bridges) is a challenge to our mental models.
Background
==========
Tor [@tor] is an anonymity network that is also used for censorship circumvention. Tor’s resistance to censorship is due not to anything inherent in the protocol itself; but to its surrounding infrastructure of *bridges* and *pluggable transports*. Bridges [@Dingledine2006a] are secret Tor servers, the addresses of which are not widely distributed, preventing easy discovery by a censor. Users are meant to learn a few bridge addresses through an out-of-band channel, like email. Pluggable transports [@pluggable-transports] are covert communications protocols that disguise Tor’s traffic signature on the wire, preventing easy online detection. Bridges that use a particular pluggable transport, called obfs4, are the main focus of our study. The properties of obfs4 are covered in Section \[sec:obfs4\].
Tor Browser [@torbrowser] is the means by which most ordinary users access the Tor network, whether for anonymity or censorship circumvention. It is a modified version of Firefox with a built-in Tor client and a special interface for the configuration of bridges and pluggable transports.
Separate from the infrastructure of secret bridges, Tor Browser also ships with a number of built-in, *default* bridges, whose addresses are baked into the source code [@torbrowser-bridgeprefs]. There is a configuration file inside the Tor Browser listing all these default bridges. Users can use a default bridge simply by selecting a pluggable transport from a menu, no out-of-band communication required. Strictly speaking, a “default bridge” is a contradiction: bridges are supposed to be secret, not easily discoverable in a configuration file. Our intuition and the common assumptions in censorship research tell us that the default bridges, which are trivially discoverable, should be quickly blocked—and yet they are not. Indeed, a study by Matic et. al [@Matic2017a] found that over 90% of bridge users use one of the default bridges. Even the uncommonly capable Great Firewall of China, before October 2016, delayed for days or weeks before blocking these bridges, in contrast to the rapidity with which it blocks other, harder-to-detect proxies. The speed with which censors block default Tor Browser bridges is the main object of our study.
Besides Tor Browser, there is also an Android version of Tor called Orbot. Orbot and Tor Browser have most of their default bridges in common, but a few appear in Orbot only. We will be looking at these set of different bridges as well.
Tor Browser releases
--------------------
Tor Browser releases changes in two different tracks: stable and alpha. The stable track is “safe” so to speak, with only small changes at any given time. For the most part, stable releases include bug fixes. On the other hand, the alpha track is much more experimental. It contains experimental features that are in “test” until it matures and can be merged with the stable track. Stable and alpha releases tend to appear around the same time since both of them are driven by Firefox releases. If a Tor Browser version number contains the letter ‘a’, it is an alpha version (for example, 6.5a3). Upon a new Firefox release, Tor Browser, too, updates any security flaws that may have been discovered. Each release is an opportunity for Tor Browser to release new bridges. During our study over a duration of approximately 12 months, we observed a total of 18 stable releases and 13 alpha releases [@torbrowser-changelog].
The lifecycle of a new bridge
-----------------------------
In the following sections, we elaborate on the lifecycle of a new bridge and the stages involved in releasing it. Each stage in the lifecycle is also a potential opportunity for censors to detect the addition of a new bridge.
1. **Ticket Filed:** Filing a ticket in Tor’s online bug tracker proposes the inclusion of new default bridges. Censors may monitor the bug tracker and discover the new default bridges here.
2. **Ticket Merged:** Bridges are added to the Tor Browser’s source code during the merging of a ticket. It is automatically included in nightly builds, and code containing the new bridge is available in executable form. Censors may learn of bridges at this stage if they are looking at the source code repository or monitoring nightly builds.
3. **Testing Release:** Preceding a public release, Tor Browser developers prepare candidate packages and send them out to the quality assurance mailing list for testing. Censors monitoring the mailing list could discover new bridges at this point.
4. **Public Release:** After bridges have been tested, the new packages are announced on the Tor Blog. The Tor Blog is publicly available, and any censors monitoring it would discover the new bridges here. Installed Tor Browsers will automatically update to include the new packages, and users will begin to actively use the new bridges. A censor could also discover the new bridges through black-box testing an auto-updating installing at this stage.
The entirety of the lifecycle usually takes a few weeks to complete. Occasionally, if the fix is small enough, the Testing Release stage is skipped. There is also a possibility that new bridges are discussed in private mailing lists beforehand, and a censor that had infiltrated the mailing lists could conceivably discover new bridges before a ticket had even been filed.
The Properties of obfs4 {#sec:obfs4}
-----------------------
In our studies, we focus primarily on a particular pluggable transport: obfs4, an advanced transport protocol that offers resistance to deep packet inspection and resistance to active probing.
- **Deep Packet Inspection** Re-encrypts a Tor stream so that it appears as a stream of random bytes that cannot be easily decrypted.
- **Active Probing Attacks** Censors scan suspected proxies in order to discover what protocols are supported. The Great Firewall is known to use active proving against predecessor protocols obfs2 and obfs3. However, this attack does not work with obfs4. Every obfs4 bridge has a per-bridge secret in which a client must prove knowledge of upon the initial message. The censor would have to shave the same out-of-band information as a legitimate client, therefore rendering the knowledge of a bridge IP address insufficient to prove the existence of a bridge.
These security features are crucial because they allow us to limit the methods of bridge discovery. They give us the confidence that censors discover our bridges in the ways that we intend them to.
obfs4 is an important bridge not only for its unique properties, but also for its applications in the real world: it is the most commonly used bridge, yielding about 35,000 concurrent users in February 2017 [@obfs4-users].
Methodology
===========
Our experiment requires us to be able to detect the moment a bridge is blocked: when the bridge transitions from being reachable to being unreachable. We do this primarily through active measurements from probe sites located in various countries.
For a little more than a year, we ran frequent TCP reachability tests of a variety of destinations from probe sites in the U.S., China, Iran, and Kazakhstan. Because of the difficulty of acquiring test machines inside countries subject to information controls, probe coverage is not continuous or complete in any country other than the U.S. The probe site in the U.S. acted as a control, allowing us to distinguish occurrences of blocking from temporary bridge outages. Figure \[fig:timespans\] shows the time periods for which we have measurements in each country. From each probe site, we attempted a TCP connection to every destination every 20 minutes, recording for each connection attempt whether the connection was successful, the time elapsed, and error message if any. The rate of probing enables us to know not only the date, but also the time of day, when each bridge became blocked. The set of destinations, a mix of fresh default obfs4 bridges and other bridges, appears in Table \[tab:destinations\].
[@r@[ : ]{}l]{} nickname & ports\
\
ndnop3 & 24215, 10527\
ndnop5 & 13764\
riemann & 443\
noether & 443\
Mosaddegh & 41835, 80, 443, 2934, 9332, 15937\
MaBishomarim & 49868, 80, 443, 2413, 7920, 16488\
GreenBelt & 60873, 80, 443, 5881, 7013, 12166\
JonbesheSabz & 80, 1894, 4148, 4304\
Azadi & 443, 4319, 6041, 16815\
Lisbeth & 443\
NX01 & 443\
LeifEricson & 50000, 50001, 50002\
\
LeifEricson & 41213\
fdctorbridge01 & 80\
\
Mosaddegh & 1984\
MaBishomarim & 1984\
JonbesheSabz & 1984\
Azadi & 1984\
\
ndnop4 & 27668\
During the latter part of our measurements, we got assistance from the established censorship measurements platforms OONI and ICLab. Razaghpanah et al. [@RazaghpanahLFNV16] describe both platforms, their similarities and different design tradeoffs. At our request, both platforms added to their repertoire of measurements active measurement of default Tor Browser bridges. Compared to our custom reachability tests, the ICLab and OONI measurements trade frequency for coverage: they run daily, rather than every 20 minutes, but they cover many more geographic locations, giving a more global view of censorship. Our OONI-derived data covers 117 ASes in 55 countries and our ICLab-derived data covers 201 ASes in 217 countries. (The actual division is by two-letter country code, of which there are more than there are countries in the world. ICLab heavily relies on measurements from VPN endpoints, including some in autonomous systems that span more than one country.)
Simple TCP reachability testing has limitations, in that a censor may make a bridge effectively useless, without directly blocking its IP address or TCP port. China blocks at the TCP/IP layer, so block are easy to detect. On the other hand, we found that Kazakhstan blocks at a higher layer. A successful TCP connection doesn’t necessarily mean a successful Tor connection. For this reason, for a limited time we also did testing of full Tor-over-obfs4 connections. Details of this experiment appear in Section \[sec:limitations\].
Throughout the measurements, we coordinated with the developers of Tor Browser to begin measurement of bridges before their introduction. During the course of the study, the Tor Project was ramping up its obfs4 capacity by adding additional bridges. We additionally ran certain controlled experiments designed to uncover specific blocking behaviors of the censor. These included changing ports on the same address, and inserting a bridge so that it is available in the same code but commented out.
Results and Observations
========================
In this section we present and interpret the results of our experiments, focusing on China, where we had the most measurements and saw the most varied behavior. Our observations in the “China 1” and “China 2” ASes were mostly in agreement; minor differences are mentioned in Sections \[sec:anomalies\] and \[sec:IPBlock\]. In Iran, we did not see any blocking of bridges; all of them were reachable all the time—though what we discovered in Kazakhstan means that there may have been blocking that TCP reachability tests would not detect. We found blocking of the default bridges in Kazakhstan, though of a qualitatively different nature than that which we observed in China. We cover the particulars of Kazakhstan in Section \[sec:limitations\]. Throughout this section, refer to Figure \[fig:timelines\] and Table \[table:timeline\], which depict the entirety of the combined “China 1” data set.
Overall, we recorded over 5.9 million individual probe results. Our high-frequency probes account for 4.9 million of these; ICLab accounts for about 800,000 and OONI for about 260,000. There are 2.1 million probe results in the “China 1” AS alone, which is the basis for Figure \[fig:timelines\].
We have organized Tor Browser releases into “batches”, where each batch contains a distinct set of fresh bridges. Figure \[fig:timelines\] and Table \[table:timeline\] are arranged by release batch. During the first part of our experiment, blocking events were distinct: when a batch contained more than one bridge, all were blocked at once (within our probing period of 20 minutes). In our first six batches, we observed blocking delays of 7, 2, 18, 10, 35, and 6 days after the first public release, and up to 57 days after the filing of the first ticket, when bridges were potentially first discoverable. The only exception to this was that in the 6.0.5/6.5a3 batch, the censor actually failed to blocked two bridges, and these two bridges were blocked only much later. This fact suggests, to us, that new default bridges are loaded into the firewall in groups, and are not, for example, detected and blocked one at a time. During the first six batches, we found that blocking in China was keyed on both IP address and port, consistent with an observation of Winter and Lindskog in 2012 [@Winter2012a]. For example, many of the bridges happened to have port 22 (SSH) open, and it remained accessible even as other ports on the same IP address were blocked. (See riemann in Figure \[fig:timelines\] for an example: its port 22 remained accessible when its port 443 was blocked in January 2016.) Per-port blocking is what enabled us to run multiple bridges on the same IP address.
In the last two batches, we noticed that the GFW seems to have altered their blocking methods. New bridges in these two batches were all blocked even before the public release. They were blocked soon after they were merged into the public Git repository. During this period, we observed that China also started blocking on whole IP address, as well as continuing blocking on IP address and port pairs. Unlike blocking bridges, block for whole IP addresses did not seem to be done all at once. By the end of December 2016, all our bridges were blocked on the whole IP. Running any more bridges on existing IPs is no longer possible.
As for Orbot bridges, we found that China did not try to block them at all. They remained accessible until late October and early November in 2016. At this point, China started blocking whole IP address of bridges in Tor Browser. Since these Orbot bridges only used different ports on the same IPs, they were blocked as a side affect.
Port Rotations
--------------
In our first few batches, we found that China blocked on IP and port pairs. This means that when we have an existing bridge that is blocked, if we just open up a new port on the same IP, the new port would still be reachable. This is an easy way of evading GFW blocks compared to setting up new bridges. We call this process port rotation. We were interested in seeing whether or not rotating the ports would give us new results. In release batches 6.0.5/6.5a3, 6.0.6/6.5a4, and 6.0.8/6.5a6, we changed the port number of certain existing bridges with each new batch, creating what appeared to the GFW to be a large set of new bridges each time. Table \[tab:portrotation\] displays the ports that we rotated to.
-------------- ------- ------- ------- -------
LeifEricson 41213 50000 50001 50002
GreenBelt 60873 5881 7013 12166
Mosaddegh 41835 2934 9332 15937
MaBishomarim 49868 2413 7920 16488
4148 4304
6041 16815
-------------- ------- ------- ------- -------
: Rotation of port numbers in successive releases. The strategy worked until the second-to-last time, when the GFW began blocking entire IP addresses. The ports in the final rotation were blocked even before they were used. []{data-label="tab:portrotation"}
The first rotation was successful. The new ports served as new unblocked bridges, and worked for a time after release before being blocked, as before. The second rotation was initially successful, but this time when the bridges were blocked, all ports on the IP address were blocked, including the ports we had reserved for the third rotation. The status of the bridges after release 6.0.5 is shown in Table \[table:blocked\].
**Bridge** **Status**
------------------- ---------------
LeifEricson:50000 Blocked
GreenBelt:5881 Unblocked$^*$
Mosaddegh:2934 Blocked
MaBishomarim:2413 Blocked
JonbesheSabz:1894 Blocked
Azadi:4319 Unblocked
: Blockage status in the days following the 6.0.5/6.5a3 releases. LeifEricson had been blocked since we started measuring it. GreenBelt had an outage on all ports during the time of blocking, which may have protected it when the other bridges were blocked. It was not blocked when it recovered from its outage. and was also unblocked once it recovered from its outage. Azadi:4319 somehow eluded discovery when the other bridges were blocked, and remained unblocked for a while even after Azadi:6041 was blocked in the following release. []{data-label="table:blocked"}
Failure to Block All New Bridges in a Single Release {#sec:differenttimes}
----------------------------------------------------
In batch 6.0.5/6.5a3, we rotated six ports, one (LeifEricson) was preemptively blocked on all ports, three were blocked on the same day, and two (GreenBelt and Azadi) were not blocked. This is the first time we have seen this phenomenon over the past year of observation. One of the unblocked one (GreenBelt) was not operational at the time, but did come back online later. We will now look at each of these two bridges in detail.
### GreenBelt
Our data shows that the U.S. probe site was not able to connect to GreenBelt for an extended period of time, namely from September 17, 2016 to September 24, 2016. After inquiring with the bridge operator, we found that GreenBelt was indeed down during this time due to an IP table configuration error. The bridge rejected any incoming traffic. The blocking of bridges for this release happened on September 22, 2016, which means that GreenBelt was not functioning when the blocking happened. This is a strong indicator that the censor used network analysis techniques rather than parsing the bridge configuration file directly.
### Azadi
Both our observation data and the bridge operator confirmed that Azadi had been working properly, unlike GreenBelt. Combined with the fact that this is the only time it happened during our observation, it shows that the censor’s method for finding new bridges would have a low probability of missing new bridges. This confirms our previous suspicion that the censor is not parsing the bridge configuration file.
### Analysis
Our speculation for this anomaly is that the censor used black-box network traffic analysis to find new bridges. In other words, they ran the released version of Tor and simply monitored what addresses the executable connected to. Since GreenBelt was down, this method would have missed it. GreenBelt being down at the time could provide a reason for it not getting blocked, were it not for the fact that Azadi did not have an outage and it also did not get blocked. Tor Browser at that point had a very large number of default bridges included. If the censor did not monitor the executable long enough, this type of traffic analysis might miss some new bridges. We believe this is what happened to Azadi.
Preemptive Blocking
-------------------
During our studies, it appears as though there was a change in the GFW’s method of bridge discovery in October 2016. Rather than wait until after a release to block bridges, it started blocking them after a ticket was merged (before release). In release 6.0.6/6.5a4, we can see in Table \[table:timeline\] that all the new bridges were blocked before the release. This behavior is drastically different from what we have seen before. Assuming that the censor has not infiltrated the private mailing list, Tor bridges appear in two places before a release. When requesting to add new bridges, a ticket has to be submitted to the Tor bug tracker. When this ticket is accepted, it is merged and the new bridges would be added to the source code. Both the Tor bug tracker and the source code repository are publicly accessible, so a censor could learn the new bridges from either of these two places. We noted though that learning new bridges from the tickets would require human inspection, since the tickets do not have specific formats and can be on any issue. On the other hand, a script can easily keep track of new bridges from the source code repository. It would be easier for them to read from the source code repository.
Batch Blocking Timing patterns {#sec:timingPattern}
------------------------------
From Table \[table:batchBlockTime\], we can see that for our eight releases, the batch blocking all happened on weekdays. Most of the blocking happened on Tuesday and Thursday. Only one blocking happened on Wednesday, and one on Friday. Furthermore, they all happened between 10:00 and 17:00 local time. There seem to be no noticeable patterns to these time. However, they all lie within working hours in China. This seems to indicate some manual effort is needed to make blocking take affect. The idea of manual effort is further supported by the delay we see in blocking for the first six batches. If the blocking is done purely automatically, we would not expect to see a varying of delay between the release and blocking.
One other thing to note is that starting from version 6.0.6/6.5a4, bridges were blocked before a release. It suggests that the blocking process might have changed as well, and the process might be automatic now.
When the GFW blocks whole IP addresses, they seem to have a different timing pattern than the batch blocking of bridges. This will be discussed in more detail in \[sec:IPBlock\].
IP Blocks {#sec:IPBlock}
---------
**Bridge** **Time** **Day** **Date**
-------------- ---------- --------- -------------
GreenBelt 14:00 Thurs 20 Oct 2016
Mosaddegh 19:40 Thurs 27 Oct 2016
MaBishomarim 20:20 Wed 19 Oct 2016
JonbesheSabz 02:00 Thurs 20 Oct 2016
Azadi 05:50 Mon 07 Nov 2016
: IP Blocking Time in CST time. Since there are interweaving unblocking after the initial IP block, we are only looking at the initial block time here.
In the first six release batches, we confirmed a finding of Winter and Lindskog [@Winter2012a] that the Great Firewall blocks bridges by their specific port number. A bridge that is discovered and blocked on one port will not cause other ports on the IP address to get blocked. This property of the firewall allowed us to rotate ports and, for example, take the MaBishomarim bridge through ports 49868, 80, 2413, in successive releases.
Since October 2016 and the 6.0.6/6.5a4 release batch, bridges have been blocked on the entire IP. On October 20, two bridges were blocked together on the entire IP address. However, we still could not account for the blocking of Mosaddegh and Azadi, which happened on a seemingly unrelated date. Blocks affected even ports not-yet-used ports that were waiting in reserve for the following release, and non-bridge ports such as 22. The all-ports blocking affected even one of the earliest bridges we had measured, riemann, whose obfs4 port was blocked in January 2016 and its SSH port 10 months later (see Figure \[fig:timelines\]).
Between November 15 and November 21, 2016, JonbesheSabz and Lisbeth were reachable from one site in the “China 1” AS, but not the other.
As mentioned in section \[sec:timingPattern\], all simultaneous blocking for bridges happened during working hours on a weekday. This suggested manual blocking instead of an automatic blocking system. One key observation we have is that blocking for IP occurred at times such as 2:00AM and 5:50AM China Standard Time, likely falling outside the range of the standard workday. This allows us to reach a few possible scenarios:
1. The limitation to working hours is just a coincidence, an artifact of the small size of the blocking event data set.
2. Individual port blocking uses a different system than IP:port blocking. Even though IP:port blocking is manual, IP address blocking is automatic.
Another observation is that there is a pattern of interweaving unblocking and blocking after an IP block. Although Mosaddegh was already blocked on October 27, it later became available again for a period of time whereas previously blocked individual ports remained blocked. It is only after November 15 that it became permanently blocked. We observe this pattern across all the blocked bridges. Also, we notice that even when the IP address was unblocked, ports that are previously blocked would still stay blocked. In the example above, we can see that port 9332 and 2934 for Mosaddegh were never unblocked even though the IP itself was unblocked multiple times. This seems to indicate a two-tier structure to the GFW. The IP:port blocking and the IP blocking are handled separately. If an IP:port pair is in either of these, it would get blocked.
One possible hypothesis is that this is an artifact resulted from GFW could not handle so much traffic. The interweaving unblocking and blocking could simply result from GFW failing under too much traffic. Currently we don’t really know whether our hypothesis is correct, but this would be part of our intended future work.
Bridge List File {#readingbridgepref}
----------------
The bridge list file is a configuration file with all the bridges written on it. Reading this file seems to be the easiest way for censors to discover new bridges.
In Section \[sec:differenttimes\], we present evidence supporting that the GFW was not reading the bridge list file at that time. It seems that it used black-box network analysis to find the bridges instead.
However, there has been recent evidence to suggest that China operators have changed their behavior and are now reading the this file either instead of or in addition to black-box testing. It appears as though they began looking at the bridge configuration file between the release of 6.0.5/6.5a3 and the blocking of Azadi:6041.
By looking at Table \[table:azadi\], we know they were not looking at the bridge list file before the release of 6.0.5/6.5a3 because some of the bridges in 6.0.5/6.5a3 did not get blocked (including Azadi:4319). We already discussed this in section \[sec:differenttimes\].
Starting from Tor 6.0.6/6.5a4, the censors started preemptively blocking bridges. Azadi:6041 was released in version 6.0.6/6.5a4. However, it got blocked after its ticket was merged and before the release came out. Furthermore, the same ticket that added Azadi:6041 also removed Azadi:4319 from the configuration file. Azadi:4319 was one of the bridges that did not get blocked in 6.0.5/6.5a3. When Azadi:6041 was blocked, Azadi:4319 remained unblocked. Even after 6.0.6/6.5a4 is released, Azadi:4319 was not blocked until after Azadi was blocked on the whole IP address, on November 7, 2016. The likely explanation is that they looked at the ticket or the Git repository for Tor. Either way, this means that they were looking at the bridge list file to block new bridges.
Therefore, we conclude that censors must have started looking at the bridge list file sometime between those two events. This is a change from past behaviors and appears to be a new action taken by China.
**Event** **Date**
-------------------------------------------- ----------
Azadi:4319 opens Aug 30
6.0.5 public release (contains Azadi:4319) Sep 16
Rotate Azadi:4319$\rightarrow$Azadi:6041 Oct 05
Azadi:6041 blocked Oct 20
6.0.6 public release (contains Azadi:6041) Nov 15
: Azadi timeline []{data-label="table:azadi"}
Commented Bridges
-----------------
From Section \[readingbridgepref\] we know that the censors changed their behavior and started reading the bridge list file to find new bridges. We wanted to know whether this process is automatic or manual.
We hypothesized the following:
- Censors parse the bridge list file automatically.
- A person manually reads the bridge list file.
In the 6.0.6/6.5a4 release batch, we incorporated two new bridges at the same time: Lisbeth:443 and NX01:443. We left NX01 commented out and added Lisbeth in as normal, as seen below.
pref(..., "obfs4 192.95.36.142:443 ...");
// Not used yet
// pref(..., "obfs4 85.17.30.79:443 ...");
We discovered that of the two, only Lisbeth was blocked while NX01 remained unblocked for a period of time. NX01 was later blocked after it was uncommented.
Our reasoning is that this would help us distinguish between human inspection and automatic blocking. If humans are processing the source code manually, they are likely to block NX01 and Lisbeth together. If the blocking process is automatic, then NX01 would be left unblocked.
Since only the uncommented bridge Lisbeth was blocked initially, this suggests an automatic parser rather than a manual parser. We believe that had a person been manually viewing the file, they would have also blocked NX01 since they would have seen that it was in the file.
Other Anomalies {#sec:anomalies}
---------------
![ Rates of reachability by time of day for two bridges from two sites, between February 1 and March 15, 2016. There is a diurnal blocking pattern in both China sites, though not the same bridges are affected at both sites. China Standard Time (CST) is UTC+08:00. []{data-label="fig:diurnal"}](data/diurnal)
There is a conspicuous on–off pattern in the reachability of certain bridges from China, for example ndnop3:24215 between January 15 and May 10, 2016. The pattern is roughly periodic with a period of 24 hours. Figure \[fig:diurnal\] averages many 24-hour periods to show the reachability against time of day of two bridges. The presence of the diurnal pattern appears to depend on both the bridge and the probing site, perhaps depending on the network path, as the same bridges do not show the pattern at both sites. The pattern can come and go, for example in riemann:443 before and after April 1, 2016.
The China sites also display what are apparently temporary failures of censorship, stretches of a few hours during which otherwise blocked bridges were reachable. Intriguingly, one of these corresponds to a known failure of the Great Firewall that was documented in the press [@scmp-gfw]. On March 27, Google services—usually blocked in China—were reachable from about 15:30 to 17:15 UTC. This time period is a subset of one in which our bridges were reachable, which went from about 10:00 to 18:00 UTC on that day.
Censorship Measurement Platforms {#sec:ooni-iclab}
--------------------------------
The Open Observatory of Network Interference (OONI) is a project that detects censorship around the globe. We contacted the the OONI team and they agreed to help us with our experiment. This afforded us more than 100 observation sites in different countries that measured Tor bridge connectivity. This measurement started in early December 2016 and we currently have measurement data for approximately three months.
One really interesting observation we have is that the firewalls of several countries might be faking connection responses. These countries includes Thailand, Indonesia, Netherlands and Bulgaria. This behavior is especially obvious in the probe data for the bridge fdctorbridge01:80. The bridge fdctorbridge01:80, at that time, was defunct, having closed down in May 2016 [@torbug-18976]. Naturally we should get a connection error (timeout) when we try to probe it. However, measurements in these few countries showed that the connection to port 80 at this bridge was successful.
Since we only had very few probe locations in Thailand, Indonesia, Netherlands, it might be just a local HTTP proxy problem for our probe sites. However, there are three probe sites in different ASes in Bulgaria, and we noticed that this problem exists in all three probe locations. This seems to indicate this isn’t just a local problem for a specific network. We decided to dig further into Bulgaria by bringing in a VPN, and attempting to make TCP connections to fdctorbridge01. We were not able to reproduce the results on OONI report, however, since all of our connections timed out, which is the expected behavior.
We also noticed that for the probe sites in Bulgaria (AS 44901), connections to port 80 and 443 for certain IP addresses have an extremely short response time, while connections to other ports takes much longer. For example, we saw that connections to port 80 and 443 for MaBishomarim only takes around one millisecond. Connections to other ports take around 500 milliseconds instead. This substantial difference makes us wonder whether a HTTP spoofing is taking place.
Retrospective Analysis {#sec:retrospective}
----------------------
 
There is a gap in our measurement data from China during August 2016. In order to see the bigger picture and capture a larger timeframe, it is preferable that we can look at logs of data even before December 2015.
CollecTor [@collector] is a public Tor data collection service. It collects network status of Tor across the entire network. It records the amount of daily traffic of each bridge. More precisely, it records the number of daily connections each bridge is receiving from each country. We believe that by looking at the number of daily connections from China, we can estimate whether a bridge was blocked. Since it is known that China does not have the same blocking rules across different areas [@Ensafi2015a], this data would contain a substantial amount of noise. Furthermore, not all bridges have data for every single day. Blocking inference from this would be a rough estimate at best, but we still think that it would be useful to see the whole picture.
We used China’s daily traffic data from CollecTor to infer the reachability of different bridges. This can be supplementary to our own measurements. CollecTor can be used to infer historical data that we were not able to see directly. The logic behind this is that when a bridge is no longer reachable from China, there would be a large drop in its daily traffic from China. Due to the fact that China has different blocking rules in different regions, the amount of traffic might never reach zero. However, we still think that the drop in traffic should be large enough for us to distinguish between when the bridge is blocked from China and when it is not.
From Figure 2, we can see CollecTor data and our own measurement for LeifEricson. There is a clear match between the two. When our measurement shows that the bridge is reachable, there would be a spike in the traffic data. Looking at the CollecTor data, we see that the last time LeifEricson was reachable from China was around July 2016. We know that LeifEricson was blocked on the whole IP even before our monitoring started in September, but were unable to observe the time. Looking at the CollecTor traffic data, we suspect the IP block may have started much earlier than we expected, potentially as early as July 2016. Traffic data for MaBishomarim demonstrates the rare case that the traffic data does not match our observation. From September to October 2016, they still match well, but since November 2016, they became contradictory. Our own measurement shows that MaBishomarim has not been reachable since November 2016, but CollecTor shows that MaBishomarim received a large burst of traffic in mid-November 2016. We speculate that this incongruity is the artifact of the different blocking rules in different regions of China. While the region our measurement site is in blocked MaBishomarim, a different region still had MaBishomarim unblocked.
Limitations {#sec:limitations}
===========
In our tests, we assume that a bridge is unblocked if we are able to make a TCP connection. It may be the case, however, that a censor effectively blocks a bridge despite allowing TCP connections. The fact that one can connect to a bridge does not always mean that the censor will allow a sustained obfs4 connection. We know of exactly such a case in Kazakhstan where TCP reachability tests underestimate the level of censorship. The firewall in Kazakhstan blocks Tor Browser’s default obfs4 bridges, but differently from the GFW: it stops transferring packets only some time after a connection is established [@torbug-20348].
We rented a VPN with an endpoint in Kazakhstan (AS 203087) and ran tests from December 18, 2016 to February 4, 2017. There is a risk that the censorship seen by a VPN censorship may not be representative of censorship elsewhere in a country; we have not been able to eliminate that possibility but we verified that the VPN saw censorship of at least some domains like tumblr.com. We ran our usual active probing experiments of all bridges every 20 minutes, as well as hourly attempts to establish a Tor connection to a selection of public and non-public bridges. The results of TCP reachability tests were uniform: all bridges were always reachable. The results of the Tor bootstrap tests told a different and ambiguous story. Tor measures its bootstrapping progress as a fraction between 0 and 100%. Some bridges, like Lisbeth:443 and Mosaddegh:9332, were essentially always able to reach 100%. Others, like Mosaddegh:80, GreenBelt:80, and GreenBelt:5881, always stalled at 10% (indicating a failure after the initial TLS handshake). Yet others, like Mosaddegh:443 and Mosaddegh:1984, reached 25%, showing that at least some Tor protocol data flowed through the obfs4 channel, but not enough to establish a full Tor circuit.
Implications for Circumvention
==============================
There are a number of implications of our findings for the practice of censorship circumvention. One is that even naive approaches like Tor Browser’s default bridges are effective against many censors.
Another implication comes from observation of how the Great Firewall discovers and blocks default bridges. Whether it is done through black-box testing, or inspection of source code, there are ways around, involving complications on the client side. For example, rather than strictly obeying the static list of bridge addresses built in, the client software could deterministically compute some function of the bridge list, perhaps varying over time, whose output is the real set of addresses to use. Suppose the bridge addresses automatically changed every day: a censor doing a black-box test would also have to test every day, else it would miss the new set of bridges. Or suppose the censor parses a file containing a list of bridges: a countermeasure is to tweak addresses so that they do not accurately reflect the addresses the client goes to; adding 1 to each octet of the IP address, for example. That would succeed until the censor devotes energy to reverse engineer the tweaking algorithm.
Of course these simple, incremental countermeasures are essentially security through obscurity, only perpetuating the lamented cat-and-mouse game of censorship and circumvention. Is it even worth pursuing such strategies, rather than looking at new circumvention techniques that are hard for a censor to block in principle? One answer is that yes, as long as censors remain relatively slow and stupid, and a little bit of investment of effort brings a large amount of effective circumvention, it is worth keeping at least a little bit ahead of the censors, even if it means tweaking conceptually broken systems. Of course, we should set our sights farther, and not allow such pursuits to fully distract us from working on fundamental advances. Ultimately, it may be worth it to play the cat-and-mouse game because by doing so and paying careful attention, we learn surprising facts about censors and their operation, revealing weaknesses that can help the development of future systems.
Ethics and Safety
=================
There are risks involved in running Tor experiments. Some risks include disrupting other measurements, disclosing bridge locations, and endangering the bridge operators. During our experiment, we consulted with the Tor Research Safety Board [@torsafetyboard], which helps researchers conduct experiments safely. The research summary we sent to the board is included in Appendix \[sec:safety\].
Tor Research Safety Board {#sec:safety}
=========================
This appendix contains a copy of the research summary we sent to the Tor Research Safety Board [@torsafetyboard], a group of researchers who provide recommendations on conducting research on Tor in a safe way. It is included here with no changes except for formatting.
Code and Data
=============
<https://www.bamsoftware.com/proxy-probe/>
[^1]: Authors are listed in alphabetical order.
|
---
abstract: 'We consider the dynamics of a periodic chain of $N$ coupled overdamped particles under the influence of noise, in the limit of large $N$. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order $N^2$), the system synchronises, in the sense that all particles assume almost the same position in their respective local potential most of the time. In a previous work, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system’s stationary configurations. We analysed, for arbitrary $N$, the behaviour for coupling intensities slightly below the synchronisation threshold. Here we describe the behaviour for any positive coupling intensity $\gamma$ of order $N^2$, provided the particle number $N$ is sufficiently large (as a function of $\gamma/N^2$). In particular, we determine the transition time between synchronised states, as well as the shape of the critical droplet to leading order in $1/N$. Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system’s potential landscape, in which the metastable behaviour is encoded.'
author:
- 'Nils Berglund, Bastien Fernandez and Barbara Gentz'
bibliography:
- '../BFG.bib'
title: |
Metastability in Interacting Nonlinear\
Stochastic Differential Equations II:\
Large-$N$ Behaviour
---
[ Spatially extended systems, lattice dynamical systems, open systems, stochastic differential equations, interacting diffusions, Ginzburg–Landau SPDE, transitions times, most probable transition paths, large deviations, Wentzell-Freidlin theory, diffusive coupling, synchronisation, metastability, symmetry groups, symplectic twist maps. ]{}
Introduction {#sec_in}
============
In this paper, we continue our analysis of the metastable dynamics of a periodic chain of coupled bistable elements, initiated in [@BFG06a]. In contrast with similar models involving discrete on-site variables, or spins, whose metastable behaviour has been studied extensively (see for instance [@denHollander04; @OlivieriVares05]), our model involves continuous local variables, and is therefore described by a set of interacting stochastic differential equations.
The analysis of the metastable dynamics of such a system requires an understanding of its $N$-dimensional potential landscape, in particular the number and location of its local minima and saddles of index $1$. In [@BFG06a], we showed that the number of stationary configurations increases from $3$ to $3^N$ as the coupling intensity $\gamma$ decreases from a critical value $\gamma_1$ of order $N^2$ to $0$. This transition from strong to weak coupling involves a sequence of successive symmetry-breaking bifurcations, and we analysed in detail the first of these bifurcations, which corresponds to desynchronisation.
In the present work, we consider in more detail the behaviour for large particle number $N$. In the limit $N\to\infty$, the system tends to a Ginzburg–Landau stochastic partial differential equation (SPDE), studied for instance in [@EckmannHairer01; @Rougemont02]. The Ginzburg–Landau SPDE describes in particular the behaviour near bifurcation points of more complicated equations, such as the stochastic Swift–Hohenberg equation [@BlomkerHairerPavliotis05]. For large but finite $N$, it turns out that a technique known as spatial map analysis allows us to obtain a precise control of the set of stationary points, for values of the coupling well below the synchronisation threshold. More precisely, given a strictly positive coupling intensity $\gamma$ of order $N^2$, there is an integer $N_0(\gamma/N^2)$ such that for all $N\geqs
N_0(\gamma/N^2)$, we know precisely the number, location and type of the potential’s stationary points. This allows us to characterise the transition times and paths between metastable states for all these values of $\gamma$ and $N$.
This paper is organised as follows. Section \[sec\_res\] contains the precise definition of our model, and the statement of all results. After introducing the model in Section \[ssec\_mod\] and describing general properties of the potential landscape in Section \[ssec\_pot\], we explain the heuristics for the limit $N\to\infty$ in Section \[ssec\_heuristics\]. In Section \[ssec\_largeN\], we state the detailed results on number and location of stationary points for large but finite $N$, and in Section \[ssec\_stoch\] we present their consequences for the stochastic dynamics. Section \[sec\_tm\] contains the proofs of these results. The proofs rely on a detailed analysis of the orbits of period $N$ of a near-integrable twist map, which are in one-to-one correspondence with stationary points of the potential. Appendix \[app\_ell\] recalls some properties of Jacobi’s elliptic functions needed in the analysis, while Appendix \[sec\_prtech\] contains some more technical proofs of results stated in Section \[sec\_tmgchi\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
Financial support by the French Ministry of Research, by way of the [*Action Concertée Incitative (ACI) Jeunes Chercheurs, Modélisation stochastique de systèmes hors équilibre*]{}, is gratefully acknowledged. NB and BF thank the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, for financial support and hospitality. BG thanks the ESF Programme [*Phase Transitions and Fluctuation Phenomena for Random Dynamics in Spatially Extended Systems (RDSES)*]{} for financial support, and the Centre de Physique Théorique (CPT), Marseille, for kind hospitality.
Model and Results {#sec_res}
=================
Definition of the Model {#ssec_mod}
-----------------------
Our model of interacting bistable systems perturbed by noise is defined by the following ingredients:
The periodic one-dimensional lattice is given by $\lattice=\Z/N\Z$, where $N\geqs2$ is the number of particles.
To each site $i\in\lattice$, we attach a real variable $x_i\in\R$, describing the position of the $i$th particle. The configuration space is thus $\cX=\R^\lattice$.
Each particle feels a local bistable potential, given by $$\label{mod1}
U(\xi) = \frac14 \xi^4 - \frac12 \xi^2\;,
\qquad
\xi\in\R\;.$$ The local dynamics thus tends to push the particle towards one of the two stable positions $\xi=1$ or $\xi=-1$.
Neighbouring particles in $\Lambda$ are coupled via a discretised-Laplacian interaction, of intensity $\gamma/2$.
Each site is coupled to an independent source of noise, of intensity $\sigma$. The sources of noise are described by independent Brownian motions $\set{B_i(t)}_{t\geqs0}$.
The system is thus described by the following set of coupled stochastic differential equations, defining a diffusion on $\cX$: $$\label{mod2}
\6x^\sigma_i(t) = f(x^\sigma_i(t))\6t
+ \frac\cng2 \bigbrak{x^\sigma_{i+1}(t)-2x^\sigma_i(t)+x^\sigma_{i-1}(t)}
\6t
+ \sigma \6B_i(t)\;,
\qquad i\in\Lambda\;,$$ where the local nonlinear drift is given by $$\label{mod3}
f(\xi) = -\nabla U(\xi) = \xi - \xi^3\;.$$ For $\sigma=0$, the system is a gradient system of the form $\dot x = -\nabla V_\gamma(x)$, with potential $$\label{mod5}
V_{\gamma}(x) =
\sum_{i\in\lattice} U(x_i) + \frac\cng4 \sum_{i\in\lattice}
(x_{i+1}-x_i)^2\;.$$
Potential Landscape and Metastability {#ssec_pot}
-------------------------------------
The dynamics of the stochastic system depends essentially on the potential landscape $V_\gamma$. As in [@BFG06a], we use the notations $$\label{mod7}
\cS = \cS(\gamma)
= \setsuch{x\in\cX}{\nabla V_{\gamma}(x)=0}$$ for the set of stationary points, and $S_k(\gamma)$ for the set of $k$-saddles, that is, stationary points with $k$ unstable directions and $N-k$ stable directions.
Understanding the dynamics for small noise essentially requires knowing the graph $\cG=(\cS_0,\cE)$, in which two vertices $x^\star, y^\star\in\cS_0$ are connected by an edge $e\in\cE$ if and only if there is a $1$-saddle $s\in\cS_1$ whose unstable manifolds converge to $x^\star$ and $y^\star$. The system behaves essentially like a Markovian jump process on $\cG$. The mean transition time from $x^\star$ to $y^\star$ is of order $\e^{2H/\sigma^2}$, where $H$ is the potential difference between $x^\star$ and the lowest saddle leading to $y^\star$ (see [@FW]).
It is easy to see that $\cS$ always contains at least the three points $$\label{pot1}
O = (0,\dots,0)\;, \qquad I^{\pm} = \pm(1,\dots,1)\;.$$ Depending on the value of $\gamma$, the origin $O$ can be an $N$-saddle, or a $k$-saddle for any odd $k$. The points $I^{\pm}$ always belong to $\cS_0$, in fact we have $$\label{pot2}
V_\gamma(x) > V_\gamma(I^+) = V_\gamma(I^-) = -\frac N4 \quad \forall
x\in\cX
\setminus\set{I^-,I^+}$$ for all $\gamma>0$, so that $I^+$ and $I^-$ represent the most stable configurations of the system. The three points $O$, $I^+$ and $I^-$ are the only stationary points belonging to the diagonal $$\label{pot2A}
\cD=\setsuch{x\in\cX}{x_1=x_2=\dots=x_N}\;.$$ On the other hand, being a polynomial of degree $4$ in $N$ variables, the potential $V_\gamma$ can have up to $3^N$ stationary points.
The potential $V_\gamma(x)$, as well as the sets $S(\gamma)$ and $S_k(\gamma)$, are invariant under the transformation group $G=G_N$ of order $4N$ ($4$ if $N=2$), generated by the following three symmetries:
the rotation around the diagonal given by $R(x_1,\dots,x_N)=(x_2,\dots, x_N,x_1)$;
the mirror symmetry $S(x_1,\dots,x_N)=(x_N,\dots,x_1)$;
the point symmetry $C(x_1,\dots,x_N)=-(x_1,\dots,x_N)$.
height 12pt depth 6pt width 0pt $N$ $x$ Type of symmetry
---------------------------------------- ----- --------------------------------------------------------------------
height 12pt depth 6pt width 0pt $4L$ $A$ $(x_1,\dots,x_L,x_L,\dots,x_1,-x_1,\dots,-x_L,-x_L,\dots,-x_1)$
height 8pt depth 6pt width 0pt $B$ $(x_1,\dots,x_L,\dots,x_1,0,-x_1,\dots,-x_L,\dots,-x_1,0)$
height 12pt depth 6pt width 0pt $4L+2$ $A$ $(x_1,\dots,x_{L+1},\dots,x_1,-x_1,\dots,-x_{L+1},\dots,-x_1)$
height 8pt depth 6pt width 0pt $B$ $(x_1,\dots,x_L,x_L\dots,x_1,0,-x_1,\dots,-x_L,-x_L,\dots,-x_1,0)$
height 12pt depth 6pt width 0pt $2L+1$ $A$ $(x_1,\dots,x_L,-x_L,\dots,-x_1,0)$
height 8pt depth 6pt width 0pt $B$ $(x_1,\dots,x_L,x_L,\dots,x_1,x_0)$
: Symmetries of the stationary points bifurcating from the origin at $\gamma=\gamma_1$. The situation depends on whether $N$ is odd (in which case we write $N=2L+1$) or even (in which case we write $N=4L$ or $N=4L+2$, depending on the value of $N \pmod4$). Points labelled $A$ are $1$-saddles near the desynchronisation bifurcation at $\gamma=\gamma_1$, those labelled $B$ are $2$-saddles (for odd $N$, this is actually a conjecture). More saddles of the same index are obtained by applying elements of the symmetry group $G_N$ to $A$ and $B$.[]{data-label="table_desync"}
In [@BFG06a], we proved the following results:
There is a critical coupling intensity $$\label{pot2B}
\gamma_1 = \frac1{1-\cos(2\pi/N)}$$ such that for all $\gamma\geqs\gamma_1$, the set of stationary points $\cS$ consists of the three points $O$ and $I^\pm$ only. The graph $\cG$ has two vertices $I^\pm$, connected by a single edge.
As $\gamma$ decreases below $\gamma_1$, an even number of new stationary points bifurcate from the origin. Half of them are $1$-saddles, while the others are $2$-saddles. These points satisfy symmetries as shown in Table \[table\_desync\]. The potential difference between $I^\pm$ and the $1$-saddles behaves like $N(1/4-(\gamma_1-\gamma)^2/6)$ as $\gamma\nearrow\gamma_1$.
New bifurcations of the origin occur for $\gamma=\gamma_M =
(1-\cos(2\pi M/N))^{-1}$, with $2\leqs M\leqs N/2$, in which saddles of order higher than $2$ are created.
The number of stationary points emerging from the origin at the desynchronisation bifurcation at $\gamma=\gamma_1$ depends on the parity of $N$. If $N$ is even, there are exactly $2N$ new points ($N$ saddles of index $1$, and $N$ saddles of index $2$). If $N$ is odd, we were only able to prove that the number of new stationary points is a multiple of $4N$, but formulated the conjecture that there are exactly $4N$ stationary points ($2N$ saddles of index $1$, and $2N$ saddles of index $2$). We checked this conjecture numerically for all $N$ up to $101$. As we shall see in Section \[ssec\_largeN\], the conjecture is also true for $N$ sufficiently large.
Heuristics for the Large-$N$ Limit {#ssec_heuristics}
----------------------------------
We want to determine the structure of the set $\cS$ of stationary points for large particle number $N$, and large coupling intensity $\gamma$. For this purpose, we introduce the rescaled coupling intensity $$\label{LargeN1}
\gammat = \frac\gamma{\gamma_1}
= \frac{2\pi^2}{N^2} \gamma
\biggbrak{1+\biggOrder{\frac1{N^2}}}\;.$$ Then, the desynchronisation bifurcation occurs for $\gammat=1$. We will consider values of $\gammat$ which may be smaller than $1$, but are bounded away from zero. The reason why the set of stationary points can be controlled in this regime is that as $N\to\infty$, the deterministic system $\dot x=-\nabla V_\gamma(x)$ behaves like a Ginzburg–Landau partial differential equation (PDE). Indeed, assume that the $N$ sites of the chain are evenly distributed on a circle of radius $1$, and that there exists a smooth function $u(\ph,t)$, $\ph\in\fS^1$, interpolating the coordinates of $x(t)$ in such a way that $$\label{heur1}
u\Bigpar{2\pi\frac{i}{N},t} = x_i(t)
\qquad \forall i\in\Lambda\;.$$ Then in the limit $N\to\infty$, the discrete Laplacian in converges to a constant times the second derivative of $u(\cdot,t)$, and we obtain the PDE $$\label{heur2}
\sdpar ut(\ph,t) = f(u(\ph,t)) + \gammat \sdpar u{\ph\ph}(\ph,t)\;.$$ Stationary solutions of satisfy the equation $$\label{heur3}
\gammat u''(\ph) = -f(u(\ph))\;,$$ describing the motion of a particle of mass $\gammat$ in the *inverted* potential $-U(\ph)$. The prefactor $\gammat$ can be removed by scaling $\ph$: Setting $u_0(\phi)=u(\sqrt{\gammat}\phi)$, we see that $u_0$ satisfies the equation $$\label{heur4}
u_0'' = -f(u_0) = u_0^3 - u_0\;.$$ All periodic solutions of this equation are known (cf. Section \[sec\_tmaa\]), and can be expressed in terms of Jacobi’s elliptic functions[^1] as $$\label{heur5}
u_0(\phi) = a(\kappa) \sn
\biggpar{\frac{\phi-\phi_0}{\sqrt{1+\kappa^2}},\kappa}\;,$$ where
$\phi_0$ is an arbitrary phase;
$\kappa\in[0,1)$ is an auxiliary parameter controlling the shape of the function: For small $\kappa$, the function is close to a sine, while it approaches a square wave as $\kappa\nearrow1$;
the amplitude $a(\kappa)$ is given by $$\label{heur6}
a(\kappa)^2 = \frac{2\kappa^2}{1+\kappa^2}\;;$$
the period of $u_0(\ph)$ is $4\sqrt{1+\kappa^2}\JK(\kappa)$, where $\JK$ denotes the complete elliptic integral of the first kind.
We are looking for periodic solutions of of period $2\pi/\sqrt{\gammat}$. Such solutions exist whenever the shape parameter $\kappa$ satisfies the condition $$\label{heur7}
4\sqrt{1+\kappa^2}\JK(\kappa) = \frac{2\pi M}{\sqrt{\gammat}}$$ for some integer $M$, which plays the rôle of a *winding number* controlling the number of sign changes of $u_0$. Equation imposes a relation between shape parameter $\kappa$ and rescaled coupling intensity $\gammat$, shown in in the case $M=1$. On the other hand, the phase $\phi_0$ is completely free.
The left-hand side of being bounded below by $2\pi$, solutions of given winding number $M$ exist provided $\gammat \leqs 1/M^2$. The smaller $\gammat$, the more different types of periodic solutions exist. A new one-parameter family of stationary solutions, parametrised by $\phi_0$, bifurcates from the identically zero solution every time $\gammat$ becomes smaller than $1/M^2$, $M=1,2,\dots$ (a).
![[**(a)**]{} Schematic bifurcation diagram of the limiting PDE . Whenever the rescaled coupling intensity $\gammat$ decreases below $1/M^2$, $M=1,2,\dots$, a one-parameter family of stationary solutions bifurcates from the identically zero solution $u\equiv0$. [**(b)**]{} Relations between the shape parameter $\alkappa$, the amplitude $a$ and the rescaled coupling intensity $\gammat$ for winding number $M=1$.[]{data-label="fig_largeN1"}](figs/fig_bifNinf "fig:"){height="45mm"} ![[**(a)**]{} Schematic bifurcation diagram of the limiting PDE . Whenever the rescaled coupling intensity $\gammat$ decreases below $1/M^2$, $M=1,2,\dots$, a one-parameter family of stationary solutions bifurcates from the identically zero solution $u\equiv0$. [**(b)**]{} Relations between the shape parameter $\alkappa$, the amplitude $a$ and the rescaled coupling intensity $\gammat$ for winding number $M=1$.[]{data-label="fig_largeN1"}](figs/fig_kappa_rev "fig:"){height="45mm"}
Finally, note that for stationary points $x$ satisfying , with $u$ given by , the value of the renormalised potential $V_\gamma(x)/N$ converges, in the limit $N\to\infty$, to an integral which can be computed explicitly (see Section \[sec\_tmaa\]) in terms of the parameter $\kappa$: $$\label{heur8}
\lim_{N\to\infty} \frac{V_\gamma(x)}{N}
= -\frac1{3(1+\alkappa^2)} \biggbrak{\frac{2+\alkappa^2}{1+\alkappa^2} -
2\frac{\JE(\alkappa)}{\JK(\alkappa)}}\;,$$ where $\JE$ denotes the complete elliptic integral of the second kind.
If we were to add noise to the PDE , we would obtain a Ginzburg–Landau SPDE. In that case, we expect that the configurations of highest energy reached in the course of a typical transition from $u\equiv-1$ to $u\equiv1$ are of the form , with winding number $M=1$. As a consequence, the potential difference in should be governing the typical time of such transitions. However, proving this would involve an infinite-dimensional version of Wentzell–Freidlin theory, moreover in a degenerate situation, which is beyond the scope of the present work. We will henceforth consider situations with large, but finite particle number.
Main Results: Stationary Points for Large but Finite $N$ {#ssec_largeN}
--------------------------------------------------------
We examine now the structure of the set $\cS(\gamma)$ for large, but finite particle number $N$. Instead of the limiting differential equation , stationary points satisfy the difference equation $$\label{largeN01}
\frac\cng2 \bigbrak{x_{n+1}-2x_n+x_{n-1}} = -f(x_n)\;,
\qquad n\in\Lambda\;.$$ The key idea of the analysis is to interpret $n$ as discrete time, and to consider as defining $x_{n+1}$ in terms of $x_n$ and $x_{n-1}$. Setting $v_n = x_n - x_{n-1}$ allows to rewrite as the system $$\label{largeN02}
\begin{split}
x_{n+1} &= x_n + v_{n+1}\;, \\
v_{n+1} &= v_n - 2\gamma^{-1} f(x_n)\;.
\end{split}$$ The map $(x_n,v_n)\mapsto(x_{n+1},v_{n+1})$ is an area-preserving twist map (twist meaning that $x_{n+1}$ is a monotonous function of $v_n$), for the study of which many tools are available [@Meiss92]. Stationary points of the potential $V_\gamma$ are in one-to-one correspondence with periodic points of period $N$ of this map. If we further scale $v$ by a factor $\eps=\sqrt{2/\gamma}$, we obtain the equivalent map $$\label{largeN03}
\begin{split}
x_{n+1} &= x_n + \eps y_{n+1}\;, \\
y_{n+1} &= y_n - \eps f(x_n)\;.
\end{split}$$ The regime of large particle number $N$ and finite rescaled coupling intensity $\gammat$ corresponds to large $\gamma$, and thus to small $\eps$. The map is a discretisation of the system of ordinary differential equations $\dot x=y$, $\dot y=-f(x)$, which is equivalent to the continuous limit equation . There should thus be some similarity between the orbits of the map and of the system . In particular, one easily checks that the energy $$\label{largeN04}
E(x,\dot x) = \frac12\dot x^2 - U(x)\;,$$ which is conserved in the continuous limit, changes only slightly, by an amount of order $\eps^2$, for the map (setting $y=\dot
x$). The map is thus close to integrable, which makes its analysis accessible to perturbation theory.
![Partial bifurcation diagram for a case where $N=4L$ is a multiple of four, and some associated graphs $\cG$. Only one stationary point per orbit of the symmetry group $G$ is shown. Dash–dotted curves with $k$ dots represent $k$-saddles. The symbols at the left indicate the zero-coupling limit of the stationary points’ coordinates, for instance $1^{2L}(-1)^{2L}$ stands for a point whose first $2L$ coordinates are equal to $1$, and whose last $2L$ coordinates are equal to $-1$. The numbers associated with the branch created at $\gammat_3$ are $L_1=\intpart{2L/3}$, $L_2=2(L-L_1)$, $L_3=\intpart{2L/3+1/2}$ and $L_4=2(L-L_3)-1$ (in case $N$ is a multiple of $12$, there are more vanishing coordinates).[]{data-label="fig_bifN"}](figs/fig_bifN1){width="142mm"}
Our main result, obtained by analysing the map , is that the bifurcation diagram looks like the one shown in . Namely,
For $\gammat > 1$, $I^+$, $I^-$ and $O$ are the only stationary points.
Below $\gammat = 1$, an explicitly known number of saddles of index $1$ and $2$ bifurcate from the origin.
For any $2\leqs M\leqs N/2$, an explicitly known number of saddles of index $2M-1$ and $2M$ bifurcate from the origin at $\gammat =
\gammat_M$, where $$\label{LargeN2}
\gammat_M = \frac{\gamma_M}{\gamma_1}
= \frac{1-\cos(2\pi/N)}{1-\cos(2\pi M/N)}
= \frac1{M^2} + \biggOrder{\frac1{N^2}}\;.$$
For any fixed $M$, if $N$ is sufficiently large compared to $M$, the above list of stationary points is complete for $\gammat>\gammat_M$. In particular, there are no secondary bifurcations of existing branches of stationary points, and no stationary points created by saddle–node bifurcation for these values of $\gammat$.
The main difficulty is to rule out the appearance of other stationary points away from the origin. Indeed, for perturbed integrable maps it is easy to obtain a lower bound on the number of periodic points, using the Poincaré–Birkhoff theorem, but it is hard to obtain an upper bound. One might imagine a scenario where stationary points appear far from the origin, which ultimately offer a more economic path for the transition from $I^-$ to $I^+$.
We now give the precise formulation of the results. We first describe the behaviour between the first two bifurcation values $\gammat_1$ and $\gammat_2$. Below, $\gcd(a,b)$ denotes the greatest common divisor of two integers $a$ and $b$, and $O_x=\setsuch{gx}{g\in G}$ denotes the group orbit of a point $x\in\cX$ under the symmetry group $G$.
\[thm\_LargeN1\] There exists $N_1<\infty$ such that when $N\geqs N_1$ and $\gammat_2<\gammat<\gammat_1=1$, the set $\cS$ of stationary points of the potential $V_\gamma$ has cardinality $$\label{LargeN3B}
\abs{\cS} = 3 + \frac{4N}{\gcd(N,2)} =
\begin{cases}
3+2N & \text{if $N$ is even\;,}\\
3+4N & \text{if $N$ is odd\;,}
\end{cases}$$ There exist points $A=A(\gammat)$ and $B=B(\gammat)$ in $\cX$ such that $\cS$ can be decomposed as[^2] $$\begin{aligned}
\nonumber
\cS_0 &= O_{I^+} = \set{I^+,I^-}\;, \\
\nonumber
\cS_1 &= O_{A} = \set{\pm A, \pm RA, \dots, \pm R^{N-1}A}\;, \\
\nonumber
\cS_2 &= O_{B} = \set{\pm B, \pm RB, \dots, \pm R^{N-1}B}\;, \\
\cS_3 &= O_{O} = \set{O}\;.
\label{LargeN4}\end{aligned}$$ The potential difference between the $1$-saddles and the well bottoms (which is the same for all $1$-saddles and well bottoms) satisfies $$\label{largeN6}
\frac{V_\gamma(A(\gammat))-V_\gamma(I^\pm)}N = \frac14 -
\frac1{3(1+\alkappa^2)}
\biggbrak{\frac{2+\alkappa^2}{1+\alkappa^2} - 2
\frac{\JE(\alkappa)}{\JK(\alkappa)}} +
\biggOrder{\frac{\alkappa^2}{N}}\;,$$ where $\alkappa=\alkappa(\gammat)$ is defined implicitly by $$\label{LargeN3}
\gammat = \frac{\pi^2}{4\JK(\alkappa)^2(1+\alkappa^2)}\;.$$
The detailed proofs are given in Section \[sec\_tm\].
The second result, which is also proved in Section \[sec\_tm\], concerns the behaviour for subsequent bifurcation values $\gammat_M$, $M\geqs 2$.
\[thm\_LargeN2\] For any $M\geqs2$, there exists $N_M<\infty$ such that when $N\geqs N_M$ and $\gammat_{M+1}<\gammat<\gammat_M$, the set $\cS$ of stationary points of the potential $V_\gamma$ has cardinality $$\label{LargeN7}
\abs{\cS} = 3 + \sum_{m=1}^M \frac{4N}{\gcd(N,2m)}\;.$$ There exist points $A^{(m)}$ and $B^{(m)}$ in $\cX$, $m=1,\dots,M$, such that $\cS$ can be decomposed as $$\begin{aligned}
\nonumber
\cS_0 &= O_{I^+} = \set{I^+,I^-}\;, \\
\nonumber
\cS_{2m-1} &= O_{A^{(m)}}\;,
&
m &= 1,\dots,M\;, \\
\nonumber
\cS_{2m} &= O_{B^{(m)}}\;,
&
m &= 1,\dots,M\;, \\
\cS_{2M+1} &= O_{O} = \set{O}\;,
\label{LargeN8}\end{aligned}$$ The potential difference between the saddles $A^{(m)}_j(\gammat)$ and the well bottoms satisfies a similar relation as , but with $\alkappa=\alkappa(m^2\gammat)$.
![[**(a)**]{} Coordinates of the $1$-saddles $A$ in the case $N=32$, shown for two different values of the coupling $\gammat'>\gammat''$. [**(b)**]{} Coordinates of the $3$-saddles $A^{(2)}$ in the case $N=32$, shown for the coupling intensities $\gammat'/4$ and $\gammat''/4$.[]{data-label="fig_symell"}](figs/fig_symell){height="40mm"}
The proof actually yields information on the coordinates of the points $A=A(\gammat)$ and $B=B(\gammat)$:
The coordinates of $A$ and $B$ satisfy the symmetries indicated in Table \[table\_desync\].
If $N$ is even, the coordinates of $A$ and $B$ are given by $$\begin{aligned}
\nonumber
A_j(\gammat) &= a(\alkappa(\gammat)) \sn
\biggpar{\frac{4\JK(\alkappa(\gammat))}N \bigpar{j-\tfrac12},
\alkappa(\gammat)} +
\biggOrder{\frac1{N}}\;, \\
B_j(\gammat) &= a(\alkappa(\gammat)) \sn
\biggpar{\frac{4\JK(\alkappa(\gammat))}N j, \alkappa(\gammat)} +
\biggOrder{\frac1{N}}\;,
\label{LargeN5}\end{aligned}$$ where the amplitude $a(\alkappa)$ is the one defined in .
If $N$ is odd, the coordinates of $A$ and $B$ are given by $$\begin{aligned}
\nonumber
A_j(\gammat) &= a(\alkappa(\gammat)) \sn
\biggpar{\frac{4\JK(\alkappa(\gammat))}N j, \alkappa(\gammat)} +
\biggOrder{\frac1{N}}\;, \\
B_j(\gammat) &= a(\alkappa(\gammat)) \cn
\biggpar{\frac{4\JK(\alkappa(\gammat))}N j, \alkappa(\gammat)} +
\biggOrder{\frac1{N}}\;.
\label{largeN7}\end{aligned}$$
The components of $A^{(m)}$ and $B^{(m)}$ are given by similar expressions, with $\gammat$ replaced by $m^2\gammat$, $j-\frac12$ replaced by $m(j-\frac12)$ and $j$ replaced by $mj$.
Note that the total number of stationary points accounted for by these results is of the order $N^2$, which is much less than the $3^N$ points present at zero coupling. Many additional stationary points thus have to be created as the rescaled coupling intensity $\cngt$ decreases sufficiently, either by pitchfork-type second-order bifurcations of already existing points, or by saddle-node bifurcations. However, the values $\gammat(N)$ for which these bifurcations occur have to satisfy $\lim_{N\to\infty}\gammat(N)=0$.
The existence of second-order bifurcations follows from stability arguments. For instance, for even $N$, the point $A(\gammat)$ converges to $(1,1,\dots,1,-1,-1,\dots,-1)$ as $\gammat\to0$, which is a local minimum of $V_\gamma$ instead of a $1$-saddle. The $A$-branch thus has to bifurcate at least once as the coupling decreases to zero (). For odd $N$, by contrast, the point $A(\gammat)$ converges to $(1,1,\dots,1,0,-1,-1,\dots,-1)$ as $\gammat\to0$, which is also a $1$-saddle. We thus expect that the point $A(\gammat)$ does not undergo any bifurcations for $0\leqs\gammat<1$ if $N$ is odd.
![Partial bifurcation diagram for a case where $N=4L$ is a multiple of four, showing the expected bifurcation behaviour of the critical $1$-saddle in the zero-coupling limit.[]{data-label="fig_bifN2"}](figs/fig_bifN2){width="120mm"}
Stochastic Case {#ssec_stoch}
---------------
We return now to the behaviour of the system of stochastic differential equations $$\label{stoch1}
\6x^\sigma_i(t) = f(x^\sigma_i(t))\6t
+ \frac\cng2 \bigbrak{x^\sigma_{i+1}(t)-2x^\sigma_i(t)+x^\sigma_{i-1}(t)}
\6t
+ \sigma \6B_i(t)\;,
\qquad i\in\Lambda\;.$$ Our main goal is to characterise the noise-induced transition from the configuration $I^-=(-1,-1,\dots,-1)$ to the configuration $I^+=(1,1,\dots,1)$. In particular, we are interested in the time needed for this transition to occur, and by the shape of the critical configuration, i.e., the configuration of highest energy reached during the transition.
In [@BFG06a Theorem 2.7], we obtained that in the synchronisation regime $\gammat>1$, for any initial condition $x_0$ in a ball $\cB(I^-,r)$ of radius $r<1/2$ around $I^-$, any particle number $N\geqs2$ and any constant $\delta>0$, the first-hitting time $\tau_+ = \tauhit(\cB(I_+,r))$ of a ball $\cB(I^+,r)$ of radius $r$ around $I^+$ satisfies $$\label{stoch4}
\lim_{\sigma\to0}
\bigprobin{x_0}{\e^{(N/2-\delta)/\sigma^2} < \tau_+ <
\e^{(N/2+\delta)/\sigma^2}} = 1$$ and $$\label{stoch4b}
\lim_{\sigma\to0} \sigma^2 \log
\expecin{x_0}{\tau_+} = \frac N2\;.$$ This means that in the synchronisation regime, the transition between $I^-$ and $I^+$ takes a time of the order $\e^{N/2\sigma^2}$. Furthermore, for any fixed radius $R\in(r,1/2)$, the first-hitting time $\tau_O =
\tauhit(\cB(O,r))$ of a ball around the origin satisfies $$\label{stoch6}
\lim_{\sigma\to0}
\bigpcondin{x_0}{\tau_O < \tau_+}{\tau_+ < \tau_-} = 1\;,$$ where $\tau_- = \inf\setsuch{t>\tauexit(\cB(I^-,R))}{x_t\in\cB(I^-,r)}$ is the time of first return to the small ball $\cB(I^-,r)$ after leaving the larger ball $\cB(I^-,R)$. This means that during a transition, the system is likely to pass close to the origin, i.e., the origin, being the only saddle of $V_\gamma$, is the critical configuration of the transition.
![Value of the rescaled potential barrier height $h(\gammat)=(V_\gamma(A)-V_\gamma(I^\pm))/N$ as a function of the rescaled coupling intensity $\cngt$. For comparison, we also show the rescaled barrier height $(V_\gamma(A^{(2)})-V_\gamma(I^\pm))/N$ for a stationary point of the higher winding number $M=2$.[]{data-label="fig_largeN2"}](figs/fig_Hofgamma_rev){height="55mm"}
We can now prove a similar result in the desynchronised regime $\gammat<1$.
\[thm\_stoch2\] For $\gammat<1$, let $$\label{stoch7}
h(\gammat) = \frac{V_\gamma(A(\gammat))-V_\gamma(I^\pm)}N = \frac14 -
\frac1{3(1+\alkappa^2)}
\biggbrak{\frac{2+\alkappa^2}{1+\alkappa^2} - 2
\frac{\JE(\alkappa)}{\JK(\alkappa)}} +
\biggOrder{\frac{\alkappa^2}{N}}\;,$$ where $\alkappa=\alkappa(\gammat)$ is defined implicitly by . Fix an initial condition $x_0\in\cB(I^-,r)$. Then for any $0<\gammat<1$, and any $\delta>0$, there exists $N_0(\gammat)$ such that for all $N>N_0(\gammat)$, $$\label{stoch8}
\lim_{\sigma\to0}
\bigprobin{x_0}{\e^{(2Nh(\gammat)-\delta)/\sigma^2} < \tau_+ <
\e^{(2Nh(\gammat)+\delta)/\sigma^2}} = 1\;$$ and $$\label{stoch8b}
\lim_{\sigma\to0} \sigma^2 \log
\expecin{x_0}{\tau_+} = 2Nh(\gammat)\;.$$ Furthermore, let $$\label{stoch9}
\tau_A = \tauhit\Bigpar{\bigcup_{g\in G}\cB(gA,r)}\;,$$ where $A=A(\gammat)$ satisfies (or if $N$ is odd). Then for any $N>N_0(\gammat)$, $$\label{stoch10}
\lim_{\sigma\to0}
\bigpcondin{x_0}{\tau_A < \tau_+}{\tau_+ < \tau_-} = 1\;.$$
The relations and mean that the transition time between the synchronised states $I^-$ and $I^+$ is of order $\e^{2Nh(\gammat)/\sigma^2}$, while relation implies that the set of critical configurations is given by the group orbit of $A$ under $G$.
The large-$N$ limit of the rescaled potential difference $h(\gammat)$ is shown in . The limiting function is increasing, with a discontinuous second-order derivative at $\gammat=1$. For small $\gammat$, $h(\gammat)$ grows like the square-root of $\gammat$. This is compatible with the weak-coupling behaviour $h=(1/4+3/2\gamma+\Order{\gamma^2})/N$ obtained in [@BFG06a], if one takes into account the scaling of $\gammat$.
The critical configuration, that is, the configuration with highest energy reached in the course of the transition from $I^-$ to $I^+$, is any translate of the configuration $A(\gammat)$ shown in a. If $N$ is even, it has $N/2$ positive and $N/2$ negative coordinates, while for odd $N$, there are $(N-1)/2$ positive, one vanishing, and $(N-1)/2$ negative coordinates. The sites with positive and negative coordinates are always adjacent. The potential difference between the $1$-saddles $A$ and the $2$-saddles $B$ is actually very small, so that transition paths become less localised as the particle number $N$ increases, reflecting the fact that the system becomes translation-invariant in the large-$N$ limit.
Proofs {#sec_tm}
======
Strategy of the Proof {#sec_tmstrat}
---------------------
The proof of Theorems \[thm\_LargeN1\] and \[thm\_LargeN2\] is based on the fact that stationary points of the potential satisfy the relation $$\label{tm1}
f(x_n) + \frac\cng2 \bigbrak{x_{n+1}-2x_n+x_{n-1}} =
0\;,$$ where $f(x)=x-x^3$. As mentioned in Section \[ssec\_largeN\], this relation can be rewritten as a two-dimensional area-preserving twist map $$\label{tm2}
\begin{split}
x_{n+1} &= x_n + v_{n+1}\;, \\
v_{n+1} &= v_n - 2\gamma^{-1} f(x_n)\;.
\end{split}$$ whose periodic points correspond to stationary points of the potential. In fact, we are going to analyse a slightly different equivalent map, which has the advantage to use the symmetries of the model in a more efficient way.
The proof is organised as follows:
In Section \[sec\_tmsym\], we introduce the alternative twist map, adapted to symmetries.
In Section \[sec\_tmaa\], we compute the expression of the map in action–angle variables, taking advantage of the existence of an almost conserved quantity.
In Section \[sec\_tmgf\], we compute the generating function of the map in action–angle variables. This reduces the problem of finding periodic orbits to a variational problem (which is simpler than the original one).
The main difficulty is that the system is almost degenerate along the translation mode. In Section \[sec\_tmgff\], we introduce Fourier variables in order to decouple the translation mode from the other, oscillating modes.
In Section \[sec\_tmgchi\], we deal with the oscillating modes, by showing with the help of Banach’s contraction principle that for each value of the translation mode, there is exactly one value of the oscillating modes yielding a stationary point.
In Section \[sec\_tmgstat\], we deal with the translation mode, by reducing the problem to one dimension, and showing that the generating function is dominated by its leading Fourier mode in this direction. This yields the exact number of stationary points.
Finally, in Section \[sec\_tmgindex\] we consider the stability of the stationary points.
Symmetric Twist Map {#sec_tmsym}
-------------------
The twist map does not exploit the symmetries of the original system in an optimal way. In order to do so, it is more advantageous to introduce the variable $$\label{tmsym1}
u_n = \frac{x_{n+1}-x_{n-1}}2$$ instead of $v_n$. Then a short computation shows that $$\label{tmsym2}
\begin{split}
x_{n+1} &= x_n + u_n - \gamma^{-1} f(x_n)\;,\\
u_{n+1} &= u_n - \gamma^{-1} \bigbrak{f(x_n)+f(x_{n+1})}\;.
\end{split}$$ The map $T_1: (x_n,u_n)\mapsto(x_{n+1},u_{n+1})$ is also an area-preserving twist map. Although it looks more complicated than the map , it has the advantage that its inverse is obtained by changing the sign of $u$, namely $$\label{tmsym3}
\begin{split}
x_n &= x_{n+1} - u_{n+1} - \gamma^{-1} f(x_{n+1})\;,\\
u_n &= u_{n+1} + \gamma^{-1} \bigbrak{f(x_{n+1})+f(x_n)}\;.
\end{split}$$ This implies that if we introduce the involutions $$\label{tmsym4}
S_1: (x,u) \mapsto (-x,u)
\qquad\text{and}\qquad
S_2: (x,u) \mapsto (x,-u)\;,$$ then the map $T_1$ and its inverse are related by $$\label{tmsym5}
T_1 \circ S_1 = S_1 \circ (T_1)^{-1}
\qquad\text{and}\qquad
T_1 \circ S_2 = S_2 \circ (T_1)^{-1}\;,$$ as a consequence of $f$ being odd. This implies that the images of an orbit of the map under $S_1$ and $S_2$ are also orbits of the map.
For large $N$, it turns out to be useful to introduce the small parameter $$\label{tm3}
\eps = \sqrt{\frac2\cng}
= \sqrt{\frac2{{\gamma_1\cngt}}}
= \frac{2\pi}{N\sqrt{\cngt}}
\biggpar{1+\biggOrder{\frac1{N^2}}}\;,$$ and the scaled variable $w = u/\eps$. This transforms the map $T_1$ into a map $T_2: (x_n,w_n)\mapsto(x_{n+1},w_{n+1})$ defined by $$\label{tm4}
\begin{split}
x_{n+1} &= x_n + \eps w_n - \frac12\eps^2 f(x_n)\;,\\
w_{n+1} &= w_n - \frac12\eps \bigbrak{f(x_n)+f(x_{n+1})}\;.
\end{split}$$ $T_2$ is again an area-preserving twist map satisfying $$\label{tm4sym}
T_2 \circ S_1 = S_1 \circ (T_2)^{-1}
\qquad\text{and}\qquad
T_2 \circ S_2 = S_2 \circ (T_2)^{-1}\;.$$
Action–Angle Variables {#sec_tmaa}
----------------------
For small $\eps$, we expect the orbits of this map to be close to those of the differential equation $$\label{tm5}
\begin{split}
\dot x &= w\;,\\
\dot w &= -f(x)\;,
\end{split}$$ which is equivalent to the second-order equation $\ddot x = -f(x)$ describing the motion of a particle in the *inverted* double-well potential $-U(x)$, compare . Solutions of can be expressed in terms of Jacobi elliptic functions. Indeed, the function $$\label{tm7}
C(x,w) = \frac12(x^2+w^2) - \frac14 x^4$$ being a constant of motion, one sees that $w$ satisfies $$\label{tm8}
w = \pm \sqrt{(a(C)^2-x^2)(b(C)^2-x^2)/2}\;,$$ where $$\begin{aligned}
\nonumber
a(C)^2 &= 1 - \sqrt{1-4C}\;,\\
b(C)^2 &= 1 + \sqrt{1-4C}\;.
\label{tm9} \end{aligned}$$ This can be used to integrate the equation $\dot x=w$, yielding $$\label{tm9b}
\frac{b(C)}{\sqrt2}t = \JF \biggpar{\ArcSin\biggpar{\frac
{x(t)}{a(C)}},\alkappa(C)}\;,$$ where $\alkappa(C)=a(C)/b(C)$, and $\JF(\phi,\kappa)$ denotes the incomplete elliptic integral of the first kind. The solution of the ODE can be written in terms of standard elliptic functions as $$\label{tm10}
\begin{split}
x(t) &= a(C) \sn\biggpar{\frac{b(C)}{\sqrt2}t,\alkappa(C)}\;,\\
w(t) &= \sqrt{2C} \cn\biggpar{\frac{b(C)}{\sqrt2}t,\alkappa(C)}
\dn\biggpar{\frac{b(C)}{\sqrt2}t,\alkappa(C)}\;.
\end{split}$$
We return now to the map $T_2$ defined in . The explicit solution of the continuous-time equation motivates the change of variables $\Phi_1:
(x,w)\mapsto(\varphi,C)$ given by $$\label{tm13}
\begin{split}
\varphi &= \frac{\sqrt2}{b(C)} \JF \biggpar{\ArcSin\biggpar{\frac
x{a(C)}},\alkappa(C)}\;,\\
C &= \frac12(x^2+w^2) - \frac14 x^4\;.
\end{split}$$ One checks that $\Phi_1$ is again area-preserving. The inverse $\Phi_1^{-1}$ is given by $$\label{tm11}
\begin{split}
x &= a(C) \sn\biggpar{\frac{b(C)}{\sqrt2}\varphi,\alkappa(C)}\;, \\
w &= \sqrt{2C} \cn\biggpar{\frac{b(C)}{\sqrt2}\varphi,\alkappa(C)}
\dn\biggpar{\frac{b(C)}{\sqrt2}\varphi,\alkappa(C)}\;.
\end{split}$$ The elliptic functions $\sn$, $\cn$ and $\dn$ being periodic in their first argument, with period $4\JK(\alkappa)$, it is convenient to carry out a further area-preserving change of variables $\Phi_2:
(\varphi,C)\mapsto(\psi,I)$, defined by $$\label{tm14}
\psi = \Omega(C) \varphi\;,
\qquad
I = h(C)\;,$$ where $$\label{tm15}
\Omega(C) = \frac{b(C)}{\sqrt2} \frac{\pi}{2\JK(\alkappa(C))}\;,
\qquad
h(C) = \int_0^C \frac{\6C'}{\Omega(C')}\;.$$ Using the facts that $C$ and $b=b(C)$ can be expressed as functions of $\alkappa=\alkappa(C)$ by $C=\alkappa^2/(1+\alkappa^2)^2$ and $b^2=2/(1+\alkappa^2)$, one can check that $$\label{tm15a}
h(C) = \frac4{3\pi}
\frac{(1+\alkappa^2)\JE(\alkappa) - (1-\alkappa^2)\JK(\alkappa)}
{(1+\alkappa^2)^{3/2}}\biggr|_{\alkappa=\alkappa(C)}
\in \biggbrak{0,\frac{2\sqrt2}{3\pi}}\;.$$ We denote by $\Phi=\Phi_2\circ\Phi_1$ the transformation $(x,w)\mapsto(\psi,I)$ and by $T = \Phi\circ T_2\circ\Phi^{-1}$ the resulting map.
\[prop\_tm1\] The map $T=T(\eps)$ has the form $$\label{tm16}
\begin{split}
\psi_{n+1} &= \psi_n + \eps \Omegabar(I_n) + \eps^3 f(\psi_n,I_n,\eps)
\qquad \pmod{2\pi}\;,\\
I_{n+1} &= I_n + \eps^3 g(\psi_n,I_n,\eps)\;,
\end{split}$$ where $\Omegabar(I) = \Omega(h^{-1}(I))$. The functions $f$ and $g$ are $\pi$-periodic in their first argument, and are real-analytic for $0\leqs I\leqs h(1/4)-\Order{\eps^3}$. Furthermore, $T$ satisfies the symmetries $$\label{tm16sym}
T \circ \Sigma_1 = \Sigma_1 \circ T^{-1}
\qquad\text{and}\qquad
T \circ \Sigma_2 = \Sigma_2 \circ T^{-1}\;,$$ where $\Sigma_1(\psi,I)=(-\psi,I)$ and $\Sigma_2(\psi,I)=(\pi-\psi,I)$.
First observe that $\Phi_1$ and $\Phi$ are analytic whenever $(x,w)$ is such that $C<1/4$. The map $T$ will thus be analytic whenever $(\psi_n,I_n)$ is such that $C(x_n,w_n)<1/4$ and $C(x_{n+1},w_{n+1})<1/4$. A direct computation shows that $$\label{tm16:1}
C(x_{n+1},w_{n+1}) - C(x_n,w_n) =
\frac{\eps^3}4 \Bigbrak{x_nw_n + 2x_nw_n^3 - 4x_n^3w_n + 3x_n^5w_n} +
\Order{\eps^4}\;.$$ This implies that $I_{n+1}=I_n+\Order{\eps^3}$, and allows to determine $g(\psi,I,0)$. It also shows that $T$ is analytic for $I_n<h(1/4)-\Order{\eps^3}$. Furthermore, writing $a_n=a(C(x_n,w_n))$, we see that implies $a_{n+1}-a_n=\Order{\eps^3}$ and similarly for $b_n$, $\kappa_n$. This yields $$\begin{aligned}
\nonumber
\varphi(x_{n+1},w_{n+1}) - \varphi(x_n,w_n)
&= \frac{\sqrt2}{b_n} \int_{x_n/a_n}^{x_{n+1}/a_n}
\frac{\6u}{\sqrt{(1-\kappa_n^2 u^2)(1-u^2)}} + \Order{\eps^3}\\
&= \eps + \Order{\eps^3}\;,
\label{tm16:2}\end{aligned}$$ which implies the expression for $\psi_{n+1}$. We remark that the fact that $T$ is area-preserving implies the relation $$\label{tm16:3}
1 = \dpar{(\psi_{n+1},I_{n+1})}{(\psi_n,I_n)}
= 1 + \eps^3 \Bigbrak{\sdpar f\psi(\psi,I,0) + \sdpar gI(\psi,I,0)} +
\Order{\eps^4}\;,$$ which allows to determine $f(\psi,I,0)$. The fact that $f$ and $g$ are $\pi$-periodic in their first argument is a consequence of the fact that $T_2(-x,-w)=-T_2(x,w)$. Finally the relations follow from the symmetries , with $\Sigma_i=\Phi \circ S_i \circ
\Phi^{-1}$.
A perturbation expansion at $I=0$ shows in particular that $$\label{tm17a}
\Omegabar(I) = 1-\frac34I+\Order{I^2}\;.$$ An important observation is that $\Omegabar(I)$ is a monotonously decreasing function, taking values in $[0,1]$. The monotonicity of $\Omegabar$ makes $T$ a *twist map* for sufficiently small $\eps$, which has several important consequences on existence of periodic orbits.
We call *rotation number* of a periodic orbit of period $N$ the quantity $$\label{tm18}
\nu = \frac1{2\pi N} \biggbrak{\sum_{n=1}^N (\psi_{n+1}-\psi_n)
\pmod{2\pi}}\;.$$ Note that because of periodicity, $\nu$ is necessarily a rational number of the form $\nu=M/N$, for some positive integer $M$. We denote by $\T^N_\nu$ the set of points $\psi$ in the torus $\T^N$ satisfying . It is sometimes more convenient to visualise $\T^N_\nu$ as the set of real $(\psi_1,\dots,\psi_N)$ such that $$\label{tm19B}
\psi_1 < \psi_2 < \dots < \psi_{N} < \psi_1 + 2\pi N\nu\;.$$ In the sequel, we shall use the shorthand *stationary point with rotation number $\nu$* instead of *stationary point corresponding to a periodic orbit of rotation number $\nu$*.
The expression for $T$ implies that $$\label{tm19}
\nu = \frac{\Omegabar(I_0)}{2\pi} \eps + \Order{\eps^2}\;.$$ The following properties follow from the Poincaré–Birkhoff theorem, whenever $\eps>0$ is sufficiently small:
For each positive integer $M$ satisfying $$\label{tm20}
M\leqs \frac{N\eps}{2\pi}(1+\Order{\eps})\;,$$ the twist map $T$ admits at least two periodic orbits of period $N$ and rotation number $\nu=M/N$. Note that Condition is compatible with the fact that $O$ bifurcates for $\gamma=\gamma_M$, $M=1,2,\dots,\intpart{N/2}$.
Any periodic orbit of period $N$ of the map $T$ is of the form $$\label{tm21}
\begin{split}
\psi_n &= \psi_0 + 2\pi\nu n + \Order{\eps^2} \;,\\
I_n &= \Omegabar^{-1} \biggpar{\frac{2\pi}\eps \nu} +
\Order{\eps^2}\;,
\end{split}$$for some $\psi_0$ and some $\nu=M/N$, where $M$ is a positive integer satisfying .
Going back to original variables, we see that these periodic orbits are of the form $$\label{tm22}
\begin{split}
x_n &= a_n \sn\biggpar{\frac{2\JK(\alkappa_n)}\pi \psi_n,\alkappa_n}\;,\\
w_n &= \sqrt{2C_n} \cn\biggpar{\frac{2\JK(\alkappa_n)}\pi
\psi_n,\alkappa_n}
\dn\biggpar{\frac{2\JK(\alkappa_n)}\pi \psi_n,\alkappa_n}\;,\\
\end{split}$$ where $a_n=a(C_n)$, $\alkappa_n = \alkappa(C_n)$ and $$\label{tm23}
C_n = \Omega^{-1}\biggpar{\frac{2\pi M}{N\eps}} + \Order{\eps}
= \Omega^{-1}\Bigpar{M\sqrt{\cngt}\,} + \Order{\eps}\;.$$ This allows in particular to compute the value of the potential at the corresponding stationary point.
\[prop\_tm3\] Let $\eps>0$ be sufficiently small, and let $x^\star$ be a stationary point of the potential $V_\gamma$, corresponding to an orbit with rotation number $\nu=M/N$. Then $$\label{tm24}
\frac{V_\gamma(x^\star)}{N}
= -\frac1{3(1+\alkappa^2)} \biggbrak{\frac{2+\alkappa^2}{1+\alkappa^2} -
2\frac{\JE(\alkappa)}{\JK(\alkappa)}} + \Order{\eps\alkappa^2}\;,$$ where $\alkappa=\alkappa(C)$, and $C$ satisfies $\Omega(C)^2 = M^2\cngt$.
The expression for the potential implies that $$\begin{aligned}
\nonumber
\frac{V_\gamma(x^\star)}N
&= \frac1N \sum_{n=1}^N \Bigpar{U(x_n)+\frac12 w_n^2 + \Order{\eps^2}}
= \frac1N \sum_{n=1}^N (w_n^2-C_n + \Order{\eps^2})\\
\nonumber
&= \frac CN \sum_{n=1}^N
\biggbrak{2 \cn^2\biggpar{\frac{2\JK(\alkappa)}\pi \psi_n,\alkappa}
\dn^2\biggpar{\frac{2\JK(\alkappa)}\pi \psi_n,\alkappa} - 1
+ \Order{\eps}}\\
&= C \biggbrak{2 \int_0^{2\pi}
\cn^2\biggpar{\frac{2\JK(\alkappa)}\pi \psi,\alkappa}
\dn^2\biggpar{\frac{2\JK(\alkappa)}\pi \psi,\alkappa)} \6\psi - 1
+ \Order{\eps}}\;,
\label{tm25}\end{aligned}$$ where $C=\Omega^{-1}(M\sqrt{\cngt})$ and $\alkappa=\alkappa(C)$. The integral can then be computed using the change of variables $2\JK(\alkappa)\psi/\pi
=
\JF(\phi,\alkappa)$. Finally, recall that $C=\alkappa^2/(1+\alkappa^2)^2$.
One can check that $V_\gamma(x^\star)/N$ is a decreasing function of $\alkappa$, which is itself a decreasing function of $M^2\cngt$. As a consequence, $V_\gamma(x^\star)/N$ is increasing in $M^2\cngt$. This implies in particular that the potential is larger for larger winding numbers $M$.
\[rem\_tmpot\] The leading term in the expression for the value of the potential is the same for all orbits of a given rotation number $\nu$. Since stationary points of the potential of different index cannot be at exactly the same height, the difference has to be hidden in the error terms. In [@BFG06a], we showed that near the desynchronisation bifurcation, the potential difference between $1$-saddles and $2$-saddles is of order $(1-\gammat)^{N/2}$. For large $N$, we expect this difference to be exponentially small in $1/N$, owing to the fact that near-integrable maps of a form similar to are known to admit adiabatic invariants to that order (cf. [@BK0 Theorem 2]).
Generating Function {#sec_tmgf}
-------------------
In this section, we transform the problem of finding periodic orbits of the near-integrable map $T$ into a variational problem. The fact that $T$ is a twist map allows us to express $I_n$ (and thus $I_{n+1}$) as a function of $\psi_n$ and $\psi_{n+1}$. A *generating function* of $T$ is a function $G(\psi_n,\psi_{n+1})$ such that $$\label{tmgf1}
\sdpar G1(\psi_n,\psi_{n+1}) = -I_n\;,
\qquad
\sdpar G2(\psi_n,\psi_{n+1}) = I_{n+1}\;.$$ It is known that any area-preserving twist map admits a generating function, unique up to an additive constant. Since $T$ depends on the parameter $\eps$, the generating function $G$ naturally also depends on $\eps$. However, we will indicate this dependence only when we want to emphasise it.
\[prop\_tmgf\] The map $T$ admits a generating function of the form $$\label{tmgf2}
G(\psi_1,\psi_2) = \eps G_0\biggpar{\frac{\psi_2-\psi_1}\eps, \eps}
+ 2\eps^3 \sum_{p=1}^\infty \Ghat_p\biggpar{\frac{\psi_2-\psi_1}\eps,
\eps} \cos\bigpar{p(\psi_1+\psi_2)}\;,$$ where the functions $G_0(u,\eps)$ and $\Ghat_p(u,\eps)$ are real-analytic for $u>\Order{1/\abs{\log\eps}}$, and satisfy $$\begin{aligned}
\nonumber
G_0'(u,0) &= \Omegabar^{-1}(u)\;,\\
\Ghat_p\bigpar{u,0}
&= \frac1{4p\pi} \int_0^{2\pi}
g\bigpar{\psi,\Omegabar^{-1}(u),0} \sin(-2p\psi)\,\6\psi\;.
\label{tmgf3}\end{aligned}$$
Fix $(\psi_2,I_2) = T(\psi_1,I_1)$. The fact that $T(\psi_1+\pi,I_1)=(\psi_2+\pi,I_2)$ implies $$\label{tmgf3:1}
G(\psi_1+\pi,\psi_2+\pi) = G(\psi_1,\psi_2) + c$$ for some constant $c$. If we set $G(\psi_1,\psi_2)=\Gtilde(\psi_2-\psi_1,\psi_1+\psi_2)$, we thus have $$\label{tmgf3:2}
\Gtilde(u,v+2\pi) = \Gtilde(u,v) + c\;.$$ This allows us to expand $G$ as a Fourier series $$\label{tmgf3:3}
G(\psi_1,\psi_2)
= \sum_{p=-\infty}^\infty \Gtilde_p(\psi_2-\psi_1,\eps) \e^{\icx
p(\psi_1+\psi_2)} + \frac c{2\pi} (\psi_1+\psi_2)\;.$$ Next we note that the symmetry implies $T(-\psi_2,I_2) = (-\psi_1,I_1)$, and thus $$\label{tmgf3:4}
\sdpar G1(\psi_1,\psi_2) = -\sdpar G2 (-\psi_2,-\psi_1)\;.$$ Plugging into this relation yields $$\label{tmgf3:5}
c = 0
\qquad\text{and}\qquad
\Gtilde_{-p}(u,\eps) = \Gtilde_p(u,\eps)\;,$$ which allows to represent $G$ as a real Fourier series as well. Computing the derivatives $I_1=-\sdpar G1(\psi_1,\psi_2)$ and $I_2=\sdpar G2(\psi_1,\psi_2)$ yields $$\label{tmgf3:6}
I_2 - I_1 = 2\sum_{p=-\infty}^\infty \icx p \Gtilde_p(\psi_2-\psi_1,\eps)
\e^{\icx p(\psi_1+\psi_2)}\;,$$ which shows in particular that $\Gtilde_p(u,\eps) = \Order{\eps^3}$ for $p\neq0$, as a consequence of . This implies $I_1=\Gtilde_0'(\psi_2-\psi_1,\eps) + \Order{\eps^3}$, and thus $u =
\Gtilde_0'(\eps\Omegabar(u)+\Order{\eps^3},\eps)$. Renaming $\Gtilde_0(u,\eps)=\eps G_0(u/\eps,\eps)$ and $\Gtilde_p(u,\eps)=\eps^3
\Ghat_p(u/\eps,\eps)$ yields . Evaluating for $\eps=0$ and taking the Fourier transform yields the expression for $\Ghat_p(u,0)$.
The relations allow to determine the expression for the generating function of the map $T$, given by . In particular, one finds $$\label{tmfg4a}
G_0(u,0) = u\Omegabar^{-1}(u) - \Omega^{-1}(u)\;,$$ so that $$\begin{aligned}
\nonumber
G_0(\Omega(C),0) &= h(C)\Omega(C)-C \\
&= -\frac1{3(1+\alkappa^2)} \biggbrak{\frac{2+\alkappa^2}{1+\alkappa^2} -
2\frac{\JE(\alkappa)}{\JK(\alkappa)}}\;,
\label{tmfg4b}\end{aligned}$$ with $\alkappa=\alkappa(C)$. Note that this quantity is identical with the leading term in the expression for the average potential per site. This indicates that we have chosen the integration constant in the generating function in such a way that $V_\gamma$ and $G_N$ take the same value on corresponding stationary points.
The main use of the generating function lies in the following fact. Consider the $N$-point function $$\label{tmgf5}
G_N(\psi_1,\dots,\psi_N) = G(\psi_1,\psi_2) + G(\psi_2,\psi_3) +
\dots + G(\psi_N,\psi_1 + 2\pi N\nu)\;,$$ defined on (a subset of) the set $\T^N_\nu$. The defining property of the generating function implies that for any periodic orbit of period $N$ of the map $T$, one has $$\label{tmgf6}
\dpar{}{\psi_n} G_N(\psi_1,\dots,\psi_N) = -I_n + I_n = 0\;,
\qquad
\text{for $n=1,\dots,N$.}$$ In other words, $N$-periodic orbits of $T$ with rotation number $\nu$ are in one-to-one correspondence with stationary points of the $N$-point function $G_N$ on $\T^N_\nu$.
The symmetries of the original potential imply that the $N$-point generating function satisfies the following relations on $\T^N_\nu$: $$\begin{aligned}
\nonumber
G_N(\psi_1,\dots,\psi_N) &= G_N(\psi_2,\dots,\psi_N,\psi_1+2\pi N\nu)\;,\\
\nonumber
G_N(\psi_1,\dots,\psi_N) &= G_N(-\psi_N,\dots,-\psi_1)\;,\\
G_N(\psi_1,\dots,\psi_N) &= G_N(\psi_1+\pi,\dots,\psi_N+\pi)\;.
\label{tmgf7}\end{aligned}$$
At this point, we are in the following situation. We have first transformed the initial problem of finding the stationary points of the potential $V_\gamma$ into the problem of finding periodic orbits of the map $T_1$, or, equivalently, of the map $T$. This problem in turn has been transformed into the problem of finding the stationary points of $G_N$. Obviously, the whole procedure is of interest only if the stationary points of $G_N$ are easier to find and analyse than those of $V_\gamma$. This, however, is the case here since the $N$-point function is a small perturbation of a function depending only on the differences $\psi_{n+1}-\psi_n$. In other words, $G_N$ can be interpreted as the energy of a chain of particles with a uniform nearest-neighbour interaction, put in a weak external periodic potential.
Fourier Representation of the Generating Function {#sec_tmgff}
-------------------------------------------------
The main difficulty in analysing the stationary points of the $N$-point generating function $G_N$ comes from the fact that it is almost degenerate under translations of the form $\psi_n\mapsto\psi_n+c$ $\forall n$. The purpose of this section is to decouple the translation mode from the other variables, by introducing Fourier variables.
We fix $\nu=M/N$. Any stationary point of $G_N$ on $\T^N_\nu$ admits a Fourier expansion of the form $$\label{tmgff1}
\psi_n = 2\pi\nu n + \sum_{q=0}^{N-1} \psihat_q \omega^{qn}\;,$$ where $\omega=\e^{2\pi\icx/N}$, and the Fourier coefficients are uniquely determined by $$\label{tmgff2}
\psihat_q = \frac1N \sum_{n=1}^{N}\omega^{-qn} (\psi_n-2\pi\nu n)
=\cc{\psihat_{-q}}\;.$$ Note that $\psihat_q=\psihat_{q+N}$ for all $q$. Stationary points of $G_N$ correspond to stationary points of the function $\Gbar_N$, obtained by expressing $G_N$ in terms of Fourier variables $(\psihat_0,\dots,\psihat_{N-1})$. In order to do so, it is convenient to write $$\begin{aligned}
\nonumber
\frac{\psi_{n+1}-\psi_n}\eps &= \Delta
+ \eps^2 \alpha_n(\psihat_1,\dots,\psihat_{N-1}) \;, \\
\psi_n + \psi_{n+1} &= 2\psihat_0 + 2\pi\nu(2n+1)
+ \eps^2 \beta_n(\psihat_1,\dots,\psihat_{N-1}) \;,
\label{tmgff3}\end{aligned}$$ where $\Delta=2\pi\nu/\eps$ and $$\begin{aligned}
\nonumber
\alpha_n(\psihat_1,\dots,\psihat_{N-1})
&= \frac1{\eps^3} \sum_{q=1}^{N-1} \psihat_q (\omega^q-1) \omega^{qn}\;,\\
\beta_n(\psihat_1,\dots,\psihat_{N-1})
&= \frac1{\eps^2} \sum_{q=1}^{N-1} \psihat_q (\omega^q+1) \omega^{qn}\;.
\label{tmgff4}\end{aligned}$$ Note that $\alpha_n$ is of order $1$ in $\eps$ for any stationary point because of the expression of the twist map. Taking the inverse Fourier transform shows that $\abs{\psihat_q(\omega^q-1)}=\Order{\eps^3}$ and $\abs{\psihat_q}=\Order{\eps^2}$ for $q\neq0$, and thus $\beta_n$ is also of order $1$.
------------------------------------- -------------------------- -------------------------------- --------------------------------------------------------------------------
height 14pt depth 6pt width 0pt $R$ $x_j \mapsto x_{j+1}$ $\psi_n \mapsto \psi_{n+1}$ $\psihat_q \mapsto \omega^q \psihat_q + 2\pi\nu\delta_{q0}$
height 8pt depth 6pt width 0pt $CS$ $x_j \mapsto -x_{N+1-j}$ $\psi_n \mapsto -\psi_{N+1-n}$ $\psihat_q \mapsto -\omega^{-q}\psihat_{N-q} - 2\pi\nu (N+1)\delta_{q0}$
height 6pt depth 8pt width 0pt $C$ $x_j \mapsto -x_j$ $\psi_n \mapsto \psi_n+\pi$ $\psihat_q \mapsto \psihat_q + \pi\delta_{q0}$
------------------------------------- -------------------------- -------------------------------- --------------------------------------------------------------------------
: Effect of some symmetries on original variables, angle variables, and Fourier variables.[]{data-label="table_Fourier_gen"}
Expressing $G_N$ in Fourier variables yields the function $$\label{tmgff5}
\Gbar_N(\psihat_0,\dots,\psihat_{N-1})
= \sum_{p=-\infty}^\infty \e^{2\icx p\psihat_0}
g_p(\psihat_1,\dots,\psihat_{N-1})\;,$$ where (we drop the $\eps$-dependence of $G_0$ and $\Ghat_p$) $$\begin{aligned}
\nonumber
g_0(\psihat_1,\dots,\psihat_{N-1})
&= \eps \sum_{n=1}^N G_0(\Delta+\eps^2 \alpha_n)\;,\\
g_p(\psihat_1,\dots,\psihat_{N-1})
&= \eps^3 \sum_{n=1}^N \Ghat_p(\Delta+\eps^2 \alpha_n) \omega^{pM(2n+1)}
\e^{\icx \eps^2 p \beta_n}
\qquad \text{for $p\neq0$\;.}
\label{tmgff6}\end{aligned}$$
We now examine the symmetry properties of the Fourier coefficients $g_p$. Table \[table\_Fourier\_gen\] shows how the Fourier variables transform under some symmetry transformations, where we only consider transformations leaving $\T^N_\nu$ invariant. As a consequence, the first two symmetries in translate into $$\begin{aligned}
\nonumber
g_p(\psihat_1,\dots,\psihat_{N-1})
&= \omega^{2pM} g_p(\omega\psihat_1,\dots,\omega^{N-1}\psihat_{N-1})\;,\\
g_p(\psihat_1,\dots,\psihat_{N-1})
&= \omega^{-2pM}
g_{-p}(-\omega^{N-1}\psihat_{N-1},\dots,-\omega\psihat_1)\;.
\label{tmgff7}\end{aligned}$$ We now introduce new variables $\chi_q$, $q\neq0$, defined by $$\label{tmgff9}
\chi_q = -\icx \omega^{-q\psihat_0/2\pi\nu} \psihat_q
= -\cc{\chi_{-q}}\;.$$ The $\chi_q$ are defined in such a way that they are real for stationary points satisfying, in original variables, the symmetry $x_j=-x_{n_0-j}$ for some $n_0$. For later convenience, we prefer to consider $q$ as belonging to $$\label{tmgff9B}
\cQ=\biggset{-\biggintpart{\frac{N-1}2},\dots,\biggintpart{\frac N2}}
\setminus\bigset{0}$$ rather than $\set{1,\dots,N-1}$. We set $\chi=\set{\chi_q}_{q\in\cQ}$ and $$\begin{aligned}
\nonumber
\Gtilde_N(\psihat_0,\chi)
&=
\Gbar_N(\psihat_0,\set{\psihat_q=\icx\omega^{q\psihat_0/2\pi\nu}\chi_q}_{
q\in\cQ}) \\
&= \sum_{p=-\infty}^\infty \e^{2\icx p\psihat_0}
\tilde g_p(\psihat_0,\chi)\;,
\label{tmgff10}\end{aligned}$$ where $$\label{tmgff11}
\tilde g_p(\psihat_0,\chi) =
g_p(\set{\psihat_q=\icx\omega^{q\psihat_0/2\pi\nu}\chi_q}_{q\in\cQ})\;.$$
\[lem\_tmgff1\] The function $\Gtilde_N(\psihat_0,\chi)$ is $2\pi\nu$-periodic in its first argument.
By , we have $$\label{tmgff11:1}
\tilde g_p(\psihat_0+2\pi\nu,\chi)
= \omega^{-2pM} \tilde g_p(\psihat_0,\chi)$$ Since $\e^{2\icx p\cdot 2\pi\nu}=\omega^{2pM}$, replacing $\psihat_0$ by $\psihat_0+2\pi\nu$ in leaves $\Gtilde_N$ invariant.
Since $\Gtilde_N$ also has period $\pi$, it has in fact period $$\label{tmgff12}
\frac{\pi}N K\;,
\qquad
K = \gcd(N,2M)\;.$$
Our strategy now proceeds as follows:
Show that for each $\psihat_0$, and sufficiently small $\eps$, the equations $\tdpar{\Gtilde_N}{\chi_q}=0$, $q\in\cQ$, admit exactly one solution $\chi=\chi^\star(\smash{\psihat_0})$. This is done in Section \[sec\_tmgchi\] with the help of Banach’s fixed-point theorem.
Show that for $\chi=\chi^\star(\psihat_0)$, the equation $\tdpar{\Gtilde_N}{\psihat_0}=0$ is satisfied by exactly $4N/K$ values of $\smash{\psihat_0}$. This is done in Section \[sec\_tmgstat\] by estimating the Fourier coefficients of $\tdpar{\Gtilde_N}{\psihat_0}$ with the help of complex analysis.
Uniqueness of $\mathbf{\chi}$ {#sec_tmgchi}
-----------------------------
In this section, we show that the equations $$\label{tmgchi001}
\dpar{\Gtilde_N}{\chi_q}=0\;,
\qquad
q\in\cQ\;,$$ admit exactly one solution $\chi=\chi^\star(\smash{\psihat_0})$ for each value of $\psihat_0$. The proof is based on a standard fixed-point argument: First we show in Lemma \[lem\_tmgff4\] that is equivalent to the fixed-point equation $\rho=\cT\rho$ for a quantity $\rho$ related to $\chi$. Then we show in Proposition \[cor\_tmgff2\] that $\cT$ is contracting in an appropriate norm, provided $\eps$ is sufficiently small.
It is useful to introduce the scaled variables $$\label{tmgffB1}
\rho_q = \rho_q(\chi) = -\frac2{\eps^3} \chi_q \sin(\pi q/N)$$ and the function $\Gamma^{(a,b)}_{\ell}(\rho)$, $\rho=\set{\rho_q}_{q\in\cQ}$, defined for $\ell\in\Z$ and $a,
b\geqs 0$ by $$\label{tmgffB2}
\Gamma^{(a,b)}_{\ell}(\rho) =
\sum_{\substack{q_1,\dots,q_a\in\cQ \\ q'_1,\dots,q'_b\in\cQ}}
\indexfct{\sum_i q_i + \sum_j q'_j = \ell}
\prod_{i=1}^a \rho_{q_i}
\prod_{j=1}^b \frac{\eps}{\tan(\pi q'_j/N)} \rho_{q'_j}\;.$$ By convention, any term in the sum for which $q'_j=N/2$ for some $j$ is zero, that is, we set $1/\tan(\pi/2)=0$. A few elementary properties following immediately from this definition are:
$\Gamma^{(0,0)}_{\ell}(\rho) = \delta_{\ell0}$;
$\Gamma^{(a,b)}_{\ell}(\rho) = 0$ for $\abs{\ell} > (a+b)N/2$;
If $\rho_q=0$ for $q\not\in K\Z$, then $\Gamma^{(a,b)}_{\ell}(\rho)=0$ for $\ell\not\in K\Z$;
If $\rho'_q=\rho_{-q}$ for all $q$, then $\Gamma^{(a,b)}_{\ell}(\rho')=(-1)^b\Gamma^{(a,b)}_{-\ell}(\rho)$.
The following result states that the conditions are equivalent to a fixed-point equation.
\[lem\_tmgff4\] Let $$\label{tmgffA3}
H_{p,q}(\Delta) = \Ghat_p'(\Delta)
- \frac{\eps p \Ghat_p(\Delta)}{\tan(\pi q/N)}\;,$$ with the convention that $H_{p,N/2}(\Delta) = \Ghat_p'(\Delta)$. Then the stationarity conditions are fulfilled if and only if $\rho=\rho(\chi)$ satisfies the fixed-point equation $$\label{tmgffB3}
\rho = \cT \rho = \rho^{(0)} + \Phi(\rho,\eps)\;,$$ where the leading term is given by $$\label{tmgffB4}
\rho^{(0)}_q =
\begin{cases}
\displaystyle
\frac{1}{G_0''(\Delta)} \sum_{k\in\Z \colon kN+q\in2M\Z}
(-1)^{k+1} \e^{\icx k\psihat_0 N/M}
H_{(kN+q)/2M,q}(\Delta)
&\text{if $q\in K\Z$\;,}\\
0
&\text{if $q\not\in K\Z$\;,}
\end{cases}$$ and the remainder is given by $\Phi_q(\rho,\eps)=\Phi^{(1)}_q(\rho,\eps)+\Phi^{(2)}_q(\rho,\eps)$, with $$\begin{aligned}
\nonumber
\Phi^{(1)}_q(\rho,\eps)
&= \frac{1}{G_0''(\Delta)} \sum_{k\in\Z} (-1)^{k+1} \e^{\icx k\psihat_0
N/M}
\sum_{a\geqs 1} \frac{\eps^{2a}}{(a+1)!}
G_0^{(a+2)}(\Delta)
\Gamma^{(a+1,0)}_{kN+q}(\rho)\;, \\
\Phi^{(2)}_q(\rho,\eps)
&= \frac{1}{G_0''(\Delta)} \sum_{k\in\Z} (-1)^{k+1} \e^{\icx k\psihat_0
N/M}
\sum_{a+b\geqs1} \frac{\eps^{2(a+b)}}{a!b!}
\sum_{p\neq0} H^{(a)}_{p,q}(\Delta) p^b
\Gamma^{(a,b)}_{kN-2pM+q}(\rho)\;.
\label{tmgffB5}\end{aligned}$$
The proof is a straightforward but lengthy computation, which we postpone to Appendix \[sec\_prtech\].
Note the following symmetries, which follow directly from the definition of $\rho^{(0)}$ and the properties of $\smash{\Gamma^{(a,b)}_\ell}$:
For all $q\in\cQ$, $\rho^{(0)}_{-q}=\rho^{(0)}_q$, because $H_{-p,-q}(\Delta)=H_{p,q}(\Delta)$, and thus $\rho^{(0)}_q\in\R$;
If $\rho_q=0$ for $q\not\in K\Z$, then $\Phi_q(\rho,\eps)=0$ for $q\not\in K\Z$;
If $\rho'_q=\rho_{-q}$ for all $q$, then $\Phi_q(\rho',\eps)=\Phi_{-q}(\rho,\eps)$;
\[rem\_tmgffB\] The condition $kN+q\in 2M\Z$, appearing in the definition of $\rho^{(0)}$, can only be fulfilled if $q\in N\Z+2M\Z = K\Z$. If this is the case, set $N=nK$, $2M=mK$, $q=\ell K$, with $n$ and $m$ coprime. Then the condition becomes $mp-kn=\ell$. By Bezout’s theorem, the general solution is given in terms of any particular solution $(p_0,k_0)$ by $$\label{tmgff17:3}
p = p_0 + n t\;,
\qquad
k = k_0 + m t\;,
\qquad
t \in \Z\;.$$ Thus there will be exactly one $p$ with $2\abs{p}< n$. If $N$ is very large, and $M$ is fixed, then $n=N/K$ is also very large. Since the $\Ghat_p(\Delta)$, being Fourier coefficients of an analytic function, decrease exponentially fast in $\abs{p}$, the sum in will be dominated by the term with the lowest possible $\abs{p}$.
We now introduce the following weighted norm on $\C^\cQ$: $$\label{tmgff20}
\norm{\rho}_\lambda = \sup_{q\in\cQ}
\e^{\lambda\abs{q}/2M}\abs{\rho_q}\;,$$ where $\lambda>0$ is a free parameter. One checks that the functions $G_0(\Delta)$ and $\Ghat_p(\Delta)$ are analytic for $\re\Delta>\Order{1/\log\abs{\eps}}$. Thus it follows from Cauchy’s theorem that there exist positive constants $L_0$, $r<\Delta-\Order{1/\log\abs{\eps}}$ and $\lambda_0$ such that $$\label{tmgff21}
\abs{G_0^{(a)}(\Delta)} \leqs L_0 \frac{a!}{r^a}
\qquad\text{and}\qquad
\abs{\Ghat_p^{(a)}(\Delta)} \leqs L_0 \frac{a!}{r^a} \e^{-\lambda_0\abs{p}}$$ for all $a\geqs0$ and $p\in\Z$. For sufficiently small $\eps$, it is possible to choose $r=\Delta/2$.
\[cor\_tmgff2\] There exists a numerical constant $c_1>0$ such that for any $\Delta>0$, any $\lambda<\lambda_0$ and any $R_0 >
c_1L_0\brak{\Delta\abs{G_0''(\Delta)}}^{-1}$, there is a strictly positive $\eps_0=\eps_0(\Delta,\lambda,\lambda_0,R_0)$ such that for all $\eps<\eps_0$, the map $\cT$ admits a unique fixed point $\rho^\star$ in the ball $\cB_\lambda(0,R_0)=\setsuch{\rho\in\C^\cQ}{\norm{\rho}_\lambda<R_0}$. Furthermore, the fixed point satisfies
$\rho^\star_q = 0$ whenever $q\not\in K\Z$;
$\rho^\star_{-q}=\rho^\star_q$, and thus $\rho^\star_q\in\R$ for all $q$.
The proof is again a straightforward but lengthy computation, so we postpone it to Appendix \[sec\_prtech\].
A direct consequence of this result is that for any $\psihat_0$, and sufficiently small $\eps$, there is a unique $\rho^\star=\rho^\star(\psihat_0)$ (and thus a unique $\chi^\star(\psihat_0)$) satisfying the equations $\tdpar{\Gtilde_N}{\chi_q}=0$ for all $q\in\cQ$. Indeed, we take $R_0$ sufficiently large that our a priori estimates on the $\chi_q$ imply that $\rho\in\cB_0(0,R_0)$. Then it follows that $\rho$ is unique. Furthermore, for any $\lambda<\lambda_0$, making $\eps$ sufficiently small we obtain an estimate on $\norm{\rho^\star}_\lambda$.
Stationary Values of $\mathbf{\psihat_0}$ {#sec_tmgstat}
-----------------------------------------
We now consider the condition $\tdpar{\Gtilde_N}{\psihat_0}=0$. As pointed out at the end of Section \[sec\_tmgff\], $\Gtilde_N(\psihat_0,\chi)$ is a $\pi K/N$-periodic function of $\psihat_0$. For the same reasons, $\chi^\star(\psihat_0)$ is also $\pi K/N$-periodic. Hence it follows that the function $\psihat_0\mapsto \Gtilde_N(\psihat_0,\chi^\star(\psihat_0))$ has the same period as well, and thus admits a Fourier series of the form $$\label{tmgstat1}
\Gtilde_N(\psihat_0,\chi^\star(\psihat_0))
= \sum_{k=-\infty}^\infty \hat g_k \e^{2\icx k \psihat_0 N/K}\;,$$ with Fourier coefficients $$\label{tmgstat2}
\hat g_k = \frac 1{2\pi} \int_0^{2\pi} \sum_{p=-\infty}^\infty
\e^{2\icx(p-kN/K)\psihat_0} \tilde
g_p(\psihat_0,\chi^\star(\psihat_0))\,\6\psihat_0$$ (we have chosen $[0,2\pi]$ as interval of integration for later convenience). Using the change of variables $\psihat_0\mapsto-\psihat_0$ in the integral, and the various symmetries of the coefficients (in particular ), one checks that $\hat g_{-k}=\hat g_{k}$. Therefore can be rewritten in real form as $$\label{tmgstat3}
\Gtilde_N(\psihat_0,\chi^\star(\psihat_0))
= \hat g_0 + 2\sum_{k=1}^\infty \hat g_k \cos\bigpar{2 k \psihat_0 N/K}\;.$$ Now $\tdpar{\Gtilde_N}{\psihat_0}$ vanishes if and only if the total derivative of $\Gtilde_N(\psihat_0,\chi^\star(\psihat_0))$ with respect to $\psihat_0$ is equal to zero. This function obviously vanishes for $\psihat_0=\ell\pi K/2N$, $\ell=1,\dots,4N/K$, and we have to show that these are the only roots.
We first observe that the Fourier coefficients $\hat g_k$ can be expressed directly in terms of the generating function , written in the form $$\label{tmgstat4}
\Gtilde(u,v,\eps) = \eps G_0(u,\eps)
+ \eps^3 \sum_{p\neq0} \Ghat_p(u,\eps) \e^{2\icx pv}\;.$$ In the sequel, $\alpha^\star_n(\psihat_0)$ and $\beta^\star_n(\psihat_0)$ denote the quantities introduced in , evaluated at $\psihat_q=\icx\omega^{q\psihat_0/2\pi\nu}\chi^\star_q(\psihat_0)$.
\[lem\_tmgstat1\] The Fourier coefficients $\hat g_k$ are given in terms of the generating function by $$\label{tmgstat5}
\hat g_k = \frac N{2\pi} \int_0^{2\pi}
\e^{-2\icx k\psihat_0N/K}
\Lambda_0(\psihat_0) \,\6\psihat_0\;,$$ where $$\label{tmgstat5B}
\Lambda_0(\psihat_0) =
\Gtilde \Bigpar{\Delta+\eps^2\alpha^\star_0(\psihat_0),
\psihat_0+\pi\nu+\frac12\eps^2\beta^\star_0(\psihat_0),\eps}\;.$$
The coefficient $\hat g_k$ can be rewritten as $$\label{tmgstat5:1}
\hat g_k = \frac1{2\pi} \sum_{n=1}^N \int_0^{2\pi}
\e^{-2\icx k\psihat_0N/K} \Lambda_n(\psihat_0) \,\6\psihat_0\;,$$ where $$\begin{aligned}
\nonumber
\Lambda_n(\psihat_0) ={}& \eps G_0(\Delta+\eps^2\alpha^\star_n(\psihat_0))
\\
&{}+ \eps^3\sum_{p\neq0} p \e^{2\icx p\psihat_0}
\Ghat_p(\Delta+\eps^2\alpha^\star_n(\psihat_0)) \omega^{pM(2n+1)}
\e^{\icx \eps^2 p \beta^\star_n(\psihat_0)} \;.
\label{tmgstat5:2}\end{aligned}$$ Using the periodicity of $\chi^\star$, one finds that $\alpha^\star_n(\psihat_0+2\pi\nu) = \alpha^\star_{n+1}(\psihat_0)$ and similarly for $\beta^\star_n$, which implies $\Lambda_n(\smash{\psihat_0}) = \Lambda_0(\smash{\psihat_0}+2\pi\nu n)$. Inserting this into and using the change of variables $\psihat_0\mapsto\psihat_0-2\pi\nu$ in the $n$th summand allows to express $\hat g_k$ as the $(2kN/K)$th Fourier coefficient of $\Lambda_0$. Finally, $\Lambda_0(\psihat_0)$ can also be written in the form .
Relation implies that the $\hat g_k$ decrease exponentially fast with $k$, like $\e^{-2\lambda_0kN/K}$. Hence the Fourier series is dominated by the first two terms, provided $N$ is large enough. In order to obtain the existence of exactly $4N/K$ stationary points, it is thus sufficient to prove that $\hat g_1$ is also bounded below by a quantity of order $\e^{-2\lambda_0N/K}$.
\[prop\_tmgstat2\] For any $\Delta>0$, there exists $\eps_1(\Delta)>0$ such that whenever $\eps<\eps_1(\Delta)$, $$\label{tmgstat6a}
\sign(\hat g_1) = (-1)^{1+2M/K}\;.$$ Furthermore, $$\label{tmgstat6b}
\frac{\abs{\hat g_k}}{\abs{\hat g_1}}
\leqs \exp\biggset{-\frac{3k-5}4 \lambda_0(\Delta)\frac NK}
\qquad
\forall k \geqs 2\;,$$ where $\lambda_0(\Delta)$ is a monotonously increasing function of $\Delta$, satisfying $\lambda_0(\Delta)=\sqrt2\pi\Delta+\Order{\Delta^2}$ as $\Delta\searrow0$, and diverging logarithmically as $\Delta\nearrow1$.
First recall that $\Delta=2\pi\nu/\eps=2\pi M/N\eps$, where $M$ is fixed. Thus taking $\eps$ small for given $\Delta$ automatically yields a large $N$. Combining the expression for the twist map and the defining property of the generating function with the relations $u=(\psi_{n+1}-\psi_n)/\eps$ and $v=\psi_n+\psi_{n+1}$, one obtains the relation $$\label{tmgstat6:1}
\sdpar{\Gtilde}v(u,v,\eps) = \frac{\eps^3}2
\Bigbrak{g\Bigpar{\tfrac12(v-\eps
u), \Omegabar^{-1}(u),\eps} + \Order{\eps^2}}\;.$$ It follows from and the definition of $h(C)$ that $$\begin{aligned}
\nonumber
g(\psi,I,0) &= \frac1{\Omegabar(I)} \frac{xw}4 \brak{1+2w^2-4x^2+3x^4}\\
&= \frac1{\Omegabar(I)} \frac{xw}4 \brak{1+4C-6x^2+4x^4}\;,
\label{tmgstat6:2}\end{aligned}$$ where $x$ and $w$ have to be expressed as functions of $\psi$ and $I$ via and . In particular, we note that $$\label{tmgstat6:3}
w = \frac{\sqrt{2C}}a \frac{\pi}{2\JK(\kappa)} \dtot x\psi
= {\Omegabar(I)} \dtot x\psi\;,$$ where we used again. This allows us to write $$\label{tmgstat6:4}
g(\psi,I,0) = \frac18 \dtot{}{\psi}
\biggbrak{(1+4C)x^2 - 3 x^4 + \frac43 x^6}\;.$$ A similar argument would also allow to express the first-order term in $\eps$ of $g(\psi,I,\eps)$ as a function of $x=x(\psi,I)$. Also note the equality $$\label{tmgstat6:5}
\Delta
= \Omegabar(I) + \Order{\eps}
= \frac{\pi b(C)}{2\sqrt2\JK(\kappa)} + \Order{\eps}
= \frac{1}{\sqrt{1+\kappa^2}\JK(\kappa)} + \Order{\eps}\;,$$ which follows from the relation between $\nu$ and $\Omegabar(I)$.
The properties of elliptic functions imply that for fixed $I$, $\psi\mapsto
x(\psi,I)$ is periodic in the imaginary direction, with period $2\lambda_0
= \pi\JK(\sqrt{1-\kappa^2})/\JK(\kappa)$, and has poles located in $\psi=n\pi + (2m+1)\icx\lambda_0$, $n,m\in\Z$. As a consequence, the definition of the map $T = \Phi\circ T_2\circ\Phi^{-1}$ implies in particular that $g(\psi,I,\eps)$ is a meromorphic function of $\psi$, with poles at the same location, and satisfying $g(\psi+2\icx\lambda_0,I,\eps)=g(\psi,I,\eps)$. These properties yield informations on periodicity and location of poles for $\Lambda_0(\psihat_0)$, in particular $\Lambda_0(\psihat_0+2\icx\lambda_0)=\Lambda_0(\psihat_0)+\Order{\eps^2}$.
![The integration contour $\Gamma$ used in the integral .[]{data-label="fig_contour"}](figs/fig_contour){height="40mm"}
Let $\Gamma$ be a rectangular contour with vertices in $-\pi/2$, $-\pi/2-2\icx\lambda_0$, $3\pi/2-2\icx\lambda_0$, and $3\pi/2$, followed in the anticlockwise direction (), and consider the contour integral $$\label{tmgstat6:6}
J = \frac1{2\pi} \oint_\Gamma \e^{-2\icx k z N/K} \Lambda_0(z)\,\6z\;.$$ The contributions of the integrals along the vertical sides of the rectangle cancel by periodicity. Therefore, by Lemma \[lem\_tmgstat1\] and the approximate periodicity of $\Lambda_0$ in the imaginary direction, $$\label{tmgstat6:7}
J = -\frac1N \Bigbrak{\hat g_k
- \e^{-2k\lambda_0N/K}\Bigpar{\hat g_k+ \Order{N\eps^5}}}\;.$$ On the other hand, the residue theorem yields $$\label{tmgstat6:8}
J = 2\pi\icx \sum_{z_j}
\e^{-2\icx kz_jN/K}\res(\Lambda_0(z),z_j)\;,$$ where the $z_j$ denote the poles of the function $\Lambda_0(z)$, lying inside $\Gamma$. There are two such poles, located in $z_1=-\icx\lambda_0-\pi\nu+\eps\Delta+\Order{\eps^2}$, and $z_2=z_1+\pi$, and they both yield the same contribution, of order $\e^{-\lambda_0kN(1+\Order{\eps^2})/K}$, to the sum. Comparing and shows that $\hat g_k/N$ is of the same order. Finally, the leading term of $\hat g_1$ can be determined explicitly using and Jacobi’s expression for the Fourier coefficients of powers of elliptic functions, and is found to have sign $(-1)^{1+2M/K}$ for sufficiently large $N$. Choosing $\eps$ small enough (for fixed $\eps N$) guarantees that $\hat g_1$ dominates all $\hat g_k$ for $k\geqs2$.
\[cor\_tmgstat\] For $\eps<\eps_1(\Delta)$, the $N$-point generating function $\Gtilde_N$ admits exactly $4N/K$ stationary points, given by $\psihat_0=\ell\pi K/2N$, $\ell=1,\dots,4N/K$, and $\chi=\chi^\star(\psihat_0)$.
In the points $\psihat_0=\ell\pi K/2N$, the derivative of the function $\psihat_0\mapsto\Gtilde_N(\psihat_0,\chi^\star(\psihat_0))$ vanishes, while its second derivative is bounded away from zero, as a consequence of Estimate . Thus these points are simple roots of the first derivative, which is bounded away from zero everywhere else.
Index of the Stationary Points {#sec_tmgindex}
------------------------------
We finally examine the stability type of the various stationary points, by first determining their index as stationary points of the $N$-point generating function $\Gbar_N$, and then examining how this translates into their index as stationary points of the potential $V_\gamma$.
\[prop\_tmgindex\] Let $(\psihat_0,\chi^\star(\psihat_0))$ be a stationary point of $\Gtilde_N$ with rotation number $\nu=N/M$. Let $x^\star=x^\star(\psihat_0)$ be the corresponding stationary point of the potential $V_\gamma$, and let $K=\gcd(N,2M)$.
If $2M/K$ is odd, then the points $x^\star(0)$, $x^\star(K\pi/N)$, …are saddles of even index of $V_\gamma$, while the points $x^\star(K\pi/2N)$, $x^\star(3K\pi/2N)$, …are saddles of odd index of $V_\gamma$.
If $2M/K$ is even, then the points $x^\star(0)$, $x^\star(K\pi/N)$, …are saddles of odd index of $V_\gamma$, while the points $x^\star(K\pi/2N)$, $x^\star(3K\pi/2N)$, …are saddles of even index of $V_\gamma$.
We first determine the index of $(\psihat_0,\chi^\star(\psihat_0))$ as stationary point of $\Gtilde_N$. Using the fact that $G_0''(\Delta)$ is negative ($\smash{\Omegabar^{-1}}(\Delta)$ being decreasing), one sees that the Hessian matrix of $\smash{\Gtilde_N}$ is a small perturbation of a diagonal matrix with $N-1$ negative eigenvalues. The $N$th eigenvalue, which corresponds to translations of $\psihat_0$, has the same sign as the second derivative of $\psihat_0\mapsto\Gtilde_N(\psihat_0,\chi^\star(\psihat_0))$, which is equal to $(-1)^{2M/K}\sign\cos(2\psihat_0 N/K)$. Thus $(\psihat_0,\chi^\star(\psihat_0))$ is an $N$-saddle of $\Gtilde_N$ if this sign is negative, and an $(N-1)$-saddle otherwise. The same is true for the index of $\psi=(\psi_1,\dots,\psi_N)$ as a stationary point of $\Gbar_N$.
Let $R$ be the so-called *residue* of the periodic orbit of $T$ associated with the stationary point. This residue is equal to $(2-\Tr(DT^N))/4$, where $DT^N$ is the Jacobian of $T^N$ at the orbit, and indicates the stability type of the periodic orbit: The orbit is hyperbolic if $R<0$, elliptic if $0<R<1$, and inverse hyperbolic if $R>1$. It is known [@MacKayMeiss83] that the residue $R$ is related to the index of $\psi$ be the identity $$\label{tmgindex1}
R = -\frac14 \frac{\det(\Hess \Gbar_N(\psi))}{\prod_{j=1}^N (-\sdpar
G{12}(\psi_j,\psi_{j+1}))}\;.$$ In our case, $-\sdpar G{12}(\psi_j,\psi_{j+1})$ is always negative, so that $R$ is positive if $\psi$ is an $(N-1)$-saddle, and negative if $\psi$ is an $N$-saddle.
Now $x^\star(\psihat_0)$ also corresponds to a periodic orbit of the map , whose generating function is $H(x_n,x_{n+1})=\frac12(x_n-x_{n+1})^2 + \frac2\gamma U(x_n)$. The corresponding $N$-point generating function is precisely $(2/\gamma)V_\gamma$. Since the residue is invariant under area-preserving changes of variables, we also have $$\label{tmgindex2}
R = -\frac1{2\gamma} \frac{\det(\Hess V_\gamma(x^\star))}{\prod_{j=1}^N
(-\sdpar
H{12}(x^\star_j,x^\star_{j+1}))}\;.$$ In this case, the denominator is positive. Therefore, $\Hess
V_\gamma(x^\star)$ has an even number of positive eigenvalues if $\psi$ is an $N$-saddle, and an odd number of positive eigenvalues if $\psi$ is an $(N-1)$-saddle.
We can now complete the proofs of Theorem \[thm\_LargeN1\] and Theorem \[thm\_LargeN2\].
We first recall the following facts, established in [@BFG06a]. Whenever $\gammat$ crosses a bifurcation value $\gammat_M$, say from larger to smaller values, the index of the origin changes from $2M-1$ to $2M+1$. Thus the bifurcation involves a centre manifold of dimension $2$, with $2M-1$ unstable and $N-2M-1$ stable directions transversal to the manifold. Within the centre manifold, the origin repels nearby trajectories, and attracts trajectories starting sufficiently far away. Therefore, all stationary points lying in the centre manifold, except the origin, are either sinks or saddles for the reduced two-dimensional dynamics. For the full dynamics, they are thus saddles of index $(2M-1)$ or $2M$ (c.f. [@BFG06a Section 4.3]), at least for $\gammat$ close to $\gammat_M$.
We now return to the twist map in action-angle variables . The frequency $\Omegabar(I)$ being maximal for $I=0$, as $\eps$ increases, new orbits appear on the line $I=0$, which corresponds to the origin in $x$-coordinates. Orbits of rotation number $\nu=M/N$ can only exist if $\eps\Omegabar(0)=\eps\geqs2\pi\nu + \Order{\eps^2}$, which is compatible with the condition $\gammat<\gammat_M$.
Consider the case of a winding number $M=1$, that is, of orbits with rotation number $\nu=1/N$, which are the only orbits existing for $\gammat_2<\gammat<\gammat_1$. We note that $\gammat>\gammat_2$ implies $\Delta=2\pi/N\eps > 1/2 - \Order{\eps}$, and thus there exists $N_1<\infty$ such that the condition $N\geqs N_1$ automatically implies that $\eps$ is small enough for Corollary \[cor\_tmgstat\] to hold. Now, Proposition \[prop\_tmgindex\] yields:
If $N$ is even, then $K=\gcd(N,2)=2$, and there are $2N$ stationary points. The points $x^\star(0), x^\star(2\pi/N), \dots$ must be $2$-saddles, while the points $x^\star(\pi/N), x^\star(3\pi/N), \dots$ are $1$-saddles;
If $N$ is odd, then $K=\gcd(N,2)=1$, and there are $4N$ stationary points. The points $x^\star(0), x^\star(\pi/N), \dots$ must be $1$-saddles, while the points $x^\star(\pi/2N), x^\star(3\pi/2N), \dots$ are $2$-saddles.
Going back to original variables, we obtain the expressions and for the coordinates of these stationary points. The fact that they keep the same index as $\gammat$ moves away from $\gammat_M$ is a consequence of Relation and the fact that the corresponding stationary points of $\smash{\Gtilde_N}$ also keep the same index. Finally, Relation on the potential difference is a consequence of Proposition \[prop\_tm3\]. This proves Theorem \[thm\_LargeN1\].
For larger winding number $M$, one can proceed in an analogous way, provided $N$ is sufficiently large, as a function of $M$, for the conditions on $\eps$ to hold. This proves Theorem \[thm\_LargeN2\].
Finally, Theorem \[thm\_stoch2\] is proved in an analogous way as Theorems 2.7 and 2.8 in [@BFG06a], using results from [@FW] (see also [@Kifer; @Sugiura96a]).
Jacobi’s Elliptic Integrals and Functions {#app_ell}
=========================================
Fix some $\kappa\in[0,1]$. The *incomplete elliptic integrals of the first and second kind* are defined, respectively, by[^3] $$\label{ell1}
\JF(\phi,\kappa) = \int_0^\phi \frac{\6t}{\sqrt{1-\kappa^2\sin^2t}}\;,
\qquad
\JE(\phi,\kappa) = \int_0^\phi \sqrt{1-\kappa^2\sin^2t}\,\6t\;.$$ The *complete elliptic integrals of the first and second kind* are given by $$\label{ell2}
\JK(\kappa) = \JF(\pi/2,\kappa)\;,
\qquad
\JE(\kappa) = \JE(\pi/2,\kappa)\;.$$ Special values include $\JK(0)=\JE(0)=\pi/2$ and $\JE(1)=1$. The integral of the first kind $\JK(\kappa)$ diverges logarithmically as $\kappa\nearrow1$.
The *Jacobi amplitude* $\am(u,\kappa)$ is the inverse function of $\JF(\cdot,\kappa)$, i.e., $$\label{ell3}
\phi = \am(u,\kappa)
\quad\Leftrightarrow\quad
u = \JF(\phi,\kappa)\;.$$ The three standard Jacobi elliptic functions are then defined as $$\begin{aligned}
\nonumber
\sn(u,\kappa) &= \sin(\am(u,\kappa))\;,\\
\label{ell4}
\cn(u,\kappa) &= \cos(\am(u,\kappa))\;,\\
\dn(u,\kappa) &= \sqrt{1-\kappa^2\sin^2(\am(u,\kappa))}\;.
\nonumber\end{aligned}$$ Their derivatives are given by $$\begin{aligned}
\nonumber
\sn'(u,\kappa) &= \cn(u,\kappa)\dn(u,\kappa)\;,\\
\label{ell7}
\cn'(u,\kappa) &= -\sn(u,\kappa)\dn(u,\kappa)\;,\\
\dn'(u,\kappa) &= -\kappa^2\sn(u,\kappa)\cn(u,\kappa)\;.
\nonumber\end{aligned}$$ The function $\sn$ satisfies the periodicity relations $$\begin{aligned}
\nonumber
\sn(u+4\JK(\kappa),\kappa) &= \sn(u,\kappa)\;,\\
\sn(u+2\icx\JK(\sqrt{1-\kappa^2}),\kappa) &= \sn(u,\kappa)\;,
\label{ell7B}\end{aligned}$$ and has simple poles in $u=2n\JK(\kappa)+(2m+1)\icx\JK(\sqrt{1-\kappa^2})$, $n,m\in\Z$, with residue $(-1)^m/\kappa$. The functions $\cn$ and $\dn$ satisfy similar relations. Since $\am(u,0)=u$, one has $\sn(u,0)=\sin u$, $\cn(u,0)=\cos u$ and $\dn(u,0)=1$. As $\kappa$ grows from $0$ to $1$, the elliptic functions become more and more squarish. This is also apparent from their Fourier series, given by $$\begin{aligned}
\nonumber
\frac{2\JK(\kappa)}\pi \sn\biggpar{\frac{2\JK(\kappa)}\pi\psi,\kappa}
&= \frac4\kappa \sum_{p=0}^\infty \frac{\nome^{(2p+1)/2}}{1-\nome^{2p+1}}
\sin\bigpar{(2p+1)\psi} \;, \\
\label{ell5}
\frac{2\JK(\kappa)}\pi \cn\biggpar{\frac{2\JK(\kappa)}\pi\psi,\kappa}
&= \frac4\kappa \sum_{p=0}^\infty \frac{\nome^{(2p+1)/2}}{1+\nome^{2p+1}}
\cos\bigpar{(2p+1)\psi} \;, \\
\nonumber
\frac{2\JK(\kappa)}\pi \dn\biggpar{\frac{2\JK(\kappa)}\pi\psi,\kappa}
&= 1 + 4 \sum_{p=0}^\infty \frac{\nome^p}{1+\nome^{2p}}
\cos\bigpar{2p\psi} \;, \end{aligned}$$ where $\nome=\nome(\kappa)$ is the *elliptic nome* defined by $$\label{ell6}
\nome = \exp\biggset{-\pi\frac{\JK(\sqrt{1-\kappa^2})}{\JK(\kappa)}}\;.$$ The elliptic nome has the asymptotic behaviour $$\label{ell6a}
\nome(\kappa) =
\begin{cases}
\vrule height 12pt depth 16pt width 0pt
\dfrac{\kappa^2}{16} + \dfrac{\kappa^4}{32} +
\bigOrder{\kappa^6}
& \text{for $\kappa\searrow0$\;,} \\
\vrule height 16pt depth 12pt width 0pt
\exp\biggset{\dfrac{\pi^2}{\log\brak{(1-\kappa^2)/16}}}
\biggbrak{1+\biggOrder{\dfrac{1-\kappa^2}{\log^2\brak{(1-\kappa^2)/16}}}}
& \text{for $\kappa\nearrow1$\;.}
\end{cases}$$ We also use the following identities, derived in [@Jacobi1892 p. 175]. For $k\geqs1$, $$\label{ell8}
\biggpar{\frac{2\JK(\kappa)}\pi}^{2k}
\sn^{2k}\biggpar{\frac{2\JK(\kappa)}\pi\psi,\kappa}
= \hat c_{2k,0} + \sum_{p=1}^\infty \hat c_{2k,p}
\frac{\nome^{p}}{1-\nome^{2p}}
\cos\bigpar{2p\psi} \;,$$ where the $\hat c_{2k,0}$ are positive constants (independent of $\psi$), and the other Fourier coefficients are given for the first few $k$ by $$\begin{aligned}
\nonumber
\hat c_{2,p} &= - \frac4{\kappa^2} (2p)\;, \\
\label{ell9}
\hat c_{4,p} &= \frac4{3!\kappa^4} \biggbrak{(2p)^3 -
4 (2p)(1+\kappa^2)\biggpar{\frac{2\JK(\kappa)}\pi}^2}\;, \\
\hat c_{6,p} &= -\frac4{5!\kappa^6} \biggbrak{(2p)^5 - 20 (2p)^3
(1+\kappa^2)\biggpar{\frac{2\JK(\kappa)}\pi}^2
+ 8 (2p) (8+7\kappa^2+8\kappa^4) \biggpar{\frac{2\JK(\kappa)}\pi}^4}\;.
\nonumber\end{aligned}$$
Proofs of the Fixed-Point Argument {#sec_prtech}
==================================
In this appendix, we give the somewhat technical proofs of the fixed-point argument given in Section \[sec\_tmgchi\]. We start by proving Lemma \[lem\_tmgff4\], stating a fixed-point equation equivalent to the stationarity conditions $\tdpar{\Gtilde_N}{\chi_q}=0$, $q\in\cQ$.
The definitions of $\alpha_n$ and $\beta_n$ imply, for any $a, b\geqs 0$, $$\begin{aligned}
\nonumber
\alpha_n^a &= \sum_{q_1,\dots,q_a\in\cQ}
\prod_{i=1}^a \rho_{q_i}
\omega^{q_i(n+1/2)} \e^{\icx\psihat_0 q_i/M} \;, \\
\beta_n^b &= \sum_{q'_1,\dots,q'_b\in\cQ}
\prod_{j=1}^b \frac{-\icx\eps\rho_{q'_j}}{\tan(\pi q'_j/N)}
\omega^{q'_j(n+1/2)} \e^{\icx\psihat_0 q'_j/M} \;.
\label{tmgff23:1}\end{aligned}$$ It is more convenient to compute $\tdpar{\Gtilde_N}{\rho_{-q}}$ rather than $\tdpar{\Gtilde_N}{\chi_q}$. We thus have to compute the derivatives of $\tilde g_p$ with respect to $\rho_{-q}$ for all $p$. For $p=0$, we have $$\label{tmgffB5:2}
\dpar{\tilde g_0}{\rho_{-q}}
= \eps^3 \sum_{n=1}^N G_0'(\Delta+\eps^2\alpha_n)
\dpar{\alpha_n}{\rho_{-q}}\;,$$ where shows that $\tdpar{\alpha_n}{\rho_{-q}} =
\omega^{-q(n+ 1/2)}\e^{-\icx\psihat_0 q/M}$. We expand $G_0'(\Delta+\eps^2\alpha_n)$ into powers of $\eps^2$, and plug in again. In the resulting expression, the sum over $n$ vanishes unless $\sum_i q_i-q$ is a multiple of $N$, say $kN$. This yields $$\label{tmgffB5:3}
\dpar{\tilde g_0}{\rho_{-q}}
= N\eps^3 \sum_{k\in\Z} (-1)^k \e^{\icx k\psihat_0 N/M}
\sum_{a\geqs0} \frac{\eps^{2a}}{a!} G_0^{(a+1)}(\Delta)
\Gamma^{(a,0)}_{kN+q}(\rho)\;.$$ We consider the terms $a=0$ and $a=1$ separately:
Since $\smash{\Gamma^{(0,0)}_{kN+q}}(\rho)=\delta_{kN,-q}$ vanishes for all $k$, the sum actually starts at $a=1$.
The fact that $\Gamma^{(1,0)}_{\ell}(\rho)$ vanishes whenever $\abs{\ell}>N/2$ implies that only the term $k=0$ contributes, and yields a contribution proportional to $-\rho_q$.
Shifting the summation index $a$ by one unit, we get $$\label{tmgffB5:4}
\dpar{\tilde g_0}{\rho_{-q}}
= N\eps^5 \biggbrak{G_0''(\Delta)\rho_q +
\sum_{k\in\Z} (-1)^k \e^{\icx k\psihat_0 N/M}
\sum_{a\geqs1} \frac{\eps^{2a}}{(a+1)!} G_0^{(a+2)}(\Delta)
\Gamma^{(a+1,0)}_{kN+q}(\rho)}\;.$$ A similar computation for $p\neq 0$ shows that $$\label{tmgffB5:5}
\dpar{\tilde g_p}{\rho_{-q}}\e^{2\icx p\psihat_0} =
N\eps^5 \sum_{k\in\Z} (-1)^k \e^{\icx k\psihat_0 N/M}
\sum_{a,b\geqs0} \frac{\eps^{2(a+b)}}{a!b!} H^{(a)}_{p,q}(\Delta)p^b
\Gamma^{(a,b)}_{kN+q-2pM}\;.$$ Solving the stationarity condition $$\label{tmgffB5:1}
0 = \dpar{\Gtilde_N}{\rho_{-q}}
= \sum_{p=-\infty}^\infty \e^{2\icx p\psihat_0} \dpar{\tilde
g_p}{\rho_{-q}}$$ with respect to $\rho_q$, and singling out the term $a=b=0$ in to give the leading term $\rho^{(0)}$ then yields the result.
The following estimates yield sufficient conditions for the operator $\cT$ to be a contraction inside a certain ball, for the norm $\norm{\cdot}_\lambda$ introduced in .
\[prop\_tmgff2\] There exist numerical constants $c_0, c_1>0$, such that for any $\lambda<\lambda_0$, and any $N$ such that $N\e^{-\lambda_0N/2M}\leqs 1/2$, the estimates $$\begin{aligned}
\label{tmgff21a}
\norm{\cT\rho}_\lambda
&\leqs \frac{c_1L_0}{\Delta\abs{G_0''(\Delta)}}
\biggbrak{1
+\frac{M}{\Delta^3}\Bigpar{\norm{\rho}_\lambda +
\eps\Delta M \eta(\lambda_0,\lambda)}\eps\norm{\rho}_\lambda}\;,
\\
\norm{\cT\rho-\cT\rho'}_\lambda
&\leqs \frac{c_1L_0}{\abs{G_0''(\Delta)}}\frac{M}{\Delta^4}
\Bigbrak{\bigpar{\norm{\rho}_\lambda\vee\norm{\rho'}_\lambda}
+ \eps\Delta M \eta(\lambda_0,\lambda)}\eps
\norm{\rho-\rho'}_\lambda
\label{tmgff21b}\end{aligned}$$ hold with $\eta(\lambda_0,\lambda) =
(\e^\lambda/\lambda_0)\vee(1/(\lambda_0-\lambda))$, provided $\rho$ and $\rho'$ satisfy $$\label{tmgff21c}
\eps \bigpar{\norm{\rho}_\lambda\vee\norm{\rho'}_\lambda}
\leqs c_0 \frac{\Delta^2}M
\biggpar{1\wedge\frac{\lambda_0-\lambda}M\wedge\frac{\lambda}M}\;.$$
The lower bound $$\label{tmgff21:1}
\frac{\abs{\tan(\pi q/N)}}{\eps} \geqs \frac{\pi\abs{q}}{N\eps}
= \frac{\Delta}{2M}\abs{q}$$ directly implies $$\label{tmgff21:2}
\abs{H^{(a)}_{p,q}(\Delta)} \leqs L_0 \frac{a!}{r^{a+1}}
\biggbrak{1+\frac{2M\abs{p}}{\abs{q}}}\e^{-\lambda_0\abs{p}}\;.$$ The assumption on $N$ allows $\abs{\rho_q}$ to be bounded by a geometric series of ratio smaller than $1/2$, which is dominated by the term $k=0$, yielding $$\label{tmgff21:3}
\norm{\rho^{(0)}}_\lambda
\leqs \frac{c_2 L_0}{\Delta\abs{G_0''(\Delta)}}
\e^{-(\lambda_0-\lambda)\abs{q}/2M}
\leqs \frac{c_2L_0}{\Delta\abs{G_0''(\Delta)}}\;,$$ where $c_2>0$ is a numerical constant. The fact that $\Gamma^{(a,b)}_\ell(\rho)$ contains less than $N^{a+b-1}$ terms, together with , implies the bound $$\label{tmgff21:4}
\abs{\Gamma^{(a,b)}_\ell(\rho)}
\leqs N^{a+b-1} \biggpar{\frac{2M}\Delta}^b
\e^{-\lambda\abs{\ell}/2M}\norm{\rho}_\lambda^{a+b}\;.$$ Assuming that $\norm{\rho}_\lambda\leqs c_0\Delta^2/M\eps$ for sufficiently small $c_0$, it is straightforward to obtain the estimate $$\label{tmgff21:5}
\norm{\Phi^{(1)}(\rho,\eps)}_\lambda
\leqs \frac{c_3L_0}{\abs{G_0''(\Delta)}} \frac{2M}{\Delta^4}
\eps\norm{\rho}_\lambda^2\;.$$ In the sequel, we assume that $q>0$, since by symmetry of the norm under permutation of $\rho_q$ and $\rho_{-q}$ the same estimates will hold for $q<0$. The norm of $\Phi^{(2)}(\rho,\eps)$ is more delicate to estimate. We start by writing $$\label{tmgff21:6}
\abs{\Phi^{(2)}_q(\rho,\eps)} \leqs
\frac{L_0}{\abs{G_0''(\Delta)}} \frac1N \sum_{a+b\geqs1}
\bigpar{\eps^2N\norm{\rho}_\lambda}^{a+b} \frac1{r^{a+1}}
\biggpar{\frac{2M}\Delta}^b S_q(b)\;,$$ where $$\label{tmgff21:7}
S_q(b) = \frac1{b!} \sum_{p\neq0} \abs{p}^{b} \biggpar{1+\frac{2M\abs{p}}q}
\e^{-(\lambda_0-\lambda)\abs{p}}
\sum_{k\in\Z} \exp\biggset{-\frac\lambda{2M}(2M\abs{p}+\abs{kN+q-2Mp})}\;.$$ We decompose $S_q(b)=S_q^+(b)+S_q^-(b)$, where $S_q^+(b)$ and $S_q^-(b)$ contain, respectively, the sum over positive and negative $p$. In the sequel, we shall only treat the term $S_q^+(b)$. The sum over $k$ in is dominated by the term for which $kN$ is the closest possible to $2Mp-q$, and can be bounded by a geometric series. The result for $p>0$ is $$\label{tmgff21:7b}
\sum_{k\in\Z} \exp\biggset{-\frac\lambda{2M}(2Mp+\abs{kN+q-2Mp})}
\leqs c_4 \bigpar{\e^{-\lambda p} \wedge \e^{-\lambda q/2M}}\;.$$ We now distinguish between two cases.
If $q\leqs2M$, we bound the sum over $k$ by $\e^{-\lambda p}$, yielding $$\label{tmgff21:9}
S^+_q(b) \leqs \frac{c_4}{b!} \frac{4M}q
\sum_{p\geqs1}p^{b+1}\e^{-\lambda_0p}
\leqs 4M c_5 \frac{b+1}{\lambda_0^b}\;.$$ Since $\e^{\lambda q/2M}\leqs\e^\lambda$, it follows that $$\label{tmgff21:10}
\abs{\Phi^{(2)}_q(\rho,\eps)} \leqs \frac{c_6L_0}{\abs{G_0''(\Delta)}}
\frac{2M^2\e^\lambda}{r^2\Delta\lambda_0} \eps^2\norm{\rho}_\lambda
\e^{-\lambda q/2M}\;.$$
If $q>2M$, we split the sum over $p$ at $q/2M$. For $2Mp\leqs q$, we bound $(1+2Mp/q)$ by $2$ and the sum over $k$ by $\e^{-\lambda q/2M}$. For $2Mp>q$, we bound the the sum over $k$ by $\e^{-\lambda p}\leqs\e^{-\lambda q/2M}$. This shows $$\label{tmgff21:11}
S^+_q(b) \leqs 2c_7M \frac{b+1}{(\lambda_0-\lambda)^b} \e^{-\lambda
q/2M}\;,$$ and thus $$\label{tmgff21:12}
\abs{\Phi^{(2)}_q(\rho,\eps)} \leqs \frac{c_8L_0}{\abs{G_0''(\Delta)}}
\frac{2M^2}{r^2\Delta(\lambda_0-\lambda)} \eps^2\norm{\rho}_\lambda
\e^{-\lambda q/2M}\;.$$
Now and , together with , imply . The proof of is similar, showing first the estimate $$\label{tmgff21:13}
\biggabs{\prod_{i=1}^a \rho_{q_i} - \prod_{i=1}^a \rho'_{q_i}} \leqs
a\bigpar{\norm{\rho}_\lambda\vee\norm{\rho'}_\lambda}^{a-1}
\e^{-\lambda\sum_{i=1}^a\abs{q_i}/2M} \norm{\rho-\rho'}_\lambda$$ by induction on $a$, and then $$\label{tmgff21:14}
\bigabs{\Gamma^{(a,b)}_\ell(\rho)-\Gamma^{(a,b)}_\ell(\rho')}
\leqs (a+b)
\bigbrak{N(\norm{\rho}_\lambda\vee\norm{\rho'}_\lambda)}^{a+b-1}
\biggpar{\frac{2M}\Delta}^b \e^{-\lambda\abs{\ell}/2M}
\norm{\rho-\rho'}_\lambda\;.$$
It is now easy to complete the proof of Proposition \[cor\_tmgff2\].
Estimate for $\norm{\cT\rho}_\lambda$ implies that if $$\label{tmgff30:1}
\eps \leqs
\frac{R_0}{\Delta M\eta(\lambda_0,\lambda)}
\wedge \frac{\Delta^3}{2MR_0^2}
\biggpar{\frac{\Delta\abs{G_0''(\Delta)}}{c_1L_0}R_0 - 1}\;,$$ then $\cT(\cB_\lambda(0,R_0))\subset\cB_\lambda(0,R_0)$. If in addition $$\label{tmgff30:2}
\eps \leqs c_0 \frac{\Delta^2}{MR_0}
\biggpar{1\wedge\frac{\lambda_0-\lambda}M\wedge\frac{\lambda}M}\;,$$ then Estimate for $\norm{\cT\rho-\cT\rho'}_\lambda$ applies for $\rho,\rho'\in\cB_\lambda(0,R_0)$. It is then immediate to check that $\cT$ is a contracting in $\cB_\lambda(0,R_0)$, as a consequence of . Thus the existence of a unique fixed point in that ball follows by Banach’s contraction lemma. Finally, the assertions on the properties of $\rho^\star$ follow from the facts that they are true for $\rho^{(0)}$, that they are preserved by $\cT$ and that $\rho^\star=\lim_{n\to\infty}\cT^n\rho^{(0)}$.
Nils Berglund\
[CPT–CNRS Luminy]{}\
Case 907, 13288 Marseille Cedex 9, France\
[*and*]{}\
[PHYMAT, Université du Sud Toulon–Var]{}\
[*Present address:*]{}\
[MAPMO–CNRS, Université d’Orléans]{}\
Bâtiment de Mathématiques, Rue de Chartres\
B.P. 6759, 45067 Orléans Cedex 2, France\
[*E-mail address:* ]{}[berglund@cpt.univ-mrs.fr]{}
Bastien Fernandez\
[CPT–CNRS Luminy]{}\
Case 907, 13288 Marseille Cedex 9, France\
[*E-mail address:* ]{}[fernandez@cpt.univ-mrs.fr]{}
Barbara Gentz\
[Weierstraß Institute for Applied Analysis and Stochastics]{}\
Mohrenstra[ß]{}e 39, 10117 Berlin, Germany\
[*Present address:*]{}\
[Faculty of Mathematics, University of Bielefeld]{}\
P.O. Box 10 01 31, 33501 Bielefeld, Germany\
[*E-mail address:* ]{}[gentz@math.uni-bielefeld.de]{}
[^1]: For the reader’s convenience, we recall the definitions and main properties of Jacobi’s elliptic integrals and functions in Appendix \[app\_ell\].
[^2]: If $N$ is even, the orbits $O_A$ and $O_B$ contain $N$ instead of $2N$ points, because $R^{N/2}=-\one$.
[^3]: One should beware of the fact that some sources use $m=\kappa^2$ as parameter.
|
---
abstract: 'We present new high resolution inelastic neutron scattering data on the candidate spin liquid Tb$_2$Ti$_2$O$_7$. We find that there is no evidence for a zero field splitting of the ground state doublet within the $0.2$ K resolution of the instrument. This result contrasts with a pair of recent works on Tb$_2$Ti$_2$O$_7$ claiming that the spin liquid behavior can be attributed to a $2$ K split singlet-singlet single-ion spectrum at low energies. We also reconsider the entropy argument presented in Chapuis [*et al.*]{} as further evidence of a singlet-singlet crystal field spectrum. We arrive at the conclusion that estimates of the low temperature residual entropy drawn from heat capacity measurements are a poor guide to the single ion spectrum without understanding the nature of the correlations.'
author:
- 'B. D. Gaulin'
- 'J.S. Gardner'
- 'P. A. McClarty'
- 'M. J. P. Gingras'
title: 'Lack of Evidence for a Singlet Crystal Field Ground State in the Tb$_2$Ti$_2$O$_7$ Magnetic Pyrochlore'
---
[*Introduction*]{} $-$ In some magnetic systems, the lattice geometry or the competition between different interactions can dramatically inhibit, or frustrate, the development of long range order. The failure of a frustrated magnetic system to exhibit magnetic order down to zero temperature, giving rise to a so-called spin liquid state, is one of the most sought after phenomena among strongly interacting condensed matter systems. Despite two decades of experimental searches, the number of candidate materials that display a spin liquid state remains small [@Balents-Nature]. The Tb$_2$Ti$_2$O$_7$ insulating material, where magnetic Tb$^{3+}$ ions sit on a pyrochlore lattice of corner-sharing tetrahedra, is one of these candidates [@Gardner-TbTO-PRL-1999]. Despite a Curie-Weiss temperature, $\theta_{\rm CW} \approx -14$ K set by the magnetic interactions [@Gingras-TbTO-PRB-2000], Tb$_2$Ti$_2$O$_7$ does not develop long range order down to at least 50 mK [@Gardner-TbTO-PRL-1999; @Gardner-TbTO-PRB-2003]. The microscopic mechanism by which Tb$_2$Ti$_2$O$_7$ fails to develop long range order at a temperature scale of approximately 1 K [@Gingras-TbTO-PRB-2000; @Hertog-PRL-2000; @Kao] is not understood [@TTOHeff]. Compounding the difficulty in understanding why Tb$_2$Ti$_2$O$_7$ does not order, one notes that Tb$_2$Sn$_2$O$_7$, seemingly closely related at the microscopic level to Tb$_2$Ti$_2$O$_7$, develops long range order at 0.87 K [@Matsuhira-TbSnO; @Mirebeau-TbSnO].
Since Tb$^{3+}$ is an even electron system (electronic configuration $^7$F$_6$, $L=3$, $S=3$, $J=6$), the existence of a magnetic ground state for an an isolated (i.e. assumed non-interacting) Tb$^{3+}$ ion in Tb$_2$Ti$_2$O$_7$ is not guaranteed by Kramers’ theorem [@Gingras-TbTO-PRB-2000]. To investigate whether Tb$^{3+}$ is magnetic, the so-called single-ion crystal field (CF) problem must first be solved [@Gingras-TbTO-PRB-2000]. For example, for a perfect cubic ionic environment, theory predicts that Tb$^{3+}$ would either have a singlet or non-magnetic doublet single-ion CF ground state [@Lea]. However, in Tb$_2$Ti$_2$O$_7$, with its Fd$\bar 3$m symmetry, the Tb$^{3+}$ environment displays a very large trigonal distortion away from cubic symmetry [@Gingras-TbTO-PRB-2000]. Early investigations found that this distortion endows Tb$^{3+}$ with a magnetic CF doublet characterized by two mutually time-reversed conjugate states, $\vert \psi_0^+\rangle $ and $\vert \psi_0^-\rangle $ [@Gingras-TbTO-PRB-2000]. The states, $\vert \psi_0^\pm \rangle$, are such that all matrix elements of the raising and lowering angular momentum operator, $J^\pm$, vanish while $\langle \psi_0^\pm \vert J^z \vert \psi_0^\pm \rangle = \pm \vert \langle J^z \rangle \vert
\approx 3.4$ [@Gingras-TbTO-PRB-2000]. Since $\langle \psi_0^\pm \vert J^\mu \vert \psi_0^\pm \rangle
= \pm \vert \langle J^z\rangle \vert \delta_{\mu,z}$, the Tb$^{3+}$ moment within its CF ground state can be described by a classical Ising spin with a moment that points “in” or “out” of the reference primitive tetrahedral unit cell to which it belongs. This makes Tb$_2$Ti$_2$O$_7$ a relative of the Ising spin ice compounds [@Gingras-TbTO-PRB-2000; @TTOHeff].
The lowest excited CF state is also a doublet, $\vert \psi_{\rm e}^{\pm} \rangle$, at an energy approximately 1.6 meV $\sim$ 18 K above the ground doublet [@Gingras-TbTO-PRB-2000; @Gardner-TbTO-PRB-2001; @Mirebeau-TbXO-2007]. Recent theoretical work [@TTOHeff] has argued that the proximity of Tb$_2$Ti$_2$O$_7$ to the zero temperature transition from an “all-in/all-out” ${\bf q}=0$ Néel to a spin ice state [@Hertog-PRL-2000], along with the exchange and dipole-dipole interaction-induced admixing of $\vert \psi_0 ^\pm \rangle$ and $\vert \psi_{\rm e}^\pm \rangle$, constitute two key ingredients as to why Tb$_2$Ti$_2$O$_7$ does not develop long range order.
In conventional (unfrustrated) magnets, spin-lattice couplings typically play an insignificant role in the development of long range order. In highly frustrated magnetic systems, however, spin-lattice couplings can lead to a combined magnetic-lattice (“spin-Peierls”) transition to long range magnetic order that reduces the magnetic frustration [@Villain-ZPhysB], as observed in the highly frustrated antiferromagnet ZnCr$_2$O$_4$ spinel compound [@Lee-ZnCr2O4]. In Tb$_2$Ti$_2$O$_7$, Mamsurova and co-workers long ago reported an unusually large anomalous field-dependent thermal expansion, indicating an important spin-lattice coupling in this material [@Mamsurova]. In more recent x-ray diffraction experiments, Ruff [*et al.*]{} found evidence for a tendency of the Tb$_2$Ti$_2$O$_7$ lattice to undergo a cubic tetragonal deformation but, down to 300 mK and in zero field, no equilibrium cubic to tetragonal transition was observed [@Ruff-tetragonal]. Only in the presence of very high magnetic fields ($\sim 29$ T), does the system show any evidence for such a structural phase transition [@Ruff-highfield]. In contrast, Chapuis and co-workers very recently argued that a tetragonal deformation does actually exist in Tb$_2$Ti$_2$O$_7$ in zero field that splits the above $\vert \psi_0^\pm \rangle$ doublet into two [*non-magnetic singlets*]{} separated by an energy scale, $\delta$, with $\delta \approx 1.8$ K [@Chapuis]. They invoke previously published inelastic neutron scattering (INS) data [@Mirebeau-TbXO-2007] to suggest that such a large splitting of the ground doublet compared to the exchange and dipolar interactions between Tb$^{3+}$ ions is responsible for inhibiting the spontaneous development of long range order. Reference \[\] also present data for the temperature dependence of the magnetic entropy, $S(T)$, which, they claim, by falling below R$\ln 2$ at low temperature, further supports the evidence for a split doublet. Very recently, Bonville and collaborators [@Bonville-singlet] have built further on Chapuis [*et al.*]{}’s split doublet picture to advocate that the failure of Tb$_2$Ti$_2$O$_7$ to order is due to the sub-critical value of the interactions compared to the singlet-singlet gap $\delta$.
In this paper, we argue that the evidence for a split doublet of energy scale as large as $\delta \sim 1.8$ K in Tb$_2$Ti$_2$O$_7$ as proposed in Refs. \[\] is not compelling. Using new high resolution INS data, we show that there is no evidence for a split doublet in this material with an energy splitting greater than 0.2 K, a factor 10 or so [*smaller*]{} than the proposed [@Chapuis; @Bonville-singlet] singlet-singlet gap $\delta$. Secondly, we argue that by neglecting correlations that develop in the collective paramagnetic (spin liquid) phase of Tb$_2$Ti$_2$O$_7$, the authors of Ref. \[\] are in principle unable to draw any conclusion about the nature of the CF state of Tb$^{3+}$ in Tb$_2$Ti$_2$O$_7$ on the basis of a measurement of $S(T)$. We illustrate that point via the calculation of $S(T)$ for a toy model which, while possessing a ground state doublet and lacking a transition to long range order, does display an $S(T)$ that falls [*below*]{} R$\ln 2$ at low temperature.
[*Inelastic neutron scattering results*]{} $-$ We first proceed to show that a singlet-singlet gap $\delta \sim 1.8$ K is inconsistent with high energy-resolution inelastic neutron scattering data and that, in fact, Tb$_2$Ti$_2$O$_7$ displays quasielastic magnetic spectral weight down to energies of at least 0.02 meV, approximately an order of magnitude lower in energy than the value $\delta \sim 1.8$ K reported in Refs. \[\].
Time-of-flight neutron scattering data was obtained on single crystal Tb$_2$Ti$_2$O$_7$ in zero and finite magnetic field applied along the $[1 {\bar 1} 0]$ (vertical) direction at T=0.4 K. These measurements used the Disk Chopper Spectrometer (DCS) at NIST and employed incident neutrons of wavelength $\lambda$=4.8 $\AA$ resulting in an energy resolution of 0.1 meV. The $H=0$, zero magnetic field data set, after integration along the \[HH0\] direction, perpendicular to the (00L) direction and within the horizontal scattering plane, is plotted in Fig. 1 a). The quasi-elastic magnetic scattering peaks at [**Q**]{}=002 (Q=1.2 $\AA^{-1}$). Previous work [@Gardner-TbTO-PRB-2003; @Gardner-TbTO-PRB-2001] has shown that the corresponding diffuse scattering is distributed in reciprocal space into a well known checkerboard pattern. Figure 1 b) shows cuts in energy of this zero magnetic field data set and also of a data set with a $H=3$ T magnetic field applied along the $[1 {\bar 1} 0]$ direction. These cuts simulate constant-[**Q**]{} scans, although they integrate the scattering data in the \[HH0\] direction (shown in Fig.1 a) and also in the \[00L\] direction between L=1.6 and 1.8. Two features are noteworthy: while there is a shoulder to this particular cut of the zero field quasi-elastic scattering near $E\sim$ 0.2 meV, there is a continuous and monotonically increasing distribution of magnetic spectral weight as the energy decreases down to zero energy. The nature of this shoulder near 0.2 meV is subtle for the zero magnetic field data set. Indeed, low energy quasi-elastic magnetic scattering is obscured by relatively strong nuclear incoherent elastic scattering which dominates the elastic signal within the 0.1 meV energy resolution of this measurement. However, this nuclear incoherent contribution to the elastic scattering can be estimated and removed by examination of the $H=3$ T data set in Fig. 1 b). Field-induced long range order [@Rule-TbTO-PRL-2006] leads to a splitting-off of the quasi-elastic magnetic scattering from the resolution-limited nuclear incoherent elastic scattering. Thus the strength of the nuclear incoherent contribution can be determined. Consider the two, otherwise identical cuts shown in Fig. 1 b), both at T=0.4 K, and taken at $H=0$ (top) and at $H=3$ T (bottom). One observes scattering at $E=0$ with an intensity of $\sim$ 9 for $H=0$ where the zero energy scattering has contributions from both nuclear incoherent elastic scattering and quasi-elastic magnetic scattering. In the bottom of panel b), for $H=3$ T, an $E=0$ intensity of $\sim$ 3.5 is measured, which has contributions from nuclear incoherent scattering alone. Hence the intensity of the elastic magnetic scattering at $H=0$ is $\sim$ 5.5 in the intensity units employed in Figs 1 b); at least a factor of two higher than the intensity associated with the shoulder near $\sim$ 0.2 meV. Therefore, the distribution of magnetic scattering intensity does indeed peak at zero energy and extends out to $\sim 0.3$ meV at low temperatures in $H=0$. In other words, there is no obvious mode centred at an energy of 1.8 K (0.16 meV) in the $H=0$ data
![\[\] a) High energy-resolution inelastic neutron scattering data taken with $\lambda$=4.8 $\AA$ neutrons on DCS [@Rule-TbTO-PRL-2006]. b) Energy cuts of the data shown in a) as described in the text. Quasi-elastic magnetic scattering at T=0.4 K and zero applied magnetic field peaks at zero energy. The crystal field excitation in zero field (top right panel) is anomalously broad in energy, compared with resolution-limited spin waves seen in the bottom right panel within the field-induced ordered phase.](TbTiO_5A.eps)
To more definitely assess whether a finite energy excitation may exist at $E\sim \delta \sim 0.16$ meV $\sim 0.18$ K, very high energy-resolution neutron scattering was carried out on the same single crystal of Tb$_2$Ti$_2$O$_7$, in zero magnetic field, as well with several different magnetic field strengths applied along the $[ 1 {\bar 1} 0]$ direction. This data, taken with $\lambda=9$ $\AA$ incident neutrons on DCS, is shown in Fig. 2. The scattering data shown in Fig. 2 integrates the raw inelastic scattering data over all $\bf Q$ surveyed, which extends out in reciprocal space to the centre of the aforementioned diffuse magnetic scattering checkerboard at ${\bf Q}=002$ [@Gardner-TbTO-PRB-2003; @Gardner-TbTO-PRB-2001]. This very high energy-resolution data clearly shows the quasi-elastic magnetic scattering to increase [*continuously*]{} with decreasing energy down to the energy resolution of the measurement, $\sim$ 0.02 meV $\sim$ 0.2 K, an order of magnitude lower than the singlet-singlet gap $\delta\sim 1.8$ K invoked in Refs. \[\].
![\[\]Very high energy-resolution inelastic neutron scattering data employing $\lambda=9$ $\AA$ incident neutrons with DCS, as described in the text. The zero field measurement shows the quasielastic magnetic distrution of scattering to extend down to at least 0.02 meV.](TbTiO_9A_fig.eps)
Figure 2 also shows that application of a sufficiently strong $[1 {\bar 1} 0]$ magnetic field ($H \sim 4$ T) moves the quasi-elastic magnetic scattering out of the field of view of the figure, revealing a low background which is energy-independent. As a function of increasing applied magnetic field, the quasi-elastic scattering is strongly depleted at low energies, and an inelastic feature is observed for field strengths above $H \sim 1$ T. This excitation moves to higher energies with increasing field strength. However, and [*crucially significant*]{} for our argument, we see that even with an applied field of $H=1$ T (red curve), the broad inelastic peak is at an energy of $\sim$ 0.1 meV $\sim 1$ K, roughly a factor of 2 lower in energy than the gap $\delta\sim 1.8$ K invoked in Refs. \[\] in zero field.
One could speculate that the quasielastic distribution of magnetic scattering in zero field out to $\sim$ 0.3 meV is a dispersive singlet-singlet excitation with a band width of twice the mean separation between the two states, such that its density-of-states fills in th quasi-elastic energy range. However, the intensity would then be the largest for the top of this band, as the density-of-states would be high where the dispersion is flat. This is not observed – the quasi-elastic scattering (Fig. 2) decreases monotonically with increasing energy.
One could also speculate a static and random Jahn-Teller distortion in which a broad distribution of singlet-singlet gaps is present in the spin liquid state. This could lead to a distribution of gaps and induce monotonically-decreasing (as a function of energy) quasielastic scattering in zero field as seen in Figs. 1 and 2. However, with decreasing temperature, these static gaps would get progressively frozen out, resulting in a larger and larger fraction of the system being in a non-magnetic state. The $1/T_1$ muon spin relaxation rate is large and flat in temperature from $\sim$ 1 K down to 0.05 K [@Gardner-TbTO-PRL-1999]. This would seem to rule out this scenario since, at 0.05 K, excited singlets with an energy gap larger than 0.005 meV would be frozen out. Similarly, neutron spin echo (NSE) results in Ref. \[\] rules out a large 1.8 K gap in the system. NSE with sub-$\mu$eV resolution and polarised neutron diffraction [@Gardner-TbTO-PRB-2003] both show an increase in the magnetic scattering below 300 mK suggesting a magnetic ground state or an extremely small gap that is thermally active at 50 mK.
Taken altogether, we conclude there is no compelling evidence for a well-defined singlet-singlet gap in Tb$_2$Ti$_2$O$_7$ in zero field at low temperatures. Its quasi-elastic, magnetic spectrum is not substantially different from that displayed by Ho$_2$Ti$_2$O$_7$ [@Clancy-HoTO-PRB] just above its frozen spin ice ground state, albeit with a higher energy scale.
[*Residual entropy*]{} $-$ Having established that there is no direct evidence for a split doublet with an energy gap larger than 0.02 meV $\sim$ 0.2 K in Tb$_2$Ti$_2$O$_7$, we now address the interpretation given in Ref. \[\] of low temperature magnetic entropy data determined from the heat capacities of [Tb$_{2}$Sn$_{2}$O$_{7}$]{} and Tb$_2$Ti$_2$O$_7$. The magnetic entropy results for [Tb$_{2}$Sn$_{2}$O$_{7}$]{} are given in Fig. 2 of Ref. \[\]. In this data, the nuclear and phonon contributions have been subtracted. One sees that roughly R$\ln 4$ is lost upon cooling the sample from $20$ K down to $40$ mK. This is consistent with having no extensive residual entropy and with magnetic ions having four states lying beneath $20$ K. The data for Tb$_2$Ti$_2$O$_7$ also indicate a similar loss of magnetic entropy of R$\ln 4$ below $20$ K [@Chapuis]. The authors of Ref. \[\] proceed to make the following argument: Suppose that there is a doublet-doublet crystal field scheme with a gap $\Delta$. Then, at high temperatures exceeding the scale of the gap, the entropy would be R$\ln 4$ whereas at low temperatures, $T\ll \Delta$, the entropy would saturate at R$\ln 2$. Therefore, in this scheme, the total entropy variation should be R$\ln 2$. Since the entropy variation is observed to be R$\ln 4$ in both [Tb$_{2}$Sn$_{2}$O$_{7}$]{} and Tb$_2$Ti$_2$O$_7$, Ref. \[\] concludes that the doublet-doublet energy level scheme must be incorrect.
In an attempt to support their argument, the authors present a calculation of the entropy for a crystal field scheme consisting of a pair of low-lying singlets separated by a tuneable energy gap of $\delta$ and one excited doublet at energy $\Delta>\delta$ above the ground singlet. Upon increasing $\delta$ from zero, the lowest temperature entropy drops to zero for $T\lesssim
\delta$ and exhibits a plateau at R$\ln 2$ for $\delta \lesssim T \lesssim \Delta$ (see Fig. 3 of Ref. \[\]). The width of the plateau decreases as $T$ increases and for $\delta\approx 1.8$ K, the entropy variation for this model roughly matches that for [Tb$_{2}$Ti$_{2}$O$_{7}$]{} (the fit is shown in Fig. $4$ of Ref. \[\]). This is presented as evidence for a singlet-singlet crystal field scheme in [Tb$_{2}$Ti$_{2}$O$_{7}$]{}. A similar conclusion is reached for [Tb$_{2}$Sn$_{2}$O$_{7}$]{}.
However, the authors of Ref. \[\] have not ruled out the possibility that the magnetic entropy is lost through the effects of interactions and the concomitant build-up of correlations as the temperature is decreased below 20 K. This is particularly evident in Tb$_2$Sn$_2$O$_7$ which exhibits a phase transition at 0.87 K to a long range ordered phase. This implies that the residual entropy for $T < 0.87$ K K should have no extensive contribution, as appears to be borne out by the analysis of the specific heat capacity data in Ref. \[\]. It follows that one must account for the details of the transition in order to extract information about the single ion level structure from entropy data obtained from the magnetic heat capacity while no attempt was made to carry out such an analysis in Ref. \[\]
We now turn to the case of [Tb$_{2}$Ti$_{2}$O$_{7}$]{}, which does not exhibit a transition to long range order down to the lowest observed temperature of $50$ mK [@Gardner-TbTO-PRL-1999; @Gardner-TbTO-PRB-2003]. Nevertheless, interactions are significant in this material as can be inferred from the Curie-Weiss temperature [@Gingras-TbTO-PRB-2000], the paramagnetic diffuse scattering [@Gardner-TbTO-PRL-1999; @Gingras-TbTO-PRB-2000; @Kao; @TTOHeff] and the dispersion on inelastic neutron scattering peaks for example [@Gardner-TbTO-PRB-2003; @Kao; @Mirebeau-TbXO-2007]. Indeed, the overall scale of exchange interactions in [Tb$_{2}$Sn$_{2}$O$_{7}$]{} and [Tb$_{2}$Ti$_{2}$O$_{7}$]{} have been estimated to be of similar magnitude [@Mirebeau-TbXO-2007]. Although the interactions in [Tb$_{2}$Ti$_{2}$O$_{7}$]{} do not manifest themselves as a phase transition, they potentially have a large effect on the entropy as we now illustrate with a toy model. We consider a spin-$3/2$ Ising model on a chain with ferromagnetic nearest neighbor exchange $J$ ($J>0$), $$H = \sum_{i} \left( -\Delta (S_{i}^{z})^{2} - J S_{i}^{z}S_{i+1}^{z} \right).$$ Of importance for our argumentation, this model has (i) a ground state doublet, (ii) no phase transition to long range order at nonzero temperature and (iii) a paramagnetic entropy of R$\ln 4$ similarly to Tb$_2$Ti$_2$O$_7$. This model can be solved exactly by splitting the partition function trace into a product of transfer matrices. Consider first the case $J=0$. The specific heat $C_{\rm mag}$ shows a single peak (Schottky anomaly) corresponding to the loss of entropy, $S_{\rm mag}$, as the excited $S^z=\pm 3/2$ levels are depopulated upon cooling \[solid lines in top panel of Fig. 3(a)\]. The entropy exhibits two plateaus – one at low temperature (R$\ln 2$) and another at high temperature (R$\ln 4$). This is the analogue of the non-interacting doublet-doublet model considered in Ref. \[\].
Switching on the exchange $J$ causes FM correlations to build up. There is no transition, but there is a peak in $C_{\rm mag}$ beneath the Schottky anomaly at $T/J \sim 0.1$. This is reflected in the magnetic entropy, $S_{\rm mag}$, dropping below the R$\ln 2$ value. The R$\ln 2$ plateau gradually shrinks as $J$ increases \[Fig. 3(b)\]. We conclude that the disappearance of an R$\ln 2$ entropy plateau can occur when interactions are introduced [*even in the absence of a phase transition*]{} and, therefore, cannot be attributed definitively to a splitting of the doublet ground state without interactions.
[*Conclusion*]{} $-$ We have reconsidered the scenario advanced in Refs. \[\] whereby long range order for Tb$_2$Ti$_2$O$_7$ is evaded because the single ion crystal field states are split in zero field, ostensibly due to a tetragonal distortion breaking the Fd$\bar{3}$m symmetry. We have presented high energy resolution neutron scattering data to re-examine the case for a singlet-singlet splitting, finding no evidence for excitations that would indicate a splitting greater than $0.2$ K. We have also argued that measurements determining the residual entropy cannot be used to draw conclusions about the single ion spectrum without considering the build-up of short-range correlations at low temperatures. In conclusion, the nature of the low temperature state of Tb$_2$Ti$_2$O$_7$ remains a remarkable and unsolved problem in the field of frustrated magnetism. One may anticipate further progress in light of the constraints that we and others are placing on possible scenarios to explain the spin liquid behavior in this material.
This research was funded by the NSERC of Canada and the Canada Research Chair program (M. G., Tier I).
[99]{}
L. Balents, Nature [**464**]{}, 199 (2010).
J. S. Gardner [*et al.*]{}, Phys. Rev. Lett. [**82**]{}, 1012 (1999);
M. J. P. Gingras [*et al.*]{}, Phys. Rev. B. [**62**]{}, 6496 (2000).
J. S. Gardner [*et al.*]{}, Phys. Rev. B. [**68**]{}, 180401(R) (2003).
B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett. [**84**]{}, 3430 (2000).
Y.-J. Kao [*et al.*]{}, Phys. Rev. B [**68**]{}, 172407 (2003).
H. R. Molavian [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 157204 (2007).
K. Matsuhira [*et al.*]{}, J. Phys. Soc. Jpn. [**71**]{}, 1576 (2002).
I. Mirebeau [*et al.*]{}, Phys. Rev. Lett. [**94**]{}, 246402 (2005).
K. R. Lea [*et al.*]{} J. Phys. Chem. Solids [**23**]{}, 1381 (1962).
J. S. Gardner [*et al.*]{}, Phys. Rev. B [**64**]{}, 224416 (2001).
I. Mirebeau [*et al.*]{}, Phys. Rev. B [**76**]{}, 184436 (2007). J. Villain, Z. Phys. B [**33**]{}, 31 (1979).
S. Ji [*et al.*]{}, Phys. Rev. Lett. [**103**]{}, 037201 (2009). L. G. Mamsurova [*et al.*]{}, JETP Lett [**43**]{}, 755 (1986).
J. P. C. Ruff [*et al.*]{}, Phys. Rev. Lett. [**99**]{}, 237202 (2007).
J. P. C. Ruff [*et al.*]{}, Phys. Rev. Lett. [**105**]{}, 077203 (2010).
Y. Chapuis [*et al.*]{} Phys. Rev. B [**82**]{}, 100402 (2010).
P. Bonville [*et al.*]{}, arXiv:1104.1584.
K. C. Rule [*et al.*]{}, Phys. Rev. Lett. [**96**]{}, 177201 (2006).
J. P. Clancy [*et al.*]{}, Phys. Rev. B [**79**]{}79, 014408 (2009).
|
---
abstract: 'Historical Functional Linear Models (HFLM) quantify associations between a functional predictor and functional outcome where the predictor is an exposure variable that occurs before, or at least concurrently with, the outcome. Current work on the HFLM is largely limited to frequentist estimation techniques that employ spline-based basis representations. In this work, we propose a novel use of the discrete wavelet-packet transformation, which has not previously been used in functional models, to estimate historical relationships in a fully Bayesian model. Since inference has not been an emphasis of the existing work on HFLMs, we also employ two established Bayesian inference procedures in this historical functional setting. We investigate the operating characteristics of our wavelet-packet HFLM, as well as the two inference procedures, in simulation and use the model to analyze data on the impact of lagged exposure to particulate matter finer than 2.5$\mu$g on heart rate variability in a cohort of journeyman boilermakers over the course of a day’s shift.'
author:
- |
Mark J. Meyer[^1]\
Department of Mathematics and Statistics, Georgetown University\
and\
Elizabeth J. Malloy\
Department of Mathematics and Statistics, American University\
and\
Brent A. Coull\
Department of Biostatistics, Harvard T. H. Chan School of Public Health
title: '**Bayesian Wavelet-packet Historical Functional Linear Models**'
---
\#1
0
[0]{}
1
[0]{}
[**Bayesian Wavelet-packet Historical Functional Linear Models**]{}
[*Keywords:*]{} Function-on-function regression; Functional data analysis; Wavelet regression; Bayesian Statistics; Functional Inference.
Introduction {#s:intro}
============
Historical Functional Linear Models (HFLMs) are used to analyze the relationship between a functional “exposure” and a functional “outcome” where only exposures occurring in time before or concurrently with the outcome can affect the outcome. HFLMs are a special case of the Function-on-Function Regression (FFR) model which fits an unconstrained surface and is therefore inappropriate for modeling functional predictors that are lagged exposures. For example, suppose that for subject $i$, $x_i(v)$ represents levels of a pollutant sampled on a grid $v \in \mathcal{V}$ and $y_i(t)$ represents measurements of heart rate variability (HRV) sampled on a grid $t \in \mathcal{T}$. A general FFR model with no constraints takes the form $$\begin{aligned}
y_{i}(t) = \alpha(t) + \int_{v \in \mathcal{V}} x_{i}(v)\beta(v,t) dv + E_{i}(t),
\label{eq:ffr}
\end{aligned}$$ where the surface $\beta(v,t)$ is the primary quantity of interest for estimation and $E_{i}(t)$ is typically assumed to be distributed as a Gaussian Process. For example, see [@Ivanescu2015], [@Meyer2015], [@Morris2015], [@Scheipl2015], [@Scheipl2016], [@Kim2018], and references therein.
However, when applied to the HRV and pollutant example, a model like would allow for HRV measurements at time $t$ to be associated with pollutants occurring both prior to time $t$, as well as after time $t$ despite the implausibility of such a relationship. The HFLM addresses this issue, constraining $\beta(v,t)$ to prevent such spurious associations by limiting the integration in to the set of coefficients such that $\{v \in \mathcal{V}, t \in \mathcal{T} : v \leq t\}$. The basic HFLM takes the form $$\begin{aligned}
y_{i}(t) = \alpha(t) + \int_{\{v \leq t\}} x_{i}(v)\beta(v,t) dv + E_{i}(t).
\label{eq:hflm}
\end{aligned}$$ The observed data is usually discrete, so that $\beta(v,t)$ can be expressed as a matrix of coefficients. Thus the problem reduces to constraining the estimate of $\beta(v,t)$ to be non-zero for the upper triangle of the matrix.
Several authors explore ways of implementing the constraint in . [@MalfaitRamsay2003] propose the use tent-shaped basis functions with support over a two-dimensional region. They estimate the surface using a multivariate linear model approximation to a finite dimensional model. Thus after dimension reduction via the basis-space expansion, they use least squares to estimate $\beta(v,t)$. [@Harezlak2007] also use basis functions defined over a two-dimensional region but specify a large number of basis functions and penalize the fit. The authors consider both the LASSO and $L_2$-norm penalty on triangular basis functions, using restricted maximum likelihood (REML) for the latter. Both [@MalfaitRamsay2003] and [@Harezlak2007] allow for a pre-defined lag beyond which the effect of exposure is zero, further constraining the surface to a trapezoidal region defined by $\{v, \in \mathcal{V}, t \in \mathcal{T} : t - \Delta \leq v \leq t\}$ for some pre-defined lag $\Delta$. [@Kim2011] take the constraint further by proposing a Recent History Functional Linear Model where the surface is constrained to a trapezoidal region defined by $\{v, \in \mathcal{V}, t \in \mathcal{T} : t - \Delta_1 \leq v \leq t - \Delta_2\}$ for $0 < \Delta_1 < \Delta_2 < T$. The authors estimate the constrained surface with a varying coefficient model representation using B-spline basis functions, although they suggest Fourier, truncated power, and Eigen basis functions can also be used.
More recently, [@Pomann2016] examine two HFLMs that allow for multiple functional predictors with estimation constrained to a fixed window similar to that used in [@Harezlak2007]. The authors implement two approaches to estimation using semi-local smoothing, which performs point-wise estimation, and global smoothing, which smooths over $\mathcal{T}$ globally. The methods select smoothing parameters via cross-validation and REML, respectively, and use B-spline basis expansions to model the functional form. In each of these methods, the covariance of the error term, $E_i(t)$, is assumed independent or to have “working” independence.
To the best of our knowledge, the existing body of work on HFLMs is largely limited to spline-based methods which can over smooth signals and peaks in spiky and irregular data. Further, the current literature does not consider inferential procedures, focusing on estimation and model fit criterion instead. As such, the performance of the proposed methods with respect to uncertainty quantification is not clear. Wavelet-based functional regression models, such as the work of [@MorrisCarroll2006] and [@Malloy2010], consider the function-on-scalar and scalar-on-function regression settings, respectively, in the Bayesian context. [@Meyer2015] extend [@MorrisCarroll2006] to the FFR case using wavelets for the basis function of the outcome and wavelet-principal components for the expansion of the predictor. One advantage of the wavelet-based framework is that it does not require the assumption of independence in the data-space. However, their approach estimates an unconstrained surface and therefore is inappropriate for modeling lagged exposures. The wavelet domain [@Meyer2015] use, which results from a discrete wavelet transformation, lacks a convenient relationship between the wavelet coefficients and the time domain, thus making it inefficient for use in an HFLM. Furthermore, the wavelet-principal components basis function does not preserve the temporal relationship between exposure and outcome in the wavelet domain. Thus to implement an HFLM using wavelets, a different basis function is warranted. Wavelet-packets, which result from a discrete wavelet-packet transformation, are a variant of the wavelet basis function that have a convenient relationship to the time domain that we can exploit to implement a wavelet-packet based HFLM.
In this work, we propose a novel use of wavelet-packets that allows us to build a Bayesian wavelet-space HFLM. We formulate our model within the framework of [@MorrisCarroll2006] and [@Meyer2015], thus our method does not require the assumption of independence in the data space. A benefit of the Bayesian context is that we can implement several Bayesian techniques for conducting multiplicity adjusted inference including joint credible intervals to quantify uncertainty and the Bayesian false discovery rate to identify exposure lags and exposure times that are associated with the outcome. We assess the operating characteristics of the methodology and inference procedures in simulation and present an application to data from a study of journeyman boilermakers exposed to varying levels of particulate matter during the course of the day. The data consists of five-minute assessments of HRV, as defined by standard deviation of the normal-to-normal intervals (SDNN) at each time-point, and particulate matter finer than 2.5 $\mu$m (PM$_{2.5}$) which results from exposure to residual oil fly ash and cigarette smoke [@Magari2001; @Cavallari2008]. [@Harezlak2007] present an analysis of part of this data and found both negative and positive time-specific associations in the morning that corresponded to the workers’ break times. Our analysis focuses on the morning hours of the work day where [@Harezlak2007] saw the largest effects. We also make available MATLAB code for the implementation of our method at <https://github.com/markjmeyer/WPHFLM>.
The remainder of the paper is organized as follows: Section \[s:dwpt\] provides a brief introduction to the discrete wavelet-packet transformation. Section \[s:BHFLM\] details the formulation of the the Bayesian wavelet-packet HFLM along with a discussion of inferential procedures in this modeling framework. Sections \[s:sim\] and \[s:app\] present the results of our simulation study and the application of our model to the Journeyman data, respectively. Finally, in Section \[s:disc\], we provide a discussion of the methodology.
Discrete Wavelet-packet Transformation {#s:dwpt}
======================================
We begin with a brief description of the partial discrete wavelet transformation (DWT). Consider a 1-dimensional function, $x(v)$, which we discretely observe as ${\textbf{x}}= \left[ \begin{array}{ccc} x_1 & \cdots & x_V \end{array} \right]'$, where $V = 2^k$ for some positive integer value $k$. For a given mother wavelet, a partial DWT to $J = 3$ levels results in a set of wavelet coefficients that can be further separated into approximation coefficients, ${\textbf{a}}$, and detail coefficients, ${\textbf{d}}$. The pyramid algorithm for performing the partial DWT is graphically depicted in Figure \[f:dwt\]. Here, ${\mathcal{G}}$ and ${\mathcal{H}}$ denote the low and high pass filters for the corresponding mother wavelet. The resulting decomposition consists of the detail coefficients from each level plus the final approximation coefficients. Post-transformation, the wavelet-space representation of ${\textbf{x}}$ is then ${\textbf{w}}= \left[ \begin{array}{cccc} {\textbf{a}}_3 & {\textbf{d}}_3 & {\textbf{d}}_2 & {\textbf{d}}_1 \end{array} \right]'$. The DWT can be expressed as a matrix multiplication by an orthogonal matrix, $W$, thus it can be shown that ${\textbf{w}}= {\textbf{x}}W$.
(0,0) rectangle (8,1) node\[midway\] [${\textbf{x}}= \left[ \begin{array}{ccc} x_1 & \cdots & x_V \end{array} \right]'$]{}; at (0,0.5) [$j = 0$]{};
(2,0) – (2,-1) node\[midway, left\] [${\mathcal{G}}$]{}; (6,0) – (6,-1) node\[midway, left\] [${\mathcal{H}}$]{};
(0,-2) rectangle (4,-1) node\[midway\] [${\textbf{a}}_1$]{}; (4,-2) rectangle (8,-1) node\[midway\] [${\textbf{d}}_1$]{}; at (0,-1.5) [$j = 1$]{};
(1,-2) – (1,-3) node\[midway, left\] [${\mathcal{G}}$]{}; (3,-2) – (3,-3) node\[midway, left\] [${\mathcal{H}}$]{};
(0,-4) rectangle (2,-3) node\[midway\] [${\textbf{a}}_2$]{}; (2,-4) rectangle (4,-3) node\[midway\] [${\textbf{d}}_2$]{}; at (0,-3.5) [$j = 2$]{};
(0.5,-4) – (0.5,-5) node\[midway, left\] [${\mathcal{G}}$]{}; (1.5,-4) – (1.5,-5) node\[midway, left\] [${\mathcal{H}}$]{};
(0,-6) rectangle (1,-5) node\[midway\] [${\textbf{a}}_3$]{}; (1,-6) rectangle (2,-5) node\[midway\] [${\textbf{d}}_3$]{}; at (0,-5.5) [$j = 3$]{};
From Figure \[f:dwt\], we see the DWT applies the filters only to the successive sets of approximation coefficients. For a given mother wavelet, the partial discrete wavelet-packet transformation (DWPT) begins in the same way, passing first the low and then high-pass filters over the original signal. However, after the first level, the DWPT applies ${\mathcal{G}}$ and ${\mathcal{H}}$ to all coefficients at each level, reversing the order when applied to detail coefficients. Figure \[f:dwpt\] illustrates the partial DWPT decomposition for $J=3$ levels. Once again, ${\mathcal{G}}$ and ${\mathcal{H}}$ denote the low and high pass filters of the corresponding mother wavelet. The resulting representation of ${\textbf{x}}$ is then the final set of coefficients from the last level, ${\textbf{w}}_P = \left[ \begin{array}{cccccccc} {\textbf{a}}_{3,0} & {\textbf{d}}_{3,1} & {\textbf{a}}_{3,2} & {\textbf{d}}_{3,3} & {\textbf{a}}_{3,4} & {\textbf{d}}_{3,5} & {\textbf{a}}_{3,6} & {\textbf{d}}_{3,7} \end{array} \right]'$. As in the DWT, the DWPT can also be expressed as a matrix multiplication of an orthogonal matrix, $W_P$. Thus ${\textbf{w}}_P$ can be shown to be ${\textbf{w}}_P = {\textbf{x}}W_P$. For both the DWT and DWPT, the resulting set of wavelet-packet coefficients will have the same length as the original signal.
(0,0) rectangle (8,1) node\[midway\] [${\textbf{x}}= \left[ \begin{array}{ccc} x_1 & \cdots & x_V \end{array} \right]'$]{}; at (0,0.5) [$j = 0$]{};
(2,0) – (2,-1) node\[midway, left\] [${\mathcal{G}}$]{}; (6,0) – (6,-1) node\[midway, left\] [${\mathcal{H}}$]{};
(0,-2) rectangle (4,-1) node\[midway\] [${\textbf{a}}_{1,0}$]{}; (4,-2) rectangle (8,-1) node\[midway\] [${\textbf{d}}_{1,1}$]{}; at (0,-1.5) [$j = 1$]{};
(1,-2) – (1,-3) node\[midway, left\] [${\mathcal{G}}$]{}; (3,-2) – (3,-3) node\[midway, left\] [${\mathcal{H}}$]{}; (5,-2) – (5,-3) node\[midway, left\] [${\mathcal{H}}$]{}; (7,-2) – (7,-3) node\[midway, left\] [${\mathcal{G}}$]{};
(0,-4) rectangle (2,-3) node\[midway\] [${\textbf{a}}_{2,0}$]{}; (2,-4) rectangle (4,-3) node\[midway\] [${\textbf{d}}_{2,1}$]{}; (4,-4) rectangle (6,-3) node\[midway\] [${\textbf{a}}_{2,2}$]{}; (6,-4) rectangle (8,-3) node\[midway\] [${\textbf{d}}_{2,3}$]{}; at (0,-3.5) [$j = 2$]{};
(0.5,-4) – (0.5,-5) node\[midway, left\] [${\mathcal{G}}$]{}; (1.5,-4) – (1.5,-5) node\[midway, left\] [${\mathcal{H}}$]{}; (2.5,-4) – (2.5,-5) node\[midway, left\] [${\mathcal{H}}$]{}; (3.5,-4) – (3.5,-5) node\[midway, left\] [${\mathcal{G}}$]{}; (4.5,-4) – (4.5,-5) node\[midway, left\] [${\mathcal{G}}$]{}; (5.5,-4) – (5.5,-5) node\[midway, left\] [${\mathcal{H}}$]{}; (6.5,-4) – (6.5,-5) node\[midway, left\] [${\mathcal{H}}$]{}; (7.5,-4) – (7.5,-5) node\[midway, left\] [${\mathcal{G}}$]{};
(0,-6) rectangle (1,-5) node\[midway\] [${\textbf{a}}_{3,0}$]{}; (1,-6) rectangle (2,-5) node\[midway\] [${\textbf{d}}_{3,1}$]{}; (2,-6) rectangle (3,-5) node\[midway\] [${\textbf{a}}_{3,2}$]{}; (3,-6) rectangle (4,-5) node\[midway\] [${\textbf{d}}_{3,3}$]{}; (4,-6) rectangle (5,-5) node\[midway\] [${\textbf{a}}_{3,4}$]{}; (5,-6) rectangle (6,-5) node\[midway\] [${\textbf{d}}_{3,5}$]{}; (6,-6) rectangle (7,-5) node\[midway\] [${\textbf{a}}_{3,6}$]{}; (7,-6) rectangle (8,-5) node\[midway\] [${\textbf{d}}_{3,7}$]{}; at (0,-5.5) [$j = 3$]{};
Similar to the DWT, the wavelet-packet coefficients comprising ${\textbf{w}}_P$ are indexed by a scale and location. The scale indexes the bin the coefficient is in at the final level of the decomposition and thus corresponds to the second subscript from the $j = 3$ level in Figure \[f:dwpt\] while the location denotes the position of the coefficient within the set of coefficients at a given scale. Since the number of elements in ${\textbf{w}}_P$ is the same as in ${\textbf{x}}$, within a scale, the time-ordering of the observed signal is preserved in the ordering of the coefficients. Suppose we have a second function, $y(t)$, that we discretely observe on a grid such that ${\textbf{y}}= \left[ \begin{array}{ccc} y_1 & \cdots & y_T \end{array} \right]'$. If we perform the DWPT on ${\textbf{y}}$, we will also obtain a set of wavelet-packet coefficients that preserve the time-ordering of the original signal within each scale. Provided the elements of ${\textbf{x}}$ are sampled in time concurrently or before the elements of ${\textbf{y}}$, we can use the location index from their respective DWPT decompositions to constrain the surface of estimation within each scale in the wavelet-packet space. When we apply the inverse DWPT (IDWPT), the constraint in then maintained in the data-space. For more details on DWPTs, see @PercivalWalden2000 [chap. 6], [@Misiti2007], and @Nason2008 [chap. 2].
Bayesian Historical Functional Linear Model {#s:BHFLM}
===========================================
We begin with the model in where we constrain the estimation to the region defined by $\{v, \in \mathcal{V}, t \in \mathcal{T} : v \leq t\}$. We assume the within-function errors come from a Gaussian process. Thus, $E_{i}(t) \sim \mathcal{GP}\left(0, \Sigma_E\right)$, where $\Sigma_E$ is an unstructured covariance matrix. Because the data, $y_{i}(t)$ and $x_i(v)$, arrive sampled on a grid of equally spaced time points $t = [t_1,\cdots,t_T]'$ and $v = [v_1,\cdots,v_V]'$, we use the discrete version of the model: ${\textbf{y}}_{i} = {\textbf{x}}_{i}{\boldsymbol{\beta}}+ {\textbf{e}}_{i}$ for the vectors ${\textbf{y}}_{i}$, ${\textbf{x}}_i$, and ${\textbf{e}}_i$ and matrix of coefficients ${\boldsymbol{\beta}}$. We recommend centering and scaling both the outcome and predictor functions first. Thus, without loss of generality, we drop the intercept function from the model. Stacking the response vectors and predictor vectors into matrices gives $${\textbf{Y}}= {\textbf{X}}{\boldsymbol{\beta}}+ {\textbf{E}},
\label{eq:dhflm}$$ where for $N$ total curves, ${\textbf{E}}$ and ${\textbf{Y}}$ are $N\times T$ while ${\textbf{X}}$ is $N\times V$. The constrained region of integration in Model restricts the form of the functional regression coefficients so that $\beta(v_k,t_{k'}) = 0$ if $v_k > t_{k'}$. If $T = V$ and $t_1 = v_1, t_2 = v_2, \ldots, t_T = v_V$, then the discrete version of ${\boldsymbol{\beta}}$ is an upper triangular matrix of the form $${\boldsymbol{\beta}}= \left(\begin{array}{cccc}
\beta(v_1,t_1) & \beta(v_1,t_2) & \cdots & \beta(v_1,t_T) \\
0 & \beta(v_2,t_2) & \cdots & \beta(v_2,t_T) \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \beta(v_V,t_T)
\end{array} \right)
\label{eq:hbeta}$$ with zeros below the main diagonal. We propose the use of wavelet-packets to enforce this constraint in the wavelet-packet space which, given that the time ordering is preserved, will ensure the constraint is maintained in the data-space as well.
Historical Constraint via Wavelet-Packets
-----------------------------------------
Working with , we apply the DWPT separately to each row of ${\textbf{Y}}$ and to each row of ${\textbf{X}}$. Performing this transformation is equivalent to the post-multiplication of the approximately orthonormal projection matrices resulting from the DWPT [@PercivalWalden2000]. The resulting decompositions have the form $\textbf{Y} = \textbf{Y}^{W_P} {\textbf{W}}_{P,Y}$ and $\textbf{X} = \textbf{X}^{W_P} {\textbf{W}}_{P,X}$ where ${\textbf{W}}_{P,Y}$ and ${\textbf{W}}_{P,X}$ are orthogonal matrices containing the wavelet packet basis functions. Then for the two dimensional decomposition on $\boldsymbol{\beta} = {\textbf{W}}_{P,X}' \boldsymbol{\beta}^{W_P} {\textbf{W}}_{P,Y}$ Model (\[eq:dhflm\]) in the wavelet-packet space is ${\textbf{Y}}^{W_P} {\textbf{W}}_{P,Y} = {\textbf{X}}^{W_P}{\textbf{W}}_{P,X} {\textbf{W}}_{P,X}'\boldsymbol{\beta}^{W_P}{\textbf{W}}_{P,Y} + {\textbf{E}}^{W_P}{\textbf{W}}_{P,Y}$ for ${\textbf{E}}= {\textbf{E}}^{W_P}{\textbf{W}}_{P,Y}$. Post-multiplying by $W_{P,Y}$ and recognizing the orthogonality of the wavelet-packet basis matrices, this model reduces to ${\textbf{Y}}^{W_P} = {\textbf{X}}^{W_P}\boldsymbol{\beta}^{W_P} + {\textbf{E}}^{W_P}$ and subject-specific model ${\textbf{y}}_i^{W_P} = {\textbf{x}}_i^{W_P}\boldsymbol{\beta}^{W_P} + {\textbf{e}}_i^{W_P}$. This model fits into the generalized basis expansion framework for function-on-function regression considered in [@Meyer2015]. The key difference, however, lies in our novel use of wavelet-packets as a basis function to induce the historical constraint whereas [@Meyer2015] is concerned only with basis functions to estimate a full, unconstrained surface or set of surfaces.
We enforce the constraint in the wavelet-packet space via our prior specification on the elements of ${\boldsymbol{\beta}}$. Let the DWP transformations be indexed by scales $j = 1, \ldots, J^y$ and $s = 1, \ldots, S^x$ and locations $k = 1, \ldots, K_j^y$ and $\ell = 1, \ldots, L_s^x$ in the ${\textbf{Y}}$ and ${\textbf{X}}$ wavelet-packet spaces respectively. Consistent with previous work on wavelet-based models in function regression, we place spike-and-slab priors on model coefficients. To restrict the surface in wavelet-packet space, we set coefficients where $\ell > k$ to zero. Thus our prior on the elements of $\boldsymbol{\beta}^{W_P} = \left[\beta^{W_P}_{s\ell,jk}\right]$ is $\beta_{s\ell,jk}^{W_P} \sim\ 1(\ell \leq k)\gamma_{s\ell,jk}N(0, \tau_{s\ell,j\cdot}) + [1-\gamma_{s\ell,jk}]d_0$, where $\gamma_{s\ell,jk} \sim Bern(\pi_{s\ell,j\cdot})$ and $d_0$ is a point-mass distribution at zero. The regularization parameters, $\tau_{s\ell,j\cdot}$ and $\pi_{s\ell,j\cdot}$, smooth over locations $k$ which we denote using the “dot” notation in the subscript. We assume inverse-gamma and beta hyper priors, respectively, for the regularization parameters with hyper-parameters fixed and based on the data.
[@MorrisCarroll2006] show that after a wavelet transformation, assuming independence in the wavelet-space does not imply independence in the data-space and therefore wavelets accommodate a wide range of covariances in the data-space. As wavelet-packets share the same whitening properties of wavelets, we assume independence in the wavelet-packet space [@PercivalWalden2000]. Thus we assume ${\textbf{e}}_i^{W_p} \sim N(0, {\boldsymbol{\Sigma}}^{W_p})$ where ${\boldsymbol{\Sigma}}^{W_p} = \text{diag}\left\{ \sigma^2_{jk} \right\}$, which varies by the scale and location of the ${\textbf{Y}}$ wavelet-packet coefficients. We place an inverse gamma prior on $\sigma^2_{jk}$. The independence assumption allows us to sample the coefficients corresponding to different $j$ and $k$ combinations separately.
Using the prior specifications, we now describe our sampling algorithm. For the $jk$th wavelet-packet space coefficient from the ${\textbf{Y}}^{W_P}$ decomposition and the $s\ell$th column of ${\textbf{X}}^{W_P}$, the conditional posterior distribution is a mixture of a point-mass at zero and a normal distribution of the form $$\begin{aligned}
\beta^{W_P}_{s\ell,jk} | {\text{rest}}\sim 1(\ell \leq k)\gamma_{s\ell,jk} N(\mu_{s\ell,jk}, \epsilon_{s\ell,jk}) + (1 - \gamma_{s\ell,jk})d_0, \label{eq:beta}\end{aligned}$$ where $\mu_{s\ell,jk} = \hat{\beta}_{s\ell,jk}^{W_P}(1 + \Lambda_{s\ell,jk}/\tau_{s\ell,j})^{-1}$ and $\epsilon_{s\ell,jk} = \Lambda_{s\ell,jk}(1 + \Lambda_{s\ell,jk}/\tau_{s\ell,j})^{-1}$ for the OLS and variance estimates $\hat{\beta}_{s\ell,jk}^{W_P}$ and $\Lambda_{s\ell, jk}$ at the current step. The conditional for $\gamma_{s\ell,jk}$ is $$\begin{aligned}
\gamma_{s\ell,jk}|{\text{rest}}\sim Bern(\alpha_{s\ell,jk}),\label{eq:gamma}\end{aligned}$$ were $\alpha_{s\ell,jk} = O_{s\ell,jk}/\left( O_{s\ell,jk} + 1 \right)$ for $O_{s\ell,jk} = \pi_{s\ell,j}/(1-\pi_{s\ell,j})\text{BF}_{s\ell,jk}$, $\text{BF}_{s\ell,jk} = \left( 1 + \frac{\tau_{s\ell,j}}{\Lambda_{s\ell, jk}} \right)^{-1/2}$ $\exp{\left\{ \frac{1}{2} \zeta^2_{s\ell,jk}\left(1 + \frac{\Lambda_{s\ell, jk}}{\tau_{s\ell,j}}\right) \right\}}$, and $\zeta_{s\ell,jk}$ equal to the ratio of the current values of $\beta^{W_P}_{s\ell,jk}$ to the current estimate of the standard deviation of $\beta^{W_P}_{s\ell,jk}$. Through the indicator function, $1(\ell \leq k)$, we enforce the historical constraint by forcing coefficients for which $\ell > k$ to come from the point-mass density $d_0$ and thus be set to zero.
The conditionals for the diagonal elements of the wavelet-packet space variance components, $\sigma^2_{jk}$, have the form $$\begin{aligned}
P\left(\sigma^2_{jk} | {\text{rest}}\right) &\propto \pi\left(\sigma^2_{jk} \right)\left( \sigma^2_{jk} \right)^{-n/2} \exp\left[ -\frac{1}{2\sigma^2_{jk}} \left({\textbf{y}}^{W_p}_{jk} - X\beta^{W_p}_{\cdot\cdot,jk}\right)'\left({\textbf{y}}^{W_p}_{jk} - X\beta^{W_p}_{\cdot\cdot,jk}\right) \right], \label{eq:sig}\end{aligned}$$ where $\pi\left(\sigma^2_{jk} \right)$ is the prior density on $\sigma^2_{jk}$ which we take to be inverse gamma with parameters $a_{\sigma^2}$ and $b_{\sigma^2}$ both set to the empirical Bayes estimates. To ensure the variance components are not too close to zero, we employ a Metropolis-Hastings step to sample them. The proposal densities are independent Gaussians, truncated at zero and centered at the previous value in each chain. The full conditionals for the regulation parameters are $$\begin{aligned}
\tau_{s\ell,j\cdot} | \text{rest} &\sim IG\left[a_{\tau} + \frac{1}{2}\gamma_{s\ell,jk}, b_{\tau} + \frac{1}{2}\gamma_{s\ell,jk}\left(\beta_{s\ell,jk}^{W_P}\right)^2 \right] \text{ and }\label{eq:tau}\\
\pi_{s\ell,j\cdot} | \text{rest} &\sim Beta\left( a_{\pi} + \gamma_{s\ell,jk}, b_{\pi} + \gamma_{s\ell,jk} \right),\label{eq:pi}\end{aligned}$$ with $a_{\tau}$, $b_{\tau}$, $a_{\pi}$, and $b_{\pi}$ set to the empirical Bayes estimates. The prior specifications are consistent with the previous work on wavelet-based functional models including [@Meyer2015], [@Malloy2010], and [@MorrisCarroll2006]. For more details on the empirical Bayes estimates, see [@MorrisCarroll2006]. Our sampler then iterates between draws from to until convergence. Upon completion of the algorithm, we apply the inverse DWPT to the posterior samples of ${\boldsymbol{\beta}}^{W_P}$ to obtain estimates in the data space of ${\boldsymbol{\beta}}$, the upper triangular matrix of historically constrained coefficients.
Thresholding and Wavelet Details
--------------------------------
Our sampler becomes computationally intensive as $V$ increases with computation time increasing almost linearly. In previous work on wavelet-based FFR, [@Meyer2015] address this issue by reducing the dimension of the wavelet transformed ${\textbf{X}}$. In their formulation, the authors use wavelet-Principal Components (wPC) retaining columns containing a large amount of the variability in ${\textbf{X}}$. The wPC decomposition involves first performing a DWT on ${\textbf{X}}$ and then performing a singular value decomposition (SVD). While such an approach reduces computation time and achieves additional denoising, it does not work in the historical framework, as performing an SVD on ${\textbf{X}}^{W_P}$ would break the time ordering maintained by the transformation. To remedy the computational concerns and simultaneously achieve additional denoising, we propose a simplified thresholding procedure where we threshold all coefficients in the larger scale levels to zero. Thus, we consider keeping the scales that comprise either 25% or 50% of the wavelet-packet coefficients. For example, using the $J = 3$ level partial DWPT, the first approach would result in retaining the coefficients ${\textbf{a}}_{3,0}$ and ${\textbf{d}}_{3,1}$ in Figure \[f:dwpt\]. The second approach, keeping 50%, would correspond to retaining the coefficients ${\textbf{a}}_{3,0}$, ${\textbf{d}}_{3,1}$, ${\textbf{a}}_{3,2}$, and ${\textbf{d}}_{3,3}$, again from Figure \[f:dwpt\]. Our simulation setting compares the performance of both approaches in terms of computational efficiency and estimation. It is important to note that we do not threshold the outcome and retain all coefficients of ${\textbf{Y}}^{W_P}$.
The choice of mother wavelet will depend largely on the data context. In our application and simulation, we use Daubechies wavelets with 3 vanishing moments for the separate DWPTs on both ${\textbf{Y}}$ and ${\textbf{X}}$. Wavelet-packets also require a choice of boundary padding, we select zero padding—padding the boundary with zeros. Finally, we must choice the number of levels to take our partial DWPT. Because of our reliance on the maintained time ordering within scales but between decompositions, we use the same number of levels of decomposition for both ${\textbf{Y}}$ and ${\textbf{X}}$. In simulation, we use $J = 3$ levels for sampling densities of $T = V = 64$ and $128$. For denser sampling rates, we recommend increasing $J$ and potentially adjusting the thresholding.
Posterior Functional Inference
------------------------------
Previous implementations of the HFLM do not discuss inferential procedures instead, in some cases, developing measures of model fit [@MalfaitRamsay2003; @Harezlak2007]. In the wavelet-based functional literature, [@Meyer2015] propose the use of a Bayesian False Discovery Rate (BFDR) procedure also used by [@MorrisEtAl2008] and [@Malloy2010], as well as joint credible bands as discussed in [@Ruppert2003]. We consider the use of both methods in the context of Bayesian wavelet-packet HFLM.
The BFDR procedure utilizes the MCMC samples to estimate the posterior probability of a given coefficient being greater than a $\delta$-fold intensity change. Once these values are determined they are ranked and a cut-off selected to control the overall FDR at a pre-specified global $\alpha$-bound. Suppose we have $M$ MCMC samples and $\beta^{(m)}(v,t)$ is $m^{\text{th}}$ draw from the posterior estimated surface. Then for $\left\{ v\in \mathcal{V} \text{ and } t\in \mathcal{T} \text{ s.t. } v \leq t \right\}$, we find the probability $P_{B}(v,t) = Pr\left\{ \left| \beta(v,t) \right| > \delta | y \right\} \approx \frac{1}{M} \sum_{m=1}^{M} 1\left\{ \left| \beta^{(m)}(v,t) \right| > \delta \right\}.$ We then flag the set of coefficients on the historical surface satisfying $\psi = \left\{ (v,t) : v \leq t \text{ and } P_{B}(v,t) \geq \phi_{\alpha} \right\}$. Here, $\phi_{\alpha}$ is determined by first ranking the values of $P_{B}$ in descending order across all coefficients to obtain the set $\left\{ P_{(r)}, r = 1,\ldots,R\right\}$ where $R$ is the total number of coefficients satisfying the historical constraint. We then define the cut-off value as $\lambda = \max\big[ r^* : \frac{1}{r^*} \sum_{r=1}^{r^*}\left\{ 1 - P_{(r)} \right\} \leq \alpha \big]$. Thus, we select coefficients with $P_B$ great than or equal to $\phi_{\alpha} = P_{(\lambda)}$ as significant.
For interval estimation, we consider both point-wise credible intervals as well as joint credible intervals. Point-wise credible intervals are constructed by taking the $\alpha/2$ and $1 - \alpha/2$ quantiles of the posterior samples taken at each coefficient for some choice of $\alpha$. As in [@Meyer2015], we construct joint credible intervals using $I_{\alpha}(v,t) = \hat{\beta}(v,t) \pm q_{(1-\alpha)}\left[ \widehat{\text{St.Dev}}\left\{ \hat{\beta}(v,t) \right\} \right]$, where $\hat{\beta}(v,t)$ and $\widehat{\text{St.Dev}}\left\{ \hat{\beta}(v,t) \right\}$ are the mean and standard deviation of the posterior samples, respectively, and $q_{(1-\alpha)}$ is the $(1-\alpha)$ quantile taken over all posterior samples of the quantity $$\begin{aligned}
q^{(m)} = \max_{(v,t)}\left| \frac{\beta^{(m)}(v,t) - \hat{\beta}(v,t)}{\widehat{\text{St.Dev}}\left\{ \hat{\beta}(v,t) \right\}} \right|, \text{ s.t. } v \leq t.
\end{aligned}$$ Such an interval satisfies $Pr\left\{ L(v,t) \leq \beta(v,t) \leq U(v,t)\ \forall\ v \in \mathcal{V}, t \in \mathcal{T} \text{ s.t. } v \leq t \right\} \geq 1 - \alpha$, where $L(v,t)$ and $U(v,t)$ are the corresponding upper and lower interval bounds. This procedure yields a joint $100(1-\alpha)$ interval for the historical association surface.
Simulation Study {#s:sim}
================
We consider four different historical surface scenarios representing plausible relationships between $x(v)$ and $y(t)$: a lagged effect of $x(v)$ on $y(t)$, a cumulative effect, a time-specific effect, and a delayed time-specific effect. Figure \[f:true\] displays the surfaces. We present their mathematical expressions in the Supplementary Material. For each scenario, we assume the data is sampled on a grid between $t_1 = v_1 = 1$ and $t_T = v_V = 64$ and vary the sampling rate such that $T = V = 64$ or $T = V = 128$. We also vary the sample size, considering $N = 50$ and $N = 200$. For the sparser, $T = 64$ setting, we vary the percent of the columns of ${\textbf{X}}^{W_P}$ we retain, first retaining 25% of coefficients and then retaining 50%. In the $T = 128$ setting, we only retain 25% of coefficients since retaining more considerably increases the computational burden. Thus in total, we present the results from 24 different simulation settings.
----- ----- ----- -- -------------------------- -------------------------- -------------------------- -------------------------- -- -------- -- --------
${\boldsymbol{\beta}}_L$ ${\boldsymbol{\beta}}_C$ ${\boldsymbol{\beta}}_T$ ${\boldsymbol{\beta}}_D$
50 64 25% 0.015 0.012 0.095 0.045 73.67% 516.2
50% 0.018 0.017 0.056 0.017 88.42% 609.6
128 25% 0.010 0.008 0.040 0.008 74.91% 1035.7
200 64 25% 0.009 0.007 0.088 0.009 73.55% 515.6
50% 0.010 0.008 0.045 0.008 88.34% 615.5
128 25% 0.004 0.004 0.036 0.003 75.63% 1032.1
----- ----- ----- -- -------------------------- -------------------------- -------------------------- -------------------------- -- -------- -- --------
: RMISEs, average percent of energy preserved in ${\textbf{X}}^{W_P}$ (PE), and computation time for the simulation settings lagged ($L$), cumulative ($C$), time-specific ($T$), and delayed time-specific ($D$). Table values represent averages taken over 200 simulated datasets. RC denotes retained coefficients. Time is averaged over all simulated datasets, over all settings.\[t:rmise\]
To evaluate each setting, we calculate the root mean integrated squared error (RMISE) and determine coverage probabilities for both point-wise and joint credible intervals. We also graphically explore the properties of the BFDR procedure. For each “true” historical surface ${\boldsymbol{\beta}}$, we generate $N$ ${\textbf{x}}_i$ curves from a mean zero Gaussian Process with a first-order auto-regressive (AR1) covariance structure. We base the variance and correlation parameters of the AR1 covariance off of the PM$_2.5$ data from the Journeyman data, setting $\sigma^2_{AR, X} = 3.5$ and $\rho_X = 0.75$. Next we generate within-subject error functions, ${\textbf{e}}_i$, from a separate mean zero Gaussian Process with an AR1 covariance structure. Once again we base the parameters of the covariance matrix off of the Journeyman data, $\sigma^2_{AR,E} = 0.1$ and $\rho_E = 0.5$. We then simulate the outcome functions using ${\textbf{y}}_i = {\textbf{x}}_i{\boldsymbol{\beta}}+ {\textbf{e}}_i$. For each setting, we repeat this data generation process 200 times and obtain 2000 posterior samples, discarding the first 1000, for each simulated dataset. For each simulated dataset we perform a $J = 3$ level DWPT on both the outcome and predictor using Daubechies wavelets with 3 vanishing moments. All computation is done using MATLAB version R2017a on a desktop with a 3.2 GHz Intel Core i5 processor and 16 GB of memory.
Table \[t:rmise\] presents RMISE averaged across all 200 simulated datasets for each setting under consideration while Figure \[f:est\] displays a single estimated surface with near average RMISE for the case when $N = 50$, $T = 64$ and only 25% of the ${\textbf{X}}^{W_P}$ coefficients are retained. From Table \[t:rmise\], we see that increasing either the sample size or the percent of retained coefficients tends to decrease RMISE, although all settings produce similar results. Denser sampling also decreases the RMISE. Regardless of sample size or sampling density, retaining 25% of the $X$-space wavelet-packet coefficients preserves roughly 75% of the energy while retaining 50% of coefficients preserves 88% of the energy in $X(v)$. To calculate average preserved energy, we find it first for each simulated subject and then average across all subjects. From Figure \[f:est\], we see that a single simulated dataset with near average RMISE accurately estimates the true relationship, even when only retaining an average of 75% of the energy in the predictor function.
We assess inference using Table \[t:cover\] and Figure \[f:pst50\]. Table \[t:cover\] contains mean point-wise coverage probabilities, calculated by first averaging over the historical surface and then over all 200 simulated datasets. We calculate coverage for both the point-wise credible interval and the joint credible interval. From Table \[t:cover\], we see that in general the joint credible intervals provide higher coverage for each setting. Coverage also tends to increase, regardless of interval type, as the percent of retained coefficients increase, though in some settings the differences are negligible. For some settings, coverage of the joint intervals is affected by sampling density and sample size. However, for the joint intervals, coverage is above the nominal level for most settings. This is not the case for the point-wise intervals where coverage sits below nominal level, in some bases considerably below. All intervals are at the 95% level.
----- ----- ----- -- -------------------------- -------------------------- -------------------------- -------------------------- -- -------------------------- -------------------------- -------------------------- --------------------------
${\boldsymbol{\beta}}_L$ ${\boldsymbol{\beta}}_C$ ${\boldsymbol{\beta}}_T$ ${\boldsymbol{\beta}}_D$ ${\boldsymbol{\beta}}_L$ ${\boldsymbol{\beta}}_C$ ${\boldsymbol{\beta}}_T$ ${\boldsymbol{\beta}}_D$
50 64 25% 0.811 0.870 0.524 0.565 0.993 0.997 0.770 0.778
50% 0.923 0.942 0.826 0.934 0.999 0.999 0.984 0.999
128 25% 0.855 0.885 0.767 0.940 1.000 1.000 0.978 1.000
200 64 25% 0.724 0.784 0.452 0.735 0.957 0.986 0.673 0.957
50% 0.865 0.890 0.718 0.930 0.999 0.999 0.933 1.000
128 25% 0.884 0.881 0.590 0.897 0.999 0.999 0.906 1.000
----- ----- ----- -- -------------------------- -------------------------- -------------------------- -------------------------- -- -------------------------- -------------------------- -------------------------- --------------------------
: Point-wise and joint credible interval coverage probabilities for the simulation settings lagged ($L$), cumulative ($C$), time-specific ($T$), and delayed time-specific ($D$). All intervals are at the 95% level. Table values represent averages taken over 200 simulated datasets. RC denotes retained coefficients. \[t:cover\]
Figure \[f:pst50\] shows the local BFDR values using a cut-off of $\delta = 0.5$ which is equal to half the largest signal, in absolute value, from each surface. Each plotted value corresponds to the posterior probability that $|\beta(v,t)| > 0.5$ for each surface from Figure \[f:est\], that is for one simulated dataset with near average RMISE. We see that using this $\delta$-intensity change, the BFDR highlights the features of each setting with elevated (or depressed) association. In the Supplementary Material, we consider one other value of $\delta$, which, while smaller, produces similar results. Choice of a specific $\delta$ is data-dependent, but several different values should be considered.
Figures \[f:est\] and \[f:pst50\] present the results from the simulation for $N = 50$, $T = 64$, and retaining 25% of the columns of ${\textbf{X}}^{W_P}$. Graphical results from simulated datasets with near average RMISE for the other settings produce similar results. Consequently, we present limited additional BFDR results in the Supplementary Material. Specifically, we only show BFDR graphs for all remaining 25% retained coefficients and $T = 64$ settings for $\delta = 0.25$ and 0.50.
Analysis of Journeyman Boilermaker Data {#s:app}
=======================================
[@Harezlak2007] and [@Cavallari2008] analyze data from 14 Journeyman boilermakers relating their five-minute SDNN to five-minute PM$_{2.5}$ levels over the course of the workday. Exposure to PM$_{2.5}$ came from two sources: residual oil fly-ash, a by-product of the manufacturing process, and cigarette smoke during mandatory breaks. [@Harezlak2007] found elevated levels association during the morning hours that corresponded to mandatory breaks which were then followed by depressions in the surface estimated over the trapezoidal region. To try to better estimate these spikes in association, we focus our analysis on the first three hours of the workday beginning at 8:30am and going until 11:30am. Further, we estimate joint intervals and calculate the BFDR to determine the significance of these peaks and troughs.
Prior to analysis, we log transform and center and scale both SDNN and PM$_{2.5}$. Thus changes in the estimated surface correspond to one unit changes in $\log($PM$_{2.5})$ and result in changes in $\log$(SDNN). To investigate the morning hours only, we take $T = V = 34$ measurements. As in the simulation, we take $J = 3$ levels of decomposition for both wavelet-packet transformations and use Daubechies wavelets with 3 vanishing moments. Given the results of the simulation, we retain only the first two levels of wavelet-packet coefficients, resulting in only 25% of the columns of ${\textbf{X}}^{W_P}$ being kept. We let this model run longer than the simulated settings, taking 2000 posterior samples after a burnin of 2000. We monitor convergence with running mean and trace plots to ensure all chains converged (see Supplementary Material).
Figure \[f:jourest\] contains the posterior estimated historical surface for the Journeyman data whereas Figure \[f:jourint\] presents the lower and upper bounds of the joint credible intervals. From Figure \[f:jourest\], moving across the $v$-axis from left to right, we see a pattern of time-specific depressions followed by elevations, which culminate in a large peak around $v = 8$ and $t = 20$. This suggests that an exposure 40 minutes into the day resulted in elevated SDNN that was sustained until, and peaked around, 100 minutes. After this peak, there is another depression that occurs around $v = 16$ and $t = 20$. These patterns are consistent with the analysis in [@Harezlak2007].
The joint bands experience edge effects along the $t$-axis, which somewhat distort the scale of the intervals in Figure \[f:jourint\]. However, from this figure, we are able to see the elevation of the lower bound around the major peak around $v = 8$ and $t = 20$ as well as the depression around $v = 16$ and $t = 20$. The ridge that runs along $v = 3$ in Figure \[f:jourest\] also appears to have an elevated lower bound. In the Supplementary Material, we present the point-wise intervals which are smaller in width, but exhibit similar features to the joint intervals. The intervals suggest regions of both increased and decreased association between SDNN and PM$_{2.5}$ throughout the morning. To more accurately pinpoint these regions, we turn to the BFDR procedure.
Figure \[f:jourfdr\] presents the local BFDR values, $P_B(v,t)$, for each coefficient (top row) as well as the flagged coefficients from the set $\psi_{\alpha}$ (bottom row) using $\alpha = 0.05$. The columns correspond to two different $\delta$ intensity cut-offs representing corresponding changes in $\log$(SDNN) of 0.5 and 0.75. The lower cut-off value highlights significant coefficients in the regions where we see the strongest effect sizes from Figure \[f:jourest\]. These regions do not extend from the diagonal nor to the edge of the surface, suggesting a series of delayed time-specific effects that result from exposures within the first 100 minutes of the workday. Some of these effects are strong, as evidenced by the sustained significance at the higher $\delta$ level. In the Supplementary Material, we present additional results which shows the main peak around $v = 8$ and $t = 20$ is significant at an intensity change of $\delta = 1.00$.
Discussion {#s:disc}
==========
One of the difficulties of estimating a historical functional effect is in maintaining the historical constraint. Previous authors have used tent shaped basis functions to achieve the constraint. When using wavelets coefficients from a DWT as a basis function, constraining the surface is difficult due to the lack of a clear relationship between the wavelet-space coefficients and the time domain. However, wavelet-packet coefficients from a DWPT have an exploitable relationship between their location indices and the original time scale that allows us to constrain the surface in the wavelet-packet space and maintain that constraint when projecting back to the data-space. In this work, we show that when performing the DWPT on both $y(t)$ and $x(v)$, we can use wavelet-packet basis functions, for a given choice of mother wavelet, to model historical effects.
The current literature on HFLMs is limited, largely focusing on the estimation of surface effects and the determination of model fit criterion. Most methods implement spline-based basis functions and, while one method does allow for different basis expansions, the authors only present results for spline-based models. Further, inference procedures are not discussed in the existing literature. To the best of our knowledge, the model we present here represents the first work in wavelet-based modeling of historical effects as well as the first Bayesian HFLM. Additionally, our method employs a novel use of wavelet-packets, which have not previously been used in functional regression models, to constrain the surface of estimation. Finally, we perform inference using two established multiplicity adjusted Bayesian inferential procedures for describing the uncertainty in the surface estimate through the use of joint credible intervals and identifying significant regions of coefficients using the BFDR.
We demonstrate in simulation that the wavelet-packet HFLM can accurately estimate several realistic historical surface settings. In particular, we show that regardless of the percent of ${\textbf{X}}^{W_P}$ coefficients we retain, the wavelet-packet HLFM has good RMISE levels that are similar across sample size and sampling density. Further, the joint credible intervals used in [@Meyer2015] for the wavelet-based FFR outperform point-wise credible intervals in terms of average coverage. And while their coverage is larger than the nominal, we prefer this to the alternative. Thus we suggest the use of the joint intervals for quantifying uncertainty in the wavelet-packet HFLM. To identify regions of significant coefficients, we implement the BFDR to show that for two reasonable choices of $\delta$ relative to the data and max signal, our model can identify regions of association in the surface. Finally, we apply the proposed model to analyze data on the association between HRV and PM$_{2.5}$ exposure in a panel of journeymen, focusing on the first three hours of exposure during the workday. Using the wavelet-packet HFLM, we are able to not only estimate regions of association but clearly identify them as representing significant changes in SDNN using the BFDR.
[**SUPPLEMENTARY MATERIAL**]{}
Supplementary Material to Bayesian HFLM with Wavelet-packets:
: Contains additional results from the simulation and application. (pdf)
MATLAB Code:
: Available at <https://github.com/markjmeyer/WPHFLM>.
Cavallari, J. M., Fang, S. C., Eisen, E. A., Schwartz, J., Hauser, R., and Herrick, R. F. (2008). Time Course of Heart Rate Variability Decline Following Particulate Matter Exposures in an Occupational Cohort. *Inhalation Toxicology* **20,** 415–422.
Harezlak, J., Coull, B. A., Laird, N. M., Magari, S. R., and Christiani, D. C. (2007). Penalized solutions to functional regression problems. *Computational Statistics & Data Analysis* **51,** 4911–4925.
Ivanescu, A.E., Staicu, A.-M., Scheipl, F., and Greven, S. (2015). Penalized function-on-function regression. *Computational Statistics* **30,** 539–568.
Kim, J. S., Staicu, A.-M., Maity, A., Carroll, R. J., and Ruppert, D. (2018). Additive Function-on-Function Regression. *Journal of Computational and Graphical Statistics* **27,** 234–244.
Kim, K., Şentürk, D., and Li, R. (2011). Recent history functional linear models for sparse longitudinal data. *Journal of Statistical Planning and Inference* **141,** 1554–1566.
Magari, S. R., Hauser, R., Schwartz, J., Williams, P. L., Smith, T. J., and Christiani, D. C. (2001). Association of Heart Rate Variability With Occupational and Environmental Exposure to Particulate Air Pollution. *Circulation* **104,** 986–991.
Malfait, N. and Ramsay, J. O. (2003). The historical functional linear model. *The Canadian Journal of Statistics* **31,** 115–128.
Malloy, E. J., Morris, J. S., Adar, S. D., Suh, H., Gold, D. R., and Coull, B. A. (2010). Wavelet-based functional linear mixed models: an application to measurement error-corrected distributed lag models. *Biostatistics* **11,** 432–452.
Meyer, M. J., Coull, B. A., Versace, F., Cinciripini, P., and Morris, J. S. (2015). Bayesian Function-on-Function Regression for Multi-Level Functional Data. *Biometrics* **71,** 563–574.
Misiti, M., Misiti, Y., Oppenheim, G., and Poggi, J.-M. (2007).*Wavelets and their Applications*. ISTE Ltd.
Morris, J. S. (2015). Functional regression. *Annual Reviews of Statistics and Its Applications* **2,** 321–359.
Morris, J. S., Brown, P. J., Herrick, R. C., Baggerly, K. A., and Coombes, K. R. (2008). Bayesian Analysis of Mass Spectrometry Proteomic Data Using Wavelet-Based Functional Mixed Models. *Biometrics* **64,** 479–489.
Morris, J. S. and Carroll, R. J. (2006). Wavelet-based functional mixed models. *Journal of the Royal Statistical Society, Series B* **68,** 179–199.
Nason, G. P. (2008). *Wavelet Methods in Statistics with R*. Springer.
Percival, D. B. and Walden, A. T. (2000). *Wavelet Methods for Time Series Analysis*. Cambridge University Press.
Pomann, G.-M., Staicu, A.-M., Lobaton, E. J., Mejia, A. F., Dewey, B. E., Reich, D. S., Sweeney, E. M., and Shinohara, R. T. (2016). A lag functional linear model for prediction of magnetization transfer ratio in multiple sclerosis lesions. *Annals of Applied Statistics* **10,** 2325–2348.
Ruppert, D., Wand, M. P., and Carroll, R. J. (2003). *Semiparametric Regression*. Cambridge University Press.
Scheipl, F. and Greven, S. (2016). Identifiability in penalized function-on-function regression models. *Electronic Journal of Statistics* **10,** 495–526.
Scheipl, F., Staicu, A.-M., and Greven, S. (2015). Functional Additive Mixed Models. *Journal of Computational and Graphical Statistics* **24,** 477–501.
[^1]: This work was supported by grants from the National Institutes of Health (ES007142, ES000002, ES016454, CA134294). The authors would like to thank Dr. David C. Christiani for use of the journeyman boilermaker data.
|
---
abstract: 'In this paper, we consider [*media-based modulation (MBM)*]{}, an attractive modulation scheme which is getting increased research attention recently, for the uplink of a massive MIMO system. Each user is equipped with one transmit antenna with multiple radio frequency (RF) mirrors (parasitic elements) placed near it. The base station (BS) is equipped with tens to hundreds of receive antennas. MBM with $m_{rf}$ RF mirrors and $n_r$ receive antennas over a multipath channel has been shown to asymptotically (as $m_{rf}\rightarrow \infty$) achieve the capacity of $n_r$ parallel AWGN channels. This suggests that MBM can be attractive for use in massive MIMO systems which typically employ a large number of receive antennas at the BS. In this paper, we investigate the potential performance advantage of multiuser MBM (MU-MBM) in a massive MIMO setting. Our results show that multiuser MBM (MU-MBM) can significantly outperform other modulation schemes. For example, a bit error performance achieved using 500 receive antennas at the BS in a massive MIMO system using conventional modulation can be achieved using just 128 antennas using MU-MBM. Even multiuser spatial modulation, and generalized spatial modulation in the same massive MIMO settings require more than 200 antennas to achieve the same bit error performance. Also, recognizing that the MU-MBM signal vectors are inherently sparse, we propose an efficient MU-MBM signal detection scheme that uses compressive sensing based reconstruction algorithms like orthogonal matching pursuit (OMP), compressive sampling matching pursuit (CoSaMP), and subspace pursuit (SP).'
author:
- |
Bharath Shamasundar and A. Chockalingam\
Department of ECE, Indian Institute of Science, Bangalore 560012
title: '[Multiuser Media-based Modulation for Massive MIMO Systems]{}'
---
–
Introduction {#sec1}
============
Media-based modulation (MBM), a promising modulation scheme for wireless communications in multipath fading environments, is attracting recent research attention [@mbm1]-[@mbm6]. The key features that make MBM different from conventional modulation are: $i)$ MBM uses digitally controlled parasitic elements external to the transmit antenna that act as radio frequency (RF) mirrors to create different channel fade realizations which are used as the channel modulation alphabet, and $ii)$ it uses indexing of these RF mirrors to convey additional information bits. The basic idea behind MBM can be explained as follows.
Placing RF mirrors near a transmit antenna is equivalent to placing scatterers in the propagation environment close to the transmitter. The radiation characteristics of each of these scatterers (i.e., RF mirrors) can be changed by an ON/OFF control signal applied to it. An RF mirror reflects back the incident wave originating from the transmit antenna or passes the wave depending on whether it is OFF or ON. The ON/OFF status of the mirrors is called as the ‘mirror activation pattern (MAP)’. The positions of the ON mirrors and OFF mirrors change from one MAP to the other, i.e., the propagation environment close to the transmitter changes from one MAP to the other MAP. Note that in a rich scattering environment, a small perturbation in the propagation environment will be augmented by many random reflections resulting in an independent channel. The RF mirrors create such perturbations by acting as controlled scatterers, which, in turn, create independent fade realizations for different MAPs.
If $m_{rf}$ is the number of RF mirrors used, then $2^{m_{rf}}$ MAPs are possible. If the transmitted signal is received through $n_r$ receive antennas, then the collection of $2^{m_{rf}}$ $n_r$-length complex channel gain vectors form the MBM channel alphabet. This channel alphabet can convey $m_{rf}$ information bits through MAP indexing. If the antenna transmits a symbol from a conventional modulation alphabet denoted by ${\mathbb A}$, then the spectral efficiency of MBM is $\eta_{{\tiny \mbox{MBM}}}=m_{rf} + \log_2|{\mathbb A}|$ bits per channel use (bpcu). An implementation of a MBM system consisting of 14 RF mirrors placed in a compact cylindrical structure with a dipole transmit antenna element placed at the center of the cylindrical structure has been reported in [@mbm4]. Early reporting of the idea of using parasitic elements for index modulation purposes (in the name ‘aerial modulation’) can be found in [@aerial1],[@aerial2].
MBM has been shown to possess attractive performance attributes, particularly when the number of receive antennas is large [@mbm1]-[@mbm6]. Specifically, MBM with $m_{rf}$ RF mirrors and $n_r$ receive antennas over a multipath channel has been shown to asymptotically (as $m_{rf}\rightarrow \infty$) achieve the capacity of $n_r$ parallel AWGN channels [@mbm2]. This suggests that MBM can be attractive for use in massive MIMO systems which typically employ a large number of receive antennas at the BS. However, the literature on MBM so far has focused mainly on single-user (point-to-point) communication settings. Our first contribution in this paper is that, we report MBM in multiuser massive MIMO settings and demonstrate significant performance advantages of MBM compared to conventional modulation. For example, a bit error performance achieved using 500 receive antennas at the BS in a massive MIMO system using conventional modulation can be achieved using just 128 antennas with multiuser MBM. Even multiuser spatial modulation (SM) and generalized spatial modulation (GSM) [@lajos]-[@mugsm] in the same massive MIMO settings require more than 200 antennas to achieve the same bit error performance. This suggests that multiuser MBM can be an attractive scheme for use in the uplink of massive MIMO systems.
The second contribution relates to exploitation of the inherent sparsity in multiuser MBM signal vectors for low-complexity signal detection at the BS receiver. We resort to compressive sensing (CS) based sparse recovery algorithms for this purpose. Several efficient sparse recovery algorithms are known in the literature[@omp]-[@romp]. We propose a multiuser MBM signal detection scheme that employs greedy sparse recovery algorithms like orthogonal matching pursuit (OMP)[@omp], compressive sampling matching pursuit (CoSaMP) [@cosamp], and subspace pursuit (SP)[@subspace]. Simulation results show that the proposed detection scheme using SP achieves very good performance (e.g., significantly better performance compared to MMSE detection) at low complexity. This demonstrates that CS based sparse signal recovery approach is a natural and efficient approach for multiuser MBM signal detection in massive MIMO systems.
The rest of the paper is organized as follows. The multiuser MBM system model is introduced in Sec. \[sec2\]. The performance of multiuser MBM with maximum likelihood detection is presented in Sec. \[sec3\]. The proposed sparsity-exploiting detection scheme for multiuser MBM signal detection and its performance in massive MIMO systems are presented in Sec. \[sec4\]. Conclusions are presented in Sec. \[sec5\].
Multiuser MBM system model {#sec2}
==========================
Consider a massive MIMO system with $K$ uplink users and a BS with $n_r$ receive antennas (see Fig. \[mbm\_mimo\]), where $K$ is in the tens (e.g., $K=16,32$) and $n_r$ is in the hundreds ($n_r=128,256$). The users employ MBM for signal transmission. Each user has a single transmit antenna and $m_{rf}$ RF mirrors placed near it. In a given channel use, each user selects one of the $2^{m_{rf}}$ mirror activation patterns (MAPs) using $m_{rf}$ information bits. A mapping is done between the combinations of $m_{rf}$ information bits and the MAPs. An example mapping between information bits and MAPs is shown in Table \[table1\] for $m_{rf}=2$. The mapping between the possible MAPs and information bits is made known a priori to both transmitter and receiver for encoding and decoding purposes, respectively.
Information bits Mirror 1 status Mirror 2 status
------------------ ----------------- -----------------
00 ON ON
01 ON OFF
10 OFF ON
11 OFF OFF
: mapping between information bits and MAPs for $m_{rf}=2$.[]{data-label="table1"}
Apart from the bits conveyed through the choice of a MAP in a given channel use as described above, a symbol from a modulation alphabet $\mathbb{A}$ (e.g., QAM, PSK) transmitted by the antenna conveys an additional $\log_2|\mathbb{A}|$ bits. Therefore, the spectral efficiency of a $K$-user MBM system is given by $$\eta_{{\tiny \mbox{MU-MBM}}}= K(m_{rf}+\log_2{|\mathbb{A}|}) \ \ \mbox{bpcu.}$$ For example, a multiuser MBM system with $K=4$, $m_{rf}=2$, and 4-QAM has a system spectral efficiency of 16 bpcu. An important point to note here is that the spectral efficiency per user increases linearly with the number of RF mirrors used at each user. To introduce the multiuser MBM signal set and the corresponding received signal vector at the BS, let us first formally introduce the single-user MBM signal set.
![Multiuser MBM in a massive MIMO system.[]{data-label="mbm_mimo"}](mu_mbm_block_diagram.eps){width="7.5cm" height="7.75cm"}
Single-user MBM channel alphabet {#sec2a}
--------------------------------
The MBM channel alphabet of a single user is the set of all channel gain vectors corresponding to the various MAPs of that user. Let us define $M {\triangleq}2^{m_{rf}}$, where $M$ is the number of possible MAPs corresponding to $m_{rf}$ RF mirrors. Let $\mathbf{h}_k^m$ denote the $n_r \times 1$ channel gain vector corresponding to the $m$th MAP of the $k$th user, where ${\mathbf h}_{k}^{m}=[h_{1,k}^m \ h_{2,k}^m \ \cdots \ h_{n_r,k}^m]^T$, $h_{i,k}^m$ is the channel gain corresponding to the $m$th MAP of the $k$th user to the $i$th receive antenna, $i=1,\cdots,n_r$, $k=1,\cdots,K$, and $m=1,\cdots,M$, and the $h_{i,k}^m$s are assumed to be i.i.d. and distributed as $\mathcal{CN}(0,1)$. The MBM channel alphabet for the $k$th user, denoted by $\mathbb{H}_k$, is then the collection of these channel gain vectors, i.e., $\mathbb{H}_k=\{\mathbf{h}_k^1,\mathbf{h}_k^2,\cdots,\mathbf{h}_k^M\}$. The MBM channel alphabet of each user is estimated at the BS receiver through pilot transmission before data transmission. The number of pilot channel uses needed for the estimation of each user’s channel alphabet grows exponentially in $m_{rf}$. It is also noted that, while the MBM channel alphabet of each user needs to be known at the BS receiver for detection purposes, the users’ transmitters need not know their channel alphabets.
Single-user MBM signal set {#sec2b}
--------------------------
Define ${\mathbb A}_0 {\triangleq}{\mathbb A}\cup 0$. The single-user MBM signal set, denoted by $\mathbb{S}_{{\tiny \mbox{SU-MBM}}}$, is the set of $M\times 1$-sized MBM signal vectors given by $$\begin{aligned}
\hspace{-4mm}
\mathbb{S}_{{\tiny \mbox{SU-MBM}}} &= \left\{\mathbf{s}_{m,q} \in {\mathbb A}_0^M : m=1,\cdots,M, \ q=1,\cdots,|\mathbb{A}| \right \} \nonumber \\
\mbox{ s.t } \ \mathbf{s}_{m,q} &= [0,\cdots,0,\hspace{-2mm}\underbrace{s_q}_{\mbox{{\scriptsize $m$th coordinate}}}\hspace{-2mm}0,\cdots,0]^T, s_q \in \mathbb{A},
\label{ss}\end{aligned}$$ where $m$ is the index of the MAP. That is, an MBM signal vector $\mathbf{s}_{m,q}$ in (\[ss\]) means a complex symbol $s_q \in \mathbb{A}$ being transmitted on a channel with an associated channel gain vector $\mathbf{h}^m$, where $\mathbf{h}^m$ is the $n_r\times 1$ channel gain vector corresponding to the $m$th MAP. Therefore, the $n_r\times 1$ received signal vector corresponding to a transmitted MBM signal vector $\mathbf{s}_{m,q}$ can be written as $$\mathbf{y}=s_q\mathbf{h}^m + \mathbf{n},$$ where $\mathbf{n} \in \mathbb{C}^{n_r}$ is the AWGN noise vector with $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$. The size of the single-user MBM signal set is $|\mathbb{S}_{{\tiny \mbox{SU-MBM}}}|= M|{\mathbb A}|$. For example, if $m_{rf}=2$ and $|{\mathbb A}|=2$ (i.e., BPSK ), then $|\mathbb{S}_{{\tiny \mbox{SU-MBM}}}|=8$, and the corresponding MBM signal set is given by
$$\hspace{-0.0mm}
\mathbb{S}_{{\tiny \mbox{SU-MBM}}}=
\left\{
\begin{bmatrix}
1 \\ 0 \\ 0\\ 0
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
-1 \\ 0 \\ 0\\ 0
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
0 \\ 1 \\ 0\\ 0
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
0 \\ -1 \\ 0\\ 0
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
0 \\ 0 \\ 1\\ 0
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
0 \\ 0 \\ -1\\ 0
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
0 \\ 0 \\ 0\\ 1
\end{bmatrix}\hspace{-1.25mm},
\begin{bmatrix}
0 \\ 0 \\ 0\\ -1
\end{bmatrix}\hspace{-1.0mm}
\right \}\hspace{-0.5mm}. \hspace{-4mm}
\label{mbm_sigset}$$
Multiuser MBM received signal {#sec2c}
-----------------------------
With the above definitions of single-user MBM channel alphabet and signal set, the multiuser MBM signal set with $K$ users is given by $\mathbb{S}_{{\tiny \mbox{MU-MBM}}}=\mathbb{S}_{{\tiny \mbox{SU-MBM}}}^K$. Let $\mathbf{x}_k \in \mathbb{S}_{{\tiny \mbox{SU-MBM}}}$ denote the transmit MBM signal vector from the $k$th user. Let $\mathbf{x} = \left[\mathbf{x}_1^T \ \mathbf{x}_2^T \ \cdots \ \mathbf{x}_K^T \right]^T \in \mathbb{S}_{{\tiny \mbox{MU-MBM}}}$ denote the vector comprising of the transmit MBM signal vectors from all the $K$ users. Let $\mathbf{H} \in \mathbb{C}^{n_r \times KM}$ denote the channel gain matrix given by $\mathbf{H}=[\mathbf{H}_1 \ \mathbf{H}_2 \ \cdots \ \mathbf{H}_K]$, where $\mathbf{H}_k=[\mathbf{h}_k^1 \ \mathbf{h}_k^2 \ \cdots \ \mathbf{h}_k^M] \in \mathbb{C}^{n_r \times M}$, and $\mathbf{h}_k^m$ is the channel gain vector of the $k$th user corresponding to $m$th MAP as defined before. The $n_r\times 1$ multiuser received signal vector at the BS is then given by $$\mathbf{y} = \mathbf{Hx} + \mathbf{n},
\label{sys}$$ where $\mathbf{n}$ is the $n_r\times 1$ AWGN noise vector with $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma ^2 \mathbf{I})$.
Performance of multiuser MBM {#sec3}
============================
In this section, we analyze the BER performance of multiuser MBM under maximum likelihood (ML) detection. We obtain an upper bound on the BER which is tight at moderate to high SNRs. We also present a comparison between the BER performance of multiuser MBM and those of other multiuser schemes that employ conventional modulation, spatial modulation, and generalized spatial modulation.
Upper bound on BER {#sec3a}
------------------
The ML detection rule for the multiuser MBM system model in (\[sys\]) is given by $$\hat{\mathbf{x}} = {\mathop{\text{argmin}}}_{\mathbf{x} \in \mathbb{S}_{{\tiny \mbox{MU-MBM}}}} \| \mathbf{y} - \mathbf{Hx} \|^2,
\label{ML}$$ which can be written as $$\hat{\mathbf{x}}={\mathop{\text{argmin}}}_{\mathbf{x} \in \mathbb{S}_{{\tiny \mbox{MU-MBM}}}}\left( \Vert\mathbf{H}\mathbf{x}\Vert^2-2\mathbf{y}^T\mathbf{H}\mathbf{x}\right) .$$ The pairwise error probability (PEP) that the receiver decides in favor of the signal vector $\mathbf{x}_2$ when $\mathbf{x}_1$ was transmitted, given the channel matrix $\mathbf{H}$ can be written as $$\begin{aligned}
\label{PEP}
PEP & = & P(\mathbf{x}_1 \rightarrow \mathbf{x}_2| \mathbf{H}) \nonumber \\
& = & P\left(2\mathbf{y}^T\mathbf{H}(\mathbf{x}_2-\mathbf{x}_1)>(\Vert\mathbf{H}\mathbf{x}_2\Vert^2-\Vert\mathbf{H}\mathbf{x}_1\Vert^2) | \mathbf{H}\right) \nonumber \\
& = & P\left(2\mathbf{n}^T\mathbf{H}(\mathbf{x}_2-\mathbf{x}_1)>\Vert\mathbf{H}(\mathbf{x}_2-\mathbf{x}_1)\Vert^2 |\mathbf{H}\right).\end{aligned}$$ Defining $z\triangleq 2\mathbf{n}^T\mathbf{H}(\mathbf{x}_2-\mathbf{x}_1)$, we observe that $z \sim \mathcal{N} \left(0, 2\sigma^2\Vert\mathbf{H}(\mathbf{x}_2-\mathbf{x}_1)\Vert^2 \right)$. Therefore, we can write $$P(\mathbf{x}_1 \rightarrow \mathbf{x}_2| \mathbf{H})=Q\left(\frac{\Vert \mathbf{H}(\mathbf{x}_2-\mathbf{x}_1)\Vert}{\sqrt{2}\sigma}\right),
\label{pep1}$$ where $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{\frac{-t^2}{2}}dt$. The conditional PEP expression in can be written as $$P(\mathbf{x}_1 \hspace{-0.5mm} \rightarrow \hspace{-0.5mm} \mathbf{x}_2 | \mathbf{H}) =
Q\hspace{-0.5mm}\left(\hspace{-1.5mm} \sqrt{ \frac{1}{2\sigma^2}\left \| \sum_{l=1}^{KM} (x_{1,l} - x_{2,l})\mathbf{h}_l \right \|^2}\right),
\label{pep2}$$ where $x_{1,l}$ and $x_{2,l}$ are $l$th entries of $\mathbf{x}_1$ and $\mathbf{x}_2$, respectively, and $\mathbf{h}_l$ is the $l$th column of $\mathbf{H}$. The argument of $Q(\cdot)$ in has the central $\chi ^2$-distribution with $2n_r$ degrees of freedom. The computation of the unconditional PEPs requires the expectation of $Q(\cdot)$ with respect to $\mathbf{H}$, which can be obtained as follows[@erroranalysis]: $$\begin{aligned}
\hspace{-3mm}
P(\mathbf{x}_1 \rightarrow \mathbf{x}_2) &= \mathbb{E}_\mathbf{H} \left[ P(\mathbf{x}_1 \rightarrow \mathbf{x}_2 | \mathbf{H}) \right] \nonumber \\
&=f(\alpha)^{n_r} \sum_{i=0}^{n_r-1} \binom{n_r-1+i}{i} (1-f(\alpha))^i, \end{aligned}$$ where $f(\alpha) \triangleq \dfrac{1}{2} \left(1-\sqrt{\dfrac{\alpha}{1+\alpha}} \right)$, $\alpha \triangleq \dfrac{1}{4\sigma^2} \sum\limits_{l=1}^{KM} \theta _l$, and $\theta _l \triangleq | x_{1,l} - x_{2,l} |^2$. Now, an upper bound on the bit error probability using union bound can be obtained as $$P_e \leq \dfrac{1}{2^{\eta_{{\tiny \mbox{MU-MBM}}}}} \hspace{-2mm} \sum_{\mathbf{x}_1 \in \mathbb{S}_{\tiny \mbox{MU-MBM}}} \sum_{\mathbf{x}_2 \in \mathbb{S}_{\tiny \mbox{MU-MBM}} \setminus \mathbf{x}_1} \hspace{-4mm} P(\mathbf{x}_1 \rightarrow \mathbf{x}_2) \dfrac{d_H(\mathbf{x}_1, \mathbf{x}_2)}{\eta_{{\tiny \mbox{MU-MBM}}}},
\label{ber}$$ where $d_H(\mathbf{x}_1, \mathbf{x}_2)$ is the Hamming distance between the bit mappings corresponding to $\mathbf{x}_1$ and $\mathbf{x}_2$.
Numerical results {#sec3b}
-----------------
We evaluated the BER performance of multiuser MBM (MU-MBM) using the BER upper bound derived above as well as simulations. For the purpose of initial comparisons with other systems, we consider a MU-MBM system with $K=2$, $n_r=8$, $m_{rf}=3$, BPSK, and 4 bpcu per user. Let $n_t$ and $n_{rf}$ denote the number transmit antennas and transmit RF chains, respectively, at each user. Note that in the considered MU-MBM system, each user uses one transmit antenna and one transmit RF chain, i.e., $n_t=n_{rf}=1$. We compare the performance of the above MU-MBM system with those of three other multiuser systems which use $i)$ conventional modulation (CM), $ii)$ spatial modulation (SM), and $iii)$ generalized spatial modulation (GSM). The multiuser system with conventional modulation (MU-CM) uses $n_t=n_{rf}=1$ at each user and employs 16-QAM to achieve the same spectral efficiency of 4 bpcu per user. The multiuser system with SM (MU-SM) uses $n_t=2$, $n_{rf}=1$, and 8-QAM, achieving a spectral efficiency of $\log_2n_t+\log_2|\mathbb A|=
\log_22+\log_28=4$ bpcu per user. The multiuser system with GSM (MU-GSM) uses $n_t=4$, $n_{rf}=2$, and BPSK, achieving a spectral efficiency of $\lfloor\log_2\binom{n_t}{n_{rf}}\rfloor+\log_2|\mathbb A|=
\lfloor\log_2\binom{4}{2}\rfloor+\log_22= 4$ bpcu per user.
Figure \[ml\] shows the BER performance of the MU-MBM, MU-CM, MU-SM, and MU-GSM systems described above. First, it can be observed that the analytical upper bound is very tight at moderate to high SNRs. Next, in terms of performance comparison between the considered systems, the following inferences can be drawn from Fig. \[ml\].
- The MU-MBM system achieves the best performance among all the four systems considered. For example, MU-MBM performs better by about 5 dB, 4 dB, 2.5 dB compared to MU-CM, MU-SM, and MU-GSM systems, respectively, at a BER of $10^{-5}$.
- The better performance of MU-MBM can be attributed to more bits being conveyed through mirror indexing, which allows MU-MBM to use lower-order modulation alphabets (BPSK) compared to other systems which may need higher-order alphabets (8-QAM, 16-QAM) to achieve the same spectral efficiency.
- MU-MBM performs better than MU-GSM though both use BPSK in this example. This can be attributed to the good distance properties of the MBM signal set [@mbm2].
![BER performance of MU-MBM, MU-CM, MU-SM, and MU-GSM with $K=2$, $n_r=8$, 4 bpcu per user, and ML detection. Analysis and simulations.[]{data-label="ml"}](ML_nr8_3.eps){width="9.0cm" height="6.25cm"}
Note that though the results in Fig. \[ml\] illustrate the performance superiority of MU-MBM over MU-CM, MU-SM, and MU-GSM, they are presented only for a small system with $K=2$ and $n_r=8$. This is because ML detection is prohibitively complex for systems with large $K$ and $n_r$ (ML detection is exponentially complex in $K$). However, massive MIMO systems are characterized by $K$ in the tens and $n_r$ in the hundreds. Therefore, low-complexity detection schemes which scale well for such large-scale MU-MBM systems are needed. To address this need, we resort to exploiting the inherent sparse nature of the MBM signal vectors, and devise a compressive sensing based detection algorithm in the following section.
Sparsity-exploiting detection of multiuser MBM signals {#sec4}
======================================================
It is evident from the example signal set in that the MBM signal vectors are inherently sparse. An MBM signal vector has only one non-zero element out of $M$ elements, leading to a sparsity factor of $1/M$. For example, consider an MBM signal set with $m_{rf}=4$ and $M=2^{m_{rf}}=16$. Out of 16 elements in a signal vector, only one element is non-zero resulting in a sparsity factor of $1/16$. Exploitation of this inherent sparsity to devise detection algorithms can lead to efficient signal detection at low complexities. Accordingly, we propose a low-complexity MU-MBM signal detection scheme that employs compressive sensing based sparse reconstruction algorithms like OMP, CoSaMP, and SP.
Proposed sparsity-exploiting detection algorithm {#sec4a}
------------------------------------------------
We first model the MU-MBM signal detection problem as a sparse reconstruction problem and then employ greedy algorithms for signal detection. Sparse reconstruction is concerned with finding an approximate solution to the following problem [@sparse]: $$\min\limits_{\mathbf{x}} \| \mathbf{x} \|_0 \mbox{ subject to } \mathbf{y} = \mathbf{\Phi x + n},
\label{sparse}$$ where $\mathbf{\Phi} \in \mathbb{C}^{m \times n}$ is called the measurement matrix, $\mathbf{x} \in \mathbb{C}^{n}$ is the complex input signal vector, $\mathbf{y} \in \mathbb{C}^{m}$ is the noisy observation corresponding to the input signal, and $\mathbf{n} \in \mathbb{C}^m$ is the complex noise vector. The MU-MBM signal detection problem at the BS in (\[sys\]) can be modeled as a sparse recovery problem in , with the measurement matrix being the channel matrix $\mathbf{H} \in \mathbb{C}^{n_r \times KM}$, the noisy observation being the received signal vector $\mathbf{y} \in \mathbf{C}^{n_r}$, and the input being the MU-MBM transmit signal vector $\mathbf{x} \in \mathbb{S}_{{\tiny \mbox{MU-MBM}}} $. The noise vector is additive complex Gaussian with $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$.
Greedy algorithms achieve sparse reconstruction in an iterative manner. They decompose the problem of sparse recovery into a two step process; recover the support of the sparse vector first, and then obtain the non-zero values over this support. For example, OMP starts with an initial empty support set, an initial solution $\mathbf{x}^0=\mathbf{0}$, and an initial residue $\mathbf{r}^0 =\mathbf{y}-\mathbf{\bf Hx}^0 =\mathbf{y}$. In each step, OMP updates one coordinate of the vector $\mathbf{x}$ based on the correlation values between the residue vector and the columns of the $\mathbf{\bf H}$ matrix. In the $k$th iteration, an element $j_0$ given by $$j_0 = {\mathop{\text{argmax}}}\limits_{j \notin \mathcal{S}^{k-1}} \frac{\mathbf{\bf h}_j^T \mathbf{r}^{k-1}}{\| \mathbf{\bf h}_j \|_2^2}$$ is added to the support set, where $\mathbf{\bf h}_j$ is the $j$th column of $\mathbf{\bf h}$, and $\mathcal{S}^{k-1}$ and $\mathbf{r}^{k-1}$ are the support set and residue after $k-1$ iterations, respectively. The entries of $\mathbf{x}$ corresponding to the obtained support set are computed using least squares. This process is iterated till the stopping criteria is met. The stopping criteria can be either a specified error threshold or a specified level of sparsity.
In the SP algorithm, instead of updating one coordinate of $\mathbf{x}$ at a time as in OMP, $K$ coordinates are updated at once. The major difference between OMP and SP is the following. In OMP, the support set is generated sequentially. It starts with an empty set and adds one element in every iteration to the existing support set. An element added to the support set can not be removed until the algorithm terminates. In contrast, SP provides flexibility of refining the support set in every iteration. CoSaMP is similar to SP except that it updates $2K$ coordinates in each iteration to the support set instead of updating $K$ coordinates as in SP. CoSaMP and SP have superior reconstruction capability comparable to convex relaxation methods [@cosamp],[@subspace]. [**Algorithm 1**]{} shows the listing of the pseudo-code of the proposed sparsity-exploiting detection algorithm for MU-MBM signals.
Inputs: $\mathbf{y}, \mathbf{H}, K $ Initialize: $j=0$ **repeat** $\mathbf{\hat{x}}_r=\mbox{SR}(\mathbf{y},\mathbf{H},K+j)$ $\mathbf{u}^j= \mbox{UAP}(\hat{\mathbf{x}}_r)$ **if** $ \| \mathbf{u}^j \| _0 = K $ **for** $k= 1 \mbox{ to } K$ $ \hat{\mathbf{x}}^k = {\mathop{\text{argmin}}}\limits_{\mathbf{s} \in \mathbb{S}_{{\tiny \mbox{SU-MBM}}}} \| \hat{\mathbf{x}}_r^k-\mathbf{s} \|^2$ **end for** **break**; **else** $j=j+1$ **end if** **until** $j < K(M-1)$ Output: The estimated MU-MBM signal vector $$\hat{\mathbf{x}} = [\hat{\mathbf{x}}^{\scriptsize{1}^T}, \hat{\mathbf{x}}^{\scriptsize{2}^T}, \cdots ,\hat{\mathbf{x}}^{\scriptsize{K}^T} ]^T$$
\[alg1\]
SR in [**Algorithm 1**]{} denotes the sparse recovery algorithm, which can be any one of OMP, CoSaMP, and SP. The signal vector reconstructed by the sparse recovery algorithm is denoted by $\hat{\mathbf{x}}_r$. Detecting the MU-MBM signal vector involves detecting the MBM signal vector transmitted by each user. An MBM signal vector from a user has exactly one non-zero entry out of $M$ entries as observed in the example MBM signal set in . Hence, SR is expected to reconstruct a MU-MBM signal vector such that the MBM signal sub-vector corresponding to a given user has only one non-zero entry. But this constraint on the expected support set is not built in the general sparse recovery algorithms. In general, a sparse recovery algorithm can output $K$ non-zero elements at any of the $KM$ locations of $\hat{\mathbf{x}}_r$. To overcome this issue, we define user activity pattern (UAP), denoted by $\mathbf{u}$, as a $K$-length vector with $k$th entry as $\mathbf{u}_k=1$ if there is at least one non-zero entry in the $k$th user’s recovered MBM signal vector, and $\mathbf{u}_k=0$ otherwise. A valid reconstructed signal vector is one which has all ones in $\mathbf{u}$. SR is used multiple times with a range of sparsity estimates starting from $K$ ($K+j$ in the algorithm listing) till the valid UAP is obtained (i.e., till the algorithm reconstructs at least one non-zero entry for each user’s MBM signal vector).
In the algorithm listing, $\mathbf{u}^j$ denotes the UAP at the $j$th iteration. On recovering an $\hat{\mathbf{x}}_r$ with valid UAP, the MBM signal vector of each user is mapped to the nearest (in the Euclidean sense) MBM signal vector in $\mathbb{S}_{{\tiny \mbox{SU-MBM}}}$. This is shown in the Step 8 in the algorithm listing, where $\hat{\mathbf{x}}_r^k$ denotes the recovered MBM signal vector of the $k$th user and $\hat{\mathbf{x}}^k$ denotes the MBM signal vector to which $\hat{\mathbf{x}}_r^k$ gets mapped to. Finally, the MU-MBM signal vector is obtained by concatenating the detected MBM signal vectors of all the users, i.e., , $\hat{\mathbf{x}} = [\hat{\mathbf{x}}^{\scriptsize{1}^T}, \hat{\mathbf{x}}^{\scriptsize{2}^T}, \cdots ,\hat{\mathbf{x}}^{\scriptsize{K}^T} ]^T$.
The decoding of information bits from the detected MBM signal vector of a given user involves decoding of mirror index bits and QAM symbol bits of that user. The mirror index bits are decoded from the MAP of the detected MBM signal vector and the QAM bits are decoded from the detected QAM symbol.
Performance results in massive MIMO system {#sec4b}
------------------------------------------
In this subsection, we present the BER performance of MU-MBM systems in a massive MIMO setting (i.e., $K$ in the tens and $n_r$ in the hundreds) when the proposed [**Algorithm 1**]{} is used for MU-MBM signal detection at the BS. In the same massive MIMO setting, we evaluate the performance of other systems that use conventional modulation (MU-CM), spatial modulation (MU-SM), and generalized spatial modulation (MU-GSM), and compare them with the performance achieved by MU-MBM. The proposed [**Algorithm 1**]{} is also used for the detection of MU-SM and MU-GSM. It is noted that the MU-SM and MU-GSM signal vectors are also sparse to some extent; the sparsity factors in MU-SM and MU-GSM are $1/n_t$ and $n_{rf}/n_t$, respectively. So the use of the proposed algorithm for detection of these signals is also appropriate. ML detection is used to detect MU-CM signals (this is possible for MU-CM with sphere decoding for $K=16$, i.e., 32 real dimensions).
[*MU-MBM performance using proposed algorithm:*]{} Figure \[fig:sparsealgos\] shows the performance of MU-MBM system using the proposed algorithm with $i)$ OMP, $ii)$ CoSaMP, and $iii)$ SP. MMSE detection performance is also shown for comparison. A massive MIMO system with $K=16$ and $n_r=128$ is considered. Each user uses $n_t=1$, $n_{rf}=1$, $m_{rf}=6$, and 4-QAM. This results in a spectral efficiency of 8 bpcu per user, and a sparsity factor of $1/64$. From Fig. \[fig:sparsealgos\], we observe that the proposed algorithm with OMP, CoSaMP, and SP achieve significantly better performance compared to MMSE. Among the the use of OMP, CoSaMP, and SP in the proposed algorithm, use of SP gives the best performance. This illustrates the superior reconstruction/detection advantage of the proposed algorithm with SP. We will use the proposed algorithm with SP in the subsequent performance results figures. It is noted that the complexity of proposed algorithm is also quite favorable; the complexity of the proposed algorithm with SP and that of MMSE are $O(K^2Mn_r)$ and $O(K^3M^3)$, respectively.
![BER performance of MU-MBM in a massive MIMO system with $K=16$, $n_r=128$, $n_t=1$, $n_{rf}=1$, $m_{rf}=6$, 4-QAM, 8bpcu per user, using the proposed detection algorithm. MMSE detection performance is also shown for comparison.[]{data-label="fig:sparsealgos"}](OMP_CoSaMP_SP3.eps){width="9cm" height="6.25cm"}
[*Performance of MU-MBM, MU-SM, MU-GSM:*]{} Figure \[fig:spdetection\] shows a BER performance comparison between MU-MBM, MU-CM, MU-SM, and MU-GSM in a massive MIMO setting with $K=16$ and $n_r=128$. The proposed algorithm with SP is used for detection in MU-MBM, MU-SM, and MU-GSM. ML detection is used for MU-CM. The spectral efficiency is fixed at 5 bpcu per user for all the four schemes. MU-MBM achieves this spectral efficiency with $n_t=1$, $n_{rf}=1$, $m_{rf}=3$, and 4-QAM. MU-CM uses $n_t=1$, $n_{rf}=1$, and 32-QAM to achieve 5 bpcu per user. To achieve the same 5 bpcu per user, MU-SM uses $n_t=4$, $n_{rf}=1$, and 8-QAM, and MU-GSM uses $n_t=5$, $n_{rf}=2$, and BPSK. The sparsity factors in MU-MBM, MU-SM, and MU-GSM are $1/8$, $1/4$, and $2/5$, respectively. It can be seen that, MU-MBM clearly outperforms MU-CM, MU-SM, and MU-GSM. For example, at a BER of $10^{-5}$, MU-MBM outperforms MU-CM, MU-GSM, and MU-SM by about 7 dB, 5 dB, and 4 dB, respectively. The performance advantage of MU-MBM can be mainly attributed to its better signal distance properties [@mbm2]. MU-MBM is also benefited by its lower sparsity factor as well as the possibility of using lower-order QAM size because of additional bits being conveyed through indexing mirrors.
{width="9cm" height="6.25cm"}
\[fig:spdetection\]
[*Effect of number of BS receive antennas*]{}: Figure \[fig:effectofnr\] shows an interesting result which demonstrates MU-MBM’s increasing performance gain compared to MU-CM, MU-SM, and MU-GSM as the number of BS receive antennas is increased. A massive MIMO system with $K=16$ and 5 bpcu per user is considered. The parameters of the four schemes are the same as those in Fig. \[fig:spdetection\] except that here SNR is fixed at 4 dB and $n_r$ is varied from 48 to 624. It is interesting to observe that a performance that could be achieved using 500 antennas at the BS in a massive MIMO system that uses conventional modulation ($3\times 10^{-3}$ BER for MU-CM at $n_r=500$ with ML detection) can be achieved using just 128 antennas when MU-MBM is used ($3\times 10^{-3}$ BER for MU-MBM at $n_r=128$ with proposed detection). MU-SM and MU-GSM also achieve better performance compared to MU-CM, but they too require more than 200 antennas to achieve the same BER. This increasing performance advantage of MU-MBM for increasing $n_r$ can be mainly attributed to its better signal distance properties particularly when $n_r$ is large [@mbm2]. This indicates that multiuser MBM can be a very good scheme for use in the uplink of massive MIMO systems.
![BER performance MU-MBM, MU-CM, MU-SM, and MU-GSM as a function of $n_r$ in a massive MIMO setting with $K=16$, 5 bpcu per user, and SNR = 4 dB.[]{data-label="fig:effectofnr"}](ber_vs_nr5.eps){width="9cm" height="6.25cm"}
Conclusions {#sec5}
===========
We investigated the use of media-based modulation (MBM), a recent and attractive modulation scheme that employs RF mirrors (parasitic elements) to convey additional information bits through indexing of these mirrors, in massive MIMO systems. Our results demonstrated significant performance advantages possible in multiuser MBM compared to multiuser schemes that employ conventional modulation, spatial modulation, and generalized spatial modulation. Motivated by the possibility of exploiting the inherent sparsity in multiuser MBM signal vectors, we proposed a detection scheme based on compressive sensing algorithms like OMP, CoSaMP, and subspace pursuit. The proposed detection scheme was shown to achieve very good performance (e.g., significantly better performance compared to MMSE detection) at low complexity, making it suited for multiuser MBM signal detection in massive MIMO systems. Channel estimation, effect of imperfect knowledge of the channel alphabet at the receiver, and effect of spatial correlation are interesting topics for further investigation.
[1]{} A. K. Khandani, “Media-based modulation: a new approach to wireless transmission,” [*Proc. IEEE ISIT’2013*]{}, pp. 3050-3054, Jul. 2013.
A. K. Khandani, “Media-based modulation: converting static Rayleigh fading to AWGN,” [*Proc. IEEE ISIT’2014*]{}, pp. 1549-1553, Jun.-Jul. 2014.
E. Seifi, M. Atamanesh, and A. K. Khandani, “Media-based modulation: a new frontier in wireless communications,” online arXiv:1507.07516v3 \[cs.IT\] 7 Oct. 2015.
Y. Naresh and A. Chockalingam, “On media-based modulation using RF mirrors,” [*Proc. ITA’2016*]{}, Feb. 2016. Online arXiv:1601.06978v2 \[cs.IT\] 22 Oct 2016. Accepted in [*IEEE Trans. Veh. Tech.*]{}
E. Seifi, M. Atamanesh, A. K. Khandani, “Media-based MIMO: outperforming known limits in wireless,” [*Proc. IEEE ICC’2016*]{}, pp. 1-7, May 2016.
O. N. Alrabadi, A. Kalis, C. B. Papadias, R. Prasad, “Aerial modulation for high order PSK transmission schemes,” [*Proc. Wireless VITAE 2009*]{}, pp. 823-826, May 2009.
O. N. Alrabadi, A. Kalis, C. B. Papadias, and R. Prasad, “A universal encoding scheme for MIMO transmission using a single active element for PSK modulation schemes,” [*IEEE Trans. Wireless Commun.*]{}, vol. 8, no. 10, pp. 5133-5142, Oct. 2009.
M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, “Spatial modulation for generalized MIMO: challenges, opportunities and implementation,” [*Proc. of the IEEE*]{}, vol. 102, no. 1, pp. 56-103, Jan. 2014.
A. Chockalingam and B. S. Rajan, [*Large MIMO Systems*]{}, Cambridge Univ. Press, Feb. 2014.
J. Wang, S. Jia, and J. Song, “Generalised spatial modulation system with multiple active transmit antennas and low complexity detection scheme,” [*IEEE Trans. Wireless Commun*]{}., vol. 11, no. 4, pp. 1605-1615, Apr. 2012.
T. Datta and A. Chockalingam, “On generalized spatial modulation,” [*Proc. IEEE WCNC’2013*]{}, pp. 2716-2721, Apr. 2013.
T. L. Narasimhan, P. Raviteja, and A. Chockalingam. “Generalized spatial modulation in large-scale multiuser MIMO systems,” [ *IEEE Trans. Wireless Commun.*]{}, vol. 14, no. 7, pp. 3764-3779, Jul. 2015.
J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” [ *IEEE Trans. Inform. Theory*]{}, vol. 53, vol. 12, pp. 4655-4666, Dec. 2007.
D. Needell and J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” [ *Applied and Computational Harmonic Analysis*]{} 26.3 (2009): 301-321.
W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing signal reconstruction,” [ *IEEE Trans. Inform. Theory*]{}, vol. 55, no. 5, pp. 2230-2249, May 2009.
N. B. Karahanoglu, and H. Erdogan, “A\* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” [*Digital Signal Processing*]{}, 22(4), pp. 555-568, 2012.
D. Donoho, Y. Tsaig, I. Drori, and J.-L. Starck, “Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit,” [*IEEE Trans. Inform. Theory*]{}, vol. 58, no. 2, pp. 1094-1121, Feb. 2012.
S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” [*SIAM review*]{}, 43(1), pp. 129-159, 2001.
B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle regression,” [*Annals of statistics,*]{} vol. 32, no. 2, pp. 407-451, 2004.
R. Tibshirani, “Regression shrinkage and selection via the lasso,” [*J. Royal Statistical Society. Series B (Methodological)*]{}, vol. 8, no. 1, pp. 267-288, 1996.
D. Needell and R. Vershynin, “Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit,” [*Foundations of computational mathematics*]{}, 9(3), pp. 317-334, 2009.
M. S. Alouini and G. Goldsmith, “A unified approach for calculating error rates of of linearly modulated signals over generalized fading channels,” [*IEEE Trans. Commun.,*]{} vol. 47, no. 9, pp. 1324-1334, Sep. 1999.
M. Elad, [*Sparse and Redundant Representations: From Theory to Applications in Signal Processing*]{}, Springer New York, 2010.
|
---
abstract: 'Sur un corps fini ${{\mathbb F}}$, toute ${{\mathbb F}}$-surface stablement ${{\mathbb F}}$-rationnelle est ${{\mathbb F}}$-rationnelle. Plus généralement, si une ${{\mathbb F}}$-surface $X$ projective et lisse et géométriquement rationnelle n’est pas ${{\mathbb F}}$-rationnelle, alors il existe une extension finie ${{\mathbb F}}''$ de ${{\mathbb F}}$ avec ${{\operatorname{Br }}}(X_{{{\mathbb F}}''}) \neq 0$. Ceci vaut plus généralement pour une telle surface $X$ sur un corps $k$ quasi-fini dès que $X(k) \neq \emptyset$.'
address: |
Université Paris Sud\
Mathématiques, Bâtiment 425\
91405 Orsay Cedex\
France
author:
- 'J.-L. Colliot-Thélène'
date: 'soumis le 27 novembre 2017; révisé le 14 juin 2018'
title: 'Surfaces stablement rationnelles sur un corps quasi-fini'
---
Introduction
============
L’énoncé suivant est essentiellement connu (cf. [@CTSRequiv §1, §2], [@CTSDesc §2.A]).
\[implgen\] Soit $k$ un corps et ${{\overline k}}/k$ une clôture séparable de $k$. Soit $X$ une $k$-variété projective, lisse. Soit ${{\overline X}}= X \times_{k} {{\overline k}}$. Supposons que ${{\overline X}}$ est ${{\overline k}}$-rationnelle, i.e ${{\overline k}}$-birationnelle à un espace projectif. Considérons les conditions suivantes.
\(i) La $k$-variété $X$ est $k$-rationnelle.
\(ii) La $k$-variété $X$ est stablement $k$-rationnelle.
\(iii) La $k$-variété $X$ est facteur direct d’une variété $k$-rationnelle, c’est-à-dire qu’il existe une $k$-variété $Y$ projective et lisse, géométriquement connexe, telle que $X \times_{k} Y$ est $k$-birationnelle à un espace projectif.
\(iv) Le module galoisien ${\operatorname{Pic}}(\X)$ est stablement de permutation, c’est-à-dire qu’il existe des modules de permutation de type fini $P_{1}$ et $P_{2}$ et un isomorphisme de modules galoisiens ${\operatorname{Pic}}(\X) \oplus P_{1} \simeq P_{2}$.
\(v) Le module galoisien ${\operatorname{Pic}}(\X)$ est un facteur direct d’un module de permutation, c’est-à-dire qu’il existe un module galoisien $M$, un module de de permutation de type fini $P$ et un isomorphisme de modules galoisiens ${\operatorname{Pic}}(\X) \oplus M \simeq P$.
\(vi) Pour toute extension finie séparable $k'/k$, on a $H^1(k', {\operatorname{Pic}}(\X))=0$.
\(vii) Pour toute extension finie séparable $k'/k$, l’application naturelle de groupes de Brauer ${{\operatorname{Br }}}(k') \to {{\operatorname{Br }}}(X_{k'})$ est surjective.
Alors : (i) implique (ii), qui implique (iii); (ii) implique (iv); (iii) implique (v); (iv) implique (v); (v) implique (vi); (vi) implique (vii).
Pour l’implication (i) implique (iv), voir [@CTSDesc Prop. 2A1, p. 461]. Le fait que (ii) implique (iv) et que (iii) implique (v) résulte alors du calcul du groupe de Picard d’un produit [@CTSRequiv Lemme 11]. L’implication (v) implique (vi) résulte du fait que pour tout module $P$ galoisien de permutation et toute extension finie séparable $k'/k$, on a $H^1(k',P)=0$. Pour $X$ une $k$-variété projective, lisse, géométriquement intègre, et toute extension finie séparable $k'/k$, on a la suite exacte (cf. [@CTSDesc (1.,5.0), p. 386]) : $${{\operatorname{Br }}}(k') \to {{\operatorname{Ker}}}[{{\operatorname{Br }}}(X _{k'}) \to {{\operatorname{Br }}}(\X)] \to H^1(k', {\operatorname{Pic}}(\X))).$$ Si de plus ${{\overline X}}$ est ${{\overline k}}$-rationnelle, alors ${{\operatorname{Br }}}(\X)=0$. Cette annulation est bien connue en caractéristique zéro, et l’argument vaut encore pour la $\ell$-torsion de ${{\operatorname{Br }}}(\X)$ pour $\ell$ un nombre premier premier à l’exposant caractéristique de $k$. Que ${{\operatorname{Br }}}(\X)=0$ vaut pour tout corps séparablement clos ${{\overline k}}$ et $\X$ ${{\overline k}}$-rationnelle se voit en combinant [@GrBr3 Cor. 5.8] et [@CTBarbara Prop. 2.1.9]. Ainsi (vi) implique (vii).
On dit qu’une $k$-surface projective, lisse, géométriquement rationnelle, est déployée par une sous-extension galoisienne $K \subset {{\overline k}}$ si $X(K) \neq \emptyset$ et si l’inclusion naturelle de réseaux ${\operatorname{Pic}}(X_{K}) \to {\operatorname{Pic}}({{\overline X}})$ est un isomorphisme. Sous l’hypothèse $X(k) \neq \emptyset$, ceci équivaut au fait que ${\rm Gal}({{\overline k}}/K)$ agit trivialement sur le réseau ${\operatorname{Pic}}({{\overline X}})$.
\[vraiprincipal\] Soient $k$ un corps et $X$ une $k$-surface projective, lisse, géométriquement rationnelle. Supposons que $X$ possède un point $k$-rationnel et que $X$ soit déployée par une extension cyclique de $k$. Si $X$ n’est pas $k$-rationnelle, alors il existe une extension finie séparable $k'/k$ telle que $H^1(k', {\operatorname{Pic}}(\X))\neq 0$, et alors $X$ n’est pas stablement $k$-rationnelle.
La démonstration sera donnée au §4 (théorème \[vraiprincipalbis\]), où l’on regroupe les résultats du §2 (surfaces fibrées en coniques) et du §3 (surfaces de del Pezzo). C’est une démonstration cas par cas, qui s’appuie de façon essentielle sur les tables établies par divers auteurs sur l’action du groupe de Galois sur le groupe de Picard des surfaces de del Pezzo de degré 3,2,1.
L’énoncé du théorème \[vraiprincipal\] est à comparer avec les exemples donnés dans [@BCTSaSD]. Si $k$ est un corps de caractéristique différente de 2, et $P(x) \in k[x]$ un polynôme séparable et irréductible de degré 3, de discriminant $a \in k^{*}$ non carré, on montre que la $k$-surface d’équation affine $ y^2-az^2=P(x)$ est stablement $k$-rationnelle mais non $k$-rationnelle. Il existe donc de telles surfaces sur tout corps $ k$ de caractéristique différente de 2 possédant une extension de corps galoisienne de groupe $\frak{S}_{3}$, par exemple le corps des rationnels ou le corps $F={{\mathbb C}}(t)$ des fractions rationnelles en une variable sur les complexes.
Par définition, un corps quasi-fini est un corps parfait dont la clôture galoisienne est le groupe procyclique $\hat{{{\mathbb Z}}}$ [@serre Chap. XIII, §2]. Les deux types d’exemples classiques sont : les corps finis et les corps de séries formelles d’une variable sur un corps algébriquement clos de caractéristique zéro.
Si $k$ est un corps fini, ou un corps de séries formelles d’une variable ${{\mathbb C}}((t))$ sur un corps algébriquement clos ${{\mathbb C}}$ de caractéristique zéro, toute $k$-surface projective, lisse, géométriquement rationnelle possède un $k$-point (Proposition \[C1\]), et toute extension finie de corps $K/k$ est cyclique. Toute conique projective et lisse sur $k$ est $k$-isomorphe à ${{\mathbf P}}^1_{k}$.
Le théorème \[vraiprincipal\] et le théorème \[implgen\] donnent alors l’énoncé suivant, qui pour un corps fini répond à une question de B. Hassett mentionnée par A. Pirutka dans [@P].
\[principal\] Soient $k$ un corps et $X$ une $k$-surface projective, lisse, géométriquement rationnelle. Sous l’une des hypothèses suivantes :
\(i) $X$ possède un $k$-point et le corps $k$ est quasi-fini;
\(ii) le corps $k$ est un corps fini;
\(iii) le corps $k={{\mathbb C}}((t))$ est le corps des séries formelles en une variable sur un corps ${{\mathbb C}}$ algébriquement clos de caractéristique zéro.
on a :
\(a) Si pour toute extension finie de corps $k'/k$ on a $H^1(k', {\operatorname{Pic}}(\X))=0$, alors $X$ est $k$-rationnelle.
\(b) Les conditions (i) à (vii) du théorème \[implgen\] sont équivalentes pour $X$. En particulier, sur un tel corps $k$, toute $k$-surface stablement $k$-rationnelle est $k$-rationnelle.
Soient $k$ un corps et $X$ une $k$-surface projective, lisse, géométriquement rationnelle. Sous l’une des hypothèses suivantes :
\(i) $X$ possède un $k$-point et le corps $k$ est quasi-fini;
\(ii) le corps $k$ est un corps fini;
\(iii) le corps $k={{\mathbb C}}((t))$ est le corps des séries formelles en une variable sur un corps ${{\mathbb C}}$ algébriquement clos de caractéristique zéro.
Sous l’une quelconque des trois hypothèses suivantes :
\(a) le pgcd des degrés des extensions finies $K/k$ telles que $X_{K}$ est stablement $K$-rationnelle est égal à 1.
\(b) $X$ est $k$-unirationnelle, et le pgcd des degrés des $k$-applications rationnelles dominantes génériquement séparables de ${{\mathbf P}}^2_{k}$ vers $X$ est égal à 1,
\(c) le groupe de Chow des zéro-cycles de $X$ est universellement trivial,
la surface $X$ est $k$-rationnelle.
On note $CH_{0}(X)$ le groupe de Chow des classes de zéro-cycles sur $X$, et $A_{0}(X)$ le sous-groupe des classes de zéro-cycles de degré zéro. Ces groupes sont des invariants $k$-birationnels des variétés projectives, lisses, géométriquement intègres. Ceci est établi dans [@CTCoray Prop. 6.3] avec quelques restrictions, par exemple en caractéristique zéro. La démonstration de [@Fulton Example 16.1.11] vaut sur un corps quelconque.
L’hypothèse (c) est que pour toute extension de corps $F/k$ l’application degré $deg_{F} : CH_{0}(X_{F}) \to {{\mathbb Z}}$ est un isomorphisme, ce qui est le cas pour $X$ projective, lisse, intègre, stablement $k$-rationnelle.
En caractéristique zéro, d’après [@CTCoray Prop. 6.4], chacune des hypothèses (a) et (b) sur la surface $X$ implique (c). Via les correspondances [@Fulton Chap. 16] on voit que ceci vaut sur un corps quelconque.
Sous l’hypothèse de (c), le module galoisien $ {\operatorname{Pic}}(\X)$ est un facteur direct d’un module de permutation. Pour la démonstration, je renvoie à [@Gille Appendix A]. Pour toute extension de corps $E/k$, on a donc $H^1(E, {\operatorname{Pic}}(\X))=0$. Le théorème \[principal\] permet de conclure.
Rappelons la classification $k$-birationnelle des $k$-surfaces projectives, lisses, géométriquement rationnelles, due à Enriques, Manin, Iskovskikh [@Isk79], et Mori.
\[classif\] Soit $k$ un corps. Toute $k$-surface projective, lisse, géométriquement rationnelle, $k$-minimale, appartient à au moins un des types suivants :
\(i) Surface fibrée en coniques relativement minimale au-dessus d’une conique lisse.
\(ii) Surface de del Pezzo de degré $d$ avec $1 \leq d \leq 9$.
Comme observé par Manin et l’auteur (cf. [@kollar Thm. IV.6.8]), ceci permet de démontrer le résultat suivant, qui pour $k$ un corps fini, admet une démonstration uniforme (A. Weil, cf. [@M Thm. 27.1, Cor. 27.1.1]).
\[C1\] Soient $k$ un corps et $X$ une $k$-surface projective et lisse géométriquement rationnelle. Si $k$ est un corps $C_{1}$, alors $X$ possède un point $k$-rationnel. Ceci vaut en particulier pour $k$ un corps fini et pour $k={{\mathbb C}}((t))$.
Fibrés en coniques
==================
Soient $k$ un corps, ${{\overline k}}$ une clôture séparable de $k$ et $g={\rm Gal}(\k/k)$. Si $X$ est une $k$-surface projective lisse géométriquement connexe munie d’un $k$-morphisme $f :X \to {{\mathbf P}}^1_{k}$ relativement minimal dont la fibre générique est une courbe lisse de genre zéro, alors les points fermés $M$ dont la fibre $X_{M}/k(M)$ est non lisse ont leur corps résiduel $k(M)$ séparable sur $k$, et $X_{M}/k(M)$ se décompose sur une extension quadratique séparable de corps $L(M)/k(M)$ en deux droites ${{\mathbf P}}^1_{L(M)}$ qui se coupent transversalement en un $k(M)$-point. On appelle ces points fermés $M \in {{\mathbf P}}^1_{k}$ les points de mauvaise réduction de la fibration. On renvoie à [@Isk79] pour la démonstration de ces faits. Ils impliquent que sur une clôture séparable $\k$ de $k$, il existe une contraction ${{\overline X}}\to Y$ au-dessus de ${{\mathbf P}}^1_{\k}$ telle que les fibres de $Y \to {{\mathbf P}}^1_{\k}$ soient toutes isomorphes à ${{\mathbf P}}^1$. Ceci définit donc un élément de ${{\operatorname{Br }}}({{\mathbf P}}^1_{\k})={{\operatorname{Br }}}(\k)=0$. L’égalité $ {{\operatorname{Br }}}({{\mathbf P}}^1_{\k})=0 $ résulte du théorème de Tsen lorsque $\k$ est algébriquement clos, et ce même théorème implique que $ {{\operatorname{Br }}}({{\mathbf P}}^1_{\k})$ est $p$-primaire pour $\k$ séparablement clos de caractéristique $p>0$. Que l’on ait $ {{\operatorname{Br }}}({{\mathbf P}}^1_{\k})=0 $ pour $\k$ séparablement clos quelconque est établi par Grothendieck [@GrBr3 Cor. 5.8]. La fibre générique de $\overline{f} : \X \to {{\mathbf P}}^1_{\k}$ admet donc un point rationnel. Comme ${{\mathbf P}}^1_{\k}$ est régulier de dimension 1 et $\overline{f}$ est un morphisme propre, tout tel point rationnel s’étend en une section de $\overline{f} : \X \to {{\mathbf P}}^1_{\k}$. La $k$-variété $X$ est déployée sur ${{\overline k}}$. La fibre générique de la fibration $f : X \to {{\mathbf P}}^1_{k}$ correspond à un élément $\beta$ de $H^2(g, \k({{\mathbf P}}^1)^*)$. En un point fermé $M$ de mauvaise réduction, la flèche diviseur définit une application $g$-équivariante $$\k({{\mathbf P}}^1)^* \to \oplus_{k(M) \subset \k} \ {{\mathbb Z}},$$ où $k(M) \subset \k$ parcourt les $k$-plongements de l’extension séparable $k(M)/k$ dans $\k$. Cet homomorphisme induit une flèche résidu $$\partial_{M} : H^2(g, \k({{\mathbf P}}^1)^*) \to H^2(g, \oplus_{k(M) \subset \k} \ {{\mathbb Z}}) = H^2(k(M), {{\mathbb Z}}) = H^1(k(M), {{\mathbb Q}}/{{\mathbb Z}}).$$ L’image de $\beta$ par cette application décrit l’extension quadratique séparable de $k(M)$ correspondant à la mauvaise fibre. Pour $k \subset L \subset \k$, avec $L/k$ finie, on a le diagramme commutatif suivant : $$\begin{array}{ccccccccc}
H^2(g_{k}, \k({{\mathbf P}}^1)^*) & \to & H^1(k(M), {{\mathbb Q}}/{{\mathbb Z}})\\
\downarrow&&\downarrow& \\
H^2(g_{L}, \k({{\mathbf P}}^1)^*) & \to &\oplus_{N \to M} H^1(k(N),{{\mathbb Q}}/{{\mathbb Z}}),
\end{array}$$ où $N$ parcourt les points fermés de ${{\mathbf P}}^1_{L}$ d’image $M$ via la projection ${{\mathbf P}}^1_{L} \to {{\mathbf P}}^1_{k}$, où les flèches horizontales sont les flèches de résidu définies ci-dessus et les flèches verticales sont les flèches de restriction.
\[utile\] Soit $f: X \to {{\mathbf P}}^1_{k}$ un fibré en coniques relativement minimal sur un corps $k$, à fibre générique lisse et espace total lisse sur $k$. Soit $M \in {{\mathbf P}}^1_{k}$ un point fermé de corps résiduel $k(M)$ où la fibration a mauvaise réduction. Soit $K=k(M)$ le corps résiduel en $M$. Soit $N$ un $K$-point of ${{\mathbf P}}^1_{K}$ au-dessus de $M \in {{\mathbf P}}^1_{k}$. La fibration $f_{K}: X_{K}\to {{\mathbf P}}^1_{K}$ a mauvaise réduction en $N$.
Soit $A/k({{\mathbf P}}^1)$ l’algèbre de quaternions associée à la fibre générique de $f : X \to {{\mathbf P}}^1_{k}$. La fibration $f$ a mauvaise réduction en $M$ si et seulement si le résidu $\gamma:=\partial_{M}(A) \in H^1(k(M),{{\mathbb Z}}/2)=H^1(K,{{\mathbb Z}}/2)$ est non trivial. D’après la compatibilité ci-dessus, on a $$\partial_{N}(A_{K})=\gamma \in H^1(K(N),{{\mathbb Z}}/2)=H^1(K,{{\mathbb Z}}/2).$$ ce qui établit le lemme.
L’énoncé suivant fut établi par Iskovskikh ([@Isk70 Thm. 2], [@Isk79 Thm. 4, Thm. 5]).
\[auplus3\] Soient $k$ un corps et $X/k$ une surface projective, lisse, géométriquement connexe sur $k$, munie d’une structure de fibré en coniques relativement minimale $X \to {{\mathbf P}}^1_{k}$. Si le nombre $r$ de fibres géométriques dégénérées est au plus égal à 3, et si $X$ possède un point $k$-rationnel, alors $X$ est une surface $k$-rationnelle.
\[fibreconiques\] Soient $k$ un corps et $X/k$ une surface projective, lisse, géométriquement connexe sur $k$, munie d’une structure de fibré en coniques relativement minimale $X \to {{\mathbf P}}^1_{k}$. Supposons $X$ déployée sur une extension cyclique de corps $K/k$. Si le nombre $r$ de fibres géométriques dégénérées de la fibration est au moins égal à 4, alors il existe une extension finie séparable $k'/k$ telle que $H^1(k',{\operatorname{Pic}}({{\overline X}})) \neq 0$.
Donnons ici quelques rappels sur le module de Picard d’une surface fibrée en coniques sur ${{\mathbf P}}^1_{k}$. Pour plus de détails, je renvoie le lecteur à [@CTSDMJ81 §2].
Soit comme ci-dessus $\k$ une clôture séparable de $k$ et ${{\overline X}}=X \times_{k}\k$. On a une suite exacte de modules galoisiens $$0 \to P \to {{\mathbb Z}}.f \oplus Q \to {\operatorname{Pic}}(\X) \to {{\mathbb Z}}\to 0,$$ où $P$ est le module de permutation sur les $\k$-points of ${{\mathbf P}}^1$ à fibre singulière, $Q$ est le module de permutation sur les composantes des fibres singulières sur $\k$, et ${{\mathbb Z}}.f$ est engendré par une fibre au-dessus d’un $k$-point de ${{\mathbf P}}^1_{k}$. L’application ${\operatorname{Pic}}(\X) \to {{\mathbb Z}}$ est induite par la restriction à la fibre générique.
Soit $M$ le noyau de cette application restriction. On a des suites exactes courtes de modules galoisiens $$0 \to P \to {{\mathbb Z}}\oplus Q \to M \to 0$$ et $$0 \to M \to {\operatorname{Pic}}(\X) \to {{\mathbb Z}}\to 0.$$ Par cohomologie galoisienne on obtient des suites exactes $$0 \to {{\mathbb Z}}/2 \to H^1(k,M) \to H^1(k, {\operatorname{Pic}}(\X)) \to 0$$ et $$0 \to H^1(k,M) \to H^2(k,P) \to H^2(k, {{\mathbb Z}}\oplus Q).$$ Cette dernière donne naissance à une suite exacte $$0 \to H^1(k,M) \to \oplus_{i=1}^r {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2),$$ où $i$ parcourt les $r\geq 1$ points fermés $P_{i}$ de ${{\mathbf P}}^1$ à fibre singulière, déployée par une extension quadratique séparable de corps, de classe $a_{i} \in H^1(k(P_{i}),{{\mathbb Z}}/2)$, et l’application $\theta_{i} : {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2)$ envoie $1$ sur la norme (de $k(P_{i})$ à $k$) de $a_{i} \in H^1(k(P_{i}),{{\mathbb Z}}/2)$.
On a en outre une relation de réciprocité [@CTSDMJ81 §2, Remark] qui implique ici que l’image de l’élément $(1,\dots,1) \in \oplus_{i} H^1(k(P_{i}),{{\mathbb Z}}/2)$ est la classe triviale dans $H^1(k,{{\mathbb Z}}/2)$.
Nous voulons montrer : si le nombre de fibres géométriques dégénérées de $X \to {{\mathbf P}}^1_{k}$ est supérieur ou égal à 4, alors il existe une extension finie séparable $k'/k$ telle que $H^1(k',{\operatorname{Pic}}({{\overline X}})) \neq 0$.
Si $E/k$ est une extension séparable de corps de degré impair, par passage à $E$, la famille $X_{E} \to {{\mathbf P}}^1_{E}$ reste relativement minimale : aucun résidu n’est annulé. On peut donc supposer que tous les points fermés de mauvaise réduction sont soit de degré 1 soit de degré pair.
Supposons qu’il existe un point fermé de mauvaise réduction $P$ de degré pair au moins égal à 4. Par le lemme \[utile\], on se ramène après extension de $k$ à $k(P)$ à la situation où il y a au moins 4 points rationnels $P_{1},P_{2},P_{3},P_{4}$ à fibre singulière. Comme la surface est par hypothèse déployée par une extension cyclique, les classes $a_{i} \in H^1(k(P_{i}),{{\mathbb Z}}/2)=H^1(k,{{\mathbb Z}}/2)$, qui sont non triviales, coïncident toutes avec une même classe non triviale $a \in H^1(k,{{\mathbb Z}}/2)$.
L’application $\oplus_{i=1}^r {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2)$ induit une application $({{\mathbb Z}}/2)^4 \to H^1(k, {{\mathbb Z}}/2)$ qui se factorise donc par $({{\mathbb Z}}/2)^4 \to {{\mathbb Z}}/2$. Le groupe $H^1(k,M)= Ker [\oplus_{i=1}^r {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2)]$ contient donc au moins le noyau d’une application $({{\mathbb Z}}/2)^4 \to {{\mathbb Z}}/2$, il est donc d’ordre au moins 8, et $H^1(k, {\operatorname{Pic}}({{\overline X}}))$ est donc d’ordre au moins 4.
Supposons désormais que les points fermés $P_{i}$ de mauvaise réduction sont tous de degré 2 ou 1.
S’il y a au moins 4 points fermés $P_{i}$ de mauvaise réduction de degré 1 sur $k$, l’argument ci-dessus permet de conclure.
Supposons qu’il y a au moins deux points fermés $P_{1}, P_{2}$ de degré 2. Comme la surface est par hypothèse déployée sur une extension cyclique, ceci impose que les extensions $k(P_{i})/k$ coïncident en ces deux points. Soit donc $L/k$ l’extension quadratique séparable de corps ainsi définie. En appliquant le lemme \[utile\], on se ramène après extension de $k$ à $L$ au cas où la fibration $X \to {{\mathbf P}}^1_{k}$ a des fibres singulières au-dessus d’au moins 4 points $k$-rationnels, cas qui a déjà été réglé.
On peut donc supposer que l’ensemble des degrés des points fermés à mauvaise réduction est soit $(2,1,1,1)$ soit $(2,1,1)$.
Considérons le cas $(2,1,1,1)$. Comme la surface est par hypothèse déployée sur une extension cyclique, ceci impose que les extensions quadratiques associées à $a_{i} \in H^1(k(P_{i}), {{\mathbb Z}}/2)$ pour $P_{i}$ $k$-rationnel coïncident avec une même classe $a \in H^1(k,{{\mathbb Z}}/2)$. Soit $R$ le point fermé de degré 2, et soit $b \in H^1(k(R),{{\mathbb Z}}/2)$ le résidu en ce point. L’application $\oplus_{i=1}^3 {{\mathbb Z}}/2 \oplus {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2)$ envoie $(x,y,z,t)$ sur $(x+y+z).a+ t.Norm_{k(R)/k}(b)\in H^1(k,{{\mathbb Z}}/2)$. On sait que la classe $(1,1,1,1)$ a une image triviale. Donc $3a+Norm_{k(R)/k}(b)$ est trivial dans $H^1(k,{{\mathbb Z}}/2)$. Ceci implique $a=Norm_{k(R)/k}(b)$, et cet élément est non trivial dans $H^1(k,{{\mathbb Z}}/2)$. L’application $\oplus_{i=1}^3 {{\mathbb Z}}/2 \oplus {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2)$ envoie donc $(x,y,z,t)$ sur $(x+y+z+t).a $ dans $H^1(k,{{\mathbb Z}}/2)$. C’est donc simplement la somme $({{\mathbb Z}}/2)^4 \to {{\mathbb Z}}/2$. Son noyau est $({{\mathbb Z}}/2)^3$, on a donc $H^1(k, {\operatorname{Pic}}({{\overline X}})) =({{\mathbb Z}}/2)^2$.
Montrons pour finir que le cas $(2,1,1)$ n’existe pas. Comme ci-dessus, les extensions quadratiques associées à $a_{i} \in H^1(k(P_{i}), {{\mathbb Z}}/2)$ pour $P_{i}$ $k$-rationnel coïncident avec une même classe non triviale $a \in H^1(k,{{\mathbb Z}}/2)$. Soit $Q$ le point fermé de degré 2, et soit $b \in H^1(k(Q),{{\mathbb Z}}/2)$ le résidu en ce point. Ceci correspond à une extension quadratique séparable $L/k(Q)$. Sous nos hypothèses, l’extension $L/k$ est cyclique de groupe de Galois ${{\mathbb Z}}/4$. Sous cette hypothèse, on vérifie que la norme $H^1(k(Q), {{\mathbb Z}}/2) \to H^1(k,{{\mathbb Z}}/2)$ envoie la classe $b$ sur la classe $a$. Par ailleurs on sait (réciprocité) que la classe $(1,1,1)$ a une image triviale par l’application $\oplus_{i=1}^2 {{\mathbb Z}}/2 \oplus {{\mathbb Z}}/2 \to H^1(k,{{\mathbb Z}}/2)$. Mais ceci dit que $3a =a \in H^1(k,{{\mathbb Z}}/2)$ est nul. Contradiction.
Surfaces de del Pezzo
=====================
On a l’énoncé connu (Châtelet, Manin, Iskovskikh, voir [@VA Thm. 2.1]):
\[granddp\] Soit $X$ une surface de del Pezzo de degré $d \geq 5$ sur un corps $k$. Si $X$ possède un point $k$-rationnel, alors $X$ est $k$-rationnelle.
\[dp4\] Soient $k$ un corps et $X \subset {{\mathbf P}}^4_{k}$ une surface de del Pezzo de degré $4$ sur $k$ déployée par une extension cyclique $K/k$. Supposons que $X$ est $k$-minimale. Alors :
\(i) Il existe une extension de corps $E/k$ telle que $H^1(E, {\operatorname{Pic}}(\X)) \neq 0$.
\(ii) Le module galoisien ${\operatorname{Pic}}(\X)$ n’est pas facteur direct d’un module de permutation.
\(iii) Si $X$ possède un $k$-point, il existe une extension finie séparable $k'/k$ telle que $H^1(k', {\operatorname{Pic}}(\X)) \neq 0$.
Soit $X$ une surface de del Pezzo de degré 4 sur un corps $k$, possédant un $k$-point, déployée sur une extension cyclique de $k$. Supposons $X$ $k$-minimale. D’après [@Isk72 Thm. 2], la surface $X$ n’est pas $k$-rationnelle.
Si $X$ possède un $k$-point $P$ non situé sur les 16 droites (géométriques) exceptionnelles, en éclatant le point $P$ on obtient une surface cubique lisse $Y$ sur $k$ équipée d’une fibration en coniques sur $Y \to {{\mathbf P}}^1_{k}$, avec 5 fibres géométriques dégénérées, déployée sur une extension cyclique de $k$. Si cette fibration n’est pas relativement minimale, soit $Z \to {{\mathbf P}}^1_{k}$ un modèle relativement minimal. Soit $r$ le nombre de fibres géométriques dégénérées. Si $r \leq 3$, alors d’après la Proposition \[auplus3\], $Y$ est $k$-rationnelle, ce qui est exclu. Ainsi on a $r\geq 4$. La Proposition \[fibreconiques\] donne alors l’existence d’une extension finie séparable $k'/k$ telle que $H^1(k', {\operatorname{Pic}}(\overline{Z})) \neq 0$. Comme on passe de $X$ à $Z$ par des séries d’éclatements en des points fermés séparables, les modules galoisiens ${\operatorname{Pic}}(\overline{X})$ et ${\operatorname{Pic}}(\overline{Z})$ sont isomorphes à addition de modules de permutation près. On a donc $H^1(k', {\operatorname{Pic}}(\overline{X})) \neq 0$.
Si $X$ possède un $k$-point, et le corps $k$ possède au moins 23 éléments, alors il existe un $k$-point sur $X$ hors des 16 droites [@M Chap. 4, §8, Teor. 8.1= Thm. 30.1]. Supposons que $k$ est fini et $X$ est déployée sur l’extension cyclique $K/k$. Il existe une extension finie $L/k$ linéairement disjointe de $K$ sur laquelle $X$ possède un point $L$-rationnel non situé hors des 16 droites. La $L$-surface de del Pezzo de degré 4 $X\times_{k}L$ est $L$-minimale et déployée par l’extension cyclique $K.L/L$. L’argument ci-dessus donne alors une extension finie $k'$ de $L$ telle que $H^1(k', {\operatorname{Pic}}(\overline{X})) \neq 0$.
Ceci établit le point (iii).
Pour établir (i), on utilise l’astuce du passage au point générique (voir [@CTSDesc Thm. 2.B.1]). Soit $F=k(X)$ le corps des fonctions de $X$. La $F$-variété $X_{F}=X\times_{k}F$ possède un $F$-point. Elle est $F$-minimale, car $k$ est algébriquement clos dans $F=k(X)$. Le module galoisien ${\operatorname{Pic}}(\overline{X})$ ne change pas par passage du corps de base de $k$ à $F$, il est déployé par l’extension $k'/k$ comme par l’extension $F'/F$, où $F':=F.K$. Comme le corps $F$ est infini, il existe un $F$-point de $X$ non situé sur les droites (géométriques) de $X$. La $F$-surface minimale $X_{F}$ est déployée par l’extension cyclique $F'/F$. Par le point (iii), il existe une extension finie séparable $E/F$ telle que $H^1(E, {\operatorname{Pic}}(\overline{X})) \neq 0$, ce qui donne (i).
Ceci implique que le module galoisien ${\operatorname{Pic}}(\overline{X}) $ n’est pas facteur direct d’un module de permutation (Théorème \[implgen\]), ce qui donne (ii).
On s’intéresse maintenant aux surfaces de del Pezzo de degré 3, 2 et 1 déployées par une extension cyclique $K/k$ du corps de base $k$. On note $Frob$ un générateur du groupe cyclique ${{\operatorname{Gal }}}(K/k)$. Les diverses actions du groupe cyclique ${{\operatorname{Gal }}}(K/k)$ sur le groupe ${\operatorname{Pic}}(X_{K})={\operatorname{Pic}}({{\overline X}})$ ont été classifiées par Frame [@Frame], puis Swinnerton-Dyer [@SwD], corrigées par Manin [@M Chapitre IV], corrigées et complétées par Urabe [@U] puis récemment par Banwait, Fité et Loughran [@BFL].
Dans [@M Chap. IV, Table I, Colonne 5] et dans [@BFL Table 7.1, Colonne 5], une surface a le symbole $\prod_{m}m^{n_{m}}$, avec tous les $n_{m}\geq 0$, si pour $m$ donné l’ensemble Galois invariant des racines primitives $m$-ièmes de l’unité parmi les valeurs propres de $Frob$ a $n_{m}$ éléments. En d’autres termes, on décompose le polynôme caractéristique de $Frob$ agissant sur ${\operatorname{Pic}}({{\overline X}})\otimes_{{{\mathbb Z}}}{\bf C}$ en regroupant les orbites des racines sous l’action du groupe de Galois. On appellera ce symbole le symbole caractéristique de $Frob$ (pour son action sur ${\operatorname{Pic}}({{\overline X}})\otimes_{{{\mathbb Z}}}{\bf C}$).
Urabe [@U Supplement] utilise le symbole de Frame [@Frame]. Le symbole de Frame $\prod_{m}m^{n_{m} }$, avec $n_{m} \in {{\mathbb Z}}$, correspond à une réécriture du polynôme caractéristique de $Frob$ pour son action sur ${\operatorname{Pic}}({{\overline X}})\otimes_{{{\mathbb Z}}}{\bf C}$ comme un produit $\prod_{m}(t^m-1)^{n_{m}}$, avec $n_{m} \in {{\mathbb Z}}$. Il y a une unique façon d’écrire le polynôme caractéristique comme un tel produit (avec des entiers $m>0$ distincts non nuls, et des $n_{m}$ non nuls). Soit $r>1$. Pour calculer le symbole de Frame de $Frob^{r}$, dans le produit $\prod_{m}(t^m-1)^{n_{m}}$ attaché au symbole de Frame $\prod_{m}m^{n_{m} }$ de $Frob$, pour chaque entier $m$ on écrit $r=uv$ et $m=uw$ ($u, v, w$ entiers positifs) avec $(v,w)=1$, et on remplace $(t^m-1)$ par $(t^w-1)^u$, puis on regroupe les termes.
Dans les tables de [@U] et [@BFL], les symboles (d’un type ou de l’autre type) associés à des surfaces différentes peuvent coïncider mais c’est rare.
La proposition suivante m’a été indiquée par K. Shramov.
(A. Trepalin) \[dp3\] Soit $X$ une surface cubique lisse sur un corps $k$, déployée par une extension cyclique $K/k$. Supposons que $X$ est $k$-minimale. Il existe alors une extension finie séparable de corps $k'/k$ telle que $H^1(k', {\operatorname{Pic}}(\X)) \neq 0.$
Nous utilisons ici la table 7.1 de l’article [@BFL]. Les actions correspondant à des surfaces $k$-minimales, c’est-à-dire d’indice $i(X)=0$, sont celles numérotées 1, 2, 3, 4, 5 dans la table 7.1 de [@BFL]. Pour les numéros 3 et 5, on a donc $H^1(k, {\operatorname{Pic}}(\overline{X})) \neq 0$. Pour les autres, on a $H^1(k,{\operatorname{Pic}}(\overline{X})) = 0$.
Pour le numéro 1, les valeurs propres de $Frob$ sont $1, 3^2, 12^4$, c’est-à-dire $1$, les deux racines cubiques primitives de l’unité et les 4 racines primitives 12-ièmes de l’unité. Si on remplace $Frob$ par $Frob^4$, c’est-à-dire si on passe à l’extension $k'/k$ de degré 4, les valeurs propres de $Frob^4$ sont $1, 3^6$. Dans la table, seul le numéro 3 a ces valeurs propres. Et à ce niveau on a $H^1(k', {\operatorname{Pic}}(\overline{X})) \neq 0$.
Pour le numéro 2, les valeurs propres de $Frob$ sont $1,3^2,6^4$. Si on remplace $Frob$ par $Frob^2$, c’est-à-dire si on passe à l’extension $k'/k$ de degré 2, les valeurs propres de $Frob^2$ sont $1, 3^6$. Dans la table, seul le numéro 3 a ces valeurs propres. Et à ce niveau on a $H^1(k', {\operatorname{Pic}}(\overline{X})) \neq 0$.
Pour le numéro 4, les valeurs propres de $Frob$ sont $1, 9^6$. Si on remplace $Frob$ par $Frob^3$, c’est-à-dire si on passe à l’extension $k'/k$ de degré 3, les valeurs propres de $Frob^3$ sont $1, 3^6$. Dans la table, seul le numéro 3 a ces valeurs propres. Et à ce niveau on a $H^1(k', {\operatorname{Pic}}(\overline{X})) \neq 0$.
\[dp2\] Soit $X$ une surface de del Pezzo de degré 2 sur un corps $k$, déployée par une extension cyclique $K/k$. Supposons que $X$ est $k$-minimale. Il existe alors une extension finie séparable de corps $k'/k$ telle que $H^1(k', {\operatorname{Pic}}(\X)) \neq 0.$
On utilise ici la table 1 de l’article [@U] de T. Urabe et les symboles de Frame.
Il suffit de discuter les surfaces numérotées de 1 à 19, qui correspondent à des surfaces $k$-minimales. Le cas 1 de la table 1, comme Daniel Loughran me l’a signalé, contient une erreur. Son indice n’est pas 0, il est au moins 2, la surface n’est pas $k$-minimale. On ne discute donc que les cas 2 à 19.
Pour les surfaces avec $H^1(k, {\operatorname{Pic}}({{\overline X}})) \neq 0$ il n’y a rien à faire.
Dans chacun des cas ci-dessous, on considère une puissance $Frob^r$ de $Frob$ et on note $k'$ le corps fixe de $Frob^r$.
Cas 5. En prenant $Frob^5$, on trouve $1^{-4}.2^6$ comme nouveau symbole de Frame. La seule possibilité est le cas 2, si $k'$ est le corps fixe de $Frob^5$, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 6. En prenant $Frob^2$, on trouve $4^2$. La seule possibilité est le cas 3, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 7. En prenant $Frob^3$, on trouve $1^{-4}.2^6$. La seule possibilité est le cas 2, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 15. En prenant $Frob^9$, on trouve $1^{-4}.2^6$. La seule possibilité est le cas 2, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 16. En prenant $Frob^7$, on trouve $1^{-6}. 2^7$. La seule possibilité est le cas 8, on a $H^1(k',{\operatorname{Pic}}(\overline{X}) ) \neq 0$.
Cas 17. En prenant $Frob^3$, on trouve $1^{-2}.2^1.4^2$. La seule possibilité est le cas 9, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 18. En prenant $Frob^3$, on trouve $1^{-1}.2^2.5^{-1}.10^1$. La seule possibilité est le cas 13, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Case 19. En prenant $Frob^3$, on trouve $1^{-6}.2^7$. La seule possibilité est le cas 8, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Dans [@L], l’auteur mentionne trois types de surfaces de del Pezzo de degré 2 dans la table 1 d’Urabe qui auraient tous leurs groupes $H^1(k',{\operatorname{Pic}}(\overline{X}))$ triviaux. Ce sont les types 1, 5 et 16. Pour le cas 1, nous avons vu que ce n’est pas une surface $k$-minimale. Pour les deux autres cas, il doit s’agir d’une erreur de calcul dans [@L].
\[dp1\] Soit $X$ une surface de del Pezzo de degré 1 sur un corps $k$, déployée par une extension cyclique $K/k$. Supposons que $X$ est $k$-minimale. Il existe alors une extension finie séparable $k'/k$ telle que $H^1(k', {\operatorname{Pic}}(\X)) \neq 0.$
On utilise ici l’article [@U] de Urabe et sa table 2. On ne considère que les surfaces d’indice 0. Si $H^1(k,{\operatorname{Pic}}(\overline{X})) \neq 0$, on a fini. Sinon, on est dans l’un des cas suivants, pour lesquels on utilise les symboles de Frame. On note $Frob$ un générateur de ${{\operatorname{Gal }}}(K/k)$. Dans chacun des cas ci-dessous, on considère une puissance $Frob^r$ de $Frob$ et on note $k'$ le corps fixe de $Frob^r$.
Cas 5. En prenant $Frob^3$, on trouve $1^1.2^{-2}.4^3$ comme nouveau symbole de Frame. La seule possibilité est le cas 3, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 6. En prenant $Frob^5$, on trouve $1^{-3}.2^4.4^1$. La seule possibilité est le cas 1, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Case 7. En prenant $Frob^3$, on trouve $1^{-1}.2^3.4^{-1}.8^1$. La seule possibilité est le cas 4, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 29. En prenant $Frob^{10}$, on trouve $1^{-3}.3^4$. La seule possibilité est le cas 9, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 30. En prenant $Frob^3$, on trouve $1^1.4^{-2}.8^2$. La seule possibilité est le cas 23, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 31. En prenant $Frob^5$, on trouve $1^1.2^{-4}.4^4$. La seule possibilité est le cas 17, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 32. En prenant $Frob^3$, on trouve $1^1.2^{-4}.4^4$. La seule possibilité est le cas 17, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 33. En prenant $Frob^2$, on trouve $9^1$. La seule possibilité est le cas 14, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 34. En prenant $Frob^3$, on trouve $1^{-1}.5^2$. La seule possibilité est le cas 11, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 35. En prenant $Frob^2$, on trouve $1^{-1}.5^2$. La seule possibilité est le cas 11, $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 36. En prenant $Frob^3$, on trouve $1^{-3}.2^2.4^2$. La seule possibilité est le cas 10, $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Cas 37. En prenant $Frob^2$, on trouve $1^{-3}.3^4$. La seule possibilité est le cas 9, on a $H^1(k',{\operatorname{Pic}}(\overline{X})) \neq 0$.
Comme le note un rapporteur, dans les propositions \[dp1\] et \[dp2\] un certain nombre de cas relèvent aussi du cas des surfaces fibrées en coniques, déjà traitées dans la proposition \[fibreconiques\].
L’énoncé du théorème 5.4.3 de [@L] est identique à l’énoncé ci-dessus. Il convient néanmoins de corriger la démonstration donnée dans [@L] pour le cas 6, car le choix fait là de $Frob^4$ mène, comme le dit l’auteur à la surface numéro 102, mais cette dernière a $H^1=0$.
Conclusion
==========
Récapitulons. Il s’agit de montrer :
\[vraiprincipalbis\] Soient $k$ un corps et $X$ une $k$-surface projective, lisse, géométriquement rationnelle. Supposons que $X$ possède un point $k$-rationnel et que $X$ soit déployée par une extension cyclique de $k$. Si $X$ n’est pas $k$-rationnelle, alors :
\(i) Il existe une extension $k'/k$ finie séparable telle que $H^1(k', {\operatorname{Pic}}(\X))\neq 0$.
\(ii) Le module galoisien ${\operatorname{Pic}}(\X)$ n’est pas un facteur direct d’un module de permutation.
\(iii) La $k$-variété $X$ n’est pas stablement $k$-rationnelle.
On peut supposer que $X$ est $k$-minimale, car si $f : Y \to X$ est un $k$-morphisme birationnel de $k$-surfaces projectives lisses géométriquement rationnelles, si $Y$ est déployée par une extension cyclique de $k$, il en est de même de $X$. D’après le théorème \[classif\], on peut en outre supposer que $X$ ou bien est une surface de del Pezzo $k$-minimale de degré $d$ avec $1 \leq d \leq 9$, ou bien est munie d’une fibration en coniques relativement minimale au-dessus de ${{\mathbf P}}^1_{k}$.
Si $X$ est une surface de del Pezzo de degré $d$ avec $5 \leq d \leq 9$, alors $X$ est $k$-rationnelle d’après les propositions \[C1\] et \[granddp\].
Si $X$ est une surface de del Pezzo $k$-minimale de degré $d=4$, resp. $d=3$, resp. $d=2$, resp. $d=1$, alors d’après la proposition \[dp4\] , resp. \[dp3\], resp. \[dp2\], resp. \[dp1\], il existe une extension finie séparable $k'/k$ avec $H^1(k', {\operatorname{Pic}}(\X))\neq 0$.
Si $X$ est munie d’une fibration en coniques $X \to {{\mathbf P}}^1_{k}$ relativement $k$-minimale, les propositions \[auplus3\] et \[fibreconiques\] assurent que soit $X$ est $k$-rationnelle, soit il existe une extension finie séparable $k'/k$ avec $H^1(k', {\operatorname{Pic}}(\X))\neq 0$.
Ceci donne (i), et les autres énoncés suivent (Théorème \[implgen\]).
On aimerait avoir une démonstration du théorème \[vraiprincipalbis\] qui ne passe pas par l’analyse cas par cas utilisée dans le présent article, et spécialement qui évite celle utilisée pour les surfaces de del Pezzo de degré $1$ à $3$. Pour les termes employés dans ce qui suit, on renvoie à [@CTSDesc]. Soit $k$ un corps quasi-fini, et soit $X$ une $k$-surface projective, lisse, géométriquement rationnelle, possédant un $k$-point. Comme le groupe de Galois absolu de $k$ est procyclique, l’hypothèse $H^1(k', {\operatorname{Pic}}(\X))=0$ pour toute extension finie $k'/k$, implique que le module galoisien ${\operatorname{Pic}}(\X)$ est un facteur direct d’un module de permutation (théorème d’Endo et Miyata, cf. [@CTSRequiv Prop. 2, p. 184]). Le corps $k$ est de dimension cohomologique 1. Soit $S$ le $k$-tore dual du module galoisien ${\operatorname{Pic}}(\X)$. Il existe alors un unique torseur universel ${\mathcal T} \to X$ sur $X$ (à isomorphisme près). C’est un torseur sous le $k$-tore $S$, lequel est un facteur direct d’un $k$-tore quasi-trivial. Ce torseur est donc génériquement scindé (théorème 90 de Hilbert). La $k$-variété ${\mathcal T}$ est donc $k$-birationnelle au produit $X \times_{k} S$. C’est une question ouverte de savoir si l’espace total d’un torseur universel ${\mathcal T}$ avec un $k$-point au-dessus d’une surface géométriquement rationnelle $X$ est une variété (stablement) $k$-rationnelle. Si c’était le cas, l’argument ci-dessus montrerait au moins que, sous l’hypothèse $H^1(k', {\operatorname{Pic}}(\X))=0$ pour toute extension finie $k'/k$, la surface $X$ est facteur direct d’une $k$-variété $k$-rationnelle.
[**Remerciements**]{}. A. Pirutka m’a signalé la question de B. Hassett. Je remercie K. Shramov de m’avoir montré la proposition \[dp3\] (A. Trepalin) pour les surfaces cubiques. Je remercie D. Loughran de m’avoir donné des précisions sur l’article [@U]. Après avoir vu une première version du présent article, il a aussi attiré mon attention sur l’article [@L], dont certains des calculs pour les surfaces de del Pezzo coïncident avec ceux des propositions \[dp2\] et \[dp1\] ci-dessus. Des erreurs dans [@L] n’ont pas permis à l’auteur d’obtenir le résultat général pour les surfaces de del Pezzo de degré 2. Les critiques de deux rapporteurs sur la version initiale de cet article m’ont permis de préciser certains points.
[99]{}
B. Banwait, F. Fité et D. Loughran, Del Pezzo surfaces over finite fields and their Frobenius traces, Mathematical Proceedings of the Cambridge Philosophical Society, à paraître. https://arxiv.org/pdf/1606.00300.pdf
A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc et Sir Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles, Ann. of Math. [**121**]{} (1985) 283–318.
J.-L. Colliot-Thélène, Birational invariants, purity, and the Gersten conjecture, in [*K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras*]{}, AMS Summer Research Institute, Santa Barbara 1992, ed. W. Jacob and A. Rosenberg, Proceedings of Symposia in Pure Mathematics [**58**]{}, Part I (1995) 1–64.
J.-L. Colliot-Thélène et D. Coray, L’équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques, Compositio Mathematica [**39**]{} no. 3 (979) 301–332.
J.-L. Colliot-Thélène et J.-J. Sansuc, La R-équivalence sur les tores, Ann. sci. Éc. Norm. Sup. (4) [**10**]{} (1977) 175–229.
J.-L. Colliot-Thélène et J.-J. Sansuc, La descente sur les variétés rationnelles, II, Duke Math. J. [**54**]{} (1987) 375–492.
J.-L. Colliot-Thélène et J.-J. Sansuc, On the Chow group of rational surfaces: a sequel to a paper of S. Bloch, Duke Math. J. [**48**]{} no. 2 (1981) 421–447.
J. S. Frame, The classes and representations of the groups of 27 lines and 28 bitangents, Ann. Math. Pura Appl. (4) [**32**]{} (1951), 83–119.
W. Fulton, [*Intersection Theory*]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 2, Springer-Verlag (1984).
S. Gille, Permutation modules and Chow motives of geometrically rational surfaces, with appendix by J.-L. Colliot-Thélène, J. Algebra [**440**]{} (2015) 443–463.
A. Grothendieck, Le groupe de Brauer, III, in [*Dix exposés sur la cohomologie des schémas*]{}, Masson, North-Holland, Paris, 1968, p. 88–188.
V. A. Iskovskikh, Surfaces rationnelles avec un pinceau de courbes rationnelles et avec carré de la classe canonique positif (en russe), Mat. Sbornik [**83**]{} (125) no. 1 (1970) 90–119. Trad. ang. Math. USSR-Sb. [**12**]{} (1970), 91–117.
V. A. Iskovskikh, Propriétés birationnelles d’une surface de degré 4 dans ${\bf P}^4_{k}$, Mat. Sbornik [**88**]{} (130) no. 1 (1972), trad. ang. Math. USSR Sbornik [**17**]{} no. 1 (1972).
V. A. Iskovskikh, Modèles minimaux des surfaces rationnelles sur un corps quelconque (en russe), Izv. Akad. Nauk SSSR Ser. Mat. [**43**]{} (1979), no. 1, 19–43, 237, trad. ang. Math. USSR Izvestija [**14**]{} no1 (1980) 17–39.
J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 3, Springer-Verlag, Berlin Heidelberg (1996).
Shuijing Li, Rational points on del Pezzo surfaces of degree 1 and 2, thèse, Rice University, https://arxiv.org/pdf/0904.3555.pdf
Yu. I. Manin, [*Formes cubiques*]{}, Nauka, Moscou, 1972 (en russe). Traduction en anglais (avec une numérotation différente), 2ème édition, North Holland 1986.
A. Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomology, in Proc. AMS 2015 Summer Conference (Salt Lake City).
J-P. Serre, Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, Actualités scientifiques et industrielles [**1296**]{}, Hermann, Paris, 1968.
J-P. Serre, Cohomological invariants, Witt invariants and trace forms, in [*Cohomological invariants in Galois Cohomology*]{}, University Lecture Series [**28**]{} (2003) Amer. Math. Soc.
H.P.F. Swinnerton-Dyer, The zeta-function of a cubic surface over a finite field, Math. Proc. Camb. Phil. Soc. [**63**]{} (1967) 55–71.
T. Urabe, Calculations of Manin’s invariant for del Pezzo surfaces, Mathematics of computations, Volume [**65**]{}, Number 213, Jan. 1996, 247–258. Supplement, [**65**]{}, no. 213 (Jan. 1996) p. S15–S23.
A. Várilly-Alvarado, Arithmetic of del Pezzo surfaces, in [*Birational geometry, rational curves, and arithmetic*]{} (F. Bogomolov, B. Hassett and Y. Tschinkel eds.) Simons Symposia [**1**]{} (2013), 293–319.
|
---
abstract: |
Interaction between spin waves (or excitons) moving in the lowest Landau level is studied using numerical diagonalization. Becuse of complicated statistics obeyed by these composite particles, their effective interaction is completely different from the dipole–dipole interaction predicted in the model of independent (bosonic) waves. In particular, spin waves moving in the same direction attract one another which leads to their dynamical binding. The interaction pseudopotentials $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$ for two spin waves with equal wavevectors $k$ and moving in the same or opposite directions have been calculated and shown to obey power laws $V(k)\propto
k^\alpha$ at small $k$. A high value of $\alpha_{\uparrow\uparrow}\approx4$ explains the occurrence of linear bands in the spin excitation spectra of quantum Hall droplets.
author:
- 'Arkadiusz Wójs,$^{1,2}$ Anna G[ł]{}adysiewicz,$^1$ Daniel Wodziński,$^1$ and John J. Quinn$^2$'
title: Interaction and dynamical binding of spin waves or excitons in quantum Hall systems
---
Introduction
============
Description of interactions and correlations between excitons [@haug93] (electron-hole pairs, $X=e+h$) is somewhat problematic because of their complicated statistics. Being pairs of fermions, the excitons obey Bose statistics under a “full” exchange and, consequently, condense into a Bose–Einsetin ground state at sufficiently low density. [@keldysh68] However, their composite nature comes into play when the excitons overlap and “partial” exchanges (of only a pair of electrons or holes) can occur. And, unlike for charged complexes (such as trions, $X^-=2e+h$) naturally separated by the Coulomb repulsion, the overlaps between neutral excitons can often be significant.
In the absence of a magnetic field $B$, exciton correlations have been discussed[@okamura02] in connection with four-wave mixing experiments that involve two-photon absorption.[@shah93; @feuerbacher91; @baars98; @borri99] Here, we will consider 2D systems in the high-$B$ limit, so-called “quantum Hall systems.”[@prange87] While the bosonization scheme for excitons confined to the lowest Landau level (LL$_0$) has recently been proposed, [@doretto04] we will concentrate on the numerical results for the $X$–$X$ interaction pseudopotential.
In LL$_0$, a well-known statistics/correlation effect is the decoupling and condensation of $k=0$ excitons in the ground state of interacting electrons and holes.[@lerner81] It can be interpreted in terms of an inter–exciton ($X$–$X$) exchange attraction exactly compensating for a decrease in the intra-exciton ($e$–$h$) attraction due to the phase space blocking for the coexisting identical constituent fermions.
The exciton condensation in LL$_0$ results from the mapping of an $e$–$h$ system onto a two–spin system with spin-symmetric interactions.[@macdonald90] The “hidden” $e$–$h$ symmetry corresponding to the conservation of the total spin and responsible for exciton condensation holds in LL$_0$ because there the electron and hole orbitals are identical despite different effective masses (in experimental systems with finite width, this also requires symmetric doping to avoid normal electric field that would split the $e$ and $h$ layers).
The mapping between $e$–$h$ and two-spin systems makes interband excitons in an empty LL$_0$ equivalent to spin waves (SW’s) in a filled LL$_0$, i.e., in the quantum Hall ground state with the filling factor $\nu=1$. A SW (or spin exciton) consists of a hole in the spin-polarized LL$_0$ and a reversed-spin electron in the same LL$_0$. Although excitons and SW’s in LL$_0$ are formally equivalent and the conclusions of Ref. and ours apply to both complexes, they are relevant for two different types of experiments (photoluminescence and spin relaxation).
Being charge-neutral, excitons move along straight lines and carry a linear wavevector $k$ even in a magnetic field $B$. The origin of their (continuous) dispersion[@gorkov68] $\varepsilon(k)$ in LL$_0$ is not the (constant) $e$ or $h$ kinetic energy, but the dependence of an average $e$–$h$ separation on $k$. A moving exciton carries an electric dipole moment $d$, proportional and orthogonal to both $k$ and $B$.
For a pair of moving excitons, one could think that the dominant contribution to their interaction $V({\mathbf k}_1,{\mathbf k}_2)$ would be the dipole–dipole term,[@olivares01] specially at small values of $k_1$ and $k_2$, when this term is too weak on the scale of $\varepsilon(k)$ to cause a significant polarization of the $X$ wavefunctions. Such assumption would lead to the repulsion between excitons moving in the same direction.
However, we show that this assumption is completely false because of the required (anti)symmetry of the wavefunction of overlapping excitons under exchange of individual constituent electrons or holes. This statistics/correlation effect is significant even at small $k$, and it reverses the sign of the $X$–$X$ interaction, compared to the dipole–dipole term. Specifically, excitons moving in the same direction attract one another, and the ground state of a pair of excitons carrying a total wavevector ${\mathbf k}$ is a (dynamically) bound state with ${\mathbf k}_1={\mathbf k}_2={1\over2}{\mathbf k}$.
The $X$–$X$ interaction pseudopotential is calculated numerically for two special cases: ${\mathbf k}_1=\pm{\mathbf k}_2$, corresponding to a pair of excitons moving with equal wavevectors $k_1=k_2\equiv k$ in the same ($\uparrow\uparrow$) and opposite ($\uparrow\downarrow$) direction. In addition to the sign reversal, we find that the inclusion of the statistics effects leads to the significant weakening of the $X$–$X$ interaction, specially at small $k$ (e.g., for the $\uparrow\uparrow$ configuration. we find a $V\propto k^4$ power-law behavior).
The near vanishing of the interaction between excitons moving in the same direction explains the occurrence of nearly linear multi-exciton bands found numerically in the spin-excitation spectra of finite-size quantum Hall droplets [@palacios94; @spectral] and of extended quantum Hall systems. [@skyrmion] And the attractive character of this interaction explains the slightly convex shape of these bands, which for a confined droplet leads to the oscillations of the total spin as a function of the magnetic field.[@palacios94; @spectral]
Model
=====
We consider spin excitations at the filling factor $\nu=1$, i.e., in a system of $N$ electrons half-filling the lowest Landau level (LL$_0$) single-particle angular momentum ($l$) shell with two-fold spin degeneracy and the orbital degeneracy $g\equiv2l+1=N$. The interaction among the electrons in the Hilbert space restricted to LL$_0$ is entirely determined by Haldane pseudopotential [@haldane87] defined as pair interaction energy $V_{ee}$ as a function of relative pair angular momentum $\mathcal{R}$ and plotted in Fig. \[fig1\](a).
The even and odd values of $\mathcal{R}$ correspond to symmetric and antisymmetric pair wavefunction, i.e., to the singlet and triplet pair spin state, respectively. Assuming large cyclotron gap $\hbar\omega_c$ between LL’s (compared to the Zeeman gap $E_Z$ and the interaction energy scale $e^2/\lambda$, where $\lambda=\sqrt{hc/eB}$ is the magnetic length), similar low-energy excitations of electrons at larger odd integral values of $\nu=2n+1$ occur only in the half-filled LL$_n$, and the only difference compared to the $\nu=1$ case is a different form of $V(\mathcal{R})$, as shown in Fig. \[fig1\](a) for $n=1$ and 3.
The two-spin system of $N=N_\downarrow+N_\uparrow$ electrons can be mapped onto that of $K_e=N_\uparrow$ spin-$\uparrow$ electrons and $K_h=N-N_\downarrow$ of spin-$\downarrow$ holes.[@macdonald90] At $\nu=1$, $K_e=K_h\equiv K$. The electrons and holes obtained through such mapping are both spin-polarized, and their (equal) $e$–$e$ and $h$–$h$ interactions are determined by the pseudopotential parameters $V_{ee}(\mathcal{R})$ corresponding only to odd values of $\mathcal{R}$. The effective $e$–$h$ interaction depends on $V_{ee}(\mathcal{R})$ at both even and odd values of $\mathcal{R}$, but it can be described more directly by an $e$–$h$ pseudopotential (pair $e$–$h$ energy $V_{eh}$ as a function of pair wavevector $k$) plotted in Fig. \[fig1\](b). In LL$_0$, both $e$–$e$ and $e$–$h$ pseudopotentials are monotonic, while in higher LL’s they have oscillations reflecting additional nodes of the single-particle wavefunctions.
Because of the exact mapping between two-spin and two-charge systems, all results discussed here are in principle applicable to systems of conduction electrons and valence holes. This equivalence is true for ideal systems (with zero layer width $w$ and no LL mixing) considered here. However, in realistic interband systems (realized e.g. by optical excitation of an electron gas) the $e$ and $h$ wavefunctions are usually different both in the plane of motion (because of mass-dependent LL mixing) and in the normal direction (because of mass-dependent density profiles $\varrho(z)$ and a spatial separation of $e$ and $h$ planes induced by an electric field produced by a charged doping layer). Therefore, the “hidden symmetry” is usually broken in experimental $e$–$h$ systems, while the equivalent conservation of the total spin $S$ is easily realized in the corresponding two-spin systems.
Spin-excitation spectrum at $\nu=1$
===================================
An intriguing feature known to occur in the spin-excitation spectrum at $\nu=1$ is the low-energy band that is linear in spin and angular momentum. It was first identified in finite size quantum Hall droplets, [@palacios94] and later discussed[@spectral] in Haldane spherical geometry,[@haldane83] convenient in modeling infinite, translationally invariant systems.
As shown in Fig. \[fig2\](a) obtained for $N=14$ electrons on a sphere, the lowest state at each total angular momentum $L$ has the total spin $S$ corresponding to $K={1\over2}N-S$ (the number of spin flips relative to the polarized ground state) equal to $L$. This band is nearly linear in $L$ and thus it can be interpreted as containing states of $K$ ordered and noninteracting SW’s, each carrying angular momentum $\ell=1$ and energy $\varepsilon_\ell=
V_{eh}(k_\ell)$, where $k_\ell=\ell/R$ (and $R$ is the sphere radius). Ordering means here that the angular momentum vectors of the $K$ SW’s are all parallel to give a total $L=K\ell$, i.e., that all SW’s move in the same direction along the same great circle of the sphere. On a plane (corresponding to $R\rightarrow\infty$), this corresponds to $K$ SW’s moving in parallel along a straight line, each with an infinitesimal wavevector $k_\ell$.
Scaling of this $L=K$ band with the size of the system is shown in Fig. \[fig2\](b), where we overlay the data for different $N\le14$. The excitation energy $E$ appears be a (nearly size-independent) linear function of “spin polarization” $\zeta=K/N$. Assuming exact decoupling of SW’s in this band, $E(\zeta)\equiv
K\varepsilon_\ell$ can be extrapolated to the planar geometry, where the SW dispersion is[@gorkov68] $$V_{eh}(k)=\sqrt{\pi\over2}\left(1-e^{-\kappa^2}I_0(\kappa^2)
\right){e^2\over\lambda},$$ with $\kappa={1\over2}k\lambda$ and $I_0$ being the modified Bessel function of the first kind. For small $k_\ell$, $$\varepsilon_\ell\equiv V_{eh}(k_\ell)\approx
\sqrt{\pi\over2}\,\kappa_\ell^2\,{e^2\over\lambda}.$$ Substituting $k_\ell\lambda=\ell/R$, $R=\sqrt{Q}\lambda$ (where $2Q$ is the magnetic monopole strength; $2Q\cdot hc/e=4\pi R^2B$), $l=Q$ for the lowest electron shell (LL), and, at $\nu=1$, $N=g
\equiv2l+1$, we have $k_\ell\lambda=\sqrt{2/N}$, and finally $$E(\zeta)=\zeta\sqrt{\pi\over8}\,{e^2\over\lambda}.$$ This slope is much smaller from the one in Fig. \[fig2\](b) due to finite-size/curvature errors on a sphere, particularly significant at small $k_\ell$. The total wavevector $k=L/R=Kk_\ell$ for the $L=K$ band scales as $$\label{eqdivk}
k\lambda=\sqrt{2N}\zeta,$$ i.e., on a plane is it divergent. Therefore, $E(\zeta)$ is a lower bound for the actual excitations at a given $\zeta$ that will have large but finite $k$.
Effective SW–SW interaction
===========================
Regardless of divergence of $k$ in Eq. (\[eqdivk\]), the (nearly) linear behavior of $E(K)$ suggests decoupling of SW’s in the $L=K$ band and invokes a more general question of interaction between SW’s in the lowest (or higher) LL’s. Unlike their number $K={1\over2}N-S$, the individual angular momenta of interacting SW’s are not conserved. For example, a pair of SW’s both with $\ell=1$ and with the total angular momentum $L=2$ are coupled to a pair with the same $L$ but with different $\ell=1$ and 2; these two configurations being denoted as $\left|1+1;2\right>$ and $\left|1+2;2\right>$. However, unless the single-SW energies $\mathcal{E}$ of such coupled configurations (here, $\mathcal{E}=2\varepsilon_1$ and $\varepsilon_1
+\varepsilon_2$) are close, this coupling can be effectively incorporated into the SW–SW interaction. In Fig. \[fig3\](a) we have made such assignment for the lowest excitations of the 14-electron spectrum.
Following this assignment, we can extract not only the (exact) single-SW energies, $\varepsilon_L=E[L]-E_0$, but also the parameters of an effective SW–SW interaction pseudopotential, $V[\ell+\ell';L]=
E[\ell+\ell';L]-\varepsilon_\ell-\varepsilon_{\ell'}-E_0$. Using these two-SW interaction parameters one can describe interactions in the states of more than two SW’s.
Let us demonstrate it on a simple example of $K$ SW’s each with $\ell=1$. In this case, there are only two pair-SW states, at $L=0$ and 2, corresponding to the relative (with respect to the center of mass of the two SW’s) angular momenta $\mathcal{R}\equiv2\ell-L=2$ and 0 (SW’s are pairs of fermions, and thus for two SW’s with equal $\ell$, ${\mathcal R}$ must be even as for two identical bosons). Thus, there are only two interaction parameters, in a 14-electron system equal to $V_2\equiv V[1+1;0]=0.0236\,e^2/\lambda$ and $V_0\equiv V[1+1;2]=-0.0026\,e^2/\lambda$ (note that for the subscripts in $V_0$ and $V_2$ we use notation $V_\mathcal{R}$ and not $V_L$).
The total energy of the state $\Psi$ of $K$ SW’s, $E=E_0+K
\varepsilon_\ell+U$, contains the inter-SW interaction energy that can be expressed as $$\label{eqv1}
U={K\choose2}\sum_\mathcal{R}\mathcal{G}_\mathcal{R}V_\mathcal{R}.$$ Here, $\mathcal{G}_\mathcal{R}$ are the pair amplitudes [@haldane87; @parentage] (pair-correlation functions) that measure the number of SW pairs with a given $\mathcal{R}$ (for brevity, we omit index $\Psi$ in $E$, $U$, and $\mathcal{G}_\mathcal{R}$). They are normalized, $\sum_\mathcal{R}\mathcal{G}_\mathcal{R}=1$, and satisfy an additional sum rule that on a sphere has the form [@sum-rule] $$\label{eqsr}
L(L+1)+K(K-2)\,\ell(\ell+1)={K\choose2}
\sum_\mathcal{R} \mathcal{G}_\mathcal{R}\,\mathcal{L}
(\mathcal{L}+1),$$ where $L$ and $\mathcal{L}\equiv 2\ell-\mathcal{R}$ are the total and pair SW angular momenta, respectively.
For $\ell=1$, there are only two pair amplitudes, $\mathcal{G}_0$ and $\mathcal{G}_2$, and hence they are independent of the SW–SW interaction and can be completely determined from Eq. (\[eqsr\]). This allows expression of $\mathcal{G}_\mathcal{R}$ and, using the values of $V_\mathcal{R}$ and Eq. (\[eqv1\]), of $U$ and $E$ as a function of $K$ and $L$, $$\begin{aligned}
\label{eqv2}
U&=&{L(L+1)+2K(K-2)\over6}(V_0-V_2)\nonumber\\
&+&{K(K-1)\over2}V_2.\end{aligned}$$ For $L=K$ this gives $\mathcal{G}_2=0$ and $U={1\over2}K(K-1)V_0$, i.e., the linearity of $E(K)$ depends on the vanishing of $V_0$. Energies $E(K,L)$ obtained from Eq. (\[eqv2\]) for all combinations of $L$ and $K$ are compared with the exact 14-electron energies in Fig. \[fig3\](b). Good agreement, especially for the $L=K$ band, justifies interpretation of the actual spin excitations in terms of $K$ SW’s with well-defined $\ell$, interacting through the effective SW–SW pseudopotentials.
SW–SW pseudopotential
=====================
This brings up the question of why are the SW’s in the $L=K$ band (nearly) noninteracting (i.e., why is $V_0$ so small compared to $V_2$ or $\varepsilon_1$). And a more general one, what is the pseudopotential describing interaction between the SW’s. The SW–SW pseudopotential $V$ depends on the pair of wavevectors, ${\mathbf k}$ and ${\mathbf k}'$. However, in extension of $V_0$ and $V_2$ in Eq. (\[eqv2\]), we will only consider two special cases: $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$, corresponding to two SW’s with equal wavevectors $k$ moving in the same and opposite directions, respectively.
Independent SW’s
----------------
A moving SW carries[@gorkov68] an in-plane dipole electric moment ${\mathbf d}$, with magnitude $d$ proportional to $k$ and oriented orthogonally to the direction of ${\mathbf k}$. For a pair of uncorrelated SW’s this implies simple dipole–dipole interaction, repulsive for the $\uparrow\uparrow$ configuration, and attractive for $\uparrow\downarrow$. Indeed, in Fig. \[fig4\](a) we plot $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$ showing such behavior.
Moreover, at small $k$ we find a very regular power-law dependence, $$\label{eqpwr1}
V_{\uparrow\uparrow}(k)\sim
0.42\,(k\lambda)^{5\over2}{e^2\over2\pi R}.$$ The curves in Fig. \[fig4\](a) have been calculated as an expectation value of the Coulomb interaction in a trial state $\left|k,k;q\right>$ describing two uncorrelated (independent) SW’s, each with the wavevector $k$ and with the total wavevector $q=2k$ ($\uparrow\uparrow$) and $q=0$ ($\uparrow\downarrow$). Such trial states have been constructed on a sphere in the basis of two electrons and two holes in a lowest LL with $l=Q$. The two electrons (and two holes) are distinguished by different isospins $\sigma=\pm{1\over2}$. A pairing hamiltonian $H_\ell$ is introduced with the $e$–$h$ pseudopotential in the form $$V_{eh}^{(\ell)}(\sigma_e,\sigma_h,\ell')
=-\delta_{\sigma_e\sigma_h}\delta_{\ell\ell'}$$ and the $e$–$e$ and $h$–$h$ interactions set to zero. At each total angular momentum $L$, there is exactly one eigenstate of $H_\ell$ corresponding to the eigenvalue $-2$. It describes two independent $e$–$h$ pairs (i.e., excitons or SW’s), one with $\sigma_e=\sigma_h={1\over2}$ and one with $\sigma_e=\sigma_h
=-{1\over2}$, each in an eigenstate of pair angular momentum $\ell$ corresponding to the pair wavevector $k_\ell=\ell/R$ (on a sphere, describing motion of a charge-neutral pair along a great circle). The total angular momentum $L$ of two pairs can also be converted into the total wavevector, $q=L/R$. We have concentrated on the trial states with $L=2\ell$ and $0$ (i.e., with $q=2k_\ell$ and 0), denoted as $\left|k_\ell,k_\ell;
2k_\ell\right>$ and $\left|k_\ell,k_\ell;0\right>$. They describe two pairs each with the same $k_\ell$ and moving in the same and opposite directions, respectively. Discrete SW–SW pseudopotentials $V_{\uparrow\uparrow}(k_\ell)$ and $V_{\uparrow\downarrow}(k_\ell)$ on a sphere have been calculated as the expectation value of the inter-SW Coulomb interaction (i.e., the total Coulomb energy of the $2e+2h$ state minus the intra-SW $e$–$h$ attraction $2\varepsilon_\ell$). When the sphere curvature $R/\lambda=Q^2$ decreases, the discrete values quickly converge to the continuous curves $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$ appropriate for a planar system. The interpolated curves for the LL degeneracy $2l+1\equiv2Q+1=30$ and 50 are compared in Fig. \[fig4\](a). Note that $V$ is plotted as a function of $e^2/2\pi R$ (rather than $e^2/\lambda$) what reflects the fact that SW’s are extended objects confined to a great circle of length $2\pi R$ (in contrast to electrons or holes that are confined to cyclotron orbits of radius $\sim\lambda$).
Coupled SW’s
------------
The SW–SW pseudopotentials obtained above describe interaction between independent SW’s (distinguished by isospins $\sigma_e$ and $\sigma_h$). However, the following two correlation effects must be incorporated into the effective SW–SW interaction to describe the actual spin excitations at $\nu\sim1$ (i.e., the interacting $e$–$h$ systems).
First, the Coulomb (charge–charge) interaction between the SW’s breaks the conservation of $\ell$ and causes relaxation of the individual SW wavefunctions and their energies $\varepsilon_\ell$. This perturbation effect mixes the SW states within the energy range $\Delta\varepsilon\sim V$, so it becomes negligible when $V$ is small, i.e., at small $k$. In particular, it does not affect the behavior of $V_{\uparrow\uparrow}
(k)$ at small $k$, responsible for the linearity of the $L=K$ band.
Second, strictly speaking, the SW’s are not bosons but pairs of fermions, and a wavefunction of two SW’s must not only be symmetric under interchange of the entire SW’s, but also antisymmetric under interchange of two constituent electrons or holes. The trial paired states $\left|k,k;q\right>$ with $H_\ell=-2$ do not obey these symmetry requirements, because $H_\ell$ is isospin-asymmetric and hence it does not commute with pair $e$ or $h$ isospins, $\Sigma_e$ and $\Sigma_h$. Therefore, the trial eigenstates of $H_\ell=-2$ are different from the properly symmetrized eigenstates of $\Sigma_e=\Sigma_h=1$. This statistics effect is generally weak for spatially separated composite particles, but for the SW’s moving along the same line (or great circle) it is large and cannot be treated perturbatively (even at small $k$ when the Coulomb SW–SW interaction is negligible). At each $L$, the exact form of the ground state in the $\Sigma_e
=\Sigma_h=1$ subspace depends on $\ell$ and on the details of the actual (Coulomb) hamiltonian, and so does the average value of $H_\ell$ (measuring the actual “degree of pairing”). However, as a reasonable approximation one can introduce the “maximally paired” states, defined at each $L$ as the lowest-energy state of the pairing interaction hamiltonian $V_{eh}^{(\ell)}$ within the $\Sigma_e=\Sigma_h=1$ subspace.
The relaxation of the wavefunctions of the overlapping SW’s is evident from the analysis of the $e$–$e$ and $h$–$h$ pair amplitudes $\mathcal{G}(\mathcal{R})$. For a pair of different particles, such as electrons or holes distinguished by isospin $\sigma$ in the trial state $\left|k,k;
q\right>$, $\mathcal{R}$ can be any integer. Therefore, $\mathcal{G}_{ee}(\mathcal{R})$ and $\mathcal{G}_{eh}
(\mathcal{R})$ calculated for the independent SW’s are positive at both even and odd $\mathcal{R}$ (in fact, there is no obvious correlation whatsoever between the parity of $\mathcal{R}$ and the value of $\mathcal{G}_{ee}$ or $\mathcal{G}_{eh}$). In contrast, for a pair of identical fermions, such as electrons or holes in an actual, interacting state of two SW’s, $\mathcal{G}_{ee}(\mathcal{R})$ and $\mathcal{G}_{eh}(\mathcal{R})$ vanish exactly at all even values of $\mathcal{R}$. The change of pair amplitudes when going from the trial states $\left|k,k;q\right>$ to the actual Coulomb ground states is quite dramatic, precluding adequacy of the pseudopotentials of Fig. \[fig4\](a) for the description of many-SW systems.
Because of the above relaxation effects, interaction between the SW’s is not purely a two-body interaction, and thus it cannot be completely described by a (pair) pseudopotential $V(k)$. In other words, a SW–SW pseudopotential taking these effects into account is not rigorously defined. However, as demonstrated in Fig. \[fig2\](b), many-SW spectra can be reasonably well approximated using an effective pseudopotential obtained for only two SW’s.
To determine such effective $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$, we calculate the $2e+2h$ Coulomb energy spectra similar to the $K\le2$ part of Fig. \[fig3\](a) and make analogous assignments for the $K=2$ states. The lowest state at each even value of $L=2$, 4, … is interpreted as one of two SW’s each with $\ell={1\over2}L$ and moving in the same direction. Similarly, consecutive states at $L=0$ contain two SW’s each with $\ell=1$, 2, … and moving in opposite directions. In both cases, $V(\ell)=E-2\varepsilon_\ell-E_0$. When $\ell$ is converted into $k_\ell=\ell/R$ and $V$ is plotted in the units of $e^2/2\pi R$, the discrete pseudopotentials $V
(k_\ell)$ fall on the continuous curves $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$ that very quickly converge to ones appropriate for a planar system when the sphere curvature $R/\lambda=Q^2$ is decreased. The interpolated curves for $2l+1\equiv2Q+1=30$ and 50 are compared in Fig. \[fig4\](b), showing virtually no size dependence. Similar curves were obtained for the “maximally paired” states used instead of actual Coulomb eigenstates.
The justification for the above assignment comes from the observation of distinct bands in the low-energy $K=2$ spectrum. The values of $L$ within each band are consistent with the addition of angular momenta of two SW’s, $|\ell-\ell'|\le L\le\ell+\ell'$ (with the additional requirement that $L-2\ell\equiv\mathcal{R}$ be even for $\ell=\ell'$). In the absence of the SW relaxation, these bands would contain the eigenstates of $\mathcal{E}\equiv\varepsilon_\ell+\varepsilon_{\ell'}$, with the intra-band dispersion reflecting interaction of the independent SW’s with $\ell$ and $\ell'$. In the actual spectrum, the bands mix, but remain separated, making the assignment possible. The interband mixing and the resulting changes in the energy spectrum are precisely the relaxation effects, effectively incorporated into $V(k)$. For $L=0$ ($\uparrow\downarrow$), the mixing is minimal, because the contributing “independent SW” configurations $\left|\ell,\ell';
L=0\right>$ must all have $\ell=\ell'$, and thus very different single-SW energies $\mathcal{E}$. For $L=2\ell$ ($\uparrow\uparrow$), mixing between configurations $\left|\ell+\delta,\ell-\delta;L=2\ell\right>$ with close values of $\mathcal{E}$ can occur, having a stronger effect on the effective $V_{\uparrow\uparrow}(k)$.
The main two findings about the effective SW–SW pseudopotentials shown in Fig. \[fig4\](b) are the following. First, the statistics effect turns out so strong as to reverse the sign of interaction. In contrast to the prediction of the model of independent SW’s with dipole–dipole interaction, the SW’s moving in the same direction decrease their total energy (what can be interpreted as attraction), while the SW’s moving in opposite direction increase their energy (i.e., repel one another). Second, the magnitude of the $\uparrow\uparrow$ attraction at small $k$ is greatly reduced compared to Eq. (\[eqpwr1\]). It can also be approximated by a power-law dispersion, but with a much higher exponent and a much smaller prefactor, $$\label{eqpwr2}
V_{\uparrow\uparrow}(k)\sim
-0.069\,(k\lambda)^4{e^2\over2\pi R}.$$ Although the near vanishing of $V_{\uparrow\uparrow}$ at small $k$ was anticipated from the linearity of the $L=K$ band in Fig. \[fig2\], the negative sign and large exponent are rather surprising and of a wider consequence. It may be worth stressing that the identified attraction between $N$ SW’s (or interband excitons) moving in the same direction is too weak to induce a stable bound ground state, with the total energy lower than $N$ times ground state energy of a single SW/exciton. Therefore, it does not contradict a well-known fact that the ground state of $N$ electrons and $N$ holes in the lowest LL is a multiplicative state[@lerner81; @macdonald90] of $N$ SW’s/excitons each with $k=0$ (in particular, a biexciton is unstable toward breaking up into two $k=0$ excitons, while the energy of $N$ SW’s is never lower than $N\varepsilon_0=0$, and so the $\nu=1$ ground state is spontaneously polarized). However, for two or more SW’s/excitons carrying a conserved total wavevector $q>0$, the convex shape of $V_{eh}(k)$ causes equal distribution of $q$ among all SW’s/excitons, and the SW–SW or $X$–$X$ attraction binds them together. Such a moving multi-SW/exciton can only break up (into separate SW’s/excitons) through an inelastic collision taking away its wavevector. This dynamical binding will affect spin relaxation (for the SW’s) or photoluminescence (for the excitons) of an electron gas, but the relevant spectra are yet to be calculated.
Conclusion
==========
We have studied interaction between moving SW’s (excitons) in the lowest LL. For a pair of SW’s with equal wavevectors $k$ and moving in the same ($\uparrow\uparrow$) or opposite ($\uparrow\downarrow$) directions, the effective interaction pseudopotentials $V_{\uparrow\uparrow}(k)$ and $V_{\uparrow\downarrow}(k)$ have been calculated numerically. They account for relaxation of overlapping SW’s due to the Fermi statistics of constituent (reversed-spin) electrons and (spin-) holes, and differ completely from the prediction for independent SW’s interacting through their dipole moments. In particular, the signs of the interactions are reversed and their magnitudes are strongly decreased. The former effect leads to a “dynamical binding” of mobile multiexcitons, and the latter explains the near decoupling of excitons in the linear $L=K$ band in the spin-excitation spectrum at $\nu=1$.
AW thanks Manfred Bayer, Leszek Bryja, Pawel Hawrylak, and Marek Potemski for helpful discussions and acknowledges support by the Polish Ministry of Scientific Research and Information Technology under grant 2P03B02424. This work was also supported by grant DE-FG 02-97ER45657 of the Materials Science Program – Basic Energy Sciences of the U.S. Dept. of Energy.
[99]{}
H. Haug and S.W. Koch, [*Quantum Theory of the Optical and Electronic Properties of Semiconductors*]{}, (World Scientific, Singapore, 1993); S. A. Moskalenko and D. W. Snoke, [*Bose-Einstein Condensation of Excitons and Biexcitons*]{}, (Cambridge University Press, Cambridge, 2000).
L.V. Keldysh and A.N. Kozlov, Zh. Eksp. Teor. Fiz. [**54**]{}, 978 (1968) \[Sov. Phys. JETP [**27**]{}, 521 (1968)\].
S. Okumura and T. Ogawa, Phys. Rev. B [**65**]{}, 035105 (2002); M. Combescot and O. Betbeder-Matibet, Europhys. Lett. [**59**]{}, 579 (2002).
J. Shah, [*Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures*]{}, (Springer-Verlag, Berlin 1996).
B. F. Feuerbacher, J. Kuhl, and K. Ploog, Phys. Rev. B [**43**]{}, 2439 (1991); K. H. Pantke, D. Oberhauser, V. G. Lyssenko, J. M. Hvam, and G. Weimann, Phys. Rev. B [**47**]{}, 2413 (1993); E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, D. Bennhardt, S. W. Koch, P. Thomas, R. Hey, and K. Ploog, Phys. Rev. B [**50**]{}, 14730 (1994); J. Ishi, H. Kunugita, K. Ema, T. Ban, and T. Kondo, Phys. Rev. B [**63**]{}, 073303 (2001).
T. Baars, W. Braun, M. Bayer, and A. Forchel, Phys. Rev. B [**58**]{}, R1750 (1998); T. Baars, M. Bayer, A. A. Gorbunov, and A. Forchel, Phys. Rev. B [**63**]{}, 153312 (2001).
P. Borri, W. Langbein, J. M. Hvam, and F. Martelli, Phys. Rev. B [**60**]{}, 4505 (1999); H. P. Wagner, A. Schätz, W. Langbein, J. M. Hvam, and A. L. Smirl, Phys. Rev. B [**60**]{}, 4454 (1999); W. Langbein and J. M. Hvam, Phys. Rev. B [**61**]{}, 1692 (2000).
, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987); [*Perspectives in Quantum Hall Effects*]{}, edited by S. Das Sarma and A. Pinczuk (John Wiley & Sons, New York, 1997).
R. L. Doretto, A. O. Caldeira, and S. M. Girvin, cond-mat/0402292.
I. V. Lerner and Yu. E. Lozovik, Zh. Eksp. Teor. Fiz. [**80**]{}, 1488 (1981) \[Sov. Phys. JETP [**53**]{}, 763 (1981)\]; Y. A. Bychkov and E. I. Rashba, Solid State Commun. [**48**]{}, 399 (1983); T. M. Rice, D. Paquet, and K. Ueda, Helv. Phys. Acta [**58**]{}, 410 (1985); D. Paquet, T. M. Rice, and K. Ueda, Phys. Rev. B [**32**]{}, 5208 (1985).
A. H. MacDonald and E. H. Rezayi, Phys. Rev. B [**42**]{}, 3224 (1990).
L. P. Gor’kov and I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. [**53**]{}, 717 (1967) \[Sov. Phys. JETP [**26**]{}, 449 (1968)\]; C. Kallin and B. I. Halperin, Phys. Rev. B [**30**]{}, 5655 (1984).
Dipole–dipole interaction studied by M. A. Olivares-Robles and S. E. Ulloa, Phys. Rev. B [**64**]{}, 115302 (2001), may become dominant in asymmetric structures (with separated electron and hole planes) in which all excitons carry large dipole moments oriented in the same (normal) direction.
J. J. Palacios, L. Martín-Moreno, G. Chiappe, E. Louis, and C. Tejedor, Phys. Rev. B [**50**]{}, 5760 (1994); J. H. Oaknin, L. Martín-Moreno, J. J. Palacios, and C. Tejedor, Phys. Rev. Lett. [**74**]{}, 5120 (1995); J. H. Oaknin, L. Martín-Moreno, and C. Tejedor, Phys. Rev. B [**54**]{}, 16850 (1996).
A. Wójs and P. Hawrylak, Phys. Rev. B [**56**]{}, 13227 (1997).
A. Wójs and J. J. Quinn, Phys. Rev. B [**66**]{}, 045323 (2002); Solid State Commun. [**122**]{}, 407 (2002).
F. D. M. Haldane, in: [*The Quantum Hall Effect*]{}, edited by R. E. Prange and S. M. Girvin (New York: Springer-Verlag, 1987), chapter 8, pp. 303–352.
F. D. M. Haldane, Phys. Rev. Lett. [**51**]{}, 605 (1983).
A. Wójs and J. J. Quinn, Philos. Mag. B [**80**]{}, 1405 (2000); Acta Phys. Pol. A [**96**]{}, 593 (1999); J. J. Quinn and A. Wójs, J. Phys.: Condens. Matter [**12**]{}, R265 (2000).
A. Wójs and J. J. Quinn, Solid State Commun. [**110**]{}, 45 (1999); A. Wójs, Phys. Rev. B [**63**]{}, 125312 (2001).
|
---
abstract: 'We propose an interferometric method to measure [$\mathbb{Z}_2~$]{}topological invariants of time-reversal invariant topological insulators realized with optical lattices in two and three dimensions. We suggest two schemes which both rely on a combination of Bloch oscillations with Ramsey interferometry and can be implemented using standard tools of atomic physics. In contrast to topological Zak phase and Chern number, defined for individual 1D and 2D Bloch bands, the formulation of the [$\mathbb{Z}_2~$]{}invariant involves at least two Bloch bands related by time-reversal symmetry which one has keep track of in measurements. In one of our schemes this can be achieved by the measurement of Wilson loops, which are non-Abelian generalizations of Zak phases. The winding of their eigenvalues is related to the [$\mathbb{Z}_2~$]{}invariant. We thereby demonstrate that Wilson loops are not just theoretical concepts but can be measured experimentally. For the second scheme we introduce a generalization of time-reversal polarization which is continuous throughout the Brillouin zone. We show that its winding over half the Brillouin zone yields the [$\mathbb{Z}_2~$]{}invariant. To measure this winding, our protocol only requires Bloch oscillations within a single band, supplemented by coherent transitions to a second band which can be realized by lattice-shaking.'
author:
- 'F. Grusdt'
- 'D. Abanin'
- 'E. Demler'
title: 'Measuring [$\mathbb{Z}_2~$]{}topological invariants in optical lattices using interferometry'
---
Introduction
============
It has been understood almost since its discovery in 1980 that the quantum Hall effect [@Vonklitzing1980] emerges from the non-trivial topology of Landau levels [@Thouless1982]. More recently it was realized that one can have topologically nontrivial states that differ from the quantum Hall effect (see [@Hasan2010; @Qi2011; @Bernevig2013] for review). Unlike the Chern number however, the topological invariants characterizing such systems are only quantized as long as certain symmetries are present. The quantum spin Hall effect (QSHE) [@Kane2005; @Bernevig2006a; @Bernevig2006] for example is protected by the time-reversal (TR) symmetry. Superconductors on the other hand are particle-hole symmetric, which allows to define a subclass of topological superconductors. Topological insulators and superconductors were completely classified for non-interacting fermions [@Ryu2010] and the QSHE (i.e. a 2D [$\mathbb{Z}_2~$]{}topological insulator) as well as 3D [$\mathbb{Z}_2~$]{}topological insulators have been observed in solid state systems [@Koenig2007; @Hsieh2008].
Cold atom experiments offer a large degree of control[@Bloch2008] and allow for measurements impossible in solid state systems [@Gericke2008; @Bakr2009; @Sherson2010]. Therefore an implementation of topological insulators in these systems would allow to investigate them from a different perspective. Theoretically, topological invariants are related to geometric Berry phases of particles moving in Bloch bands. Recently, Berry phases and corresponding topological invariants were directly measured in a cold atomic system in an optical lattice [@Atala2012] thus allowing a direct experimental investigation of the topology of Bloch band wavefunctions.
While realizing quantum Hall like systems of cold atoms has been a longstanding challenge [@Jaksch2003; @Schweikhard2004; @Cooper2008], there was considerable progress in the implementation of artificial gauge fields [@Dalibard2011; @Cooper2011; @Cooper2011a; @Lin2009; @Aidelsburger2011; @Kolovsky2011a; @Jimenez2012] and recently two experimental groups reported on the realization of the Hofstadter Hamiltonian in optical lattices [@Aidelsburger2013; @Miyake2013]. For the simulation of the QSHE (or, more generally, a [$\mathbb{Z}_2~$]{}topological insulator) with ultra-cold atoms artificial spin-orbit coupling (SOC) is required which has also been demonstrated experimentally [@Lin2011]; Different SOC schemes have lead to several proposals for the implementation of two [@Liu2010; @Goldman2010; @Beri2011; @Mei2012] and three dimensional [@Beri2011] TR invariant topological insulators. In the recent experiment of the Munich group [@Aidelsburger2013] Abelian SOC has successfully been implemented, which is sufficient for a realization of the QSHE. Also the recent MIT experiment [@Miyake2013] allows an implementation of Abelian SOC [@Kennedy2013].
In this paper we propose measurement schemes for [$\mathbb{Z}_2~$]{}topological invariants in TR invariant topological insulators in two and three dimensions. Our method uses one of the most important technical strengths of cold atom experiments, the ability to perform interferometric measurements. This goes to the heart of topological states, whose topological nature is encoded in the overlaps of Bloch wavefunctions. We discuss formulas relating the [$\mathbb{Z}_2~$]{}invariant to simple non-Abelian Berry phases and show how the latter can be measured.
We now provide a brief overview of the main idea of our method and put it in the context of earlier studies. Topological properties of 1D Bloch bands are chacterized by the so-called Zak phase [@Zak1989]. This is essentially Berry’s phase [@Berry1984] for a a trajectory enclosing a 1D Brillouin zone (BZ). Recent experiments with optical superlattices used a combination of Bloch oscillations and Ramsey interferometry to measure the Zak phase of the dimerized lattice[@Su1979]. In these experiments momentum integration was achieved with Bloch oscillations of atoms in momentum space and Berry’s phase was measured using Ramsey’s interferometric protocol (see [@Atala2012] and discussion below for more details). Zak phase measurement in 1D is shown schematically in FIG.\[fig:sketchIntro\] (a). This approach can be extended to measure the Chern number of two-dimensional Bloch bands (the idea is illustrated in Fig. 1(b)) [@Abanin2012]. The key is to measure Zak phases for fixed values of momenta $k_y$, and their winding in the BZ $k_y=0...2\pi$ yields the Chern number (in the entire paper we set the lattice constant $a=1$). Alternatively the geometric Zak phases can be read out from semi-classical dynamics, which also allows one to measure the Chern number [@Price2012].
In this paper, we generalize the ideas of Refs.[@Atala2012; @Abanin2012] for interferometric measurement of [$\mathbb{Z}_2~$]{}invariants in TR-symmetric optical lattices. The key challenge in this case is to keep track of *two* Kramers degenerate bands, required by TR invariance. Defining the topological properties of such bands requires understanding how Bloch eigenstates in the two bands relate to each other. We argue that the Bloch/Ramsey sequence should be supplemented by band switching as shown schematically in FIG.\[fig:sketchIntro\] (c). The obtained interferometric signal not only depends on the phase accumulated when adiabatically moving within a single band but also on the phase picked up during the transition from one band to the other. Experimentally band switching can be achieved by applying oscillating force at the frequency matching the band energy difference. We show that when applying this particular band switching protocol, a geometric phase for the Bloch cycle is obtained, the winding of which (over half the BZ) yields the [$\mathbb{Z}_2~$]{}invariant.
We also present an alternative approach based on measurements of the so-called Wilson loops, which are essentially non-Abelian generalizations of the Zak phase. Their eigenvalues are directly related to the [$\mathbb{Z}_2~$]{}invariant, as was shown by Yu et.al. [@Yu2011]. The measurement of Wilson loops requires moving atoms non adiabatically in the BZ in two directions and relies on keeping track of two-band dynamics of atoms. We show how this can be achieved using currently available experimental techniques.
Other methods suggested to detect topological properties of cold atom systems mostly focused on detecting characteristic gapless edge states [@Stanescu2009; @Stanescu2010; @LiuLiu2010; @Goldman2012; @Buchhold2012]. Even for typical smooth confinement potentials present in cold atom systems, theoretical analysis showed [@Buchhold2012] that these edge states should still be observable. To detect [$\mathbb{Z}_2~$]{}topological phases of cold atoms, a spin-resolved version of optical Bragg spectroscopy was suggested [@Goldman2010]. A different approach to measure Chern numbers makes use of the Streda formula, relating them to the change in atomic density when a finite magnetic field is switched on [@Umucalilar2008; @Shao2008]. Extensions of this method for detection of [$\mathbb{Z}_2~$]{}topological phases were suggested [@Liu2010; @Goldman2010], however they only work when the Chern numbers for individual spins are well-defined (which is generally not the case [@Sheng2006]). Recently also an interferometric method has been suggested to measure the [$\mathbb{Z}_2~$]{}invariant of inversion-symmetric TR invariant topological insulators [@Liu2013]. Our method in contrast does not make any assumptions about the system’s symmetry (except TR of course).
The paper is organized as follows. In section \[subSec:IntroSummary\] we explain the basic idea of our measurement schemes. To this end we review different formulations of the [$\mathbb{Z}_2~$]{}invariant in terms of simple Zak phases, which are at the heart of our interferometric schemes. In section \[sec:AdiabaticScheme\] the first of our two measurement schemes (twist scheme) is presented. The experimental realization of this scheme is discussed and we show that it can easily be implemented in the experimental setup proposed in [@Goldman2010]. In section \[sec:WilsonScheme\] we present the Wilson loop scheme and discuss its experimental feasibility. Finally in section \[sec:Summary\] we conclude and give an outlook how our scheme can easily be applied also to 3D topological insulators.
Interferometric measurement of the [$\mathbb{Z}_2~$]{}invariant {#subSec:IntroSummary}
===============================================================
In the following we will review how topological invariants can be formulated in terms of geometrical Zak phases. After a short discussion of the Chern number case, we move on to [$\mathbb{Z}_2~$]{}invariants. This allows us to introduce the basic ideas of our measurement protocols.
Zak phases {#subsec:ZakPhasesDiscussion}
----------
We start by discussing Zak phases in 1D Bloch bands. Let us consider some eigenstate $u_k(x) = \psi_k(x) e^{-i k x}$ of a Bloch Hamiltonian ${\hat{\mathcal{H}}}(k)$ which continuously depends on quasi momentum $k$, and where $k$ is varied from $k=-\pi$ to $k=\pi$ over some time $T$. Thereby the wavefunction generally picks up a dynamical phase that depends on $T$ as well as a *geometric* phase which only depends on the path in momentum space [@Berry1984; @Zak1989]. This so-called Berry or Zak phase is given by $$\varphi_{\text{Zak}}= \int_{-\pi}^\pi dk ~ {\mathcal{A}}(k),
\label{eq:defZak}$$ where the *Berry connection* is defined as $${\mathcal{A}}(k) = {\left\langleu(k)\right|} i \partial_k {\left|u(k)\right\rangle}.
\label{eq:BerryConnectionDef}$$
As mentioned in the introduction, Zak phases of optical lattices have been measured using a combination of Bloch oscillations and Ramsey interferometry [@Atala2012].
For later purposes we will now shortly discuss the issue of dynamical phases, which read $$\varphi_\text{dyn} = - \frac{\int_0^{2 \pi} dk ~ \epsilon(k) }{ \frac{dk}{dt} }.$$ Here $\epsilon(k)$ is the band energy. One can always get rid of dynamical phases by driving Bloch oscillations extremely fast (i.e. $dk/dt \rightarrow \infty$), as long as non-adiabatic transition are prohibited by a sufficiently large energy gap to other bands.
Chern numbers and Zak phases {#subSec:ChernZak}
----------------------------
To understand how Zak phases of 1D systems constitute topological invariants in higher dimensions, we start by reviewing the Chern number case. To this end we note that there is a fundamental relation between the Zak phase and the polarization $P$ of a 1D system [@Kingsmith1993; @Ortiz1994], $$\frac{1}{2\pi} \varphi_{{\text{Zak}},\alpha} = {\left\langlew_\alpha(0)\right|} \hat{x} {\left|w_\alpha(0)\right\rangle} =: P_\alpha.
\label{eq:KingsmithVanderbilt}$$ Here ${\left|w_\alpha(0)\right\rangle}=(2\pi)^{-1} \int_{-\pi}^\pi dk ~ \psi_{k,\alpha}(x)$ denotes the Wannier function of band $\alpha$ localized at lattice site $j=0$ and $\hat{x}$ is the position operator in units of the lattice constant $a$.
The Chern number (${\text{Ch}}$) describes the Hall response of a filled band, which is quantized at integer multiples of $e^2/h$, $$\sigma_{xy} = \frac{J_x}{E_y} = {\text{Ch}}\frac{e^2}{h}.
\label{eq:defCh}$$ Here $E_y$ denotes an electric field along $y$-direction and $J_x$ the perpendicular Hall current density along $x$-direction. Since the electric field $E_y$ leads to transport of electrons (or atoms) along $k_y$ through the BZ, the corresponding current density $J_x$ perpendicular to the field is related to the change of polarization $\partial_{k_y} P$ (polarization is measured in $x$-direction as in Eq.). Using Eq., one easily derives from this simple physical consideration the well-known relation between Zak phases and the Chern number (see [@Xiao2010] for review) $${\text{Ch}}= \frac{1}{2 \pi} \int_{-\pi}^\pi dk_y ~ \partial_{k_y} \varphi_{\text{Zak}}(k_y).
\label{eq:ChWindingZak}$$ A more detailed discussion of this argument can be found in Appendix \[sec:Apdx:ZakChern\].
A simple physical picture illustrating Eq. is given in FIG. \[fig:WannierCenters\] (a) following [@Fu2006]. There the Wannier centers (i.e. the polarizations $P(k_y)$ of the Wannier functions at different sites $j$) are shown as a function of $k_y$. The case when a Wannier center reconnects with its $n$th nearest neighbor after going from $k_y=-\pi$ to $k_y=\pi$ corresponds to a non-trivial Chern number of ${\text{Ch}}=n$.
Relation indicates that the Chern number can be measured in an optical lattice by measuring the gradient of the Zak phase [@Abanin2012].
[$\mathbb{Z}_2~$]{}invariant and time-reversal polarization
-----------------------------------------------------------
The quantum spin Hall phase was constructed by Kane and Mele [@Kane2005] starting from two time reversed copies (spin $\uparrow$ and $\downarrow$) of Chern insulators realizing the quantum Hall effect. Since time-reversal inverts $k_y$ but not $x$, the Wannier centers of the second spin are obtained from those in FIG.\[fig:WannierCenters\] (a) by reflecting on the $x$-axis, see FIG.\[fig:WannierCenters\] (b). Consequently the Chern numbers have opposite signs and cancel to give a vanishing total Chern number. The underlying topology of the system however can be classified by the *difference* of the two Chern numbers, $${\nu_{2\text{D}}}= \frac{1}{2} {\left(}{\text{Ch}}_\uparrow - {\text{Ch}}_\downarrow {\right)}.$$
In the generic case with SOC mixing the spins $\uparrow, \downarrow$, spin is no longer a good quantum number and two bands labeled ${\text{I}},{\text{II}}$ emerge. As a consequence of TR symmetry they are related by $${\left|u^{\text{II}}(-{\textbf{k}})\right\rangle}=e^{i \chi({\textbf{k}})} \hat{\theta} {\left|u^{\text{I}}({\textbf{k}})\right\rangle}.
\label{eq:TRconnection}$$ Here $\hat{\theta} = K i \hat{\sigma}^y$ is the TR operator with $K$ denoting complex conjugation and the phase $\chi({\textbf{k}})$ describes the independent gauge degree of freedom at $\pm {\textbf{k}}$ in the BZ.
The two bands ${\text{I}}$ and ${\text{II}}$ are characterized by a [$\mathbb{Z}_2~$]{}topological invariant ${\nu_{2\text{D}}}$ [@Kane2005]. Fu and Kane pointed out in [@Fu2006] that, like the Chern number, ${\nu_{2\text{D}}}$ can be understood from the topology of the Wannier centers. To see how this works, let us first discuss a generic TR invariant band structure as it is sketched in FIG.\[fig:sketchIntro\] (c).
TR invariance requires the Bloch Hamiltonian ${\hat{\mathcal{H}}}({\textbf{k}})$ to fulfill $$\hat{\theta}^\dagger {\hat{\mathcal{H}}}({\textbf{k}}) \hat{\theta} = {\hat{\mathcal{H}}}(-{\textbf{k}}).$$ As a consequence there are two 1D subsystems at fixed $k_y^{\text{TRIM}}= 0,\pi$ (referred to as time-reversal invariant momenta, TRIM) which are TR invariant *as 1D systems*, i.e. $\hat{\theta}^\dagger {\hat{\mathcal{H}}}(k_x) \hat{\theta} = {\hat{\mathcal{H}}}(-k_x)$. Within these two 1D systems there are in total four momenta ${\textbf{k}}^{\text{TRIM}}=(k_x^{\text{TRIM}},k_y^{\text{TRIM}})$ (also referred to as TRIM) where the Bloch Hamiltonian is TR invariant itself, $\hat{\theta}^\dagger {\hat{\mathcal{H}}}({\textbf{k}}^{\text{TRIM}}) \hat{\theta} = {\hat{\mathcal{H}}}({\textbf{k}}^{\text{TRIM}})$.
At these four points Kramers theorem requires eigenvalues to come in degenerate pairs. Therefore the generic TR invariant band structure consists of two valence bands with degeneracies at the four ${\textbf{k}}^{\text{TRIM}}$, separated from the conduction bands by an energy gap. Cuts through such a generic band structure are sketched in FIG.\[fig:ad&twist\]. In principle there can be additional accidental degeneracies of the two bands ${\text{I}}, {\text{II}}$. However in the rest of the paper we will restrict ourselves to the simpler case without any further degeneracies besides the four Kramers degeneracies.
FIG. \[fig:WannierCenters\] (c) illustrates the corresponding Wannier centers for a generic – but topologically non-trivial – case. The underlying TR symmetry requires Wannier centers to come in *Kramers pairs* at TRIM $k^{\text{TRIM}}_y=0,\pi$, again as a consequence of Kramers theorem. When these Kramers pairs switch partners upon going from $k_y=0$ to $k_y=\pi$ the system is topologically non-trivial, while it is trivial otherwise [@Fu2006].
Using the change of polarizations of the two states $\Delta P^{{\text{I}},{\text{II}}}$ as indicated in FIG.\[fig:WannierCenters\] (c), we see that the topology is described by the integer invariant $\Delta P_\theta = \Delta P^{\text{I}}- \Delta P^{\text{II}}$. Fu and Kane [@Fu2006] coined the name *time-reversal polarization* (TRP) for the quantity $$P_\theta(k_y) = P^{\text{I}}(k_y) - P^{\text{II}}(k_y).
\label{eq:defTRP}$$ Using their language, the [$\mathbb{Z}_2~$]{}invariant is given by the change of TRP over half the BZ, i.e. $${\nu_{2\text{D}}}= P_\theta(\pi) - P_\theta(0) \mod 2.
\label{eq:ntDTRP}$$ A more detailed, pedagogical derivation of this formula can be found in the Appendix \[sec:apdx:ZtAndTRP\].
Discontinuity of time-reversal polarization
-------------------------------------------
Naively one might think that, with the formulation of ${\nu_{2\text{D}}}$ Eq. entirely in terms of polarizations (i.e. due to in terms of Zak phases), we have an interferometric scheme at hand. According to Eqs., one would only have to measure the difference of Zak phases $\varphi_{\text{Zak}}^{\text{I}}(0)$ at $k_y=0$ and $\varphi_{\text{Zak}}^{\text{I}}(\pi)$ at $k_y=\pi$ and repeat the protocol for the second band ${\text{II}}$.
Zak phases, however, can only be measured up to $2\pi$. Typically the problem of $2\pi$ ambiguities of Zak phases can be circumvented by rewriting their *difference* as a *winding* over some continuous parameter. As pointed out above, this strategy works out for the case of Chern numbers, see Eq..
However we can not simply replace the change $\Delta P_\theta$ of TRP by its winding $\int dk_y \partial_{k_y} P_\theta(k_y)$, because TRP *is not continuous over the BZ*. This discontinuity is a direct consequence of Kramers degeneracies: Let us consider the Zak phase $\varphi_{\text{Zak}}^{\text{I}}(0)$ at $k_y^{\text{TRIM}}=0$, see FIG.\[fig:ad&twist\] (a). According to Eqs., $\varphi_{\text{Zak}}^{\text{I}}(0)$ is determined by the Berry connection $\mathcal{A}_{\text{I}}(k_x,0)$ within band ${\text{I}}$ (note that band ${\text{I}}$ crosses band ${\text{II}}$ at the two Kramers degeneracies.) Now let us imagine going to some slightly larger $0 < k_y \ll 2 \pi$ and measure the Zak phase of band ${\text{I}}$ here, see FIG.\[fig:ad&twist\] (b). Because there is no longer any true band crossing, we now always have to follow the energetically upper band. This means however, that the Zak phase $\varphi_{\text{Zak}}^{\text{I}}(k_y)$ is determined by the Berry connection $\mathcal{A}_{{\text{I}}}(k_x,k_y) \approx \mathcal{A}_{{\text{I}}}(k_x,0)$ from $k_x < 0$ and by $\mathcal{A}_{{\text{I}}}(k_x,k_y) \approx \mathcal{A}_{{\text{II}}}(k_x,0)$ (note the exchanged index!) from $k_x > 0$ [^1]. Then, because in general $\mathcal{A}_{\text{I}}({\textbf{k}}) \neq \mathcal{A}_{\text{II}}({\textbf{k}})$, we obtain a very different result, $\varphi_{\text{Zak}}^{\text{I}}(k_y\rightarrow 0) \nrightarrow \varphi_{\text{Zak}}^{\text{I}}(0)$ in general.
Let us add that as a consequence of the discontinuity of TRP, the meaning of Wannier centers in FIG.\[fig:WannierCenters\] (b)-(d) has to be taken with care. What is shown is a non-Abelian generalization of simple Zak phases , as will be discussed in detail at the end of Sec.\[subsubSec:WilsonVsTRP\].
The twist scheme {#subsec:IntrodAd&Twist}
----------------
The basic idea of our first (out of two) interferometric scheme for the measurement of the [$\mathbb{Z}_2~$]{}invariant is to circumvent the discontinuity of TRP discussed above, while keeping all Bloch oscillations completely adiabatic. To do so, we want to add band switchings at the end and in the middle of the sequence. Then close to the Kramers degeneracy at $k_x=0$, instead of staying in the energetically upper band ${\text{I}}$, atoms will be transferred to the energetically lower band ${\text{II}}$. These band switchings correspond to applying Ramsey $\pi$ pulses, as indicated in FIG.\[fig:ad&twist\](b).
After finishing the entire Bloch cycle and applying a second Ramsey $\pi$-pulse, the atoms will finally return to the band they initially started from. The two possible *twisted* paths through energy-momentum space will be labeled ${\text{i}}$ and ${\text{ii}}$ and they are illustrated in FIG.\[fig:ad&twist\]. Path ${\text{i}}$ corresponds to atoms starting in band ${\text{I}}$, while ${\text{ii}}$ corresponds to atoms starting in ${\text{II}}$.
In this process atoms pick up geometrical Zak phases $\tilde{\varphi}_{\text{Zak}}^{{\text{i}},{\text{ii}}}$. We will refer to these as *twisted* Zak phases, because they consist of Zak phases from the movement within bands ${\text{I}},{\text{II}}$ as well as additional geometric phases from the Ramsey $\pi$-pulses. The key idea of the *twist scheme* is to measure these twisted Zak phases.
We note that for TR invariant $k_y^{\text{TRIM}}=0,\pi$ no band switchings are required and twisted Zak phases coincide with their conventional counterparts, $$\varphi^{{\text{I}}({\text{II}})}_{\text{Zak}}(k_y^{\text{TRIM}}) = \tilde{\varphi}^{{\text{i}}({\text{ii}})}_{\text{Zak}}(k_y^{\text{TRIM}}).
\label{eq:twistedTRZakPhase}$$ Moreover we will see that twisted Zak phases $\tilde{\varphi}_{\text{Zak}}(k_y)$ are *continuous* as a function of $k_y$; This is because we added band switchings by hand right where conventional Zak phases fail to follow the desired path. Like all geometric phases, twisted Zak phases are by definition gauge invariant up to integer multiples of $2\pi$.
Twisted Zak phases thus allow us to define a *continuous* version to TRP (which we will refer to as cTRP) by $$\tilde{P}_\theta (k_y) = \frac{1}{2 \pi} \left[ \tilde{\varphi}_{\text{Zak}}^{\text{i}}(k_y) - \tilde{\varphi}_{\text{Zak}}^{\text{ii}}(k_y) \right].
\label{eq:defcTRP}$$ For TR invariant momenta, cTRP reduces to TRP see . Thus, starting from the definition of the [$\mathbb{Z}_2~$]{}invariant as *difference* of TRP Eq. and using continuity of cTRP, we can express ${\nu_{2\text{D}}}$ as the *winding* of cTRP: $${\nu_{2\text{D}}}= \int_0^\pi dk_y ~\partial_{k_y} \tilde{P}_\theta(k_y) \mod 2.
\label{eq:ntDWindingcTRP}$$ This formulation is fully gauge invariant.
[$\mathbb{Z}_2~$]{}invariant and Wilson loops {#subSec:Z2invAndWilsonLoops}
---------------------------------------------
In this subsection we discuss non-Abelian generalizations of Zak phases – so-called Wilson loops. Yu et. al. [@Yu2011] showed that Wilson loops provide a natural way of defining the [$\mathbb{Z}_2~$]{}invariant in terms of their eigenvalues. We will describe a second method for measuring the [$\mathbb{Z}_2~$]{}invariant which relies on the Wilson-loop formulation. As we shall see below, this method allows one to circumvent the difficulties related to band crossings at the TRIM
The authors of [@Yu2011] derived various formulas for the [$\mathbb{Z}_2~$]{}invariant. For our interferometric scheme we will focus on one particular relation which reads $${\nu_{2\text{D}}}= \frac{1}{\pi} {\left(}\Delta \varphi_W - \frac{1}{2} \int_0^\pi dk_y ~\partial_{k_y} \Phi(k_y) {\right)}\mod 2,
\label{eq:nu2Dresult}$$ where the terms on the right hand side are related to eigenvalues of Wilson loop operators; They will be precisely defined below (in \[subsubsec:RelationWilsonZ2\]), after discussing Wilson loops (in \[subsubsec:DefWilsonLoops\]). A rigorous proof of Eq. can be found in the Appendix \[subsecAppdx:WilsonLoops\] and a simple explanation will be given in the following subsection \[subsubSec:WilsonVsTRP\].
### Wilson loops {#subsubsec:DefWilsonLoops}
A natural question to ask, from our interferometric point of view, is what happens in the limit of very strong driving when the Bloch oscillation frequency exceeds all energy spacings between bands ${\text{I}},{\text{II}}$. Let us still assume a large energy gap separating ${\text{I}},{\text{II}}$ from other bands, such that non-adiabatic transitions into the latter can be neglected.
The multi-band Bloch dynamics in the strong driving limit (period $T \rightarrow 0$) is characterized by a geometric quantity depending solely on the path within the BZ. Since there is generally strong mixing between bands ${\text{I}}$ and ${\text{II}}$, the $U(1)$ Zak phase we encountered in the single-band case generalizes to a $U(2)$ unitary matrix acting in ${\text{I}}-{\text{II}}$ space, the so-called *$U(2)$ Wilson loop* [^2] $$\hat{W} = \mathcal{P} \exp {\left(}-i \int_{-\pi}^\pi dk ~ \hat{\mathcal{A}}(k) {\right)}.
\label{eq:defWilsonLoop}$$ Here $\mathcal{P}$ denotes the path ordering operator [^3] and the *non-Abelian Berry connection* [@Wilczek1984] generalizing Eq. is defined by $${\mathcal{A}}_\mu^{s,s'} = {\left\langleu^s({\textbf{k}})\right|} i \partial_{k_\mu} {|u^{s'}({\textbf{k}})\rangle}, \qquad \mu =x,y.
\label{eq:defnonAbBerryCon}$$ $s,s'$ label the two bands ${\text{I}},{\text{II}}$ in our case. In the rest of the paper, without loss of generality, we will typically consider the Berry connection along $x$ and drop the index $\mu=x$. We also note that Wilson loops have proven useful as a tool to classify other symmetry protected topology [@Alexandradinata2012].
In Appendix \[sec:ApdxB\] we derive the general propagator $\hat{U}$ describing Bloch oscillations within a restricted set of $N$ bands. From that derivation one can easily show that Wilson loops indeed emerge as the propagators describing Bloch oscillations in the limit of infinite driving force, $\hat{U}_{F=\infty} = \hat{W}$.
For the discussion of the [$\mathbb{Z}_2~$]{}invariant, TR invariant Wilson loops play a special role. (With TR invariant Wilson loops we mean Wilson loops at TRIM.) Such TR invariant $U(2)$ Wilson loops reduce to $U(1)$ phase factors [@Yu2011], $$\hat{W}_{\text{TR}} = e^{-i \varphi_W} ~ \hat{\mathbb{I}}_{2\times 2}
\label{eq:TRWU1}$$ as a consequence of Kramers theorem. $\varphi_W$ will be referred to as the *Wilson loop phase*.
Since Eq. will be important later on, we quickly prove it here. To this end we choose a special gauge where $\chi(k)=0$ in Eq. (known as the TR constraint [@Fu2006]). In this gauge one has $\hat{\theta}^\dagger \hat{\mathcal{A}}(k) \hat{\theta} = \hat{\mathcal{A}}(-k)$ which leads to $\hat{\theta}^\dagger \hat{W} \hat{\theta} = \hat{W}^\dagger$. Since Wilson loops are gauge invariant this holds for an arbitrary gauge. Moreover it implies doubly degenerate eigenvalues: Assume $\hat{W} {\left|u\right\rangle} = e^{-i \varphi_W} {\left|u\right\rangle}$ and thus also $\hat{W}^\dagger {\left|u\right\rangle} = e^{i \varphi_W} {\left|u\right\rangle}$. Therefore $\hat{W} \hat{\theta} {\left|u\right\rangle} = \hat{\theta} \hat{W}^\dagger {\left|u\right\rangle} = e^{-i \varphi_W} \hat{\theta} {\left|u\right\rangle}$ and besides ${\left|u\right\rangle}$ also $\hat{\theta} {\left|u\right\rangle}$ is eigenvector of $\hat{W}$. These two eigenvectors can not be parallel however, i.e. we can not write $\hat{\theta} {\left|u\right\rangle} = \tau {\left|u\right\rangle}$ with a complex number $\tau \in \mathbb{C}$, since this would imply $-{\left|u\right\rangle} = \hat{\theta}^2 {\left|u\right\rangle} = \tau^* \hat{\theta} {\left|u\right\rangle} = |\tau|^2 {\left|u\right\rangle} \neq - {\left|u\right\rangle}$.
### Relation to [$\mathbb{Z}_2~$]{}invariant {#subsubsec:RelationWilsonZ2}
As pointed out in the beginning, Wilson loops are related to the [$\mathbb{Z}_2~$]{}invariant by Eq.. Now we will explain the different terms in this equation.
For the first term in Eq. we recall that the unitary Wilson loops at TRIM $k_y^{\text{TRIM}}=0,\pi$ reduce to simple $U(1)$ phase factors, see Eq., and we can write $$\hat{W}(k_y^{\text{TRIM}}) = e^{-i \varphi_W(k_y^{\text{TRIM}})} ~ \hat{\mathbb{I}}_{2\times 2}.$$ In Eq. the Wilson loop phase difference $\Delta \varphi_W$ appears, which is defined as $$\Delta \varphi_W := \varphi_W(\pi) - \varphi_W(0).$$ In our interferometric scheme this difference of Wilson loop phases has to be measured.
The second term is the winding of the *total Zak phase*, $$\Phi(k_y) := {\text{tr}}\int_{-\pi}^{\pi} dk_x ~ \hat{\mathcal{A}}_x({\textbf{k}}) \equiv \varphi^{\text{I}}_{\text{Zak}}(k_y) + \varphi^{\text{II}}_{\text{Zak}}(k_y),
\label{eq:PhiSumZak}$$ across *half* the BZ. Importantly, unlike TRP, the total Zak phase is *continuous* throughout the BZ because the *sum* of Zak phases appears. The idea for our second interferometric protocol is to measure the windings of the Zak phases $\varphi_{\text{Zak}}^{{\text{I}},{\text{II}}}(k_y)$ individually.
The Wilson loop scheme
----------------------
Our second interferometric scheme (*Wilson loop scheme*) is based on Eq. from the previous subsection. The basic idea is to measure both terms, the Wilson loop phase $\Delta \varphi_W$ and the total Zak phases $\Phi$ separately. Both these quantities can be obtained from measurements of simpler Zak phases.
To obtain the winding of total Zak phase $\Phi(k_y)$ we suggest to use the tools developed for the measurement of the Chern number, see \[subSec:ChernZak\]. The only complication is that now *two* bands have to be treated. This can be done by adiabatically moving within only a single band (say ${\text{I}}$) and repeating the same measurement for the second band ${\text{II}}$. An alternative protocol allowing non-adiabatic transitions between bands ${\text{I}}$ and ${\text{II}}$ will also be presented in \[subsubsec:TotalZakPhaseRealization\].
To obtain the difference of Wilson loop phases $\Delta \varphi_W = \varphi_W(\pi) -\varphi_W(0) \mod 2 \pi$ we suggest to use a direct spin-echo type measurement. Like any interferometric phase, the obtained result is only known up to integer multiples of $2 \pi$. The key to the Wilson loop scheme is that knowledge of $\Delta \varphi_W \mod 2 \pi$ is sufficient in Eq.. I.e. if $\Delta \varphi_W$ is replaced by $\Delta \varphi_W + 2 \pi$ in that equation, the resulting [$\mathbb{Z}_2~$]{}invariant ${\nu_{2\text{D}}}\rightarrow {\nu_{2\text{D}}}+ 2 = {\nu_{2\text{D}}}\mod 2$ does *not* change.
Relation between Wilson loops and TRP {#subsubSec:WilsonVsTRP}
-------------------------------------
Before proceeding to the detailed discussion of our two interferometric protocols, we want to point out the relation between the corresponding formulations of the [$\mathbb{Z}_2~$]{}invariant. This will also shed more light on the relation between [$\mathbb{Z}_2~$]{}invariant and Wilson loops given in Eq..
Let us start by rewriting the winding of total Zak phase in terms of polarizations. Using Eq. we obtain $$\frac{1}{2 \pi} \int_0^\pi dk_y ~\partial_{k_y} \Phi(k_y) = P^{\text{I}}(\pi) + P^{\text{II}}(\pi) - P^{\text{I}}(0) - P^{\text{II}}(0).
\label{eq:windingTotZakPols}$$ Meanwhile the formulation of the [$\mathbb{Z}_2~$]{}invariant in terms of TRP reads $${\nu_{2\text{D}}}= P^{\text{I}}(\pi) - P^{\text{II}}(\pi) - P^{\text{I}}(0) + P^{\text{II}}(\pi) \mod 2,$$ see Eq.. After clever adding and subtracting terms in the last equation we can write $$\begin{gathered}
{\nu_{2\text{D}}}= 2 {\left(}P^{\text{I}}(\pi) - P^{\text{I}}(0) {\right)}\\ - \sum_{s={\text{I}},{\text{II}}} {\left(}P^s(\pi) - P^s(0) {\right)}\mod 2.
\label{eq:ntDintermediate}\end{gathered}$$
In the second line of this equation we recognize the winding of total Zak phase discussed before. The term in the first line on the other hand denotes the difference of Zak phases at $k_y=0$ and $\pi$, $$P^{\text{I}}(\pi) - P^{\text{I}}(0) = \frac{1}{2 \pi} {\left(}\varphi_{\text{Zak}}^{\text{I}}(\pi) - \varphi_{\text{Zak}}^{\text{I}}(0) {\right)}.$$ Here, as a consequence of TR invariance, the Zak phases of the two bands ${\text{I}},{\text{II}}$ are equal, explaining why only the polarization $P^{\text{I}}$ appears. What’s more, these Zak phases are given by the Wilson loop phase $\varphi_W$, i.e. we obtain $$P^{\text{I}}(\pi) - P^{\text{I}}(0) = \frac{1}{2 \pi} {\left(}\varphi_W^{\text{I}}(\pi) - \varphi_W^{\text{I}}(0) {\right)}= \frac{\Delta \varphi_W}{2 \pi}.
\label{eq:DeltaPdeltaphiW}$$ Combining Eqs., in Eq. we have thus derived Eq..
Now the two terms in Eq. have a clear physical meaning: The winding of total Zak phase is related to the translation of the center of mass of the two Wannier centers, i.e. $\Delta {\left(}P^{\text{I}}+ P^{\text{II}}{\right)}$. (Here $\Delta$ denotes the difference of the quantity across half the BZ.) The difference of Wilson loop phases meanwhile stands for the change of polarization of a single band, $\Delta \varphi_W / 2 \pi = \Delta P^{\text{I}}= \Delta P^{\text{II}}\mod 1$.
In FIG.\[fig:WannierCenters\] (a)-(d) these changes of polarization can easily be read off from the plotted Wannier centers. A word of caution is in order, however. As a consequence of the discontinuity of TRP, FIG.\[fig:WannierCenters\](c) has to be taken with a grain of salt: Although appealing, the idea that each line (solid/dashed) shows the polarization of a *single band* is wrong. As explained by Yu et.al.[@Yu2011], what is shown are the eigenvalues of the position *operator* $\hat{X}$ projected on the two bands ${\text{I}},{\text{II}}$ and its non-commutative quantum mechanical nature plays a crucial role in resolving the discontinuity of TRP. Yu et.al. showed that the eigenvalues of $\hat{X}$ are given by the angle (in the complex plane) of the $U(1)$ Wilson loop eigenvalues. Because Wilson loops include non-adiabatic band-mixings they are in general continuous as a function of $k_y$ - and so is their spectrum.
Twist scheme {#sec:AdiabaticScheme}
============
In this section we discuss the twist scheme in detail. We start by introducing the concrete protocol and show how to get rid of dynamical phases. We proceed by giving the theoretical derivation of the phases to be measured; Then we show their relation to the [$\mathbb{Z}_2~$]{}invariant and present a mathematical formulation of continuous time-reversal polarization (cTRP). We close the section by discussing cTRP using the example of the Kane-Mele model [@Kane2005].
Interferometric sequence {#subsec:InterferometricSeq}
------------------------
As discussed in Sec.\[subsec:IntrodAd&Twist\], the basic idea of the twist scheme is to measure twisted Zak phases using a combination of Bloch oscillations and Ramsey interferometry. Twisted Zak phases were defined by introducing band-switchings in the middle ($k_x=0$) and at the end ($k_x=\pi$) of the interferometric sequence, see FIG.\[fig:ad&twist\] (b). These band switchings correspond to Ramsey $\pi$ pulses between the bands, and along with them come additional geometric phases which will be discussed at the end of this section.
Note that since only a continuous function interpolating between TRP $P_{\theta}(\pi)$ and $P_\theta(0)$ is required, the two band switchings (labeled $1,2$) can be performed at *any* intermediate $k_x=f_{1,2}(k_y)$. The only requirements are that $f_1(0)=f_1(\pi) = 0$ and $f_2(0)=f_2(\pi) = \pi$ as well as continuity of $f_{1,2}(k_y)$. This most general case only leads to a redefinition of twisted Zak phases, while keeping their relation to the [$\mathbb{Z}_2~$]{}invariant Eq. unchanged. We will therefore not discuss it in the following.
### Band-switchings
To realize the Ramsey $\pi$ pulses between the bands we suggest to drive Bloch oscillations with a time-dependent force, see FIG.\[fig:RamseyPulses\] (a), described by a Hamiltonian $$\hat{\text{H}}_{\text{rf}}(t) = \int d^2 {\textbf{r}} ~ \hat{\Psi}^\dagger({\textbf{r}}) ~ \cos(\omega_{\text{rf}}~ t) {\textbf{F}}_0 \cdot {\textbf{r}} ~ \hat{\Psi}({\textbf{r}}).
\label{eq:HrfNonFC}$$ Here $\hat{\Psi}({\textbf{r}})$ is a pseudo-spinor (components $\uparrow,\downarrow$) annihilating a particle at position ${\textbf{r}}$ and $\omega_{\text{rf}}$ is the (typically radio-frequency, rf) driving frequency. Note that in this way only motional degrees of freedom are coupled, independent of the (pseudo) spin state of the atoms. This turns out to be crucial for the scheme to work. For simpler realizations with a direct coupling between the pseudospins, additional information about the Bloch wave functions is required. We discuss this issue in detail in Appendix \[sec:AppdxNonUniversalFCPh\].
The equations of motion for the Hamiltonian Eq. are derived in Appendix \[sec:ApdxB\]. According to Eq. in that Appendix we obtain a modulation of momentum $${\textbf{k}}(t) = {\textbf{k}}(0)- \sin(\omega_{\text{rf}}~ t) {\textbf{F}}_0 / \omega_{\text{rf}}.$$ Dynamics of this kind have been studied before, see e.g. [@Dunlap1986]. FIG.\[fig:RamseyPulses\](b) illustrates the effect of this driving in momentum space: particles undergo Bloch oscillations within a restricted area $\pm \frac{|{\textbf{F}}_0|}{\omega_{\text{rf}}}$ around their mean position.
Therefore, when $|{\textbf{F}}_0| \ll \omega_{\text{rf}}$ (with lattice spacing $a=1$), we may approximate the Berry connection (and equivalently the Bloch Hamiltonian) by $\mathcal{A}{\left(}{\textbf{k}}(t){\right)}\approx \mathcal{A}{\left(}{\textbf{k}}(0) {\right)}$. Taking into account only the two Kramers partners ${\text{I}},{\text{II}}$ and applying the rotating wave approximation we obtain the Hamiltonian in the frame rotating at frequency $\omega_{\text{rf}}$ $${\hat{\mathcal{H}}}_{\text{rf}}({\textbf{k}}) = \left( \begin{array}{cc}
0 & {\textbf{F}}_0 \cdot \mathcal{A}^{u,l}({\textbf{k}}) \\
{\textbf{F}}_0 \cdot \mathcal{A}^{l,u}({\textbf{k}}) & \Delta({\textbf{k}}) - \omega_{\text{rf}}.
\end{array} \right).
\label{eq:Hdriving}$$ The basis of the rotating frame is defined as ${\left|l,{\textbf{k}}\right\rangle} e^{- i E^l t}$ and ${\left|u,{\textbf{k}}\right\rangle} e^{-i {\left(}E^l + \omega_{\text{rf}}{\right)}t}$, and $\Delta = E^u- E^l$ denotes the band-gap between the upper ($u$) and lower ($l$) of the two bands. For the rotating wave approximation to be valid, we require $$|{\textbf{F}}_0\cdot \mathcal{A}^{u,l}({\textbf{k}})| \ll \omega_{{\text{rf}}} \sim \Delta.
\label{eq:RWAcond}$$
We note that the phase of the effective driving field, $$\varphi_{\mathcal{A}}({\textbf{k}}) := \arg \mathcal{A}^{l,u}({\textbf{k}}) = - \arg \mathcal{A}^{u,l}({\textbf{k}}),
\label{eq:varphiA}$$ is determined by the non-Abelian Berry connection (where in the second step we employed $\hat{\mathcal{A}}^\dagger=\hat{\mathcal{A}}$). This is important because the latter encodes information about the underlying topology of the two bands ${\text{I}},{\text{II}}$. We will come back to this point below.
One might be afraid that the resulting Rabi frequency is too small for the method to be practically applicable. However we find e.g. for the Kane-Mele model [@Kane2005] (which will be discussed in more detail below in \[subsec:KaneMele\]) that $|\mathcal{A}^{u,l}|$ takes substantial values in the entire BZ, see FIG. \[fig:Axlu\].
Note that the edges of the BZ are not shown in FIG.\[fig:Axlu\] since $|\mathcal{A}^{u,l}|$ diverges around the Kramers degeneracies. (The reason is that the lower-band Bloch function continuously evolves into the upper one at the Kramers degeneracy, such that ${\left\langle l,-\delta k_x |l, \delta k_x \right\rangle} \rightarrow 0$ for $\delta k_x \rightarrow 0$ and thus $|{\left\langleu,k_x\right|} \partial_{k_x} {\left|l,k_x\right\rangle}| \rightarrow \infty$ at $k_x=0$.) In this case of too large $|\mathcal{A}^{u,l}|$, according to Eq. rotating wave approximation is not applicable, but the band switching protocol can be replaced by a quick Landau-Zener sweep across the avoided crossing.
### Sequence
Now we introduce the interferometric sequence which allows one to measure twisted Zak phases $\tilde{\varphi}^{{\text{i}},{\text{ii}}}_{\text{Zak}}$, and therefore cTRP Eq. directly. To this end we assume that atoms are located initially in the upper band at $k_x=-\pi$ and some fixed $k_y$, i.e. ${\left|\psi_0\right\rangle}={\left|u,-\pi\right\rangle}$ and start by applying a $\pi/2$-pulse, see FIG.\[fig:ad&twistScheme1\]. In the following we will ignore all dynamical phases which will be discussed below in \[subSec:Ad&Twist\_dyn\].
The $\pi/2$ pulse creates a superposition state of atoms in the upper and lower band, $${\left|\psi_1\right\rangle}=\frac{1}{\sqrt{2}} \bigl( {\left|u,-\pi\right\rangle} -i e^{i \varphi_{\mathcal{A}}(\pi)} {\left|l,-\pi\right\rangle} \bigr).$$ In this step atoms in lower and upper band pick up the relative phase $\varphi_\mathcal{A}(\pi)$ of the driving field, see Eqs. and .
Next, a Bloch oscillation half-cycle transports the atoms from $k_x=-\pi$ to $k_x=0$ and each component picks up geometric phases $\varphi_{{\text{Zak}},-}^{u,l}$. These *incomplete Zak phases* are defined for the lower ($s=l$) and upper ($s=u$) band as $$\varphi_{\text{Zak},\pm}^s(k_y) = \pm \int_{0}^{\pm \pi} dk_x ~ \mathcal{A}^{ss}({\textbf{k}}), \quad s=u,l.
\label{eq:partialZak}$$
Note that incomplete Zak phases are not gauge invariant, and thus not physical observables. However the interferometric signal we obtain at the end of our sequence will be fully gauge invariant and observable.
The resulting state now reads $${\left|\psi_2\right\rangle}=\frac{1}{\sqrt{2}} {\left(}e^{i \varphi_{{\text{Zak}},-}^{u} } {\left|u,0\right\rangle} - i e^{i {\left(}\varphi_{\mathcal{A}}(\pi) + \varphi_{{\text{Zak}},-}^{l} {\right)}} {\left|l,0\right\rangle} {\right)}.$$ A $\pi$-pulse at $k_x=0$ then exchanges populations of the upper and lower band such that the corresponding wave function reads $$\begin{gathered}
{\left|\psi_3\right\rangle}=\frac{1}{\sqrt{2}} \Bigl( e^{i {\left(}\varphi_{\mathcal{A}}(\pi) + \varphi_{{\text{Zak}},-}^{l} - \varphi_{\mathcal{A}}(0) {\right)}} {\left|u,0\right\rangle} \\ -i e^{i {\left(}\varphi_{\mathcal{A}}(0) + \varphi_{{\text{Zak}},-}^{u} {\right)}} {\left|l,0\right\rangle} \Bigr).\end{gathered}$$ After a second Bloch oscillation half-cycle the atoms reach $k_x=\pi = - \pi \mod 2 \pi$ and pick up incomplete Zak phases $\varphi_{{\text{Zak}},+}^{u,l}$.
Finally another $\pi/2$-pulse is applied to read out the relative phase of the two components ${\left|u,\pi\right\rangle}$, ${\left|l,\pi\right\rangle}$. This is achieved by a phase shift of the driving frequency, $\omega_{\text{rf}}~t \rightarrow \omega_{\text{rf}}~t - \varphi_E^{\pi}$ in Eq.. As a function of this shift the population in the upper band yields Ramsey fringes $$|\psi_u(\varphi_E^\pi)|^2 = \cos^2 \left[ \frac{1}{2} {\left(}2 \pi \tilde{P}_\theta(k_y) - \varphi_E^\pi - \Phi_{\text{dyn}} {\right)}\right].
\label{eq:RamseySignal}$$ Here $\Phi_{\text{dyn}}$ contains all dynamical phases from the Bloch oscillations as well as Ramsey pulses. Most importantly, the incomplete Zak phases in combination with the phases $\varphi_\mathcal{A}$ yield a full expression for cTRP, $$\begin{gathered}
2 \pi \tilde{P}_\theta(k_y) = \varphi_{\text{Zak},-}^u(k_y) + \varphi_{\text{Zak},+}^l(k_y) - \varphi_{\text{Zak},-}^l(k_y) \\ -\varphi_{\text{Zak},+}^u(k_y) - 2 \bigl( \varphi_{\mathcal{A}}(\pi,k_y) - \varphi_{\mathcal{A}}(0,k_y) \bigr).
\label{eq:cTRPsequenceRes}\end{gathered}$$
At the end of this section we will give an explicit proof that the above equation has all desired properties of cTRP. In particular, it reduces to TRP at $k_y=0,\pi$ and is continuous throughout the BZ; therefore its winding yields the [$\mathbb{Z}_2~$]{}invariant, see Eq..
Dynamical-phase-free sequence {#subSec:Ad&Twist_dyn}
-----------------------------
Now we turn to the discussion of dynamical phases and present a scheme that completely eliminates them. When performing Bloch oscillations, to move the atoms from e.g. $k_x(0)=-\pi$ to $k_x(T)=+ \pi$ in time $T$, additional dynamical phases $$\Phi_{\text{dyn},s}^\text{BO}(k_y) = \int_0^T dt ~ E^s {\left(}k_x(t),k_y {\right)}$$ contribute to $\Phi_\text{dyn}$ in Eq. . Here $s=u,l$ denotes the band index and $E^s$ the corresponding energy.
To cancel them we use the opposite transformation properties of geometrical and dynamical phases when inverting the path taken in the BZ. From $\frac{dk}{dt}=F$ we see that dynamical phases do not depend on the orientation of the path, $$\int_0^T dt E(k(t)) = \int_{-\pi}^\pi dk ~ \frac{E(k)}{F} = \int_{\pi}^{-\pi} dk ~ \frac{E(k)}{-F}.$$ Geometric phases on the other hand acquire a negative sign upon path inversion, $$\int_{-\pi}^\pi dk ~ \mathcal{A}(k) = - \int_{\pi}^{-\pi} dk ~ \mathcal{A}(k).$$
Therefore, when reversing the interferometric sequence ($F \rightarrow -F$) after reaching $k_x=\pi$ (as indicated in FIG. \[fig:noDyn\]), the Ramsey signal yields *twice* the continuous TR polarization while dynamical phases are canceled.
Experimentally, phases can only be measured up to $2 \pi$. As we argued above, the [$\mathbb{Z}_2~$]{}invariant can be written as *winding* of cTRP, see Eq.. This winding is measured by summing up small changes $\delta \tilde{P}_\theta = \tilde{P}_\theta(k_y+\delta k_y) - \tilde{P}_\theta(k_y)$. By choosing $\delta k_y$ sufficiently small we may always assume $2 \delta \tilde{P}_\theta \ll 1$ and doubling the interferometric sequence still allows to infer the winding of cTRP.
The complete sequence is summarized in FIG.\[fig:noDyn\]. The Ramsey signal in this case reads $$|\psi_u(\varphi_E^\pi)|^2 = \cos^2 \left[ 2 \pi \tilde{P}_\theta - \varphi_E^\pi - \Phi_{\text{dyn}}^{(0)} \right],$$ where the remaining dynamical phase is picked up when applying Ramsey pulses. It only depends on the known driving parameters, $\Phi_{\text{dyn}}^{(0)} = \pi {\left(}\frac{3\omega_{\text{rf}}(\pi)}{4 \Omega_{\text{rf}}(\pi)} - \frac{\omega_{\text{rf}}(0)}{\Omega_{\text{rf}}(0)} {\right)}$.
Experimental realization and limitations {#subsec:expRealAdAndTwist}
----------------------------------------
Our scheme is readily applicable in the proposal [@Goldman2010] where nano-wires on an atom-chip are used to generate state-dependent potentials for different magnetic hyperfine states. These could also be used to realize the band-switching Hamiltonian and for driving Bloch oscillations. In more conventional setups without atom chips, like e.g. the experiment [@Aidelsburger2013] and the proposals [@Liu2010; @Beri2011; @Kennedy2013], Bloch oscillations can e.g. be driven using magnetic field gradients [@Atala2012] or optical potentials. This would also allow the realization of Hamiltonian for band-switchings.
The main advantage of the twist scheme is that - although it makes use of interferometry - no additional degrees of freedom are required besides the pseudospins $\uparrow,\downarrow$ needed for the realization of the QSHE. This is of practical relevance, since already the realization of two pseudospins for the QSHE is a non-trivial task.
The applicability of our scheme is somewhat limited in that we did not consider accidental degeneracies besides the four Kramers degeneracies. If such additional degeneracies are present, the definition of cTRP has to be modified. The scheme for the Ramsey pulses presented in subsection \[subsec:InterferometricSeq\] is also not applicable when the off-diagonal Berry connections become too small. Let us also add however, that cTRP contains more information about the band structure than only the [$\mathbb{Z}_2~$]{}invariant, since it resolves the two TR partners individually.
Formal definition and calculation of cTRP
-----------------------------------------
In this section we will give a formal proof that our scheme presented above does indeed measure the [$\mathbb{Z}_2~$]{}invariant; I.e. we will derive Eq.. Instead of starting from this explicit expression for cTRP however, we will introduce the concept of cTRP in a formal way and derive it independently.
### Definition of cTRP {#subsubsec:DefcTRP}
We will now formally define a generalization of TRP $P_\theta(k_y)$ that we will refer to as $\tilde{P}_\theta(k_y)$; We require this quantity to fulfill the following properties, making it suitable for an interferometric measurement of the [$\mathbb{Z}_2~$]{}invariant. It has to
- reduce to TRP at the end points $k_y^{\text{TRIM}}= 0,\pi$, i.e. $\tilde{P_\theta}(k_y^{\text{TRIM}}) = P_\theta(k_y^{\text{TRIM}})$, and
- be continuous as a function of $k_y$.
Any such function $\tilde{P}_\theta(k_y)$ will be called *continuous time-reversal polarization* (cTRP). To assure that cTRP constitutes a physical observable it should furthermore
- be gauge-invariant, at least up to an integer at each $k_y$.
Finally, from a practical point of view, we want cTRP to
- be measurable in an interferometric setup consisting of a combination of Bloch oscillations and Ramsey interferometry.
In the following subsection we will explicitly construct cTRP and subsequently prove all its desired properties (i)-(iv). We will always consider a generic 2D TR invariant band structure consisting of two time reversed Kramers partners, see FIG.\[fig:ad&twist\].
Our construction of cTRP is motivated by the experimental sequence described earlier in this section. It will reproduce the expression obtained from our interferometric protocol and thus (iv) follows naturally. Let us add that as a direct consequence of the properties (i) and (ii) the winding of cTRP yields the [$\mathbb{Z}_2~$]{}invariant, see Eq..
### Discretized version of continuous time-reversal polarization
We start by discretizing momentum space for fixed $k_y$ into $N$ equally spaced (spacing $\delta k$) points $k_x^0,...,k_x^{N-1}$. The discrete version of the Zak phase in a single gapped band ${\left|u,k_x\right\rangle}$ is then given by $$\varphi_{\text{Zak}}= -\lim_{N \rightarrow \infty} \text{arg} \biggl\{ \prod_{j=0}^{N-2} {\left\langle u,k_x^j |u,k_x^{j+1} \right\rangle} {\left\langle u,k_x^{N-1} |u,k_x^{0} \right\rangle} \biggr\}.$$ Here $\text{arg} z$ denotes the polar angle of the complex number $z$. One obtains the continuum expression Eq. for the Zak phase by using that $${\left\langle s,k_x^j |s',k_x^{j+1} \right\rangle} \approx \delta_{s,s'} - i \delta k_x \mathcal{A}^{s,s'}(k_x^j).
\label{eq:contBerryConn}$$ Here $s$ and $s'$ denote band indices (the single band above was labeled $s=s'=u$) and the Berry connection $\mathcal{A}$ was defined in Eq..
For $k_y^{\text{TRIM}}=0,\pi$ TRP is given by the difference of the Zak phases of bands ${\text{I}}$ and ${\text{II}}$ which – unlike $u$ and $l$ – are defined continuously at the Kramers-degenerate points, see Eqs. and . Due to the presence of Kramers degeneracies the discretized versions of these Zak phases contain cross terms between the energetically upper ($u$) and lower ($l$) band,
$$\varphi_{\text{Zak}}^{\text{I}}= -\lim_{N \rightarrow \infty} \text{arg} \biggl\{ \prod_{j=1}^{N/2-2} {\left\langle u,k_x^j |u,k_x^{j+1} \right\rangle} {\langle u,k_x^{N/2-1} |l,k_x^{N/2+1} \rangle}
\prod_{j=N/2+1}^{N-2} {\left\langle l,k_x^j |l,k_x^{j+1} \right\rangle} {\left\langle l,k_x^{N-1} |u,k_x^{1} \right\rangle} \biggr\},
\label{eq:discreteZakI}$$
and equivalently for $ \varphi_{\text{Zak}}^{\text{II}}$. This discrete product is shown in a graphical form in FIG. \[fig:giTRpol\] with the mid point $M=N/2$ assumed to be integer. Note that in order to avoid ambiguities in the definition of the wavefunctions at the Kramers degeneracies we did not include $k_x^{\text{TRIM}}= 0,\pi$ in the product. This is justified when taking the limit $N\rightarrow \infty$.
The above discrete expression can readily be generalized to non-TRIM $0<k_y<\pi$. To this end we introduce a discrete version of twisted Zak phases $\tilde{\varphi}_{\text{Zak}}$ (twisted polarization $\tilde{P}$) for given $k_y$ in the BZ as $$\tilde{\varphi}_{\text{Zak}}^{\text{i}}= 2 \pi \tilde{P}^{\text{i}}(k_y) = - \lim_{N \rightarrow \infty \atop M/N \text{const.}} \text{arg} \biggl\{ \prod_{j=1}^{M-2} {\left\langle u,k_x^j |u,k_x^{j+1} \right\rangle} {\left\langle u,k_x^{M-1} |l,k_x^{M+1} \right\rangle} \prod_{j=M+1}^{N-2} {\left\langle l,k_x^j |l,k_x^{j+1} \right\rangle} {\left\langle l,k_x^{N-1} |u,k_x^{1} \right\rangle} \biggr\}.
\label{eq:giTRpol}$$
Here ${\text{i}}$ is a the band index labeling the twisted contour introduced in Sec.\[subsec:IntrodAd&Twist\], see also FIGs. \[fig:giTRpol\] and \[fig:ad&twist\]; $M$ denotes the index of some intermediate band switching point, see FIG.\[fig:giTRpol\]. Analogously we can define twisted polarization $\tilde{P}^{\text{ii}}(k_y)$ (twisted Zak phase $\tilde{\varphi}_{\text{Zak}}^{{\text{ii}}}(k_y)$ of the second band ${\text{ii}}$, which is obtained from ${\text{i}}$ by exchanging energetically upper ($u$) and lower ($l$) band indices.
Like in Sec.\[subsec:IntrodAd&Twist\] we can now define the discretized version of cTRP using twisted polarizations, see Eq., $$\tilde{P}_{\theta}(k_y) = \tilde{P}^{\text{i}}(k_y) - \tilde{P}^{\text{ii}}(k_y).
\label{eq:defcTRPtheory}$$ In the following we will check all its desired properties (i)-(iv) listed above.
By construction it is clear that (i) $\tilde{P}_{\theta}(k_y^{\text{TRIM}})$ reduces to standard TRP provided that $M=N/2$ is chosen, cf. . To check (ii), i.e. continuity of $\tilde{P}_\theta(k_y)$, we notice that all scalar products are continuous as a function of $k_y$ for fixed discretization into $N$ points along $k_x$. Therefore the discrete version of cTRP is continuous as a function of $k_y$, assuming that also the band switching point labeled by $M$ changes continuously with $k_y$. Finally $\tilde{P}^{{\text{i}},{\text{ii}}}(k_y)$ – and thus $\tilde{P}_\theta(k_y)$ – are gauge invariant up to an integer. This can be seen by considering $U(1)$ gauge transformations in momentum space, ${\left|s,k_x\right\rangle} \rightarrow {\left|s,k_x\right\rangle} e^{i \vartheta_s(k_x)}$. Since all wavefunctions appear twice in , once as a bra ${\left\langles,k_x\right|}$ and once as a ket ${\left|s,k_x\right\rangle}$, all $U(1)$ phases drop out. A $2 \pi \mathbb{Z}$ ambiguity of $\tilde{\varphi}_{\text{Zak}}$ remains since $\text{arg}$ is only well-defined up to $2 \pi$ (unless Riemann surfaces are considered).
We point out that cTRP can also be used for numerical evaluation of the [$\mathbb{Z}_2~$]{}invariant. In subsection \[subsec:KaneMele\] we demonstrate this for the specific example of the Kane-Mele model [@Kane2005].
### Incomplete Zak phases and continuum version of continuous time-reversal polarization
To derive a continuum version of cTRP Eq. constructed above, we use Eq. to replace scalar products by Berry connections. Between the band switching points, for simplicity assumed to be located at $k_x=0,\pi$, we obtain e.g. $$\prod_{j=1}^{M-2} {\left\langle u,k_x^j |u,k_x^{j+1} \right\rangle} \rightarrow \exp \left[ -i \varphi_{\text{Zak},-}^u(k_y) \right]$$ with the incomplete Zak phase $\varphi_{\text{Zak},-}^u$ defined in Eq..
We are now in a position to formulate the discontinuity problem discussed in the introduction in a more precise way. For TRIM $k_y^{\text{TRIM}}$ there are two band-crossings right where we switch from one ($\varphi_{\text{Zak},-}$) to the other ($\varphi_{\text{Zak},+}$) incomplete Zak phase, see FIG. \[fig:ad&twist\] (a). Here TRP can be written in terms of incomplete Zak phases, $$P_\theta(k_y^{\text{TRIM}}) = \varphi_{\text{Zak},-}^u + \varphi_{\text{Zak},+}^l - \varphi_{\text{Zak},-}^l - \varphi_{\text{Zak},+}^u.$$ Away from TR invariant lines, $k_y \neq 0,\pi$, gaps open in the vicinity of the Kramers degeneracies, see FIG. \[fig:ad&twist\] (b). Consequently the incomplete Zak phases belong to bands that no longer cross, and their relation to TRP is strikingly different, $$P_\theta(k_y^{\text{TRIM}}) = \varphi_{\text{Zak},-}^u + \varphi_{\text{Zak},+}^u - \varphi_{\text{Zak},-}^l - \varphi_{\text{Zak},+}^l.$$
To obtain a complete continuum description of cTRP, we note that cross terms like ${\left\langle l,k_x^{N-1} |u,k_x^{1} \right\rangle}$ between energetically upper and lower band are related to *off-diagonal* elements of the non-Abelian Berry connections according to Eq.. (Note that care has to be taken in the case $k_y=k_y^{\text{TRIM}}=0,\pi$ where ${\left\langle s,k_x^{N-1} |s',k_x^{1} \right\rangle} \propto (1-\delta_{s,s'})$ for $s,s'=u,l$ as a consequence of the Kramers degeneracies.) For non-TRIM $k_y \neq k_y^{\text{TRIM}}$ we thus have $$\arg {\left\langle l,k_x^{M-1} |u,k_x^{M+1} \right\rangle} \rightarrow \arg \Bigl( -i \delta k_x \mathcal{A}^{l,u}(0,k_y) \Bigr).$$
In terms of the phase $\varphi_{\mathcal{A}}$ of $\mathcal{A}^{l,u}$ introduced in Eq. we obtain the continuum expression of twisted polarization, $$\begin{gathered}
\tilde{P}^{{\text{i}}} = \frac{1}{2 \pi} \Bigl[ \varphi_{\text{Zak},-}^u(k_y) + \varphi_{\text{Zak},+}^l(k_y) \\ - \varphi_{\mathcal{A}}(\pi,k_y) + \varphi_{\mathcal{A}}(0,k_y) \Bigr],
\label{eq:contTwistPol}\end{gathered}$$ and analogously for $\tilde{P}^{{\text{ii}}}$. This finally leads to the continuum description of cTRP, $$\begin{gathered}
\tilde{P}_\theta(k_y) =\frac{1}{2 \pi} \Bigl[ \varphi_{\text{Zak},-}^u(k_y) + \varphi_{\text{Zak},+}^l(k_y) - \varphi_{\text{Zak},-}^l(k_y) \\ -\varphi_{\text{Zak},+}^u(k_y) - 2 \bigl( \varphi_{\mathcal{A}}(\pi,k_y) - \varphi_{\mathcal{A}}(0,k_y) \bigr) \Bigr],\end{gathered}$$ which coincides with the Ramsey signal of our interferometric protocol, see Eq..
All desired properties of $\tilde{P}_\theta(k_y)$ listed in \[subsubsec:DefcTRP\] carry over from its discretized version. To get a better understanding of the physical meaning of the different terms, we now show that twisted polarization Eq. is gauge invariant up to an integer. To this end we consider a gauge-transformation, $${\left|s,k_x\right\rangle} \rightarrow e^{-i \chi_s(k_x)} {\left|s,k_x\right\rangle} \qquad s=l,u.$$ Under this transformation the diagonal of the Berry connection obtains additional *summands*, $\mathcal{A}^{s,s}(k_x) \rightarrow \mathcal{A}^{s,s}(k_x) + \partial_{k_x} \chi_s(k_x)$, whereas off-diagonal terms in the Berry connection obtain additional *factors*, $ \mathcal{A}^{u,l} \rightarrow \mathcal{A}^{u,l} e^{ i {\left(}\chi_u - \chi_l {\right)}}$, as can be seen from $$\begin{gathered}
\mathcal{A}^{u,l}(k_x) = {\left\langleu,k_x\right|} i \partial_{k_x} {\left|l,k_x\right\rangle} \rightarrow \\
\rightarrow \Bigl( \mathcal{A}^{u,l}(k_x) + \underbrace{{\left\langle u,k_x |l,k_x \right\rangle}}_{=0} {\left(}\partial_{k_x} \chi_l(k_x) {\right)}\Bigr) \times \\ \times e^{ i {\left(}\chi_u(k_x) - \chi_l(k_x) {\right)}}
= \mathcal{A}^{u,l}(k_x) e^{ i {\left(}\chi_u(k_x) - \chi_l(k_x) {\right)}} .\end{gathered}$$ Incomplete Zak phases from Eq. alone or $\varphi_{\mathcal{A}}$ from Eq. alone are *not* gauge-invariant because e.g.
\_[,-]{}\^u &\_[,-]{}\^u + \_u(0) - \_u(-) 2 ,\
\_(0) &\_(0) + ł\_u(0) - \_l(0) [)]{}2 .
However using $\chi_s(-\pi) = \chi_s(\pi) \mod 2 \pi$ ($s=u,l$) we find that twisted polarization Eq. is a gauge invariant quantity, transformations of incomplete Zak phases and phases $\varphi_{\mathcal{A}}$ cancel out.
Example: Kane-Mele model {#subsec:KaneMele}
------------------------
We will now illustrate that the winding of cTRP indeed gives the [$\mathbb{Z}_2~$]{}invariant by explicitly calculating it for the Kane-Mele model [@Kane2005]. The physical system described by this model is sketched in FIG.\[fig:KaneMele\] and its Hamiltonian reads $$\begin{gathered}
\hat{H} = t \sum_{\langle i, j \rangle} {\hat{c}^\dagger}_i {\hat{c}}_j + i \lambda_{SO} \sum_{\langle \langle i,j \rangle \rangle} \nu_{ij} {\hat{c}^\dagger}_i s^z {\hat{c}}_j
\\ + i \lambda_R \sum_{\langle i,j \rangle} {\hat{c}^\dagger}_i {\left(}{\textbf{s}} \times {\textbf{d}}_{ij} {\right)}\cdot {\textbf{e}}_z {\hat{c}}_j + \lambda_v \sum_i \xi_i {\hat{c}^\dagger}_i {\hat{c}}_i,
\label{eq:HkaneMele}\end{gathered}$$ with the same notations as in [@Kane2005]; The spin indices of ${\hat{c}^\dagger}_i,{\hat{c}}_j$ were suppressed and ${\textbf{s}}$ denotes the vector of Pauli matrices for the spins. Moreover $\nu_{ij}=2/\sqrt{3}{\left(}{\textbf{d}}_1 \times {\textbf{d}}_2 {\right)}\cdot {\textbf{e}}_z = \pm 1$ with ${\textbf{e}}_z$ the unit vector along $z$-direction and ${\textbf{d}}_{1}, {\textbf{d}}_2$ being unit vectors along the two bonds which have to be traversed when hopping between next nearest neighbor sites $j$ and $i$.
Kane and Mele started from a Hamiltonian describing two copies $\uparrow,\downarrow$ of the Haldane model [@Haldane1988] on a honeycomb lattice (first line in Eq.). Importantly, the magnetic flux seen by $\uparrow$ is opposite to that seen by $\downarrow$ which is realized by a spin-dependent next nearest neighbor hopping with amplitude $\pm i \lambda_{SO}$. They also included TR invariant Rashba SOC terms $\propto \lambda_R$ as well as a staggered sublattice potential $\propto \pm \lambda_v$ characterized by $\xi_i=\pm 1$.
In order to define cTRP we use a non-orthogonal basis in $k$-space labeled by $\kappa_x, \kappa_y$, see FIG. \[fig:KaneMele\] (b). In this basis the unit cell is given by $\kappa_x \times \kappa_y = [0,2\pi] \times [0,2\pi]$ and TRIM are found at $\kappa_x=0,\pi$ and $\kappa_y=0,\pi$. The fact that we use a non-orthogonal basis does not affect the definition of 1D Zak phases nor their relation to the [$\mathbb{Z}_2~$]{}invariant.
Using Eq. we calculate cTRP $\tilde{P}_\theta(\kappa_y)$ for band switchings at $\kappa_x=0$ as indicated in FIG.\[fig:KaneMele\](b). The result is shown in FIG. \[fig:windingKaneMele\] for $\lambda_v=0.1 t$ ($\lambda_v=0.4 t$) corresponding to a topologically non-trivial (trivial) phase. As predicted by Eq. $\tilde{P}_\theta$ does not wind in the topologically trivial case whereas it does so in the topologically non-trivial case. The example also demonstrates that the derivative $\partial_{\kappa_y} \tilde{P}_\theta(\kappa_y)$ generally takes finite values which is important to make measurements of the winding experimentally feasible.
Wilson loop scheme {#sec:WilsonScheme}
==================
As we discussed in Sec.\[subSec:Z2invAndWilsonLoops\], Wilson loops are related to the [$\mathbb{Z}_2~$]{}invariant [@Yu2011] by Eq., i.e. $${\nu_{2\text{D}}}= \frac{1}{\pi} {\left(}\Delta \varphi_W - \frac{1}{2} \Delta\Phi {\right)}\mod 2.$$ We identified two terms, the difference of Wilson loop phases $\Delta \varphi_W$ and the winding of the total Zak phase $\Delta \Phi = \int_0^\pi dk_y ~ \partial_{k_y} \Phi(k_y)$ constituting the [$\mathbb{Z}_2~$]{}invariant.
Our second interferometric scheme (Wilson loop scheme) for the measurement of the [$\mathbb{Z}_2~$]{}invariant consists of treating these two terms ($\Delta \varphi_W$ and $\Delta \Phi$) separately. The basic idea of our protocol is to express them in terms of simple Zak phases which can be measured using Ramsey interferometry in combination with Bloch oscillations [@Atala2012; @Abanin2012].
In the entire section we will assume that, when driving Bloch oscillations, non-adiabatic transitions from the valence bands ${\text{I}},{\text{II}}$ to conduction bands are suppressed. From the adiabaticity condition (given in Appendix \[sec:ApdxB\], Eq. ) we find that this is justified as long as the band gap $\Delta_{\text{band}}$ [^4] is smaller than the Bloch oscillation frequency $a F$ (with $a$ the lattice constant), $$a F \ll \Delta_{\text{band}}.$$
We start this section by discussing the relation of TR Wilson loops (\[subsec:TRWilsonLoops\]) and total Zak phase (\[sec:totalZaks\]) to simpler geometric Zak phases. Then we show in \[subsec:expRealWilsonLoops\] how this leads to a realistic experimental scheme and discuss necessary requirements.
TR Wilson loops and their phases {#subsec:TRWilsonLoops}
--------------------------------
As we pointed out in Sec.\[subSec:Z2invAndWilsonLoops\], $U(2)$ Wilson loops correspond to propagators describing completely non-adiabatic (i.e. infinitly fast) Bloch oscillations within the two bands ${\text{I}},{\text{II}}$, $$\hat{U}_{F=\infty} = \hat{W}.$$ This can be seen directly by comparing the general propagator $\hat{U}$ derived in Appendix \[sec:ApdxB\] Eq. with the definition of the Wilson loop $\hat{W}$ Eq..
An infinite driving force corresponds to the condition $\Delta_{{\text{I}}-{\text{II}}} \ll a F$ that the energy spacing $\Delta_{{\text{I}}-{\text{II}}}$ of the two bands ${\text{I}},{\text{II}}$ is always much smaller than the Bloch oscillation frequency. If this condition can be met, the Wilson loop phase can directly be measured experimentally, see Eq.. We will show below however that *even when this condition is violated* the Wilson loop phase $\varphi_W$ can still be measured, provided that TR symmetry is present.
To this end we consider TR invariant Bloch oscillations of *finite* speed within the two valence bands. With TR invariant Bloch oscillations we mean that the driving forces at momenta $\pm {\textbf{k}}(T/2\pm t)$ related by TR coincide, ${\textbf{F}}(T/2-t) = {\textbf{F}}(T/2+t)$. For simplicity we will further restrict ourselves to a homogeneous movement through the BZ in the following calculations, $${\textbf{k}}(t) = {\left(}F ~ t, 0 {\right)}^T + {\textbf{k}}(0),$$ which is TR invariant in the above sense.
The effect of TR invariant Hamiltonian dynamics within the two bands ${\text{I}},{\text{II}}$ is just a $U(1)$ *phase* $\varphi_U$, without any residual band mixing between ${\text{I}},{\text{II}}$. I.e. the propagator describing one Bloch oscillation cycle reads $$\hat{U}(k_y^{\text{TRIM}}) = e^{i \varphi_U(k_y^{\text{TRIM}})} \hat{\mathbb{I}}_{2\times 2}.
\label{eq:WilsonLoopOneSpin}$$
For an exact proof, which is a generalization of the calculation performed by Yu et.al.[@Yu2011], we refer the reader to the Appendix \[sec:ApdxC\] while here we only outline the basic idea. The propagator for propagation from $k_x$ to $k_x+\delta k_x$ is given by $\delta \hat{U}(k_x) = \exp {\left(}- i \delta k_x \hat{\mathcal{B}}_x(k_x) {\right)}$, see Eq. in Appendix \[sec:ApdxB\], with $$\hat{\mathcal{B}}_x(k_x) = \hat{\mathcal{A}}(k_x) + \frac{{\hat{\mathcal{H}}}(k_x)}{F}.$$ From TR symmetry it follows that the corresponding propagator from $-k_x-\delta k_x$ to $-k_x$ is given by $\delta \hat{U}(-k_x)= \exp {\left(}+ i \delta k_x \hat{\mathcal{B}_x}(k_x) -2 i \delta k_x \mathcal{B}_x^{U(1)}(k_x) {\right)}$ up to a gauge-dependent phase factor. (Following Yu et.al. [@Yu2011] we used that $\hat{\theta}^\dagger \hat{\sigma}^j \hat{\theta} = - \hat{\sigma}^j$ for $j=x,y,z$ while $\hat{\theta}^\dagger \hat{\mathbb{I}}_{2\times 2} \hat{\theta} = + \hat{\mathbb{I}}_{2\times 2}$. Here $\hat{\theta} = K i \hat{\sigma}^y$ denotes the TR operator.) This shows that band mixings at $-k_x$ are reversed at $+k_x$, while phases at $\pm k_x$ add up. This is depicted in FIG. \[fig:TRWilsonIdea\].
For the $U(1)$ phase $\varphi_U$ characterizing the propagator in Eq. we obtain (see Eq. in Appendix \[sec:ApdxC\]) $$\varphi_U(k_y^{\text{TRIM}}) = - \varphi_W(k_y^{\text{TRIM}}) + \frac{1}{2 F} \int_{-\pi}^\pi dk_x ~ {\text{tr}}{\hat{\mathcal{H}}}({\textbf{k}}),
\label{eq:wilsonLoopSignal}$$ which can be measured in an interferometric setup. The last term on the right hand side $\propto 1/F$ is a dynamical phase [^5] and can in principle be inferred by comparing $\varphi_U$ taken at different driving forces $F$.
Before turning to a more detailed discussion of a possible experimental protocol in subsection \[subsec:expRealWilsonLoops\], let us comment on the relation between the Wilson loop phase $\varphi_W$ and the Zak phases $\varphi_\text{Zak}$ of the time reversed bands ${\text{I}},{\text{II}}$. Since the geometric phase $\varphi_W$ in the propagator Eq. is *independent* of the speed $F$ of Bloch oscillations, we can consider the case of infinitesimal driving force $F \rightarrow 0$. In this limit, as a consequence of the adiabatic theorem, an atom starting in say band ${\text{I}}$ remains in this band. The geometric phase it picks up in this process is therefore given by the Zak phase $\varphi_{\text{Zak}}^{\text{I}}$ of the corresponding band. At the same time we can calculate this phase using the general result Eq. from which we conclude that the geometric phase picked up by the atoms is given by the Wilson loop phase $\varphi_W$. Because these two phases must coincide we have $$\varphi_W = \varphi_{\text{Zak}}^{\text{I}}= \varphi_{\text{Zak}}^{\text{II}}\mod 2 \pi.
\label{eq:ZakWilsonPhase}$$
We note that since there is a priori no fixed relation between the Zak phases at $k_y=0$ and $\pi$, Wilson loop phases $\varphi_W$ may take any value between $0$ and $2 \pi$ in general. A particular example is sketched in FIG.\[fig:WannierCenters\] (b). In [@Yu2011] it was claimed that TR Wilson loops “ are proportional to unity matrix, up to a sign”; This statement is not correct (already the Kane-Mele model [@Kane2005] provides counter examples), and in general $\Delta \varphi_W$ can take arbitrary values.
Let us furthermore mention that the results Eqs. - are relevant for the twist scheme presented in section \[sec:AdiabaticScheme\]: *To measure the Zak phase $\varphi_{\text{Zak}}^{{\text{I}}} = \varphi_{\text{Zak}}^{\text{II}}$ at TR invariant momenta $k_y$ of the two time reversed partners ${\text{I}},{\text{II}}$, adiabaticity is only required with respect to the conduction bands. The gap $\Delta_{{\text{I}}-{\text{II}}} = |E^{\text{I}}- E^{\text{II}}|$ may be arbitrarily small compared to the Bloch oscillation frequency $a F$.*
Zak phases {#sec:totalZaks}
----------
In the following we will discuss how to measure the change of total Zak phase $\Delta \Phi= \Phi(\pi)-\Phi(0)$ which is required (besides the Wilson loop phases $\Delta \varphi_W$) to obtain the [$\mathbb{Z}_2~$]{}invariant from Eq.. The basic idea is, as in the Chern number protocol [@Abanin2012], to express it as a winding (which is well-defined not only up to $2\pi$): $$\begin{gathered}
\Delta \Phi = \int_0^\pi dk_y ~\partial_{k_y} \Phi(k_y) \approx \\
\approx \sum_{k_y} \Phi(k_y+\delta k_y) - \Phi(k_y).
\label{eq:summedChange}\end{gathered}$$ Since $\Phi(k_y)$ is the sum of two Zak phases $\varphi_{\text{Zak}}^{{\text{I}},{\text{II}}}$, see Eq. , the latter can simply be measured independently, provided that the bands of interest are separated by a sufficiently large energy gap from each other. However, when accidental degeneracies are present or the gap is simply too small to follow adiabatically (which is always the case close to the Kramers degeneracies at the four TRIM), we can still infer the *total Zak phase* from non-Abelian loops.
For this purpose let us consider the general propagator $\hat{U}(T)$ within the (restricted) set of bands to which the dynamics is constrained. In practice these will be the two Kramers partners ${\text{I}},{\text{II}}$ and non-adiabatic transitions to the conduction bands can be neglected. Like in the case of a *single* band, a geometric and a dynamical $U(1)$ Berry phase can be identified, $$\begin{gathered}
i \log \det \hat{U}(T) =-\oint d{\textbf{k}} \cdot {\text{tr}}{\textbf{$\hat{\mathcal{A}}$}}({\textbf{k}}) + \int_0^T dt ~ {\text{tr}}~{\hat{\mathcal{H}}}({\textbf{k}}(t)),
\label{eq:sumZakPh}\end{gathered}$$ when the time-dependent parameter ${\textbf{k}}(t)$ returns to its initial value after time $T$. The proof of this statement is a simple non-Abelian generalization of Berrys calculation [@Berry1984] for the (Abelian) Berry phase.
When ${\textbf{k}}$ denotes quasi-momentum we will call the corresponding geometric phase the *total Zak phase*, $$\Phi = \oint d{\textbf{k}} \cdot {\text{tr}}{\textbf{$\hat{\mathcal{A}}$}}({\textbf{k}}).$$ This, of course, is exactly the definition we gave in Eq. already. Therefore we see that it is sufficient to measure the determinant of the propagator, $$\Phi(k_y) = -i \log \det \hat{U}(k_y) + \int_0^T dt ~ {\text{tr}}~{\hat{\mathcal{H}}}(k_x(t),k_y).$$
For a generic two-band model the propagator is given by a generic unitary matrix $$\hat{U} = e^{i \eta} \cdot \left( \begin{array}{cc}
\alpha & -\beta^*\\
\beta & \alpha^* \end{array} \right), \quad |\alpha|^2 + |\beta|^2 = 1,
\label{eq:2by2Unitary}$$ such that $-i \log \det \hat{U} = 2 \eta$. We will discuss below how $\eta$ can be measured using a combination of interferometry and Bloch oscillations.
Experimental realization {#subsec:expRealWilsonLoops}
------------------------
We begin this subsection by commenting on the necessary degrees of freedom to realize the Wilson loop scheme. In general, to perform interferometry one needs (at least) two auxiliary “interferometric” pseudospin degrees of freedom. The first one (referred to as ${\left|\Uparrow\right\rangle}$) picks up a phase $\varphi_\Uparrow$ that is to be measured while the second one (${\left|\Downarrow\right\rangle}$) picks up $\varphi_\Downarrow$ and serves for comparison afterwards. The interferometric signal is $\varphi_\Uparrow - \varphi_\Downarrow$. Therefore $\varphi_\Downarrow$ has to be known (it may also be a suitable known function of $\varphi_\Uparrow$).
Note that the interferometric pseudospin degrees of freedom ${\left|\Uparrow\right\rangle},{\left|\Downarrow\right\rangle}$ have to be distinguished from the “spin” pseudospin degrees of freedom ${\left|\uparrow\right\rangle},{\left|\downarrow\right\rangle}$ which mimic the electron spin of the QSHE. Therefore the Hilbert space in general consist of $${\left|\Uparrow\right\rangle} \otimes {\left|\uparrow\right\rangle}, \quad {\left|\Uparrow\right\rangle} \otimes {\left|\downarrow\right\rangle}, \quad {\left|\Downarrow\right\rangle} \otimes {\left|\uparrow\right\rangle}, \quad {\left|\Downarrow\right\rangle} \otimes {\left|\downarrow\right\rangle}.$$ Each of these sectors also contains motional degrees of freedom and we assume that the QSHE is at least realized in the sector ${\left|\Uparrow\right\rangle} \otimes \left\{ {\left|\uparrow\right\rangle}, {\left|\downarrow\right\rangle} \right\} $.
We note that the twist scheme presented in section \[sec:AdiabaticScheme\] relies only on interferometry between the bands. Therefore in this case linear combinations of ${\left|\uparrow\right\rangle}$, ${\left|\downarrow\right\rangle}$ yield the interferometric pseudospins ${\left|\Uparrow\right\rangle}$ and ${\left|\Downarrow\right\rangle}$, which are exactly the eigenstates of the Bloch Hamiltonian.
In the following we will discuss the case of *two equivalent copies* of the QSHE realized in the two sectors defined by ${\left|\Uparrow\right\rangle}$ and ${\left|\Downarrow\right\rangle}$.
### Wilson loop phase
We start by discussing the measurement of the Wilson loop phase $\Delta \varphi_W = \varphi_W(\pi) - \varphi_W(0)$. The essential idea of this part is based on the schemes [@Abanin2012; @Atala2012] for measuring Zak phases within a single band. To make the measurement more robust, we suggest a spin-echo type measurement as depicted in FIG. \[fig:WilsonMeasurement\]. In the movements along $k_y$, $\Uparrow$ ($\Downarrow$) atoms pick up geometric Wilson loop phases $\varphi_W(\pi)$ ($\varphi_W(0)$), while geometric phases corresponding to movements along $k_x$ cancel.
We assume an initial wavepacket of atoms in some superposition state ${\left|\psi_0,{\textbf{k}}\right\rangle}$ of bands ${\text{I}},{\text{II}}$ at quasi-momentum ${\textbf{k}}=(-\pi,\pi/2)$, and in the internal state ${\left|\Uparrow\right\rangle}$. A $\pi/2$-pulse between the internal states ${\left|\Uparrow\right\rangle}$, ${\left|\Downarrow\right\rangle}$ then creates a superposition $${\left|\Psi_1\right\rangle} = \frac{1}{\sqrt{2}} {\left(}{\left|\Uparrow\right\rangle} + {\left|\Downarrow\right\rangle} {\right)}\otimes {\left|\psi_0,{\textbf{k}}\right\rangle}.$$ A Zeeman field gradient for interferometric spins ${\left|\Uparrow\right\rangle}$, ${\left|\Downarrow\right\rangle}$, $$\hat{\text{H}}_Z = \int d^2 {\textbf{r}} ~ {\textbf{f}}_0 \cdot {\textbf{r}} {\left(}\hat{\Psi}^\dagger_{\Uparrow}({\textbf{r}}) \hat{\Psi}_\Uparrow({\textbf{r}}) - \hat{\Psi}^\dagger_\Downarrow({\textbf{r}}) \hat{\Psi}_\Downarrow({\textbf{r}}) {\right)}\label{eq:Hzeeman}$$ with ${\textbf{f}}_0 \propto {\textbf{e}}_y$ moves $\Uparrow$ ($\Downarrow$) atoms to $k_y=\pi$ ($k_y=0$) at fixed $k_x=-\pi$ and the state is given by $$\begin{gathered}
{\left|\Psi_2\right\rangle} = \frac{1}{\sqrt{2}} \Bigl( {\left|\Uparrow\right\rangle} \hat{U}_{\Uparrow}^{(+)} {\left|\psi_0,(-\pi,\pi)\right\rangle} \\ + {\left|\Downarrow\right\rangle} \hat{U}_{\Downarrow}^{(-)} {\left|\psi_0,(-\pi,0)\right\rangle} \Bigr) .\end{gathered}$$ Here $\hat{U}_{\Uparrow,\Downarrow}^{(\pm)}$ denote the propagators of the corresponding paths, see FIG.\[fig:WilsonMeasurement\].
Next, an equal potential gradient along ${\textbf{e}}_x$ is applied such that atoms move from $k_x=-\pi$ at time $t_1$ to $k_x=\pi$ at time $t_2$. We assume this to be done in a TR invariant fashion, i.e. $$k_x{\left(}\frac{t_2 - t_1}{2} - \delta t {\right)}= k_x{\left(}\frac{t_2 - t_1}{2} + \delta t {\right)},$$ where $k_x(t)$ is a function of time $t$. Thereby atoms only pick up the $U(1)$ phases $\varphi_U(k_y^{\text{TRIM}})$ from Eq. as discussed in subsection \[subsec:TRWilsonLoops\] and their quantum state is described by $$\begin{gathered}
{\left|\Psi_3\right\rangle} = \frac{1}{\sqrt{2}} \Bigl( e^{i \varphi_U(\pi)} {\left|\Uparrow\right\rangle} \hat{U}_{\Uparrow}^{(+)} {\left|\psi_0,(\pi,\pi)\right\rangle} \\ + e^{i \varphi_U(0)} {\left|\Downarrow\right\rangle} \hat{U}_{\Downarrow}^{(-)} {\left|\psi_0,(\pi,0)\right\rangle} \Bigr).\end{gathered}$$ As pointed out in Sec.\[subsec:TRWilsonLoops\] adiabaticity is only required with respect to the conduction band in this step.
Finally, reversing the first part of the protocol and moving the atoms back to ${\textbf{k}} = (\pi,\pi/2) = (-\pi,\pi/2) \mod 2 \pi$ yields the final state $$\begin{gathered}
{\left|\Psi_4\right\rangle} =\frac{1}{\sqrt{2}} \Bigl( e^{i \varphi_U(\pi) } {\left|\Uparrow\right\rangle} \hat{U}_{\Uparrow}^{(-)} \hat{U}_{\Uparrow}^{(+)} {\left|\psi_0,(\pi,\pi)\right\rangle} \\ + e^{i \varphi_U(0)} {\left|\Downarrow\right\rangle} \hat{U}_{\Downarrow}^{(+)} \hat{U}_{\Downarrow}^{(-)} {\left|\psi_0,(\pi,0)\right\rangle} \Bigr).
\label{eq:WLpsi4}\end{gathered}$$ Note that dynamical Zeeman-phases due to the different Zeeman fields felt by $\Uparrow$, $\Downarrow$ Eq. cancel when the protocol applied at $k_x=\pi$ reverses that at $k_x=-\pi$.
To realize a Ramsey interferometer, we have to make sure that $\hat{U}_{\Uparrow}^{(-)} \hat{U}_{\Uparrow}^{(+)} = e^{i \varphi_{y,\Uparrow}}$ and $\hat{U}_{\Downarrow}^{(+)} \hat{U}_{\Downarrow}^{(-)} = e^{i \varphi_{y,\Downarrow}}$ only constitute *dynamical phases* but not geometric phases or band-mixing between ${\text{I}}$ and ${\text{II}}$. This can be realized either by a completely non-adiabatic protocol (with $a F \gg \Delta_{{\text{I}}-{\text{II}}}$) or a completely adiabatic protocol (with $a F \ll \Delta_{{\text{I}}-{\text{II}}}$). In the former case dynamical phases are negligible while non-Abelian geometric $U(2)$ propagators cancel, i.e. $\varphi_{y,\Uparrow / \Downarrow} \approx 0$. In the latter case in contrast, there is no band-mixing between ${\text{I}},{\text{II}}$ and geometric Zak phases cancel while non-vanishing *dynamical* $U(1)$ phases $\varphi_{y,\Uparrow / \Downarrow} \propto 1/F$ are picked up.
The Ramsey signal $\Phi_R$, given by the phase difference between the $\Downarrow$ and $\Uparrow$ components in Eq., thus yields $\Phi_R = \varphi_U(0) - \varphi_U(\pi) +\varphi_{y,\Downarrow} - \varphi_{y,\Uparrow}$. Using Eq. we find that the geometric part of the Ramsey signal is given by the Wilson loop phases, $$\Phi_R = \Delta \varphi_W + \underbrace{\varphi_{\text{dyn}}}_{ \propto 1/F}.$$
Here $\varphi_{\text{dyn}}$ summarizes all dynamical phases, and they are inversely proportional to the driving force $F$. Therefore repeating the whole cycle after rescaling the time-scale by some factor allows to measure the dynamical phases, as long as adiabaticity with respect to the conduction band is still fulfilled. Moreover we can see that symmetries of the band structure might be helpful to minimize these dynamical phases and should be considered in a concrete setup.
### Total Zak phase {#subsubsec:TotalZakPhaseRealization}
Next we turn to the measurement of total Zak phase winding Eq. . We will discuss spin-echo type measurements which directly yield the difference $\Phi(k_y+\delta k_y) - \Phi(k_y)$ while canceling all dynamical phases. The sequence described in the following is depicted in FIG. \[fig:totZakMeas\].
We assume starting with atoms in the upper band ${\left|u\right\rangle}$ at ${\textbf{k}}=(0,k_y)$ in the state $${\left|\Psi_1\right\rangle}={\left|u,(0,k_y)\right\rangle} \otimes {\left(}{\left|\Uparrow\right\rangle} + {\left|\Downarrow\right\rangle} {\right)}/\sqrt{2}.$$ Then a Zeeman field gradient Eq. along ${\textbf{f}}_0 \propto {\textbf{e}}_x$ for $\Uparrow$, $\Downarrow$ can be used to move the $\Uparrow$ atoms in positive $k_x$ direction to ${\textbf{k}} = (2 \pi, k_y)$ and the $\Downarrow$ atoms in opposite direction to ${\textbf{k}} = (-2 \pi, k_y)$. After a displacement by $\delta k_y$ using a potential gradient (equal for both interferometric spins $\Uparrow$, $\Downarrow$) the sequence is reversed at $k_y+\delta k_y$. The final state is given by $${\left|\Psi_2\right\rangle} = \frac{1}{\sqrt{2}} {\left(}{\left|\Uparrow\right\rangle} \otimes \hat{U}_\Uparrow {\left|u\right\rangle} + {\left|\Downarrow\right\rangle} \otimes \hat{U}_\Downarrow {\left|u\right\rangle} {\right)}.
\label{eq:totZakfinalState}$$
From Eq. we find that dynamical phases vanish (including Zeeman phases from the different potential gradients) and the total accumulated phase yields *twice* the change of the total Zak phase, $$i \log \det {\left(}\hat{U}_\Downarrow^\dagger \hat{U}_\Uparrow {\right)}= - {\text{tr}}\oint_{\mathcal{C}} d {\textbf{k}} \cdot \hat{\mathcal{A}} \equiv 2 \Delta \Phi.$$ Here $\mathcal{C}$ denotes the (counterclockwise) contour through the BZ shown in FIG. \[fig:totZakMeas\]. Consequently it is sufficient to measure only $\det \bigl( \hat{U}_\Downarrow^\dagger \hat{U}_\Uparrow \bigr)$, and according to Eq. we have $$i \log \det {\left(}\hat{U}_\Downarrow^\dagger \hat{U}_\Uparrow {\right)}= 2 \eta_\Downarrow - 2 \eta_\Uparrow.$$
Next we assume that the two bands ${\left|u,l\right\rangle}$ are individually addressable experimentally; This is feasible with current experimental technology, see e.g. [@Cheuk2012]. The population in the upper band of the final state Eq. is described by the wave function $${\left|\psi_u\right\rangle} = \frac{1}{\sqrt{2}} {\left(}e^{i \eta_\Uparrow} \alpha_\Uparrow {\left|\Uparrow\right\rangle} +e^{i \eta_\Downarrow} \alpha_\Downarrow {\left|\Downarrow\right\rangle} {\right)}.$$ After measuring the populations $|\alpha_{\Uparrow,\Downarrow}|^2$ standard Ramsey pulses between the spin states ${\left|\Uparrow\right\rangle}$, ${\left|\Downarrow\right\rangle}$ can be used to obtain the phase-difference, $$\Delta \phi_u = \eta_\Uparrow + \arg(\alpha_\Uparrow) - \eta_\Downarrow - \arg(\alpha_\Downarrow).$$ Analogously one finds for the populations in the lower band *when also starting in the lower band* $${\left|\psi_l\right\rangle} = \frac{1}{\sqrt{2}} {\left(}e^{i \eta_\Uparrow} \alpha^*_\Uparrow {\left|\Uparrow\right\rangle} +e^{i \eta_\Downarrow} \alpha^*_\Downarrow {\left|\Downarrow\right\rangle} {\right)}$$ and the corresponding phase difference is given by $$\Delta \phi_l = \eta_\Uparrow - \arg(\alpha_\Uparrow) - \eta_\Downarrow + \arg(\alpha_\Downarrow).$$ Finally combining these equations, we find that the change of the total Zak phase is $$2 \Delta \Phi = \Delta \phi_u + \Delta \phi_l.$$
Note that if $\alpha$ is too small one may use a protocol which starts from atoms in the lower band again but detects the resulting wave function *in the upper band*. A similar calculation as above can be done and one can again infer the total Zak phase $2 \Delta \Phi$.
Summary and outlook {#sec:Summary}
===================
Summarizing, we have shown that the [$\mathbb{Z}_2~$]{}invariant classifying time-reversal invariant topological insulators can be measured using a combination of Bloch oscillations and Ramsey interferometry. The interferometric signal yields direct information about the topology of the bulk wavefunctions. We presented two schemes which are both applicable to realizations of topological insulators in ultra-cold atoms in optical lattices without the need of introducing sharp boundaries and resolving any edge states. Similar schemes have already been realized experimentally [@Atala2012] in 1D systems and discussed theoretically for 2D Chern numbers [@Abanin2012]. Unlike these situations the measurement of the [$\mathbb{Z}_2~$]{}invariant requires *non-Abelian* Bloch oscillations (i.e. some form of band switchings) and makes the interferometric protocol more involved.
Our first scheme (“twist scheme”) uses the fact that the [$\mathbb{Z}_2~$]{}invariant is the difference of time-reversal polarization at $k^{\text{TRIM}}_y=0$ and $k_y^{\text{TRIM}}=\pi$, which itself is a difference of Zak phases. Since standard time-reversal polarization is discontinuous however, its difference can not be formulated as a winding. To circumvent this issue we developed a continuous generalization of time-reversal polarization $\tilde{P}_\theta$ , the winding of which gives the [$\mathbb{Z}_2~$]{}invariant, $${\nu_{2\text{D}}}= \int_0^\pi dk_y ~\partial_{k_y} \tilde{P}_\theta(k_y) \mod 2.$$
We further laid out a measurement protocol for continuous time-reversal polarization, employing a combination of Abelian (i.e. adiabatic) Bloch oscillations with Ramsey pulses between the two valance bands required by TR symmetry. Such Ramsey pulses can easily be realized by shaking the optical lattice and using the coupling of the bands through non-Abelian Berry connections. We also pointed out that a general coupling scheme realizing the required Ramsey pulses does not work since the phases of the corresponding coupling constants at different points in the BZ are generally unknown. Our scheme is readily applicable in the suggested experimental setup [@Goldman2010]. Most importantly, it does not require any additional degrees of freedom to perform Ramsey interferometry.
The second scheme (”Wilson loop scheme“) uses a formulation of the [$\mathbb{Z}_2~$]{}invariant in terms of non-Abelian Wilson loops. In particular our protocol relies on an expression which involves eigenvalues of Wilson loops along with total Zak phases, $${\nu_{2\text{D}}}= \frac{1}{\pi} {\left(}\Delta \varphi_W - \frac{1}{2} \Delta\Phi {\right)}\mod 2.$$
The *Wilson loop phase* $\Delta \varphi_W$ is the difference of polarizations at $k_y=\pi$ and $k_y=0$. We showed that to measure the polarization of a band at time-reversal invariant momentum $k_y$, the existence of the second (partly degenerate) Kramers partner can be ignored. This is a direct consequence of TR symmetry.
Secondly the winding $\Delta\Phi$ of the *total Zak phase* is required. The total Zak phase is the *sum* of the Zak phases of the two Kramers partners and therefore continuous throughout the BZ. When the bands are separated by a sufficiently large energy gap they can be measured independently, but we also showed how one can still reliably measure their sum when Abelian Bloch oscillations are not applicable e.g. due to accidental degeneracies. The experimental realization of the Wilson loop scheme requires a second copy of the quantum spin Hall effect that can independently be controlled, making it harder to implement in some of the existing proposals.
Although for the formulation of the two protocols we restricted ourselves to two spatial dimensions, our scheme is applicable to 3D TR invariant topological insulators as well. The reason is that the 3D [$\mathbb{Z}_2~$]{}invariants (one strong and three weak ones) can be expressed as products of 2D [$\mathbb{Z}_2~$]{}invariants corresponding to specific 2D planes within the 3D Brillouin zone [@Fu2007] (see Appendix \[subsec:3DTIs\]). These constituting 2D invariants can straightforwardly be measured with our scheme.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank M. Fleischhauer, I. Bloch, M. Aidelsburger and M. Atala for stimulating discussions. F.G. wants to thank the physics department of Harvard University for hospitality during his visit. He is a recipient of a fellowship through the Excellence Initiative (DFG/GSC 266).
[70]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1126/science.1148047) [****, ()](\doibase 10.1038/nature06843) @noop [****, ()]{} [****, ()](\doibase 10.1038/nphys1102) [****, ()](\doibase
10.1038/nature08482) [****, ()](\doibase
10.1038/nature09378) @noop [****, ()]{} [****, ()](\doibase 10.1088/1367-2630/5/1/356) @noop [****, ()]{} @noop [****, ()]{} @noop [**** ()]{} [****, ()](\doibase 10.1103/PhysRevLett.106.175301) [****, ()](\doibase 10.1209/0295-5075/95/66004) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevLett.107.255301) [****, ()](\doibase 10.1209/0295-5075/93/20003) [****, ()](\doibase 10.1103/PhysRevLett.108.225303) @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1038/nature09887) [****, ()](\doibase
10.1103/PhysRevA.82.053605) [****, ()](\doibase
10.1103/PhysRevLett.105.255302) [****, ()](\doibase 10.1103/PhysRevLett.107.145301) [****, ()](\doibase
10.1103/PhysRevA.85.013638) [****, ()](\doibase 10.1103/PhysRevLett.111.225301) [****, ()](\doibase 10.1103/PhysRevLett.62.2747) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevLett.42.1698) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevA.85.033620) [****, ()](http://www.nature.com/nmat/journal/v12/n9/full/nmat3682.html) [****, ()](\doibase 10.1103/PhysRevA.85.063614) [****, ()](\doibase
10.1103/PhysRevA.79.053639) [****, ()](\doibase 10.1103/PhysRevA.82.013608) [****, ()](\doibase
10.1103/PhysRevA.81.033622) [****, ()](\doibase 10.1103/PhysRevLett.108.255303) [****, ()](\doibase 10.1103/PhysRevLett.100.070402) [****, ()](\doibase
10.1103/PhysRevLett.101.246810) [****, ()](\doibase 10.1103/PhysRevLett.97.036808) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevA.84.063629) [****, ()](\doibase 10.1103/PhysRevLett.110.166802) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevB.47.1651) [****, ()](\doibase 10.1103/PhysRevB.49.14202) [****, ()](\doibase 10.1103/RevModPhys.82.1959) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevB.84.075119) [****, ()](\doibase 10.1103/PhysRevLett.52.2111) @noop [**** ()]{} [****, ()](\doibase 10.1103/PhysRevB.34.3625) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1134/S106377881005011X) @noop [****, ()]{}
Relation between Zak phase and Chern number {#sec:Apdx:ZakChern}
===========================================
In the main text we mentioned that the Chern number is related to the winding of the Zak phase across the BZ, $${\text{Ch}}= \frac{1}{2 \pi} \int_{-\pi}^\pi dk_y ~ \partial_{k_y} \varphi_{\text{Zak}}(k_y).$$ Here we present a simple proof of this formula by starting from the definition of the Chern number as the (quantized) Hall conductivity, see Eq..
To this end we consider the 2D system as a collection of 1D systems labeled by their lattice momentum $k_y$. Applying an electric field $E_y$ corresponds to driving Bloch oscillations, i.e. the momentum $k_y$ changes in time according to $e E_y = \hbar \partial_t k_y$. At the same time the polarization of each 1D system, $P(k_y)$, changes accordingly, $P(t)=P(k_y(t))$. If in time $T$, $k_y$ changes by $2\pi$, we have $k_y(t) = k_y^0 + \frac{t}{T} 2 \pi$. The change of the total polarization gives the current density $$J_x = \frac{e}{T L_y} \sum_{k_y^0} \int_0^T dt ~ \partial_t P {\left(}k_y^0 + \frac{t}{T} 2 \pi {\right)},$$ where $L_y$ is the length of the sample in $y$-direction. Since $\partial_{k_y} P(k_y)$ is $2\pi$-periodic in $k_y$ this simplifies and we obtain the relation to the Chern number: $$\sigma_{xy} = \frac{e^2}{h} \int_{-\pi}^{\pi} dk_y ~\partial_{k_y} P(k_y) = \frac{e^2}{h} {\text{Ch}}.$$ Importantly, we use windings $\int dk_y \partial_{k_y}$ rather than differences because gauge-transformations can change the polarization by an integer. Note that due to the periodicity of $P(k_y)$ in $k_y$, ${\text{Ch}}$ is quantized. Finally using Eq. we can express the Chern number as the winding of the Zak phase, as we wanted to show.
[$\mathbb{Z}_2~$]{}topological invariant {#sec:Z2inv}
========================================
In this Appendix we give a more rigorous but pedagogical introduction to the different formulations of the [$\mathbb{Z}_2~$]{}invariant used in the main text. It is written as self-contained as possible and some results mentioned already in the main text will thus be repeated.
[$\mathbb{Z}_2~$]{}invariant and time-reversal polarization {#sec:apdx:ZtAndTRP}
-----------------------------------------------------------
Our starting point are two copies (spin $\uparrow$ and $\downarrow$) of the quantum Hall effect, where spin is conserved $[{\hat{\mathcal{H}}},\hat{\sigma}^z]=0$. In this case the [$\mathbb{Z}_2~$]{}invariant is defined as the difference of the spin up and down Chern numbers [@Kane2005], $${\nu_{2\text{D}}}= \frac{1}{2} {\left(}{\text{Ch}}_\uparrow - {\text{Ch}}_\downarrow {\right)}.
\label{eq:ntDszCons}$$ The Chern number is defined as the integral of the Berry curvature $\mathcal{F}$ over the entire BZ [@Thouless1982], $${\text{Ch}}= \frac{1}{2 \pi} \int_{\text{BZ}} d^2k ~\underbrace{\epsilon_{\mu,\nu} \partial_\mu \mathcal{A}_\nu}_{= \mathcal{F}({\textbf{k}})}, \quad \mu,\nu=x,y,$$ where $\epsilon_{\mu,\nu}$ is the totally antisymmetric tensor.
The Chern number can be written as a winding of polarization across the BZ, $${\text{Ch}}= \int_{-\pi}^\pi dk_y ~\partial_{k_y} P(k_y),$$ see Appendix \[sec:Apdx:ZakChern\] and recall that polarization and Zak phase are related through Eq.. Therefore we can write the [$\mathbb{Z}_2~$]{}invariant for the case of conserved spin Eq. in terms of polarizations. When doing so we also use that $\mathcal{F}_\downarrow(-{\textbf{k}}) = - \mathcal{F}_\uparrow({\textbf{k}})$ as a consequence of TR invariance. Then we can express ${\nu_{2\text{D}}}$ as a winding over only *half the BZ*, $${\nu_{2\text{D}}}= \int_{0}^{\pi} dk_y ~\partial_{k_y} {\left(}P_\uparrow(k_y) - P_\downarrow(k_y) {\right)}.
\label{eq:ntDwinding}$$ Motivated by this expression and following Fu and Kane [@Fu2006] we can introduce the *time-reversal polarization* $P_\theta$ (TRP) of two bands $\uparrow,\downarrow$ as $$P_{\theta}(k_y) = P_\uparrow(k_y) - P_\downarrow(k_y).$$ Thus the last equation for the [$\mathbb{Z}_2~$]{}invariant Eq. states that ${\nu_{2\text{D}}}$ is given by the winding of TRP *when spin is conserved*.
Fu and Kane [@Fu2006] realized however that TRP is integer quantized for TR invariant $k_y^{\text{TRIM}}=0,\pm \pi$ even in the presence of arbitrary SOC. In this case the emerging bands ${\text{I}}$ and ${\text{II}}$ can no longer be labeled by their spin quantum number. It can easily be checked that in TR constrained gauge, where $\chi({\textbf{k}}) =0$ is chosen in Eq. , $P^{\text{I}}= P^{\text{II}}$ at $k_y^{\text{TRIM}}=0,\pm \pi$ as a direct consequence of TR symmetry Eq.. Since gauge transformations can only change polarizations by an integer amount it follows that in a general gauge $$P_\theta( k_y^{\text{TRIM}}) \in \mathbb{Z}, \quad k_y^{\text{TRIM}}=0,\pi.$$ Therefore one can construct an integer quantized topological invariant defined as the difference of TRP at TR invariant momenta, ${\nu_{2\text{D}}}=P_\theta(\pi)-P_\theta(0) \in \mathbb{Z}$. (We discuss below why only *two* values are topologically distinct, which leads to the [$\mathbb{Z}_2~$]{}classification.) Importantly for this definition a *continuous* gauge has to be used in the entire BZ, since otherwise $P_\theta(\pi)$ and $P_\theta(0)$ could independently be changed by discontinuous gauge transformations. We note that such a gauge choice is always possible when the total Chern number vanishes [@KOHMOTO1985]. This is indeed the case here, since we may conclude from TR symmetry that ${\text{Ch}}_{\text{I}}+ {\text{Ch}}_{\text{II}}=0$.
Finally we discuss why only a [$\mathbb{Z}_2~$]{}classification survives. To this end we note that for a general Hamiltonian without accidental degeneracies, TRP can only change by $\Delta P_\theta =0,\pm 1$ between $k_y=0,\pi$. This is because otherwise there exists some intermediate $k_y\neq 0,\pm \pi$ with $P^{\text{I}}= P^{\text{II}}$, and as pointed out by Yu et.al.[@Yu2011] small TR invariant perturbations can split this degeneracy (of polarizations) away from Kramers degeneracies, see FIG.\[fig:WannierCenters\](d). Moreover since $\Delta P=-1$ and $\Delta P=+1$ only differ by exchanging up and down spins, they should be topologically equivalent. Therefore the topological invariant can only take two topologically distinct values $\Delta P = 0,1$ and we end up with $${\nu_{2\text{D}}}= P_{\theta}(\pi) - P_{\theta}(0) \mod 2.$$
Wilson loops {#subsecAppdx:WilsonLoops}
------------
In the main text Sec.\[subSec:Z2invAndWilsonLoops\] we motivated $U(2)$ Wilson loops as natural generalizations of Abelian Zak phases (single band) to multiple bands. We also mentioned their relation to the [$\mathbb{Z}_2~$]{}invariant Eq. which we will prove in this subsection.
To this end we first summarize the formulation of the [$\mathbb{Z}_2~$]{}invariant derived by Fu and Kane [@Fu2006]. They assumed the most general gauge Eq. which can be characterized by the so-called sewing matrix, $$w_{s,s'}({\textbf{k}})={\left\langleu_s(-{\textbf{k}})\right|} ~\hat{\theta}~ {\left|u_{s'}({\textbf{k}})\right\rangle},
\label{eq:sewingCond}$$ where $s,s'$ are band indices (${\text{I}},{\text{II}}$). Their expression for ${\nu_{2\text{D}}}$ reads $${\left(}-1 {\right)}^{{\nu_{2\text{D}}}} = \prod_{j=1}^4 \frac{\sqrt{\det w(\Gamma_j)}}{{\text{Pf }}w(\Gamma_j)} \equiv \prod_{j=1}^4 \delta_{\Gamma_j}.
\label{eq:ntDFuKane}$$ Here ${\text{Pf }}$ denotes the Pfaffian of an antisymmetric matrix, ${\textbf{k}}=\Gamma_j$ denote the four TRIM in the 2D BZ $$\Gamma_1=(0,0),~ \Gamma_2=(\pi,0),~ \Gamma_3=(\pi,\pi),~ \Gamma_4=(0,\pi),$$ and the branch of the square root in Eq. has to be chosen correctly, see [@Fu2006]. Yu et.al. [@Yu2011] calculated TR invariant two-by-two Wilson loops (time reversed bands ${\text{I}}$ and ${\text{II}}$) at $k_y^{\text{TRIM}}=0,\pi$ and found at $k_y^{\text{TRIM}}=0$ $$\hat{W}(0) = e^{-\frac{i}{2} \Phi(0)} ~ \delta_{\Gamma_1} ~\delta_{\Gamma_2} ~ \hat{\mathbb{I}}_{2\times 2} = e^{-i \varphi_W(0)} ~ \hat{\mathbb{I}}_{2\times 2}.
\label{eq:TRinvWilsonLoops}$$ Here $\Phi(k_y)$ denotes the total Zak phase, see Eq.. A similar formula holds for $k_y^{\text{TRIM}}=\pi$ with $ \Phi(0) \rightarrow \Phi(\pi)$ and $ \delta_{\Gamma_1} ~\delta_{\Gamma_2} \rightarrow \delta_{\Gamma_3} ~\delta_{\Gamma_4} $. (We generalized the proof given by these authors from Wilson loops to arbitrary TR invariant propagators, and our generalized result can be found in Eq. in Appendix \[sec:ApdxC\].)
To proceed we note that since the determinant of an anti-symmetric matrix is given be the square of its Pfaffian, $\det w(\Gamma_j)= {\text{Pf }}^2 w(\Gamma_j) $, $\delta_\Gamma$ can only take the two values $\pm 1$, $$\delta_{\Gamma_j} = \frac{\sqrt{{\text{Pf }}^2 w(\Gamma_j) }}{{\text{Pf }}w(\Gamma_j)} \in \left\lbrace \pm 1 \right\rbrace.
\label{eq:FuKanesqrtDetPf}$$ Therefore we may rewrite Eq. as $$e^{ i \pi {\nu_{2\text{D}}}} = {\left(}-1 {\right)}^{{\nu_{2\text{D}}}} =\frac{\delta_{\Gamma_1} ~\delta_{\Gamma_2}}{ \delta_{\Gamma_3} ~\delta_{\Gamma_4} }.$$ Taking the product of the Wilson loop at $k_y^{\text{TRIM}}=0$ and the inverse Wilson loop at $k_y^{\text{TRIM}}= \pi$ we get according to Eq. $$e^{i {\left(}\varphi_W(\pi) - \varphi_W(0) {\right)}} = e^{- i {\left(}\Phi(0) - \Phi(\pi) {\right)}/ 2 } ~ e^{ i \pi {\nu_{2\text{D}}}} .$$ Therefore we have $${\nu_{2\text{D}}}= \frac{1}{\pi} {\left(}\Delta \varphi_W - \frac{1}{2} {\left(}\Phi(\pi) - \Phi(0) {\right)}{\right)}\mod 2,$$ from which our previously claimed equation immediately follows using continuity of the total Zak phase $\Phi(k_y)$.
We conclude this subsection by commenting on alternative formulations of the [$\mathbb{Z}_2~$]{}invariant. In [@Fu2006] the [$\mathbb{Z}_2~$]{}invariant was expressed as an obstruction to continuously defining a gauge in the BZ. This lead to a formulation of ${\nu_{2\text{D}}}$ entirely in terms of Berry’s connection and Berry’s curvature which is valid however only when TR invariant gauge (i.e. $\chi({\textbf{k}})=0$ in Eq.) is used. We emphasize that the formula Eq. we employ in this paper also only involves Berry’s connections, but without any restriction of the gauge. The relation between the two expressions is shown in the Appendix \[sec:ApdxA\_B\]. Finally the [$\mathbb{Z}_2~$]{}invariant is also related to the systems response to spin dependent twisted boundary conditions which lead to the classification in terms of a Chern number matrix [@Sheng2006].
The 3D case {#subsec:3DTIs}
-----------
In 3D two kinds of topological invariants exist [@Fu2007]. There is one strong topological invariant, which is protected against TR invariant (non-magnetic) disorder. It can be written as a product of 2D invariants for subsystems at different $k_z=0,\pi$: $$(-1)^{\nu_{3\text{D}}} = (-1)^{{\nu_{2\text{D}}}(k_z=0)} \cdot (-1)^{{\nu_{2\text{D}}}(k_z=\pi)}.$$ On the other hand, there are also 3 additional weak topological invariants which are not protected against any kind of disorder. They as well may be formulated in terms of 2D invariants of different subsystems: $$(-1)^{\nu_i} = (-1)^{{\nu_{2\text{D}}}(k_i=\pi)}, \qquad i=x,y,z.$$ Consequently, measuring 3D [$\mathbb{Z}_2~$]{}invariants only requires the measurement of the [$\mathbb{Z}_2~$]{}invariants of different 2D subsystems within the 3D BZ.
Bloch oscillation’s equations of motion {#sec:ApdxB}
=======================================
Atoms in optical lattices undergo Bloch oscillations when a constant force ${\textbf{F}}(t)$ is applied. They can be described by the following Schrödinger equation, $${\left|\psi({\textbf{r}},t)\right\rangle} = \underbrace{{\left(}H \pm {\textbf{F}} \cdot {\textbf{r}} {\right)}}_{=H_B} {\left|\psi({\textbf{r}},t)\right\rangle}.
\label{eq:defHBforBOEOM}$$ We assume that non-adiabatic conduction-band mixing is negligible. Using the Landau-Zener probability for band-mixing one finds the following adiabaticity condition: $$\omega_\text{B} = a |{\textbf{F}}| \ll \frac{\Delta_{\text{band}}^2 2\pi}{\Delta_{{\text{I}}-{\text{II}}}}
\label{eq:adiabCond}$$ with $\Delta_{\text{band}}$ the band gap, $\Delta_{{\text{I}}-{\text{II}}}$ the energy spacing between valence bands ${\text{I}}, {\text{II}}$, $a$ the size of the unit cell and $\omega_\text{B}$ the Bloch oscillation frequency. We may now decompose the wavefunction into Bloch states ${\left|\Phi_{s,{\textbf{k}}}({\textbf{r}})\right\rangle}$: $${\left|\psi({\textbf{r}},t)\right\rangle} = \sum_{s={\text{I}},{\text{II}}} \int_{\text{BZ}} d^2 {\textbf{k}} ~\psi_{s,{\textbf{k}}}(t) {\left|\Phi_{s,{\textbf{k}}}({\textbf{r}})\right\rangle}.$$ For simplicity we only consider the case of two bands $s={\text{I}},{\text{II}}$ here. Using orthogonality $$\int d^2 {\textbf{r}} {\left\langle\Phi_{s,{\textbf{k}}}({\textbf{r}})\right|} \left. \Phi_{s',{\textbf{k}}'}({\textbf{r}}) \right\rangle = \delta ({\textbf{k}}-{\textbf{k}}') \delta_{s,s'},$$ one obtains equations of motion for the amplitudes $\psi_{s,{\textbf{k}}}(t)$: $$\begin{gathered}
i \partial_t \psi_{s,{\textbf{k}}}(t) = \sum_{s'={\text{I}},{\text{II}}} \int_{\text{BZ}} d^2 {\textbf{k}}' ~ \psi_{s',{\textbf{k}}'}(t) \times \\
\int d^2{\textbf{r}} ~ {\left\langle\Phi_{s,{\textbf{k}}}({\textbf{r}})\right|} H_B {\left|\Phi_{s',{\textbf{k}}'}({\textbf{r}})\right\rangle}.\end{gathered}$$ With the Bloch theorem, ${\left|\Phi_{s,{\textbf{k}}}({\textbf{r}})\right\rangle} = e^{i {\textbf{k}} \cdot {\textbf{r}}} {\left|u_{s,{\textbf{k}}}({\textbf{r}})\right\rangle}$, we find $$\begin{gathered}
\sum_{s'={\text{I}},{\text{II}}} \int_{\text{BZ}} d^2 {\textbf{k}} ~ \psi_{s',{\textbf{k}}'}(t) {\textbf{F}} \cdot {\textbf{r}} e^{i {\textbf{k}}' \cdot {\textbf{r}}} {\left|u_{s',{\textbf{k}}'}({\textbf{r}})\right\rangle} = \\
= i \sum_{s'={\text{I}},{\text{II}}} \int_{\text{BZ}} d^2 {\textbf{k}}' ~ {\textbf{F}} \cdot \nabla_{{\textbf{k}}'} {\left(}\psi_{s',{\textbf{k}}'}(t) {\left|u_{s',{\textbf{k}}'}({\textbf{r}})\right\rangle} {\right)}e^{i {\textbf{k}}' \cdot {\textbf{r}}}.\end{gathered}$$ After defining the time-dependent quasi-momentum $${\textbf{k}}(t) = {\textbf{k}}_0 \mp \int^t_0 {\textbf{F}} d\tau
\label{eq:koft}$$ and introducing the amplitudes at these ${\textbf{k}}$ components, $$\phi_{s,{\textbf{k}}}(t) :=\psi_{s,{\textbf{k}}(t)}(t),$$ it is easy to derive their equations of motion: $$\begin{gathered}
i \partial_t \phi_{s,{\textbf{k}}}(t) = \sum_{s'} \left[ \pm {\textbf{F}}(t) \cdot \mathcal{A}^{s,s'}{\left(}{\textbf{k}}(t) {\right)}\right. \\ \left. + H^{s,s'}{\left(}{\textbf{k}}(t){\right)}\right] \phi_{s,{\textbf{k}}}(t).\end{gathered}$$ Now each ${\textbf{k}}$-component sees a different $t$-dependent Hamiltonian but there is no mixing *between* different ${\textbf{k}}$. This is a direct consequence of the translational symmetry of the problem. Formally these equations can be solved by a time-ordered exponential, which translates into a path-ordered one when using Eq..
The full propagator is thus given by $$U_{{\textbf{k}}_2,{\textbf{k}}_1} = \mathcal{P} \exp \left[ -i \int_{{\textbf{k}}_1}^{{\textbf{k}}_2} d k ~ \underbrace{{\left(}\mathcal{A}({\textbf{k}}) \pm \frac{1}{F} H({\textbf{k}}) {\right)}}_{=\mathcal{B}({\textbf{k}})} \right].
\label{eq:diabPropagator}$$
Non-universal Franck-Condon factor phases {#sec:AppdxNonUniversalFCPh}
=========================================
In this appendix we discuss general interferometric sequences realizing the twist scheme presented in Sec. \[sec:AdiabaticScheme\]. To this end we consider the most general coupling scheme realizing Ramsey pulses between the two bands ${\text{I}},{\text{II}}$. We show that, in general, additional phases are picked up in the cycle which depend on the intrinsic properties of the Bloch functions. This rules out many simpler schemes realizing Ramsey pulses between the two bands for the measurement of cTRP.
We start by formalizing the idea of a band-switching, which is realized by some time-dependent microscopic Hamiltonian $$\hat{\text{H}}_{\text{rf}}(t) = e^{i {\left(}\varphi_E + \omega_{\text{rf}}t {\right)}} \hat{p},$$ with $\omega_{\text{rf}}$ the frequency of the (typically radio-frequency, rf) transition, $\varphi_E$ the phase of the driving field and $\hat{p}$ some microscopic operator coupling the two bands (called $\hat{p}$ in analogy to an atomic dipole operator in quantum optics).
In a rotating frame and in the Bloch function basis this Hamiltonian may generally be described by $$\label{eq:rfHam}
{\hat{\mathcal{H}}}_{{\text{rf}}}({\textbf{k}}) = \Omega_{\text{rf}}({\textbf{k}}) {\left|u,{\textbf{k}}\right\rangle}{\left\langlel,{\textbf{k}}\right|} + \text{h.c.},$$ where $\Omega({\textbf{k}},t) = e^{i \varphi_\Omega({\textbf{k}})} \cos {\left(}\omega_{\text{rf}}({\textbf{k}}) ~ t {\right)}$ is the Rabi frequency for atoms at quasi-momentum ${\textbf{k}}$. The phase $\varphi_\Omega=\varphi_E+\tilde{\varphi}_{{\text{FC}}}$ of the driving is then determined by the phase of the driving field $\varphi_E$ relative to the phase $\tilde{\varphi}_{{\text{FC}}}$ of the corresponding Franck-Condon (FC) factors $$\tilde{\varphi}_{{\text{FC}}} = \arg {\left\langleu,{\textbf{k}}\right|} \hat{\text{H}}_{\text{rf}}(0) {\left|l,{\textbf{k}}\right\rangle},
\label{eq:FCphaseDef}$$ with $\text{arg}$ denoting the polar angle of a complex number.
When transitions take place between given atomic (e.g. hyperfine) states one can make use of the freedom in the choice of the global $U(1)$ phase in order to eliminate the appearance of FC factor phases. In our case however *two* FC phases appear at the two band switching points $k_x=0,\pi$ and only *one* of them may be eliminated using the global $U(1)$ gauge freedom.
The difference between FC phases at *different* momenta however carries information about the band structure and can not be eliminated. In fact, it contains exactly those terms we need to connect incomplete Zak phases from different bands $u$ and $l$ in a meaningful way. To see this we decompose $\tilde{\varphi}_{{\text{FC}}}$ into the gauge-dependent term $\varphi_{\mathcal{A}}$ from Eq. and a gauge invariant remainder $$\varphi_{\text{FC}}:= \tilde{\varphi}_{\text{FC}}-\varphi_{\mathcal{A}}.
\label{eq:giFCphase}$$
To prove gauge invariance of $\varphi_{\text{FC}}$ we note that the Hamiltonian is invariant under local $U(1)$ gauge-transformations in momentum space, ${\left|u,{\textbf{k}}\right\rangle} \rightarrow e^{i \vartheta^{u}({\textbf{k}})} {\left|u,{\textbf{k}}\right\rangle}$ and analogously for the lower band $l$. Therefore $\Omega_{{\text{rf}}}$ transforms as $\Omega_{{\text{rf}}} \rightarrow \Omega_{{\text{rf}}} e^{i{\left(}\vartheta^{l}-\vartheta^{u} {\right)}}$, and one easily checks that this is also how $\mathcal{A}_x^{u,l}$ transforms. Since $\varphi_E$ is gauge-invariant this shows that so is $\varphi_{\text{FC}}$, and from now on we can forget about $\tilde{\varphi}_{\text{FC}}$. Summarizing we have $$\quad \varphi_\Omega({\textbf{k}}) = \varphi_{\text{FC}}({\textbf{k}}) + \varphi_\mathcal{A}({\textbf{k}}) + \varphi_E({\textbf{k}}).$$
It is crucial for our measurement scheme to consider FC factor phases $\varphi_{{\text{FC}}}$, which in general take non-universal values. Let us illustrate this for a simple example. In experimental schemes [@Liu2010; @Goldman2010; @Beri2011] the spin states $\uparrow, \downarrow$ are typically proposed to be realized as hyperfine states. In general the spins will be coupled in some way by the Bloch Hamiltonians ${\hat{\mathcal{H}}}({\textbf{k}})$ (realizing SOC) and the FC phases depend on the spin-mixture in the Bloch eigenfunctions. We will consider a toy model of a two dimensional Hilbert space with the two orthogonal bands ${\left|u\right\rangle}=\alpha e^{i \phi_\alpha} {\left|\uparrow\right\rangle}+\beta e^{i \phi_\beta} {\left|\downarrow\right\rangle}$ and ${\left|l\right\rangle}=\beta e^{-i \phi_\beta} {\left|\uparrow\right\rangle}-\alpha e^{-i \phi_\alpha} {\left|\downarrow\right\rangle}$. Here the amplitudes $\alpha,\beta$ as well as the phases $\phi_\alpha, \phi_\beta$ are chosen to be real numbers.
The simplest rf Hamiltonian flips the spins but leaves spatial coordinates unchanged, $$\hat{\text{H}}_{\text{rf}}= \Omega_{\text{rf}}{\left|\uparrow\right\rangle}{\left\langle\downarrow\right|} + \Omega_{\text{rf}}^* {\left|\downarrow\right\rangle}{\left\langle\uparrow\right|}.
\label{eq:HrfSpinFlip}$$ According to Eqs. and we thus have $\varphi_{\text{FC}}= \arg {\left(}- \alpha^2 e^{-2i\phi_\alpha} \Omega_{\text{rf}}+ \beta^2 e^{-2i\phi_\beta} \Omega_{\text{rf}}^* {\right)}- \varphi_{\mathcal{A}}$. We note that $\Delta \phi = \phi_\alpha-\phi_\beta$ is gauge invariant (up to $2\pi$) and from the last equation we conclude that the FC phase $\varphi_{\text{FC}}$ generally depends on $\Delta \phi$. Therefore a simple Ramsey pulse using rf transition between internal spin states Eq. can generally *not* be used to realize the band switchings required for the measurement of cTRP, unless for some reason the intrinsic FC phases $\varphi_{\text{FC}}$ at the band switching points are known.
The scheme presented in Sec.\[subsec:InterferometricSeq\] yields universal FC phases, i.e. $\varphi_{\text{FC}}=0$ for the Hamiltonian given in Eq.. This was achieved by coupling only to the motional degrees of freedom but not to the (pseudo) spins $\uparrow,\downarrow$.
TR non-adiabatic loops {#sec:ApdxC}
======================
In this appendix we derive formulas for the propagators describing Bloch oscillations in 1D TR invariant band structures. Our calculations straightforwardly generalize the results obtained by Yu et.al. [@Yu2011].
The generic form of the propagator describing Bloch oscillations within two bands ${\text{I}},{\text{II}}$ between quasi momenta $k_{1,2}$ is derived in Appendix \[sec:ApdxB\], and it is given by (see eq) $$\hat{U}(k_2;k_1) = \mathcal{P} \exp {\left(}- i \int_{k_1}^{k_2}dk ~\hat{\mathcal{B}}(k) {\right)}, \quad k_2 > k_1.$$ Here $\hat{B}(k)$ describes geometrical as well as dynamical contributions, $$\hat{\mathcal{B}}(k) = \hat{\mathcal{A}}(k) \pm \frac{{\hat{\mathcal{H}}}(k)}{F(k)},$$ and the sign $\pm$ corresponds to the direction of the driving force $F$, cf. Eq.. We will consider a single Kramers pair, i.e. $\hat{\mathcal{A}},\hat{\mathcal{B}},{\hat{\mathcal{H}}},\hat{U}$ are all two-by-two matrices in the band indices ${\text{I}},{\text{II}}$ and $\hat{\theta}=K (i \hat{\sigma}^y)$ denotes TR. Furthermore we assume TR invariant driving of the Bloch oscillations, i.e. forces at $\pm k$ are related by $F(-k) = F(k)$.
In the context of the QSHE these propagators correspond to non-adiabatic generalizations of Zak phases along $k_x$ at $k_y=0,\pi$. More specifically, for infinite driving $F\rightarrow \infty$ (or equivalently $\|{\hat{\mathcal{H}}}\| \rightarrow 0$) they correspond to the non-Abelian $U(2)$ Wilson loops, $\hat{U}=\hat{W}$. For this case results were obtained in [@Yu2011], and in the following we will generalize the latter to finite $F$. Generally one expects $F \neq 0$ to cause qualitative changes of the propagators since the commutator $[{\hat{\mathcal{H}}},\hat{\mathcal{A}}] \neq 0$ in general. However, as will be shown below, when TR invariant loops are considered, non-zero $F$ only yields a dynamical $U(1)$ phase factor (instead of a $U(2)$ rotation).
In the following we will consider a general gauge characterized by $\chi(k)$, see Eq.. Starting from ${\left|u_{\text{I}}(k)\right\rangle}$ defined in some continuous gauge on the entire BZ $-\pi < k \leq \pi$, $\chi(k)$ fixes $${\left|u_{\text{II}}(k)\right\rangle} = e^{i \chi(-k)} \hat{\theta} {\left|u_{\text{I}}(-k)\right\rangle}
\label{eq:chiDefIIfromI}$$ for all $k$. We will without loss of generality assume a continuous gauge choice on the patches $-\pi< k < 0$ as well as $0 < k <\pi$, whereas discontinuities of $\chi(k)$ are allowed at the sewing points $k=0,\pm \pi$. We note that in the construction of the Bloch eigenfunctions the gauge-choice $${\left|u(k+G)\right\rangle} = e^{-i G x} {\left|u(k)\right\rangle}$$ was made with $G \in 2 \pi \mathbb{Z}$ a reciprocal lattice vector, see [@Zak1989]. This imposes a constraint on the possible discontinuities of $\chi(k)$ at $k=0,\pi$ since
[|u\_()]{} &= e\^[-i 2x]{} [|u\_(-]{}\
&= - e\^[-i 2x]{} e\^[i (-)]{} [|u\_()]{}\
&= - e\^[-i 2x]{} e\^[i (-)]{} e\^[-i 2x]{} [|u\_(-)]{}\
&= - e\^[i (-)]{} e\^[i ()]{} [|u\_()]{}\
&= e\^[i ]{} [|u\_()]{}.
Therefore $\chi(-\pi) - \chi(\pi) \in 2\pi \mathbb{Z}$, and similarly around $k=0$. Defining the difference $$\eta(k) := \chi(k) - \chi(-k)
\label{eq:defEtak}$$ we thus obtain $$\eta(0), \eta(\pi) \in 2\pi \mathbb{Z}.
\label{eq:intQuantEta}$$ Using the relation we find by an explicit calculation $$\hat{\theta}^\dagger \hat{\mathcal{A}}(-k) \hat{\theta} = \hat{\Xi}^\dagger_k \hat{\mathcal{A}}(k) \hat{\Xi}(k) + \partial_k ~\text{diag}{\left(}\chi(k), \chi(-k) {\right)},$$ where the gauge choice enters in the definition of the following unitary matrix $$\hat{\Xi}_k = \text{diag} {\left(}e^{-i \eta(k) / 2} , e^{i \eta(k) / 2} {\right)}.$$ Using TR invariance, $\hat{\theta}^{\dagger} {\hat{\mathcal{H}}}(-{\textbf{k}}) \hat{\theta} = {\hat{\mathcal{H}}}({\textbf{k}})$ and $F(-k)=F(k)$, together with the fact that ${\hat{\mathcal{H}}}(k) = \text{diag} {\left(}E_{\text{I}}, E_{\text{II}}{\right)}$ such that $[{\hat{\mathcal{H}}},\hat{\Xi}_k]=0$, we find that also $$\hat{\theta}^\dagger \hat{\mathcal{B}}(-k) \hat{\theta} = \hat{\Xi}^\dagger_k \hat{\mathcal{B}}(k) \hat{\Xi}_k + \partial_k ~\text{diag}{\left(}\chi(k), \chi(-k) {\right)}$$ This can be rewritten as $$\hat{\theta}^\dagger \hat{\mathcal{B}}(-k) \hat{\theta} = \hat{\Xi}^\dagger_k \hat{\mathcal{B}}(k) \hat{\Xi}_k + i \hat{\Xi}^\dagger_k \partial_k \hat{\Xi}_k + \frac{1}{2} \partial_k {\left(}\chi(k) + \chi(-k) {\right)},$$ where the first two terms on the right hand side describe a gauge transformation of the effective connection $\hat{\mathcal{B}}$ when $\hat{\Xi}_k$ is a continuous unitary matrix. This condition is indeed fulfilled on the two patches $(0,\pm \pi)$ since $\eta(k)$ Eq. was chosen continuously there. From the transformation properties of Wilson loops under this gauge transformation [@Makeenko2010] we obtain: $$\hat{\theta}^\dagger \hat{U}(0;-k) \hat{\theta} = \hat{\Xi}^\dagger_0 \hat{U}(k;0)^\dagger \hat{\Xi}_k ~ e^{-i \Lambda}
\label{eq:UTR1}$$ where $\Lambda=\frac{1}{2} \bigl( \chi(0-) + \chi(0+) - \chi(-k) - \chi(k) \bigr)$.
Now we will derive a second expression for the transformation properties of $\hat{U}(0;-k)$ under TR. Since $\hat{\mathcal{B}}^\dagger = \hat{\mathcal{B}}$ we may write it as $$\hat{\mathcal{B}} = \mathcal{B}^{U(1)} \hat{\mathbb{I}}_{2\times 2} + \sum_{j=1,2,3} \mathcal{B}^{SU(2),j} \hat{\sigma}^j,
\label{eq:Bexpansion}$$ with $\mathcal{B}^{SU(2),j}$ and $\mathcal{B}^{U(1)}$ real numbers. From Eq. and using $\hat{\theta}^\dagger \hat{\sigma}^j \hat{\theta} = - \hat{\sigma}^j$ for $j \neq 0$ we obtain $$\hat{\theta}^{\dagger} {\left(}- i \hat{\mathcal{B}}(k) {\right)}\hat{\theta} = -i \hat{\mathcal{B}}(k) + 2 i \mathcal{B}^{U(1)}(k) \hat{\mathbb{I}}_{2\times 2},$$ and therefore we also find $$\hat{\theta}^{\dagger} \hat{U}(k;0) \hat{\theta} = \hat{U}(k;0) \exp {\left(}2i \int_0^k dk ~ \mathcal{B}^{U(1)}(k) {\right)}.
\label{eq:UTR2}$$
Combining the results from Eqs. and , we obtain for TR symmetric propagators from $-k$ to $k$:
(k & ;-k) = (k;0) (0;-k)\
&= \^\
&= (k;0) e\^[ł2i \_0\^k dk \^[U(1)]{}(k) -i [)]{}]{} \^\_0 (k;0)\^\_k \^\
&= ł-2i \_0\^k dk \^[U(1)]{}(k) + i [)]{} \_0\^\_k \^.
In the last step we used the fact that $\hat{U}(k;0)$ is unitary, as well as the integer quantization of $\eta(0)$ Eq. $$\hat{\Xi}_0 = \text{diag} {\left(}e^{-i \eta(0)/2} , e^{i \eta(0)/2} {\right)}= {\left(}-1 {\right)}^{\eta(0)/2 \pi} \hat{\mathbb{I}}_{2\times 2}.
\label{eq:Xi0Quant}$$ The result can be further simplified by noting that
\^[U(1)]{}(-k) &= (-k)\
&= łK (-k) K [)]{}(\^= )\
&= ł\^(-k) [)]{}\
&= ł\_k\^(k) \_k [)]{}+ \_k ( (k)+(-k) )\
&= \^[U(1)]{}(k) + \_k ( (k)+(-k) ).
Using this we have $$-2i \int_0^k dk ~\mathcal{B}^{U(1)}(k) = -i \int_{-k}^k dk~ \mathcal{B}^{U(1)}(k) - i \Lambda,$$ and we thus obtain $$\hat{U}(k;-k) = e^{{\left(}-i \int_{-k}^k dk ~ \mathcal{B}^{U(1)}(k) {\right)}} ~ \hat{\theta} \hat{\Xi}_0^\dagger \hat{\Xi}_k \hat{\theta}^\dagger.$$ Note that until here even the phases of the matrices are well defined (i.e. the above calculations can be thought of being performed on a Riemann surface in the complex plane). We will now drop this additional constraint and using Eq. we finally obtain the full propagator as $$\hat{U}(\pi;-\pi) = {\left(}-1 {\right)}^{\frac{\eta(0) + \eta(\pi)}{2 \pi}} ~ e^{{\left(}-i \int_{-\pi}^\pi dk ~ \mathcal{B}^{U(1)}(k) {\right)}}~ \hat{\mathbb{I}}_{2\times 2}.$$ The factor ${\left(}-1 {\right)}^{\frac{\eta(0) + \eta(\pi)}{2 \pi}}$ can be related to the Pfaffian-expressions Eq.. Therefore we note that $$w(k) = \left( \begin{array}{cc}
0 & -e^{-i \chi(-k)} \\
e^{-i \chi(-k)} & 0
\end{array} \right)$$ and thus $\det w(k) = e^{i {\left(}\chi(k) + \chi(-k) {\right)}}$ as well as ${\text{Pf }}w(k) = -e^{-i \chi(k)}$. To evaluate Eq. it is important to choose the branch cut of the square root correctly [@Fu2006]. To avoid these difficulties we use the simpler but lengthy formula $\delta_0 \delta_\pi = (-1)^{P_\theta}$ with the expression for TRP [@Fu2006]
P\_&=\
&=\
&= + 2 .
Therefore we end up with $$\hat{U}(\pi;-\pi) = \delta_0 \delta_\pi \exp {\left(}-i \int_{-\pi}^\pi dk_x~\mathcal{B}^{U(1)}(k_x) {\right)}\hat{\mathbb{I}}_{2\times 2}.
\label{eq:AppBwilson}$$
By taking the limit $F\rightarrow \infty$ in Eq. we recover the Wilson loop phase $$e^{-i \varphi_W} = \delta_0 \delta_\pi \exp {\left(}-i \int_{-\pi}^\pi dk_x~\mathcal{A}^{U(1)}(k_x) {\right)}$$ derived in [@Yu2011]. Thus our final result for the propagator of general TR invariant Bloch oscillations within a single Kramers pair reads $$\hat{U}(\pi;-\pi) = e^{-i \varphi_W} \exp {\left(}\mp i \frac{1}{2F} {\text{tr}}\int_{-\pi}^\pi dk_x~{\hat{\mathcal{H}}}(k_x) {\right)}.
\label{eq:UTRBOsViaWilsonLoops}$$
Relation to to the TR constraint formula for ${\nu_{2\text{D}}}$ {#sec:ApdxA_B}
================================================================
Fu and Kane [@Fu2006] identified the [$\mathbb{Z}_2~$]{}invariant as an obstruction for a continuous definition of the gauge respecting TR symmetry, i.e. where $\chi({\textbf{k}})=0$ in Eq.. If such a gauge is chosen, they showed that the [$\mathbb{Z}_2~$]{}invariant can be written as $${\nu_{2\text{D}}}= \frac{1}{2\pi} {\left(}\int_{\partial \tau_{1/2}} d\ell ~ {\text{tr}}\mathcal{A} - \int_{\tau_{1/2}} d \tau_{1/2} ~ {\text{tr}}\mathcal{F} {\right)}\mod 2,
\label{eq:OstructionFormula}$$ where $\mathcal{F} = d \mathcal{A}+ \mathcal{A} \wedge \mathcal{A}$ denotes the Berry curvature and $\tau_{1/2}$ half the BZ. Importantly, the gauge is generally not continuous on $\partial \tau_{1/2}$. If it is however, Stokes theorem immediately gives ${\nu_{2\text{D}}}=0$. The second term in may be rewritten as $$- \frac{1}{2 \pi} \int d \tau_{1/2} ~ {\text{tr}}\mathcal{F} = \frac{1}{2\pi} {\left(}\Phi(\pi) - \Phi(0) {\right)},$$ see Appendix \[sec:Apdx:ZakChern\]. This is exactly the second, gauge-invariant term in Eq. . Since the TR invariant gauge was used, the Zak phases of different Kramers partners are equal. Identifying points in the BZ at $k_x=\pm \pi$ we can thus write: $$\frac{1}{2\pi} \int_{\partial \tau_{1/2}} d\ell ~ {\text{tr}}\mathcal{A} = - \frac{1}{\pi}\left[ \varphi_{\text{Zak}}^s(\pi) - \varphi_{\text{Zak}}^s(0) \right],$$ where $s={\text{I}},{\text{II}}$. Since Wilson loop phases coincide with Zak phases, see Eq., $$\frac{1}{2\pi} \int_{\partial \tau_{1/2}} d\ell ~ {\text{tr}}\mathcal{A} = - \frac{1}{\pi} \Delta \varphi_W \mod 2.$$ We therefore recover the gauge-invariant formulation involving TR Wilson loop phases.
[^1]: We can assume $\mathcal{A}({\textbf{k}})$ to be continuous on the small patch $[-\pi,\pi) \times [0, k_y]$ in the BZ, with $0 < k_y \ll 2 \pi$.
[^2]: For properties of Wilson loops, see e.g. [@Makeenko2010].
[^3]: The path ordering operator $\mathcal{P}$ is defined similar to the time-ordering operator. For $k_2 > k_1$ ($k_2 < k_1$) and acting on an operator valued function $\hat{{\mathcal{A}}}(k)$ it is defined by $\mathcal{P} \hat{{\mathcal{A}}}(k_2) \hat{{\mathcal{A}}}(k_1) = \hat{{\mathcal{A}}}(k_2) \hat{{\mathcal{A}}}(k_1)$ ($= \hat{{\mathcal{A}}}(k_1) \hat{{\mathcal{A}}}(k_2)$).
[^4]: The band gap $\Delta_{\text{band}}$ is defined as the minimum energy spacing from the two bands ${\text{I}}$ or ${\text{II}}$ to any further (conduction) bands. Here we assume that the band gap is larger or comparable to the width of the valence band $\Delta_{{\text{I}}-{\text{II}}}$, i.e. $\Delta_{{\text{I}}- {\text{II}}} \lesssim \Delta_{\text{band}}$.
[^5]: When a single band is considered the dynamical phase reduces to the well-known result; Taking ${\hat{\mathcal{H}}}({\textbf{k}}) = \text{diag} {\left(}E({\textbf{k}}), E({\textbf{k}}) {\right)}$ we obtain for the dynamical phase in Eq. $\varphi_U(k_y^{\text{TRIM}}) + \varphi_W(k_y^{\text{TRIM}}) = \frac{1}{F} \int_{-\pi}^\pi dk_x ~ E({\textbf{k}}) = \int_{0}^{2 \pi / ( a F )} dt ~ E({\textbf{k}}(t))$
|
---
abstract: 'Current status and future prospects of the structure functions and parton distribution studies are presented.'
author:
- 'J. C. Peng'
bibliography:
- 'sample.bib'
title: 'Structure Functions - Status and Prospect'
---
[ address=[University of Illinois, Urbana, Illinois, 61801, U.S.A.]{} ]{}
Introduction
============
The study of nucleon’s structure functions and parton distributions is an active area of research in nuclear and particle physics. The parton distributions address both the perturbative and nonperturbative aspects of QCD, and they also provide an essential input for describing hard processes in high-energy hadron collisions. As a result of several decades’s intense effort, the unpolarized proton structure functions have been well mapped out over a broad range of $Q^2$ and Bjorken-$x$. While these data are invaluable for testing QCD and for extracting various parton distributions, several questions remain unanswered. For example, the unexpected finding of the flavor asymmetry of the light-quark sea ($\bar u,
\bar d$) suggests that other aspects of the flavor structure, such as possible asymmetry between the $s$ and $\bar s$ sea quark distributions and the bahavior of valence $d/u$ ratio at large $x$, need to be examined. The issue of quark-hadron duality, as reflected in the intriguing similarity between the structure functions measured at the resonance region and at the DIS region, also requires further studies.
Remarkable progress in the study of spin-dependent structure functions has been made since the discovery of the “proton spin puzzle" in the late 1980’s. Very active spin-physics programs have been pursued at many facilities including SLAC, CERN, HERA, JLab, and RHIC. The polarized DIS data now cover a sufficiently broad $Q^2$ range for scaling-violation to be observed. In recent years, new experimental tools such as semi-inclusive polarized DIS, polarized proton-proton collision, and deeply exclusive reactions have been employed to address the major unresolved question in spin physics: How is the proton’s spin distributed among its various constituents?
On the theory front, the formulation of the generalized parton distributions as well as the identification of various $k_T$ (intrinsic transverse momentum of partons)-dependent structure and fragmentation functions have opened exciting new directions of research. Furthermore, important progress in the Lattice calculations for the moments of various parton distributions and in the extrapolations to their chiral limits has been made.
In this review I will focus on recent progress in the following areas:
- Flavor structure of parton distributions
- Transition from high-$Q^2$ to low-$Q^2$
- Novel distribution and fragmentation functions; Generalized parton distributions
Flavor structures of parton distributions
=========================================
$\bar d / \bar u$ flavor asymmetry
----------------------------------
The earliest parton models assumed that the proton sea was flavor symmetric, even though the valence quark distributions are clearly flavor asymmetric. The flavor symmetry assumption was not based on any known physics, and it remained to be tested. Under the assumption of a $\bar u$, $\bar d$ flavor-symmetric sea in the nucleon, the Gottfried Sum Rule [@gott], $I_G =
\int_0^1 (F^p_2 (x,Q^2) - F^n_2 (x,Q^2))/x~ dx = 1/3$, is obtained. The NMC collaboration determined the Gottfried integral to be $ 0.235\pm 0.026$, significantly below 1/3. This surprising result can be explained by a large flavor asymmetry between the $\bar u$ and the $\bar d$.
The $x$ dependence of $\bar d / \bar u$ asymmetry has been determined by proton-induced Drell-Yan (DY) as well as semi-inclusive DIS measurements. Figure 1 shows that the Fermilab E866 [@towell01] DY cross section per nucleon for $p + d$ clearly exceeds $p + p$, and it indicates an excess of $\bar d$ with respect to $\bar u$ over an appreciable range in $x$. In contrast, the $\sigma(p+d)/2\sigma(p+p)$ ratios for $J/\Psi$ and $\Upsilon$ production, also shown in Fig. 1, are very close to unity. This reflects the dominance of gluon-gluon fusion process for quarkonium production and the expectation that the gluon distributions in the proton and in the neutron are identical.
Many theoretical models, including meson cloud model, chiral-quark model, Pauli-blocking model, instanton model, chiral-quark soliton model, and statistical model, have been proposed to explain the $\bar d/ \bar u$ asymmetry. For recent reviews, see [@kumano98; @garvey02]. These models can describe the $\bar d - \bar u$ data very well, as shown in Fig. 1. However, they all have difficulties explaining the $\bar d / \bar u$ data at large $x$ ($x>0.2$). The new 120 GeV Fermilab Main Injector and the proposed 50 GeV Japanese Hadron Facility present opportunities for extending the $\bar d/ \bar u$ measurement to larger $x$ ($0.25 < x < 0.7$).
Models in which virtual mesons are admitted as degrees of freedom have implications that extend beyond the $\bar d, \bar u$ flavor asymmetry addressed above. They create hidden strangeness in the nucleon via such virtual processes as $p \to \Lambda + K^+, \Sigma + K$, etc. Such processes are of considerable interest as they imply different $s$ and $\bar s$ parton distributions in the nucleon, a feature not found in gluonic production of $s \bar s$ pairs.
A difference between the $s$ and $\bar s$ distribution can be made manifest by direct measurements of the $s$ and $\bar s$ parton distribution functions in neutrino DIS. A fit to the CDHS neutrino charged-current inclusive data together with charged lepton DIS data found evidence for $\int_0^1 s(x) dx > \int_0^1 \bar s(x) dx$ [@barone00]. However, an analysis [@zeller02a] of the recent CCFR and NuTeV $\nu (\bar \nu) N \to \mu^+ \mu^- x$ dimuon production data [@goncharov01] favored $\int_0^1 s(x) dx < \int_0^1 \bar s(x) dx$ ($\int_0^1 (s(x) - \bar s(x)) dx = -0.0027 \pm 0.0013$). To better determine the $s/\bar s$ asymmetry, an NLO analysis is currently underway [@olness03]. Violation of the $s/\bar s$ symmetry would have impact on the recent extraction [@zeller02b] of sin$^2\theta_W$ from the CCFR/NuTeV $\nu N$ scattering data.
Asymmetry in the $s, \bar s$ distributions can also be revealed in the measurements of the strange quark’s contribution to the nucleon’s electromagnetic and axial form factors. These “strange” form factors can be measured in neutrino elastic scattering [@garvey93] from the nucleon, or by selecting the parity-violating component of electron-nucleon elastic scattering. Two completed parity-violating experiments [@spayde00; @aniol01] suggest small contributions of strange quarks to nucleon form factors. Several new experiments are underway at JLab and MAMI to measure parity-violating asymmetry at various kinematic regions.
Flavor structure of polarized nucleon sea
-----------------------------------------
The flavor structure and the spin structure of the nucleon sea are closely connected. Many theoretical models originally proposed to explain the $\bar d / \bar u$ flavor asymmetry also have specific implications for the spin structure of the nucleon sea. In the meson-cloud model, for example, a quark would undergo a spin flip upon an emission of a pseudoscalar meson ($
u^\uparrow \to \pi^\circ (u \bar u, d \bar d) + u^\downarrow,~u^\uparrow
\to \pi^+ (u \bar d) + d^\downarrow,~u^\uparrow \to K^+ + s^\downarrow$, etc.). The antiquarks ($\bar u, \bar d, \bar s$) are unpolarized ($\Delta \bar u = \Delta \bar d = \Delta \bar s = 0$) since they reside in spin-0 mesons. The strange quarks ($s$), on the other hand, would have a negative polarization.
In the chiral-quark soliton model [@diakonov96; @wakamatsu98], the polarized isovector distributions $\Delta \bar u(x) - \Delta \bar d(x)$ appears in leading-order ($N_c^2$) in a $1/N_c$ expansion, while the unpolarized isovector distributions $\bar u(x) - \bar d(x)$ appear in next-to-leading order ($N_c$). Therefore, this model predicts a large flavor asymmetry for the polarized sea $[\Delta \bar u (x) - \Delta \bar d(x)] > [\bar d(x) - \bar u(x)]$.
The HERMES collaboration has recently reported the extraction of $\Delta \bar u(x)$, $\Delta \bar d(x)$, and $\Delta \bar s(x) (=\Delta s(x))$ using polarized semi-inclusive DIS (SIDIS) data [@hermes03a]. Although the statistics are still limited, the HERMES results for $\Delta \bar u, \Delta \bar d, \Delta
\bar u - \Delta \bar d$, as shown in Fig. 2, are all consistent with being zero. In particular, there is no evidence for a large positive $\Delta \bar u(x) - \Delta \bar d(x)$ asymmetry as was predicted [@dressler00] by the chiral quark soliton model. Figure 2 also shows that $\Delta s$ tends to be positive, in contrast to the predictions of a negative polarization of the strange sea in the analysis of inclusive DIS and hyperon decay data assuming SU(3) symmetry. However, the HERMES result of $\Delta s =
0.03 \pm 0.03 \pm 0.01$ over $0.023 < x < 0.3$ is not in disagreement with the inclusive DIS result of $(\Delta s + \Delta \bar s)/2 \simeq -0.02$ [@adeva98].
Another promising technique for measuring sea-quark polarization is $W$-boson production [@bunce00] at RHIC. The longitudinal single-spin asymmetry for $W$ production in polarized $ p + p \to W^{\pm} + x$ gives a direct measure of sea-quark polarization. The RHIC $W$-production and the HERMES SIDIS measurements are clearly complementary tools for determining polarized sea quark distributions.
$d/u$ ratio at large $x$
------------------------
Another quantity related to the flavor symmetry of the proton is the $d/u$ ratio at large $x$. Assuming SU(2)$_{spin} \times$ SU(2)$_{flavor}$ symmetry, the proton wave function is given as $$\begin{aligned}
|p>\uparrow & = & \frac{1}{\sqrt{2}} u\uparrow (ud)_{S=0,S_Z=0}
+ \frac{1}{\sqrt{18}}u\uparrow (ud)_{S=1,S_Z=0}
- \frac{1}{3} u\downarrow (ud)_{S=1,S_Z=1} \nonumber \\
& & -\frac{1}{3} d\uparrow
(uu)_{S=1,S_Z=0} + \frac{\sqrt{2}}{3} d\downarrow (uu)_{S=1,S_Z=1}
\label{Eq:proton}\end{aligned}$$ The neutron wave function is readily obtained from $u \leftrightarrow d$ interchange. In nature, the SU(2)$_{spin} \times$ SU(2)$_{flavor}$ symmetry is clearly broken, as evidenced by the large $N-\Delta$ mass splitting. The dynamic origins of this symmetry breaking remains unclear. Close and Carlitz [@close73; @carlitz75] argued that the dominance of the $S=0$ diquark configuration over the $S=1$ configuration would account for the $N-\Delta$ mass splitting as well as the SU(2) $\times$ SU(2) symmetry breaking. An alternative suggestion, based on perturbative QCD, was offered by Farrar and Jackson [@farrar75]. They pointed out that the spin-aligned diquark configuration with $S_Z=1$ is suppressed since only longitudinal gluons can be exchanged. A similar result was also obtained by Brodsky et al. [@brodsky95] using counting rule argument. It is straightforward to show that in the $x \to 1$ limit, the different models predict the folllowing values for various ratios:
- SU(2)$_{spin} \times$ SU(2)$_{flavor}$ symmetry: $\frac{d}{u} = \frac{1}{2},~\frac{\Delta u}{u} = \frac{2}{3},~
\frac{\Delta d}{d} = - \frac{1}{3},~\frac{F^n_2}{F^p_2} = \frac{2}{3}$.
- $S=0$ diquark dominance: $\frac{d}{u} = 0,~\frac{\Delta u}{u} = 1,~
\frac{\Delta d}{d} = - \frac{1}{3},~\frac{F^n_2}{F^p_2} = \frac{1}{4}$.
- $S_Z = 0$ diquark dominance: $\frac{d}{u} = \frac{1}{5},~\frac{\Delta u}{u} = 1,~
\frac{\Delta d}{d} = 1,~~~~\frac{F^n_2}{F^p_2} = \frac{3}{7}$.
The distinct predictions for $F^n_2/F^p_2$ from various models could be tested against DIS experiments. However, there exist considerable uncertainties in the extraction of $F^n_2$ from the measurement of $F^d_2$. Depending on the treatment of the nuclear effects in the deuteron, very different values for $F^n_2/F^p_2$ (and $d/u$) were obtained at large $x$ [@wally96]. It is clearly desirable to measure $d/u$ without the need to model nuclear effects in the deuteron. One method is to measure the charge asymmetry of $W$ production in $p-\bar p$ collision. Indeed, the CDF data [@cdf98] on the $W$ charge asymmetry have already provided useful constraints on the $d/u$ ratio.
The $d/u$ ratio can also be probed by measuring the $e^- p \to \nu_e x$ and $e^+ p \to \bar \nu_e x$ charged-current DIS, where the underlying processes are $e^- u \to \nu_e d$ and $e^+ d \to \bar \nu_e u$, respectively. The recent H1 charged-current data [@zhang01], shown in Fig. 3, indicate that the $u$ quark density at large $x (x=0.65)$ is smaller than expected from the current PDF parametrization. Very recently, the Fermilab E866/NuSea collaboration reported the absolute Drell-Yan cross sections of 800 GeV $p + p$ and $p + d$ [@webb03]. As shown in Fig. 3, the data fall below the PDF predictions at large $x$ (up to $x=0.8$). The H1 and the E866 results suggest that $u$ quark density at large $x$ might be smaller than expected from current PDFs. This clearly would impact on the $d/u$ ratio at large $x$, as shown in a recent global PDF analysis [@tung03].
The uncertainties involved in the extraction of $F^n_2$ from $F^d_2$ data can be greatly reduced using the technique of neutron-tagging. A new experiment [@bonus03] has been proposed at the JLab Hall-B to detect $e^- d \to e^- p x$, where a low-energy recoiled proton will be measured in coincidence with the $(e,e^{\prime})$ scattering. Using this method, the $F^n_2/F^p_2$ ratio over the range $0.2 < x < 0.7$ could be determined with small systematic uncertainties.
Transition from high-$Q^2$ to low-$Q^2$
=======================================
Quark-hadron duality
--------------------
The recent studies at JLab of the spin-averaged and spin-dependent structure functions at low $Q^2$ region have shed new light on the subject of quark-hadron duality. Thirty years ago, Bloom and Gilman [@bloom70] noticed that the structure functions obtained from deep-inelastic scattering experiments, where the substructures of the nucleon are probed, are very similar to the averaged structure functions measured at lower energy, where effects of nucleon resonances dominate. This surprising similarity between the resonance electroproduction and the deep inelastic scattering suggests a common origin for these two phenomena, called local duality.
Recently, high precision data [@niculescu00] from JLab have verified the quark-hadron duality for spin-averaged scattering on proton and deuteron targets. For $Q^2$ as low as 0.5 GeV$^2$, the resonance data are within 10% of the DIS results. When the mean $F_2$ curve from the resonance data is plotted as a function of the Nachtmann variable, $\xi = 2x/(1+\sqrt{1+4M^2x^2/Q^2})$, it resembles the $xF_3$ structure function obtained in neutrino scattering experiments. Since $xF_3$ is a measure of the valence quark distributions, this suggests that the $F_2$ structure function at low $Q^2$ originates from valence quarks only.
The study of quark-hadron duality was recently extended to other structure functions. Results from HERMES [@hermes03b] show that duality is also observed for the spin-dependent quantity $A^p_1$. Another recent result from JLab shows that the nuclear modifications to the unpolarized structure functions in the resonance region are in surprisingly good agreement with those measured in DIS [@arrington03].
$\Gamma_1(Q^2)$ at low $Q^2$ and the generalized GDH integral
-------------------------------------------------------------
The extensive data on $g_1(x,Q^2)$ allow accurate determinations of the integrals $\Gamma_1^{p,n}(Q^2) = \int_{0}^{1} g_1^{p,n}(x,Q^2)dx$ for the proton and the neutron, as well as $\Gamma_1^p(Q^2) - \Gamma_1^n(Q^2)$. While the values of $\Gamma_1^p$ and $\Gamma_1^n$ are different from the predictions of Ellis and Jaffe who assumed SU(3) flavor symmetry and an unpolarized strange sea, the data are in good agreement with the prediction of the Bjorken sum rule.
How does $\Gamma_1(Q^2)$ evolve as $Q^2 \to 0$? This question is closely related to the Gerasimov-Drell-Hearn (GDH) sum rule:
$$\int_{\nu_0}^{\infty} [\sigma_{1/2}(\nu) - \sigma_{3/2}(\nu)] \frac{d\nu}{\nu}
= -\frac{2\pi^2\alpha}{M^2} \kappa^2.
\label{Eq:GDH}$$
The GDH sum rule, based on general physics principles (causality, unitarity, Lorentz and gauge invariances) and dispersion relation, relates the total absorption cross sections of circularly polarized photons on longitudinally polarized nucleons to the static properties of the nucleons. In Eq. \[Eq:GDH\], $\sigma_{1/2}$ and $\sigma_{3/2}$ are the photo-nucleon absorption cross sections of total helicity of $1/2$ and $3/2$, $\nu$ is the photon energy and $\nu_0$ is the pion production threshold, $M$ is the nucleon mass and $\kappa$ is the nucleon anomalous magnetic moment. The GDH integral in Eq. \[Eq:GDH\] can be generalized from real photon absorption to virtual photon absorption with non-zero $Q^2$:
$$\begin{aligned}
I_{GDH}(Q^2) \equiv \int_{\nu_0}^{\infty} [\sigma_{1/2}(\nu,Q^2)
- \sigma_{3/2}(\nu,Q^2)] \frac{d\nu}{\nu} = \frac{16\pi^2\alpha}{Q^2}
\Gamma_1(Q^2).
\label{Eq:GGDH}\end{aligned}$$
Eq. \[Eq:GGDH\] shows that the $Q^2$-dependence of the generalized GDH integral is directly related to the $Q^2$-dependence of $\Gamma_1$. The GDH sum rule (Eq. \[Eq:GDH\]) predicts $\Gamma_1^p = 0$ at $Q^2=0$ with a negative slope for $d\Gamma_1^p(Q^2)/dQ^2$ and $\Gamma_1^p$ is known to be positive at high $Q^2$, therefore, $\Gamma_1^p(Q^2)$ must become negative at low $Q^2$.
The GDH integrals at low $Q^2$ have recently been measured in several experiments at JLab [@amarian02; @fatemi03] and HERMES [@hermes03c]. Results from a JLab Hall-B measurement [@fatemi03] of $\Gamma_1^p(Q^2)$ are shown in Fig. 4. These data indeed show that $\Gamma_1^p$ changes sign around $Q^2 = 0.3$ GeV$^2$. The origin of the sign-change can be attributed to the competition between $\Delta(1232)$ and higher nucleon resonances. At the lowest $Q^2$, the $\Delta(1232)$ has a dominant negative contribution to $\Gamma_1^p$. However, at larger $Q^2$, higher mass nucleon resonances take over to have a net positive $\Gamma_1^p$.
Results [@amarian02] from a JLab Hall-A measurement of the generalized GDH integral for neutron using a polarized $^3$He target are shown in Fig. 4. In contrast to the proton case, the strong negative contribution to the GDH integral from the $\Delta(1232)$ resonance now dominates the entire measured $Q^2$ range. Future experiments at JLab will extend the measurements down to $Q^2=0.02$ GeV$^2$ in order to map out the low $Q^2$ behavior of the neutron and proton generalized GDH integrals.
Novel distribution and fragmentation functions
==============================================
In addition to the unpolarized and polarized quark distributions, $q(x,Q^2)$ and $\Delta q(x,Q^2)$, a third quark distribution, called transversity, is the remaining twist-2 distribution yet to be measured. This helicity-flip quark distribution, $\delta q(x,Q^2)$, can be described in quark-parton model as the net transverse polarization of quarks in a transversely polarized nucleon. Due to the chiral-odd nature of the transversity distribution, it can not be measured in inclusive DIS experiments. In order to measure $\delta q(x,Q^2)$, an additional chiral-odd object is required. For example, the double spin asymmetry, $A_{TT}$, for Drell-Yan cross section in transversely polarized $p p$ collision, is sensitive to transversity since $A_{TT} \sim \sum_{i} e_i^2 \delta q_i(x_1) \delta \bar q_i(x_2)$. Such a measurement could be carried out at RHIC [@bunce00], although the anticipated effect is small, on the order of $1-2$%.
Several other methods for measuring transversity have been proposed for semi-inclusive DIS. In particular, Collins suggested [@collins93] that a chiral-odd fragmentation function in conjunction with the chiral-odd transversity distribution would lead to a single-spin azimuthal asymmetry in semi-inclusive pion production.
The HERMES collaboration recently reported [@hermes03d] observation of single-spin azimuthal asymmetry for charged and neutral hadron electroproduction. Using unpolarized positron beam on a longitudinally polarized hydrogen and deuterium targets, the cross section was found to have a sin$\phi$ dependence correlating with the target spin direction. $\phi$ is the azimuthal angle between the pion and the $(e, e^\prime)$ scattering plane. This Single-Spin-Asymmetries (SSA) can be expressed as the analyzing power in the sin$\phi$ moment, and the result is shown in Fig. 5. The sin$\phi$ moment for an unpolarized (U) positron scattered off a longitudinally (L) polarized target contains two main contributions $$\begin{aligned}
\langle sin \phi \rangle & \alpha & S_L \frac{2 (2-y)}{Q\sqrt{1-y}}
\sum_{q} e_q^2 x h_L^q(x) H_1^{\bot,q}(z)
+ S_T (1-y) \sum_{q} e_q^2 x h_1^q(x) H_1^{\bot,q}(z),
\label{Eq:ssa1}\end{aligned}$$ where $S_L$ and $S_T$ are the longitudinal and transverse components of the target spin orientation with respect to the virtual photon direction. For the HERMES experiment with a longitudinally polarized target, the transverse component is nonzero with a mean value of $S_T \approx 0.15$. The observed azimuthal asymmetry could be a combined effect of the $h_1$ transversity and the twist-3 $h_L$ distribution. Recently, another mechanism involving a chiral-even T-odd Sivers distribution function [@sivers90] was shown to contribute to azimuthal asymmetry [@brodsky02; @collins02]. For a longitudinally polarized target the Collins and the Sivers mechanisms can not be distinguished.
If the azimuthal asymmetry observed by HERMES is indeed caused by the $h_1$ transversity, a much larger asymmetry is expected for a transversely polarized target. The HERMES and COMPASS collaborations have collected polarized SIDIS using transversely polarized hydrogen and $^6$LiD targets, respectively. These data would shed much light on the origins of the SSA and could also disentangle the Sivers effect from the Collins effect. The Collins effect has a sin$(\phi^l_h + \phi^l_s)$ dependence while the Sivers effect is proportional to sin$(\phi^l_h - \phi^l_s)$, where $\phi^l_s = \phi_s - \phi^l$ is the angle between target spin and the lepton scattering plane. For longitudinally polarized target $\phi^l_s = 0$ and the two effects have identical $\phi$ dependence. For transversely polarized target, however, $\phi^l_s \neq 0$ and the two effects can be separated.
The Collins fragmentation function represents a correlation between the quark’s transverse spin and the transverse momentum of the leading hadron formed in the fragmentation process. The Sivers distribution function reflects the correlation between the quark’s transverse spin and its transverse momentum within the proton. It has been shown [@anselmino95; @anselmino98] that both the Collins and the Sivers effects can contribuite to the analysing power $A_N$ observed in the Fermilab E704 $p\uparrow p \to \pi x$ reaction [@adams91]. Very recently, $A_N$ was measured [@bland02] at RHIC at a much higher energy of $\sqrt s =$ 200 GeV using transversely polarized proton beams. The RHIC data could provide new information on the Collins and Sivers functions.
Generalized parton distributions
================================
There has been intense theoretical and experimental activities in recent years on the subject of Generalized Parton Distribution (GPD). In the Bjorken scaling regime, exclusive leptoproduction reactions can be factorized into a hard-scattering part calculable in QCD, and a non-perturbative part parameterized by the GPDs. The GPD takes into account dynamical correlations between partons with different momenta. In addition to the dependence on $Q^2$ and $x$, the GPD also depends on two more parameters, the skewedness $\xi$ and the momentum transfer to the baryon, $t$. Of particular interest is the connection between GPD and the nucleon’s orbital angular momentum [@ji97].
The deeply virtual Compton scattering (DVCS), in which an energetic photon is produced in the reaction $e p \to e p \gamma$, is most suitable for studying GPD. Unlike the exclusive meson productions, DVCS avoids the complication associated with mesons in the final state and can be cleanly interpreted in terms of GPDs. An important experimental challenge, however, is to separate the relatively rare DVCS events from the abundant electromagnetic Bethe-Heitler (BH) background. From the collision of 800 GeV protons with 27.5 GeV positrons, both the ZEUS [@chekanov03] and the H1 [@adloff01] collaborations at DESY observed an excess of $e^+ + p \to e^+ + \gamma + p$ events in a kinematic region where the BH cross section is largely suppressed. The excess events were attributed to the DVCS process and the ZEUS collaboration further determined [@chekanov03] the DVCS cross section over the kinematic range $5 < Q^2 < 100$ GeV$^2$, $40 < W < 140$ GeV. Both the $W$ and $Q^2$ depndences of the ZEUS DVCS cross section data are well described by calculations based on GPD and on the color-dipole model.
At lower c.m. energies, the HERMES [@airapetian01] and the CLAS [@stepanyan01] collaborations observed the interference between the DVCS and the BH processes, which manifests itself as a pronounced sin$\phi$ azimuthal asymmetry correlated with the beam helicity. Another observable sensitive to the interference between the DVCS and the BH processes is the azimuthal asymmetry between unpolarized $e^+$ and $e^-$ beams. In contrast to the Beam Spin Asymmetry (BSA) which is sensitive to the imaginary part of the DVCS amplitudes, the Beam Charge Asymmetry (BCA) is probing the real part of the DVCS amplitudes [@diehl97]. Analysis of the HERMES $e^-$ data in 98-99 and the $e^+$ data in 99-00 has shown a positive effect for BSA [@bianchi03].
QCD factorization was proved to be valid for exclusive meson production with longitudinal virtual photons [@collins97]. Such factorization allowed new means to extract the unpolarized and polarized GPD. In particular, unpolarized GPDs can be measured with exclusive vector meson production, while polarized GPDs can be probed via exclusive pseudoscalar meson production. A broad program of DVCS and hard exclusive processes has been proposed [@burkert03] for the 12 GeV upgrade at JLab.
I would like to thank V. Burkert, J. P. Chen, C. Keppel, N. Makins, and W. K. Tung for helpful discussion.
|
---
abstract: 'For an $m$-dimensional multivariate extreme value distribution there exist $2^{m}-1$ exponent measures which are linked and completely characterise the dependence of the distribution and all of its lower dimensional margins. In this paper we generalise the inequalities of [@schltawn02] for the sets of extremal coefficients and construct bounds that higher order exponent measures need to satisfy to be consistent with lower order exponent measures. Subsequently we construct nonparametric estimators of the exponent measures which impose, through a likelihood-based procedure, the new dependence constraints and provide an improvement on the unconstrained estimators.'
author:
- Ioannis Papastathopoulos
- 'Jonathan A. Tawn'
title: 'Dependence Properties of Multivariate Max-Stable Distributions'
---
**Keywords:** max-stable distributions; multivariate extremes; exponent measure; inequalities; constrained estimators
Introduction
============
Max-stable distributions arise naturally from the study of limiting distributions of appropriately scaled componentwise maxima of independent and identically distributed random variables. Here and throughout the vector algebra is to be interpreted as componentwise. A vector random variable $Y=(Y_1,\hdots,Y_m)$ with unit Fréchet margins, i.e., $G_i(y):={\mbox{$\mathbb{P}$}}(Y_i < y)=\exp(-1/y)$, $y>0$, $i \in
M_m=\{1,\hdots,m\}$, is called max-stable if its distribution function is max-stable, i.e., if $$G_{M_m}(y_{M_m}):={\mbox{$\mathbb{P}$}}\left(Y<y_{M_m}\right) = \exp\left\{- \int_{S_m} \max_{i\in
M_m}\left(\frac{w_i}{y_i}\right)dH(w_1,\hdots,w_m)\right\},
\label{eq:max_stable_distribution}$$ where $y_{M_m} = (y_1,\hdots,y_m) \in {\mbox{$\mathbb{R}$}}_+^m$, $S_m =
\left\{(w_{1},\hdots,w_m)\in {\mbox{$\mathbb{R}$}}_{+}^{m}: \sum_{i=1}^m w_i =
1\right\}$ is the $(m-1)$-dimensional unit simplex and $H$ is an arbitrary finite measure that satisfies $$\int_{S_m}w_idH(w_1,\hdots,w_m)=1\quad \text{for any}\quad i\in M_m.$$
The last condition is necessary for $G_{M_m}$ to have unit Fréchet margins and representation (\[eq:max\_stable\_distribution\]) is due to [@pick81]. There is no loss of generality in assuming unit Fréchet margins since our focus is placed on the dependence structure of max-stable distributions, i.e., we are interested in the copula function [@ne:99] which is invariant to strictly monotone marginal transformations and in practice we can standardise random variables to unit Fréchet margins.
The dependence properties of max-stable distributions have received attention in the multivariate extreme value literature. Dating back to [@sib60] and [@tiag62], it has been known that max-stable distributions are necessarily positively quadrant dependent, i.e., $$G_{M_m}(y_{M_m}) \geq \prod_{i=1}^m G_i(y_i)\quad y_{M_m} \in {\mbox{$\mathbb{R}$}}_+^m,
\label{eq:pqd}$$ which implies that no pair of random variables can be negatively dependent. Additionally, max-stable distributions satisfy even stronger forms of dependence. [@marsolki83] show that $\text{Cov}\{g(Y),h(Y)\}\geq 0$ for every pair of non-decreasing real functions $g$ and $h$ on ${\mbox{$\mathbb{R}$}}^m$, i.e., they are associated. For a review of the dependence properties of max-stable distributions we refer the reader to [@beiretal04] and the references therein.
Although all of the aforementioned properties exhibit characteristics for the dependence structure of the class of max-stable distributions, they are far too general to be either tested or implemented in practice. In this paper, we introduce additional constraints for the dependence structure that can be incorporated, through a likelihood-based procedure, into the estimation of max-stable distributions from observed componentwise maxima. The new constraints are in essence the generalisation of the [@schltawn02; @schltawn03] inequalities for the extremal coefficients which correspond to the dependence properties of max-stable distributions for the special case of $G_{M_m}(y,\hdots,y)$, $y>0$. As such, our notation and strategy are influenced by the work of [@schltawn02; @schltawn03]. The new inequalities presented in this paper are related to the general case of $G_{M_m}(y_{M_m})$, $y_{M_m}\in {\mbox{$\mathbb{R}$}}_+^m$.
In Section \[sec:dependence\_properties\] we introduce the class of max-stable distributions along with the [@schltawn02] inequalities for the extremal coefficients. Subsequently, we present the general result of the paper that gives rise to inequalities for the exponent measures. In Section \[sec:inference\] we consider the [@halltajv00] nonparametric estimator for the exponent measure and extend it, through a likelihood-based procedure, to satisfy the new inequalities. Finally, in Section \[sec:simulation\] a simulation study is conducted to assess the performance of the constrained estimator.
Dependence Properties {#sec:dependence_properties}
=====================
Background {#sec:back_and_not}
----------
The class of max-stable distributions arises naturally from the study of appropriately scaled component-wise maxima of random variables. Consider a set of independent and identically distributed random vectors $X^j=(X_1^{j},\hdots,X_m^{j})$, $j=1,\hdots,n$, with unit Fréchet margins. Under weak conditions [@resn87] it follows that $$\lim_{n\rightarrow
\infty}{\mbox{$\mathbb{P}$}}\left(\bigcap_{i=1,\hdots,m}\left\{\max_{j=1,\hdots,n}X_{i}^{j}/n
< y_i\right\}\right) = G_{M_m}(y_{M_m}), \quad y_{M_m}\in
{\mbox{$\mathbb{R}$}}_+^m.
\label{eq:limiting_law}$$ The distribution function $G_{M_m}$ can be completely characterised by the following representations
[rCl]{} V\_[M\_m]{}(y\_[M\_m]{})=-G\_[M\_m]{}(y)&=&\_[S\_m]{}\_[i M\_m]{}()dH(w\_1,,w\_m), \[eq:Gspmeas\]\
\
& = &{\_[i=1]{}\^[m]{} 1/y\_i} A\_[M\_m]{}(,,), \[eq:pickands\_A\]
where the function $V_{M_m}$ is known as the exponent measure of the multivariate extreme value distribution $G_{M_m}$ and $A_{M_m}$, called the *Pickands’ dependence function*, is a convex function that satisfies $\max\{w_1,\hdots,w_m\}\leq A_{M_m}(w_1,\hdots,w_m)
\leq 1$, $(w_1,\hdots,w_m) \in S_m$. This condition implies that $A_{M_m}(e_j)=1$, $j \in M_m$, where $e_j$ is the $j$-th unit vector in ${\mbox{$\mathbb{R}$}}^m$.
Let $C_m=2^{M_m}\setminus \{\emptyset\}$ and denote also by $y_B=\{y_i: i\in B\}$ for $B \in C_m$. Then we can define $2^m - 1$ exponent measures for an $m$-dimensional max-stable random vector $Y$, where each one characterises completely the distribution function of a marginal random variable $Y_B$ of $Y$, i.e., $$V_B(y_B) = -\log\left\{{\mbox{$\mathbb{P}$}}(Y_B < y_B)\right\} =
-\log\left\{\lim_{y_{M_m\setminus B}\rightarrow \boldsymbol{\infty}
}G_{M_m}(y_{M_m})\right\}, \quad B \in C_m.$$ The set of exponent measures $\{V_B: B \in C_m\}$ describes completely the dependence structure of a max-stable distribution given by equation (\[eq:max\_stable\_distribution\]) and all of its lower dimensional margins. It is also trivial to see that with each exponent measure $V_B$ there is an associated Pickands’ dependence function $A_B$. Additionally, $V_B$, $B \in C_m$, is homogeneous of order $-1$, i.e., $V_B\left(y,\hdots,y\right)=y^{-1} V_B\left(1,\hdots,1\right)$, $y>0$.
The importance of the homogeneity property is mostly illustrated through one widely used measure of extremal dependence for the variables indexed by a set $B\in C_m$. More specifically, the quantity defined by $$\theta_B = V_B(1,\hdots,1) = \int_{S_m}\max_{i \in B}w_i
dH(w_1,\hdots,w_m), \quad 1\leq \theta_B \leq |B|,
\label{eq:ext_coeff}$$ describes the effective number of independent variables in the set $B$ and arises naturally from the distribution of the maximum of all the variables indexed by the set $B$, i.e., $${\mbox{$\mathbb{P}$}}\Big\{\max_{i\in B}Y_i < y\Big\}= {\mbox{$\mathbb{P}$}}\left\{Y_i <
y\right\}^{\theta_B}, \quad y>0.
\label{eq:ext_coeff_power}$$ The measure $\theta_B$ is termed the extremal coefficient and complete dependence and independence corresponds to $\theta_B=1$ and $\theta_B=|B|$ respectively. Also, from expression (\[eq:ext\_coeff\_power\]) it follows trivially that $\theta_{B}=1$ for any $B \in C_m$ with $|B|=1$. Due to its simple interpretation, the set of extremal coefficients $\{\theta_B: B\in
C\}$ has been used as a dependence measure in various applications [@tawn90; @schltawn03].
[@schltawn02; @schltawn03] inequalities for the extremal coefficients {#sec:schltawn02}
---------------------------------------------------------------------
[@schltawn02; @schltawn03] constructed bounds for the set of extremal coefficients $\{\theta_{B}:B \in C_m\}$ of max-stable distributions that characterise the dependence structure for the special case of $G_{M_m}(y,\hdots,y)$, $y>0$. Here we use the terminology of [@schltawn02] and for non-empty distinct subsets $B_1,\hdots,B_s$ of $M_m$, $s \in {\mbox{$\mathbb{N}$}}$, we refer to the set of extremal coefficients $\{\theta_{B_1},\hdots,\theta_{B_s}\}$ as complete and consistent if $s=2^m - 1$ and $\theta_{B_i}$ is given by expression (\[eq:ext\_coeff\]), respectively. Their main result is given in the following theorem.
\[th:schltawn02\] A complete set of extremal coefficients $\{\theta_B: B \in
C_m\}$, where $M_m$ is a finite set of indices, is consistent if and only if $$\sum_{B \in C_m, B \supseteq M_m\setminus L} (-1)^{|B \cap L|+1} \theta_B \geq 0, \quad \text{for all $L\in C_m$}.
\label{eq:schltawn02_inequalities}$$
Theorem \[th:schltawn02\] yields bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. For example consider the inequalities (\[eq:schltawn02\_inequalities\]) for the cases $m=2$ and $m=3$ and let for ease of notation $\theta_{\{i,j\}}$ and $\theta_{\{i,j,k\}}$ be $\theta_{ij}$ and $\theta_{ijk}$, for $i,j,k
\in M_m$ and $i \neq j \neq k$. These are respectively
[CC]{} &1 \_[12]{},\_[13]{},\_[23]{} 2\
&\
&{\_[12]{},\_[13]{},\_[23]{},\_[12]{}+\_[13]{}+\_[23]{} - 3} \_[123]{} {\_[12]{}+\_[13]{}-1,\_[12]{}+\_[23]{}-1,\_[13]{}+\_[23]{}-1}.
The first set of inequalities represents the well known bounds of the extremal coefficients that come from the positively quadrant dependence property (\[eq:pqd\]) of max-stable distributions. However, the second set of inequalities gives tighter bounds for the higher order extremal coefficient $\theta_{123}$. This can be seen easily since the combined inequalities for the cases $m=2$ and $m=3$ reduce to $1\leq \theta_{123}\leq 3$.
Inequalities for the exponent measures of max-stable distributions {#sec:inequalities}
------------------------------------------------------------------
It transpires that similar inequalities as with those in expression (\[eq:schltawn02\_inequalities\]) can be obtained for the exponent measures $\{V_B:B\in C_m\}$ of max-stable distributions. Analogously with the terminology for the extremal coefficients in Section \[sec:schltawn02\] we introduce the following definition.
\[def:consistent\_V\] Let $s$ be an integer, for $i=1,\hdots,s$ $B_i$ are distinct non-empty subsets of $M_m=\{1,\hdots,m\}$ and $y_{M_m}=(y_1,\hdots,y_m) \in {\mbox{$\mathbb{R}$}}_+^{m}$. An ensemble $\left\{V_{B_1}(y_{B_1}),\hdots,V_{B_s}(y_{B_s})\right\}$ of exponent measures, where $y_{B_i}=\{y_j:j\in B_i\}$, is called consistent if $$V_{B_i}(y_{B_i}) =\int_{S_m} \max_{j\in B_i}
\left(\frac{w_j}{y_j}\right) dH(w_1,\hdots,w_m),$$ for $i=1,\hdots,s$ and $H$ is an arbitrary finite measure that satisfies $\int_{S_m}w_idH(w_1,\hdots,w_m)=1$ for any $i\in M_m$.
If $s=2^m-1$ then the set of exponent measures is called complete. The following theorem provides a new representation of the exponent measures of multivariate extreme-value distributions in terms of non-negative and uniquely defined real functions.
\[th:theorem1\] Let $\{V_B:B\in C_m\}$ be a complete and consistent set of exponent measures. Then, there exist $2^m-1$ non-negative functions $d_L:{\mbox{$\mathbb{R}$}}_+^{m}\rightarrow {\mbox{$\mathbb{R}$}}_+$, $L\in C_m$, such that, for any $B \in C_m$ $$V_{B}(y_B) = \sum_{L\in M_m, L\cap B \neq \emptyset} d_L\left(y_{M_m}\right),
\label{eq:drepresentation}$$ and the functions $d_L$ are uniquely given by $$d_L\left(y_{M_m}\right) = \sum_{B\in C_m, B \supseteq M_m \setminus L}
\left(-1\right)^{|B\cap L|+1} \int_{S_m} \max_{j\in B}
\left(\frac{w_j}{y_j}\right) dH(w_1,\hdots,w_m).
\label{eq:d_L}$$
### Proof {#proof .unnumbered}
The proof of equation (\[eq:drepresentation\]) of Theorem \[th:theorem1\] follows along the lines of [@schltawn02] proof of Theorem 5 for the simpler case of the extremal coefficients by replacing the constants $\alpha_{k}^{i}(n)$ of [@deheuv83] representation of max-stable distributions with $\alpha_{k}^{i}(n)/y_i$, $i\in M_m$, $k\in
\mathbb{Z}$. Equation (\[eq:d\_L\]) is the Möbius inversion of equation (\[eq:drepresentation\]).The characterisation of a consistent set of exponent measures is obtained from the following corollary.
\[cor:nonneg\] A complete set of exponent measures $\{V_B: B\in C_m\}$ is consistent if and only if $$\label{eq:ineqgenerator}
\sum_{B\in C_m, B \supseteq M_m \setminus L} \left(-1\right)^{|B\cap
L|+1} \int_{S_m} \max_{j\in B} \left(\frac{w_j}{y_j}\right)
dH(w_1,\hdots,w_m) \geq 0,$$ for all $y_{M_m}\in {\mbox{$\mathbb{R}$}}_+^{m}$ and $L\in C_m$.
Inference {#sec:inference}
=========
The [@halltajv00] estimator of the exponent measure {#sec:ex_ests}
---------------------------------------------------
The fundamental premise in all statistical extreme value modelling is that the observed extremes of a stochastic process are well modelled by the limiting theoretical extreme-value distributions. Let for example $X^j=(X_1^{j},\hdots,X_m^{j})$, $j=1,\hdots,N$, be a set of independent and identically distributed $m$-dimensional random vectors with unit Fréchet margins. Here and throughout we assume that the normalised componentwise block maxima $$Y^j:= \bigvee_{r = (j-1) d + 1}^{j d} \frac{X^r}{d}, \quad j =
1,\hdots,n,$$ where $n d = N$, follow exactly the law $G_{M_m}$ of the limiting expression (\[eq:limiting\_law\]).
Let now $w_{B}\in S_{|B|} = \left\{w_{B}\in {\mbox{$\mathbb{R}$}}^{|B|}_+ :
\sum_{i\in B}w_{B,i}=1\right\}$, $B \in C_m$, and define $Z_{B}^j
= w_{B} Y_{B}^j$, for $j=1,\hdots,n$. It then follows that the cumulative distribution function of $\max_{i \in B} Z_{i}^j$ is Fréchet with scale parameter equal to the Pickands’ dependence function $A_{B}(w_{B})$ of $G_{B}$, i.e., $${\mbox{$\mathbb{P}$}}\left\{\max_{i \in B} Z_{i}^{j} < y\right\} = \exp\left\{-
\frac{A_{B}(w_{B})}{y}\right\}, \quad y>0.$$ A natural consistent estimator of $A_{B}$ then is the [@halltajv00] corrected version of Pickands’ estimator [@pick81] which maximises the likelihood $$\ell_B\left\{A_{B}(w_B)\right\} = n
\log\left\{A_{B}(w_B)\right\} - 2 \sum_{j=1}^{n} \log W_{B}^j -
A_{B}(w_B)\sum_{j=1}^{n} \frac{1}{W_{B}^j},
\label{eq:llik}$$ where $W_{B}^j = \max_{i\in B} \left\{w_{B,i} Y_{i}^j
\left[\sum_{j=1}^n (1/Y_{i}^j)/n\right]\right\}$, $j=1,\hdots,n$, is the [@halltajv00] correction which ensures that $\max
w_B\leq\hat{A}_{B}(w_B)$, for all $w_B\in S_{|B|}$, as well as $\hat{A}_B(e_j) = 1$, for any $j \in M_m$, where $e_j$ is the $j$-th unit vector in ${\mbox{$\mathbb{R}$}}^m$. The maximum likelihood estimator is given by $\hat{A}_{B}(w_B) = \left\{ n^{-1 }\sum_{j=1}^{n}
(1/W_B^j)\right\}^{-1}$ which is subsequently corrected by $$\tilde{A}_{B}(w_B)=\min\left\{\hat{A}_{B}(w_B),1\right\}$$ to satisfy $\tilde{A}_{B}(w_B)\leq 1$, for all $w_{B} \in S_{|B|}$. On combining the estimator $\tilde{A}_{B}$ with equation (\[eq:pickands\_A\]), the following consistent estimator of the exponent measure $V_{B}$ is obtained, $$\tilde{V}_{B}(y_{B}) = \left\{\sum_{i \in B} 1/y_i\right\}
\tilde{A}_{B}\left(\frac{1/y_B}{\sum_{i\in B} 1/y_{i}}\right), \quad y_B\in {\mbox{$\mathbb{R}$}}_+^{|B|},\quad B \in C_m.
\label{eq:V_est}$$ Other types of estimators exist in the literature such as the non-parametric estimators proposed by [@deuh91] and [@capetal97] for the bivariate case. [@zhangetal08] gives a detailed overview of the existing estimators and extends them to the multivariate case. In this paper though we use the [@halltajv00] estimator since it arises as the maximum of a log-likelihood function based on which the new inequalities of Section \[sec:inequalities\] can be imposed.
Constrained estimators {#sec:constr_estimators}
----------------------
It transpires that the aforementioned nonparametric estimators of the exponent measures do not necessarily ensure that the resulting estimated set of exponent measures satisfy inequalities (\[eq:ineqgenerator\]). The focus here is placed on incorporating these additional constraints in the estimation procedure so that the resulting complete set of estimated exponent measures $V_B(y_B)$, $B\in C_m$, is consistent in the sense of Definition \[def:consistent\_V\] for fixed $y_{M_m}\in {\mbox{$\mathbb{R}$}}_+^m$. To incorporate the inequalities we construct similarly with [@schltawn03] a joint log-likelihood function $\ell$ of $\left\{A_B; B\in C_m\right\}$ by falsely assuming independence between the observations for all different $B$ to give the pseudo-log-likelihood $$\ell\left(\left\{A_B\left(\frac{1/y_B}{\sum_{i\in B} 1/y_{i}}\right):
B\in C_m\right\}\right) = \sum_{B \in C_m, |B|\geq 2 } \ell_{B}
\left\{ A_{B}\left(\frac{1/y_B}{\sum_{i\in B} 1/y_{i}}\right)
\right\}.
\label{eq:pseudo_likelihood}$$ The maximum pseudo-likelihood estimators are consistent [@lianself96] and the constrained estimators are obtained by maximising the pseudo-log-likelihood (\[eq:pseudo\_likelihood\]) subject to $$\sum_{B\in C_m, B \supseteq M_m \setminus L} \left(-1\right)^{|B\cap
L|+1} \left\{\sum_{i \in B}1/y_i\right\}
A_{B}\left(\frac{1/y_B}{\sum_{i \in B} 1/y_i}\right) \geq 0,\quad \text{for
all $L\in C_m$}\quad$$
and
$$A_{B}\left(\frac{1/y_B}{\sum_{i \in B} 1/y_i}\right)\leq 1, \quad
\text{for all $B\in C_m$}.$$ The resulting constrained estimators are denoted by $\tilde{A}_B^c$ which in turn yield the estimators $\tilde{V}_B^c$ as in equation (\[eq:V\_est\]). The joint estimation of the exponent measures ensures that all estimators are self-consistent. Note that the resulting estimates of lower order exponent measures are affected by higher order measures, i.e., estimates of $V_{B_0}(y_{B_0})$ are affected by estimates of $V_{B_1}(y_{B_1})$, where $B_0 \subset
B_1$. The major benefit of this feature is that this guarantees the existence of higher order measures which are self-consistent with the lower order measures.
An alternative way of obtaining a set of estimated exponent measures is via sequential estimation, i.e., the lower order exponent measures are estimated firstly and then are used as constraints in the estimation of the higher order exponent measures, see also [@schltawn03]. Although this method is faster than the joint optimization problem described by equation (\[eq:pseudo\_likelihood\]), it does not have the desirable feature described above.
Simulation Study {#sec:simulation}
================
Design {#sec:design}
------
We illustrate the impact of constraining the [@halltajv00] estimators to satisfy the new inequalities (\[eq:ineqgenerator\]) over the unconstrained estimators of the set of exponent measures using simulated data from a 3-dimensional max-stable distribution, i.e., the extreme value logistic distribution with dependence parameter $\alpha \in (0,1]$ and set of exponent measures given by $$\left\{ V_{B}\left(y_B\right)= \left(\sum_{i\in B} y_i^{-1/\alpha}\right)^\alpha: \quad B\in C_3\right\},\quad y_B\in {\mbox{$\mathbb{R}$}}_+^{|B|}.
\label{eq:logistic}$$ The values $\alpha=1$ and $\alpha=0$, taken as $\alpha\rightarrow 0$, correspond to independence and complete dependence, respectively.
All comparisons are based on the root mean square error () performance of the exponent measure estimators for a range of dependence parameters $\alpha$ and a cube grid of values, say $\mathbb{L}^3\subseteq {\mbox{$\mathbb{R}$}}_+^3$, for $y_{M_3}$. Specifically, the values chosen for the dependence parameter and the sample size are $\alpha \in \{0.2, 0.5, 0.8\}$ and $n=50$, respectively. Results from larger sample sizes are not reported in the paper since they are unrealistic for applications and also, the efficiency of the estimators $\tilde{V}_{B}$ and $\tilde{V}_{B}^c$ is similar, a fact that comes from the consistency property of the [@halltajv00] estimator. The set $\mathbb{L}$ was chosen to be the discrete set $\{x_{p_1},\hdots,x_{p_7}\}$ with $x_p$ denoting the $p$-th quantile of the unit Fréchet distribution. We chose $p_1 = 0.05$, $p_7=0.95$ and step size $p_j - p_{j-1} = 0.15$. The Monte Carlo size used to compute estimates of the RMSE is 500.
To obtain an aggregated measure of performance, we also report the Monte Carlo estimates of the integrated square deviation of the estimators from the theoretical function, i.e., $$\tilde{T}_{B} = \int_{C\left(\mathbb{L}^{|B|}\right)}\left\{\tilde{V}_B(y_{B}) -
V_B(y_{B})\right\}^2d y_{B},\quad \text{for all}\quad B \in C_3,
\label{eq:T}$$ where $C(\mathbb{L}^{|B|})$ is the smallest $|B|$-hypercube that contains the set $\mathbb{L}^{|B|}$. The integral in expression (\[eq:T\]) is approximated in each Monte Carlo iteration by the quadrature mid-point numerical integration technique on the grid $\mathbb{L}^{|B|}$ and the measure $\tilde{T}_B^c$ is defined analogously by replacing $\tilde{V}_B$ in expression (\[eq:T\]) with $\tilde{V}_B^c$.
Results {#sec:results}
-------
Figure \[fig:ratio\_rmse\] shows the histograms of the ratio of RMSEs between $\tilde{V}_B^c$ and $\tilde{V}_B$, $B \in C_3$, for all grid points in ${\mbox{$\mathbb{R}$}}_+^3$ for the extreme value logistic distribution with $\alpha = 0.2,0.5$ and $0.8$. The figures indicate either similar or better performance of the constrained estimators.
In particular, for the $\alpha=0.8$ case, the constrained estimators are more efficient than the unconstrained estimators especially for the higher order exponent measure $V_{123}$ and improvement in RMSE, although lower in magnitude, can be also seen in the bivariate exponent measures $V_{B}$, $B\in C_3 \setminus
M_3$. Also, the percentage of Monte Carlo samples where the constrained estimates changed with respect to the [@halltajv00] estimates is $62\%$. Regarding the $\alpha=0.5$ case, we found better performance of the constrained estimators for $V_{123}$, although lower in magnitude than the $\alpha=0.8$ case, and similar performance for the bivariate exponent measures. This feature is also supported by the smaller percentage of change in estimates which is $30\%$. For the case of strong dependence, i.e., $\alpha=0.2$, the percentage of change in estimates is very low and equal to $6\%$ which results in similar efficiency of the estimators for all exponent measures as is also shown from Figure \[fig:ratio\_rmse\].
------------- --------------- ----------------- --------------- ----------------- --------------- -------------------------
$B$ $\tilde{T}_B$ $\tilde{T}_B^c$ $\tilde{T}_B$ $\tilde{T}_B^c$ $\tilde{T}_B$ $\tilde{T}_B^c$
$\{1,2\}$ 0.02 0.02 0.36 0.35 0.76 0.72
$\{1,3\}$ 0.02 0.02 0.34 0.34 0.76 0.74
$\{2,3\}$ 0.02 0.02 0.37 0.36 0.74 0.72
$\{1,2,3\}$ 0.66 0.66 11.15 10.63 26.80 22.14 \[table:TtildeB\]
------------- --------------- ----------------- --------------- ----------------- --------------- -------------------------
: Monte Carlo estimates of $\tilde{T}_B$ and $\tilde{T}_{B}^c$, $B \in C_3$, for the extreme value logistic case with $\alpha=0.2,0.5$ and $0.8$.
Table \[table:TtildeB\] shows the Monte Carlo estimates of the integrated square deviation of the estimators from the theoretical function. For the case of strong dependence there is no practical benefit of $\tilde{V}_B^c$ over $\tilde{V}_B$. However, in all other cases the constrained estimators are more efficient than the unconstrained estimators. This shows that not only does the imposition of the constraints improve the performance of the estimators for the higher order exponent measures, but so does for the bivariate level of dependence.
To conclude, the performance of the estimators $\tilde{V}_B$ and $\tilde{V}_B^c$ is similar as the dependence increases and becomes identical in the limiting case of $\alpha \rightarrow 0$. This feature is explained by the increase in performance of the [@halltajv00] estimators $\tilde{V}_B$ as dependence increases which yields a consistent set of estimated exponent measures. Overall, we found the imposition of the new constraints to be beneficial for the simplest max-stable distribution, i.e., the extreme value logistic, and superior in efficiency, especially for the case of moderate or weak dependence. The largest improvement is observed for higher order exponent measures which is promising for implementations in higher than 3 dimensions.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I. Papastathopoulos acknowledges financial support from AstraZeneca and EPSRC.
[xx]{} Ballani, F. Schlather, M. 2011 , ‘A construction principle for multivariate extreme value distributions’, [*Biometrika*]{} [**98**]{}, 633–645.
Beirlant, J., Goegebeur, Y., Segers, J. Teugels, J. 2004, [*Statistics of Extremes, Theory and Applications*]{}, Wiley.
Cap[é]{}ra[à]{}, P., Foug[è]{}res, A.-L. Genest, C. 1997, ‘A nonparametric estimation procedure for bivariate extreme value copulas’, [*Biometrika*]{} [ **84**]{}, 567–577.
Coles, S. G. Tawn, J. A. 1991, ‘Modelling extreme multivariate events’, [*J. Roy. Statist. Soc., B*]{} [ **53**]{}, 377–392.
Deheuvels, P. 1983, ‘Point processes and multivariate extremes’, [*J. Multivariate Anal.*]{} [**13**]{}, 257–272.
Deheuvels, P. 1991, ‘On the limiting behavior of the [P]{}ickands estimator for bivariate extreme-value distributions’, [*Statist. Probab. Lett.*]{} [**12**]{}, 429–439.
Hall, P. Tajvidi, N. 2000, ‘Distribution and dependence-function estimation for bivariate extreme-value distributions’, [*Bernoulli*]{} [**6**]{}, 835–44.
Liang, K.-Y. Self, S. G. 1996, ‘On the asymptotic behaviour of the pseudolikelihood ratio test statistic’, [*J. Roy. Statist. Soc. Ser. B*]{} [**58**]{}, 785–796.
Marshall, A. W. Olkin, I. 1983 , ‘Domains of attraction of multivariate extreme and value distributions’, [*Ann. Probab.*]{} [**11**]{}, 168–177.
Nelsen, R. B. 1999, [*An Introduction to Copulas*]{}, Springer-Verlag, New York.
Pickands, J. 1981, ‘Multivariate extreme value distributions’, [*Proc. 43rd Sess. Int. Statist. Inst.*]{} [ **49**]{}, 859–878.
Resnick, S. I. 1987, [*Extreme Values, Regular Variation and Point Processes*]{}, Springer–Verlag, New York.
Schlather, M. Tawn, J. A. 2002 , ‘Inequalities for the extremal coefficients of multivariate extreme value distributions’, [*Extremes*]{} [**87**]{}, 87–102.
Schlather, M. Tawn, J. A. 2003 , ‘A dependence measure for multivariate and spatial extreme values: properties and inference’, [*Biometrika*]{} [**90**]{}, 139–156.
Sibuya, M. 1960, ‘Bivariate extreme statistics’, [*Ann. Inst. Statist. Math.*]{} [**11**]{}, 195–210.
Tawn, J. A. 1990, ‘Modelling multivariate extreme value distributions’, [*Biometrika*]{} [**77**]{}, 245–53.
Tiago de Oliveira, J. 1962/63, ‘Structure theory of bivariate extremes; extensions’, [*Est. Mat., Estat. e. Econ.*]{} [**7**]{}, 165–295.
Zhang, D., Wells, M. T. Peng, L. 2008, ‘Nonparametric estimation of the dependence function for a multivariate extreme value distribution’, [*J. Multivariate Anal.*]{} [**99**]{}, 577–588.
|
---
address: 'Max-Planck-Institut für Astrophysik, P.O. Box 1317, D–85741 Garching, Germany'
author:
- Klaus Dolag
title: 'PROPERTIES OF SIMULATED MAGNETIZED GALAXY CLUSTERS\'
---
Introduction
============
Observations consistently show that clusters of galaxies are pervaded by magnetic fields of $\sim\mu{\rm G}$ strength. Coherence of the observed Faraday rotation across large radio sources demonstrates that there is at least a field component that is smooth on cluster scales. The origin of such fields is largely unclear. Models invoking individual cluster galaxies for field generation and amplification generally yield field strengths too low by an order of magnitude.
We used the cosmological MHD code described in Dolag et al. (1999) to simulate the formation of magnetised galaxy clusters from an initial density perturbation field. Our main results can be summarised as follows: (i) Initial magnetic field strengths are amplified by approximately three orders of magnitude in cluster cores, one order of magnitude above the expectation from flux conservation and spherical collapse. (ii) Vastly different initial field configurations (homogeneous or chaotic) yield results that cannot significantly be distinguished. (iii) Micro-Gauss fields and Faraday-rotation observations are well reproduced in our simulations starting from initial magnetic fields of $\sim10^{-9}\,{\rm G}$ strength at redshift 15. Our results show that (i) shear flows in clusters are crucial for amplifying magnetic fields beyond simple compression, (ii) final field configurations in clusters are dominated by the cluster collapse rather than by the initial configuration, and (iii) initial magnetic fields of order $10^{-9}\,{\rm G}$ are required to match Faraday-rotation observations in real clusters.
We used these magnetized clusters to study the final magnetic field structure, the dynamical importance of magnetic fields for interpretation of observed X-Ray properties and start to constrain further processes in galaxy clusters like the population of relativistic particles causing the observed radio halos or the behavior of magnetized cooling flows.
GrapeSPH+MHD
============
The code combines the merely gravitational interaction of a dark-matter component with the hydrodynamics of a gaseous component. The gravitational interaction of the particles is evaluated on GRAPE boards (Sugimoto et al. 1990), while the gas dynamics is computed in the SPH approximation. It was also supplemented with the magneto-hydrodynamic equations to trace the evolution of the magnetic fields which are frozen into the motion of the gas because of its assumed ideal electric conductivity. The back-reaction of the magnetic field on the gas is included. It is based on GrapeSPH (Steinmetz 1996) and solves the following equations:
- [ Equation of motion:]{} $$\begin{aligned}
\frac{{{\rm d}}{\vec{v}}_a}{{{\rm d}}t}= &-& \sum_b m_b\left(\frac{P_b}{\rho_b^2}+
\frac{P_a}{\rho_a^2}+\Pi_{ab}
\right)\nabla_a{ W({\vec{r}}_a-{\vec{r}}_b,h) }\nonumber \\
&+& \sum_b m_b\left[\left(\frac{{\cal M}_{ij}}{\rho^2}\right)_a+
\left(\frac{{\cal M}_{ij}}{\rho^2}\right)_b
\right]\nabla_{a,j}{ W({\vec{r}}_a-{\vec{r}}_b,h) }\nonumber \\
&-& \sum_i\frac{m_i}{(|{\vec{r}}_a-{\vec{r}}_i|^2+\epsilon_a^2)^{1.5}}
({\vec{r}}_a-{\vec{r}}_i) \nonumber \\
&+& \Omega_\Lambda^0H_0^2{\vec{r}}_a \nonumber\end{aligned}$$
- [ Energy equation:]{} $$\frac{{{\rm d}}u_a}{{{\rm d}}t} = \frac{1}{2}\sum_b m_b\left(
\frac{P_a}{\rho_a^2}+ \frac{1}{2}\Pi_{ab}
\right)({\vec{v}}_a-{\vec{v}}_b)\nabla_a{ W({\vec{r}}_a-{\vec{r}}_b,h) }$$
- [ Ideal gas equation:]{} $ P_i=(\gamma-1)u_i\rho_i. $
- [ Induction equation:]{} $$\frac{{{\rm d}}{\vec{B}}_{a,j}}{{{\rm d}}t}=\frac{1}{\rho_a}\sum_b m_b(
{\vec{B}}_{a,j}{\vec{v}}_{ab}-{\vec{v}}_{ab,j}{\vec{B}}_a)\nabla_a{ W({\vec{r}}_a-{\vec{r}}_b,h) }$$
- [ Cooling:]{}\
Non-equilibrium solver, 6 Species H,H$^+$, He, He$^+$, He$^{++}$, e$^-$ (Cen 1992 , Katz et al. 1996)
- [ Heating:]{}\
UV background (Haardt & Madau 1996)\
Initial Conditions
==================
We need two types of initial conditions for our simulations, namely (i) the cosmological parameters and initial density perturbations, and (ii) the properties of the magnetic seed field. Two different kinds of cosmological models are used, EdS and FlatLow. For each cosmology, we calculate ten different realisations which result in clusters of different final masses and different dynamical states at redshift $z=0$. We simulate each of these clusters with different initial magnetic fields, yielding a total of more than 100 cluster models. Since the origin of magnetic fields on cluster scales is unknown, we use either completely homogeneous or chaotic initial magnetic field structures. An overview of the initial conditions is given in Figure \[fig:ic\].
The Simulation
==============
The simulations consist of a dissipation-free dark matter component interacting only through gravity, and a dissipational, gaseous component. The surroundings of the clusters are dynamically important because of tidal forces and the details of the merger history. To account for that, the cluster simulation volumes are surrounded by a layer of boundary particles which accurately represent accurately the sources of the tidal fields in the cluster neighborhood. Figure \[fig:sim\] shows the structure of one of our simulations, figure \[fig:cat\] shows the whole clusters catalog.
Results
=======
Faraday Rotation Measurements
-----------------------------
We found that the synthetic Faraday-rotation measurements produced by the clusters in our simulations match very well those measured in individual clusters like Coma (Figure \[fig:coma\]) or A119 (Figure \[fig:a119\]). For details, see Dolag et al. 1999.
The statistics of the synthetic Faraday-rotation measurements produced by our simulated cluster sample also match the observations quite well, as demonstrated in figure \[fig:comp\_all\] for both cosmologies. For detail see Dolag, Bartelmann & Lesch (1999, 2000a). The conclusions drawn from the comparison of synthetic and observed rotation measurements can be summarized as follows:
- While simple collapse models for motivated initial magnetic fields only predict final field strengths of $\sim0.1\,\mu{\rm G}$, additional field amplification by shear flows indeed produce the observed $\sim\mu{\rm G}$ fields.
- The final field configuration in the clusters is dominated by the cluster collapse rather than the initial field configuration. Simulations starting with either chaotic or homogeneous initial fields lead to indistinguishable Faraday rotation measures.
- Synthetic RM observations obtained from our simulations agree very well with collections of real observations. The best agreement is reached when starting with $\sim10^{-9}\,{\rm G}$ fields at z=15.
- The RM statistic of the best-observed clusters, Coma and A 119, are well reproduced by simulated clusters with comparable masses and temperatures.
Dynamical Importance
--------------------
The magnetic fields affect the balance between the gravitational force and the total (magnetic plus thermal) pressure in the cluster and therefore can change the temperature of the inter- cluster medium. Figure \[fig:tprof\] shows the change of the temperature in the inter-cluster medium due to the presence of magnetic fields in our simulation. Figure \[fig:tsig\] shows how this affects the temperature-mass relation in our simulated cluster samples. For details see Dolag, Evrard & Bartelmann (2000).
The magnetic pressure is not taken into account in the X-ray mass-determination methods and therefore potentially leads to an underestimation of the mass. Figure \[fig:mxray\] focusses on the effect on the mass reconstructed via the X-ray method. For details see Dolag & Schindler (2000). The conclusions drawn from synthetic X-ray observations can be summarized as follows:
- Non-thermal pressure support reaches 5% at most.
- The core temperatures of clusters drop by about 5% due to the non-thermal pressure support. The induced spread in the mass-temperature relation can be up to 15%.
- The mean effect on the mass reconstruction of relaxed clusters via the X-ray method is negligible compared to the uncertainties of the widely used $\beta$-model. Nevertheless, the additional effect due to the magnetic field in merging clusters can lead to wrong reconstructed masses up to a factor of two.
Additional Processes
--------------------
We demonstrated that a simple model for hadronic electron injection in a realistic magnetic field configuration taken from our simulated cluster sample leads to radio halos which reproduce several types of observations: the profile of the radially decreasing radio emission as shown in figure \[fig:radiomap\], the low radio polarization, the correlation between radio luminosity an x-ray surface brightness and the cluster radio halo luminosity-temperature relation as shown in figure \[fig:radiocorr\]. For details see Dolag & Ensslin (2000).
It is known from observations that strong magnetic fields appear in cooling-flows. Turning cooling on in our simulations, the collapse of the cool regions strongly amplifies the magnetic field. The magnetic field reaches the regime, where the magnetic pressure exceeds the thermal pressure and stops the collapse of the gas. The synthetic rotation measures in these cool regions are well in agreement with the observed values of thousands of rad/m$^2$. For details see Dolag, Bartelmann & Lesch (2000b). The conclusions drawn for additional processes in our simulations can be summarized as follows.
- The energy content of relativistic protons needed to produce enough relativistic electrons to get typical radio luminosities for the simulated clusters lies between 4% and 15% of the thermal energy content of the gas (in the range of magnetic field strength suggested by Faraday measurements).
- The synthetic radio halo of one simulated cluster with comparable mass and temperature reproduces the radial profile observed in Coma very well.
- Using one normalization for the whole set of simulations the simulation predicts the observed, strong correlation between the temperature and the radio luminosity of galaxy clusters.
- For simulations allowing the ICM to cool, the magnetic pressure becomes important for the dynamics of the regions with strong cooling. The temperature drops less and the cool regions get less dense in the presence of magnetic fields.
- The synthetic Faraday rotation measurements in the cooling-flow regions reach the observed extreme values.
References {#references .unnumbered}
==========
[99]{}
Dolag, K., Bartelmann, M., Lesch, H., 1999: [\
[*“SPH simulations of magnetic fields in galaxy clusters”*]{}\
]{} Sugimoto, D., Chikada, Y., Makino,J., Ito, T., Ebisuzaki, T., Umemura, M., 1990: [\
[*“A Special-Purpose Computer for Gravitational Many-Body Problems”*]{}\
]{} Steinmetz, M., 1996: [\
[*“GRAPESPH: cosmological smoothed particle hydrodynamics simulations with the special-purpose hardware GRAPE”*]{}\
]{} Rees, M.J., 1994: [\
[*“Origin of sees magnetic field for a galactic dynamo”*]{}\
]{} in [*Cosmic Magnetism*]{}, ed. D. Lyden-Bell (Kluwer Academic Publishers, 1994). Kim, K.T., Kronberg, P.P., Dewdney, P.E., Landecker, T.L., 1990: [\
[*“The halo and magnetic field of the Coma cluster of galaxies”*]{}\
]{} Ferretti, L., Dallacasa, D., Govoni, F., Giovannini, G., Taylor, G. B., Klein, U., 1999: [\
[*“The radio galaxies and the magnetic field in Abell 119”*]{}\
]{} Kim, K.T., Kronberg, P.P., Tribble, P.C., 1991: [\
[*“Detection of excess rotation measure due to intracluster magnetic fields in clusters of galaxies”*]{}\
]{} Dolag, K., Bartelmann, M., Lesch, H., 2000a: [\
[*“Evolution and structure of magnetic fields in simulated galaxy clusters”*]{}\
]{} In preperation. Dolag, K., Evrard, A., Bartelmann, M., 2000: [\
[*“The temperature-mass relation in magnetized galaxy clusters”*]{}\
]{} Submitted to [**A&A**]{}. Dolag, K. & Schindler, S., 2000: [\
[*“The effect of magnetic fields on the mass determination of clusters of galaxies”*]{}\
]{} Accepted for publication in [**A&A**]{}. Deiss, B.M., Reich, W., Lesch, H., Wieblebinski, R., 1997: [\
[*“The large-scale structure of the diffuse radio halo of the Coma cluster at 1.4 GHz”*]{}\
]{} Dolag, K. & Ensslin, T., 2000: [\
[*“Radio Halos of Galaxy Clusters from Hadronic Secondary Electron Injection in Realistic Magnetic Field Configurations”*]{}\
]{} Accepted for publication in [**A&A**]{}. Dolag, K., Bartelmann, M., Lesch, H., 2000b:\
In preperation.
|
---
abstract: 'Using the highly detuned interaction between three-level $\Lambda$-type atoms and coherent optical fields, we can realize the C-NOT gates from atoms to atoms, optical fields to optical fields, atoms to optical fields and optical fields to atoms. Based on the realization of the C-NOT gates we propose an entanglement purification scheme to purify a mixed entangled states of coherent optical fields. The simplicity of the current scheme makes it possible that it will be implemented in experiment in the near future.'
author:
- Ming Yang
- 'Zhuo-Liang Cao'
title: Quantum Information Processing using coherent states in cavity QED
---
Entanglement plays a key role not only in refuting the local ”hidden variable” theory [@Einstein:1935; @Bell:1965] but in quantum information processing also, such as quantum teleportation [@Bennett:1993; @Bouwmeester:1997], quantum dense coding [@Bennett:1992; @Mattle:1996], quantum cryptography [@Ekert:1991; @Bennett:1994] and so on.
The preparation of entangled states becomes a vital step in quantum information processing (QIP). Recently, the generation of entanglement has been realized by NMR [@generation:nmr1; @generation:nmr2], SPDC [@Dik:1999; @generation:spdc], Cavity QED [@generation:qed1; @generation:qed2], and Ion Trap [@generation:ion1] schemes. The generation scheme for polarization entangled photon state using SPDC has been reported [@Dik:1999], and it has been realized in experiment. Now, the techniques for generating entangled photon pairs have become rather mature. At the same time, quantum teleportation of unknown photon state and quantum cryptography process are all realized in experiment [@Bouwmeester:1997]. But, for the entangled atomic state, it is not the case. Although many theoretical schemes for the generation of entangled atomic state have been proposed, the number of the schemes that can really be realized in experiment is very small. Hitherto, only the entanglement of two atoms has been realized experimentally [@generation:qed2; @generation:qed3]. The teleportation and cryptography schemes can not yet been implemented in experiment for atoms. To realize the atom-based quantum information processing, we must present more experimentally efficient atom-base QIP schemes.
In view of the previous QIP schemes, we found that the quantum controlled-not (C-NOT) gate is the key part of a total scheme. C-NOT gate can not only generate entangled states but realize the teleportation process through Bell state measurement also. In the cavity QED domain, the schemes for C-NOT gate have been proposed. By far the most efficient scheme is the one proposed by Zheng [@generation:qed2], where the interaction between two atoms induced by a dispersive cavity mode plays a key role and the C-NOT operation from atom to atom has been realized.
In this contribution, we will propose an efficient scheme to realize the C-NOT gates between atom and field of coherent light. The C-NOT operations involve four kinds: C-NOT gate from atom to field of coherent light, C-NOT gate from coherent light to atom, C-NOT gate from one atom to another and C-NOT gate from one field to another. Being more efficient than the proposal of Zheng [@generation:qed2], our scheme not only can generate multi-atom entangled states but also can generate multi-mode entangled coherent states. The scheme is mainly based on the dispersive interaction between atoms and coherent optical fields.
Consider the interaction between an $\Lambda$-type three-level atom and a coherent optical field. Here the two lower levels of the atom are degenerate, and the frequency of the coherent optical field $\omega_{c}$ is largely detuned from the atomic transition frequency $\omega_{0}$ between the degenerate lower levels and the upper level. In this large detuning limit, the upper level $|i\rangle$ can be adiabatically eliminated during the interaction.
Then the effective Hamiltonian for the system can be expressed as follow [@xu]:
$$\hat{H}=-\lambda a^{+}a(|e\rangle \langle g|+|g\rangle \langle
e|)-a^{+}a(\beta _{1}|e\rangle \langle e|+\beta _{2}|g\rangle
\langle g|) \label{hamiltonian}$$
where$\lambda =g_{1}g_{2}/\Delta, \beta_{1}=g_{1}^{2}/\Delta,
\beta_{2}=g_{2}^{2}/\Delta,$ with $\Delta=\omega_{0}-\omega_{c} $ being the detuning between atomic transition frequency and the frequency of the coherent light, $g_{1},g_{2}$ being the coupling constant between the cavity mode and the transitions $|i\rangle
\rightarrow |e\rangle, |i\rangle \rightarrow |g\rangle$ respectively. Suppose $g=g_{1}=g_{2}$, $\lambda
=\beta_{1}=\beta_{2}=g^{2}/\Delta$.
Suppose that the atom is initially prepared in $|e\rangle $ state, and the optical field is in coherent state. Then the interaction between atom and coherent optical field will lead to the following evolution:
$$|e\rangle |\alpha \rangle \overset{U(t)}{\longrightarrow
}(1/2)[(|\alpha \rangle +|\alpha e^{2i\lambda t}\rangle)|e\rangle
-(|\alpha \rangle -|\alpha e^{2i\lambda t}\rangle)|g\rangle]
\label{evolution1}$$
Similarly, if the atom is initially prepared in $|g\rangle $ state, the evolution takes a new form:
$$|g\rangle |\alpha \rangle \overset{U(t)}{\longrightarrow
}(1/2)[(|\alpha \rangle +|\alpha e^{2i\lambda t}\rangle)|g\rangle
-(|\alpha \rangle -|\alpha e^{2i\lambda t}\rangle)|e\rangle]
\label{evolution2}$$
If we select the atomic velocity to realize the interaction time $t=\pi/(2\lambda)$, then Eqs (\[evolution1\], \[evolution2\])will become:
$$|e\rangle |\alpha \rangle \overset{U(t)}{\longrightarrow
}(1/2)[(|\alpha \rangle +|-\alpha \rangle)|e\rangle-(|\alpha
\rangle -|-\alpha \rangle)|g\rangle]$$
$$|g\rangle |\alpha \rangle \overset{U(t)}{\longrightarrow
}(1/2)[(|\alpha \rangle +|-\alpha \rangle)|g\rangle -(|\alpha
\rangle -|-\alpha \rangle)|e\rangle]$$
Let$|\alpha_{+}\rangle =(1/\sqrt{2})(|\alpha \rangle +|-\alpha
\rangle)$, $|\alpha_{-}\rangle =(1/\sqrt{2})(|\alpha \rangle
-|-\alpha \rangle) $, then we get
$$|e\rangle |\alpha \rangle \overset{\lambda
t=\pi/2}{\longrightarrow }(1/\sqrt{2})[|\alpha _{+}\rangle
|e\rangle -|\alpha _{-}\rangle |g\rangle] \label{detailed
evolution1a}$$
$$|g\rangle |\alpha \rangle \overset{\lambda
t=\pi/2}{\longrightarrow }(1/\sqrt{2})[|\alpha _{+}\rangle
|g\rangle -|\alpha _{-}\rangle |e\rangle] \label{detailed
evolution1b}$$
From the analysis of coherent state [@van], we get that : $|u\rangle =1/\sqrt{2(1+e^{-2\|\alpha|^{2}})}(|\alpha \rangle
+|-\alpha \rangle)$, $|v\rangle =1/\sqrt{2(
1-e^{-2|\alpha|^{2}})}(|\alpha \rangle -|-\alpha \rangle)$ are two orthogonal basis. Here we use $|\alpha_{+}\rangle $ and $|\alpha_{-}\rangle $ to replace $|u\rangle $ and $|v\rangle $. In fact, when $|\alpha|=3$, the approximation is rather perfect. So the entangled states in Eqs (\[detailed evolution1a\], \[detailed evolution1b\]) are maximally entangled states between the atom and the coherent optical field.
Next, we will give the evolution of the system for different initial state:
$$|e\rangle |-\alpha \rangle \overset{\lambda
t=\pi/2}{\longrightarrow}(1/\sqrt{2})[|\alpha_{+}\rangle |e\rangle
+|\alpha_{-}\rangle |g\rangle] \label{detailed evolution2a}$$
$$|g\rangle |-\alpha \rangle \overset{\lambda
t=\pi/2}{\longrightarrow }(1/\sqrt{2})[|\alpha_{+}\rangle
|g\rangle +|\alpha_{-}\rangle |e\rangle] \label{detailed
evolution2b}$$
From Eqs (\[detailed evolution1a\], \[detailed evolution1b\], \[detailed evolution2a\], \[detailed evolution2b\]), we can give the following operations:
$$|\alpha_{+}\rangle |e\rangle \longrightarrow |\alpha_{+}\rangle
|e\rangle \label{cnot1a}$$
$$|\alpha_{+}\rangle |g\rangle \longrightarrow |\alpha_{+}\rangle
|g\rangle \label{cnot1b}$$
$$|\alpha_{-}\rangle |e\rangle \longrightarrow -|\alpha_{-}\rangle
|g\rangle \label{cnot1c}$$
$$|\alpha_{-}\rangle |g\rangle \longrightarrow -|\alpha_{-}\rangle
|e\rangle \label{cnot1d}$$
which are C-NOT operations from optical field to atom, and the operations:
$$|-\rangle |\alpha \rangle \longrightarrow |-\rangle |\alpha \rangle
\label{cnot2a}$$
$$|-\rangle |-\alpha \rangle \longrightarrow |-\rangle |-\alpha \rangle
\label{cnot2b}$$
$$|+\rangle |\alpha \rangle \longrightarrow |+\rangle |-\alpha \rangle
\label{cnot2c}$$
$$|+\rangle |-\alpha \rangle \longrightarrow |+\rangle |\alpha \rangle
\label{cnot2d}$$
which are C-NOT operations from atom to optical field. Where $|+\rangle =(1/\sqrt{2})(|e\rangle +|g\rangle)$, $|-\rangle
=(1/\sqrt{2})(|e\rangle -|g\rangle)$.
With the C-NOT gates being realized, we can realize the generation of entangled atomic states and entangled coherent states. In addition, we also can realize the purification of the mixed entangled atomic states and mixed entangled coherent states.
Firstly, we will consider the generation of maximally entangled atomic states. After the first atom $|e_{1}\rangle $ has interacted with the coherent optical field for $t_{1}=\pi/(2\lambda)$, the evolution of the system can be described by Eq (\[detailed evolution1a\]). Then the second atom, initially prepared in $|e_{2}\rangle $ state, will be sent through the field area. If the interaction time is still $t_{2}=\pi/(2\lambda)$, the total evolution reads:
$$\begin{aligned}
&&|\alpha \rangle |e_{1}\rangle |e_{2}\rangle \overset{\lambda
t_{1}=\pi/2}{\longrightarrow}(1/\sqrt{2})[|\alpha_{+}\rangle
|e_{1}\rangle |e_{2}\rangle -|\alpha _{-}\rangle
|g_{1}\rangle |e_{2}\rangle]\nonumber\\
&&\overset{\lambda t_{2}=\pi/2}{\longrightarrow
}(1/\sqrt{2})[|\alpha_{+}\rangle |e_{1}\rangle |e_{2}\rangle
+|\alpha _{-}\rangle |g_{1}\rangle |g_{2}\rangle]
\label{geneatom1}\end{aligned}$$
That is to say, after interactions the state of total system becomes: $$|\Psi _{total}\rangle \longrightarrow (1/\sqrt{2})[|\alpha \rangle
|\Phi _{12}^{+}\rangle +|-\alpha \rangle |\Phi _{12}^{-}\rangle]
\label{geneatom2}$$ where $|\Phi_{12}^{+}\rangle =(1/\sqrt{2})[|e_{1}\rangle
|e_{2}\rangle +|g_{1}\rangle |g_{2}\rangle]$, $|\Phi_{12}^{-}\rangle=(1/\sqrt{2})[|e_{1}\rangle
|e_{2}\rangle-|g_{1}\rangle |g_{2}\rangle]$ are two Bell states for atoms $1$ and $2$.
Then we will detect the optical field. If the result is $|\alpha\rangle$, the two atoms will be left in Bell state $|\Phi
_{12}^{+}\rangle $; If the result is $|-\alpha \rangle $, the two atoms will be left in Bell state $|\Phi_{12}^{-}\rangle$.
In fact, if we do not detect the optical field after the second atom flying out of the cavity, multi-atom entangled states can be created. We will send the next atom $(|e_{n}\rangle)$ through the field area after the previous one flying out of it. Then after the last atom flying out of the cavity field, the optical field will be detected. Conditioned on different results, the $n$ atoms will be left in different maximally entangled states:
$$\begin{aligned}
|\Psi _{n}\rangle
&&\longrightarrow(1/\sqrt{2})[(1/\sqrt{2})(|e_{1}\rangle
|e_{2}\rangle \cdots |e_{n}\rangle +(-1)^{n}|g_{1}\rangle
|g_{2}\rangle \cdots
|g_{n}\rangle)|\alpha \rangle \nonumber \\
&&+(1/\sqrt{2})(|e_{1}\rangle |e_{2}\rangle \cdots |e_{n}\rangle
+(-1)^{(n-1)}|g_{1}\rangle |g_{2}\rangle \cdots
|g_{n}\rangle)|-\alpha \rangle] \label{geneatomn}\end{aligned}$$
Secondly, we will consider the generation of maximally entangled coherent states. Let an atom, initially prepared in $|e\rangle$ state, interact with the first coherent optical field. Let the interaction time satisfy $t_{1}=\pi/(2\lambda)$. After flying out of the first cavity, the atom will be sent through the second cavity field, and the interaction time is still $t_{2}=\pi/(2\lambda)$. The evolution of the total system is:
$$|e\rangle |\alpha _{1}\rangle |\alpha _{2}\rangle \overset{\lambda
t_{1}=\pi/2}{\longrightarrow }(1/\sqrt{2})[( |-\rangle |\alpha
_{1}\rangle +|+\rangle |-\alpha _{1}\rangle) |\alpha
_{2}\rangle]\overset{\lambda
t_{2}=\pi/2}{\longrightarrow}(1/\sqrt{2})[|-\rangle|\alpha_{1}\rangle
|\alpha _{2}\rangle +|+\rangle |-\alpha _{1}\rangle |-\alpha
_{2}\rangle]\label{genefield1}$$
which can be expressed in another form:
$$|\Psi _{total}\rangle \longrightarrow
(1/\sqrt{2})[(1/\sqrt{2})(|\alpha _{1}\rangle |\alpha _{2}\rangle
+|-\alpha _{1}\rangle |-\alpha _{2}\rangle) |e\rangle
+(1/\sqrt{2})(|\alpha _{1}\rangle |\alpha _{2}\rangle -|-\alpha
_{1}\rangle |-\alpha _{2}\rangle) |g\rangle] \label{genefield2}$$
If we measure the state of the atom in basis ${|e\rangle
,|g\rangle}$, we can get the maximally entangled state of the two cavity fields: $(1/\sqrt{2})(|\alpha_{1}\rangle |\alpha
_{2}\rangle+|-\alpha _{1}\rangle |-\alpha _{2}\rangle)$ for result $|e\rangle$, $(1/\sqrt{2})(|\alpha _{1}\rangle |\alpha _{2}\rangle
-|-\alpha _{1}\rangle |-\alpha _{2}\rangle)$ for $|e\rangle$,
Like the generation of $n$-atom maximally entangled state, after the atom flying out of the second cavity, we will not measure the atomic state. Instead, we will send it through other coherent optical fields one by one. In each cavity, the interaction time are all equal to $t=\pi/(2\lambda)$. Then we can get the $n$-cavity maximally entangled states:
$$\begin{aligned}
|\Psi_{n}\rangle &\longrightarrow
&(1/\sqrt{2})[(1/\sqrt{2})(|\alpha_{1}\rangle |\alpha_{2}\rangle
\cdots |\alpha_{n}\rangle+|-\alpha_{1}\rangle
|-\alpha_{2}\rangle \cdots |-\alpha_{n}\rangle) |e\rangle \nonumber\\
&&+(1/\sqrt{2})(|\alpha_{1}\rangle|\alpha_{2}\rangle \cdots
|\alpha_{n}\rangle-|-\alpha_{1}\rangle|-\alpha_{2}\rangle \cdots
|-\alpha _{n}\rangle) |g\rangle]\label{genefieldn}\end{aligned}$$
Due to cavity decay, the two-mode maximally entangled coherent state in Eq (\[genefield2\]) more easily evolve into a mixed state. So next we will consider the purification of the mixed entangled coherent state [@bennett; @pan; @me].
Suppose we have generated two pairs of the two-mode entangled states of optical fields, and cavities $1$, $3$ are in the access of one user Alice, cavities $2$, $4$ are in the access of the other user Bob. At each side, there will be an auxiliary atom, denoted by $a$ or $b$. With the help of the Ramsey Zones between the two cavities. We can realize the C-NOT operations from cavity $1$ to cavity $3$, and from cavity $2$ to cavity $4$.
Here atoms $a$, $b$ are all prepared at $|e\rangle $ state, and the Ramsey Zones all have the same function, i.e. it can realize the following rotations: $|e\rangle\rightarrow|+\rangle
=(1/\sqrt{2})(|e\rangle +|g\rangle)$, $|g\rangle\rightarrow|-\rangle =(1/\sqrt{2})(|e\rangle -|g\rangle)
$. Then consider the purification procedure. The auxiliary atom $a$ will be sent through the cavity $1$, Ramsey Zone $R_{a}$ , and cavity $3$ one after another. The interaction times between atom $a$ and cavity $1$, cavity $3$ are all equal to $t=\pi/(2\lambda)$. Then we find that, if the state of cavity $1$ is expressed in $\{|\alpha_{+}\rangle ,|\alpha_{-}\rangle\}$ basis, and the state of cavity $3$ is expressed in $\{|\alpha
\rangle ,|-\alpha \rangle\}$ basis, this interaction sequence will realize the C-NOT operations from cavity $1$ to cavity $3$:
$$|\alpha_{+}\rangle _{1}|e_{a}\rangle |\pm \alpha_{3}\rangle
\overset{\lambda t_{1}=\pi/2}{\longrightarrow }|\alpha _{+}\rangle
_{1}|e_{a}\rangle |\pm \alpha _{3}\rangle
\overset{R_{a}}{\longrightarrow } |\alpha _{+}\rangle
_{1}|+_{a}\rangle |\pm \alpha _{3}\rangle \overset{\lambda
t_{3}=\pi/2}{\longrightarrow }|\alpha _{+}\rangle
_{1}|+_{a}\rangle |\mp \alpha _{3}\rangle \label{cnot3a}$$
$$|\alpha_{-}\rangle _{1}|e_{a}\rangle |\pm \alpha_{3}\rangle
\overset{\lambda t_{1}=\pi/2}{\longrightarrow }|\alpha
_{-}\rangle_{1}|g_{a}\rangle |\pm
\alpha_{3}\rangle\overset{R_{a}}{\longrightarrow } |\alpha
_{+}\rangle _{1}|-_{a}\rangle |\pm \alpha _{3}\rangle \overset{
\lambda t_{3}=\pi/2}{\longrightarrow }|\alpha _{+}\rangle
_{1}|-_{a}\rangle |\pm \alpha _{3}\rangle \label{cnot3b}$$
The two mixed states to be purified are:
$$\rho_{12}=F|\Phi^{+}\rangle_{12}\langle
\Phi^{+}|+(1-F)|\Psi^{+}\rangle_{12}\langle \Psi^{+}|
\label{rho12}$$
$$\rho_{34}=F|\Phi^{+}\rangle_{34}\langle
\Phi^{+}|+(1-F)|\Psi^{+}\rangle_{34}\langle \Psi^{+}|
\label{rho34}$$
where Eq (\[rho12\]) is expressed in basis $\{|\alpha
_{+}\rangle_{1}|\alpha _{+}\rangle _{2}$, $|\alpha _{+}\rangle
_{1}|\alpha _{-}\rangle _{2}$, $|\alpha _{-}\rangle _{1}|\alpha
_{+}\rangle _{2}$, $|\alpha _{-}\rangle _{1}|\alpha _{-}\rangle
_{2}\}$, and Eq (\[rho34\]) is expressed in basis $\{|\alpha
_{3}\rangle |\alpha _{4}\rangle$, $|\alpha _{3}\rangle |-\alpha
_{4}\rangle$, $|-\alpha _{3}\rangle |\alpha _{4}\rangle$, $|-\alpha_{3}\rangle |-\alpha _{4}\rangle\}$. Here the fidelity of the mixed state relative to the initial maximally entangled state is $F=\langle \Phi ^{+}|\rho |\Phi ^{+}\rangle$.
After the procedure described in Eqs (\[cnot3a\], \[cnot3b\]), we can measure the atoms $a$, $b$. There will be four possible results, $|e_{a}\rangle |e_{b}\rangle$, $|g_{a}\rangle
|g_{b}\rangle$, $|e_{a}\rangle |g_{b}\rangle$, and $|g_{a}\rangle
|e_{b}\rangle$. These four results can be divided into two kinds, $|e_{a}\rangle |e_{b}\rangle$, $|g_{a}\rangle |g_{b}\rangle$ and $|e_{a}\rangle |g_{b}\rangle$, $|g_{a}\rangle |e_{b}\rangle$. Corresponding to each kind, we all can get the purified entangled coherent state of cavity $1$, $2$ conditioned on the measurement result on cavity $3$, $4$ in the $\{|\alpha _{3}\rangle |\alpha
_{4}\rangle$, $|\alpha _{3}\rangle |-\alpha _{4}\rangle$, $|-\alpha _{3}\rangle |\alpha _{4}\rangle$, $|-\alpha _{3}\rangle
|-\alpha _{4}\rangle\}$ basis.
For the first kind of result $|e_{a}\rangle |e_{b}\rangle$, $|g_{a}\rangle |g_{b}\rangle$, we can get the new state of cavity fields $1$, $2$ provided that cavity fields $3$, $4$ are all in the same coherent state.
$$\rho _{12new}=F_{new}|\Phi^{+}\rangle _{12}\langle \Phi
^{+}|+(1-F_{new})|\Psi ^{+}\rangle _{12}\langle \Psi^{+}|
\label{rho12a}$$
$$F_{new}=\frac{F^{2}}{F^{2}+(1-F)^{2}}.
\label{fidelity1}$$
For the second kind of result $|e_{a}\rangle |g_{b}\rangle
,|g_{a}\rangle |e_{b}\rangle $ we can get that:
$$\rho^{'}_{12new}=F^{'}_{new}|\Phi^{-}\rangle _{12}\langle \Phi
^{-}|+(1-F^{'}_{new})|\Psi ^{-}\rangle _{12}\langle \Psi^{-}|
\label{rho12b}$$
$$F^{'}_{new}=F_{new}.
\label{fidelity2}$$
From the result in Eqs (\[rho12a\], \[rho12b\], \[fidelity1\], \[fidelity2\]), we find that the mixed entangled state in Eq (\[rho12\]) has been purified through the C-NOT operations form cavities $1$, $2$ to cavities $3$, $4$ plus the measurements on atoms and cavities. Consider the first result as example. When $F>\frac{1}{2}$, $F_{new}>F$. So the mixed state in Eq (\[rho12\]) has been purified. Because the initial Fidelity $F$ is an arbitrary number between $0.5$ and $1.0$, the iteration of the above scheme can extract a entangled coherent state with an degree of entanglement arbitrarily close to $1.0$. The same analysis applies to the second result.
In the above scheme, the C-NOT operations from one cavity to another has been realized using the highly detuned interaction between three-level $\Lambda$-type atoms and coherent optical fields. In fact, using this kind of interaction we also can realize the C-NOT operations from one atom to anther.
Let the first atom ($1$) through a cavity, initially prepared in coherent state. The interaction is governed by the Hamiltonian expressed in Eq (\[hamiltonian\]). After the first atom flying out of the cavity, we can complete the rotational operation on the coherent state of the cavity: $|\alpha\rangle\rightarrow|\alpha_{+}\rangle,
|-\alpha\rangle\rightarrow|\alpha_{-}\rangle$ by using nonlinear Kerr medium [@Yurke:1986]. Then we will sent the second atom ($2$) through the cavity. If the interaction times between the cavity and atoms $1$, $2$ are all equal to $t=\pi/(2\lambda)$, the total evolution of the system can be expressed as:
$$|+\rangle_{1}|\alpha\rangle|e_{2}\rangle\overset{\lambda
t_{1}=\pi/2}{\longrightarrow}|+\rangle_{1}|-\alpha\rangle|e_{2}\rangle\overset{R}{\longrightarrow}|+\rangle_{1}|\alpha_{-}\rangle|e_{2}\rangle\overset{\lambda
t_{2}=\pi/2}{\longrightarrow}|+\rangle_{1}|\alpha_{-}\rangle|g_{2}\rangle
\label{cnot4a}$$
$$|+\rangle_{1}|\alpha\rangle|g_{2}\rangle\overset{\lambda
t_{1}=\pi/2}{\longrightarrow}|+\rangle_{1}|-\alpha\rangle|g_{2}\rangle\overset{R}{\longrightarrow}|+\rangle_{1}|\alpha_{-}\rangle|g_{2}\rangle\overset{\lambda
t_{2}=\pi/2}{\longrightarrow}|+\rangle_{1}|\alpha_{-}\rangle|e_{2}\rangle
\label{cnot4b}$$
$$|-\rangle_{1}|\alpha\rangle|e_{2}\rangle\overset{\lambda
t_{1}=\pi/2}{\longrightarrow}|-\rangle_{1}|\alpha\rangle|e_{2}\rangle\overset{R}{\longrightarrow}|-\rangle_{1}|\alpha_{+}\rangle|e_{2}\rangle\overset{\lambda
t_{2}=\pi/2}{\longrightarrow}|-\rangle_{1}|\alpha_{+}\rangle|e_{2}\rangle
\label{cnot4c}$$
$$|-\rangle_{1}|\alpha\rangle|g_{2}\rangle\overset{\lambda
t_{1}=\pi/2}{\longrightarrow}|-\rangle_{1}|\alpha\rangle|g_{2}\rangle\overset{R}{\longrightarrow}|-\rangle_{1}|\alpha_{+}\rangle|g_{2}\rangle\overset{\lambda
t_{2}=\pi/2}{\longrightarrow}|-\rangle_{1}|\alpha_{+}\rangle|g_{2}\rangle
\label{cnot4d}$$
That is to say, we also realize the atom-to-atom C-NOT operations.
Then, because we have realized the C-NOT operations: atom-to-atom, field-to-field, atom-to-field and field-to-atom, the teleportation schemes for unknown coherent superposition state of cavity and unknown atomic states can be easily realized [@Bennett:1993]. That is to say, the joint Bell state measurement will be carried out by sending an atom through a detuned optical field, i.e. the Bell state measurement will be converted into the product single atom measurement and single field measurement.
The detection of coherent field has been realized by B.Yurke *et al* [@Yurke:1986], and they can distinguish $|\alpha\rangle$ and $|-\alpha\rangle$ by homodyne detection. So the detection of coherent field in our scheme is realizable, and the detection of atom can be realized by the field-induced ionization [@Raimond:2001].
We now consider the implementation of the above-mentioned scheme. The atoms used in our scheme are all $\lambda$-type three level atoms with one excited level and two degenerate ground levels, which can be achieved by using Zeeman sublevels. From the discussion in [@zheng:song], by using Rydberg atom of long lifetime and superconducting microwave cavities with an enough high-Q, there is sufficient time to achieve our schemes in experiment.
In conclusion, Using the highly detuned interaction between three-level $\lambda$-type atoms and coherent optical fields, the C-NOT operations from atoms to atoms, optical fields to optical fields, from atoms to optical fields and from optical fields to atoms all have been realized in cavity QED. Our scheme not only can generate multi-atom entangled states but also can generate multi-mode entangled coherent states. Based on the C-NOT gates, the entanglement purification for mixed entangled coherent states and teleportation of unknown states of atoms or fields all have been proposed. In the previous quantum information processing proposals, many atoms are required to interact with single mode field simultaneously. In our scheme, only the interaction between single atom and single mode field is needed, which avoids the problem of the synchronization of many atoms in the previous quantum information processing proposals.
This work is supported by the Natural Science Foundation of the Education Department of Anhui Province under Grant No: 2004kj005zd and Anhui Provincial Natural Science Foundation under Grant No: 03042401 and the Talent Foundation of Anhui University.
[99]{} A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1965). C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). D. Bouwmeester, J-W. Pan, et al, Nature, 390, 575-579(1997). C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881(1992). K. Mattle, H. Weinfurter, P. G. Kwiat, [&]{}A. Zeilinger, Phys. Rev. Lett. 76, 4656-4659 (1996). A. Ekert, Phys. Rev. Lett. 67, 661 (1991). C. H. Bennett, G. Brassard, and N. D. Mermin. Phys. Rev. Lett. 68, 557(1992). D. Bouwmeester, J-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999). N. Gershenfeld and I. L. Chuang, Science 275, 350(1995). S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys.Rev.Lett. 83 1054-1057(1999). A. Lamas-Linares et. al. Nature 412, 887 (2001). A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond, and S. Haroche, Science 288, 2024 (2000). S-B. Zheng and G-C. Guo, Phys. Rev. Lett. 85, 2392 (2000). Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 3631¨C3634 (1998) S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 87, 037902 (2001) L. Xu, Z. M. Zhang, Z. Phys. B 95, 507 (1994). S. J. van Enk, O. Hirota, Phys. Rev. A 64, 022313 (2001). C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smoin, [&]{} W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996). J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs [&]{} A. Zeilinger, Nature 423, 417 (2003). Ming Yang, Wei Song, Zhuo-Liang Cao, Phys. Rev. A 71, 012308 (2005). B. Yurke and D. Stoler, Phys. Rev. Lett.57, 13(1986). J. M. Raimond, M. Brune, and S. Haroche, Reviews of Modern Physics, 73 565(2001). S. B. Zheng, Quant. Semiclass. Opt. B 10, 691 (1998); K.-H. Song1, W.-J. Zang, and G.-C. Guo, Eur. Phys. J. D 19, 267(2002).
|
---
abstract: 'This review discusses the dynamics of negative ion reactions with neutral molecules in the gas phase. Most anion-molecule reactions proceed via a qualitatively different interaction potential than cationic or neutral reactions. It has been and still is the goal of many experiments to understand these reaction dynamics and the different reaction mechanisms they lead to. We will show how rate coefficients and cross sections for anion-molecule reactions are measured and interpreted to yield information on the underlying dynamics. We will also present more detailed approaches that study either the transient reaction complex or the energy- and angle-resolved scattering of negative ions with neutral molecules. With the help of these different techniques many aspects of anion-molecule reaction dynamics could be unravelled in the last years. However, we are still far from a complete understanding of the complex molecular interplay that is at work during a negative ion reaction.'
author:
- Jochen Mikosch
- 'Matthias Weidem[ü]{}ller'
- Roland Wester
title: On the dynamics of chemical reactions of negative ions
---
Introduction
============
Questions about the nature of chemical reactions, why and how they proceed and how this can be used to form certain desired chemical products are already very old. In fact they are older than most other of the current research topics in atomic and molecular physics or physical chemistry. The efforts to answer these questions have lead to numerous achievements over the centuries, starting maybe with the re-discovery of the discreet atomic structure of matter, leading to the invention of chemical catalysis, and including the understanding of the quantum mechanical nature of the chemical bond. Several technological advances have fertilised experimental research on the dynamics of chemical reactions. Besides the development of versatile tunable laser source, one notes supersonic single and crossed molecular beams [@lee1986:nl], multi-dimensional momentum imaging and coincidence detection [@heck1995:annu; @whitaker:book; @sanov2008:irpc], ultrafast time-resolved spectroscopy [@stolow2004:cr], and, most recently, the preparation of cold and ultracold atoms and molecules [@doyle2004:epd; @krems2005].
Today the study of the reaction dynamics of molecules has advanced to precise quantum-state resolved scattering experiments in quantitative agreement with high-level scattering calculations – at least for reactive complexes that involve no more than four atoms. If more atoms contribute to a chemical reaction the dimensionality of the scattering process, i. e. the number of degrees of freedom, increases beyond what quantum scattering calculations can do on current computers. Also experiments are challenged by the growing complexity, because individual quantum states are increasingly difficult to separate in the initial as well as the final state of a scattering event. However, it is precisely this “complexity limit” in chemical dynamics that is driving a lot of research, because one would like to understand the details of reactions that are of relevance in organic chemistry, in living cells, or in the Earth’s atmosphere. In such many-atom reactions new phenomena may occur that lead to different reaction mechanisms than in triatomic reactions.
Within the large field of reaction dynamics an important division follows the electric charge of the collision partners that participate in the reaction. Besides the neutral-neutral reaction processes, the second largest class are ion-molecule reactions. As will be shown later, the much stronger ion-neutral interaction than the neutral-neutral van-der-Waals interaction leads to qualitatively different behaviour. Ion-molecule reactions have often very high rate coefficients and are thus important as soon as ions are present in an environment. These environments include discharge plasmas [@armentrout2000:adv], combustion processes, the Earth’s ionosphere [@smith1995:msr], and many other stellar and interstellar regions [@herbst1998:adv; @petrie2007:msr]. Also in the condensed phase, more specifically in liquid solutions, ion-molecule reactions are important. Reactions of ions are consequently studied almost as long as neutral-neutral reactions. They require, however, significantly different experimental techniques.
A large portion of the research on ion-molecule reactions is devoted to positively charged ions. In dilute plasmas, ionisation by electron impact or by radiation will directly lead to the formation of cations and they should therefore play an important role there. Nevertheless, also negatively charged ions are important constituents of gas phase environments, for example in the lower ionosphere of the Earth. Here the negative charge density is mostly represented by atomic and molecular anions (such as O$^-$, O$_2^-$, OH$^-$, HCO$_3^-$, NO$_2^-$ and their clusters with water molecules) and not by free electrons [@smith1995:msr]. Similarly negative ions are omnipresent in liquid solutions and have a strong impact on reactions there. Furthermore, negative ions are precursors for solvated electrons in liquids, which are responsible for strong optical absorption and have been the subject of many studies.
In recent years negative ions have also been detected in other planetary atmospheres in our solar system, such as the atmosphere of Saturn’s satellite titan, and they have been found in the interstellar medium, in the dark molecular cloud TMC-1 and in a circumstellar shell around the carbon rich star IRC+10216 [@mccarthy2006:apj; @bruenken2007:apj]. The understanding of the complex chemistry that leads to anions in these areas is still fragmentary. The questions here are not only related to the formation of the anions, but also if they can add new formation pathways to the complex neutral molecules found in the interstellar medium, such as long unsaturated carbon chains or possibly polyaromatic hydrocarbon molecules.
The study of the reaction dynamics of anion-molecule reactions are the topic of this review. Comparisons with cation-molecule reactions will be made if suitable. Chemical reactions of negative ions have been studied extensively for many decades. Several excellent reviews have appeared a few years ago on this topic [@depuy2000:ijm; @gronert2001:cr; @ervin2001:cr]. Here, we will concentrate specifically on the reaction dynamics in contrast to the mere kinetics of anion-molecule reactions. “Molecular reaction dynamics is the study of elementary processes \[of molecular collisions\] and the means of probing them, understanding them and controlling them.” [@levine2005]. In this sense we focus on processes where isolated molecules collide in a well-controlled reactive event with a negative ion. We give an overview over all the major experimental techniques that are used to study anion-molecule reaction dynamics. In particular, we include the recent experimental developments, on the one hand cryogenic ion traps and on the other hand ion-molecule crossed-beam imaging, and show how they provide new insights.
Negative ions are also subject to spectroscopic studies in the infrared and optical range, using bound-bound absorption and bound-free photoelectron spectroscopy. This reveals the stability, i. e. the electron affinity of the neutral, as well as structural information about the anion. Since negative ions are still today challenging for quantum chemical calculations, due to the spatial extension of the wavefunction of the excess electron and its interaction with the atomic core, spectroscopy and detachment studies can provide benchmark data [@hammer2004:sci; @sanov2008:irpc; @neumark2008:jpc; @wang2009:annu; @hlavenka2009:jcp; @gerardi2010:jpc]. Photoelectron spectroscopy of negative ions can also reveal the energy level structure of the neutral system that is reached after photodetaching the excess electron. If the neutral system is not a bound molecule, but a reactive complex, such as for the FH$_2^-$ anion, one can even study neutral reactions near the transition state between reactants and products [@neumark2005:pccp]. For more information on the spectroscopy of negative ions see Refs. [@pegg2004:rpp; @simons2008:jpc; @wester2010:book].
This article will start with the discussion of a few general considerations of anion-molecule collisions and reactions in the next section. Then it will be discussed how measurements of integral cross sections or total rate coefficients, respectively, are performed. They are carried out either at a given collision energy or a given sample temperature and can already be used to infer information on the underlying reaction dynamics. In the next section, experimental approaches are discussed that allow the investigation of the transient reaction complex. Then in section \[crossed-beams:sect\] experiments with crossed beams are presented, which allow the determination of energy- and angle-differential cross sections, under optimum conditions even quantum-state resolved. From these measurements a wealth of information about the collision and reaction dynamics can be extracted. In the final section some of the future directions for the study of anion-molecule reaction dynamics are highlighted.
\[interaction:sect\] Anion-molecule interactions
================================================
The understanding of the dynamics of an anion-molecule reaction starts with the description of the interaction potential, usually in the Born-Oppenheimer approximation. The long-range and short-range properties of the interaction potential are discussed in the next sections. The proper theoretical treatment of the dynamics of the colliding molecules and ions on the interaction potential is a quantum scattering calculation [@clary1998:sci; @althorpe2003:annu; @schmatz2004:cpc], which is usually, however, limited to a reduced set of dimensions by the computational effort. Alternatively, the atomic motion is treated with classical dynamics [@hase1994:sci; @hase1998:adv_gas_chem], which is generally a good assumption because of the small de-Broglie wavelength of the atoms. In this case all dimensions can be included, but such effects as the zero-point motion, tunnelling through potential barriers or quantum scattering resonances can usually not be accounted for.
Long-range and short range potentials
-------------------------------------
At large distances the force between an ion and a neutral molecule is determined by the electrostatic interaction potential of the ion with the induced, and possibly also permanent, electric dipole moment of the molecule. $$V(R) = - \frac{q^2 \alpha}{2 R^4} - \frac{q \mu}{R^2} \cos\theta
\label{long-range-pot:eq}$$ $\alpha$ is the orientationally averaged polarisability and $\mu$ the permanent dipole moment of the neutral molecule. $q$ is the charge of the colliding ion. In certain cases also the ion-quadrupole interaction and other higher order multipole terms need to be added.
If the molecule carries a permanent dipole moment its interaction with the charge, which scales as $R^{-2}$ typically dominates over the induced-dipole interaction, which scales as $R^{-4}$. However, at large relative separation the free rotation of the molecule leads to a vanishing time-averaged permanent dipole moment. At the largest inter-atomic distances the ion-induced dipole interaction is therefore most important, which is always attractive. Once the interaction potential then becomes comparable to the rotational energy “locking” of the permanent molecular dipole to the incoming ion may occur and will thereby enhance the attractive interaction.
![Schematic view of the Born-Oppenheimer potential of an anion-molecule nucleophilic substitution reaction along the reaction path. This path is defined as the lowest-energy trajectory across the Born-Oppenheimer hypersurface that connects the reactants with the products. The presented potential exhibits two minima and an intermediate reaction barrier between them, which is a typical configuration for many anion-neutral reactions. Often the barrier is found below the energy of one of the asymptotes.[]{data-label="reaction-pot:fig"}](anion-potential.eps){width="\columnwidth"}
At short distances the electronic wavefunctions of the ion and the molecule interact. Determining the short range interaction potential therefore requires solving the Schr[ö]{}dinger equation for the collision system. This is often particularly difficult for negative ions, because the excess electron is so delocalized that a large basis set is required in the numerical solution of the Schr[ö]{}dinger equation. However, it works as an advantage that many negative ions of interest are “closed-shell” systems with a spin singlet configuration of the electronic wavefunction, because these are the most stable negative ions with respect to electron detachment. As a consequence, only a single Born-Oppenheimer potential hypersurface governs the reactions of such negative ions with closed-shell neutral molecules.
For many negative ion-neutral collision systems a repulsive potential is obtained at very short range. This may be attributed to the electron-electron repulsion by electrostatic forces and the Pauli exclusion principle. Together with the strong attractive interaction this leads to potential energy landscapes with two characteristic minima along the reaction coordinate, as shown in Fig. \[reaction-pot:fig\]. The path along the reaction coordinate is defined as the path of minimum potential energy that connects the reactants with the products. The potential energy minima typically lie several hundred meV below the asymptotic energies. Consequently, substantial short-range energy barriers may have negative transition state energies, as shown in Fig.\[reaction-pot:fig\]. But even if the transition state energy is negative, it can have a profound influence on the reactivity as we will see in the next paragraph. This is in contrasts to most reactions of cations, because there is usually no or at least no substantial short-range barrier.
Capture model
-------------
The kinetics of a chemical reaction, which describes the rate at which products are formed as a function of time, is usually characterised by the reaction rate coefficient. For a bimolecular reaction that occurs in a collision of two species the second order rate equation $$\frac{d\,n_{\rm product}}{dt} = k\,n_{\rm reactant 1}\,n_{\rm reactant 2}$$ describes the time-dependent increase of the product particle density $n_{\rm
product}$ for given reactant densities $n_{\rm reactant 1,2}$. This equation is typically applied to thermal ensembles. The second order rate coefficient $k$ is then dependent on the absolute temperature $T$. It is related to the scattering cross section $\sigma(v_{\rm rel})$ for the individual collision events, which depends on the relative velocity, by the thermal average $$k
=
\langle \sigma(v_{\rm rel}) v_{\rm rel} \rangle_T
=
\int_0^\infty \sigma(v_{\rm rel}) p_T(v_{\rm rel}) v_{\rm rel} dv_{\rm rel}.$$ $p_T(v_{\rm rel})$ is the thermal probability distribution for the relative velocities at the absolute temperature $T$.
The standard model to estimate collision rates of ions with neutral atoms or molecules is the Langevin or capture model [@levine2005]. The assumption of this model is that a collision occurs with 100% probability if the two collision partners come closer to each other than a critical distance. Only the longest-range attractive interaction, the ion-induced dipole potential (first term in Eq. \[long-range-pot:eq\]), and the repulsive centrifugal potential are taken into account. This critical distance is then given by the location of the maximum of the centrifugal barrier. The largest impact parameter for a scattering trajectory that reaches this critical distance and can surmount the centrifugal barrier determines the scattering cross section. For larger impact parameters the ion-neutral interaction is neglected. With an additional thermal averaging, this assumption yields the Langevin rate coefficient (in SI units) $$k = \frac{|q|}{2 \epsilon_0} \sqrt{\frac{\alpha}{m_{\rm r}}},
\label{langevin:eq}$$ which turns out to be temperature independent. $\alpha$ is again the orientation-averaged polarisability of the neutral molecule while $m_{\rm r}$ denotes the reduced mass of the two-body system ($q$ is the ion charge and $\epsilon_0$ the electric constant). Typical Langevin rate coefficients range between $5\times10^{−10}$cm$^3$/s and $5\times10^{−9}$cm$^3$/s.
A correction to the Langevin rate coefficient has to be introduced if the neutral target carries a permanent dipole moment $\mu$. Then the second term in Eq. (\[long-range-pot:eq\]) needs to be included. The “average dipole orientation” approach [@su1975:ijm] introduced an effective “dipole locking constant” from which an improved collision rate coefficient was derived. In a more accurate approach, the rate coefficient for ion-polar molecule collisions is calculated from a series of classical trajectories that are calculated to numerical precision for the exact interaction potential. The parameterisation of these trajectories leads to a pre-factor $K(T)>1$ to the Langevin rate constant Eq. \[langevin:eq\], which is a function of $\mu$ and $\alpha$ [@su1982:jcp; @su1988:jcp; @lim1994:qcpe]. As an effect the ion-polar molecule capture rate coefficient is typically a factor of two to four larger than the prediction for the Langevin rate coefficient.
Dynamics inferred from kinetics
===============================
The Langevin rate constant has proven to be extremely valuable as a guideline and often describes fairly accurately measured reaction rates for cations with nonpolar neutral molecules [@ferguson1975:annu]. When the observed rate coefficient for an ion-molecule reaction is smaller than the capture rate coefficient, which occurs when the reaction probability after crossing the centrifugal barrier is smaller than 100%, temperature- or energy-dependent rate coefficient measurements can be used to infer information on the reaction dynamics at short internuclear separation.
In the next sections this approach will be illustrated with several studies from recent years. Most major experimental techniques that are used for these studies, drift and flow tubes, guided ion beams, free jet expansions and low-temperature ion traps will be discussed. Ion cyclotron resonance experiments and high-pressure mass spectrometry (see e.g.[@gronert2001:cr]), however, will not be covered in this article.
Reaction rate coefficients from drift tubes
-------------------------------------------
Flow and drift tubes have been the classic workhorse for the acquisition of ion-molecule reaction rate coefficients. They have produced a wealth of atmospherically and astrophysically relevant data over the almost 50 years of their existence [@ferguson1969:cjc; @fehsenfeld1974:jcp; @depuy1981:acr; @lindinger1998:ijms; @viggiano2001:agc]. Today they are found in many different variants e.g.[@adams1976:ijms; @vandoren1987:ijms; @arnold2000:jpca; @korolov2008:cpp]. These instruments usually operate at room temperature, but may also be heated up to more than 1000K or cooled down to near liquid nitrogen temperatures (77K). Ions are created typically by electron impact, eventually chemically transformed and mass selected, and then injected into a constant flow of buffer gas of several m/s velocity. The pressure of this buffer gas, which is usually helium, is of the order of a few millibar. Consequently, the mean free path of the ions is smaller than the dimension of the tube and the flowing buffer gas transports the thermalised ions downstream through the flow tube. A distance away from the ion source, neutral reactant gas is injected. Due to reactions of the ions with the neutral gas their density decreases with flow time, which now corresponds to the flow distance. At the same time, product ions are generated. By sampling ion yield and composition with a quadrupole mass spectrometer as a function of flow distance absolute reaction rate coefficients can be extracted. Their temperature dependence may be obtained by heating or cooling the buffer gas inlet and flow section. An electric field gradient may be applied along the flow section turning the flow tube into a drift tube. This allows studies at elevated kinetic energy rather than temperature.
The rate coefficient gives a first hint at the dynamics of an ion-molecule reaction. Barrier-less proton transfer reactions usually exhibit large rate coefficients in accordance with the capture model [@boehme2000:ijm]. In contrast many anion-molecule reactions feature orders of magnitude lower rate coefficients, pointing at dynamical bottlenecks and intermediate barriers on the potential energy hypersurface. Chemically versatile flow and drift tubes are well suited to study the effects of chemical substitution on the rate coefficient [@depuy1990:jacs]. Different reaction mechanisms leading to the same (ionic) products may be distinguished by studying kinetic isotope effects [@eyet2008:jasms; @villano2009:jacs]. More direct insight into the dynamics can be obtained from a temperature dependent flow tube measurement of the rate coefficient [@seeley1997:jacs; @kerkines2010:jcp]. Moreover, in drift tubes internal and translational degrees of freedom of the reaction partners are decoupled and can be separately controlled. This allows to investigate how the reaction rate changes if the same amount of energy is provided in different forms, challenging a statistical model of the reaction dynamics [@viggiano1992:jacs].
Probably the most important type of anion-molecule reaction that has been studied up to now is the nucleophilic substitution (S$_{\rm N}$2) reaction [@laehrdahl2002:ijms; @uggerud2006:jpo]: $${\rm X}^- + {\rm CH}_3{\rm\,Y} \rightarrow {\rm Y}^- + {\rm CH}_3{\rm\,X},
\label{sn2:reaction}$$ where X and Y can anything from simple halogen atoms to large macro-molecular systems. The assumption of statistical energy redistribution was a matter of fierce debate in the 1990s in the framework of S$_{\rm N}$2 reactions. A variable temperature flow tube study had shown that the Cl$^-$ + CH$_3$Br reaction rate is strongly dependent on the relative translational energy, while being at the same time insensitive to the internal temperature of the reactants [@viggiano1992:jacs]. Such behaviour is in clear contradiction with the statistical assumption of rapid randomisation of all available energy in the \[Cl $\cdots$ CH$_3$Br\]$^-$ entrance channel complex. This observation prompted a large number of theoretical studies, which uncovered a dynamical bottleneck for energy transfer between internal modes of CH$_3$Br and the intermolecular low-frequency modes [@laehrdahl2002:ijms].
In a recent application of an ion flow tube instrument, the competition between nucleophilic substitution (S$_{\rm N}$2) and base-induced elimination (E2) was investigated via chemical substitution and the deuterium kinetic isotope effect [@villano2006:jacs; @villano2009:jacs]. Villano et al. studied the reaction of BrO$^-$ and ClO$^-$ with methyl chloride (CH$_3$Cl) and its partially and fully methylated form CXYZCl (where X,Y,Z can be either H or CH$_3$). S$_{\rm N}$2 and E2 mechanisms lead to the same ionic reaction product, a challenge for all experiments, which rely on charged particle detection. The authors used that deuteration of the neutral reactant changes the rate of the reaction. The deuterium kinetic isotope effect is defined as the ratio of perprotio to perdeuterio rate coefficient (KIE=$k_H/k_D$). Whereas for CH$_3$Cl an inverse KIE was determined (i.e. KIE$<$1), the KIE was found to become increasingly more normal (KIE$>$1) as the extend of methyl-substitution in the neutral reactant is increased. For the reaction of BrO$^-$ with the fully methylated neutral species, an about a factor of three larger reaction rate coefficient was measured for (CH$_3$)$_3$CCl as compared to its deuterated form (CD$_3$)$_3$CCl. Villano et al. referred to a marked effect of deuteration on the vibrational dynamics near the respective transition states of the S$_{\rm N}$2 and the E2 mechanism [@hu1996:jacs]. They argued that the E2 pathway is a minor channel for the small neutral reactant CH$_3$Cl, whereas it becomes gradually more important with increasing methylation of the neutral, which sterically hinders and finally impedes nucleophilic substitution.
The oxidation of the trichlorooxyphosphorus anion POCl$_3^-$, which occurs in combustion flames, has been recently studied by Kerkines [*et al.*]{} [@kerkines2010:jcp]. Despite its very low rate coefficient of only around 1x10$^{-14}$ cm$^3$/s (at 300K), such oxidations can change the chemistry of flames due to the high abundance of O$_2$. The authors employed a turbulent ion flow tube, where orders of magnitude higher neutral gas densities can be applied as compared to the conventional laminar flow tubes [@arnold2000:jpca]. They measured the rate coefficient over the range of 300-626K and found it to increase slowly with increasing temperature. An Arrhenius fit yielded an activation energy of 50meV. Since the oxidation is exothermic by about 1.8eV, the presence of an intermediate potential barrier was concluded. Examination of the reaction pathways at different levels of molecular orbital theory led to the proposal of a multistep reaction mechanism. It involves product formation via the transformation of the entrance channel ion-dipole complex into a four-membered P $\cdots$ O-O $\cdots$ Cl ring transition state, the highest point on the potential energy surface.
The highest temperature studies of ion-molecule reactions under fully thermalised conditions have been undertaken in a high temperature flowing afterglow apparatus at up to 1800K [@viggiano2001:agc]. For an anion-molecule reaction, 1440K has been reached for CO$_3^-$ + SO$_2$ $\rightarrow$ SO$_3^-$ + CO$_2$ [@miller2006:jcp]. The main driving force for these technically challenging experiments is to model the chemical environment of the earth’s ionosphere and other planetary atmospheres at high altitude. However, these studies are also interesting from the reaction dynamics point of view. At these high temperatures even small molecules carry significant amounts of rotational and vibrational excitation. It can be explored how energy supplied in different forms - translational or internal - affects the reaction rate. For the reaction of CO$_3^-$ with SO$_2$, Miller et al. found by comparison with drift tube data that the total energy alone controls the reactivity [@miller2006:jcp]. They concluded that the independence of the rate coefficient on the form of energy implies that the reaction is governed by long lived intermediates in which energy equilibrates, even at the very high temperatures.
Integral cross sections from guided ion beams
---------------------------------------------
Guided ion beam (GIB) studies are the method of choice for precise measurements of integral reaction cross sections of ion-molecule reactions for collision energies of the millielectronvolt to the tens of electronvolt range. These measurements also allow for the determination of the opening and competition of different reactive channels in ion-molecule collisions [@armentrout2000:ijm]. Teloy and Gerlich introduced GIBs into gas-phase chemistry in pioneering experiments in Freiburg in the 1970s [@teloy1974:cp]. In this technique mass selected ions are passed into a long radio frequency (rf) multipole ion guide - typically an octupole - with a selected kinetic energy, controlled by the dc potential difference between the ion source and the ion guide. Importantly, the guiding multipole electric rf field contains the ions radially, while at the same time minimising alteration of their kinetic energy. The latter is crucial to the technique and impedes the use of quadrupole rf fields: Micro motion of the ions driven by the oscillating field results in energetic collisions with buffer and reaction gas. These effects are referred to as radio frequency heating and are minimised in a multipole rf field [@gerlich1992:adv]. The created effective potential guides the ions through a scattering cell located at the centre of the long guide.
The neutral reaction partner is introduced into the scattering cell - which may be temperature-variable [@levandier1997:rsi] - at a well characterised density. Ions collide with the neutrals as they pass through the scattering cell and undergo chemical reactions. Note that the multipole guide also contains the ionic reaction products. All ions are mass analysed when they reach the end of the guide, typically with a quadrupole mass selector. Counting the number of reactant and product ions allows one to determine absolute integral cross sections for ion-molecule reactions and collision induced dissociation. These measurements can be done as a function of the relative kinetic energy of the reactants over an extended range of energies from about 0.1 to hundreds of eV. GIB measurements achieve high resolution since they ensure single collision conditions in contrast to drift tubes and employ better defined, narrow ion velocity distributions [@deturi1997:jpc]. At the same time the range of relative collision energies is extended since ions are radially contained and can not get lost by drifting to the walls of the tube. To further minimise the energy spread of the reactants, sophisticated ion sources such as flow tubes and multipole traps are used [@haufler1997:jpc; @deturi1997:jpc; @gerlich2004:jams].
The bulk of GIB studies have focused on positively charged ions. Negative ion chemistry has been investigated in particular by the group of Kent Ervin, with special emphasis on nucleophilic substitution. To uncover reaction dynamics and mechanisms, the group heavily employs ab initio calculations and density functional theory in the interpretation of their measurements. The high resolution of GIB measurements allows to reveal the threshold behaviour of endoergic reactions [@rempala2000:jcp] and of exothermic reactions, which feature an intermediate potential barrier [@deturi1997:jpc]. The wide tunability of the relative collision energy enables studies of the opening and interplay of reaction mechanisms leading to different ionic products. Absolute integral cross sections of competing nucleophilic substitution and abstraction reactions were obtained [@angel2001:jpc; @angel2002:jacs]. Collision induced dissociation of anion-dipole complexes accesses the dynamics at the transition state [@akin2006:jpc]. In an advanced mode of operation, GIB experiments may be run in a pulsed manner and the dwell time of the ionic reaction products in the long guide may be used to analyse their axial kinetic energy distribution. This allows to some degree to distinguish between forward and backward scattering for a more direct insight into ion-molecule reaction dynamics [@haufler1997:jpc; @angel2003:jacs].
Haufler, Schlemmer and Gerlich carried out a pioneering GIB study for anions, investigating the fundamental molecular reaction H$^-$ + D$_2$ $\rightarrow$ D$^-$ + HD and its isotopic variant D$^-$ + H$_2$ $\rightarrow$ H$^-$ + HD [@haufler1997:jpc]. Integral cross sections have been obtained as a function of collision energy between 0.1 and 10eV translational energy in the centre-of-mass frame. They show an onset at around 0.3eV, a maximum at around 1eV and a decrease at larger collision energies. Barriers of 350 and 330meV were deduced for H$^-$ + D$_2$ and D$^-$ + H$_2$, respectively. The decrease of the cross section was attributed to the competition with collisional electron detachment of the reactant anion. Interestingly, it was found that the integral cross section of the heavy ion colliding with the light molecule is larger by a factor of two, reaching 2.7Å$^2$ at its maximum. Axial product time-of-flight distributions have been measured by recording arrival times and converted to differential cross sections d$\sigma$/d$v_p$ ($v_p$ denotes the velocity component of the products along the guide). Preferred forward scattering was found for low collision energies just above threshold. By variation of the depth of the effective potential via the rf amplitude, which probes the transversal velocity distribution, the authors concluded that internal excitation of the product molecule is not significant at these energies. Thus they were able to deconvolute d$\sigma$/d$v_p$ to obtain angle-differential cross sections d$\sigma$/d$\theta$. These could be cross-calibrated against crossed-beam data [@mueller1996:jpb] (see section \[conv-crossed-beams:sect\]). The angle differential cross sections were found to be quite similar for the two variants of the reaction, despite the big isotope effect observed in the integral cross section. The authors concluded that the region of the potential energy surface probed in the exit channel has to be similar and the increased reactivity of D$^-$ + H$_2$ has to stem from dynamics in the entrance channel.
DeTuri [*et al.*]{} investigated the symmetric S$_{\rm N}$2 reaction Cl$^-$ + CH$_3$Cl $\rightarrow$ ClCH$_3$ + Cl$^-$ [@deturi1997:jpc] by isotopic labelling. For this system, a translational energy threshold of 2eV had been determined 10 years before in a pioneering and well-recognised drift tube measurement by the Bierbaum group [@barlow1988:jacs]. Based on that finding, anionic attack on the chlorine side of chloromethane (“frontside attack”) had been proposed as the reaction mechanism (potential barrier 2.0eV), and subsequently been backed by theoretical studies [@deng1994:jacs; @glukhovtsev1996:jacs2]. In contrast, DeTuri [*et al.*]{} determined a translational energy threshold of 470$\pm$160meV in their GIB experiment. The authors argued that the previous measurement had been affected by the skewed reactant kinetic energy distribution in the drift tube as well as by collisions with its high density helium buffer gas during the lifetime of the entrance channel complex. The adjusted threshold, however, energetically excludes frontside attack as the reaction mechanism. On the other hand the conventional S$_{\rm N}$2 back-side attack mechanism, with inversion of the carbon centre, features a potential barrier height of only around 120meV, substantially lower than the measured threshold by DeTuri et al.
This puzzle prompted a lot of studies on the theoretical side. Quasiclassical trajectory simulations by the Hase group both on an analytical potential energy surface [@mann1998:jpc] and with the ab initio direct dynamics method [@li1999:jacs] as well as reduced-dimensionality quantum dynamical calculations [@hennig2005:pccp; @hennig2004:jcp] were undertaken. It was confirmed that frontside attack is not the reaction mechanism and backside attack prevails. However, in contrast to the conventional S$_{\rm N}$2 mechanism, it was found that only a very small fraction of the reactive trajectories are indirect, with transient trapping in the ion-dipole wells. This is assigned to poor coupling of the provided translational excitation to internal degrees of molecular freedom. A direct pathway of product formation, on the other hand, seems to feature strict dynamical constraints for low collision energy such as a very restricted geometry of approach for passing over the minimum energy barrier.
The power of the GIB technique is nicely demonstrated in complementary cross section studies of the exothermic S$_{\rm N}$2 reaction F$^-$ + CH$_3$Cl and its counterpart, the endothermic reverse reaction Cl$^-$ + CH$_3$F [@angel2001:jpc; @angel2002:jacs]. The data from Angel and Ervin are reproduced in Fig. \[angelervin\]a and b. As can been seen, the measured reaction cross sections range over an impressive dynamic range from more than 10$^{2}$Å$^2$ down to 5$\times$10$^{-4}$Å$^2$. The exothermic S$_{\rm
N}$2 reaction leading to Cl$^-$ product formation is most efficient at the lowest centre of mass collision energies, while its cross section decreases rapidly over the range 0.1-2eV. The reason for this behaviour derives from the S$_{\rm N}$2 reaction mechanism. A more impulsive collision allows for less alignment of the reaction partners along the backside attack coordinate by their ion-dipole interaction, i.e. along the near collinear reaction path. Fig. \[angelervin\]b shows that the endothermic S$_{\rm N}$2 reaction leading to F$^-$ product formation features an energy threshold as expected. Angel and Ervin determined a threshold energy of 1.88eV, which is 0.54eV in excess of the reaction endothermicity.
![Absolute integral cross section from a GIB study of F$^-$ + CH$_3$Cl (a) and the inverse reaction Cl$^-$ + CH$_3$F (b) for different product ions as a function of the relative collision energy. The calculated capture cross section is shown as solid line. Reprinted with permission from [@angel2001:jpc] and [@angel2002:jacs]. Copyright (2001), American Chemical Society.[]{data-label="angelervin"}](angelervin.eps){width="\columnwidth"}
Inspecting the pseudo-collinear PES, the authors interpreted their finding in terms of the Polanyi rules [@polanyi1972:acr]. An early transition state in the exothermic direction for the collinear reaction pathway renders translational energy efficient in promoting the reaction. This accounts for the high cross section in excess of 100Å$^2$ for low collision energies. After translational passage over the barrier trajectories need to pass the “bend” along the reaction coordinate to form products. Hence much of the reactant translation is expected to be converted into excitation of the C-Cl stretch vibration of the product molecule. Microscopic reversibility suggests in turn that vibrational excitation of CH$_3$Cl is needed in the endothermic direction to efficiently pass the initial tight bend on the PES before surmounting the late transition state. Translational activation such as in the GIB experiment should hence be inefficient at promoting the endothermic reaction. This is in accordance with the finding of an excess barrier by Angel and Ervin and a low cross section of maximal 0.6Å$^2$. Fig.\[angelervin\] shows that at higher collision energies other reaction channels open up and compete with nucleophilic substitution. For the exothermic direction (Fig. \[angelervin\]a), both proton transfer and chlorine abstraction reaction were observed. At collision energies above 2eV, the Cl$^-$ cross section was found to increase again. Dissociation of the CH$_2$Cl$^-$ and FCl$^-$ reaction products was proposed as an explanation. For the endothermic direction (Fig. \[angelervin\]b), methylene and fluorine abstraction as well as a subsequent threebody dissociation were observed. Some of these reactions exhibit dual rising features. Reaction mechanisms have been proposed based on molecular structure calculations.
While GIB techniques are strong at determining integral reaction cross sections, the very detailed and insightful study by Angel and Ervin also clearly shows their limitations. An experimental method is needed, which can provide more direct insight into the reaction dynamics and is capable of dissecting the partitioning of energy into translational and internal degrees of molecular freedom (see Section \[crossed-beams:sect\]).
As first demonstrated by Gerlich [@gerlich1992:adv] and already mentioned above, GIB methods can in fact provide to some degree more direct insight into the reaction dynamics by recording the arrival time of the product ions. Angel and Ervin applied this technique in a study of the exothermic S$_{\rm N}$2 reaction Cl$^-$ + CH$_3$Br [@angel2003:jacs]. They reported the relative product velocity distribution along the axis of the ion guide for a series of relative collision energies between 0.1 and 4.0eV. At 0.1eV, the S$_{\rm
N}$2 product velocity was found to be symmetrically distributed around the centre of mass velocity and to agree well with the prediction of a statistical phase space theory (PST) model. This can be regarded as strong sign that the reaction is complex-mediated. At 0.25eV, the velocity distribution becomes asymmetric with a preference for forward scattering of the CH$_3$Cl product. This shows that the reaction becomes more direct, with reactant interaction on a time scale less than the rotational period of the complex. At 0.5eV, backward scattering arises and becomes dominant above 1.0eV. The authors interpreted this as introduction of a direct rebound mechanism.
Low-temperature reactions in jets and traps
-------------------------------------------
At lower and lower temperatures the dynamics of ion-molecule reactions become more sensitive to details of the interaction potential and to quantum dynamical effects such as tunnelling. Changes in the rate coefficients at low temperature also affect the abundance of ions in atmospheric and interstellar plasmas, which are often characterised by temperatures well-below room temperature. This has raised substantial interest in low-temperature studies of ion-molecule reactions.
Very low temperatures, down to about 0.1K, are reached in free jet supersonic flows employing adiabatic expansion [@smith1998:irpc]. However, the complex cooling process that is induced by the transient flow dynamics of an expanding free jet does lead to temperature disequilibria for the different molecular degrees of freedom and non-thermal rotational state distributions [@zacharias1984:jcp; @belikov1998:cp]. Despite these difficulties supersonic jets have been used to study a few reactions of cations, produced by electron impact or laser ionisation. Two examples are the study of vibrational quenching of NO$^+$($\nu$=1) via helium by complex formation below 3Kelvin [@hawley1991:jcp] and the reopening of the bimolecular C$_2$H$_3^+$ channel in the hydrogen transfer of H$_2$ to C$_2$H$_2^+$ for low temperatures, which is attributed to tunnelling of a collision complex through the reaction barrier [@hawley1992:jcp]. An extension of this technique to negative ion reactions is not straight forward and has not been carried out up to now.
Uniform expansions from precisely designed Laval nozzles can overcome some of the difficulties of free jet flows by maintaining parallel stream lines at constant Mach number [@smith1998:irpc]. Constant high densities are reached at a fixed temperature in the travelling frame over the entire flow, which makes equilibration of the molecules degrees of freedom more likely. By terminating the supersonic expansion through parallelisation of the stream lines in the nozzle, the exceedingly low temperatures of free expansions are unfortunately lost. This CRESU (a French acronym for Cinétique de Réaction en Ecoulement Supersonique) technique is applicable to negatively charged ions, created by dissociative electron attachment in the isentropic flow.
CRESU was employed by Le Garrec [*et al.*]{} to study the S$_{\rm N}$2 reaction Cl$^-$ + CH$_3$Br from 180 down to 23K [@legarrec1997:jcp]. Combining their results with previous measurements obtained from other experimental techniques, the authors demonstrated a dramatic increase in the rate coefficient by a factor of 400 upon reduction of the temperature from 500 to 23K. A quantum scattering method, the “rotating bond approximation” has been employed in this work. It was found that at the lowest temperatures the reaction follows the potential energy profile with no activation energy present. The excitation of intermolecular bending modes of the transition state then induces an activation barrier for increasing temperature.
With the exploration of radio-frequency heating in guided ion beams, it became clear that for the efficient cooling and complete thermalisation of molecular ions with a buffer gas, trapping in a quasi field-free environment is indispensable [@gerlich1992:adv]. Gerlich pioneered the development of cryogenic storage devices based on electric multipole fields, which are capable of cooling molecular ions to a few Kelvin in all degrees of molecular freedom. The most popular design is the 22pole ion trap [@gerlich1995:ps], which is nowadays used in a wide range of applications [@wester2009:jpb].
In such a temperature-variable 22-pole ion trap, the lowest-temperature for an anion-molecule reaction up to now has been achieved in a recent experiment in our group [@otto2008:prl]. The specific system under investigation was the proton transfer reaction to the negatively charged amide ion $${\rm NH}_2^- + {\rm H}_2 \rightarrow {\rm NH}_3 + {\rm H}^-.
\label{nh2:reaction}$$ For this reaction a room temperature reaction probability, given by the ratio of the measured rate coefficient to the calculated Langevin rate coefficient, of about 2% had been found many years ago [@bohme1973:jcp]. Thus, in 98% of the collision events the reactants are reproduced and no products are being formed. This occurs despite the fact that the reaction is exothermic and also the intermediate potential barrier lies below the energy of the entrance channel. It shows that for this reaction the reaction dynamics at short range are very important.
![Reaction probability, given by the ratio of the measured rate coefficient and the constant Langevin rate coefficient, for the reaction of NH$_2^-$ with H$_2$ (Eq. \[nh2:reaction\]) as a function of temperature [@otto2008:prl]. The data show that the probability for reaction increases with decreasing temperature but stays well below the Langevin limit. Below 20K the data show an unexpected decrease of the reaction probability that can not be explained by a classical statistical model (dashed line).[]{data-label="reactions:fig"}](nh2_rate.eps){width="\columnwidth"}
At lower temperatures the probability for reaction (\[nh2:reaction\]) increases strongly [@otto2008:prl]. The data are shown in the right panel of Fig. \[reactions:fig\]. At 20K the probability has increased by a factor of six. This increase is a manifestation of the complex-mediated reaction dynamics [@troe1994:far]: the intermediate NH$_4^-$ complex, which is transiently formed during a collision that surmounts the centrifugal barrier, has a longer lifetime with respect to decay back to reactants at lower temperatures, because the number of available decay channels decreases. The probability to cross the intermediate potential barrier and form products, however, remains approximately constant. Therefore the overall probability to react increases. The observed decrease of the reaction probability for temperatures lower than 20K (see Fig. \[reactions:fig\]) can not be explained within the classical dynamics picture of a complex-mediated reaction mechanism. Instead it is expected to represent a signature of quantum mechanical reaction dynamics in low temperature ion-molecule reactions.
Probing the collision complex
=============================
In the following section we discuss experiments that investigate the transient complex that the reactants form along the path to reaction products. Experimentally, this is either achieved by analysing ternary, complex-forming reactions, by studying unimolecular decay of anion-molecule compounds or by applying direct time-resolved spectroscopy, as detailed below.
Three-body association and dissociation of reaction intermediates
-----------------------------------------------------------------
Average lifetimes of transient entrance channel complexes formed upon collision of the reaction partners can be determined indirectly from ternary rate coefficients for their formation and collisional stabilisation. For reactions of cations with neutral molecules, the extraction of complex lifetimes from ternary association has a long history using flow and drift tubes at high pressure. For reactions of anions, work has focused on the symmetric S$_{\rm N}$2 reaction Cl$^-$ + CH$_3$Cl, specifically the formation and unimolecular decay time of the metastable \[Cl$^-\cdots\,$CH$_3$Cl\]$^*$ entrance channel complex [@li1996:jacs; @mikosch2008:jpc].
Ternary association is studied in a thermal environment, where the density of the neutral reaction partner and a buffer gas is well defined and controllable. It can be understood as a two-step process. $$\label{3body_equations}
\begin{array}{c c c c c}
& k_{f} n_{\rm CH_3Cl} & & k_{s} n' & \\
\mathrm{Cl^- + CH_3Cl} & \stackrel{\rightharpoonup}{\leftharpoondown} &
\mathrm{[Cl^-\cdots CH_3Cl]^*} & \rightarrow & \mathrm{[Cl^-\cdots CH_3Cl]}\\
& k_{\Gamma} & & &\\
\end{array}
\label{ThreeBody}$$ In the first step collisions of the reactants Cl$^-$ and CH$_3$Cl form the metastable ion-dipole complex under investigation, where the relative kinetic energy is transiently transferred into internal energy. The complex is highly excited with respect to its ground state and decays back to reactants with the unimolecular rate k$_{\Gamma}$. Stabilisation of the complex can occur in a second step if a third particle impact removes more internal excitation from the complex than the initial relative translational and internal energy of its constituents. This is only a small fraction of its internal excitation given the binding energy of ground state \[Cl$^-\cdots\,$CH$_3$Cl\] of about 450meV with respect to the asymptote. In subsequent collisions the internal energy is thermalised with the environment. Re-excitation of the complex above the asymptote for dissociation is then strongly suppressed by a Boltzmann factor. Conditions are arranged such that the stabilisation step occurs with a high probability by the buffer gas, which is provided in large excess as compared to the reaction partners. At the same time the low-pressure limit of the steady state approximation has to be valid, which practically means that it has to be ensured that stabilisation of the complex is a rare event. Then the rates for the association and stabilisation step in (\[ThreeBody\]) are well defined and the overall rate of \[Cl$^-\cdots\,$CH$_3$Cl\] formation is R = k$_3$ n$_{\rm CH_3Cl}$n, where k$_3$ is the ternary rate coefficient k$_3$ = k$_f$k$_s$/k$_{\Gamma}$. The ternary rate coefficient is experimentally determined by measuring R as a function of the densities of the reactant (n$_{\rm CH_3Cl}$) and the buffer gas (n). The rate coefficients k$_f$ and k$_s$ describe barrier-free ion-molecule reactions; the method relies on the assumption that these rate coefficients are very close to the capture-limited values. With this, the unimolecular dissociation rate can be extracted, the inverse of which is the average lifetime of the excited collision complex \[Cl$^-\cdots\,$CH$_3$Cl\]$^*$.
Li [*et al.*]{} studied ternary association with high pressure mass spectrometry and obtained an average lifetime of 12 to 16ps for \[Cl$^-\cdots\,$CH$_3$Cl\]$^*$ at a fixed temperature of 296K [@li1996:jacs] (see also [@mikosch2008:jpc]). In a recent experiment in our group, a 22-pole radiofrequency ion trap has been employed, which provides the advantage of temperature-variability and long interaction times [@mikosch2008:jpc]. This allows to study the complex lifetime as a function of reactant collision temperature down to 150K, at which point the vapour pressure of the neutral reactant becomes too low to be bearable. We have found a strong inverse temperature dependence of the average lifetime of the transient \[Cl$^-\cdots\,$CH$_3$Cl\]$^*$ complex in disagreement with a simple statistical model. Longer lifetimes of the entrance channel complex at lower temperatures give S$_{\rm N}$2 reaction systems more time to exploit the available phase space, randomise energy, cross the central reaction barrier and finally form reaction products. This substantiates further that enhanced lifetimes of the transient entrance channel complex are likely to underlie the strong inverse temperature dependence of rate coefficients for exothermic S$_{\rm N}$2 reactions with a submerged barrier.
Unimolecular dissociation of the metastable reactant complex into products $$[{\rm X}^-\cdots\,{\rm CH}_3{\rm\,Y}]^* \rightarrow {\rm Y}^- + {\rm CH}_3{\rm\,X}
\label{UnimolecDiss}$$ was studied in a series of experiments on asymmetric S$_{\rm N}$2 reactions in the Bowers group [@graul1991:jacs; @graul1994:jacs; @graul1998:jacs]. Since entrance channel complexes were prepared, the formation of products involves at least one crossing of the intermediate reaction barrier. Relative kinetic energy distributions for the released products were recorded by means of ion kinetic energy spectroscopy. The measurements were compared to statistical phase space theory. Somewhat controversial in these decomposition experiments is the initial state of the metastable reactant complexes under investigation. They have to have sufficient lifetime to survive the transport from the high pressure ion source to the field-free high vacuum region of the employed mass spectrometer and still undergo unimolecular dissociation. This renders partial collisional stabilisation in the ion source likely (see also [@seeley1997:jacs]). The kinetic energy release distributions for the investigated S$_{\rm N}$2 reactions all peak at zero relative kinetic energy of the products and rapidly decrease for increasing energy. Interestingly, despite the long complex lifetimes the experimental distributions are much narrower than calculated ones based on phase space theory. This means that less energy is partitioned to relative translation of the dissociating products than predicted for a statistical redistribution of energy. The phase space calculation was brought into agreement with the experimental results only if a significant amount of energy was made unavailable for energy redistribution. This effect remains if partial collisional stabilisation is taken into account. The energy missing in translation has to be trapped in internal degrees of freedom of the reaction products. The authors argued that significant rotational excitation is not present and that hence the neutral reaction product has to be vibrationally hot. Surprisingly, the same result was found for “simple” S$_{\rm N}$2 reactions of methylhalides [@graul1994:jacs] as well as for S$_{\rm N}$2 reactions involving reaction partners with many more internal degrees of freedom [@graul1998:jacs]. While for the methylhalide reactions this observation is in agreement with classical trajectory studies [@wang1994:jacs], it challenges the notion that statistical theories become valid for more complex S$_{\rm N}$2 reactions with extended complex lifetimes, whereas non-statistical dynamics is restricted to small or highly energised systems with short complex lifetimes on the order of tens of picoseconds [@craig1999:jacs; @laehrdahl2002:ijms]. For extended complex lifetimes, it was speculated that tunnelling through the central barrier might be an alternative way of product formation, leading to a reduced effective barrier height [@seeley1997:jacs].
Dissociation of reaction complexes is ideally studied upon state-specific excitation of an initially cold complex. For Cl$^-\cdots$CH$_3$Br it was found that excitation of high-frequency intramolecular vibrations in the CH$_3$Br moiety by a CO$_2$ laser leads exclusively to the product formation Br$^-$ + CH$_3$Cl [@tonner2000:jacs]. This was predicted for this system by classical trajectory calculations from Hase and coworkers [@wang1994:jacs], which demonstrated an enhancement of central barrier crossing upon selective excitation. Similarly, excitation of the doubly degenerate C-H stretch modes in the same complex induces central barrier crossing and the formation of Br$^-$ products [@ayotte1999:jacs]. Craig et al. created the S$_{\rm N}$2 intermediate \[CF$_3$CO$_2$CH$_3\cdots$Cl$^-$\]$^*$ in a highly vibrationally excited state by means of a precursor exothermic association reaction [@craig1998:jacs]. This resulted in an at least four-fold enhancement of the branching ratio for the S$_{\rm N}$2 reaction pathway as compared to unexcited complexes. The observation is in disagreement with statistical RRKM calculations and was interpreted as manifestation of a bottleneck for energy transfer between intra- and intermolecular modes on the timescale of the complex lifetime [@hase1994:sci].
Time-resolved photoelectron spectroscopy
----------------------------------------
Chemical reactions involve the nuclear motion from reactants to products as well as the coupled structural and energetic transformation of molecular orbitals. Ultrafast laser pulses allow to follow half-reactions in real time by photoinitiating the dissociation of a transition state and probing the evolution to products with a second photon in a pump-probe experiment [@scherer1990:jcp; @williamson1997:nat]. In time-resolved photoelectron spectroscopy (TRPES) the probe laser generates free electrons by photoionisation or photodetachment, whose kinetic energy and eventually angular distribution is measured. Since TRPES is sensitive to both the electronic configurations and the vibrational dynamics, it has been a particularly successful tool for real-time insight into molecular photodynamics (see recent reviews [@stolow2004:chemrev; @stolow2008:adv; @mabbs2009:csr]).
In pioneering TRPES studies of negative ions, Neumark and coworkers time-resolved the photodissociation of the I$_2^-$ anion [@greenblatt1996:cpl; @zanni1999:jcp]. An ultrashort laser pulse at 780nm pumps ground state I$_2^-$ ($X ^2\Sigma_u^+$) to the first excited electronic state ($A'$ $^2\Pi_{g,1/2}$). Following this excitation, I$_2^-$ dissociates into ground state products I$^-$ + I($^2P_{3/2}$) with a 0.6eV kinetic energy release. At variable time delay with respect to photoinitiation an ultrashort probe pulse is employed, which produces a photoelectron spectrum (PES). For a 260nm probe wavelength, two photoelectron bands are observed, which asymptotically correspond to photodetachment of the I$^-$ photo-reaction product. For short pump-probe delays, the photoelectron bands shift to smaller kinetic energy for increasing delay, while at the same time their width narrows (see also [@sanov2008:irpc]). This effect was attributed to the change in character of the orbital from which the photoelectron is derived. By tracing the transition from the molecular to atomic orbital, the PES uncovers that the dissociation is complete within the first 320fs after photoinitiation. Following dissociation, a subtle shift of the photoelectron kinetic energy in the opposite direction is observed, which continues for another 400fs. This behaviour was assigned to the interaction of the separating fragments, in particular the polarisation-induced charge-dipole attraction between the anion and the neutral atom. It corresponds to a shallow well on the long-range part of the potential surface, which was characterised in the TRPES experiment and determined to be 17meV deep [@zanni1999:jcp]. Measured photoelectron angular distributions reveale that the localisation of the excess charge on one of the atoms is only complete after about 800fs [@davis2003:jcp]. The observed dynamics in the exit channel of I$_2^-$ photodissociation becomes upon time-reversal entrance channel dynamics for an I$^-$ + I collision.
Mabbs [*et al.*]{} [@mabbs2005a:jcp; @sanov2008:irpc] extended this work to the related system IBr$^-$. Here excitation at 780nm - to the lowest optically bright excited electronic state - correlates to the second lowest product channel, I$^-$ + Br($^2P_{3/2}$). Also here, a well is found in the long range part of the potential, but compared to I$_2^-$ it is with about 60meV considerably deeper. Based on the result of a classical trajectory calculation, the authors transformed the time-axis to the intermolecular distance R - thus providing an “image” of the potential as a function of reaction coordinate. Sheps [*et al.*]{} [@sheps2010:sci] showed very recently that for the same photoexcitation the presence of a single solvent molecule introduces a new product channel, the formation of I + Br$^-$. This was traced back to a non-adiabatic transition to one of the lower-energy electronic states driven by the solvent molecule, whose vibrational temperature was found to play a critical role in the process.
![Signal intensity of I$^-$CH$_3$I (upper panel) and I$^-$ (lower panel) in the femtosecond photoelectron spectra. Biexponential decay or growth curves, shown as solid lines, are found reproduce these signals [@wester2003:jcp].[]{data-label="sn2-fpes:fig"}](sn2-fpes.eps){width="\columnwidth"}
One of the authors [*et al.*]{} performed the only time-resolved investigation of a bimolecular S$_{\rm N}$2 reaction in the gas phase by studying I$^-$ + CH$_3$I [@wester2003:jcp]. The reactants were derived from the precursor ion-dipole-bound cluster I$_2^-\,\cdot\,$CH$_3$I. Start trigger for the reaction is a femtosecond pump pulse, which dissociates the I$_2^-$ chromophore. Upon its photodissociation, the neutral iodine leaves the cluster, while the S$_{\rm N}$2 reactants I$^-$ and CH$_3$I start to interact due to their charge-dipole interaction. A probe pulse creates a PES, which identifies reaction transients and products as a function of interaction time. In the experiment two different pump photon energies have been employed, which result in different kinetic energies of the I$^-$ reactant. At $\lambda_{\rm pump}$ = 790nm, this leads to a relative kinetic energy between I$^-$ and CH$_3$I of 0.15eV, where the interaction of CH$_3$I with I$^-$ during dissociation has been neglected. The PES reveal that in this case the dominant process is the production of the vibrationally excited entrance channel complex I$^-\,\cdot\,$CH$_3$I on a time scale of 600fs. In contrast for $\lambda_{\rm pump}$ = 395nm, the relative kinetic of the reactants I$^-$ and CH$_3$I is 0.32eV (see Fig. \[sn2-fpes:fig\]). Contributions of I$^-\,\cdot\,$CH$_3$I and I$^-$ are now identified in the PES and can be separated. The complex contribution increases rapidly after the pump pulse with a time constant indistinguishable from the laser cross correlation of 200fs. Interestingly, it then shows an exponential decay on two different time scales, which is mirrored in the I$^-$ contribution. Since the reactants do not have sufficient energy to cross the central reaction barrier, this is interpreted as decay of the complex back to reactants. The fast time constant was determined to be 0.75ps, which compares to the vibrational period of the I$^-\cdots\,$CH$_3$I stretching mode. It suggests fairly direct dissociation dynamics in which the I$^-$ undergoes a quasi-elastic collision with CH$_3$I before dissociation. The longer time constant of about 10ps indicates that the complex is stabilised by energy flow from the reaction coordinate into the modes of the complex. It is comparable to the lifetime of the Cl$^-\,\cdot\,$CH$_3$Cl entrance channel complex determined in the collisional stabilisation experiments featured above.
\[crossed-beams:sect\] Dynamics from differential scattering
============================================================
In the previous sections we have discussed measurements of the total cross section or the thermally averaged rate coefficient and illustrated how information on the reaction dynamics can be inferred from these measurements. A much more direct approach to the intrinsic dynamics of anion-molecule reactions is based on the measurement of the angle- and energy-differential scattering cross section $d\sigma/d\Omega/dE$. For these experiments collisions of atoms and molecules with well-defined momentum vectors are prepared. Angle- and energy- or velocity-resolved detection schemes are employed to obtain the differential cross section. This approach of crossed-beam measurements of the differential scattering cross section is widely used for neutral reactions [@casavecchia2000:rpp; @liu2001:annu]. We will discuss in the following two sections that it also proves to be valuable for anion-neutral reactions.
\[conv-crossed-beams:sect\] Conventional crossed beam reactive scattering
-------------------------------------------------------------------------
The classical approach to crossed-beam reactive scattering is to cross two reactant beams at 90$^\circ$ and use a rotatable detector to measure the flux and the arrival time of products under a selected set of scattering angles in the laboratory frame of reference. From these laboratory flux data the differential cross section in the centre-of-mass frame of the reaction is reconstructed, often using numerical simulations of the apparatus function. This approach has been developed for neutral-neutral reactions and has been used with great success to study the F + H$_2$ reaction (see e.g.[@lee1986:nl]). In combination with the Rydberg tagging technique to measure the H atom product, the rotatable-detector setup is still the method of choice for this reaction [@qiu2006:sci].
The same detector concept has been applied to study elastic, inelastic and reactive channels in cation-molecule reactions (see Refs.[@futrell1992:adv; @farrar1995:annu] for reviews of the early work on cation-molecule reactive scattering). Here the main difference to neutral-neutral scattering experiments is, naturally, the production of the ion beam. Where the neutral beams are usually produced in supersonic expansions with a well defined narrow range of velocities, the production of ion beams with a narrow velocity spread represents a major experimental challenge. For the study of chemical reactions the most interesting relative collision energies in the centre-of-mass frame range from millielectronvolts to a few electronvolts. Therefore ion beams have to be produced with a low kinetic energy and a correspondingly small energy spread. This has been achieved for the first time in the 1970s for continuous beams of cations [@vestal1976:rsi]. A high degree of control over the electric fields in the experimental setup, including contact potentials between different materials, are important to achieve a low energy spread. Even then, the Coulomb repulsion between the reactant ions in the beam ultimately limits the energy resolution for a given reactant density and thus scattering rate. For ion-molecule reactions this typically leads to a reduction in the ion reactant density by many orders of magnitude in comparison with a neutral supersonic beam. This is only to a small extend compensated by the higher detection efficiency for product ions in a charged-particle detector when compared to neutral products of neutral-neutral reactions which have to be ionised in the detector. Note that once the control over the ion-beam energy is achieved in an experimental setup, the relative collision energy of the crossed beams can be continuously tuned by means of the ion acceleration potential. Such a tunable scattering energy is much more difficult to achieve for neutral-neutral reactions. There, different seed gases for the molecules in the supersonic beams and different intersection angles between the supersonic beams are required for the same purpose.
It was only in the 1990s that precise low-energy crossed-beam measurements with negative ions in a rotatable detector setup have been carried out [@zimmer1995:jpb; @farrar1995:annu]. In the years before the first detailed scattering experiments on anion-molecule reactions have been carried out using a complex multiparticle coincidence scattering setup. With this setup reactions of heavy, mostly atomic, negative ions (F$^-$, Cl$^-$, Br$^-$, I$^-$, S$^-$, CN$^-$) with light-weight hydrogen could be studied [@barat1985:cp; @fayeton1989:cp; @brenot1994:cp; @goudjil1994:cp]. For example for the Cl$^-$ + H$_2$ collision, the kinematics of this combination have made it possible to study collisions down to 6eV relative energy with an ion kinetic energies of 110eV [@barat1985:cp], which is significantly easier to handle experimentally than ion beams with only few electronvolt kinetic energy. Furthermore, under these conditions the kinetic energy of the neutral product in the lab frame is large enough to be detected with a microchannel plate detector. With this technique the different channels for the collision of Cl$^-$ with H$_2$, reactive proton transfer, forming HCl + H$^-$, reactive detachment, forming HCl + H + e$^-$, simple detachment, leading to Cl + H$_2$ + e$^-$, and dissociative detachment, leading to Cl + H + H + e$^-$, could be distinguished and their differential and absolute cross sections could be measured as a function of the relative energy [@barat1985:cp]. Also some information on the vibrational state population of the product molecule could be inferred. From these data the similarity of the dynamics of reactive proton transfer and reactive detachment could be deduced, where the branching between the two channels depends on the coupling to an intermediate autodetaching HCl$^-$ state. Different scattering dynamics was found for the simple detachment channel, bearing similarity with anion-rare gas collisions.
With the first crossed-beam measurements on anion-molecule reactions using low-energy ion sources and rotatable detectors, relative collision energies down to the millielectronvolt range and an energy resolution sufficient to resolve product vibrational states became accessible. In a benchmark experiment on the reaction of H$^-$ with D$_2$ Zimmer and Linder could determine the vibrational state-to-state differential and integral scattering cross section [@zimmer1992:cpl; @zimmer1995:jpb]. The dominate forward scattering of the D$^-$ product was attributed to a collinear reaction mechanism that requires scattering at small impact parameter. Wider angular distributions were observed for higher vibrational excitation of the HD product. From the measured product kinetic energy spectra also rotational state information could be extracted, studied in more detail for the inelastic scattering in H$^-$ + H$_2$ collisions in the same laboratory [@mueller1996:jpb].
![Newton diagram and extracted product flux contour map for the reaction OH$^-$ + D$_2$ at 89kJ/mol relative energy. The orthogonal black lines indicate the velocities of the incoming beams. The point in the centre of the circles represents zero velocity in the centre of mass frame. Around this point the circles denote different product velocities. Smaller velocities indicate internal excitation of the HOD product molecule due to vibrational excitation of the bending mode (v$_3$) or the OD stretching mode (v$_2$). The measured flux shows that internal excitation, attributed to the bending mode of HOD, does occur during the reaction. Reprinted with permission from [@li2005:jpc]. Copyright (2005), American Chemical Society.[]{data-label="farrar-data:fig"}](farrar-data.eps){width="\columnwidth"}
Also in the 1990s, Farrar and coworkers started crossed-beam reactive scattering experiments of atomic oxygen with small closed-shell molecules, such as water, ammonia and hydrogen (see e.g.[@varley1992:jcp; @levandier1992:jcp; @farrar1995:annu]). For the O$^-$ + H$_2$O reaction, which had already been studied before with crossed beams at higher collision energy and with lower resolution [@karnett1981:cpl], they observed in the differential cross section both a direct and an indirect reaction mechanism for the reactive channel of OH$^-$ formation [@varley1992:jcp]. In addition they could study inelastic scattering via the O$^-$(H$_2$O) reaction complex. More recently, non-reactive and reactive collisions of OH$^-$ anions with D$_2$ molecules have been studied by the same group [@lee2000:jcp; @li2005:jpc]. In the isotopic exchange reaction channel, the formation of OD$^-$ + HD, no energy dependence was found between 0.27 and 0.69eV relative collision energy [@lee2000:jcp]. In the measured narrow angular distribution it was found that the reaction occurs fast on the time scale of rotation of the OH$^-$(H$_2$) reaction complex. In a study of the proton transfer reaction channel, forming HOD + D$^-$, also a very narrow velocity distribution was extracted [@li2005:jpc]. The reconstructed velocity distribution in the scattering plane is reproduced in Fig. \[farrar-data:fig\]. These data also indicate a direct and fast reaction mechanism. The flux at smaller product ion velocities is caused by vibrational excitation of the bending mode in the HOD product. This can be attributed to a significant change of the water bond angle during the D$^+$ transfer.
Velocity map imaging with crossed beams
---------------------------------------
Compared to the conventional crossed-beam experiments with a rotatable detector, higher angular resolution and a much more rapid data acquisition is achieved with an imaging spectrometer. Moreover, such a spectrometer detects products irrespective of their velocity or scattering angle, i.e. it represents a detector with $4\pi$ solid angle of acceptance. Ion imaging has been combined with neutral reactive scattering starting with a study of the H + D$_2$ reaction [@kitsopoulos1993:sci] and is being successfully employed in a number of laboratories for reactive and inelastic crossed-beam scattering experiments [@heck1995:annu; @elioff2003:sci; @lin2003:sci; @zhang2009:sci]. The current spectrometers are based on the technique of velocity map imaging [@eppink1997:rsi], which projects ions with the same velocity parallel to the detector surface onto the same spot on the detector. It thereby avoids broadening of the product ion images due to the finite size of the reaction volume.
To study ion-molecule reactions with crossed beam imaging, we have constructed a velocity map imaging spectrometer and a versatile low-energy ion source [@mikosch2006:pccp]. This approach is in contrast with an early exploration of ion-molecule crossed beam imaging, where an in-situ production of the reactant ions was used [@reichert2002:jcp], an approach that is not usable for negative ions. In our experiment, slow ions with between 0.5 and 5eV kinetic energy are brought to collision with neutral molecules in a supersonic molecular beam in the centre of the velocity map imaging electrode stack. Both reactant beams are pulsed to avoid a heavy gas load in the vacuum system. Once the two beams have crossed, the electric field of the imaging spectrometer is rapidly pulsed on and any product ions are projected onto the position sensitive imaging detector. Ion impact positions, which are proportional to the velocity components in the scattering plane parallel to the detector surface, are recorded with a CCD camera. Recently, we improved the ion imaging electrodes to enhance the velocity resolution and we included a photomultiplier tube to measure the third, vertical, component of the product velocity vector [@trippel2009:jpb].
With the ion-molecule crossed beam imaging spectrometer we have studied the elementary S$_{\rm N}$2-reaction of Cl$^-$ with methyl iodide (CH$_3$I) at relative scattering energies between 0.4 and 2eV [@mikosch2008:sci]. The potential energy curve for this reaction along the reaction coordinate is closely resembled in Fig. \[reaction-pot:fig\]. As discussed already in section \[interaction:sect\] the characteristic deep potential minima, separated by an intermediate barrier, are found. In the reaction studied here this barrier lies submerged below the energy of the reactants. Nevertheless it strongly influences the reactivity in that the reaction occurs at only 10% of the Langevin or capture rate.
The imaging data, which is represented by an event list of impact velocity vectors in the laboratory frame, needs only little data processing. On the one hand, the transformation to the centre of mass frame has to be performed for each scattering event, which involves a translation and a rotation of the image parallel to the relative velocity vector. On the other hand, the loss of fast moving reaction products needs to be corrected. Product ions with high laboratory velocities have a chance to leave the spectrometer volume that can be imaged onto the position-sensitive detector. This effect is referred to as ’density-to-flux correction’ in neutral crossed beam scattering experiments, where it also includes the correction for the spatial and velocity dependence of the ionisation efficiency. For ion-molecule reactive imaging, the same correction function is used for all relative collision energies. It only depends on the magnitude of the laboratory velocity and not on its direction in the lab frame.
![\[sn2-images:fig\] Measured differential scattering cross section for the reaction of Cl$^-$ + CH$_3$I giving ClCH$_3$ + I$^-$. Shown are two images at 0.39 (left panel) and 1.9eV (right panel) relative collision energy of the I$^-$ velocity vector in the scattering plane obtained by velocity map slice imaging. The centre of each image denotes zero velocity in the centre of mass frame. The circles represent constant product velocities with the largest circle showing the maximum possible product velocity based on the known total energy in the reaction system. At 0.39eV isotropic scattering is observed, indicative of an indirect reaction mechanism via a long-lived complex. At 1.9eV most of the flux shows direct scattering with large product velocities (peak near $v_x \sim +1000$m/s, but with about 10% probability, small product velocities both forward and backward scattered, are observed. These events are attributed to the indirect “roundabout” mechanism (Taken from [@mikosch2008:sci]).](sn2_images.eps){width="\columnwidth"}
Two measured images of the differential cross section in the centre-of-mass frame are shown in Fig. \[sn2-images:fig\] for two relative energies [@mikosch2008:sci]. At 0.39eV isotropic scattering of the I$^-$ product ion is observed, indicative of an indirect reaction mechanism via a long-lived complex. Here, also much more energy is partitioned into internal excitation of the neutral CH$_3$Cl product. This indicates an indirect reaction mechanism with trapping of the collision partners in the minima of the intermolecular reaction potential. In contrast, at 1.9eV the I$^-$ ions scatter preferentially backward with respect to the direction of the incoming CH$_3$I. Also their velocity is found very near the maximum possible velocity. This is explained by a fast and direct reaction mechanism where the I$^-$ leaves the reaction approximately co-linearly with the incoming Cl$^-$ anion.
In order to understand the details of the measured differential cross sections theoretical calculation are employed. However, the theoretical description of polyatomic reactions that involve more than four atoms is very difficult. The present reaction involves six atoms and therefore twelve internal degrees of freedom. Such a large system can not be calculated quantum mechanically and one has to resort to significant approximations. These are either quantum scattering calculations in reduced dimensions (typically four) [@schmatz2004:cpc] or calculations that treat the electronic structure quantum mechanically but propagate the nuclei classically on the Born-Oppenheimer surface [@hase1994:sci].
Calculated trajectories show that at 1.9eV collision energy a direct reaction mechanism governs the nucleophilic substitution reaction. The Cl$^-$ ions moves into the umbrella of the hydrogen atoms and forms a bond with the central carbon atom. Roughly co-linearly to this motion the I$^-$ product ion is moving away after the three hydrogen atoms have inverted to form the ClCH$_3$ product molecule. This numerical results corresponds directly to the back-scattering observed in the experiment. The trajectory calculations also revealed another reaction mechanism that occurs with about 10% probability. In this mechanism the CH$_3$I reactant undergoes a single 360$^\circ$ revolution about an axis perpendicular to the C-I bond. Only after this revolution the substitution occurs. This mechanism, which we named the “roundabout” mechanism, is found to go along with a large energy partitioning into internal degrees of freedom of the neutral product molecule. This agrees with the observed structures in the measured differential cross sections at small I$^-$ velocity (see Fig. \[sn2-images:fig\]), which have therefore been attributed to the roundabout mechanism [@mikosch2008:sci].
Perspectives
============
Negative ion reactions have been studied for a long time owing to their importance in many Earth-bound, planetary or astrophysical plasmas. In the last two decades research on the detailed dynamics of this class of reactions has flourished, owing to more and more precise techniques to measure both integral and differential scattering studies. In this article we have presented an overview of these experimental approaches and how much they have been teaching us on the different, and sometimes peculiar, aspects of anion-molecule reaction dynamics.
We have shown that flow and drift tube measurements are well suited to study the thermal kinetics of anion-molecule reactions. At low temperature these studies are complemented and extended using cryogenic ion traps, which also can measure much smaller rate coefficients than drift tubes due to the long attainable interaction times. Guided ion beam experiments are ideal for precise measurements of integral cross sections at well-defined relative energy over a large dynamic range. The dynamics of the transient reaction complex can be studied with direct time-resolved spectroscopy and its lifetime can also be indirectly inferred from ternary collision rates. When it comes to direct imaging the full reaction dynamics, crossed-beam experiments are very revealing. In particular the new opportunities of crossed-beam velocity map ion imaging should be stressed here.
Despite a wealth of theoretical studies, detailed insight into the flow of energy via the coupling of different vibrational modes during the reaction continues to be not accessed experimentally. Of particular interest is the regime of breakdown of ergodicity at the transition from a complex mediated, statistical to an impulsive reaction mechanism. It is in this range of relative collision energies, where crossed beam imaging provides the best resolution. Reactive scattering with vibrationally excited reactants - as successfully used to shed light on chemical reactions of neutrals [@zhang2009:sci] - is not yet explored for anion-molecule reactions. The role of non-reactive degrees of freedom in a reaction, referred to as the “spectator modes”, on the integral cross section and the product branching is a matter of debate that needs answers from experiments. Reactive scattering with spatially aligned or oriented molecules would directly access the stereodynamics on the molecular level. Importantly, anion-molecule reactions have very different kinetics in solution. A bottom-up approach, introducing the solvent molecule per molecule in scattering experiments with microsolvated anions, would highlight the role of the environment on the dynamics. Questions about the role of quantum effects such as Feshbach scattering resonances, tunnelling, zero-point vibrational motion or decoherence free subspaces can be dared to be asked now and demand innovative experimental approaches.
We expect that guided ion beam and ion trap studies as well as ion-molecule crossed-beam imaging will be most helpful to address some of these questions in the future. The combination of an ion trap as a source for internally cold molecular ions and clusters with a scattering experiment will allow for studies of the reactions of complex molecular systems while maintaining good control over their internal quantum states. Also merged ion and neutral beams represent a useful approach to study low-energy reactions, as demonstrated most recently for the reaction of H$^-$ with H [@bruhns2010:rsi], which is considered to be important for the formation of H$_2$ in the early universe. Laser-cooling of the reactants may eventually provide even lower collision temperatures than buffer gas cooling, an approach that has been demonstrated for cation-neutral reactions [@willitsch2008:pccp]; Os$^-$ is a possible candidate for anion laser cooling [@bilodeau2000:prl; @warring2009:prl]. All in all, there is also a lot of room for further research on anion-molecule reactions, that challenges the “complexity limit” and ultimately allows us to understand and control all the subtleties of the chemical dynamics of multi-atom molecular systems.
Acknowledgments
===============
We would like to thank our collaborators in the experiments on anion-molecule reaction dynamics, in particular Sebastian Trippel, Rico Otto, and Christoph Eichhorn. We also wish to thank Dan Neumark and Bill Hase for the fruitful collaborations on reaction dynamics in general and nucleophilic substitution reactions in particular. Our research is supported by the Elitef[ö]{}rderung der Landesstiftung Baden-W[ü]{}rttemberg, the Alexander von Humboldt Foundation, the Deutsche Forschungsgemeinschaft and the EU Marie Curie Initial Training Network ICONIC.
[140]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ** ().
, ****, ().
, ed., ** (, ).
, ****, (), ISSN .
, , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ** (, ).
, , , , , , , ****, ().
, ****, ().
, ****, ().
, , , , , , ****, ().
, , , , , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ** (, ), chap. .
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , , ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , **** ().
, , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, , , , , , ****, ().
, , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, , , , , , , , ****, ().
|
---
abstract: 'We present a search for long-period variable (LPV) stars among giant branch stars in M15 which, at \[Fe/H\] $\sim$ –2.3, is one of the most metal-poor Galactic globular clusters. We use multi-colour optical photometry from the 0.6-m Keele Thornton and 2-m Liverpool Telescopes. Variability of $\delta$V $\sim$ 0.15 mag is detected in K757 and K825 over unusually-long timescales of nearly a year, making them the most metal-poor LPVs found in a Galactic globular cluster. K825 is placed on the long secondary period sequence, identified for metal-rich LPVs, though no primary period is detectable. We discuss this variability in the context of dust production and stellar evolution at low metallicity, using additional spectra from the 6.5-m Magellan (Las Campanas) telescope. A lack of dust production, despite the presence of gaseous mass loss raises questions about the production of dust and the intra-cluster medium of this cluster.'
author:
- |
I. McDonald$^{1,2}$[^1], J.Th. van Loon$^{1}$, A.K. Dupree$^{3}$, M.L. Boyer$^{4}$\
$^{1}$Astrophysics Group, Lennard-Jones Laboratories, Keele University, Staffordshire, ST5 5BG, UK\
$^{2}$Jodrell Bank Centre for Astrophysics, Alan Turing Building, Manchester University, M13 9PL, UK\
$^{3}$Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA\
$^{4}$STScI, 3700 San Martin Drive, Baltimore, MD 21218, USA
date: 'Accepted 9999 December 31. Received 9999 December 31; in original form 9999 December 31'
title: 'Discovery of long-period variable stars in the very-metal-poor globular cluster M15'
---
\[firstpage\]
stars: AGB and post-AGB — stars: late-type — stars: Population II — stars: variables: other — stars: winds, outflows — globular clusters: individual: M15
Introduction
============
The onset of radial pulsation on the red giant branches is important in stellar evolution: it is one of several linked phenomena (including dust production and substantial mass loss) that control the endpoint of stellar evolution and injection of mass into the interstellar medium. While these phenomena are linked, the relative timing of their appearance is poorly understood [@MvL07]. In particular, it is debatable whether pulsation can provide enough energy to assist mass loss in these evolved stars (e.g. @Bowen88).
Optical photometric variability in highly-evolved stars is known to be less pronounced in metal-poor systems. @FW98 considered Long-Period Variable stars (LPVs) in both the Galactic Disc and globular clusters in order to prove this dependence, though they only consider stars showing large-amplitude variability as LPVs and do not consider semi-regular variables (SRVs, which are included in the definition of LPVs for the purposes of this work). This raises the question of whether pulsation is capable, or required, to drive mass loss from metal-poor stars.
We describe here the search for LPVs in one of the most metal-poor Galactic globular clusters, M15. This cluster is the only one known to harbour a dusty and/or gaseous interstellar medium and has several infrared-excessive giant stars (@ESvL+03; @vLSEM06; @BWvL+06), giving the strong implication that dusty stellar winds are present in the cluster and, by further implication, pulsation-driven winds.
No LPVs have so-far been found in M15, though several candidates were identified by @MW75 (Table \[M15CudworthTable\]). These were followed-up by @Welty85, who could not find photometric variability in three targets (K169, K288 and K709) and retained K757 and K825 as candidate variables (identifiers from @Kustner21).
Observations
============
Liverpool Telescope
-------------------
The first dataset for this work comes from the two-metre Liverpool Telescope (LT; @SSR+04), situated on La Palma. A total of 38 observation blocks were taken, spanning 462 days, from 2007 April 23 to 2008 July 27. A gap is present between 2007 December 04 and 2008 April 23 when the cluster was viewed in too close proximity to the Sun. Observations were taken roughly every eight days, with the exception of a one-week block of daily observations in August 2007, to build in redundancy against shorter-period variability. Scheduling and weather constraints mean the observations are randomly distributed enough not to form strong aliases in a Fourier spectrum.
Each observation consisted of 3 $\times$ 6s $g^\prime$ (477 nm) and 3 $\times$ 5s $i^\prime$ (762.5 nm) exposures, thus minimising stochastic effects from cosmic rays, etc. An additional 2s image was taken in both filters in case the stars saturated the detector. Preliminary analysis showed this was not necessary and these were not taken in the 2008 observing season. Image calibration was performed automatically before receipt.
Keele Thornton Telescope
------------------------
Additional images were taken with Keele’s 24-inch (60-cm) Thornton Telescope (KT) sited on Keele University campus, with elevated (205m above sea level) views over the Staffordshire countryside. Although these images have poorer photometric accuracy and astrometric resolution, they cover a longer timebase. The data were taken using the Santa Barbara Instrument Group ST7 CCD camera (at Newtonian focus). Several features were found that have the potential to affect the photometry: namely tracking, coolant issues and random bias and dark current problems. In practice, however, we find that the statistical scatter in photometry from this telescope is smaller than that from the LT.
With these issues in mind, observation blocks were devised of multiple 10-second observations, taken in $V$-, $R$- and $I$-bands (550, 700 and 880 nm, respectively). Observations taken at low altitude have not been corrected for differential reddening (due to its night-to-night variation): some of the scatter in the resultant photometry may be attributable to it.
In total, 76 observation blocks were taken, spanning from 2006 January 14 to 2008 November 24, though most observations were taken after 2007 August 02. Weather constraints at Keele are significantly worse than those on La Palma, leading to even less-regular sampling.
Results
=======
Candidate selection
-------------------
The presence of variability can be determined via difference imaging. Here, an image (*A*) is scaled and convolved to match the point-spread function of a second image (*B*, taken at a different epoch). The resulting image is subtracted from image *B* to create a difference image. Variable stars show up as non-zero flux.
We performed difference imaging on pairs of images from both telescopes, taken at various epochs. Variability was immediately identifiable in K825 and suggested in K757. Of the other stars in the cluster, the only obvious suggestion of variability came from the cluster’s core, which is entirely unresolved in our images. We therefore do not consider variability in stars other than those candidate variables listed in @MW75, which include K757 and K825.
Photometric reduction
---------------------
### Comparison star selection {#PRCS}
Example images from both telescopes are shown in Fig. \[M15ImageFig\]. Aperture photometry was extracted from these images using the software [AIP4Win]{} v.1.3.5 [@BB05]. Comparison stars (Table \[ComparisonTable\], Fig. \[M15ImageFig\]) were chosen to be:
[$\bullet$]{}
isolated (resolved from any visible neighbours by at least 2.5$\times$ the full-width half-maximum seeing), so that seeing has minimal effect on photometry due to blended stellar images;
bright ($i^\prime < 14$ mag), to provide good signal-to-noise, but not saturated;
red ($B-V > 0.75$ mag), to avoid problems with differential reddening at high airmass;
invariable, nor variable candidates themselves (we retain K1040 as its known short-period variability of 0.04 mag is well below our sensitivity — @Bao-An90; @YZQT93).
[llllllcr@l@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c]{} & & Membership & & $V$ & ($B-V$) & Variability & Period\
(1) & (2) & (3) & (4) & & & probability & &(mag) & (mag) & found? & (days)\
K134 & III-8 & 416 & & 212949.07 & 120902.0 & $\sim$0% & & & 14.02 & 0.99 & No & \
K147 & III-34& 442 & & 212950.01 & 120843.4 & $\sim$0% & +19&$^5$ & 12.62 & 0.75 & Maybe & \
K169 & II-64 & 212 & & 212950.81 & 121130.0 & 60% & & & 13.48 & 1.12 & No &\
K288 & II-16 & & & 212953.78 & 121020.1 & 90% & –107&.2$^6$ & 13.59 & 1.06 & No & \
K709 & IV-58 & & & 213000.39 & 120736.0 & 98% & –100&.7$^7$ & 13.52 & 1.07 & Maybe & \
K757 & IV-38 & & & 213000.91 & 120856.8 & 95% & –113&.5$^6$ & 12.58 & 1.36 & Yes & $\sim$250?\
K825 & I-12 & &S4 & 213002.23 & 121121.5 & 67% & –98&.87$^8$ & 12.69 & 1.37 & Yes & 350$\pm$20\
\
[l@[ ]{}l@[ ]{}l@[ ]{}l@[ ]{}l@[ ]{}l@[ ]{}l@[ ]{}l@[ ]{}]{} & & $V$ & – $V$)\
(1) & (2) & (3) & (4) & RA & Dec & (5) & (5)\
\
K87 & III & 438 & S29 & 212945.81& 120845.1& 13.81 & 1.01\
K112 & II & 213 & & 212947.90& 121131.3& 15.12 & 0.88$^3$\
K114 & III & 437 & & 212947.86& 120845.0& 13.87 & 1.06\
K129 & II & 187 & & 212948.61& 121145.6& 14.27 & 0.96\
K144 & II-75 & 248 & & 212949.78& 121105.6& 13.00 & 1.25\
K158 & III-33& 414 & & 212950.27& 120902.5& 14.13 & 0.78\
K731 & I-63 & & & 213000.52& 121136.9& 14.24 & 0.96\
K912 & IV-48 & 457 & & 213004.20& 120827.4& 14.40 & 0.91\
K1040 & I & 319 & S6 & 213010.49& 121006.2& 13.40 & 1.18\
\
K224 & II-30 & & & 212952.30 & 121051.3 & 14.51 & 1.07$^7$\
K238 & II-29 & & & 212952.63 & 121043.8 & 14.46 & 1.17$^7$\
K319 & II & & & 212954.42 & 121102.4 & & \
K497 & III & & & 212957.17 & 120917.3 & 14.42 & 1.29$^7$\
K508 & III & & & 212957.49 & 120906.1 & & \
K589 & IV & & & 212958.71 & 120855.8 & & \
K928 & I-43 & & & 213004.61 & 121032.6 & 13.83 & 1.03\
K943 & I-38 & & & 213005.14 & 121004.1 & 14.24 & 0.94\
K961 & I-41 & & & 213005.69 & 121015.7 & 14.09 & 1.03\
\
### Keele Thornton Telescope {#PRTT}
The obvious problem is that bright, red stars tend to be other cluster giants, which may also vary. Our comparison stars show no intrinsic variation, though a systematic offset in several of the comparison stars (K129, K144 and K912) appears depending on the orientation of the telescope. The reason for this is unclear, though seeing and airmass may play a rôle: the telescope must be inverted at this declination to observe at higher airmasses in the east, where light pollution is greatest. There may also be a weak correlation between calculated differential magnitude and seeing quality. Due to concerns that this offset is also present in the candidate stars, we have discounted the frames taken when the telescope was inverted from further analysis.
Images were calibrated and stacked using [AIP4Win]{} to increase signal-to-noise. Though not a lossless procedure, tests on individual and stacked images shows that the likely effect on the results is $<$1 milli-mag (mmag). A standard photometric reduction was performed: i.e. bias and dark frame subtraction, and flatfield division. Aperture photometry was performed using [AIP4Win]{}. Photometric errors (not provided by [AIP4Win]{}) were calculated manually by adding in quadrature the relative Poissonian errors in the photometric signal and calibration frames.
The absence of *long-*period variability of the comparison stars K144 (Arp II-75) and K1040 is interesting in itself, as these stars are identified as being among the brightest, reddest variables in the cluster by @FPC83. (Although K1040 (S6) is a known short-period variable (§\[PRCS\]), the 4.3-hour period is too short and the $\delta$V = 0.04 mag amplitude too small to affect our data.)
An identical photometric reduction was done on the candidate variable stars. The flux of each of the candidate variables was divided by the sum of the fluxes of the comparison stars to yield a differential magnitude.
### Liverpool Telescope
A similar reduction was performed on the LT data. The field here is much smaller, though the seeing is generally better. A different set of comparison stars (Table \[ComparisonTable\]) was used, following similar selection criteria. This builds in redundancy against comparison star variation as we have two completely independent data sets.
Fluxes and differential magnitudes were calculated on each processed image. These were summed over each night to improve fidelity. Here again, the non-variability of K238 (Arp II-29) is interesting, as it is suggested to be a comparatively-bright AGB star [@FPC83].
Internal comparisons between comparison stars show possible small-amplitude variations in K497, K508 and K589 on the order of 20–30 mmag. The residual long-term variation in the other six stars is $<$10 mmag. When constructing a summed comparison flux for each image, these three stars were discounted.
The combined results from both the KT and LT are shown in Fig. \[M15PhotFig\]. Differential magnitudes of the target stars are listed in Tables \[KTObsTable\] & \[LTObsTable\].
[l@llllllll]{}
&\
& K169 & K825 & K288 & K147 & K757 & K288 & K709\
\
245&3962.43 & 1.408 & 1.231 & 2.385 & 1.145 & 1.464 & 2.554 & 1.211\
& 3986.36 & 1.441 & 1.223 & 2.306 & 1.109 & 1.457 & 2.722 & 1.206\
& 3994.32 & 1.462 & 1.208 & 2.321 & 1.110 & 1.449 & 2.626 & 1.182\
& ... & ... & ... & ... & ... & ... & ... & ...\
[l@llllllll]{}
&\
& K169 & K825 & K288 & K147 & K757 & K288\
\
245&4213.71 & 2.300 & 1.611 & 2.496 & 1.108 & 1.694 & 2.692\
& 4220.69 & 2.263 & 1.613 & 2.477 & 1.092 & 1.717 & 2.666\
& 4237.64 & 2.278 & 1.620 & 2.467 & 1.078 & 1.683 & 2.661\
& ... & ... & ... & ... & ... & ... & ...\
Colour–magnitude diagrams {#M15CMDs}
-------------------------
Colour–magnitude diagrams (CMDs) were generated using [DAOPhot]{} [@Stetson87]. In contrast to the earlier aperture photometry, PSF-fitting was used to extract stellar magnitudes from the images here in order to increase coverage toward the very crowded cluster centre.
The errors calculated from PSF-fitting are correspondingly larger than those from aperture photometry: in the LT data, these are typically $\pm$9–15 mmag for PSF-fitting, compared to $\pm$1–2 mmag for aperture photometry. However, the aperture photometry errors do not take into account close stellar blends: the scatter of magnitudes for non-variable stars suggests that the actual errors in our aperture photometry may be larger, though still smaller than the PSF-fitting errors.
After PSF-fitting, an offset was applied to the pixel co-ordinates from the $i^\prime$ image to match those in the $g^\prime$ image, and the closest object to each detection matched between the bands, within a maximum distance of five pixels (1.3$^{\prime\prime}$).
Due to the lack of literature observations with $g^\prime$-, $i^\prime$- and $R$-band magnitudes, and lack of good colour transformations to obtain these from $B$- $V$- and $I$-band data alone, we have no absolute flux measurements to calibrate the $g^\prime$-, $i^\prime$- and $R$-bands. To circumvent this, we have computed theoretical AB magnitudes of K238 in these bands, using the Sloan DSS filter transmissions[^2] and the spectral energy distribution (SED) parameter-estimation technique used in [@MvLD+09], to give an approximate physical magnitude. We find the star has a temperature of 4583 K and luminosity of 805 L$_\odot$, assuming $E(B-V) = 0.10$ mag [@Harris96], and calculate $g^\prime$ = 13.783 mag, $i^\prime$ = 12.506 mag and $R$ = 12.532 mag. Note these values may be in error by $\sim$0.1 mag. The final CMD is shown in Fig. \[M15LTCMD\].
The CMD clearly resolves the red giant branch as a well-denoted sequence extending from a clearly-defined RGB-tip at $i^\prime \sim 12$ mag, down to the main-sequence turnoff at $i^\prime \approx 19$ mag. The horizontal branch (mainly around $i^\prime \sim 16$ mag) can also be seen extending to very blue colours. The dip at around $(g^\prime - i^\prime) = -0.3$ mag has also been found in earlier studies (e.g. @Stetson94), as has the kink and gap at the very top of the giant branch ($i^\prime \approx 12.3$ mag). Much of the scatter away from the giant and horizontal branches comes from poor magnitude determination or cross-matching near the crowded cluster centre.
Variability {#VarSect}
-----------
Although variability is clearly seen in Fig. \[M15PhotFig\] in several stars, the strength, periodicity and regularity of variation is difficult to determine due to the long-period nature of the variations.
In order to determine where variability occurs, we have created the index $\mu$. We calculate this as follows: for each star and filter, we take the stellar magnitude time series and remove any points that deviate from the last observation by more than 0.15 mag. This removes any bad data deviating by $\gtrsim 5 \times$ the standard photometric error from the average observation. For this clipped dataset, we take the standard deviation, $\sigma_{\rm S}$ and divide it by the standard deviation of the point-to-point differences, $\sigma_{\rm P}$, giving $\mu = \sigma_{\rm S} / \sigma_{\rm P}$. For a non-variable star, $\mu$ should be very near-unity. Slight departures from unity may be due to remaining bad data, but values significantly above unity should be due to variability. The index is equally sensitive to both regular and irregular variability.
For each star, we have listed in Table \[MuTable\] two variability statistics: $\mu_{\rm K}$ for the KT, given by the average of $\mu$ in the $V$-, $R$- and $I-$bands; and $\mu_{\rm L}$ for the LT, which is the average of $\mu$ in the $g^{\prime}$ and $i^{\prime}$ bands. We also give the r.m.s. variation, which demonstrates the stability of the KT data: the only star with a higher r.m.s. in these data is K288, which is close to the core and suffers from some blending. Finally, we also include the Pearson product-moment correlation co-efficient, $r$, between binned KT and LT data (positive values approaching unity denote good correlation). Here, each telescope’s data is represented by a 30-day bin of the deviation from average magnitude, over all filters. This assists differentiation between intrinsic variability and variability caused by changing observing conditions.
[l@c@c@c@c@c@l]{} & & & & & Variability\
ID & $\mu_{\rm K}$ & $\mu_{\rm L}$ & & $r$& detected?\
K134 & 1.19 & — & 30 & — & — & No\
K147 & 1.36 & 2.23 & 15 & 24 &–0.35 & Indefinite$^\ast$\
K169 & 1.14 & 0.94 & 18 & 20 &–0.36 & No\
K288 & 1.28 & 1.33 & 31 & 11 &–0.40 & No\
K709 & 1.35 & — & 19 & — & — & Maybe\
K757 & 1.65 & 2.13 & 44 & 46 & +0.73 & Yes\
K825 & 1.51 & 1.98 & 34 & 41 & +0.91 & Yes\
\
From Table \[MuTable\], we can see that both telescopes show high values of $\mu$ — indicating the presence of variability — in K757 and K825. Marginal detections of variability are possible in K709, for which there is no LT data, and K147. In the latter case, however, there appears no correlation between the trends observed from the KT and those from the LT, so we neither claim nor refute variability here.
Discussion
==========
Individual stars
----------------
The results for individual stars are summarised in Table \[M15CudworthTable\].
### K134 (Arp III-8)
Although not always in the field of view of the LT, there is nevertheless sufficient data from both telescopes to make two independent tests of variability in this star. There appears to be no variation above the scatter in the data, and we can thus conclude that K134 does not show any substantial long period variations. Note that this star is not thought to be a cluster member [@Cudworth76].
### K147 (Arp III-34) {#K147Sect}
One of the brightest stars in the cluster, K147 is a relatively blue star: @BBC+83 show it to have a near-zero ($U-B$) colour. Its proper motion and radial velocity show it to be a field star (Table \[M15CudworthTable\]). While our observations show some variability may be present (§\[VarSect\]), data from the two telescopes are discordant about the trend of this variability, suggesting the observed variability is spurious. It is possible that these variations in our data are due to seasonal instrumental effects. The sensitivity of our data means that we cannot rule out variability in this star at low ($\lesssim \pm$1%) amplitudes.
### K169 (Arp II-64)
The photometry of K169 has a relatively low scatter. It clearly shows no large-amplitude variations. Our photometry does not suggest variability in this star and we therefore conclude that it does not pulsate significantly on timescales from days to up to a year.
### K288 (Arp II-16)
K288 shows relatively low-noise photometry in the LT data, but shows a larger scatter in the KT photometry due to its proximity to the cluster core. @Welty85 could not determine variability in this star because of this, though our data show low enough scatter to determine that there appears to be no coherent variation in this star of $\delta V \gtrsim 0.01$ mag. We conclude that this star is also not variable on long timescales.
### K709 (Arp IV-58)
K709 may show variability of $\delta V \sim 0.08-0.10$ magnitudes, with maxima at JD 2454400 and 2454600, and a minimum near JD 2454500. The star also has a moderately-high variability index in the KT data, comparable with that of K147 (§\[VarSect\]). We do not have data from the LT to determine if this variability is real (it lies beyond the LT field of view), and the variations are close to the noise level in the KT data. Variability in this star would be interesting, as @Welty85 find this star to be non-variable, with an r.m.s. of 0.054 mag: the variations seen in our data would seem to be slightly larger. It is possible that Welty missed variability here due to sparsity of data coverage.
Fig. \[M15FFTFig\] shows a Fast Fourier Transform (FFT) in each filter. The left-most peak near 0 cycles day$^{-1}$, most obvious in K709 in the $R$-band, is the timebase of our photometry, but K709’s power spectrum shows no other peaks of statistical significance. The data therefore appear to be of insufficient coverage to determine any periodicity, should it exist.
### K757 (Arp IV-38)
As one of the brightest and reddest stars in the cluster, and also as a confirmed radial velocity member (Table \[M15CudworthTable\]), this star is a prime candidate for being an LPV. Indeed, the data from both telescopes shows a concurrent dimming around JD 2454340 for $\sim$100 days. This drop is wavelength-dependent, showing a clear increase with decreasing wavelength, decreasing from $\sim$0.2 magnitudes for the $g^\prime$ data to $<$0.05 magnitudes in $I$-band. Further, smaller drops may be seen near JD 2454100 and JD 2454600, though they are less-well covered.
The light curve is very similar to a typical binary star showing a single eclipse and ellipsoidal variations, though the star would have to be in a contact or near-contact binary to show variation at this (approximate) period. We consider it more likely that this star is showing pulsation. Again, the data are of insufficient coverage to determine whether a regular period exists to this variability. The FFT (Fig. \[M15FFTFig\]) shows a broad peak in all bands (except $i^\prime$) between 100 and 500 days, though visual examination of the lightcurve (Fig. \[M15PhotFig\]) suggests that it is probably $\sim$250 days. Further data are needed to see if this variation is truly periodic.
### K825 (Arp I-12) {#K825Sect}
K825 is another particularly bright and red star in the cluster which is also a confirmed radial velocity member (Table \[M15CudworthTable\]). Physically, the star is also the most-luminous star in the cluster, and one of the coolest. While we cannot distinguish between AGB and RGB stars at this luminosity, K825 is likely to define the RGB tip. It is also a known H$\alpha$ variable (see §\[Halpha\] and @MDS+08), suggesting either chromospheric activity or pulsation.
Variations from this star were immediately visible in the data from an early stage of the analysis. A clear dip is seen near JD 2454300 and another near JD 2454650 in all five bands. A further possible dip at JD 2454050 may be present, but there is only one epoch of observation here. Again, these dips are wavelength-dependent, decreasing from $\sim$0.17 mag in $g^\prime$ to $\sim$0.08 mag in $I$-band.
The light curve shows a classic long-period variation attributable to stellar pulsation, with a rapid drop in brightness at bluer wavelengths as the star expands and cools, then a slower recovery as the star contracts and warms. FFT analysis (Fig. \[M15FFTFig\]) shows peaks in all bands between 330 and 370 days. The strong peak in the LT data at 200 days is an alias of this caused by insufficient data coverage. The lack of a clear drop at JD 2453950 suggests that the amplitude of variability is irregular and thus that this star is an SRV.
Variability in the context of stellar evolution
-----------------------------------------------
### Pulsation as the source of variability {#HRDSect}
We have found variability in two of the cluster’s most evolved members (K757 and K825). These stars seem to share the metallicity of the cluster: both are within 0.12 dex of the the cluster’s average value, \[Fe/H\] = –2.4 (@SKS+97, 2000). These therefore represent the most metal-poor long-period variables known in our Galaxy. Outside our Galaxy, there is only an unconfirmed member of the Boötes I dwarf galaxy that may be more metal-poor [@DOCK+06]. This star has a period of $P \approx 85$ days, $\delta V \approx 0.4$ and an optical magnitude which would place it on the AGB, were it confirmed a member.
Notably, @SPK00 also find both K757 and K825 to be barium-enhanced by \[Ba/Fe\] $\approx$ +0.37, among the highest enhancement in the cluster, suggesting convective mixing of $s$-process elements (see also @vLvLS+07). In Sneden et al.’s earlier work, they further suggest that both may be double stars, on the basis of “weak, blue-displaced extra absorption components”. Two-epoch observations of K757 by @MDS08, which show a radial velocity shift of 6.2 km s$^{-1}$, suggest this may be the case. If pulsation is present, the “extra absorption components” visible in the spectra may be pulsation-induced line doubling, as seen in, e.g., 47 Tucanae [@LWH+05].
The location of K757 and K825 on the giant branch, along with the variation of these stars in observed colours are shown in Fig. \[M15LTCMD3\], and physical units in Fig. \[M15HRDVarFig\]. It is clear from these figures that not only are these two stars among the most-evolved in the cluster, but that their variability arises from significant changes in both temperature *and* luminosity. Stellar parameters for K757 and K825 were calculated using the same method as for K238 (§\[M15CMDs\]), with data from various optical catalogues, 2MASS and @BWvL+06. For K757, this analysis gives a temperature of $4489 \pm 201 K$, log($g$) = 0.746, and $L$ = 1426 $\pm$ 255 L$_\odot$ for K757; for K825, we find $T = 4411 \pm 155$ K, log($g$) = 0.662, $L$ = 1615 $\pm$ 227 L$_\odot$, which are broadly consistent with literature values (@FPC83; @SKS+97; @ASA+00; @MDS+08).
Variation in these parameters was estimated by refitting the SED using 50-day averages of the optical ($g^\prime, V, R, i^\prime, I$) magnitudes. Fig. \[M15HRDVarFig\] shows both stars exhibit a variation between a high-temperature, high-luminosity state, and a low-temperature, low-luminosity state, with $\delta T_{\rm eff} \approx \pm50$ K, and $\delta L \approx \pm 30$ L$_\odot$[^3]. (We presume that, since variability declines with increasing wavelength, there is negligable variation at longer wavelengths than $I$-band.) If we assume that the total optical depth of any dust envelope is largely constant through these changes, and that the stellar spectrum is close to a blackbody, this corresponds to a radius change of $\pm$1.1% around an average of 62 R$_\odot$ for K757, and $\pm$0.9% around an average of 69 R$_\odot$ for K825.
This type of variation — where maximum brightness occurs near-simultaneously for wavelengths longer than $\approx$1 $\mu$m — is also seen in stronger metal-poor pulsators in globular clusters (cf. $\omega$ Cen V42 — @DFLE72; @MW85). Similar changes can occur on spotted stars, though a rotation period of 350 days requires a rotation rate of 11 km s$^{-1}$. This is somewhat large for a star this evolved [@CSRB+09]. Furthermore, it would require the spots to be relatively constant in size and strength over a period of $\sim$1 year; it also cannot explain the changes seen in the H$\alpha$ line. A 350-day sinusoidal variation in radius with $\sim$1% amplitude yields a peak pulsation amplitude of only $\pm$0.1 km s$^{-1}$ at the photosphere for both K757 and K825. This value is very small compared to both the $\sim$60 km s$^{-1}$ escape velocity, and the speed with which we might expect a dusty wind to be accelerated via photon pressure alone. However, the speed may increase as the pulsation travels outward through the more rarified atmosphere.
For comparison, we can calculate the speed one might expect of a dusty wind. Assuming canonical values of 10 km s$^{-1}$ for a 10000 L$_\odot$ solar metallicity star, and $v \propto 10^{\rm 0.5 [Fe/H]} L^{0.25}$ (@vanLoon00; @MvLM+04), we might expect this speed to be $v \approx 450$ ms$^{-1}$ if gas-dust coupling is maintained. If gas-dust coupling is not maintained, and a driving mechanism is coupled to *either* the gas or dust, then we might expect the driven medium to escape, and the other to be left behind and fall back onto the star.
### K825: H$\alpha$ line and gaseous mass loss {#Halpha}
*Spitzer* photometry [@BWvL+06] shows no infrared excess around K757 and K825, suggesting a lack of circumstellar dust production. It appears that these stars are of too low a metallicity to sustain adequate dust production to drive a stellar wind. In the absence of a dust-driven wind, we turn our attention to the stellar chromosphere: blue-shifted absorption cores, often coupled with line emission wings, have been observed in several notable lines and are taken as tracers of the mass-loss rate and velocity of an outflow. Such lines include Mg [ii]{} k, Na [i]{} D, Ca [ii]{} K, H$\alpha$, He [i]{} and various UV lines (e.g. @DHA84; @DWP99; @CBR+04; @DLY+05; @DLS07; @MvL07; @DSS09).
We can compare our light curve to recent spectroscopic observations of K825 taken as part of an unrelated programme (K757 has already been presented in @MAD09). Fig. \[HalphaFig\] presents spectra taken with the MIKE double-echelle spectrograph mounted on the Magellan/CLAY 6.5m telescope at Las Campanas Observatory using a 0.75$^{\prime\prime}$ $\times$ 5$^{\prime\prime}$ slit, giving a resolution of $R \approx 40\,000$. The spectra were reduced with bias subtraction, flat-fielding and sky subtraction using the MIKE-IDL pipeline updated by J. Prochaska[^4]. Th-Ar arc exposures bracketting the stellar targets were used to determine the wavelength scale. Four epochs are available: two fortuitously near subsequent photometric minima, the other two epochs being shortly after adjecent photometric maxima. For each spectrum, the line bisector is also shown and the velocity shift of the line core with respect to the photospheric rest velocity indicated. These velocities are generally related to the wind outflow velocity but do not necessarily approximate them — modelling suggests that actual outflow velocities are higher than indicated by the H$\alpha$ profiles (@MCP06; @MvL07; @MAD09).
Variation seen in our lightcurve of K825 is to some extent repeated in its H$\alpha$ and Ca [ii]{} K line profiles: the spectra taken at photometric minimum show a moderate asymmetry in the H$\alpha$ emission, while the spectra taken near maximum light show either zero or reversed asymmetry in the H$\alpha$ wings. The Ca [ii]{} K lines show similar variations in the strength of their emission, though the red emission component is always dominant here. The behaviour is not entirely reproducable between epochs — there is considerably greater asymmetry at phases 1.32 and 2.16, presumably due to a more massive outflow. This suggests a more chaotic variation of the chromosphere, partially but not entirely coupled to the pulsation period.
It is interesting to note that the H$\alpha$ and Ca [ii]{} line cores are permanently blue-shifted. There is no obvious correlation with pulsation period here either, though the velocities of H$\alpha$ and Ca [ii]{} correlate well with each other. The permanency of this blueshift, despite substantial changes in the emission from the (lower altitude) chromosphere, strongly suggests that there is a permanent bulk outflow from the star. As Ca [ii]{} K absorption occurs higher in the atmosphere than H$\alpha$, and the Ca [ii]{} blueshifts are larger, we can also state that the wind is strongly accelerating in this region. This means that star’s mass loss may be enhanced by pulsation, and there may well be other driving mechanisms (e.g. magnetic reconnection emitting Alfvén waves) which operate within the line-forming region of the extended atmosphere.
@MAD09 find that K757 shows similarly permanent blueshifts in its H$\alpha$ line cores, despite changes in the line emission from the chromosphere. By modelling the H$\alpha$ line, they imply that K757’s gaseous mass-loss rate changed by a factor of $\sim$6 over their observations. Given both stars show a permanent outflow, with velocities that change by a factor of $<2$, we may speculate that the wind driving mechanism remains relatively constant. The strong changes in emission strengths, however, suggest that the rate at which material is injected into the wind is strongly variable. This further suggests that pulsations may vary the mass-loss rate from these systems, but that the kinetic energy that subsequently accelerates the wind may be provided by another, more-constant source.
### K757, K825 and the $P-L$ diagram
As a populous cluster, we may therefore naïvely expect M15 to contain a relatively large number of high-amplitude LPVs and SRVs. For comparison, $\omega$ Cen — which is five times as massive and 0.7 dex more metal rich than M15 — has $\sim$20 variables with periods $>$20 days and $V$-band amplitudes $>$0.2 mag (@Clement97; @MHB04; @PdMPB04; @vdVvdBVdZ06). By this estimate, we might expect 4$\pm$2 LPVs in M15. Finding two LPVs is therefore not unexpected, especially given our incomplete coverage. The absence of any *high amplitude* LPVs in M15 is notable, however. This confirms that stellar pulsation plays less of a rôle in stellar evolution at substantially sub-solar metallicities [@FW98], but is still present among the oldest, most-metal-poor giants we see.
The long periods of K757 and K825 are unusual. On a $P-L$ diagram (Fig. \[M15DKBFig\]), they fall neatly onto Wood’s sequence D (@WAA+99; @Wood00). This sequence denotes pulsators with long, *secondary* periods (LSPs), though we find no shorter-period pulsations that would fit sequence C (at $\sim$90 days) or higher harmonics (cf. Fig. \[M15FFTFig\]). However, the two stars also fit sequence E from Derekas et al., which is attributed to ellipsoidal variation in red giant (contact) binaries. Note, however, that K825 shows a more temporally asymmetric variability than many similar stars on sequence E (cf. 2.5873.24 in Derekas et al., Fig. 2). For comparison, the unconfirmed Böotes I dwarf galaxy member would lie near sequence C.
Pulsation period is general thought to decrease towards low metallicity among red giants of similar age in Galactic globular clusters [@FW98]. Were this the case, it would be extremely surprising to find K757 and K825 placed so neatly on one Wood’s sequences. Furthermore, K825 shows a period as long as the star with the longest period in Frogel & Whitelock’s sample (NGC 5927 V3, at 312 days), but at a metallicity where no variables this long are found; the closest analogue in @Clement97 is NGC 2419 V10, a semi-regular variable with $P = 81.3$ days and \[Fe/H\] = –2.12. Frogel & Whitelock’s data do not include all globular cluster variables, and they only claim their correlation of pulsation period and metallicity exists for \[Fe/H\] $> -1.3$. It appears that, while pulsation amplitude declines with metallicity, pulsation period does not change appreciably.
In Fig. \[M15FWFig\] we reproduce Frogel & Whitelock’s Fig. 4 with K825 and NGC 2419 V10 included. From these data, it appears as though no variables with $P > 200$ days and \[Fe/H\] $< -1.3$ had been discovered prior to @FW98. We appear to now have found stars in this missing region. The apparent lower luminosity of the RGB tip and lack of high-luminosity AGB stars in M15 (M15 has no obvious AGB stars (§\[HRDSect\]), in contrast with intermediate-metallicity clusters, e.g., @vLMO+06; @BMvL+09; @McDonald09) means that these stars are on the LSP sequence D, rather than appearing in sequence C like their metal-intermediate counterparts.
### Intracluster medium and dust production
High stellar temperatures and low metallicity make it hard to form circumstellar dust around stars in M15. Despite this, M15 remains the only cluster in which intracluster gas and dust have been detected (@ESvL+03; @vLSEM06; @BWvL+06; @vLSP+09). These intracluster media must have been produced in the (astronomically-)recent past ($\sim$1 Myr). However, while K757 and K825 both show evidence for gaseous mass loss (@MDS08; @MAD09), neither star appears to be producing dust [@BWvL+06] suggesting dust production in M15 may not be dominated by stars near and above the RGB tip, as seen in other, more-metal-rich clusters (e.g. $\omega$ Centauri, at \[Fe/H\] $\sim$ –1.6; and 47 Tuc at \[Fe/H\] = –0.7 — @vLMO+06; @LPH+06; @ORFF+07; @MvLD+09). Two likely possibilities remain to explain the presence of intra-cluster dust in M15. Dust production in individual stars could be episodic (e.g. @OFFPR02, 2007; @MvLD+09). Alternatively, relatively-massive ($M_{\rm H} \sim $0.1 M$_\odot$) amounts of dust-containing medium could be produced in a superwind at the AGB (and possibly RGB) tips, or in creating a post-AGB star. Indeed, Boyer et al. observe dust in M15’s planetary nebula, Pease 1. However, the amount of intra-cluster medium ($M_{\rm H} \sim $0.3 M$_\odot$; @vLSEM06) would require this short phase to have recently occurred near-simultaneously (within $\sim$1 Myr) for at least three stars. As a result, we expect that episodic dust production around stars with more-sustained gaseous outflows seems the more-likely explanation for the presence of intra-cluster dust, though both factors could contribute.
If dust production is episodic, would we expect to see dust in the most-evolved stars? Of the 17 most-evolved stars in M15 (Fig. \[M15LTCMD3\]), only K479, K421 and K373 show clear infrared excess in @BWvL+06. K204 may also show excess, but it is blended with K570 in the *Spitzer* Space Telescope 3–70-$\mu$m photometry. Assuming these stars have luminosities of 1400–1600 L$_\odot$ (based on §\[K825Sect\]), we can compare this to Fig. 20 of @MvLD+09, which shows the fraction of dusty stars as a function of luminosity in $\omega$ Cen. Assuming a detection limit for circumstellar dust corresponding approximately to the 6$\sigma$ line in McDonald et al. (this approximately takes into account the metallicity difference between the clusters), we might presume that $\sim$30% (five) of these 17 stars would have circumstellar dust, which is statistically indifferent from the three or four observed.
The lack of dust production in K757 and K825 is puzzling, however, when one considers that the above H$\alpha$ observations (§\[Halpha\]; @MAD09) indicate gaseous mass loss does occur from these stars, and indeed from stars further down the giant branch (@MvL07; @MAD09). The pulsation period is long enough for dust to form between pulsations, but not long enough for dust to become so cool as to be undetectable in its \[8\]–\[24\] colour (assuming $v_\infty \lesssim 50$ km s$^{-1}$). The pulsations also appear not to generate strong shock waves (which show up as sharp peaks in H$\alpha$ — e.g. @MvL07), which may assist dust production. The constant, moderate-velocity outflow provided by the chromosphere may also prevent the formation of a quasi-static molecular layer required to seed dust formation [@TOAY97]; the formation of such a layer being additionally hindered by the hotter photospheric temperatures and reduced availability of metals in such low-metallicity stars. Furthermore, it may be that the low amplitude of the pulsations means the scaleheight of the atmosphere is not sufficiently raised, meaning the wind density beyond the dust-formation radius is too low to efficiently produce dust, while the increased effective temperature of the metal-poor stars means the dust-formation radius is too far from the star.
It is clear from observations of more metal-rich clusters (@RJ01; @vLMO+06; @LPH+06; @MvLD+09) that efficient dust production occurs mainly in strongly variable stars at the upper end of the RGB, and that the amount of mass loss roughly correlates with the strength of the pulsation as traced by the optical photometric amplitude. Our observations here confirm that correlation, finding no detectable dust production in stars which (while at the RGB tip) exhibit only weak pulsation.
Conclusions
===========
This study presents a search for long-period variability among giants in the globular cluster M15 and has confirmed the most metal-poor variables known in our Galaxy: K757 and K825. These stars are very close to the RGB-tip and show lightcurves characteristic of pulsation, albeit with very low amplitude. Their periods place them on the ‘long secondary period’ (LSP) sequence (@WAA+99; @Wood00), though no ‘primary’ period has been found.
The pulsational velocities are similar to those measured at the base of the wind, but it is not certain that pulsation is required to launch or drive the wind. In any case, these pulsations appear not to lead to dust production, which must be caused by a different mechanism, possibly episodically. Although we do find evidence for acceleration in the singly-ionised calcium line profiles, radiation pressure on grains cannot therefore be held responsible for driving the wind. Instread, Alfvén waves may couple to a weakly-ionised gas and thus drive the wind, something which could be facilitated at lower metallicity [@vLOG+10].
Acknowledgments {#acknowledgments .unnumbered}
===============
I.M. was supported by a PPARC/STFC studentship for the initial stages of this work. This paper includes data gathered with the 6.5m Magellan Telescopes located at Las Campanas Observatory, Chile; the Liverpool Telescope, operated on the island of La Palma by John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrophysica de Canarias with financial support from the UK Science and Technology Facilities Council; and the Keele Thornton Telescope operated by and located at Keele University, with dedicated support from local volunteers.
[65]{} natexlab\#1[\#1]{}
A., [Salaris]{} M., [Arribas]{} S., [Mart[í]{}nez-Roger]{} C., [Asensio Ramos]{} A., 2000, A&A, 355, 1060
H. C., 1955, AJ, 60, 317
Y., 1990, IBVS, 3431, 1
R., [Burnell]{} J., 2005, [The handbook of astronomical image processing]{}, 2nd ed. Richmond, VA: Willmann-Bell, 2005
G. H., 1988, in [Stalio]{} R., [Willson]{} L. A., eds., ASSL Vol. 148: Pulsation and Mass Loss in Stars, p. 3
M. L., [et al.]{}, 2009, ApJ, 705, 746
M. L., [Woodward]{} C. E., [van Loon]{} J. T., [Gordon]{} K. D., [Evans]{} A., [Gehrz]{} R. D., [Helton]{} L. A., [Polomski]{} E. F., 2006, AJ, 132, 1415
A., 1951, ApJ, 113, 344
R., [Buscema]{} G., [Corsi]{} C. E., [Iannicola]{} G., [Fusi Pecci]{} F., 1983, A&AS, 51, 83
C., [Bragaglia]{} A., [Rossetti]{} E., [Fusi Pecci]{} F., [Mulas]{} G., [Carretta]{} E., [Gratton]{} R. G., [Momany]{} Y., [Pasquini]{} L., 2004, A&A, 413, 343
C., 1997, VizieR Online Data Catalog, 5097, 0
C., [Silva]{} J. R. P., [Recio-Blanco]{} A., [Catelan]{} M., [Do Nascimento]{} J. D., [De Medeiros]{} J. R., 2009, ApJ, 704, 750
K. M., 1976, AJ, 81, 519
M., [Clementini]{} G., [Kinemuchi]{} K., [Ripepi]{} V., [Marconi]{} M., [di Fabrizio]{} L., [Greco]{} C., [Rodgers]{} C. T., [K[ü]{}hn]{} C., [Smith]{} H. A., 2006, ApJ, 653, L109
A., [Kiss]{} L. L., [Bedding]{} T. R., [Kjeldsen]{} H., [Lah]{} P., [Szab[ó]{}]{} G. M., 2006, ApJ, 650, L55
R. J., [Feast]{} M. W., [Lloyd Evans]{} T., 1972, MNRAS, 159, 337
A. K., [Hartmann]{} L., [Avrett]{} E. H., 1984, ApJ, 281, L37
A. K., [Whitney]{} B. A., [Pasquini]{} L., 1999, ApJ, 520, 751
A. K., [Lobel]{} A., [Young]{} P. R., [Ake]{} T. B., [Linsky]{} J. L., [Redfield]{} S., 2005, ApJ, 622, 629
A. K., [Li]{} T. Q., [Smith]{} G. H., 2007, AJ, 134, 1348
A. K., [Smith]{} G. H., [Strader]{} J., 2009, AJ, 138, 1485
A., [Stickel]{} M., [van Loon]{} J. T., [Eyres]{} S. P. S., [Hopwood]{} M. E. L., [Penny]{} A. J., 2003, A&A, 408, L9
J. A., [Persson]{} S. E., [Cohen]{} J. G., 1983, ApJS, 53, 713
J. A., [Whitelock]{} P. A., 1998, AJ, 116, 754
K., [Pryor]{} C., [Williams]{} T. B., [Hesser]{} J. E., [Stetson]{} P. B., 1997, AJ, 113, 1026
Harris W. E., 1996, ApJ, 112, 1487
F., 1921, Veröffentlichungen des Astronomisches Institute der Universität Bonn, 15, 1
T., [Posch]{} T., [Hinkle]{} K., [Wood]{} P. R., [Bouwman]{} J., 2006, ApJ, 653, L145
T., [Wood]{} P. R., [Hinkle]{} K. H., [Joyce]{} R. R., [Fekel]{} F. C., 2005, A&A, 432, 207
J. R., [van Loon]{} J. T., [Matsuura]{} M., [Wood]{} P. R., [Zijlstra]{} A. A., [Whitelock]{} P. A., 2004, MNRAS, 355, 1348
P. J. D., [Cacciari]{} C., [Pasquini]{} L., 2006, A&A, 454, 609
I., 2009, Ph.D. Thesis, Keele Univ.
I., [van Loon]{} J. T., 2007, A&A, 476, 1261
I., [van Loon]{} J. T., [Decin]{} L., [Boyer]{} M. L., [Dupree]{} A. K., [Evans]{} A., [Gehrz]{} R. D., [Woodward]{} C. E., 2009, MNRAS, 394, 831
B. J., [Harrison]{} T. E., [Baumgardt]{} H., 2004, ApJ, 602, 264
J. W., [Whitelock]{} P. A., 1985, MNRAS, 212, 783
S., [Dupree]{} A. K., [Szentgy[ö]{}rgyi]{} A., 2008, AJ, 135, 1117
—, 2008, AJ, 135, 1117
S., [Avrett]{} E. H., [Dupree]{} A. K., 2009, AJ, 138, 615
D. R., [White]{} R. E., 1975, in Bull. Amer. Astron. Soc., Vol. 7, p. 535
C. P., [Wood]{} P. R., [Cioni]{} M., [Soszy[ń]{}ski]{} I., 2009, MNRAS, 399, 2063
L., [Ferraro]{} F. R., [Fusi Pecci]{} F., [Rood]{} R. T., 2002, ApJ, 571, 458
L., [Rood]{} R. T., [Fabbri]{} S., [Ferraro]{} F. R., [Fusi Pecci]{} F., [Rich]{} R. M., 2007, ApJ, 667, L85
A., [de Marchi]{} G., [Pulone]{} L., [Brigas]{} M. S., 2004, A&A, 428, 469
C. A., [Sneden]{} C., [Kraft]{} R. P., [Harmer]{} D., [Willmarth]{} D., 2000, AJ, 119, 2895
A., [Jorissen]{} A., 2001, A&A, 372, 85
A., 1970, ApJ, 162, 841
J. V., [Keenan]{} F. P., [Lehner]{} N., [Trundle]{} C., 2002, A&A, 387, 1057
C., [Kraft]{} R. P., [Shetrone]{} M. D., [Smith]{} G. H., [Langer]{} G. E., [Prosser]{} C. F., 1997, AJ, 114, 1964
C., [Pilachowski]{} C. A., [Kraft]{} R. P., 2000, AJ, 120, 1351
I. A., [et al.]{}, 2004, in [J. M. Oschmann Jr.]{}, ed., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5489, p. 679
P. B., 1987, PASP, 99, 191
—, 1994, PASP, 106, 250
T., [Ohnaka]{} K., [Aoki]{} W., [Yamamura]{} I., 1997, A&A, 320, L1
G., [van den Bosch]{} R. C. E., [Verolme]{} E. K., [de Zeeuw]{} P. T., 2006, A&A, 445, 513
J. T., 2000, A&A, 354, 125
J. T., [McDonald]{} I., [Oliveira]{} J. M., [Evans]{} A., [Boyer]{} M. L., [Gehrz]{} R. D., [Polomski]{} E., [Woodward]{} C. E., 2006, A&A, 450, 339
J. T., [Stanimirovi[ć]{}]{} S., [Evans]{} A., [Muller]{} E., 2006, MNRAS, 365, 1277
J. T., [van Leeuwen]{} F., [Smalley]{} B., [Smith]{} A. W., [Lyons]{} N. A., [McDonald]{} I., [Boyer]{} M. L., 2007, MNRAS, 382, 1353
J. T., [Stanimirovi[ć]{}]{} S., [Putman]{} M. E., [Peek]{} J. E. G., [Gibson]{} S. J., [Douglas]{} K. A., [Korpela]{} E. J., 2009, MNRAS, 396, 1096
J. T., [et al.]{}, 2010, AJ, 139, 68
D. E., 1985, AJ, 90, 2555
P. R., 2000, 17, 18
P. R., [et al.]{}, 1999, in [le Bertre]{} T., [Lebre]{} A., [Waelkens]{} C., eds., IAU Symp. 191: Asymptotic Giant Branch Stars, p. 151
B.-A., [Zhang]{} C.-S., [Qin]{} D., [Tong]{} J.-H., 1993, Ap&SS, 210, 163
\[lastpage\]
[^1]: E-mail: mcdonald@jb.man.ac.uk
[^2]: http://www.sdss.org/dr3/instruments/imager/\#filters
[^3]: Though the absolute errors are large, relative errors can be considered much smaller. Ranges also include a small (estimated $\sim $10 – 20%) noise contribution.
[^4]: http://web.mit.edu/$\sim$burles/www/MIKE/
|
---
author:
- 'D.-Y. Yang'
- 'L.-F. Li'
- 'Q.-W. Han'
bibliography:
- 'bib.bib'
title: Absolute physical parameters of three poorly studied detached eclipsing binaries
---
Introduction {#sect:intro}
============
Various questions about stellar structure and evolution remain to be answered and also require precise and detailed information on the physical parameters, such as mass, radius, luminosity and effective temperature . Meanwhile, the mass of a star is the most critical parameter, which determines the way it evolves and what is left over after its death [@2015NewA...41...42B]. By far, the accurate method to derive the masses of the stars is through the photometric and spectroscopic observations of eclipsing binaries. Current observational techniques can produce masses to a precision of better than $3\%$, accurate enough to provide strong constraint for stellar models with inadequate physics to be rejected [@2009ApJ...700.1349T]. Particularly, detached eclipsing binaries contain two components, which have not filled their Roche lobe yet (i.e. they have not been contaminated by the mass transfer between them) and thus evolve as single stars [@2015NewA...41...42B]. Therefore, the two components of detached eclipsing binaries with well determined parameters can provide a stringent test for the stellar evolutionary models.
Short-period binaries usually have an orbital eccentricity close to zero due to the strong tidal friction, and some long-period ones sometimes have an eccentric orbit. In addition, double-lined spectroscopic binaries, especially those with a low mass ratio, are rare, since the secondary star is too faint to be spectroscopically observed at present [@2019AJ....158..198F]. However, the double-lined spectroscopic binaries can give us the opportunity to obtain their orbital parameters, and to determine their precise masses together with other physical parameters . Moreover, detached binaries are the progenitors of many peculiar objects , and detailed observation investigation of detached binaries are also important to understand the formation of these peculiar objects.
In this work, we selected three detached eclipsing binaries from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope Media-Resolution Spectra Survey[LAMOST-MRS, @1996ApOpt..35.5155W; @1997ASSL..212...67C; @1999oaaf.conf....1Z; @2012RAA....12..723Z; @2006ChJAA...6..265Z; @2015RAA....15.1095L], and determined their mass ratio based on the analysis of their radial velocity curves. The photometric observations in $V$ pass-band for these binaries were collected from the All-Sky Automated Survey[ASAS, @1997AcA....47..467P], the All-Sky Automated Survey for supernova [ASAS-SN, @2014ApJ...788...48S] and the Wide Angle Search for Planets . The spectroscopic and photometric data were analyzed simultaneously by using the pyWD2015 code [@2020CoSka..50..535G]. In this process, we determined the accurate orbital and physical parameters for the three objects and discussed their evolution. The basic information of the three objects are listed in Table \[table:table1\].
This paper is organized as follows. In Sect \[sec:two\], the origin of the photometric and spectroscopic data used in this work are stated. In Sect \[sec:three\], the method used for our study and the results of the derived accurate parameters are described. Our discussions and conclusions are shown in Sect \[sec:five\].
Observation data {#sec:two}
================
Photometry data {#subsec:tables}
---------------
The photometric data for three detached eclipsing binaries were collected from various photometric surveys, and the orbital periods of these binaries are too long to obtain the complete light curve(s) easily. Only data points with a relatively high precision were used.
For ASASSN-V J063123.82+192341.9, a total of 111 measurements in $V$ band were collected from ASAS-SN [@2017PASP..129j4502K; @2018MNRAS.477.3145J], and all measurements were used in deriving the photometric solution for this eclipsing binary since they have a relatively high observational accurancy (better than 0.03 mag). For ASAS J011416+0426.4, a total of 528 observations in $V$ band were obtained from ASAS [@2001ASPC..246...53P; @2008AcA....58..405S], only 469 measurements were employed to obtain a light curve solution for this object after 59 scattered data points were removed. In addition, a total of 3067 observations in $V$ band for MW Aur were obtained from WASP , but only 2723 data points were used for analysis after the scattered points and those with an error higher than 0.1 mag were removed.
Spectroscopy data {#subsec:tables}
-----------------
LAMOST is a special reflecting Schmidt telescope with an effective aperture of 3.6$\times$4.9m, a focal length of 20m and a field of view (FOV) of $5^\circ$, which locates at Xinglong station, Hebei Province, China [@1996ApOpt..35.5155W; @2012RAA....12..723Z]. The focal surface has 4000 precisely positioned fibers connected to 16 spectrographs with a distributive parallel-controllable fiber positioning system, each spectrograph equipped with a 4K $\times$ 4K Charge-Coupled Devices (CCD) for blue and red channels [@1998SPIE.3352..839X]. By the end of 2017, all 16 media$-$resolution spectrographs were in operation, and had prepared for a new five-year medium-resolution spectroscopic survey (started in September 2018). The MRS operates at 4950$\AA < \lambda < 5350\AA$ (B band) and 6300$\AA < \lambda < 6800\AA$ (R band) with a spectral resolution of R $\thicksim$ 7500 [@liu2020lamost]. In this study, the media-resolution spectra were all collected from LAMOST-MRS.
Data analysis and results {#sec:three}
=========================
Radial Velocities (RVs) {#subsec:tables}
-----------------------
In the process of calculating the RVs for both components of the binary systems, we employed the RaveSpan software [@2013MNRAS.436..953P; @2015ApJ...806...29P; @2017ApJ...842..110P], in which includes three major velocity extraction methods: cross-correlation function, two dimensional cross-correlation [TODCOR; @1994ApJ...420..806Z] and the broadening function technique [BF; @2002AJ....124.1746R]. For media-resolution spectra, we chose the CCF pattern of the RaveSpan software to derive RVs. However, this method relies heavily on the sign-to-noise ratio (S/N) of the observed spectra, so we made use of the co-added LAMOST DR7 [^1] B band spectra, which produced by combing the single exposure spectra with a relatively low S/N. Meanwhile, the template spectra used in our study were selected from . The values of CCF for double-lined spectrum usually present the double peaks, which represent the RVs of the primary and secondary respectively. During the eclipse, usually only one peak can be measured, which represents the RV of the mass center. We chose “4th-order polynomial” method of RaveSpan software to fit each CCF peak, then obtained RVs of each component of these binary systems. An example is shown in Figure \[fig:genera1\]. The derived RVs of all objects are shown in Table \[rvs\].
![An example of calculating RVs for ASAS J011416+0426.4, the blue and read line represent the RVs of primary and secondary, respectively. \[fig:genera1\]](2020-0213Fig1.eps){height="7cm" width="100.00000%"}
The simultaneous solution of light and RV curves for three objects {#subsec:four}
------------------------------------------------------------------
The light curves in $V$ band and radial velocity curves of three eclipsing binaries were analyzed through 2015 version of WD code , which provides a convenient interface to input parameters and run DC and LC programs for the users [@2020CoSka..50..535G].
The input fixed parameters for the DC subroutine would decide whether the appropriate solution can be obtained for a binary system or not. At first, the mean effective temperature of the primary component of each binary system was obtained from Gaia DR2 [@2019AJ....158...93B]. The rotation parameters ($F_{1,2}$) are defined as a ratio of the axial rotation rate to the mean orbital rate for both components of a binary system. Therefore, they were set to be 1.0 for ASASSN-V J063123.82+192341.9 and ASAS J011416+0426.4 since the orbital periods of two binaries are relatively short and their components should rotate synchronously with orbital motion because of tidal friction. The rotation parameters of both components of MW Aur were set to be 5.0 (given by $F = \frac{P_{\rm orb}}{P_{\rm rot}}$, where $P_{\rm orb}$ and $P_{\rm rot}$ denote orbital period and rotation period, respectively) since this binary has an elliptical orbit [$P_{\rm rot}$ = 3.068, @2018AJ....155...39O] and a relatively long orbital period. In addition, two binary systems (ASASSN-V J063123.82+192341.9 and ASAS J011416 +0426.4) should have a convective atmosphere since their primary components have a low surface effective temperature (see Table \[table:table1\]), thus the bolometric albedos and the gravity darkening coefficients were taken to be 0.5 and 0.32 for both components of two binaries, respectively [@1969AcA....19..245R]. Since the binary system MW Aur should have a radiative atmosphere according to the effective temperature of its primary component, then the two coefficients mentioned above were all taken to be 1.0 for two components of MW Aur [@1924MNRAS..84..665V]. Meanwhile, we chose a logarithmic law [@1970AJ.....75..175K] to determine the limb darkening coefficients for both components of these eclipsing binaries \[i.e. set LD1 (LD2) = $\pm2$\]. At last, a simple reflection treatment (with parameters MREF = 1, NREF = 1) was chosen. The orbital eccentricities of our targets except for MW Aur were set to be 0, since the eclipsing system MW Aur exhibits an asymmetric light curve in V-band evidently. Therefore, the adjustable parameters used in WD models are as the followings: the orbital semi-major axis ($a$), the systemic velocity ($V_\gamma$), the orbital inclination ($i$), the mean surface temperature of secondary star ($T_2$), the modified surface potential of both components ($\Omega_1$ and $\Omega_2$), the mass ratio ($q$), the bandpass luminosity of primary star ($L_1$), the third light ($l_3$), the epoch of primary minimum ($T_0$) and orbital period ($P$). In addition, the orbital eccentricity ($e$), the argument of periastron ($\omega$) and the phase shift ($\phi_0$) were also set as free parameters for MW Aur.
The calculation for each binary system always started at mode 2 (detached mode), then the best solutions for $V$-band light and radial velocity curves were gotten from multiple iterations using automated differential correction (DC) optimizing subroutine of the WD code until they converged. In the calculation, we found that the solutions for all three eclipsing binaries were converged at mode 2, suggesting that three eclipsing binaries all have a detached configuration at present. We also checked whether the third light ($l_3$) has influence on these solutions, and found that all cases show an unphysical value for the third light ($l_3 < 0$), and therefore we adopted $l_3 = 0$ in our final solution. The convergent solutions derived from the V-band light curves and the radial-velocity curves are presented in Table \[table:table3\], and the results are shown in Figure \[mode1\], Figure \[mode2\] and Figure \[mode3\], respectively.
![The observed radial-velocity (top panel) and V-band light (bottom panel) curves (solid points) and the best fits carried out by WD code in dashed or solid lines for MW Aur. The fitting residuals are presented at the bottom of each panel. \[mode3\]](2020-0213Fig4.eps){width="\textwidth"}
Absolute dimensions {#sec:four}
-------------------
The standard errors of the effective temperatures of secondary components were derived to be 53 K for ASASSN-V J063123.82+192341.9, 30 K for ASAS J011416 +0426.4 and 34 K for MW Aur (see Table \[table:table3\]), which are the formal $1\sigma$ errors arising from the WD light curve solution. The corrected standard errors of the effective temperatures of secondary components following $\sqrt{(err_1)^2+(err_2)^2}$, where $err_1$ and $err_2$ are standard errors of the effective temperatures of the primary and the secondary, respectively. The values of the radii of components are obtained according to a relation: $R = ra$, where $r$ is the mean fractional radius getting from WD code and $a$ is the orbital separation. The mass of the components follows the functions:
$$M_1[M_{\odot}]=1.34068 \times 10^{-2}\frac{1}{1+q}\frac{a^3[R_{\odot}]}{P^2[d]} ,$$
$$M_2[M_{\odot}]= M_1 \cdot q ,$$
where $q$ is the mass ratio and $P$ is the orbital period in days. The individual magnitudes in $V$-band of the stars were derived based on the following equations:
$$V_{1}=V-2.5{\rm log}\frac{1}{1+(L_2/L_1)_V},$$
$$V_{2}=V-2.5{\rm log}\frac{(L_2/L_1)_V}{1+(L_2/L_1)_V} ,$$
where $V_1$ and $V_2$ are the magnitudes of the primary and secondary components and $(L_2/L_1)_V$ is the luminosity ratio in $V$ band. The derived absolute parameters for the three binary systems are presented in Table \[abs\] .
Discussion and conclusions {#sec:five}
==========================
In this work, the V-band light curves and RV curves for each eclipsing binary were simultaneously analyzed by using the WD code. It was found that all of the binary systems we studied had a relatively high mass ratio ($\ga 0.8$). It might be a result of observational selection effects, which require the spectral signals of both components presented in a binary spectrum. Meanwhile, the surface potential of each component of our targets is much higher than the potential of their inner Roche lobe. This indicates that the components of these binary systems have not filled the inner Roche lobe and thus they are all well detached eclipsing binaries. In order to estimate the evolutionary status and age of these binary systems, we compared their absolute physical parameters with the theoretical PARSEC stellar evolutionary tracks and isochrones, which produced from the latest version (v1.2S) of PAdova and TRieste Stellar Evolution Code[^2] . The best isochrones (age) was selected to minimise the $\chi^2$ function including effective temperature and luminosities of the two components.
![Comparison between physical parameters of ASASSN-V J063123.82+192341.9 and PARSEC isochrones in log HR diagram. Filled and open circles represent primary and secondary components, respectively. Red solid line is the isochrone for age = 3.2 Gyr and $Z = 0.0176$, black dashed and solid lines represent Zero-age main-sequence (ZAMS) and terminal-age main-sequence (TAMS), blue lines are the evolutionary tracks were taken from PARSEC models for the stars with different initial masses. \[evo1\]](2020-0213Fig5.eps){width="\textwidth"}
![Comparison between physical parameters of ASAS J011416+0426.4ASAS J011416+0426.4 and PARSEC isochrones in log HR diagram. Filled and open circles represent primary and secondary components, respectively. Red solid line is the isochrone for age = 8.8 Gyr and $Z=0.0200$, black dashed and solid lines represent ZAMS and TAMS, blue lines are the evolutionary tracks were taken from PARSEC models for stars with different initial masses. \[evo2\]](2020-0213Fig6.eps){width="\textwidth"}
![Comparison between physical parameters of MW Aur and PARSEC isochrones in log HR diagram. Filled and open circles represent the primary and secondary components, respectively. Red solid line is the isochrone for age = 1.2 Gyr and $Z=0.0199$, black dashed and solid lines represent ZAMS and TAMS, blue lines are the evolutionary tracks were taken from PARSEC models for the stars with different initial masses. \[evo3\]](2020-0213Fig7.eps){width="\textwidth"}
Figure \[evo1\] shows the locations of two components of ASASSN-V J063123.82+192341.9 in the Hertzsprung-Russell (HR; ${\rm log}T_{\rm eff}-{\rm log} L$) diagram. It was found that the two components are evolved on the main sequence (MS) stage with an age of about 3.2 Gyr and a metallicity of $Z=0.0176$. According the age and the HR diagram of this binary system, it was inferred that both components in this binary system have not yet undergone the mass transfer and their evolutions should be similar to those of single stars. The same character can be found in Figure \[evo2\], where showed the results of the binary system ASAS J011416 +0426.4. It was found in Figure \[evo2\] that both components of ASAS J011416 +0426.4 are also evolved on the MS stage and have an age of 8.8 Gyr and a metallicity of $Z=0.0200$. The results of binary MW Aur are shown in Figure \[evo3\]. As seen from Figure \[evo3\], it was found that the more massive and hotter primary component in this abnormal binary system is evolving on the MS stage, while its secondary component has just evolved away from MS stage, implying that the less massive component seemly evolves more rapidly than its more massive one do. The most likely explanation for this situation might be that some mass had been periodically lost through the Lagrangian point L$_1$ from the present less massive component close to periastron for a binary system with large eccentricity [@2005MNRAS.358..544R]. Therefore, the present less massive component shrank and divorced from its inner Roche lobe, and thus the mass transfer is stopped and this binary show a detached configuration at present. Finally, the age of MW Aur was estimated as 1.2 Gyr and a metallicity of $Z=0.0199$. If the mass transfer had indeed been taken place in this binary, and thus the age and metallicity estimates for this object should be inaccurate, we will use the Modules for Experiments in Stellar Astrophysics (MESA) evolution code [@2011ApJS..192....3P; @2013ApJS..208....4P] to investigate the evolutionary status of MW Aur in detailed in our future work.
We found that the best fitted metallicity are consistent with those shown in the Table \[table:table1\] and the studied systems are all have a solar metallicity. We also compared our results with the classical MS mass-luminosity relation [MLR, @2018MNRAS.479.5491E], the results are shown in Figure \[M\_l\]. It was found in Figure \[M\_l\] that the three eclipsing binaries are evolving on or near the MS stage, which correspond to the result of the above analysis.
![Comparison between the derived parameters and the classical MLR. The black line is the MLR, black and blue symbols represent primary and secondary components, green points represent the data from DEBCat catalogue [@2015ASPC..496..164S]. \[M\_l\]](2020-0213Fig8.eps){width="\textwidth"}
The orbital and physical parameters for three well-detached eclipsing binaries were derived from their light curves and RV curves based on WASP, ASAS-SN, ASAS and LAMOST-MRS data. The accurate parameters are very important for testing the stellar structure and evolution models [@2009ApJ...700.1349T]. LAMOST-MRS survey lasts for five years, it will find more and more double-lined spectroscopic binaries and play an important role in determining the accurate parameters of both components of eclipsing binaries. We will study other eclipsing binaries found from the five-year LAMOST-MRS survey and determine their absolute parameters in the future.
We thank the anonymous referee for the helpful comments and suggestions which improvement this work greatly. This work was supported by the National Natural Science Foundation of China (NSFC) under No. 11773065.
[^1]: <http://dr7.lamost.org/doc/mr-data-production-description>
[^2]: <http://stev.oapd.inaf.it/cgi-bin/cmd>
|
---
abstract: 'We trace the evolution of the theory of stochastic partial differential equations from the foundation to its development, until the recent solution of long-standing problems on well-posedness of the KPZ equation and the stochastic quantization in dimension three.'
address: |
Laboratoire de Probabilités, Statistique et Modélisation\
Sorbonne Université, Université de Paris, CNRS\
4 Place Jussieu, 75005 Paris, France
author:
- Lorenzo Zambotti
bibliography:
- 'DCDS.bib'
title: A brief and personal history of stochastic partial differential equations
---
[*Keywords:* Stochastic partial differential equations]{}\
[*MSC classification:* 60H15]{}
Introduction
============
In September 2017 I attended a meeting in Trento in honor of Luciano Tubaro, who was retiring. Mimmo Iannelli gave a humorous and affectionate talk whose title was [*Abstract stochastic equations: when we used to study in Rome’s traffic jams*]{}. He talked about the ’70s, when he and Luciano were the first students of Giuseppe Da Prato’s, who around 1975 proposed them to work on a brand new topic: stochastic partial differential equations. Since I was myself a PhD student of Da Prato’s in the late ’90s, on that day in Trento I was being told the story of the beginning of our scientific family.
Then, a month later, I was at the Fields Institute in Toronto for a conference in honor of Martin Hairer, who had been awarded in 2014 a Fields medal [*“for his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations”*]{} (the official citation of the International Mathematical Union).
Within a few weeks I was therefore confronted with a vivid representation of the beginning of SPDEs and with a celebration of their culminating point so far. I realised that, because of Hairer’s Fields medal, the mathematical community was suddenly aware of the existence of SPDEs, although very little was commonly known about them.
For example, during his laudatio which introduced Hairer’s talk at the 2014 International Congress of Mathematics in Korea, Ofer Zeitouni felt the need to say to the audience [*“I guess that many of you had never heard about stochastic partial differential equations”*]{}. The other three Fields medals in 2014 were awarded for work in, respectively, dynamical systems, Riemann surfaces and number theory. Certainly there was no need to introduce these topics to the mathematicians attending the ICM. However, after forty years of work, with thousands of published papers and hundreds of contributors, SPDEs were still unknown to a large portion of the mathematical world.
I decided to dedicate my talk in Toronto not just to Hairer’s achievements, but to the whole community that had formed and nurtured him. In the last two years I have given several times this talk in different occasions. This special issue of DCDS gives me the opportunity to write down the few thoughts I have to share about this topic, in the hope that someone else may continue this work and enrich this tale with other points of view. I will make no claim to exhaustivity: the topic is vast and I know only a fraction of the literature. I wish to explain the origin and the development of SPDEs from my personal point of view, and I apologise in advance for the aspects of this story that I will fail to explain properly or even mention. I encourage anyone wishing to see this tale completed or told differently and better to do so and continue the work I am starting.
The beginnings
==============
In principle, a Stochastic Partial Differential Equation (SPDE) is a Partial Differential Equations (PDE) which is perturbed by some random external force. This definition is however too general: if we have a PDE with some random coefficients, where the randomness appears as a parameter and the equation can be set and solved with classical analytic arguments, then one speaks rather of a *random PDE*; this is the case for example of a (deterministic) PDE with a random initial condition.
A SPDE is, more precisely, a PDE which contains some stochastic process (or field) and cannot be defined with standard analytic techniques; typically such equations require some form of stochastic integration. In most of the cases, the equation is a classical PDE perturbed by adding a random external forcing. One of the first examples is the following *stochastic heat equation with additive noise* $$\label{she}
\frac{\partial u}{\partial t} = \Delta u +\xi$$ where $u=(u(t,x))_{t\geq 0,x\in{\mathbb{R}}^d}$ is the unknown solution and $\xi=(\xi(t,x))$ is the random external force. Then one can add non-linearities and, in some cases, multiply the external force by a coefficient which depends on the unknown solution, for example $$\label{she2}
\frac{\partial u}{\partial t} = \Delta u +f(u)+\sigma(u)\,\xi$$ where $f,\sigma:{\mathbb{R}}\to{\mathbb{R}}$ are smooth. The product $\sigma(u)\, \xi$ is not always well-defined, since in many cases of interest $\xi$ is a *generalised function* and $u$ is not expected to be smooth; in this case one writes the equation in an integral form and uses Itô integration to give a sense to the stochastic term.
The idea of associating PDEs and randomness was already present in the physics literature in the ’50s and ’60s, see for example [@spiegel52; @lyon60; @chen64; @gibson67]. In the mathematical literature, several authors extended Itô’s theory of stochastic differential equations (SDE) to a Hilbert space setting, see for example Daleckiĭ [@dalecki66] and Gross [@gross67]. In a paper published in 1969 [@zakai69], Zakai wrote that the unnormalised conditional density in a filtering problem satisfies a linear SPDE.
However, to my knowledge, the first papers which studied explicitly a SPDE as a problem in its own appeared in the ’70s. In 1970 Cabaña [@cabana70] considered a linear wave equation $$\frac{\partial^2 u}{\partial t^2} + 2b\, \frac{\partial u}{\partial t}= \Delta u +\xi$$ with a *space-time white noise* $\xi$ and one-dimensional space variable $x$. This is a very important particular choice for the random external force: it is given by a random generalised function $\xi$ which is *Gaussian* and has very strong independence properties, namely the “values” at different points in space-time are independent.
In 1972 three papers were published on the topic: two in France (Bensoussan-Temam [@bt72] and Pardoux [@pardoux72]) and one in Canada (Dawson [@dawson72]). The French school was strongly influenced by the PDE methods of the time, championed by Jacques-Louis Lions and his collaborators. Bensoussan and Temam [@bt72] considered an evolution equation driven by a monotone non-linear operator $A_t$ $$\frac{{\rm d}y}{{\rm d}t} + A_t(y)=\xi$$ and with an external forcing $\xi$ which we can call now *white in time and coloured in space*; this means that values of the noise on points with different time-coordinate are independent, but there is a non-trivial correlation in space. In [@pardoux72] Pardoux considered a similar problem with multiplicative noise $$\frac{{\rm d}y}{{\rm d}t} + A_t(y)=B_t(y)\,\xi$$ where $B_t$ is a non-linear operator and the stochastic term is treated with the Itô integration theory. In 1975 Pardoux defended his PhD thesis written under the supervision of Bensoussan and Temam, which is considered the first extended work on the topic.
Dawson’s paper [@dawson72] has a more probabilistic flavour. It treats the stochastic heat equations and with one-dimensional space variable $x$ and space-time white noise $\xi$; it shows that the solution $u$ to the linear equation is almost-surely continuous in $(t,x)$ (this is false in higher dimension, as we are going to see below); moreover, it introduces the non-linear equation with the coefficient $\sigma(u)=\sqrt{u}$, which will soon become famous as the equation of the Super-Brownian motion (for $f=0$).
In the following years more and more researchers got interested in SPDEs. In particular, the Italian and Russian schools were founded, respectively, in 1976 with Da Prato’s first paper [@dpit76] on the topic (together with his students Iannelli and Tubaro) and between 1974 and 1977 with Rozovskiĭ’s papers [@rozovski74; @rozovski75] and Krylov-Rozovskiĭ’s [@kr77].
The physical models
===================
In the ’80s some theoretical physicists published a few very influential papers based on applications of SPDEs to several important physical problems: Parisi-Wu’s [@pw81] and Jona Lasinio-Mitter’s [@jlm85] on the *stochastic quantization*, and the Kardar-Parisi-Zhang model for the *dynamical scaling of a growing interface* [@kpz86]. All these papers would be, thirty years later, an important motivation for the theory of regularity structures, see below.
The Stochastic Quantization
---------------------------
The 1981 paper [@pw81] by Parisi and Wu proposed a dynamical approach to the construction of probability measures which arise in Euclidean Quantum Field Theory. The difficulty with such measures is that they are supposed to be supported by spaces of *distributions* (generalised functions) on ${\mathbb{R}}^d$, which makes the definition of *non-linear* densities problematic. For example one would like to consider a measure on the space of distributions ${\mathcal D}'([0,1]^d)$ of the form $$\mu({\rm d}\phi)=\frac1Z \exp\left(-\int_{[0,1]^d} V(\phi(x))\d x\right) {\mathcal N}(0,(1-\Delta)^{-1})({\rm d}\phi)$$ where ${\mathcal N}(0,(1-\Delta)^{-1})$ is a Gaussian measure with covariance operator $(1-\Delta)^{-1}$, with $\Delta$ the Laplace operator on $[0,1]^d$ with suitable boundary conditions, and $V:{\mathbb{R}}\to{\mathbb{R}}$ is some potential. If $d>1$ then ${\mathcal N}(0,(1-\Delta)^{-1})$-a.s. $\phi$ is a distribution and not a function, and the non-linearity $V(\phi)$ is therefore ill-defined. Parisi-Wu introduce a stochastic partial differential equation $$\label{eq:pw}
\frac{\partial u}{\partial t}= \Delta u -u-\frac12\,V'(u) + \xi, \qquad x\in[0,1]^d$$ which has $\mu$ as invariant measure, namely if $u(0,\cdot)$ has law $\mu$, then so has $u(t,\cdot)$ for all $t\geq 0$. This is an infinite-dimensional analog of the classical *Langevin dynamics*. By the ergodic theorem, for a generic initial condition $u(0,\cdot)$, the distribution of $u(t,\cdot)$ converges to $\mu$ as $t\to+\infty$. Therefore one can use the stochastic dynamical system $(u(t,\cdot))_{t\geq 0}$ in order to obtain useful information on $\mu$.
We note however that, for $d>1$, the solution to is expected to be again a distribution on space-time, at least this is the case for the linear equation with $V'\equiv 0$. Therefore a rigorous study of this equation is also problematic, since $V'(u)$ is again ill-defined.
The first rigorous paper on the Parisi-Wu programme was by Jona Lasinio-Mitter [@jlm85], where the authors chose the non-linearity $V(\phi)=\phi^4$ and the space dimension $d=2$, in order to construct the continuum $\phi^4_2$ model of Euclidean Quantum Field Theory [@simon74; @gj87], and called this equation the *stochastic quantization*. Jona Lasinio-Mitter studied a modified version of equation and obtained probabilistically weak solutions via a Girsanov transformation; strong solutions to were obtained in a later paper by Da Prato-Debussche [@dpd03], see below. The case of space dimension $d=3$ remained however open until the inception of regularity structures.
The KPZ equation
----------------
The Kardar-Parisi-Zhang (KPZ) equation [@kpz86] is the following SPDE $$\label{eq:kpz}
\frac{\partial h}{\partial t}= \nu\Delta h +\lambda|\nabla h|^2 + \xi, \qquad x\in{\mathbb{R}}^d$$ and describes the fluctuations around a deterministic profile of a randomly growing interface, where $\nabla$ is the gradient with respect to the space variable $x$.
From an analytic point of view, even if $d=1$ the KPZ equation is very problematic: if we consider the case $\lambda=0$ then we are back to the stochastic heat equation with additive white noise , for which it is known that the solution $u$ is not better than Hölder-continuous in $(t,x)$ and certainly not differentiable; we expect $h$ in to have at best the same regularity as $u$. In particular the gradient in space $\nabla h$ is defined only as a distribution and the term $(\nabla h)^2$ is ill-defined. We restrict ourselves for simplicity to the case $\nu=\lambda=1/2$.
In the original KPZ paper [@kpz86] it was noticed that one can *linearize* by means of the *Cole-Hopf transformation*: if we define $\psi=(\psi(t,x))_{t\geq 0,x\in{\mathbb{R}}}$ as the unique solution to the equation $$\label{eq:colehopf}
\frac{\partial \psi}{\partial t}= \frac12\frac{\partial^2 \psi}{\partial x^2} +\psi \, \xi, \qquad x\in{\mathbb{R}},$$ which is called the *stochastic heat equation with multiplicative noise*, then $h:=\log\psi$ (formally) solves .
In the first mathematical paper on KPZ, Bertini-Cancrini [@bc95] studied in 1995 the stochastic heat equation in the Itô sense for $d=1$. Since Mueller [@mueller91] had proved that a.s. $\psi(t,x)>0$ for all $t>0$ and $x\in{\mathbb{R}}$, then the Cole-Hopf solution $h=\log\psi$ is indeed well-defined. Bertini-Giacomin [@bg97] proved in 1997 that the stationary Cole-Hopf solution is the scaling limit of a particle system, the weakly-asymmetric simple exclusion process (WASEP); this celebrated result was the first example of the *KPZ universality class*, see below.
Since is to be interpreted in the Itô sense, one can apply the Itô formula to $h=\log\psi$ and the result is, at least formally, that $h$ solves $$\label{eq:kpz2}
\frac{\partial h}{\partial t}= \frac12\,\frac{\partial^2 h}{\partial x^2} +\frac12\left[(\partial_x h)^2-\infty\right] + \xi, \qquad x\in{\mathbb{R}},$$ which is almost , apart from the appearance of the famous infinite constant which is supposed to *renormalize* the ill-defined term $(\partial_x h)^2$. Making sense of this renormalization and constructing a well-posedness theory for such equations were however open problems for over 15 years until Hairer’s breakthrough [@hairer13], see below.
We note that the KPZ equation, and in particular its *universality class*, has been one of the most fertile topics in probability theory of the last decade, with connections to particle systems, random matrices, integrable probability, random polymers and much else. See the surveys by Quastel [@quastel12] and Corwin [@corwin16] for more details.
Superprocesses
--------------
SPDEs have also been applied to *biological systems*, in particular in the context of the so-called *superprocesses* introduced by Watanabe and Dawson in the ’70s. Superprocesses are limits of discrete population models of the following type: particles evolve in a ${\mathbb{R}}^d$ space following some Markovian dynamic, typically Brownian motion, independently of each other; at random exponential times each particle dies and is replaced by a random number of identical particles, which become new elements of the population and behave as all other particles. We refer to the Saint-Flour lecture notes by Dawson [@dawson93] and Perkins [@perkins02] for pedagogical introductions to this topic.
The total number of members of the population which are alive at time $t\geq 0$ follows a standard branching process and is independent of the motion of the particles. Therefore there are three situations, depending on the value $m$ of the average number of descendants that a particle has when it dies: if $m>1$ the population grows at an exponential rate, if $m<1$ it dies after a finite and integrable time, if $m=1$ it dies after a finite but non-integrable time. The three situations are called, respectively, *supercritical, subcritical* and *critical*.
The critical case, with Brownian spatial motion, has a scaling limit which is a Markov process with values in the space of measures on the state space ${\mathbb{R}}^d$; this process is called the *super-Brownian motion*. If $d=1$, then Konno-Shiga [@ks88] proved in 1988 that a.s. this random measure has a continuous density $X_t(x)$ with respect to the Lebesgue measure ${\rm d}x$ on ${\mathbb{R}}$, and $(X_t(x))_{t\geq 0,x\in{\mathbb{R}}}$ solves the SPDE $$\label{eq:sbm}
\frac{\partial X}{\partial t}= \frac12\,\frac{\partial^2 X}{\partial x^2} + \sqrt{X}\,\xi.$$ The diffusion coefficient of this equation, already introduced by Dawson in [@dawson72], does not satisfy the usual Lipschitz condition and, indeed, *pathwise uniqueness* for is still an open problem, see the papers by Mytnik-Perkins [@mp11] and Mueller-Mytnik-Perkins [@mmp14]. More precisely, the situation is the following: we consider the SPDE $$\label{eq:sigmaholder}
\frac{\partial X}{\partial t}= \frac12\,\frac{\partial^2 X}{\partial x^2} + \sigma(X)\,\xi,$$ with $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ a Hölder function with exponent $\gamma\in\,]0,1[$, namely $|\sigma(x)-\sigma(y)|\leq C|x-y|^\gamma$ and one looks in general for solutions with values in ${\mathbb{R}}$, rather than in ${\mathbb{R}}_+$; in particular, for equation one would have $\sigma(u)=\sqrt{|u|}$. Then:
- if $\gamma>3/4$ we have pathwise uniqueness, namely if we have two solutions $(X^1,\xi)$ and $(X^2,\xi)$ to driven by the same noise $\xi$ with $X^1(0,\cdot)=X^2(0,\cdot)$ a.s., then $X^1\equiv X^2$ almost surely
- if $\gamma<3/4$ then pathwise uniqueness fails in general and there are counterexamples
- if $\sigma(0)=0$ and one is interested only in the class of *non-negative* solutions, then it is not known whether pathwise uniqueness holds or fails in this class for $\gamma<3/4$. This leaves in particular the hope that the equation for super-Brownian motion may satisfy pathwise uniqueness. However for the related equation of super-Brownian motion *with immigration* the pathwise non-uniqueness was proved by Chen in [@chen15].
If the state space ${\mathbb{R}}^d$ has dimension greater or equal to 2, then a.s. the measure $X_t({\rm d}x)$ is singular with respect to the Lebesgue measure (see [@dh79]), but the equation is still well-defined as a *martingale problem*, since the diffusion coefficient $\sigma(x)=\sqrt{x}$ has the special property that $\sigma^2(x)=x$ is linear. Remarkably, this martingale problem is well-posed and one can prove uniqueness in law of these superprocesses using a technique called *duality* due to Watanabe [@watanabe68], see the cited paper by Konno-Shiga [@ks88]; duality can also be applied to prove uniqueness for other processes, see the works of Shiga [@shiga81; @shiga87] and Mytnik [@mytnik96].
Finally, we mention that superprocesses are related to Le Gall’s Brownian snake, see [@legall99], which also plays a crucial role in the context of planar random maps, see e.g. Miermont’s lecture notes [@miermont].
The theory
==========
During the ’80s and the ’90s, several monographs were published with the aim of presenting a systematic theory of SPDEs.
The first major monograph was Walsh’s Saint-Flour lecture notes [@walsh86], which were published in 1986. In this course Walsh proposed a general approach to SPDEs which has been very influential; his point of view has a very probabilistic flavour, since it consists in regarding the solution $u=u(t,x)$ of a (parabolic or hyperbolic) SPDE as a *multi-parameter process*, or more generally a *multi-parameter random field*. The stochastic integration with respect to space-time white noise is developed according to this point of view, considering $t\mapsto \xi(t,\cdot)$ as a so-called *martingale measure*, thus generalizing the Itô theory. We have used Walsh’s notations for the equations numbered from to above, and for others below.
In 1992 the first book by Da Prato-Zabczyk [@dpz1] was published. This monograph, also known as the *red book* among Da Prato’s students, is still the reference text for the so-called *semigroup approach* to SPDEs. Da Prato-Zabczyk’s point of view is to treat a SPDE as an-infinite dimensional SDE, and the solution $u=u(t,\cdot)$ as a function-valued process with a single parameter, the time $t$. The notations are different from those of Walsh; for example the stochastic heat equation with additive space-time white noise is written as $$\d X=AX\d t+\d W$$ where $X_t=u(t,\cdot)\in L^2({\mathbb{R}})= H$, $A:D(A)\subset H\to H$ is the realization of $\partial^2_x$ in $H$, $(W_t)_{t\geq 0}$ is a *cylindrical Wiener process*. The solution to this equation is called the *stochastic convolution* and is written explicitly as $$X_t=e^{tA}X_0+\int_0^t e^{(t-s)A}\d W_s, \qquad t\geq 0.$$ The general SPDE with non-linear coefficients is written as $$\d X=(AX+F(X))\d t+\Sigma(X)\d W$$ where $F:D(F)\subseteq H\to H$ is some non-linear function and $\Sigma$ is a map from $H$ to the linear operators in $H$. This approach has a more functional-analytical flavour, and is based mainly on the study of the properties of the semigroup $(e^{tA})_{t\geq 0}$ generated by $A$ in $H$, and their interplay with the properties of the cylindrical Wiener process $W$. This non-linear equation is usually written in its *mild formulation* $$X_t=e^{tA}X_0+\int_0^t e^{(t-s)A}\,F(X_s)\d s+\int_0^t e^{(t-s)A}\,\Sigma(X_s)\d W_s.$$
During the ’90s there was also an important activity on *infinite-dimensional analysis*, namely on elliptic and parabolic PDEs where the space-variable belongs to a Hilbert space. The connection with SPDEs is given by the notion of *infinitesimal generator* which is associated with a Markov process with continuous paths. As for finite-dimensional diffusions, the transition semigroup of the solution to a SPDE solves a parabolic equation, known as *Kolmogorov equation*. One can find a systematic theory of these operators in the third book by Da Prato-Zabczyk [@dpz3]. Much work was dedicated to existence and uniqueness of invariant measures, see the next section; the second Da Prato-Zabczyk book was entirely dedicated to this topic [@dpz2].
It can be recalled that Itô introduced his notion of stochastic differential equations in order to give a probabilistic representation of the solution to Kolmogorov equations. Viceversa, if the Kolmogorov equation is well-posed, then it is possible to construct the law of the associated Markov process. This allows to construct *weak* (in the probabilistic sense) solutions, especially in the form of *martingale solutions*, see the 1979 monograph by Stroock-Varadhan [@sv79] on the theory for finite dimensional diffusions.
The construction of the transition semigroup of a Markov process in a locally compact space can be done also with another analytical tool, a *Dirichlet Form*, for which a theory was developped in particular by Fukushima, see the monographs [@fukushima80; @fot11]. The state space of a SPDE is however always a function space, and therefore infinite-dimensional. The extension of Fukushima’s theory to non locally compact spaces was a project of Albeverio-H[ø]{}egh-Krohn [@ah77] since the ’70s and was finally obtained by Ma-Röckner [@mr92]. Although Dirichlet forms allow to construct only weak solutions, they are a powerful tool in very singular situations, where pathwise methods are often ineffective.
Another approach to SPDEs is given by Krylov’s $L^p$-theory, see for example [@krylov94].
Ergodicity of Navier-Stokes
===========================
The Navier-Stokes equation for the flow of an incompressible fluid is one of the most prominent PDEs and it is therefore not surprising that its stochastic version was among the first SPDEs to be studied, starting from the 1973 paper [@bt73] by Bensoussan-Temam. The equation has the form (in Walsh’s notation) $$\frac{\partial u}{\partial t}+(\nabla u)\cdot u= \nu\Delta u -\nabla p + \xi, \qquad {\rm div}\ u=0,$$ where $u(t,x)\in{\mathbb{R}}^d$ denotes the value of the velocity of the fluid at time $t\geq 0$ and position $x\in{\mathbb{R}}^d$, $p(t,x)$ is the pressure, $\nu>0$ and $\xi$ is an external noise whose structure will be made precise below.
The statistical approach to hydrodynamics is based on the assumption that the fluid has a stationary state (invariant measure) on the phase space; by the ergodic theorem, the time average of an observable computed over the dynamics converges for large time to the average of the observable with respect to the invariant measure. This ergodicity property must however be proved, and in the case of the Stochastic Navier Stokes equation in 2D this has been a very active area of research, at least between the 1995 paper by Flandoli-Maslowski [@fm95] and the 2006 paper by Hairer-Mattingly [@hm06].
Ellipticity versus hypoellipticity
----------------------------------
For stochastic differential equations in general, the choice of the external noise plays a very important role. In most of the literature on SPDEs, the space-time noise $\xi$ is realised as the following series $$\xi(t,x) = \sum_{k=1}^\infty \lambda_k\, e_k(x)\, \dot{B}_k(t), \qquad t\geq 0, \ x\in {\mathcal O}\subseteq {\mathbb{R}}^d,$$ where $(\lambda_k)_k$ is a sequence of real numbers, $(e_k)_k$ an orthonormal basis of $L^2({\mathcal O},{\rm d}x)$ and $(B_k)_k$ an independent family of standard Brownian motions. If $\lambda_k=1$ for all $k$ then we have space-time white noise, which has the property that for all $\varphi\in L^2({\mathcal O},{\rm d}x)$ the random variable $$\int_{[0,T]\times{\mathcal O}} \varphi(t,x)\, \xi(t,x)\d t\d x:=
\sum_{k=1}^\infty \langle \varphi,e_k\rangle_{L^2({\mathcal O},{\rm d}x)}\, {B}_k(T)$$ has normal law ${\mathcal N}\left(0,T\,\|\varphi\|^2_{L^2({\mathcal O},{\rm d}x)}\right)$.
In analogy with the finite-dimensional case, if $\lambda^2_k\geq {\varepsilon}>0$ for all $k$, then we are in the *elliptic* case. In finite dimension, we are in a degenerate case as soon as $\lambda_k=0$ for some $k$; in infinite dimension, however, we can have $\lambda_k>0$ for all $k$ but $\lambda_k\to 0$ as $k\to+\infty$. This situation is neither degenerate nor elliptic.
The paper by Flandoli-Maslowski proved for the first time ergodicity for a stochastic Navier-Stokes equation in 2D, under the assumption that $\lambda_k>0$ for all $k$ but $\lambda_k\to 0$ as $k\to+\infty$ with two (different) power-law controls from above and from below. This article sparked an intense activity and a heated debate which revolved around the following question: what is the most relevant choice of the noise structure, which allows to prove ergodicity?
If, as in Flandoli-Maslowski [@fm95], the noise is sufficiently non-degenerate, namely if $\lambda_k>0$ and $\lambda_k\to 0$ not too fast as $k\to+\infty$, then it is often possible to prove ergodicity using an argument due to Doob and based on two ingredients: the *Strong-Feller property* and *irreducibility*; the former means that the transition semigroup of the dynamics maps bounded Borel functions on the state space into continuous functions, the latter that all non-empty open sets of the state space are visited with positive probability at any positive time. The Strong-Feller property is proved with ideas coming from Malliavin calculus, in particular on an integration by parts on the path space which is now known as the Bismut-Elworthy-Li formula, see the paper by Elworthy-Li [@el94] and the monograph by Cerrai [@cerrai01]; irreducibility is based on control theory for PDEs. These techniques were explored and applied to a number of examples in the second Da Prato-Zabczyk book [@dpz2] of 1996.
However, it soon appeared clear that it was possible to consider a degenerate noise and still obtain uniqueness of the invariant measure. Here by degenerate we mean that $\lambda_k=0$ for all $k>N$, where $N$ is a deterministic integer. The main idea behind this line of research was that, if the noise acted on a sufficiently large but finite number of *modes* (i.e. the functions $e_k$), then the noise is elliptic on the modes which determine the long-time behavior of the dynamics: we can call this the *essentially elliptic* case. These results, together with exponential convergence to equilibrium, were proved independently (for Gaussian or for discrete noise) by three groups of authors during the same years: Mattingly [@matt99] and E-Mattingly-Sinai [@ems01], Kuksin-Shirikyan [@ks00; @ks01], Bricmont-Kupiainen-Lefevere [@bkl01; @bkl02].
However in these works the number $N$ of randomly forced modes is not universal but depends on the parameters $\nu$ and $\sum_k \lambda^2_k$ of the equation. This was dramatically improved in the paper by Hairer-Mattingly [@hm06] published in 2006 in Annals of Mathematics, which proved that it is enough to inject randomness only in *four* well-chosen modes, then the non-linearity propagated the randomness to the whole system for any $\nu>0$: the so-called *hypoelliptic* case, for which it is possible to derive uniqueness of the invariant measure for the 2D stochastic Navier-Stokes. One of the main novelties in this paper was the notion of the *asymptotic Strong-Feller property*, which could be proved in the hypoelliptic case, while the standard Strong-Feller property requires much stronger non-degeneracy properties of the noise.
Let us mention here that the Malliavin Calculus, see e.g. Nualart’s monograph [@nualart], has played an important role for Navier-Stokes like for many other SPDEs.
My SPDEs
========
The results on the ergodicity of the stochastic Navier-Stokes equation seemed at the time to make SPDEs with degenerate noise particularly prominent. Now that singular SPDEs with space-time white noise and regularity structures have become so famous, this may seem even strange. In fact, since the very first papers that I have mentioned, see Cabaña [@cabana70] and Dawson [@dawson72], the research activity on SPDEs with genuinely infinite-dimensional noise has always been intensive and most of the problems I have mentioned above concern space-time white noise.
The case of degenerate noise is certainly more difficult if one wants to prove ergodicity, as we have seen. However, if the noise is spatially finite-dimensional, then the solution to the SPDE are typically smooth in space, although still Brownian-like in time. In the case of space-time white noise, on the contrary, the solution are rather Brownian-like *in space* if the space dimension is $d=1$, and even less regular in time; if $d>1$, as we have already seen, solutions are rather distributions.
Therefore, SPDEs driven by space-time white noise are particularly strange objects: even the solutions to the simplest equation, as the stochastic heat equation with additive space-time white noise, are far too irregular for any of the derivatives which appear in to make any sense as a function. The KPZ equation has almost an explicit solution given by the Cole-Hopf transform $h=\log\psi$, with $\psi$ solution to the stochastic heat equation with multiplicative space-time white noise ; however the KPZ equation itself makes no sense as it is written in !
It is in this topic that I made my first steps as a researcher. I did my PhD at Scuola Normale in Pisa under the supervision of Giuseppe Da Prato (also known as Beppe) from 1997 to 2001. Like Da Prato himself and many of his students, I started as an analyst but felt increasingly attracted by probability theory, in particular stochastic calculus and SDEs. In the shelves of Beppe’s office I found the Revuz-Yor monograph, which became one of my favourite mathematics books. I started to dream of unifying two worlds: the classical Itô theory of stochastic calculus based on martingales, and SPDEs.
Chapter 5 in the book by Revuz-Yor on local time and reflecting Brownian motion was one of the topics which most intrigued me. At that time Da Prato was studying equations of the form $$\label{eq:sdi}
\d X\in (AX-\partial U(X))\d t+{\rm d} W$$ with $U:H\to{\mathbb{R}}$ a *convex* lower semi-continuous but not necessarily differentiable function. In the deterministic setting, this is a classical problem and the set $\partial U(x)$ is the *subdifferential* at a point $x\in H$, namely the set of all directions $h\in H$ such that the affine subspace $U(x)+\{z\in H:\langle z,h\rangle =0\}$ lies below the graph of $U$. For a simple example, think of the function ${\mathbb{R}}\ni x\mapsto|x|\in{\mathbb{R}}_+$, which is convex and has as subdifferential the set $\{1\}$ for all $x>0$, the set $\{-1\}$ for all $x<0$ and the set $[-1,1]$ for $x=0$. Then equation is rather a *stochastic differential inclusion*, and if $U$ is differentiable at $x$ then $\partial U(x)=\{\nabla U(x)\}$. There is an extensive literature on this problem in the finite-dimensional case, see e.g. Cépa [@cepa], much less so in infinite dimension where many problems remain open.
The case of $U$ being equal to $0$ on a closed convex set $K\subseteq H$ and to $+\infty$ on $H\setminus K$ seemed to be outside the scope of Da Prato’s techniques. I convinced myself that this case had to be related with reflection on the boundary of $K$, but I was unable to make this precise. Then Samy Tindel pointed out to me a 1992 paper by Nualart and Pardoux [@nupa] on the following SPDE with reflection at 0 $$\label{eq:nupa}
\frac{\partial u}{\partial t} = \frac12\frac{\partial^2 u}{\partial x^2} +\xi +\eta, \qquad t\geq 0, \ x\in[0,1],$$ where $\eta$ is a Radon measure on $]0,+\infty[\,\times\,]0,1[$, $u$ is *continuous* and non-negative, and the support of $\eta$ is included in the zero set $\{(t,x): u(t,x)=0\}$ of $u$, or equivalently $$\label{eq:nupa2}
u\geq 0, \qquad \eta\geq 0, \qquad \int_{]0,+\infty[\,\times\,]0,1[} u\d\eta=0.$$ This is a *stochastic obstacle problem*, the obstacle being the constant function equal to $0$, which can be formulated in the abstract setting of the stochastic differential inclusion . Continuity of $(t,x)\mapsto u(t,x)$ here is essential in order to make sense of the condition ; in this setting the Walsh approach is clearly necessary, since continuity of $t\mapsto u(t,\cdot)$ in $L^2(0,1)$ would not be sufficient. In higher space dimension, $u$ is not expected to be continuous and indeed it remains an open problem to define in this case a notion of solution to -. We note also that this equation arises as the scaling limit of interesting microscopic models of random interfaces: see Funaki-Olla [@fo01] and Etheridge-Labbé [@el15].
The Nualart-Pardoux paper was motivated by stochastic analysis but it was an entirely deterministic work, which pushed the PDE techniques to cover a situation of minimal regularity for the solution; a probabilistic interpretation of this result remained elusive. This is what I tried to give with the results of my PhD thesis. First I identified in [@lz01] the unique invariant measure of - as the 3-d Bessel bridge (also known as the normalized Brownian excursion), an important process which plays a key role in the study of Brownian motion and its excursion theory, see [@reyo]. Then I proved in [@lz02] an infinite-dimensional integration by parts with respect to the law of the 3-d Bessel bridge, which gave a powerful probabilistic tool to study the reflection measure $\eta$ (it provides its *Revuz measure*). Then I set out to study the fine properties of the solution, in particular of the contact set $\{(t,x): u(t,x)=0\}$ between the solution $u$ and the obstacle $0$, see [@lz04] and the paper [@dmz06] in collaboration with Dalang and Mueller.
In these papers I tried to realize my dream, by showing that solutions to SPDEs display very rich and new phenomena with respect to finite-dimensional SDEs, and that it was possible to go much beyond results on existence and uniqueness. I found some interesting link between classical stochastic processes arising in the study of Brownian motion and SPDEs. For a more recent account, see my Saint-Flour lecture notes [@lz15].
However it does not seem that this point of view has been followed by many others. As we are going to see, the SPDE community would soon be heading in a very different direction.
Rough paths and regularity structures
=====================================
In 1998 T. Lyons published a paper [@lyons98] on a new approach to stochastic integration. Lyons was an accomplished probabilist and an expert of stochastic analysis. Therefore it may seem puzzling that the aim of his most famous contribution to mathematics, the invention of *rough paths*, is to give a deterministic theory of stochastic differential equations!
The classical Itô theory of stochastic calculus, see again [@reyo], is a wonderful tool to study stochastic processes (more precisely continuous semimartingales). Not only does it allow to prove existence and uniqueness of solutions to stochastic differential equations, but it also allows to compute the law of a great variety of random variables and stochastic processes. The key tool is that of martingales, which allow explicit computations of expectations and probabilities with often deep and surprising results.
In particular one obtains well-posedness of SDEs in ${\mathbb{R}}^d$ of the form $$\label{eq:SDE1}
\d X_t=b(X_t)\d t+\sigma(X_t)\d W_t,$$ with $b:{\mathbb{R}}^d\to{\mathbb{R}}^d$ and $\sigma:{\mathbb{R}}^d\to {\mathbb{R}}^d\otimes{\mathbb{R}}^d$ smooth coefficients and $(W_t)_{t\geq 0}$ a Brownian motion in ${\mathbb{R}}^d$. However, in general $X$ is not better than a *measurable* function of $W$. This fact is rarely mentioned in courses of stochastic calculus, and probabilists seem used to it. Nevertheless, a physicist may point out that Brownian motion or its derivative, white noise, are an approximation of a real noise, not the other way round; an analyst may found this lack of continuity disturbing. Therefore a theory which is too sensitive on the structure of the noise is not so satisfactory after all. A *robust* theory would be more convincing from this point of view. In the late ’70s, the works of Doss [@doss77] and Sussmann [@sussmann78] gave sufficient conditions on the coefficient $\sigma$ for continuity of the maps $W\mapsto X$ in the sup-norm topology on $C([0,T];{\mathbb{R}}^d)$. These conditions were however very restrictive for $d>1$.
Following an early intuition by Föllmer [@follmer81], Lyons constructed a deterministic (*pathwise*) approach to stochastic integration. The main result is the construction of a topology that makes the map $W\mapsto X$ continuous. However, there is a very important twist: the topology is not just on $W$ or $X$, but on a richer object which contains more information. If for example $W:[0,T]\to{\mathbb{R}}^d$ is a deterministic smooth path, then one needs to consider a finite number of *iterated integrals* of $W$, which take the form $${\bf W}^{n}_{s,t}=\int_{s<u_1<\cdots<u_n<t} \d{W}_{u_1}\otimes\cdots\otimes\d{W}_{u_n}, \qquad n\in{\mathbb{N}}, \ 0\leq s\leq t\leq T,$$ where $\d W_u=\dot{W}_{u}\d u$. For a fixed $\gamma\in\,]0,1[$, one takes $N\in{\mathbb{N}}$ such that $N\gamma\leq 1<(N+1)\gamma$ and for every smooth $W:[0,T]\to{\mathbb{R}}^d$ $${\bf W}^{(N)}_{s,t}:=1+\sum_{n=1}^N {\bf W}^{n}_{s,t},\qquad 0\leq s\leq t\leq T,$$ which belongs to the truncated tensor algebra $T^{(N)}=\oplus_{n=0}^N ({\mathbb{R}}^d)^{\otimes n}$. We note that ${\bf W}^{1}_{s,t}=W_t-W_s$, so that ${\bf W}^{(N)}_{s,t}$ *contains* the increments of the original process, plus additional information. We can now define a distance between two such objects ${\bf W}^{(N)}_{s,t}$ and ${\bf V}^{(N)}_{s,t}$, for smooth $W,V:[0,T]\to{\mathbb{R}}^d$ $$\d_\gamma\left({\bf W}^{(N)},{\bf V}^{(N)}\right):=\sup_{n=1,\ldots,N} \sup_{s\ne t} \frac{\left|{\bf W}^{n}_{s,t}-{\bf V}^{n}_{s,t}\right|}{|t-s|^{n\gamma}}.$$ Then Lyons’ result was that the map ${\bf W}^{(N)}\mapsto {\bf X}^{(N)}$, where $W,X:[0,T]\to{\mathbb{R}}^d$ are smooth processes which satisfy , is *continuous* with respect to the metric ${\rm d}_\gamma$.
Lyons’ paper [@lyons98] was astounding for its novelty: it introduced in stochastic analysis a number of concepts which were unknown to many probabilists, in particular the algebraic language based on the work of Chen [@chen57] on iterated integrals. Moreover it presented a radically different approach to the pillar of modern probability theory, the Itô stochastic calculus. For these reasons, it seems that Lyons’ ideas took some time before being widely accepted by the community and became really famous only fifteen years later, when Hairer proved their power in the context of SPDEs. See the book of Friz-Hairer [@fh14] for a pedagogical introduction.
Singular SPDEs and regularity structures
----------------------------------------
As we have seen above, several interesting physical models were described in the ’80s with SPDEs such as the *dynamical $\phi^4_d$ model*, recall the stochastic quantization , $$\label{phi4d}
\frac{\partial \phi}{\partial t}= \Delta \phi -\phi^3 + \xi, \qquad x\in{\mathbb{R}}^d,$$ for $d=2,3$ and the KPZ equation . In both equations there are ill-defined non-linear functionals of some distribution. Equations of this kind are now commonly known as *singular SPDEs*.
In 2003 Da Prato-Debussche [@dpd03] solved the stochastic quantization in $d=2$ with the following idea: they wrote $\phi=z+v$, where $z$ is the solution to the linear stochastic heat equation with additive white noise $$\frac{\partial z}{\partial t}= \Delta z + \xi, \qquad x\in{\mathbb{R}}^2,$$ and they wrote an equation for $v=\phi-z$ $$\frac{\partial v}{\partial t}= \Delta v -z^3-3z^2v-3zv^2-v^3, $$ which is now random only through the explicit Gaussian process $z$. We note that $z$ is still a distribution, so that the terms $z^2$ and $z^3$ are still ill-defined; however it turns out that it is possible to give a meaning to these terms as distributions with the classical *Wick renormalization*. Then, the products $z^2v$ and $zv^2$ are defined using *Besov spaces*. This allows to use a fixed point argument for $v$ and obtain existence and uniqueness for the original (renormalized) equation. However this technique does not work for $d=3$, since in this case the products $z^2v$ and $zv^2$ are still ill-defined.
Since Lyons’ foundational paper of 1998, rough paths have been based on *generalised Taylor expansions*, with standard monomials replaced by iterated integrals of the driving noise. In 2004 Gubinelli built on this idea a new approach to rough integration based on the notion of *controlled paths* [@gubi04] and started to work on the project of a rough approach to SPDEs, see for example the 2010 paper [@gt10] with Tindel.
In 2011 Hairer [@hairer11] considered the equation $$\frac{\partial u}{\partial t} = \frac12\frac{\partial^2 u}{\partial x^2} +g(u)\,\frac{\partial u}{\partial x}+ \xi , \qquad t\geq 0, \ x\in{\mathbb{R}}$$ with $u$ and $\xi$ taking values in ${\mathbb{R}}^d$ with $d>1$, and $g$ taking values in ${\mathbb{R}}^{d\times d}$. Although this is less frightening than KPZ, the product $g(u)\frac{\partial u}{\partial x}$ is ill-defined for the usual reason: the partial derivative of $u$ is a distribution, the function $g(u)$ is not smooth, and therefore the product cannot be defined by an integration by parts or other classical tools (the fact that $u$ is vector valued prevents in general this product from being written as $\frac{\partial}{\partial x}G(u)$). The idea was to treat the solution $u(t,x)$ as a rough path *in space*.
In 2013 Hairer managed to apply the same techniques to KPZ [@hairer13], thus giving a well-posedness theory for this equation first introduced in 1986. The importance of this result was amplified by the explosion of activity around the KPZ universality class following the 2011 papers by Balázs-Quastel-Seppäläinen [@bqs11] and Amir-Corwin-Quastel [@acq11], which proved that the Cole-Hopf solution proposed by Bertini-Cancrini has indeed the scaling computed in the original KPZ paper [@kpz86] with non-rigorous renormalization group techniques.
In order to solve the stochastic quantization in $d=3$, and many other equations, Hairer [@hairer14] expanded the theory of rough paths to cover functions of space-time. Da Prato-Debussche [@dpd03] had solved the case $d=2$ with the *global* expansion $\phi=z+v$ of the solution, in terms of an explicit term $z$ and a *remainder* $v$. Hairer’s idea was to use rather *local* expansions at each point $(t,x)$ in space-time, with a far-reaching generalization of the classical notion of Taylor expansion. The theory has been developed and expanded in three subsequent papers: Bruned-Hairer-Zambotti [@bhz], Chandra-Hairer [@ch16], Bruned-Chandra-Chevyrev-Hairer [@BCCH].
In the meantime, Gubinelli-Imkeller-Perkowski [@gip] constructed a different approach to singular SPDEs based on *paracontrolled distributions*, combining the *paradifferential calculus* coming from harmonic analysis and the ideas of rough paths. This approach is effective in many situations like KPZ and the stochastic quantization, see also the paper [@mw17] by Mourrat-Weber on the convergence of the two-dimensional dynamic [I]{}sing-[K]{}ac model to the dynamical $\phi^4_2$, see , but not in all cases which are covered by regularity structures. In my personal opinion it is Hairer’s theory which transposes in the most faithful way Gubinelli’s ideas on rough paths from SDEs to SPDEs.
Another interesting approach to the KPZ equation is that of energy solutions by Gonçalves-Jara [@gj14] and Gubinelli-Jara [@gj13], which is particularly effective in order to prove convergence under rescaling of a large class of particle systems to a martingale problem formulation of KPZ. Uniqueness for such a martingale problem was proved in [@gp18] by Gubinelli-Perkowski. Other construction of the $\phi^4_3$ dynamical model are due to Kupiainen [@kupiainen], using renormalization group methods, and to Albeverio-Kusuoka [@alku], using finite-dimensional approximations.
Conclusions
===========
In this brief and personal history of SPDEs I have left aside many topics that would deserve more attention, for example
- *regularization by noise*, see Flandoli-Gubinelli-Priola [@fgp10]
- the stochastic FKPP equation, see Mueller-Mytnik-Quastel [@mmq11]
- stochastic dispersive equations, stochastic conservation laws and viscosity solutions for fully non-linear SPDEs
- numerical analysis of SPDEs.
I hope that I have at least managed to express my enthousiasm for this topic. The last seven years have been particularly exciting: Gubinelli and Hairer have clearly influenced each other in a number of occasions, and their work has spurred an exceptional activity in this area. Rough paths and regularity structures tend to make relatively little use of classical probability theory, and my project of combining stochastic calculus and SPDEs went exactly in the opposite direction. However in the years before 2013 I felt somewhat discouraged by the lack of progress of this project, and Hairer’s paper on KPZ came as a revelation to me. What came afterwards was one of those rare situations when reality surpasses our own dreams.
The message that I wished to convey is that the ground for the success of today was prepared by a considerable amount of work by a whole community, in particular on equations driven by space-time white noise. I am convinced that this activity has produced many ideas which could and should be of interest for other communities and there are already encouraging signs in this direction.
|
---
address:
- ' ETH - Zürich'
- University of Iowa
author:
- Tom Ilmanen
- Dan Knopf
date: 'Version 10.02.02'
title: 'A lower bound for the diameter of solutions to the Ricci flow with nonzero $H^{1}(M^{n};\mathbb{R})$'
---
[^1]
Introduction
============
Consider the Ricci flow$$\frac{\partial}{\partial t}g=-2\operatorname{Rc}(g),\qquad0\leq t<T,$$ of a metric $g$ on $S^{1}\times S^{2}$. Our intuition suggest that no matter how wild the metric is, the Ricci curvature in the $S^{1}$ direction should more or less average out to zero, so that the distance around the $S^{1}$ should tend not to decrease.
In this paper, we substantiate this idea by proving the following more general theorem. Given a Riemannian manifold $(M^{n},g)$ and a homology element $\alpha\in H_{1}(M^{n};\mathbb{Z)}$, let $L_{\alpha}(g)$ denote the infimum of the lengths measured with respect to $g$ of all curves representing $\alpha$.
\[Main\]If $(M^{n},g(t):0\leq t<T)$ is a compact solution of the Ricci flow and $\alpha\in H_{1}(M^{n};\mathbb{Z)}$ is an element of infinite order, there exists $c=c(\alpha,g(0))\ $such that$$L_{\alpha}(g(t))\geq c>0$$ for all $t\in\lbrack0,T)$.
A particular consequence is that the diameter of $(M^{n},g(t))$ is bounded from below independently of $t$. As a result, we can resolve a conjecture made by Hamilton in §26 of [@H-95a]. Suppose that $(M^{n},g(t):0\leq t<T)$ is a solution of the Ricci flow on a maximal time interval. For $x_{j}\in
M^{n}$, $t_{j}\in\lbrack0,T)$, and $\lambda_{j}>0$, define the dilations$$g_{j}(t):=\lambda_{j}g(t_{j}+\frac{t}{\lambda_{j}}),\qquad-\lambda_{j}%
t_{j}\leq t<\lambda_{j}(T-t_{j}). \label{blowups}%$$ If $t_{j}\nearrow T$ as $j\rightarrow\infty$ and $(M^{n},g_{j}(t),x_{j})$ converges locally smoothly to a limit $(M_{\infty}^{n},g_{\infty}%
(t),x_{\infty})$, we call the latter a *final time limit flow* of $(M^{n},g(t):0\leq t<T)$. The following result answers Hamilton’s conjecture affirmatively.
\[Hamilton\]$(S^{1}\times S^{n-1},\bar{g}(t))$ cannot arise as a final time limit flow.
Here $\bar{g}(t)$ is the Ricci soliton$$\bar{g}(t):=ds\otimes ds+2(n-1)(\bar{T}-t)g_{\operatorname*{can}},
\label{soliton}%$$ where $g_{\operatorname*{can}}$ is a round metric on $S^{n-1}$. Note that the only possible final time limit flow of $(S^{1}\times S^{n-1},\bar{g}(t))$ is $(\mathbb{R}\times S^{n-1},\bar{g}(t))$. The main content of the corollary concerns the case that only a subsequence is known to converge.
In order to put our results into context, recall that we should not be surprised if a solution $(M^{n},g(t))$ of the Ricci flow starting from an arbitrary Riemannian manifold encounters a finite time singularity. Indeed, this must be the case if the scalar curvature ever becomes everywhere positive. (See the proof of Corollary \[Hamilton\], below.) To study a finite time singularity, it is often useful to construct a sequence of dilations (\[blowups\]), sometimes called a blowup sequence. In certain cases (namely if the $\lambda_{j}$ are comparable to the suprema of the curvatures and if an injectivity radius estimate is available for the sequence $g_{j}$) one can apply Gromov-type compactness arguments such as those in [@H-95b] in order to show $C^{\infty}$ convergence to a final time limit flow. (Such flows are also called singularity models in the literature.) Final time limit flows have special properties which aid analysis of the original solution $(M^{n},g(t))$. With sufficient knowledge of the limit, one can draw useful conclusions about the analytic, geometric, and topological character of a singular solution just prior to the formation of the singularity. The analysis of singularities via the formation of final time limit flows is an integral part of Hamilton’s well-developed program to resolve Thurston’s Geometrization Conjecture for closed $3$-manifolds [@T-82] by means of the Ricci flow. (See for example [@H-95a] and the survey [@CaoChow].)
Theorem \[Main\] is essentially a monotonicity result. We shall offer two proofs which are dual to one another. The first proof (Section \[Cohomology\]) uses cohomology, is simpler, and is more direct. The second proof (Section \[Homology\]) uses homology. Its value lies in better revealing the geometry; in particular; we hope that it is instructive in showing how the ideas introduced here might be generalized. We briefly discuss such potential applications in Section \[Conclusion\].
The cohomology proof\[Cohomology\]
==================================
Our starting point is the following observation, which was pointed out to the first author by Sun-Chin (Michael) Chu. Let $(M^{n},g(t))$ be a solution of the Ricci flow, and let $\phi(t)$ be a $1$-parameter family of $1$-forms evolving by$$\frac{\partial}{\partial t}\phi=\Delta_{d}\phi,$$ where $-\Delta_{d}:=d\delta+\delta d$ is the Hodge–de Rham Laplacian. Recalling that$$\Delta_{d}\phi_{i}=\Delta\phi_{i}-R_{i}^{j}\phi_{j},$$ where $\Delta$ is the rough Laplacian, one computes that$$\begin{aligned}
\frac{\partial}{\partial t}\left\vert \phi\right\vert ^{2} & =\frac
{\partial}{\partial t}(g^{ij}\phi_{i}\phi_{j})\\
& =2R^{ij}\phi_{i}\phi_{j}+2\phi^{i}\frac{\partial}{\partial t}\phi_{i}\\
& =2\phi^{i}\Delta\phi_{i}\\
& =\Delta\left\vert \phi\right\vert ^{2}-2\left\vert \nabla\phi\right\vert
^{2}.\end{aligned}$$ Applying the parabolic maximum principle, one concludes that $$\left\Vert \phi(t)\right\Vert _{g(t)}\leq\left\Vert \phi(0)\right\Vert _{g(0)}
\label{key}%$$ for as long as the solution $g(t)$ exists, where $\left\Vert \phi\right\Vert
_{g}$ denotes the supremum norm $\left\Vert \phi\right\Vert _{g}:=\sup_{x\in
M}\left\vert \phi(x)\right\vert _{g(x)}$.
We use this observation to establish a key monotonicity property. Given a Riemannian manifold $(M^{n},g)$ and an element $\Phi$ of the first de Rham cohomology group $H_{dR}^{1}(M^{n};\mathbb{R})$, define$$N_{g}(\Phi):=\inf_{\phi\in\Phi}\left\Vert \phi\right\Vert _{g}.$$
\[Monotone1\]If $(M^{n},g(t):0\leq t<T)$ is a solution of the Ricci flow, $N_{g(t)}(\Phi)$ is a non-increasing function of time.
For any $\varepsilon>0$, there is a smooth representative $\phi_{0}\in\Phi$ such that $$\left\Vert \phi_{0}\right\Vert \leq N_{g(0)}(\Phi)+\varepsilon.$$ Define $\phi(t)$ by
\[Hodge\]$$\begin{aligned}
\frac{\partial}{\partial t}\phi & =\Delta_{d}\phi\\
\phi(0) & =\phi_{0},\end{aligned}$$ noting that a solution $\phi(t)$ exists for as long as $g(t)$ exists. Note too that $\phi(t)\in\Phi$. Indeed, if we define a smooth function $F(t)$ by
$$\begin{aligned}
\frac{\partial F}{\partial t} & =\Delta F-\delta\phi_{0}\\
F(0) & =0,\end{aligned}$$
we have $$\phi=\phi_{0}+dF$$ for all $t\in\lbrack0,T)$, because$$\frac{\partial}{\partial t}(\phi_{0}+dF)=\Delta_{d}(\phi_{0}+dF).$$ Hence by (\[key\]), we obtain$$N_{g(t)}(\Phi)\leq\left\Vert \phi\right\Vert _{g(t)}\leq\left\Vert
\phi\right\Vert _{g(0)}\leq N_{g(0)}(\Phi)+\varepsilon.$$
The following observation is of independent interest.
\[Norm\]If $(M^{n},g)$ is a compact Riemannian manifold, then $N_{g}$ is a norm on $H_{dR}^{1}(M^{n};\mathbb{R})$.
Homogeneity and the triangle inequality are readily verified. To show positivity, suppose that $N_{g}(\Phi)=0$. Then there is a sequence $\left\{
\phi_{j}:j\in\mathbb{N}\right\} \subset\Phi$ of smooth $1$-forms such that $\left\Vert \phi_{j}\right\Vert _{g}\rightarrow0$. Fix any $\phi\in\Phi$. We may write$$\phi-\phi_{j}=dF_{j},$$ where each $F_{j}$ is smooth. Since $M^{n}$ is compact, $\sup_{j\in\mathbb{N}%
}\left\Vert dF_{j}\right\Vert <\infty$. So after adding a locally constant function to $F_{j}$, we may by Arzela–Ascoli select a subsequence $F_{j_{k}}$ that converges uniformly to a Lipschitz function $F$. Then$$\operatorname*{ess}\sup\left\vert \phi-dF\right\vert _{g}\leq\limsup
_{k\rightarrow\infty}\left\Vert \phi-dF_{j_{k}}\right\Vert _{g}=\limsup
_{k\rightarrow\infty}\left\Vert \phi_{j_{k}}\right\Vert _{g}=0.$$ So $dF=\phi$ almost everywhere, which implies in particular that $F$ is smooth. Hence $\Phi=0$.
We can now obtain a lower bound for the diameter of a solution of the Ricci flow on a compact manifold $M^{n}$ with $H^{1}(M^{n};\mathbb{R})\neq\left\{
0\right\} $.
\[First proof of Theorem \[Main\]\]Let $(M^{n},g(t):0\leq t<T)$ be a solution of the Ricci flow. Consider the natural map $\rho:H_{1}(M^{n};\mathbb{Z}%
)\rightarrow H_{1}(M^{n};\mathbb{R})$ and note that $\beta\in H_{1}%
(M^{n};\mathbb{Z})$ is a torsion element if and only if $\rho(\beta)=0$, hence if and only if $\left\langle \Psi,\beta\right\rangle =0$ for all $\Psi\in
H^{1}(M^{n};\mathbb{R})$. So if $\alpha\in H_{1}(M^{n};\mathbb{Z})$ is an element of infinite order, then there exists $\Phi\in H^{1}(M^{n};\mathbb{R})$ such that $\left\langle \Phi,\alpha\right\rangle >0$.
Fix any $t\in\lbrack0,T)$, and let $a$ be any curve representing $\alpha$. Then for all $\phi\in\Phi$, we have$$0<\left\langle \Phi,\alpha\right\rangle =\int_{a}\phi\leq\left\Vert
\phi\right\Vert _{g(t)}\cdot\operatorname{length}_{g(t)}(a).$$ Taking the infimum over $\phi\in\Phi$, we get $$\left\langle \Phi,\alpha\right\rangle \leq N_{g(t)}(\Phi)\cdot
\operatorname{length}_{g(t)}(a)\leq N_{g(0)}(\Phi)\cdot\operatorname{length}%
_{g(t)}(a)$$ by Lemma \[Monotone1\]. Taking the infimum over all $a\in\alpha$, we obtain$$L_{\alpha}(g(t))\geq\frac{\left\langle \Phi,\alpha\right\rangle }%
{N_{g(0)}(\Phi)}>0. \label{lower-bound}%$$
\[Proof of Corollary \[Hamilton\]\]Let $(M^{n},g(t):0\leq t<T\leq\infty)$ be a solution of the Ricci flow on a maximal time interval, and let $g_{j}%
(t)=\lambda_{j}g(t_{j}+t/\lambda_{j})$ be a sequence of dilations such that$$(M^{n},g_{j}(t))\rightarrow(S^{1}\times S^{n-1},\bar{g}(t)),
\label{smooth-convergence}%$$ where $\bar{g}(t)$ is defined by (\[soliton\]). Then there exists $j_{0}$ such that $g(t_{j_{0}})$ has positive scalar curvature $R>0$. Because$$\frac{\partial}{\partial t}R=\Delta R+2\left\vert \operatorname{Rc}\right\vert
^{2}\geq\Delta R+\frac{2}{n}R^{2},$$ the maximum principle implies that the solution must fail to exist at a finite time $T<\infty$. By Theorem 8.1 of [@H-95a], a finite time singularity implies that$$\limsup_{t\nearrow T}\left( \sup_{x\in M}\left\vert \operatorname{Rm}%
(x,t)\right\vert \right) =\infty.$$ Then because there is $C=C(n)$ such that $$\frac{\partial}{\partial t}\left\vert \operatorname{Rm}\right\vert ^{2}%
\leq\Delta\left\vert \operatorname{Rm}\right\vert ^{2}+C\left\vert
\operatorname{Rm}\right\vert ^{3},$$ the maximum principle further implies a lower bound for the curvature blowup rate,$$\sup_{x\in M}\left\vert \operatorname{Rm}(x,t)\right\vert \geq\frac{2/C}%
{T-t}.$$ But then smooth convergence (\[smooth-convergence\]) is possible only if$$\lim_{j\rightarrow\infty}\lambda_{j}=\infty.$$ On the other hand, since $S^{1}\times S^{n-1}$ is compact, (\[smooth-convergence\]) also implies that$$H_{1}(M^{n};\mathbb{Z})\cong H_{1}(S^{1}\times S^{n-1};\mathbb{Z}%
)\cong\mathbb{Z}.$$ Let $\alpha$ generate $H_{1}(M^{n};\mathbb{Z})$. By Theorem \[Main\], we have$$L_{\alpha}(g(t))\geq c>0.$$ Hence$$L_{\alpha}(g_{j}(0))\geq\lambda_{j}c\rightarrow\infty$$ as $j\rightarrow\infty$. This contradicts (\[smooth-convergence\]) and establishes Corollary \[Hamilton\].
The homology proof\[Homology\]
==============================
We now seek a monotone quantity dual to the metric norms $N_{g(t)}$ defined above on $H_{dR}^{1}$. Let $(M^{n},g)$ be a Riemannian manifold. For each free homotopy class $\Gamma\in\operatorname*{Free}(M^{n})$, define$$\begin{aligned}
\ell_{g}(\Gamma) & :=\inf_{\gamma\in\Gamma}\operatorname{length}_{g}%
(\gamma),\\
m_{g}(\Gamma) & :=\liminf_{k\rightarrow\infty}\frac{\ell_{g}(k\Gamma)}{k},\end{aligned}$$ where $k\Gamma$ denotes the $k$-fold cover of $\Gamma$.
We first obtain a lower bound on the decay of $\ell_{g(t)}(\Gamma)$ during the Ricci flow.
\[DecayBound\]Let $(M^{n},g(t):0\leq t<T)$ be a solution of the Ricci flow and $\Gamma\in\operatorname*{Free}(M^{n})$ a free homotopy class. Then there exists $C>0$ depending only on $n$ such that$$(\ell_{g(t)}(\Gamma))^{2}\geq(\ell_{g(0)}(\Gamma))^{2}-Ct$$ for all $t\in\lbrack0,T)$.
We may assume $\Gamma$ is nontrivial. Fix $t\in\lbrack0,T)$. There is a nontrivial smooth closed geodesic $\gamma\in\Gamma$ such that$$\operatorname{length}_{g(t)}(\gamma)=\ell_{g(t)}(\Gamma)>0.$$ Let $V$ denote the unit tangent vector field along $\gamma$. Stability implies that$$\int_{\gamma}(\left\vert \nabla_{V}X\right\vert ^{2}-\left\langle
R(V,X)X,V\right\rangle )\,ds\geq0 \label{Stability}%$$ for any smooth vector field $X$ along $\gamma$. Because of holonomy, there may not exist a parallel orthonormal frame along $\gamma$; but we can choose an orthonormal frame $(e_{1},\ldots,e_{n})$ along $\gamma$ such that $e_{n}=V$ and $$\left\vert \nabla_{V}e_{i}\right\vert \leq\frac{C_{n}}{\operatorname{length}%
_{g(t)}(\gamma_{t})}=\frac{C_{n}}{\ell_{g(t)}(\Gamma)}%$$ for $1\leq i\leq n-1$, where $C_{n}>0$ depends only on $n$. Taking $X=e_{i}$ in (\[Stability\]) and summing over $i=1,\dots,n-1$ yields$$0\leq(n-1)\left( \frac{C_{n}}{\operatorname{length}_{g(t)}(\gamma)}\right)
^{2}\cdot\operatorname{length}_{g(t)}(\gamma)-\int_{\gamma}\operatorname{Rc}%
(V,V)\,ds.$$ Thus$$\left. \frac{d}{ds}(\operatorname{length}_{g(s)}(\gamma))\right\vert
_{s=t}=-\int_{\gamma}\operatorname{Rc}(V,V)\,ds\geq-\frac{(n-1)C_{n}^{2}%
}{\operatorname{length}_{g(t)}(\gamma)}. \label{bound-on-length-decrease}%$$
Now define$$\begin{aligned}
f & :\Gamma\times\lbrack0,T)\rightarrow\mathbb{R},\\
f(\beta,t) & :=\operatorname{length}_{g(t)}(\beta).\end{aligned}$$ Note that $f$ is continuous in $(\beta,t)$ and is $C^{1}$ in $t$ for each fixed $\beta\in\Gamma$. Moreover, for each $u<T$, there is a compact set $K_{u}\subseteq\Gamma$ such that$$F(T):=\min_{\beta\in\Gamma}f(\beta,t)\equiv\ell_{g(t)}(\Gamma)$$ is attained in $K_{u}$ for $0\leq t\leq u$. It follows therefore from (\[bound-on-length-decrease\]) that the lower derivate$$\bar{D}F(t):=\liminf_{s\rightarrow t}\frac{F(s)-F(t)}{s-t}%$$ satisfies$$\bar{D}F(t)\geq-\frac{(n-1)C_{n}^{2}}{F(t)},\qquad0\leq t<T.$$ Hence as in §3 of [@H-86], we conclude that$$(F(t))^{2}+(n-1)C_{n}^{2}t$$ is nondecreasing, as required.
The preceding lemma yields a monotonicity result dual to Lemma \[Monotone1\].
\[Monotone2\]If $(M^{n},g(t):0\leq t<T)$ is a solution of the Ricci flow, then $m_{g(t)}(\Gamma)$ is non-decreasing.
By Lemma \[DecayBound\], we have$$(m_{g(t)}(\Gamma))^{2}=\liminf_{k\rightarrow\infty}\frac{(\ell_{g(t)}%
(k\Gamma))^{2}}{k^{2}}\geq\liminf_{k\rightarrow\infty}\frac{(\ell
_{g(s)}(k\Gamma))^{2}-Ct}{k^{2}}=(m_{g(s)}(\Gamma))^{2}%$$ whenever $0\leq s\leq t<T$.
To exploit Lemma \[Monotone2\], we need to know when $m_{g(0)}(\Gamma)$ is nonzero. Let $\eta(\Gamma)\ $denote the image of $\Gamma$ in $H_{1}%
(M^{n};\mathbb{R}).$
\[Nontorsion\]If $(M^{n},g)$ is a Riemannian manifold and $\Gamma
\in\operatorname*{Free}(M^{n})$ is a free homotopy class such that $\eta(\Gamma)$ is nonzero, then $m_{g}(\Gamma)>0$.
Since $\eta(\Gamma)\neq0$, there exists $\Phi\in H^{1}(M^{n};\mathbb{R})$ such that $\left\langle \Phi,\eta(\Gamma)\right\rangle >0$. For any $\phi\in\Phi$ and any curve $\gamma\in k\Gamma$, we have$$\left\langle \Phi,\eta(k\Gamma)\right\rangle =\int_{\gamma}\phi\leq\left\Vert
\phi\right\Vert \cdot\operatorname{length}_{g}(\gamma).$$ Taking the infimum over $\phi$ and $\gamma$ yields$$\left\langle \Phi,\eta(k\Gamma)\right\rangle \leq N_{g}(\Phi)\cdot\ell
_{g}(k\Gamma).$$ Hence$$m_{g}(\Gamma)=\liminf_{k\rightarrow\infty}\frac{\ell_{g}(k\Gamma)}{k}%
\geq\liminf_{k\rightarrow\infty}\frac{\left\langle \Phi,\eta(k\Gamma
)\right\rangle }{kN_{g}(\Phi)}=\frac{\left\langle \Phi,\eta(\Gamma
)\right\rangle }{N_{g}(\Phi)}>0.$$
These observations lead to another proof of the main result of this paper.
\[Second proof of Theorem \[Main\]\]Let $(M^{n},g(t):0\leq t<T)$ be a solution of the Ricci flow, and let $\alpha\in H_{1}(M^{n};\mathbb{Z})$ be an element of infinite order. Then there exists a free homotopy class $\Gamma
\in\operatorname*{Free}(M^{n})$ whose image in $H_{1}(M^{n};\mathbb{Z})$ is $\alpha$. Clearly, $L_{\alpha}(g(t))=\ell_{g(t)}(\Gamma)\geq m_{g(t)}(\Gamma
)$. Since $\alpha$ is of infinite order, $\eta(\Gamma)\in H_{1}(M^{n}%
;\mathbb{R})$ is nonzero. So we can apply Lemmas \[Monotone2\] and \[Nontorsion\] to conclude that$$L_{\alpha}(g(t))\geq m_{g(t)}(\Gamma)\geq m_{g(0)}(\Gamma)>0.$$
Concluding remarks\[Conclusion\]
================================
Although one expects Ricci flow evolutions to encounter finite-time singularities for a large class of initial Riemannian manifolds, the main result of this paper shows that there are topological restrictions on the geometry of such singularities. Motivated by this observation, we pose the following problems.
**Problem 1.** Suppose that $(M^{n},g_{j}(t))$ is a blowup sequence converging smoothly (in the pointed category) to a solution $(M_{\infty}^{n},g_{\infty}(t))$ of the Ricci flow. Show that the image of $H^{1}(M_{\infty}^{n};\mathbb{Z})$ in $H^{1}(M^{n};\mathbb{Z})$ under the natural map is finite.
**Problem 2.** The lens spaces $L(p,q)$ demonstrate that there can be no lower bound for the length of a torsion element, hence no torsion analogue of Theorem \[Main\]. If there is torsion in $H_{1}(M^{n}%
;\mathbb{Z})$, is it true that any solution $(M^{n},g(t))$ of the Ricci flow must become singular in finite time?
[9]{}
Cao, Huai-Dong and Chow, Bennett. *Recent developments on the Ricci flow.* Bull. Amer. Math. Soc. (N.S.) **36** (1999), no. 1, 59–74.
Hamilton, Richard S. *Four-manifolds with positive curvature operator.* J. Differential Geom. **24** (1986), no. 2, 153–179.
Hamilton, Richard S. *The formation of singularities in the Ricci flow.* Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Internat. Press, Cambridge, MA, 1995.
Hamilton, Richard S. *A compactness property for solutions of the Ricci flow.* Amer. J. Math. **117** (1995), no. 3, 545–572.
Hamilton, Richard S. *Non-singular solutions of the Ricci flow on three-manifolds.* Comm. Anal. Geom. **7** (1999), no. 4, 695–729.
Thurston, William P. *Three-dimensional manifolds, Kleinian groups and hyperbolic geometry.* Bull. Amer. Math. Soc. (N.S.) **6** (1982), no. 3, 357–381.
[^1]: Second author partially supported by NSF grant DMS - 0202796.
|
---
abstract: 'Nous présontons une preuve et une extension de deux formules de Frobenius et Stickelberger ainsi que des développements basés sur la formule d’inversion de Lagrange.'
address: 'Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 cours de la Libération F-33405 Talence cedex'
author:
- 'Roger Gay & Marcel Grangé & Ahmed Sebbar'
title: |
Sur deux formules de Frobenius et Stickelberger\
et inversion de Lagrange
---
Le point de départ du présent travail est les deux formules, utilisées par Frobenius et Stickelberger dans leur important travail sur les fonctions elliptiques [@F1]. Ces deux formules relèvent du calcul différentiel pur et s’énoncent ainsi: Soient $U,V$ deux fonctions $n$-fois continûment différentiable sur un intervalle $J\subset \BR$, on a: $$\label{1}
D^{(n)} (V)= \sum_{0 \leq p \leq n} {n \atopwithdelims() p} \frac{1}{p+1} D^p (U^{p+1}) D^{(n-p)} \left(\frac{V}{U^{p+1}} \right)$$ et $$\label{2}
D^{(n)}(V)= \sum_{0 \leq p \leq n} (-1)^p {n \atopwithdelims() p} \frac{1}{p+1} U^{-p-1} D^{(n-p)} (VD^p U^{p+1} ).$$
Le calcul différentiel comporte diverses formules intéressantes [@F0], notamment faisant intervenir le produit de fonctions. La plus connue est la formule de Leibniz donnant la dérivée d’ordre $n$ d’un produit. La formule d’inversion de Lagrange, [@F3] et [@F5], possède de nombreuses applications dont la plus emblématique est la fonction arbre $$a(x)= \sum_{n=1}^{\infty} n^{n-1}\frac{x^n}{n!}$$ qui résout l’équation $ \displaystyle a(x)= xe^{a(x)} $. Cette fonction a beaucoup d’applications combinatoires et est souvent donnée à l’aide de la classique fonction de Weber $$\displaystyle W(x)= -a(-x),\, W(x)e^{W(x)}=x.$$ De la formule d’inversion de Lagrange on a pu déduire ce qu’il est convenu d’appeler la formule du produit de Lagrange, qui généralise la formule de Leibniz. Par ailleurs Frobenius et Stickelberger indiquent les égalités et (en bas de page de [@F1], sans démonstration) et d’autres dont l’aspect rappelle encore la formule de Leibniz, sans toutefois pouvoir se réduire à cette dernière.
Le travail présenté ici propose de démontrer ces diverses formules en utilisant un même outil: les applications bilinéaires $\Phi$, introduites et étudiées dans la première section. La deuxième section est dévolue à deux généralisations des formules de Frobenius-Stickelberger [@F1] et une application aux fonctions entières de type exponentiel. La troisième section reprend donc la formule du produit de Lagrange à partir des applications bilinéaires $\Phi$, et grâce à des calculs algébriques sur des fonctions et leurs dérivées successives, évidemment sans utiliser la formule d’inversion de Lagrange. Enfin la dernière et quatrième section traite d’un résultat de P. J. Olver qui a été, nous semble t-il, succinctement avancé dans [@F4], qui est ici intégralement démontré, notamment à l’aide de la formule du produit de Lagrange. Ainsi on constate que rien de ce qui est avancé ici dans le cadre de la variable complexe, ne dépend de la formule intégrale de Cauchy, contrairement à [@F5]: seule la théorie des séries entières est utilisée.
Les concepts et notations sont assez courants, toutefois il est peut-être utile de donner les précisions qui suivent. Tous les intervalles considérés dans cette étude contiennent au mois deux points distincts. La fonction dérivée d’une fonction complexe $f$ définie et dérivable sur un intervalle est notée $Df$, et les éventuelles dérivées successives sont notées $D^{k}f$. Un espace vectoriel complexe est aussi un espace vectoriel réel, et on désigne par $E_{n}$ l’espace de Banach réel des fonctions complexes de classe $\mathcal{C}^{n}$ sur le segment réel $\bigl[a,b\bigr]$. La norme étant : $$\left\Vert f\right\Vert =\sum_{j=0}^{n}\left\Vert D^{j}f\right\Vert _{\left[a,b\right]}$$ et par $\mathcal{U}_{n}$ l’ouvert de $E_{n}$ $$\mathcal{U}_{n}=\bigl\{ f\in E_{n}\;;\;\forall t\in\bigl[a,b\bigr],\; f\left(t\right)\neq0\bigr\}.$$ Cet ouvert est aussi connexe car les fonctions de $E_{n}$ sont à valeurs complexes. Étant donnée une application $\mathbf{F}$ définie et différentiable sur un ouvert $\Omega$ d’un espace normé réel $E$, à valeurs dans un espace normé réel, sa différentielle en un point $x$ de $\Omega$ est notée dans ce contexte $\mathfrak{D}\mathbf{F}\left(x\right)$ de manière à éviter la confusion avec l’opération $D$ ci-dessus décrite.
Pour tout couple $\left(n,p\right)\in \BN \times \BN^*$ on désigne par $E\left(n,p\right)$ l’ensemble $$\bigl\{\alpha=\left(\alpha_{1},\ldots,\alpha_{p}\right)\in\mathbb{N}^{p}\;;\;\alpha_{1}+\cdots+\alpha_{p}=n\bigr\}.$$ Si le couple d’entiers vérifie en outre la condition $0\leq p\leq n$, on désigne, comme d’habitude, par $\dbinom{n}{p}$ le nombre entier $\dfrac{n!}{p!\left(n-p\right)!}$, qui est inférieur ou égal à $2^{n}$ en vertu de la formule du binôme. L’ensemble $E\left(n,p\right)$ est de cardinal $\dbinom{n+p-1}{p-1}$.
Les notations de la quatrième section, plus spécifiques à celle-ci, sont rappelées ou introduites au début de cette dernière section.
Les applications bilinéaires $\Phi$
====================================
Étant donnés un entier naturel $n$ et un intervalle $J\subset \BR$, on considère une fonction complexe $u$ de classe $\mathcal{C}^{n}$ et ne s’annulant pas sur $J$. À cette fonction $u$ est attachée l’application bilinéaire $\Phi_{n,u}$ définie sur l’espace vectoriel complexe des fonctions de classe $\mathcal{C}^{n}$ sur $J$, à valeurs dans l’espace vectoriel complexe des fonctions continues sur $J$: $$\label{eq}
\Phi_{n,u}\left(f,g\right)=\sum_{p=0}^{n}\dbinom{n}{p}D^{p}\left(u^{p}f\right)D^{n-p}\left(u^{-p}g\right).$$
\[t1\] Pour tout couple $\left(n,u\right)$ comme ci-dessus et pour tout entier naturel $q$ on a: $$\label{2.1}
\Phi_{n,u}\left(u^{q},u^{-q}\right)=\Phi_{n,u}\left(1,1\right).$$ Pour tout couple $\left(n,u\right)$ comme ci-dessus on a: $$\Phi_{n,u}\left(f,g\right)=\sum_{r=0}^{n}\binom{n}{r}\Phi_{n-r,u}\left(1,1\right)D^{r}\left(fg\right)\label{eq2}.$$
En développant $D^{p}\left(u^{p}f\right)$ et $D^{n-p}\left(u^{-p}g\right)$ par la formule de Leibniz, après avoir interverti la sommation ${\displaystyle \sum_{p=0}^{n}}$ avec les sommations ${\displaystyle \sum_{0\leq j\leq p}}$ et ${\displaystyle \sum_{0\leq k\leq n-p}}$ de la formule de Leibniz, on obtient: $$\Phi_{n,u}\left(f,g\right)={\displaystyle \sum_{0\leq j,k\leq n}Q_{n}\left(u,j,k\right)D^{j}f\, D^{k}g}$$ où l’on a posé: $$Q_{n}\left(u,j,k\right)=\left\{ \begin{array}{ccc}
0 & \; si\;\: j+k>n\\
\\
{\displaystyle \sum_{j\leq p\leq n-k}\binom{n}{p}\binom{p}{j}\binom{n-p}{k}}D^{p-j}u^{p}\, D^{n-k-p}u^{-p} & \; si\;\: j+k\leq n.
\end{array}\right.$$ Par suite on obtient: $$\Phi_{n,u}\left(f,g\right)=\sum_{r=0}^{n}\biggl(\,\sum_{j+k=r}Q_{n}\left(u,j,k\right)D^{j}f\, D^{k}g\biggr).$$ Or, notant $r=j+k$, et supposant $r\leq n$, on a l’égalité: $$\binom{n}{p}\binom{p}{j}\binom{n-p}{k}=\binom{n}{r}\binom{r}{j}\binom{n-r}{p-j},$$ d’où l’on tire: $$\begin{aligned}
Q_{n}\left(u,j,k\right)&= \binom{n}{r}\binom{r}{j}\sum_{q=0}^{n-r}\binom{n-r}{q}D^{q}u^{q+j}\, D^{n-r-q}u^{-q-j}\\
&= \binom{n}{r}\binom{r}{j}\Phi_{n-r,u}\left(u^{j},u^{-j}\right).\end{aligned}$$ En conséquence: $$\label{eq3}
\Phi_{n,u}\left(f,g\right)=\sum_{r=0}^{n}\binom{n}{r}\biggl(\,\sum_{j+k=r}\binom{r}{j}\Phi_{n-r,u}\left(u^{j},u^{-j}\right)D^{j}f\, D^{k}g\biggr).$$
[*Preuve de la formule* ]{}: Pour tout entier naturel $q$ on a immédiatement: $$\Phi_{0,u}\left(u^{q},u^{-q}\right)=u^{q}u^{-q}=1=\Phi_{0,u}\left(1,1\right).$$ Soit un entier naturel $n$ supérieur ou égal à $1$, et supposons la propriété vraie jusqu’au rang $n-1$. Compte-tenu de la formule , pour tout entier naturel $q$ on a: $$\Phi_{n,u}\left(u^{q},u^{-q}\right)=\sum_{r=0}^{n}\binom{n}{r}\biggl(\,\sum_{j+k=r}\binom{r}{j}\Phi_{n-r,u}\left(u^{j},u^{-j}\right)D^{j}u^{q}\, D^{k}u^{-q}\biggr).$$ Puis, en vertu de l’hypothèse de récurrence $$\Phi_{n,u}\left(u^{q},u^{-q}\right)=\Phi_{n,u}\left(1,1\right)u^{q}u^{-q}+\sum_{r=1}^{n}\binom{n}{r}\Phi_{n-r,u}\left(1,1\right)\biggl(\,\sum_{j+k=r}\binom{r}{j}D^{j}u^{q}\, D^{k}u^{-q}\biggr)$$ $$=\sum_{r=0}^{n}\binom{n}{r}\Phi_{n-r,u}\left(1,1\right)\biggl(\,\sum_{j+k=r}\binom{r}{j}D^{j}u^{q}\, D^{k}u^{-q}\biggr)=\sum_{r=0}^{n}\binom{n}{r}\Phi_{n-r,u}\left(1,1\right)D^{r}\left(u^{q}u^{-q}\right),$$ ce qu’il fallait montrer.
[ *Preuve de la formule* ]{}: En vertu des formules et et de la formule de Leibniz, on obtient: $$\Phi_{n,u}\left(f,g\right)=\sum_{r=0}^{n}\binom{n}{r}\biggl(\,\sum_{j+k=r}\binom{r}{j}\Phi_{n-r,u}\left(1,1\right)D^{j}f\, D^{k}g\biggr)=\sum_{r=0}^{n}\binom{n}{r}\Phi_{n-r,u}\left(1,1\right)D^{r}\left(fg\right).$$
\[r1\] Pour tout couple de couples $\left(f_{1},g_{1}\right)$ et $\left(f_{2},g_{2}\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ et vérifiant $f_{1}g_{1}=f_{2}g_{2}$ on a: $$\Phi_{n,u}\left(f_{1},g_{1}\right)=\Phi_{n,u}\left(f_{2},g_{2}\right)=\Phi_{n,u}\left(1,f_{1}g_{1}\right).$$
Une extension des formules de Frobenius-Stickelberger
======================================================
Les formules
------------
La proposition suivante établit une extension de l’identité de Frobenius et Stickelberger [@F1]
\[t2\] Pour tout nombre entier $\lambda$ supérieur ou égal à $1$, pour tout triplet $\left(u,v,w\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$, la fonction $u$ ne s’annulant pas, on a la formule: $$\label{eq4}
\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}\left(u^{p+\lambda}w\right)D^{n-p}\left(u^{-p-\lambda}v\right)=\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}\left(w\right)D^{n-p}\left(v\right).$$
Il suffit de démontrer le résultat lorsque l’intervalle $J$ est un segment $\bigl[a,b\bigr]$.
Étant donné un couple $\left(v,w\right)$ de fonctions de l’espace de Banach $E_{n}$, on considère l’application $$\begin{split}
\mathbf{G}_{n,v,w}&:\mathcal{U}_{n}\longrightarrow E_{0}\\
\mathbf{G}_{n,w,v}\left(u\right)&=\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}\left(u^{p+\lambda}w\right)D^{n-p}\left(u^{-p-\lambda}v\right).
\end{split}$$ Or, pour tout entier $\alpha$, l’application $u\longmapsto u^{\alpha}$ est différentiable sur l’ouvert $\mathcal{U}_{n}$, à valeurs dans $E_{n}$, et sa différentielle est $h\longmapsto\alpha\, u^{\alpha-1}h$. Par ailleurs, pour tout entier naturel $q\leq n$ l’application linéaire $D^{q}$ est continue de $E_{n}$ dans $E_{0}$. Donc l’application $\mathbf{G}_{n,v,w}$ est différentiable sur l’ouvert $\mathcal{U}_{n}$ et on a: $$\mathfrak{D}\mathbf{G}_{n,v,w}\left(u\right)=
\sum_{p=0}^{n}\dbinom{n}{p}\Bigl(D^{p}\left(u^{p+\lambda-1}hw\right)D^{n-p}\left(u^{-p-\lambda}v\right)-D^{p}\left(u^{p+\lambda}w\right)D^{n-p}\left(u^{-p-\lambda-1}hv\right)\Bigr).$$ Introduisant l’application bilinéaire $\Phi_{n,u}$, considérée définie sur $E_{m}\times E_{m}$ à valeurs dans $E_{0}$, on observe l’égalité: $$\mathfrak{D}\mathbf{G}_{n,v,w}\left(u\right)\cdot h=\Phi_{n,u}\left(hu^{\lambda-1}w,u^{-\lambda}v\right)-\Phi_{n,u}\left(u^{\lambda}w,hu^{-\lambda-1}v\right)$$ soit, compte-tenu du théorème \[t1\]: $$\mathfrak{D}\mathbf{G}_{n,v,w}\left(u\right)\cdot h=\Phi_{n,u}\left(1,hu^{-1}vw\right)-\Phi_{n,u}\left(1,hu^{-1}vw\right)=0.$$ L’application $\mathbf{G}_{n,v,w}$ est donc constante sur l’ouvert connexe $\mathcal{U}_{n}$. Mais il est clair qu’on a: $$\mathbf{G}_{n,v,w}\left(1\right)=\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}\left(w\right)D^{n-p}\left(v\right)$$ l’application $\mathbf{G}_{n,v,w}$ est ainsi contante de valeur ${\displaystyle \sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}\left(w\right)D^{n-p}\left(v\right)}$.
Par dualité, à partir de la formule de la proposition \[t2\] ci-dessus, on obtient une seconde formule qui étend l’identité de Frobenius-Stickelberger. La proposition suivante précise cette deuxième formule, et en avance aussi une troisième, différente malgré les apparences.
Pour tout nombre entier $\lambda\geq1$, pour tout triplet $\left(u,v,w\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$, la fonction $u$ ne s’annulant pas, on a: $$\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{\left(-1\right)^{p}}{p+\lambda}u^{-p-\lambda}D^{n-p}\left(v\, D^{p}\left(u^{p+\lambda}w\right)\right)=
\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{\left(-1\right)^{p}}{p+\lambda}D^{n-p}\left(v\, D^{p}w\right)\label{eq5}.$$ $$\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{\left(-1\right)^{p}}{p+\lambda}u^{p+\lambda}D^{p}\left(v\, D^{n-p}\left(u^{-p-\lambda}w\right)\right)=
\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{\left(-1\right)^{p}}{p+\lambda}D^{p}\left(v\, D^{n-p}w\right).\label{eq6}$$
Seule est donnée une esquisse de la preuve de la formule . Pour toute fonction $\varphi$ de classe $\mathcal{C}^{\infty}$ à support compact dans l’intervalle $J$, par $n-p$ intégrations par parties et en vertu de la formule on a: $$\int_{J}\varphi\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{\left(-1\right)^{p}}{p+\lambda}u^{-p-\lambda}D^{n-p}\left(v\, D^{p}\left(u^{p+\lambda}w\right)\right)\, dt=\left(-1\right)^{n}\int_{J}v\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}wD^{n-p}\varphi\, dt$$ À nouveau $n-p$ intégrations par parties conduisent à l’égalité $$\left(-1\right)^{n}\int_{J}v\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{1}{p+\lambda}D^{p}wD^{n-p}\varphi\, dt=\int_{J}\varphi\sum_{p=0}^{n}\dbinom{n}{p}\dfrac{\left(-1\right)^{p}}{p+\lambda}D^{n-p}\left(v\, D^{p}w\right) dt.$$ Enfin on conclut grâce au lemme de du Bois-Reymond.
Une application de la première formule de Frobenius-Stickelberger
-----------------------------------------------------------------
Choisissons les trois fonctions $u$, $v$ et $w$ comme suit: $$u\left(t\right)=\exp\left(xt\right);\quad v\left(t\right)=\exp\left(yt\right);\quad w\left(t\right)=\exp\left(zt\right)$$ où $x$, $y$ et $z$ désignent des nombres complexes. Pour tout entier $\lambda\geq1$ on obtient: $$\sum_{p=0}^{n}\binom{n}{p}\dfrac{1}{p+\lambda}\left(z+\left(p+\lambda\right)x\right)^{p}\left(y-\left(p+\lambda\right)x\right)^{n-p}=\sum_{p=0}^{n}\binom{n}{p}\dfrac{1}{p+\lambda}z^{p}y^{n-p}.$$ La différence des deux membres de cette relation est, pour $\left(x,y,z\right)$ fixé, une fonction *fraction rationnelle* en $\lambda$, qui s’annule sur l’ensemble infini $\mathbb{N}^{*}$. Donc cette fonction fraction rationnelle est la fonction nulle et on obtient ainsi l’identité: $$\label{eq51}
\sum_{p=0}^{n}\binom{n}{p}\dfrac{1}{p+\lambda}\left(z+\left(p+\lambda\right)x\right)^{p}\left(y-\left(p+\lambda\right)x\right)^{n-p}=\sum_{p=0}^{n}\binom{n}{p}\dfrac{1}{p+\lambda}z^{p}y^{n-p}$$ valable pour tout $\left(x,y,z\right)\in {\BC}^{3}$ et tout $\lambda \in {\BC}\setminus\left\{ 0,-1,\ldots,-n\right\} $.
Soit une fonction entière $f$ de type exponentiel. Il existe un voisinage symétrique convexe compact $A$ de $x=0$ dans $\mathbb{C}$ vérifiant: Pour tout $\left(x,y,z\right)$ de $A\times\mathbb{C}^{2}$ et tout $\lambda$ de $\mathbb{C}\setminus\left\{ 0,-1,\ldots,-n,\ldots\right\} $ on a: $$\label{eq6}
\sum_{p=0}^{\infty}\dfrac{1}{p+\lambda}\dfrac{\left(z+\left(p+\lambda\right)x\right)^{p}}{p!}D^{p}f\left(y-\left(p+\lambda\right)x\right)=
\sum_{p=0}^{\infty}\dfrac{1}{p+\lambda}\dfrac{z^{p}}{p!}D^{p}f\left(y\right).$$ Pour tout $\left(x,\lambda\right)$ de $A\times\mathbb{C}$ on a: $$\label{eq7}
f\left(\lambda x\right)=f\left(0\right)+\lambda\sum_{p=1}^{\infty}\left(\lambda-p\right)^{p-1}\dfrac{x^{p}}{p!}D^{p}f\left(px\right).$$ Pour tout $x$ de $A$ et pour tout entier $m$ supérieur ou égal à $1$ on a: $$\label{eq8}
\dfrac{1}{m!}D^{m}f\left(0\right)=\sum_{p=m}^{\infty}\dbinom{p-1}{m-1}\dfrac{\left(-1\right)^{p-m}p^{p-m}}{p!}x^{p-m}D^{p}f\left(px\right).$$
[*Démontrons la formule* ]{}: Par hypothèse, il existe deux nombres strictement positifs $C$ et $K$ tels que pour tout entier naturel $n$ on ait l’inégalité $\bigl|D^{n}f\left(0\right)\bigr|\leq K^{n}C$. De là, on effectue les majorations suivantes: $$\begin{aligned}
\sum_{n=0}^{\infty}\sum_{p=0}^{n}&\left|\dfrac{1}{n!}D^{n}f\left(0\right)\right|\dbinom{n}{p}\dfrac{1}{\left|p+\lambda\right|}\bigl|z+\left(p+\lambda\right)x\bigr|^{p}\bigl|y-\left(p+\lambda\right)x\bigr|^{n-p}
\\
&\leq C\sum_{p=0}^{\infty}\dfrac{\bigl|z+\left(p+\lambda\right)x\bigr|^{p}}{p!\left|p+\lambda\right|}\left(\sum_{n=p}^{\infty}K^{n}\dfrac{\bigl|y-\left(p+\lambda\right)x\bigr|^{n-p}}{\left(n-p\right)!}\right)
\\
&\leq C\sum_{p=0}^{\infty}K^{p}\dfrac{\bigl|z+\left(p+\lambda\right)x\bigr|^{p}}{p!\left|p+\lambda\right|}\exp\left(K\left|y-\left(p+\lambda\right)x\right|\right)
\\
&\leq C\Bigl(\exp\left(K\left|y-\lambda x\right|\right)\Bigr)\sum_{p=0}^{\infty}K^{p}\dfrac{\bigl|z+\left(p+\lambda\right)x\bigr|^{p}}{p!\left|p+\lambda\right|}\left(\exp\left(K\left|x\right|\right)\right)^{p}.\end{aligned}$$ Dans le cas $x=0$, la série ci-dessus est, évidemment, convergente. Dans le cas $x\neq0$, on peut écrire: $$\sum_{p=1}^{\infty}K^{p}\dfrac{\bigl|z+\left(p+\lambda\right)x\bigr|^{p}}{p!\left|p+\lambda\right|}\left(\exp\left(K\left|x\right|\right)\right)^{p}\leq\sum_{p=1}^{\infty}
\left(K^{p}\dfrac{p^{p}\left(\left|x\right|\exp\left(K\left|x\right|\right)\right)^{p}}{p!\left|p+\lambda\right|}\left(1+\dfrac{\left|z+\lambda x\right|}{p\left|x\right|}\right)^{p}\right).$$ Grâce à l’inégalité de Stirling : $p!\geq p^{p}e^{-p}\sqrt{2\pi p}$, on obtient la majoration: $$\sum_{p=1}^{\infty}\left(K^{p}\dfrac{p^{p}\left(\left|x\right|\exp\left(K\left|x\right|\right)\right)^{p}}{p!\left|p+\lambda\right|}
\left(1+\dfrac{\left|z+\lambda x\right|}{p\left|x\right|}\right)^{p}\right)\lesssim\exp\left(\dfrac{\left|z+\lambda x\right|}{\left|x\right|}\right)
\sum_{p=1}^{\infty}\left(eK\left|x\right|\exp\left(K\left|x\right|\right)\right)^{p}p^{-\tfrac{3}{2}}.$$ En conséquence, dans le voisinage symétrique convexe compact $$A=\bigl\{ x\in \BC, \,eK\left|x\right|\exp K\left|x\right|\leq1\bigr\}$$ du point $x=0$, la série double $$\sum_{n=0}^{\infty}\left(\sum_{p=0}^{n}\dfrac{1}{n!}D^{n}f\left(0\right)\dbinom{n}{p}\dfrac{1}{p+\lambda}\left(z+\left(p+\lambda\right)x\right)^{p}\left(y-\left(p+\lambda\right)x\right)^{n-p}\right)$$ est absolument convergente, et en intervertissant les sommes, elle s’écrit d’une part: $$\sum_{p=0}^{\infty}\dfrac{1}{p+\lambda}\dfrac{\left(z+\left(p+\lambda\right)x\right)^{p}}{p!}D^{p}f\left(y-\left(p+\lambda\right)x\right).$$ D’autre part, grâce à l’identité , et en intervertissant les sommes elle s’écrit aussi: $$\sum_{n=0}^{\infty}\left(\sum_{p=0}^{n}\dfrac{1}{n!}D^{n}f\left(0\right)\dbinom{n}{p}\dfrac{1}{p+\lambda}z^{p}y^{n-p}\right)=\sum_{p=0}^{\infty}\dfrac{1}{p+\lambda}\dfrac{z^{p}}{p!}D^{p}f\left(y\right)$$ d’où la conclusion.
[*Démontrons la formule* ]{}: Faisant $z=0$ dans l’identité et multipliant par $\lambda$, on obtient: $$f\left(y\right)=f\left(y-\lambda x\right)+\lambda\sum_{p=1}^{\infty}\dfrac{\left(p+\lambda\right)^{p-1}}{p!}x^{p}D^{p}f\left(y-\left(p+\lambda\right)x\right).$$ Mais le terme général de cette série est majoré comme suit: $$\begin{aligned}
&\left|\dfrac{\left(p+\lambda\right)^{p-1}}{p!}x^{p}D^{p}f\left(y-\left(p+\lambda\right)x\right)\right|\\
&\leq C\Bigl(\exp\left(K\left|y-\lambda x\right|\right)\Bigr)K^{p}\dfrac{\left|p+\lambda\right|^{p-1}}{p!}\left(K\left|x\right|\exp\left(K\left|x\right|\right)\right)^{p}
\\
&\lesssim\exp\left(\left|\lambda\right|+K\left|y-\lambda x\right|\right)\left(K\left|x\right|\exp\left(K\left|x\right|\right)\right)^{p}p^{-\tfrac{3}{2}}.\end{aligned}$$ La série ci-dessus est donc normalement convergente par rapport à $\lambda$ sur tout compact de $\mathbb{C}$. En conséquence la formule ci-dessus exprimant $f\left(y\right)$ s’étend à tout triplet $\left(x,y,\lambda\right)$ appartenant à $A\times\mathbb{C}\times\mathbb{C}$. La formule s’obtient en faisant $y=\lambda x$ et en remplaçant $\left(x,\lambda\right)$ par $\left(-x,-\lambda\right)$.
[*Démontrons la formule* ]{}: En développant $\left(\lambda-p\right)^{p-1}$ par la formule du binôme, et grâce aux majorations effectuées pour démontrer , il résulte aussi que pour tout $\left(x,\lambda\right)$ appartenant à $A\times\mathbb{C}$: $$f\left(\lambda x\right)=f\left(0\right)+\sum_{m=1}^{\infty}\left(-1\right)^{m}\lambda^{m}\left(\sum_{p=m}^{\infty}\dbinom{p-1}{m-1}\dfrac{\left(-1\right)^{p}p^{p-m}}{p!}x^{p-m}D^{p}f\left(px\right)\right).
\qedhere$$
Formule du produit de Lagrange
==============================
Dans cette section et la suivante, pour tout couple $\left(\psi,f\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$, la fonction désignée par $D^{m-1}\left(\psi^{m}D\left(f\right)\right)$ est, pour $m=0$, la fonction $f=D^{-1}\left(Df\right)$. Mais tout d’abord il convient de s’assurer du lemme suivant sur les applications bilinéaires $\Phi$.
\[lem1\] Pour tout couple $\left(m,N\right)$ d’entiers naturels et pour toute fonction complexe $\psi$ de classe $\mathcal{C}^{n}$ sur un intervalle $J$ et ne s’y annulant pas, on a: $$\label{eq9}
\Phi_{m,\psi}\left(1,\psi^{N}D\psi\right)=\dfrac{1}{m+1}\Bigl(\Phi_{m+1,\psi}\left(1,\psi^{N+1}\right)-D^{m+1}\left(\psi^{N+1}\right)\Bigr)$$ $$\label{eq10}
\Phi_{m,\psi}\Bigl(1,\psi^{N}\left(D\psi\right)^{2}\Bigr)=\dfrac{1}{m+1}\Bigl(\Phi_{m+1,\psi}\left(1,\psi^{N+1}D\psi\right)-D^{m+1}\left(\psi^{N+1}D\psi\right)\Bigr)$$ $$=\dfrac{1}{\left(m+1\right)\left(m+2\right)}\Bigl(\Phi_{m+2,\psi}\left(1,\psi^{N+2}\right)-D^{m+2}\left(\psi^{N+2}\right)\Bigr)-\dfrac{1}{m+1}D^{m+1}\left(\psi^{N+1}D\psi\right).$$
[*Démontrons la formule* ]{}: Compte-tenu de la remarque on a successivement: $$\begin{aligned}
\Phi_{m,\psi}&\left(1,\psi^{N}D\psi\right)=\Phi_{m,\psi}\left(D\psi,\psi^{N}\right)=\sum_{p=0}^{m}\dbinom{m}{p}D^{p}\left(\psi^{p}D\psi\right)D^{m-p}\left(\psi^{N-p}\right)
\\
&=\sum_{p=0}^{m}\dbinom{m}{p}\dfrac{1}{p+1}D^{p+1}\left(\psi^{p+1}\right)D^{m-p}\left(\psi^{N-p}\right)
\\
&=\dfrac{1}{m+1}\sum_{p=0}^{m}\dbinom{m+1}{p+1}D^{p+1}\left(\psi^{p+1}\right)D^{m+1-\left(p+1\right)}\left(\psi^{N-p}\right)
\\
&=-\dfrac{1}{m+1}D^{m+1}\left(\psi^{N+1}\right)+\dfrac{1}{m+1}\sum_{q=0}^{m+1}\dbinom{m+1}{q}D^{q}\left(\psi^{q}\right)D^{m+1-q}\left(\psi^{N+1-q}\right)
\\
&=\dfrac{1}{m+1}\Bigl(\Phi_{m+1,\psi}\left(1,\psi^{N+1}\right)-D^{m+1}\left(\psi^{N+1}\right)\Bigr).\end{aligned}$$ [*Montrons la formule* ]{}: Compte-tenu de la remarque on a successivement: $$\begin{aligned}
\Phi_{m,\psi}&\Bigl(1,\psi^{N}\left(D\psi\right)^{2}\Bigr)= \Phi_{m,\psi}\left(D\psi,\psi^{N}D\psi\right)=\sum_{p=0}^{m}\dbinom{m}{p}D^{p}\left(\psi^{p}D\psi\right)D^{m-p}\left(\psi^{N-p}D\psi\right)
\\
&= \sum_{p=0}^{m}\dbinom{m}{p}\dfrac{1}{p+1}D^{p+1}\left(\psi^{p+1}\right)D^{m-p}\left(\psi^{N-p}D\psi\right)
\\
&= \dfrac{1}{m+1}\sum_{p=0}^{m}\dbinom{m+1}{p+1}D^{p+1}\left(\psi^{p+1}\right)D^{m+1-\left(p+1\right)}\left(\psi^{N-p}D\psi\right)
\\
&= -\dfrac{1}{m+1}D^{m+1}\left(\psi^{N+1}D\psi\right)+\dfrac{1}{m+1}\sum_{q=0}^{m+1}\dbinom{m+1}{q}D^{q}\left(\psi^{q}\right)D^{m+1-q}\left(\psi^{N+1-q}D\psi\right)
\\
&= \dfrac{1}{m+1}\Bigl(\Phi_{m+1,\psi}\left(1,\psi^{N+1}D\psi\right)-D^{m+1}\left(\psi^{N+1}D\psi\right)\Bigr)
\\
&= \dfrac{1}{\left(m+1\right)\left(m+2\right)}\Bigl(\Phi_{m+2,\psi}\left(1,\psi^{N+2}\right)-D^{m+2}\left(\psi^{N+2}\right)\Bigr)-\dfrac{1}{m+1}D^{m+1}\left(\psi^{N+1}D\psi\right)\end{aligned}$$ en vertu de la formule ci-dessus démontrée.
Pour tout triplet $\left(\psi,f,g\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$, on a la formule: $$\dfrac{1}{n!}D^{n-1}\left(\psi^{n}D\left(fg\right)\right)=\sum_{k=0}^{n}\dfrac{1}{k!}D^{k-1}\left(\psi^{k}D\left(f\right)\right)\dfrac{1}{\left(n-k\right)!}D^{n-k-1}\left(\psi^{n-k}D\left(g\right)\right).\label{eq:3-1}$$ Plus généralement, pour entier naturel $p$ supérieur ou égal à $2$ et tout système $\left(\psi,f_{1},\ldots,f_{p}\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$, on a la formule: $$\dfrac{1}{n!}D^{n-1}\left(\psi^{n}D\left(f_{1}\cdots f_{p}\right)\right)=
\sum_{\alpha_{1}+\cdots+\alpha_{p}=n}\left(\prod_{j=1}^{p}\dfrac{1}{\alpha_{j}!}D^{\alpha_{j}-1}\left(\psi^{\alpha_{j}}D\left(f_{j}\right)\right)\right).$$
La seconde formule s’obtient immédiatement à partir de la première, par récurrence sur l’entier $p$. La démonstration ne concerne donc que la première formule, qui est clairement vraie dans les cas $n=0$ et $n=1$ ; aussi l’entier $n$ est supposé supérieur ou égal à $2$ dans ce qui suit. L’idée consiste à exprimer le membre de droite de la formule de Lagrange en fonction des dérivées successives $D^{j}\left(fg\right)$ où $j$ appartient à $\left\{ 0,\ldots,n\right\} $. Si on suppose que la fonction $\psi$ ne s’annule pas, on constate qu’apparaissent les applications bilinéaires $\Phi$ introduites dans la section 2.
[*Réduction au cas où la fonction $\psi$ ne s’annule pas*]{}:\
Ainsi, on suppose la formule de Lagrange pour tout triplet $\left(\varphi,u,v\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $I$ lorsque la fonction $\varphi$ ne s’annule pas. Soit un triplet $\left(\psi,f,g\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$ et notons $F$ le sous-ensemble fermé $\left\{ \psi=0\right\} $ de $J$. Grâce à l’hypothèse, la formule de Lagrange est acquise sur l’intérieur de $F$ qui est une réunion au plus dénombrable d’intervalles, ouverts dans $J$, et pour la même raison sur le complémentaire $\Omega$ de $F$. La formule de Lagrange est donc vraie en tout point de l’adhérence $\overline{\Omega}$ dans $J$ de $\Omega$. Enfin, sachant qu’on a $J=\overset{\circ}{F}\cup\overline{\Omega}$ la formule est vraie sur tout l’intervalle $J$. On suppose dorénavant que la fonction $\psi$ ne s’annule pas. Compte-tenu des relations $$\psi^{k}Df=D\left(\psi^{k}f\right)-\left(k\psi^{k-1}D\psi\right)f,\quad\psi^{n-k}Dg=D\left(\psi^{n-k}g\right)-\left(\left(n-k\right)\psi^{n-k-1}D\psi\right)g$$ le membre de droite de la formule de Lagrange s’écrit comme la somme de quatre termes $$T_{1}\left(f,g\right)+T_{2}\left(f,g\right)-T_{3}\left(f,g\right)-T_{4}\left(f,g\right)$$ qui sont exprimés et traités dans ce qui suit. [*Expression du premier terme*]{}: $$\begin{aligned}
&T_{1}\left(f,g\right)=\sum_{k=0}^{n}\binom{n}{k}D^{k}\left(\psi^{k}f\right)D^{n-k}\left(\psi^{n-k}g\right)=\Phi_{n,\psi}\left(f,\psi^{n}g\right)
\\
&=\sum_{r=0}^{n}\binom{n}{r}\Phi_{n-r,\psi}\left(1,1\right)D^{r}\left(\psi^{n}fg\right)=\sum_{r=0}^{n}\binom{n}{r}\left(\sum_{j=0}^{r}\binom{r}{j}D^{r-j}\left(\psi^{n}\right)D^{j}\left(fg\right)\right)\Phi_{n-r,\psi}\left(1,1\right)
\\
&=\sum_{j=0}^{n}\left(\sum_{r=j}^{n}\binom{n}{r}\binom{r}{j}\Phi_{n-r,\psi}\left(1,1\right)D^{r-j}\left(\psi^{n}\right)\right)D^{j}\left(fg\right)
\\
&=\sum_{j=0}^{n}\binom{n}{j}\left(\sum_{q=0}^{n-j}\binom{n-j}{q}\Phi_{n-j-q,\psi}\left(1,1\right)D^{q}\left(\psi^{n}\right)\right)D^{j}\left(fg\right)=
\sum_{j=0}^{n}\binom{n}{j}\Phi_{n-j,\psi}\left(1,\psi^{n}\right)D^{j}\left(fg\right).\end{aligned}$$
[*Expression du deuxième terme*]{}: $$\begin{aligned}
T_{2}\left(f,g\right)&= \sum_{k=1}^{n-1}\binom{n}{k}k\left(n-k\right)D^{k-1}\left(\psi^{k-1}fD\psi\right)D^{n-k-1}\left(\psi^{n-k}gD\psi\right)
\\
&= n\left(n-1\right)\sum_{k=1}^{n-1}\binom{n-2}{k-1}D^{k-1}\left(\psi^{k-1}fD\psi\right)D^{n-k-1}\left(\psi^{n-k}gD\psi\right)
\\
&= n\left(n-1\right)\sum_{j=0}^{n-2}\binom{n-2}{j}D^{j}\left(\psi^{j}fD\psi\right)D^{n-2-j}\left(\psi^{n-2-j}gD\psi\right)
\\
&= n\left(n-1\right)\Phi_{n-2,\psi}\left(fD\psi,\psi^{n-2}gD\psi\right)
\\
&= n\left(n-1\right)\sum_{r=0}^{n-2}\binom{n-2}{r}\Phi_{n-2-r,\psi}\left(1,1\right)D^{r}\Bigl(\psi^{n-2}fg\left(D\psi\right)^{2}\Bigr)
\\
&= n\left(n-1\right)\sum_{r=0}^{n-2}\binom{n-2}{r}\left(\sum_{j=0}^{r}\binom{r}{j}D^{r-j}\Bigl(\psi^{n-2}\left(D\psi\right)^{2}\Bigr)D^{j}\left(fg\right)\right)\Phi_{n-2-r,\psi}\left(1,1\right)
\\
&= n\left(n-1\right)\sum_{j=0}^{n-2}\left(\sum_{r=j}^{n-2}\binom{n-2}{r}\binom{r}{j}\Phi_{n-2-r,\psi}\left(1,1\right)D^{r-j}\Bigl(\psi^{n-2}\left(D\psi\right)^{2}\Bigr)\right)D^{j}\left(fg\right)
\\
&= n\left(n-1\right)\sum_{j=0}^{n-2}\binom{n-2}{j}\left(\sum_{q=0}^{n-2-j}\binom{n-2-j}{q}\Phi_{n-2-j-q,\psi}\left(1,1\right)D^{q}\Bigl(\psi^{n-2}\left(D\psi\right)^{2}\Bigr)\right)D^{j}\left(fg\right)
\\
&= n\left(n-1\right)\sum_{j=0}^{n-2}\binom{n-2}{j}\Phi_{n-2-j,\psi}\Bigl(1,\psi^{n-2}\left(D\psi\right)^{2}\Bigr)D^{j}\left(fg\right).\end{aligned}$$ [*Expression du troisième terme*]{}: $$\begin{aligned}
T_{3}\left(f,g\right)&= \sum_{k=1}^{n}\binom{n}{k}kD^{k-1}\left(\psi^{k-1}fD\psi\right)D^{n-k}\left(\psi^{n-k}g\right)
\\
&= n\sum_{k=1}^{n}\binom{n-1}{k-1}D^{k-1}\left(\psi^{k-1}fD\psi\right)D^{n-k}\left(\psi^{n-k}g\right)
\\
&= n\sum_{j=0}^{n-1}\binom{n-1}{j}D^{j}\left(\psi^{j}fD\psi\right)D^{n-1-j}\left(\psi^{n-1-j}g\right)=n\Phi_{n-1,\psi}\left(fD\psi,\psi^{n-1}g\right)
\\
&= n\sum_{r=0}^{n-1}\binom{n-1}{r}\Phi_{n-1-r,\psi}\left(1,1\right)D^{r}\left(\psi^{n-1}fgD\psi\right)
\\
&= n\sum_{r=0}^{n-1}\binom{n-1}{r}\left(\sum_{j=0}^{r}\binom{r}{j}D^{r-j}\left(\psi^{n-1}D\psi\right)D^{j}\left(fg\right)\right)\Phi_{n-1-r,\psi}\left(1,1\right)
\\
&= n\sum_{j=0}^{n-1}\left(\sum_{r=j}^{n-1}\binom{n-1}{r}\binom{r}{j}\Phi_{n-1-r,\psi}\left(1,1\right)D^{r-j}\left(\psi^{n-1}D\psi\right)\right)D^{j}\left(fg\right)
\\
&= n\sum_{j=0}^{n-1}\binom{n-1}{j}\left(\sum_{q=0}^{n-1-j}\binom{n-1-j}{q}\Phi_{n-1-j-q,\psi}\left(1,1\right)D^{q}\left(\psi^{n-1}D\psi\right)\right)D^{j}\left(fg\right)
\\
&= n\sum_{j=0}^{n-1}\binom{n-1}{j}\Phi_{n-1-j,\psi}\left(1,\psi^{n-1}D\psi\right)D^{j}\left(fg\right).\end{aligned}$$
[*Expression du quatrième terme*]{}: $$T_{4}\left(f,g\right)=\sum_{k=0}^{n-1}\binom{n}{k}\left(n-k\right)D^{k}\left(\psi^{k}f\right)D^{n-k-1}\left(\psi^{n-k-1}gD\psi\right)$$ $$=\sum_{p=1}^{n}\binom{n}{p}pD^{p-1}\left(\psi^{p-1}gD\psi\right)D^{n-p}\left(\psi^{n-p}f\right)=\Sigma_{3}\left(g,f\right)=\Sigma_{3}\left(f,g\right).$$ En conséquence le membre de droite de la formule de Lagrange s’écrit comme suit: $$\sum_{j=0}^{n}C_{n}\left(j,\psi\right)D^{j}\left(fg\right)$$ où les fonctions coefficients $C_{n}\left(j,\psi\right)$ sont données par: $$C_{n}\left(n,\psi\right)=\Phi_{0,\psi}\left(1,\psi^{n}\right)$$ $$C_{n}\left(n-1,\psi\right)=n\Phi_{1,\psi}\left(1,\psi^{n}\right)-2n\Phi_{0,\psi}\left(1,\psi^{n-1}D\psi\right)$$ et pour tout entier naturel $j\leq n-2$: $$C_{n}\left(j,\psi\right)=\dbinom{n}{j}\Phi_{n-j,\psi}\left(1,\psi^{n}\right)+n\left(n-1\right)\dbinom{n-2}{j}\Phi_{n-2-j,\psi}\Bigl(1,\psi^{n-2}\left(D\psi\right)^{2}\Bigr)$$ $$-2n\dbinom{n-1}{j}\Phi_{n-1-j,\psi}\left(1,\psi^{n-1}D\psi\right).$$ Comme on a en général $\displaystyle \Phi_{0,\psi}\left(F,G\right)=FG$ et $\displaystyle \Phi_{1,\psi}\left(F,G\right)=D\left(FG\right)+\dfrac{D\psi}{\psi}FG$, il vient: $$C_{n}\left(n,\psi\right)=\psi^{n}$$ $$C_{n}\left(n-1,\psi\right)=n\left(n-1\right)\psi^{n-1}D\psi=\left(n-1\right)D\left(\psi^{n}\right)$$ et pour tout entier naturel $j\leq n-2$, en vertu du lemme précédent: $$C_{n}\left(j,\psi\right)=\dbinom{n}{j}\Phi_{n-j,\psi}\left(1,\psi^{n}\right)+n\left(n-1\right)\dbinom{n-2}{j}\dfrac{1}{\left(n-j-1\right)\left(n-j\right)}\Phi_{n-j,\psi}\left(1,\psi^{n}\right)$$ $$-n\left(n-1\right)\dbinom{n-2}{j}\dfrac{1}{\left(n-j-1\right)\left(n-j\right)}D^{n-j}\left(\psi^{n}\right)$$ $$-n\left(n-1\right)\dbinom{n-2}{j}\dfrac{1}{n-j-1}D^{n-j-1}\left(\psi^{n-1}D\psi\right)$$ $$-2n\dbinom{n-1}{j}\dfrac{1}{n-j}\Phi_{n-j,\psi}\left(1,\psi^{n}\right)+2n\dbinom{n-1}{j}\dfrac{1}{n-j}D^{n-j}\left(\psi^{n}\right)$$ $$=\left\{ \begin{array}{cc}
0 & \quad si\quad j=0\\
\\
\dbinom{n-1}{j-1}D^{n-j}\left(\psi^{n}\right) & \quad si\quad1\leq j\leq n-2
\end{array}\right.$$ En conséquence, on aboutit à: $$\sum_{k=0}^{n}\binom{n}{k}D^{k-1}\Bigl(\psi^{k}D\left(f\right)\Bigr)D^{n-k-1}\left(\psi^{n-k}D\left(g\right)\right)=\sum_{j=0}^{n}C_{n}\left(j,\psi\right)D^{j}\left(fg\right).$$ $$=\sum_{j=1}^{n}\dbinom{n-1}{j-1}D^{n-j}\left(\psi^{n}\right)D^{j}\left(fg\right)=\sum_{k=0}^{n-1}\dbinom{n-1}{k}D^{n-1-k}\left(\psi^{n}\right)D^{k+1}\left(fg\right)=D^{n-1}\left(\psi^{n}D\left(fg\right)\right)$$
\[t10\] Pour tout couple $\left(\psi,f\right)$ de fonctions complexes de classe $\mathcal{C}^{n}$ sur un intervalle $J$, et pour tout polynôme $P\in\mathbb{C}\left[X\right]$ on a la formule: $$\label{eq11}
P\left(\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(\psi^{n}Df\right)X^{n}\right)=\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(\psi^{n}D\left(P\left(f\right)\right)\right)X^{n}.$$
Une autre formule de Frobenius-Stickelberger {#une-autre-formule-de-frobenius-stickelberger .unnumbered}
--------------------------------------------
Dans ce paragraphe la fonction complexe $\psi$ est de classe $\mathcal{C}^{\infty}$ sur un segment $J$. Frobenius et Stickelberger [@F1] ont avancé une autre formule qu’il est possible d’établir à l’aide de la formule du produit de Lagrange, en remplaçant les fonctions $f$ et $g$ par la fonction $\psi^{-1}$ si la fonction $\psi$ ne s’annule pas, et plus généralement on peut remplacer les fonctions $f$ et $g$ par la fonction $\psi^{-p}$ ou $\psi^{p}$ où $p$ désigne un entier naturel non nul.
Supposant que la fonction $\psi$ ne s’annule pas, pour tout entier naturel $m$ et tout entier rationnel $q$ on définit la fonction de classe $\mathcal{C}^{\infty}$ sur le segment $J$: $$F_{m}\left(\psi,q\right)=D^{m-1}\left(\psi^{m}D\left(\psi^{q}\right)\right)$$ Bien entendu $F_{0}\left(\psi,q\right)=\psi^{q}$.
Il est immédiat de constater, par récurrence sur l’ordre $q$ de dérivation, la formule: $$\label{eq12}
q\, D^{p}F_{m}\left(\psi,q+p\right)=\left(q+p\right)F_{m+q}\left(\psi,q\right).$$ Tenant compte de la formule du produit de Lagrange et de la formule de dérivation ci-dessus, pour tout couple $\left(n,p\right)$ d’entiers naturels vérifiant $1\leq p\leq n$ on obtient: $$2D^{p}F_{n-p}\left(\psi,-p\right)=\sum_{k=0}^{n}\binom{n}{k}F_{k}\left(\psi,-p\right)F_{n-k}\left(\psi,-p\right)\label{eq:5}$$ formule plus générale que celle de Frobenius-Stickelberger, écrite par ces auteurs dans le cas particulier $p=1$. Pour tout couple $\left(n,p\right)$ d’entiers non nuls on peut disposer aussi de la formule : $$2F_{n+p}\left(\psi,p\right)=D^{p}\left(\,\sum_{k=0}^{n}\binom{n}{k}F_{k}\left(\psi,p\right)F_{n-k}\left(\psi,p\right)\right)\label{eq:6-1}.$$
Sur un théorème de P. J. Olver
==============================
Notations {#notations .unnumbered}
----------
On désigne par $\mathcal{A}\left(J\right)$ la $\BC$-algèbre des fonctions réelle-analytiques sur $J$, à valeurs complexes. Le *théorème de Pringsheim* stipule qu’une fonction complexe $f\in \mathcal{C}^{\infty}\left(J\right)$ est réelle-analytique si et seulement s’ il existe $R>0$ tel qu’on ait $${\displaystyle \sup_{p\in\mathbb{N}}\dfrac{\left\Vert D^{p}g\right\Vert }{p!\, R^{p}}<+\infty}.$$ Un tel nombre sera appelé niveau de la fonction $g$.
Pour tout nombre réel $R>0$, on introduit le sous-espace vectoriel de l’algèbre $\mathcal{A}\left(J\right)$: $$\mathcal{A}_{R}\left(J\right)=\left\{ g\in\mathcal{A}\left(J\right)\;;\;\sup_{p\in\mathbb{N}}\dfrac{\left\Vert D^{p}g\right\Vert }{p!\, R^{p}}<+\infty\right\}$$ qu’on munit de la norme ${\displaystyle N_{R}\left(g\right)=\sup_{p\in\mathbb{N}}\dfrac{\left\Vert D^{p}g\right\Vert }{p!\, R^{p}}}$. L’espace normé $\mathcal{A}_{R}\left(J\right)$ est complet. Pour tous nombres réels $R$ et $S$ vérifiant $0<R<S$, on a l’inclusion $\mathcal{A}_{R}\left(J\right)\subset\mathcal{A}_{S}\left(J\right)$ et l’injection canonique est continue: pour toute fonction $g$ appartenant à l’espace normé $\mathcal{A}_{R}\left(J\right)$ on a l’inégalité $N_{S}\left(g\right)\leq N_{R}\left(g\right)$. L’algèbre $\mathcal{A}\left(J\right)$ est munie de la structure *limite-inductive de la suite croissante des sous-espaces de Banach $\left(\mathcal{A}_{k}\left(J\right)\right)_{k\geq1}$*.
Étant donnés une suite $\left(a_{n}\right)_{n\in\mathbb{N}}$ d’éléments de l’algèbre $\mathcal{A}\left(J\right)$ et $\zeta\in\mathbb{C}^{*}$, la série ${\displaystyle \sum_{n=0}^{\infty}\zeta^{n}a_{n}}$ est convergente dans l’algèbre $\mathcal{A}\left(J\right)$ si, par définition, elle l’est dans un espace de Banach $\mathcal{A}_{R}\left(J\right)$, ce qui suppose en particulier l’appartenance des sommes partielles à l’espace de Banach $\mathcal{A}_{R}\left(J\right)$ ; ainsi toutes les fonctions coefficients $a_{n}$ ont un même niveau $R$, car le nombre complexe $\zeta$ est distinct de $0$. La suite de terme général $\zeta^{n}a_{n}$ est donc bornée dans l’espace de Banach $\mathcal{A}_{R}\left(J\right)$, de sorte que l’ensemble $$\bigl\{ r>0\;;\;\mathfrak{A}\left(r\right)=\sup_{n}N_{R}\left(a_{n}\right)r^{n}<+\infty\bigr\}$$ n’est pas vide puisqu’il contient $\left|\zeta\right|$, et est un intervalle dont la borne supérieure $\rho$ est strictement positive, éventuellement infinie. Ce nombre $\rho$ est le *rayon de convergence de la série entière* ${\displaystyle \sum_{n=0}^{\infty}z^{n}a_{n}}$: pour tout nombre complexe $z$ vérifiant $\left|z\right|<\rho$, la série ${\displaystyle \sum_{n=0}^{\infty}z^{n}a_{n}}$ est absolument convergente dans l’espace de Banach $\mathcal{A}_{R}\left(J\right)$, de plus on a la formule de Hadamard: $$\rho=\Bigl(\limsup_{n}\left(\sqrt[n]{N_{R}\left(a_{n}\right)}\right)\Bigr)^{-1}>0.$$ Ainsi on dispose de l’application somme de série entière à valeurs dans $\mathcal{A}\left(J\right)$: $$\overset{\circ}{D}\left(0,\rho\right)\overset{f}{\longrightarrow}\mathcal{A}\left(J\right),\qquad f\left(z\right)={\displaystyle \sum_{n=0}^{\infty}z^{n}a_{n}}$$ qui satisfait, pour tout $z\in \BC,\;\left|z\right|<\rho$, à l’inégalité $$N_{R}\left(f\left(z\right)\right)\leq\sum_{n=0}^{\infty}N_{R}\left(a_{n}\right)\left|z\right|^{n}$$ et la série de fonctions ${\displaystyle \sum_{n=0}^{\infty}z^{n}a_{n}}$ est normalement convergente sur le segment $J$ et ainsi pour tout point $t$ du segment $J$ on a $f\left(z\right)\left(t\right)={\displaystyle \sum_{n=0}^{\infty}a_{n}\left(t\right)z^{n}}$. La condition d’appartenance des coefficients $a_{n}$ à l’un des sous-espaces de Banach $\mathcal{A}_{R}\left(J\right)$ et l’inégalité stricte $\rho>0$ sont suffisantes pour assurer que pour tout $z$ de module strictement inférieur à $\rho$, la somme de la série de fonctions ${\displaystyle \sum_{n=0}^{\infty}a_{n}\left(t\right)z^{n}}$ de la variable réelle $t$, est analytique sur le segment $J$. On peut observer que la convergence normale sur le segment $J$ de cette série et de toutes ses séries dérivées, ne conduit pas à l’analyticité de la somme, comme le montre l’exemple ${\displaystyle \sum_{n=0}^{\infty}\dfrac{1}{1+nt}z^{n}}$ sur le segment $\bigl[0,1\bigr]$.
On va énoncer à présent quelques lemmes qui nous seront utiles par la suite
\[t3\] Soit une suite $\left(f_{n}\right)_{n\in\mathbb{N}}$ d’éléments de l’algèbre $\mathcal{A}\left(J\right)$. On suppose qu’il existe trois nombres strictement positifs $R$, $A$ et $S$, un nombre entier $q\geq1$, tels pour tout entier naturel $n\geq1$ on ait $N_{R}\left(f_{n}\right)\leq A\, n!\, S^{n}$.
Alors la série entière ${\displaystyle \sum_{n=0}^{\infty}\dfrac{1}{n!}\, z^{n}f_{n}}$ définit l’application $f$ somme de série entière à valeurs dans **** $\mathcal{A}\left(J\right)$, précisément dans $\mathcal{A}_{R}\left(J\right)$ $$\overset{\circ}{D}\left(0,S^{-1}\right)\overset{f}{\longrightarrow}\mathcal{A}\left(J\right),\qquad f\left(z\right)=\sum_{n=0}^{\infty}\dfrac{1}{n!}\, z^{n}f_{n}.$$
Pour tout entier naturel $n\in \BN^*$ on a $\displaystyle N_{R}\left(\dfrac{1}{n!}\, f_{n}\right)\leq S^{n}A$. De ce fait l’ensemble $\displaystyle \left\{ r>0\;;\;\sup_{n}N_{R}\left(\dfrac{1}{n!}f_{n}\right)r^{n}<+\infty\right\} $ contient l’intervalle $\bigl]0,S^{-1}\bigr[$ et est contenu dans l’intervalle $\bigl]0,S^{-1}\bigr]$. Observons qu’on a l’inégalité $\displaystyle N_{R}\left(f\left(z\right)\right)\leq\dfrac{A}{1-S\left|z\right|}$.
\[t4\] Pour tout entier naturel $p\geq1$, pour tout $p$-uple $\left(g_{1},\ldots,g_{p}\right)$ de fonctions réelle-analytiques de niveau $R$ sur le segment $J$, on a l’inégalité: $$N_{2R}\left(g_{1}\cdots g_{p}\right)\leq2^{p-1}N_{R}\left(g_{1}\right)\cdots N_{R}\left(g_{p}\right).$$
En vertu de la formule de Leibniz, pour tout entier naturel $n$, on a: $$D^{n}\left(g_{1}\cdots g_{p}\right)=\sum_{\alpha\in E\left(n,p\right)}\dfrac{n!}{\alpha_{1}!\cdots\alpha_{p}!}D^{\alpha_{1}}g_{1}\cdots D^{\alpha_{p}}g_{p}.$$ Par suite $$\begin{aligned}
&\left\Vert D^{n}\left(g_{1}\cdots g_{p}\right)\right\Vert \leq n!\sum_{\alpha\in E\left(n,p\right)}\dfrac{1}{\alpha_{1}!}\left\Vert D^{\alpha_{1}}g_{1}\right\Vert \cdots\dfrac{1}{\alpha_{p}!}\left\Vert D^{\alpha_{p}}g_{p}\right\Vert
\\
&\leq n!\, N_{R}\left(g_{1}\right)\cdots N_{R}\left(g_{p}\right)\sum_{\alpha\in E\left(n,p\right)}R^{\alpha_{1}+\cdots+\alpha_{p}}= \dbinom{n+p-1}{p-1}\, n!\, R^{n}N_{R}\left(g_{1}\right)\cdots N_{R}\left(g_{p}\right)
\\
&\leq2^{n+p-1}n!\, R^{n}N_{R}\left(g_{1}\right)\cdots N_{R}\left(g_{p}\right)= n!\,\left(2R\right)^{n}2^{p-1}N_{R}\left(g_{1}\right)\cdots N_{R}\left(g_{p}\right).\end{aligned}$$
\[t5\] Pour toute fonction réelle-analytique $g$ de niveau $S$ sur le segment $J$, et pour tout entier naturel $q$ on a l’inégalité: $$N_{2S}\left(D^{q}g\right)\leq N_{S}\left(g\right)q!\left(2S\right)^{q}.$$
Comme on a l’inégalité $\left(p+q\right)!\leq p!\, q!\,2^{p+q}$, on a successivement: $$\bigl\Vert D^{p}\left(D^{q}g\right)\bigr\Vert=\bigl\Vert D^{p+q}g\bigr\Vert\leq N_{S}\left(g\right)\left(p+q\right)!\, S^{p+q}\leq\left(N_{S}\left(g\right)q!\left(2S\right)^{q}\right)p!\left(2S\right)^{p}.$$
Soient un couple $\left(u,g\right)$ de fonctions réelle-analytiques sur $J$, $u$ étant de niveau $R$ et $g$ de niveau $S$. On définit le nombre $T=2\,\max\left(R,2\, S\right)$. Alors la suite $\left(D^{n-1}\left(u^{n}Dg\right)\right)_{n\geq1}$ d’éléments de l’algèbre $\mathcal{A}\left(J\right)$ satisfait l’hypothèse du lemme , à savoir pour tout entier naturel $n\geq1$: $$N_{2T}\left(D^{n-1}\left(u^{n}Dg\right)\right)\leq\dfrac{S}{T}N_{S}\left(g\right)n!\left(4N_{R}\left(u\right)T\right)^{n}\leq\dfrac{1}{4}N_{S}\left(g\right)n!\left(4N_{R}\left(u\right)T\right)^{n}.$$
D’après le lemme la fonction réelle-analytique $Dg$ appartient à $\mathcal{A}_{2S}\left(J\right)$, précisément $N_{2S}\left(Dg\right)\leq2SN_{S}\left(g\right)$. Ensuite, d’après le lemme , on a: $$N_{T}\left(u^{n}Dg\right)\leq2^{n}N_{R}\left(u\right)^{n}N_{2S}\left(Dg\right)\leq2^{n+1}S\, N_{R}\left(u\right)^{n}N_{S}\left(g\right).$$ Puis à nouveau en vertu du lemme , pour tout entier naturel $n\geq1$ on conclut: $$N_{2T}\left(D^{n-1}\left(u^{n}Dg\right)\right)\leq N_{T}\left(u^{n}Dg\right)\left(n-1\right)!\left(2\, T\right)^{n-1}\leq\dfrac{S}{T}N_{S}\left(g\right)n!\left(4N_{R}\left(u\right)T\right)^{n}.$$
\[t6\] On dispose donc de l’application somme de série entière à valeurs dans $\mathcal{A}\left(J\right)$, précisément dans $\mathcal{A}_{2T}\left(J\right)$: $$\overset{\circ}{D}\left(0,\left(4N_{R}\left(u\right)T\right)^{-1}\right)\longrightarrow\mathcal{A}\left(J\right),\qquad z\longmapsto\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right).$$
Rappelons qu’une fonction somme de série entière à coefficients complexes ${\displaystyle f\left(z\right)=\sum_{n=0}^{\infty}c_{n}z^{n}}$, de rayon de convergence strictement positif $\rho$, est telle que pour tout nombre complexe $z$ vérifiant $\left|z\right|<\rho$, pour tout nombre réel $r$ appartenant à l’intervalle $\bigl[\left|z\right|,\rho\bigr[$ on a l’inégalité: $$\sum_{k=0}^{\infty}\dfrac{1}{k!}\left|f^{\left(k\right)}\left(z\right)\right|\left(r-\left|z\right|\right)^{k}\leq\sum_{n=0}^{\infty}\left|c_{n}\right|r^{n}=f^{*}\left(r\right).$$
Soient une fonction $g$ réelle-analytique de niveau $S$ sur le segment $J$, un point $t$ de $J$ et une série entière à coefficients complexes ${\displaystyle \psi\left(w\right)=\sum_{n=0}^{\infty}c_{n}w^{n}}$, de rayon de convergence $\rho >0$. La fonction somme de série entière $$\zeta\longmapsto\varphi_{\, t}\left(\zeta\right)=\psi\left(\zeta-g\left(t\right)\right)$$ est définie dans le disque ouvert $\overset{\circ}{D}\left(g\left(t\right),\rho\right)$.
Pour tout segment $J_{t}$ contenant le point $t$, contenu dans le segment $J$ et tel que l’image $g\left(J_{t}\right)$ soit contenue dans le disque ouvert $\overset{\circ}{D}\left(g\left(t\right),\rho\right)$, la fonction composée $\varphi_{\, t}\circ g$ est réelle-analytique de niveau $S_{J_{t}}\left(r\right)$ sur le segment $J_{t}$ où: $$S_{J_{t}}\left(r\right)=\left(1+\dfrac{N_{S}\left(g\right)}{r-\left\Vert g-g\left(t\right)\right\Vert _{J_{t}}}\right)S$$ et de plus $$\quad\left\Vert g-g\left(t\right)\right\Vert _{J_{t}}<r<\rho$$
En vertu de la formule de Faà di Bruno exprimée avec les polynômes de Bell [@F2] (ainsi que les références qui s’y trouvent), pour tout entier naturel $p\geq1$, on peut écrire: $$D^{p}\left(\varphi_{\, t}\circ g\right)=\sum_{i=1}^{p}\left(\varphi_{\, t}^{\left(i\right)}\circ g\right)B_{p,i}\left(Dg,\ldots,D^{p-i+1}g\right).$$ Compte-tenu de l’expression des polynômes de Bell: $$B_{p,i}=\dfrac{1}{i!}\sum_{\alpha\in F\left(p,i\right)}\dfrac{p!}{\alpha_{1}!\cdots\alpha_{i}!}X_{\alpha_{1}}\cdots X_{\alpha_{i}}$$ où $F\left(p,i\right)$ est l’ensemble $$\bigl\{\alpha=\left(\alpha_{1},\ldots,\alpha_{i}\right)\in\left(\mathbb{N}^{*}\right)^{i}\;;\;\alpha_{1}+\cdots+\alpha_{i}=p\bigr\}$$ dont le cardinal est ${\displaystyle \binom{p-1}{i-1}}$. On obtient: $$\left\Vert B_{p,i}\left(Dg,\ldots,D^{p-i+1}g\right)\right\Vert _{J}\leq\binom{p-1}{i-1}\dfrac{1}{i!}\, N_{S}\left(g\right)^{i}\, p!\, S^{p}.$$ Pour tout $s$ appartenant à $J_{t}$ on tire l’inégalité: $$\bigl|D^{p}\left(\varphi_{\, t}\circ g\right)\left(s\right)\bigr|
\leq p!\, S^{p}\sum_{j=0}^{p-1}\binom{p-1}{j}\dfrac{1}{\left(j+1\right)!}\bigl|\varphi_{\, t}^{\left(j+1\right)}\left(g\left(s\right)\right)\bigr|N_{S}\left(g\right)^{j+1}.$$ Or, pour tout $s$ appartenant à $J_{t}$ et pour tout nombre $r$ vérifiant $\left|g\left(s\right)-g\left(t\right)\right|<r<\rho$, on dispose des inégalités: $$\dfrac{1}{\left(j+1\right)!}\bigl|\varphi_{\, t}^{\left(j+1\right)}\left(g\left(s\right)\right)\bigr|=\dfrac{1}{\left(j+1\right)!}\bigl|\psi^{\left(j+1\right)}\left(g\left(s\right)-g\left(t\right)\right)\bigr|\leq\dfrac{\psi^{*}\left(r\right)}{\left(r-\left|g\left(s\right)-g\left(t\right)\right|\right)^{j+1}}$$ desquelles on déduit: $$\bigl|D^{p}\left(\varphi_{\, t}\circ g\right)\left(s\right)\bigr|\leq\psi^{*}\left(r\right)p!\, S^{p}\sum_{j=0}^{p-1}\binom{p-1}{j}\dfrac{N_{S}\left(g\right)^{j+1}}{\left(r-\left|g\left(s\right)-g\left(t\right)\right|\right)^{j+1}}$$ ou $$\label{eq13}
\bigl|D^{p}\left(\varphi_{\, t}\circ g\right)\left(s\right)\bigr|
\leq\psi^{*}\left(r\right)\, p!\, S^{p}\left(1+\dfrac{N_{S}\left(g\right)}{r-\left|g\left(s\right)-g\left(t\right)\right|}\right)^{p}.$$ La conclusion s’ensuit immédiatement.
Formulation du problème
------------------------
Soient un couple $\left(u,g\right)$ de fonctions réelle-analytiques sur $J$, $u$ étant de niveau $R$ et $g$ de niveau $S$, un point $t$ de $J$ et une série entière à coefficients complexes ${\displaystyle \psi\left(w\right)=\sum_{n=0}^{\infty}c_{n}w^{n}}$, de rayon de convergence $\rho >0$, à laquelle est associée la fonction somme de série entière, $\zeta\longmapsto\varphi_{\, t}\left(\zeta\right)=\psi\left(\zeta-g\left(t\right)\right)$, définie dans le disque ouvert $\overset{\circ}{D}\left(g\left(t\right),\rho\right)$.
D’après la remarque , notant $T_{J_{t}}=2\max\left(R,2S_{J_{t}}\left(\rho\right)\right)$, comme on a $T_{J_{t}}\geq T$, on obtient les applications somme de série entière à valeurs dans $\mathcal{A}\left(J_{t}\right)$ $$\begin{aligned}
\overset{\circ}{D}\left(0,\left(4N_{R}\left(u\right)T_{J_{t}}\right)^{-1}\right)\longrightarrow\mathcal{A}\left(J_{t}\right),& \qquad z\longmapsto\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)
\\
\overset{\circ}{D}\left(0,\left(4N_{R}\left(u\right)T_{J_{t}}\right)^{-1}\right)\longrightarrow\mathcal{A}\left(J_{t}\right),& \qquad z\longmapsto\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}
\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right).\end{aligned}$$ La formule suggère la question: pour quels nombres complexes $z$ de module éventuellement plus petit que $\left(4N_{R}\left(u\right)T_{J_{t}}\right)^{-1}$, peut-on, pour tout $s$ du segment $J_{t}$, considérer la composée $\varphi_{\, t}\left({\displaystyle \sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(s\right)}\right)$ et avoir l’égalité$ $ $$\varphi_{\, t}\left(\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(s\right)\right)=\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right)\left(s\right).$$
Étude et résolution du problème
-------------------------------
*Afin de simplifier les énoncés ultérieurs on s’attache dorénavant au cas particulier du segment $J_{t}=\left\{ t\right\} $, sans que cela modifie la substance de l’étude.* De ce fait, notons qu’on a : $S_{\left\{ t\right\} }\left(\rho\right)=\left(1+\dfrac{N_{S}\left(g\right)}{\rho}\right)S=S_{0}$, on écrit $T_{\left\{ t\right\} }=T_{0}=2\max\left(R,2S_{0}\right)$. On remarque que si le nombre complexe $z$ satisfait l’inégalité $\left|z\right|<\dfrac{1}{4N_{R}\left(u\right)T_{0}}$ alors $$\sum_{n= 0}^{\infty}\frac{|z|^n}{n!} |D^{n-1}\left( u^n Dg \right)(t)|< \infty$$ et $$\sum_{n=0}^{\infty}\dfrac{\left|z\right|^{n}}{n!}\bigl|\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right)\left(t\right)\bigr|<+\infty.$$ On suppose donc que le nombre complexe $z$ satisfait à cette l’inégalité. Devant disposer de l’expression $\varphi_{\, t}\left({\displaystyle \sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(s\right)}\right)$ on souhaite l’inégalité $$\left|{\displaystyle \sum_{n=1}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)}\right|<\rho.$$ Comme on a $${\displaystyle \sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)=g\left(t\right)+\sum_{n=1}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)}$$ l’inégalité souhaitée est impliquée par l’inégalité ${\displaystyle \sum_{n=1}^{\infty}\dfrac{\left|z\right|^{n}}{n!}\bigl|D^{n-1}\left(u^{n}Dg\right)\left(t\right)\bigr|<\rho}$. Grâce à la formule de Leibniz on a
$$\dfrac{1}{n!}\bigl|D^{n-1}\left(u^{n}Dg\right)\left(t\right)\bigr|\leq\dfrac{1}{n}\sum_{\alpha\in E\left(n-1,n+1\right)}\dfrac{1}{\alpha_{1}!}\bigl|D^{\alpha_{1}}u\left(t\right)\bigr|\cdots\dfrac{1}{\alpha_{n}!}\bigl|D^{\alpha_{n}}u\left(t\right)\bigr|\dfrac{1}{\alpha_{n+1}!}\bigl|D^{1+\alpha_{n+1}}g\left(t\right)\bigr|$$ $$\leq N_{R}\left(u\right)^{n}N_{S}\left(g\right)R^{n}\sum_{\alpha\in E\left(n-1,n+1\right)}\left(\dfrac{S}{R}\right)^{1+\alpha_{n+1}}.$$ Mais en général pour tout nombre réel $x$ strictement positif on a $$\sum_{\alpha\in E\left(n-1,n+1\right)}x^{1+\alpha_{n+1}}=x\sum_{j=0}^{n-1}\dbinom{2n-2-j}{n-1}\, x^{j}\leq x\sum_{j=0}^{n-1}2^{2n-2-j}x^{j}=2^{2n-1}\sum_{j=1}^{n}\left(\dfrac{x}{2}\right)^{j}$$ Il vient alors $$\dfrac{1}{n!}\bigl|D^{n-1}\left(u^{n}Dg\right)\left(t\right)\bigr|\leq\dfrac{1}{2}\left(4N_{R}\left(u\right)R\right)^{n}N_{S}\left(g\right)\sum_{j=1}^{n}\left(\dfrac{S}{2R}\right)^{j}.$$ Par suite $$\sum_{n=0}^{\infty}\dfrac{\left|z\right|^{n}}{n!}\bigl|D^{n-1}\left(u^{n}Dg\right)\left(t\right)\bigr|\leq
\dfrac{N_{S}\left(g\right)}{2}\left(\sum_{j=1}^{\infty}\left(2SN_{R}\left(u\right)\left|z\right|\right)^{j}\right)\left(\sum_{m=0}^{\infty}\left(4RN_{R}\left(u\right)\left|z\right|\right)^{m}\right).$$ L’hypothèse faite sur le nombre $z$ conduit aux deux inégalités : $$2SN_{R}\left(u\right)\left|z\right|<\dfrac{1}{8}\left(1+\dfrac{N_{S}\left(g\right)}{\rho}\right)^{-1},\qquad 4RN_{R}\left(u\right)\left|z\right|<\dfrac{1}{2}.$$
De là on tire: $$\sum_{n=0}^{\infty}\dfrac{\left|z\right|^{n}}{n!}\bigl|D^{n-1}\left(u^{n}Dg\right)\left(t\right)\bigr|<\dfrac{N_{S}\left(g\right)}{8}\left(1+\dfrac{N_{S}\left(g\right)}{\rho}\right)^{-1}<\dfrac{\rho}{8}<\rho.$$ En conclusion, pour tout nombre complexe $z$ satisfaisant l’inégalité $\left|z\right|<\dfrac{1}{4N_{R}\left(u\right)T_{0}}$, se pose alors la question de l’égalité: $$\varphi_{\, t}\left(\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)\right)=\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right)\left(t\right).$$ Désignons par $P_{m}$ la fonction polynôme somme partielle d’ordre $m$ de la série entière de somme $\varphi_{\, t}$, à savoir $$P_{m}\left(\zeta\right)=\sum_{k=0}^{m}c_{k}\left(\zeta-g\left(t\right)\right)^{k}.$$ En vertu de l’égalité , On peut écrire: $$\varphi_{\, t}\left(\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)\right)=\lim_{M}\, P_{M}\left(\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)\right)$$ $$=\lim_{M}\,\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(P_{M}\circ g\right)\right)\left(t\right)=
\lim_{M}\left(\lim_{N}\,\sum_{n=0}^{N}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(P_{M}\circ g\right)\right)\left(t\right)\right).$$ Le lemme suivant montre qu’on peut intervertir les limites.
\[t7\] Dans les conditions ci-dessus énoncées, uniformément par rapport au couple $\left(N,z\right)$ appartenant à $\mathbb{N}\times\Delta$, où $\Delta$ désigne le disque ouvert centré en $z=0$ et de rayon $\dfrac{1}{4N_{R}\left(u\right)T_{0}}$, on a: $$\sum_{n=0}^{N}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right)\left(t\right)=\lim_{M}\,\sum_{n=0}^{N}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(P_{M}\circ g\right)\right)\left(t\right).$$
Comme $\left(\varphi_{\, t}-P_{M}\right)\left(g\left(t\right)\right)=0$, il s’agit de démontrer qu’on a: $$\lim_{M}\left(\sup_{N\geq1\: et\: z\in\Delta}\left|\sum_{n=1}^{N}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(\left(\varphi_{\, t}-P_{M}\right)\circ g\right)\right)\left(t\right)\right|\right)=0.$$ Tout d’abord on définit les fonctions $\psi_{M}$ et $\phi_{M}$: $${\displaystyle \psi_{M}\left(w\right)=\sum_{m=M+1}^{\infty}c_{m}w^{m}}\qquad\phi_{M}\left(\zeta\right)=\psi_{M}\left(\zeta-g\left(t\right)\right)=\left(\varphi_{\, t}-P_{M}\right)\left(\zeta\right).$$ Pour tout $r$ vérifiant $0<r<\rho$, on a: $$\dfrac{1}{n!}\bigl|D^{n-1}\left(u^{n}D\left(\phi_{M}\circ g\right)\right)\left(t\right)\bigr|$$ $$\leq\dfrac{1}{n}\sum_{\alpha\in E\left(n-1,n+1\right)}\dfrac{1}{\alpha_{1}!}\bigl|D^{\alpha_{1}}u\left(t\right)\bigr|\cdots\dfrac{1}{\alpha_{n}!}
\bigl|D^{\alpha_{n}}u\left(t\right)\bigr|\dfrac{1}{\alpha_{n+1}!}\bigl|D^{1+\alpha_{n+1}}\left(\phi_{M}\circ g\right)\left(t\right)\bigr|.$$ Or, la fonction $\phi_{M}$ étant à $\psi_{M}$ ce que la fonction $\varphi_{\, t}$ est à $\psi$, en vertu de l’inégalité , pour tout entier naturel $p$ on a: $$\bigl|D^{p}\left(\phi_{M}\circ g\right)\left(t\right)\bigr|\leq\psi_{M}^{\;*}\left(r\right)p!\, S^{p}\left(1+\dfrac{N_{S}\left(g\right)}{r}\right)^{p}.$$ Donc: $$\dfrac{1}{n!}\bigl|D^{n-1}\left(u^{n}D\left(\phi_{M}\circ g\right)\right)\left(t\right)\bigr|\leq\psi_{M}^{\;*}
\left(r\right)N_{R}\left(u\right)^{n}R^{n}\sum_{\alpha\in E\left(n-1,n+1\right)}\left(\dfrac{S}{R}\left(1+\dfrac{N_{S}\left(g\right)}{r}\right)\right)^{1+\alpha_{n+1}}.$$ Or, pour tout nombre réel $x$ strictement positif: $$\sum_{\alpha\in E\left(n-1,n+1\right)}x^{1+\alpha_{n+1}} \leq 2^{2n-1}\sum_{j=1}^{n}\left(\dfrac{x}{2}\right)^{j}.$$ Il vient alors: $$\dfrac{1}{n!}\bigl|D^{n-1}\left(u^{n}D\left(\phi_{M}\circ g\right)\right)\left(t\right)\bigr|\leq\dfrac{1}{2}\psi_{M}^{\;*}\left(r\right)\left(4N_{R}\left(u\right)R\right)^{n}
\sum_{j=1}^{n}\left(\dfrac{S}{2R}\left(1+\dfrac{N_{S}\left(g\right)}{r}\right)\right)^{j}$$ puis l’inégalité: $$\sum_{n=1}^{N}\dfrac{\left|z\right|^{n}}{n!}\bigl|D^{n-1}\left(u^{n}D\left(\phi_{M}\circ g\right)\right)\left(t\right)\bigr|\leq\dfrac{\psi_{M}^{\;*}\left(r\right)}{2\left(1-4\left|z\right|N_{R}
\left(u\right)R\right)}\sum_{j=1}^{N}\left(2\left|z\right|N_{R}\left(u\right)S\left(1+\dfrac{N_{S}\left(g\right)}{r}\right)\right)^{j}.$$ L’hypothèse sur le nombre complexe $z$ conduit aux deux inégalités: $$2\left|z\right|SN_{R}\left(u\right)<\dfrac{1}{8}\left(1+\dfrac{N_{S}\left(g\right)}{\rho}\right)^{-1}, \quad 4RN_{R}\left(u\right)\left|z\right|<\dfrac{1}{2}.$$ De là on tire: $$\sum_{n=0}^{\infty}\dfrac{\left|z\right|^{n}}{n!}\bigl|D^{n-1}\left(u^{n}D\left(\phi_{M}\circ g\right)\right)\left(t\right)\bigr|<\psi_{M}^{\;*}\left(r\right)\sum_{j=1}^{\infty}\dfrac{1}{8^{j}}
\left(1+\dfrac{N_{S}\left(g\right)}{r}\right)^{j}\left(1+\dfrac{N_{S}\left(g\right)}{\rho}\right)^{-j}.$$ Choisissant convenablement le nombre $r$, par exemple $\dfrac{N_{S}\left(g\right)\rho}{\rho+2N_{S}\left(g\right)}\leq r<\rho$, on obtient: $$\sup_{N\geq1\: et\: z\in\Delta}\,\sum_{n=1}^{N}\dfrac{\left|z\right|^{n}}{n!}\bigl|D^{n-1}\left(u^{n}D\left(\phi_{M}\circ g\right)\right)\left(t\right)\bigr|\leq\psi_{M}^{\;*}\left(r\right)
\sum_{j=1}^{\infty}\dfrac{1}{4^{j}}=\dfrac{1}{3}\sum_{m=M+1}^{\infty}\left|c_{m}\right|r^{m}.$$
En conséquence, les limites ayant été interverties, on aboutit à: $$\varphi_{\, t}\left(\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)\right)=\lim_{N}\left(\lim_{M}\,\sum_{n=0}^{N}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(P_{M}\circ g\right)\right)\left(t\right)\right)=$$ $$=\lim_{N}\,\sum_{n=0}^{N}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right)\left(t\right)=\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}D\left(\varphi_{\, t}\circ g\right)\right)\left(t\right)$$ de sorte qu’on énonce le
\[t8\] Soient un couple $\left(u,g\right)$ de fonctions réelle-analytiques sur le segment $J$, $u$ de niveau $R$, $g$ de niveau $S$, et une série entière à coefficients complexes ${\displaystyle \psi\left(w\right)=\sum_{n=0}^{\infty}c_{n}w^{n}}$, de rayon de convergence $\rho$ strictement positif. La fonction somme de série entière, $$\zeta\longmapsto\varphi_{\, t}\left(\zeta\right)=\psi\left(\zeta-g\left(t\right)\right)$$ est définie dans le disque ouvert $\overset{\circ}{D}\left(g\left(t\right),\rho\right)$. On note $$S_{0}=\left(1+\dfrac{N_{S}\left(g\right)}{\rho}\right)S,\quad
T_{0}=2\max\left(R,2S_{0}\right).$$ Pour tout point $t$ du segment $J$, et pour tout nombre complexe $z$ vérifiant $\left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T_{0}}$ le nombre complexe ${\displaystyle \sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)z^{n}}$ appartient au disque ouvert $\overset{\circ}{D}\left(g\left(t\right),\rho\right)$ et on a la relation de commutation $$\varphi_{\, t}\left(\,\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)z^{n}\right)=\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}D\left(\varphi_{\, t}
\circ g\right)\right)\left(t\right)z^{n}.\label{eq:1-4}$$
\[t9\] Soit un couple $\left(u,g\right)$ de fonctions réelle-analytiques sur le segment $J$, $u$ de niveau $R$, $g$ de niveau $S$, et on note $T=2\max\left(R,2S\right)$.Soit une série entière à coefficients complexes ${\displaystyle \phi\left(w\right)=\sum_{n=0}^{\infty}c_{n}w^{n}}$, de rayon de convergence infini.
Pour tout point $t$ du segment $J$, et pour tout nombre complexe $z$ vérifiant $\left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$ on a la relation: $$\phi\left(\,\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)z^{n}\right)=\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}D\left(\phi\circ g\right)\right)\left(t\right)z^{n}.\label{eq:2-2}$$
Dans le contexte présent le résultat de P. J. Olver [@F4] s’écrit brièvement comme suit. Pour tout point $t$ du segment $J$, et pour tout nombre complexe $z$ vérifiant $\left|z\right|<\dfrac{1}{16\, N_{R}\left(u\right)R}$ on a la relation $$\phi\left(\,\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}Du\right)\left(t\right)z^{n}\right)=\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(u^{n}D\left(\phi\circ u\right)\right)\left(t\right)z^{n}.$$
Soient $\left(u,g\right)$ un couple de fonctions réelle-analytiques sur le segment $J$, $u$ de niveau $R$, $g$ de niveau $S$. On peut désigner par $L_{u}\left(g\right)$ l’application somme de la série entière ****** à valeurs dans $\mathcal{A}\left(J\right)$ $$\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)$$ dont le rayon de convergence est, selon le corollaire , supérieur ou égal à $\dfrac{1}{4\, N_{R}\left(u\right)T}$. Dans la suite on préfère adopter l’écriture $L_{u}\left(g,z\right)$ au lieu de $L_{u}\left(g\right)\left(z\right)$. Le résultat qui suit n’est autre que la formule d’inversion de Lagrange dans le cadre des fonctions réelle-analytiques , obtenue donc à partir de la formule du produit de Lagrange, de l’extension du résultat de Olver, et sans passer par la formule intégrale de Cauchy. De plus, notons que la formule d’inversion de Lagrange dans le cadre des fonctions holomorphes peut se déduire assez facilement de la proposition qui suit.
1. Soit un couple $\left(u,g\right)$ de fonctions réelle-analytiques sur le segment $J$, $u$ de niveau $R$, $g$ de niveau $S$. On note $T=2\max\left(R,2S\right)$. Pour tout point $t$ du segment $J$ et pour tout nombre complexe $z$ vérifiant $\left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$, si le nombre complexe $t+z\, L_{u}\left(u,z\right)\left(t\right)$ appartient au segment $J$, alors on a: $$L_{u}\left(g,z\right)\left(t\right)=g\left(t+z\, L_{u}\left(u,z\right)\left(t\right)\right).$$
2. Pour tout point $t$ du segment $J$ et pour tout nombre réel $z$ vérifiant $\left|z\right|<\dfrac{1}{16\, N_{R}\left(u\right)R}$, si le nombre complexe $t+z\, L_{u}\left(u,z\right)\left(t\right)$ appartient au segment $J$, alors on a: $$L_{u}\left(u,z\right)\left(t\right)=u\left(t+z\, L_{u}\left(u,z\right)\left(t\right)\right).$$ Autrement dit, si le nombre $x=t+z\, L_{u}\left(u,z\right)\left(t\right)$ appartient au segment $J$, alors il satisfait l’équation $t+z\, u\left(x\right)=x$. La formule $$x=t+\sum_{n=1}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}\right)\left(t\right)$$ est la formule d’inversion de Lagrange pour l’équation $t+z\, u\left(x\right)=x$ et relativement aux fonctions réelle-analytiques. De plus, pour tout nombre réel $z$ vérifiant $\left|z\right|<\dfrac{1}{8\, N_{R}\left(u\right)T}$, l’expression de $g\left(x\right)$, où le nombre $x=t+z\, L_{u}\left(u,z\right)\left(t\right)$ est solution réelle de l’équation $t+z\, u\left(x\right)=x$, est: $$g\left(x\right)=g\left(t\right)+\sum_{n=1}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)$$ formule qui complète la formule d’inversion de Lagrange.
<!-- -->
1. Supposant le nombre complexe $z$ vérifier $\displaystyle \left|z\right|<\dfrac{1}{4\, TN_{R}\left(u\right)}$, on transforme l’expression de $L_{u}\left(g,z\right)$ ainsi: $$L_{u}\left(g,z\right)=g+\sum_{n=1}^{\infty}\left(\,\sum_{k=0}^{n-1}\dfrac{z^{n}}{n!}\binom{n-1}{k}\, D^{n-k-1}\left(u^{n}\right)D^{k+1}g\right).$$ Afin de justifier l’interversion dans $\mathcal{A}\left(J\right)$ des sommes ci-dessus, on effectue les majorations suivantes: $$\begin{aligned}
{}&\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\dfrac{\left|z\right|^{n}}{n!}\binom{n-1}{k}\, N_{2T}\left(D^{n-k-1}\left(u^{n}\right)\right)N_{2T}\left(D^{k+1}g\right)\\
&\leq\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\dfrac{\left|z\right|^{n}}{n}\left(k+1\right)\left(2T\right)^{n-k-1}N_{T}\left(u^{n}\right)\left(2T\right)^{k+1}N_{T}\left(g\right)\\
&\leq\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\dfrac{\left|z\right|^{n}}{n}\left(k+1\right)\left(2T\right)^{n}2^{n-1}\left(N_{\frac{T}{2}}\left(u\right)\right)^{n}N_{T}\left(g\right).\end{aligned}$$ Comme on a simultanément $T\geq2R$ et $T\geq4S>S$, on aboutit à: $$\begin{aligned}
&\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\dfrac{\left|z\right|^{n}}{n!}\binom{n-1}{k}\, N_{2T}\left(D^{n-k-1}\left(u^{n}\right)\right)N_{2T}\left(D^{k+1}g\right)
\\
&\leq\dfrac{N_{S}\left(g\right)}{2}\sum_{n=1}^{\infty}\dfrac{\left|z\right|^{n}}{n}\left(4\, T\, N_{R}\left(u\right)\right)^{n}\left(\,\sum_{k=0}^{n-1}\left(k+1\right)\right)\\&= \dfrac{N_{S}\left(g\right)}{4}\sum_{n=1}^{\infty}\left(n+1\right)\left|z\right|^{n}\left(4TN_{R}\left(u\right)\right)^{n}<+\infty.\end{aligned}$$ L’hypothèse $\left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$ permet donc l’interversion, et on obtient: $$\begin{aligned}
L_{u}\left(g,z\right)&= g+\sum_{k=0}^{\infty}\dfrac{1}{k!}D^{k+1}g\left(\,\sum_{n=k+1}^{\infty}\dfrac{z^{n}}{\left(n-k-1\right)!\, n}D^{n-k-1}\left(u^{n}\right)\right)
\\
&= g+\sum_{k=0}^{\infty}\dfrac{z^{k+1}}{k!}D^{k+1}g\left(\,\sum_{j=0}^{\infty}\dfrac{z^{n}}{j!}D^{j}\left(\dfrac{u^{j+k+1}}{j+k+1}\right)\right)
\\
&= g+\sum_{k=0}^{\infty}\dfrac{z^{k+1}}{k!}D^{k+1}g\left(\,\sum_{j=0}^{\infty}\dfrac{z^{n}}{j!}D^{j-1}\left(u^{j}D\left(\dfrac{u^{k+1}}{k+1}\right)\right)\right)
\\
&= g+\sum_{m=1}^{\infty}\dfrac{z^{m}}{m!}D^{m}g\left(\,\sum_{j=0}^{\infty}\dfrac{z^{n}}{j!}D^{j-1}\left(u^{j}D\left(u^{m}\right)\right)\right)=\sum_{m=0}^{\infty}L_{u}\left(u^{m},z\right)\dfrac{z^{m}}{m!}D^{m}g.\end{aligned}$$ En vertu du corollaire , comme on a $\left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}\leq\dfrac{1}{8\, N_{R}\left(u\right)R}$, pour tout entier naturel $m$ on obtient $L_{u}\left(u^{m},z\right)=\left(L_{u}\left(u,z\right)\right)^{m}$. De sorte que pour tout $z$ vérifiant $\left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$ et pour tout point $t$ du segment $J$: $$L_{u}\left(g,z\right)\left(t\right)=\sum_{m=0}^{\infty}\dfrac{1}{m!}\left(z\, L_{u}\left(u,z\right)\left(t\right)\right)^{m}D^{m}g\left(t\right).$$ Notant $\rho_{g}$ le rayon de convergence de la série entière ${\displaystyle \sum_{m=0}^{\infty}\dfrac{w^{m}}{m!}D^{m}g\left(t\right)}$. On constate donc que, pour tout $z$ vérifiant $\displaystyle \left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$ et pour tout point $t$ du segment $J$, on a $\displaystyle \left|z\, L_{u}\left(u,z\right)\left(t\right)\right|\leq\rho_{g}$. Or, la fonction $z\longmapsto z\, L_{u}\left(u,z\right)\left(t\right)$, qui est somme d’une série entière à coefficients complexes, est une application ouverte, d’où l’inégalité stricte $\left|z\, L_{u}\left(u,z\right)\left(t\right)\right|<\rho_{g}$ pour tous $z$ et $t$ comme ci-dessus. En conséquence, pour tout $z$ vérifiant $\displaystyle \left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$ et pour tout point $t$ du segment $J$, et si le nombre complexe $\displaystyle t+z\, L_{u}\left(u,z\right)\left(t\right)$ appartient au segment $J$, on obtient: $$L_{u}\left(g,z\right)\left(t\right)=\sum_{m=0}^{\infty}\dfrac{1}{m!}\Bigl(\left(t+z\, L_{u}\left(u,z\right)\left(t\right)\right)-t\Bigr)^{m}D^{m}g\left(t\right)=g\left(t+z\, L_{u}\left(u,z\right)\left(t\right)\right).$$
2. Si $g=u$, on a $T=4R$. Remarquant l’égalité $\displaystyle z\, u\left(x\right)=z\, L_{u}\left(u,z\right)\left(t\right)=x-t$, on obtient: $$x=t+z\, L_{u}\left(u,z\right)\left(t\right)=t+\sum_{n=0}^{\infty}\dfrac{z^{n+1}}{n!}D^{n-1}\left(u^{n}Du\right)\left(t\right)=t+\sum_{n=1}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}\right)\left(t\right)$$ Les conditions $\displaystyle \left|z\right|<\dfrac{1}{4\, N_{R}\left(u\right)T}$ et $\displaystyle \left|z\right|<\dfrac{1}{16\, N_{R}\left(u\right)R}$ sont simultanément réalisées dès qu’on a $\left|z\right|<\dfrac{1}{8\, N_{R}\left(u\right)T}$, de sorte qu’on peut appliquer la première partie pour, à la fois tenir une solution $x=t+z\, L_{u}\left(u,z\right)\left(t\right)$ de l’équation $t+z\, u\left(x\right)=x$ lorsque ce nombre $x$ appartient au segment $J$, et obtenir une expression de $g\left(x\right)$, à savoir: $$\begin{aligned}
g\left(t+zL_{u}\left(u,z\right)\left(t\right)\right)&= L_{u}\left(g,z\right)\left(t\right)\\
&=\sum_{n=0}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right)=g\left(t\right)+\sum_{n=1}^{\infty}\dfrac{z^{n}}{n!}D^{n-1}\left(u^{n}Dg\right)\left(t\right).\end{aligned}$$
Rappelons que les séries entières à coefficients complexes ${\displaystyle \phi\left(w\right)=\sum_{k=0}^{\infty}c_{k}w^{k}}$, de rayon de convergence infini opèrent par composition sur les séries formelles ${\displaystyle T=\sum_{n=0}^{\infty}f_{n}X^{n}}$ à coefficients dans l’anneau $\mathcal{C}^{\infty}\left(J\right)$ selon la formule: $$\phi\left(T\right)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}c_{k}\left(\dfrac{1}{n!}D^{n}T^{k}\right)\left(0\right)\right)X^{n}.$$ La formule du corollaire , conséquence directe de la formule du produit de Lagrange , et quelques considérations relativement élémentaires de convergence dans l’espace de Fréchet $\mathcal{C}^{\infty}\left(J\right)$ montrent que pour tout couple $\left(\psi,f\right)$ de fonctions de classe $\mathcal{C}^{\infty}$ sur le segment $J$ et pour toute fonction $\phi$ somme de série entière à coefficients complexes de rayon de convergence infini, on a la formule: $$\phi\left(\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(\psi^{n}Df\right)X^{n}\right)=\sum_{n=0}^{\infty}\dfrac{1}{n!}D^{n-1}\left(\psi^{n}D\left(\phi\circ f\right)\right)X^{n}.$$
[Références]{} Adams E. P., Hippisley, R. L., Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Miscellaneous CoIlections, Vol. 74, No. 1, Smithsonian Inst., Washington, D.C., 1922.
Frobenius, F. G. und Stickelberger, L. Üeber die Differentiation der elliptischen Functionen nach den Perioden und Invarianten, 92, 311-327 (1882).
Johnson, W. P. The Curious History of Faà di Bruno’s Formula, 109, 217-234 (2002)
Lagrange, J-L. Oeuvres complètes, tome 8, Gauthier-Villars, Paris (1869) http://gallica.bnf.fr/ark:/12148/bpt6k229943n/f272.image
Olver, P. J. A Nonlinear Differential Operator Series That Commutes with Any Function. 23, 209-221 (1992)
Whittaker, E. T. and Watson, G. N. An Elementary Treatise in Modern Analysis, 4th ed., Cambridge, England : Cambridge University Press.
$\,$
|
---
abstract: |
Using a new analysis approach, we establish a general convergence theory of the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple eigenvalue nearest to a given target $\sigma$ and the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimics the exact SIRA well, that is, the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with [*low*]{} or [*modest*]{} accuracy during outer iterations. Based on the theory, we design practical stopping criteria for inner solves. Our analysis is on one step expansion of subspace and the approach applies to the Jacobi–Davidson (JD) method with the fixed target $\sigma$ as well, and a similar general convergence theory is obtained for it. Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.
[**Keywords.**]{} Subspace expansion, expansion vector, inexact, low or modest accuracy, the SIRA method, the JD method, inner iteration, outer iteration.
[**AMS subject classifications.**]{} 65F15, 15A18, 65F10.
author:
- 'Zhongxiao Jia[^1]'
- 'Cen Li[^2]'
title: 'On Inner Iterations in the Shift-Invert Residual Arnoldi Method and the Jacobi–Davidson Method[^3]'
---
Introduction
============
Consider the large and possibly sparse matrix eigenproblem $${\mathbf{A}}{\mathbf{x}}=\lambda{\mathbf{x}}, \label{problem}$$ with ${\mathbf{A}}\in\mathcal{C}^{n\times n}$, the 2-norm $\|{\mathbf{x}}\|=1$ and the eigenvalues labeled as $$\begin{aligned}
0<|\lambda_1-\sigma|<|\lambda_2-\sigma|\leq\cdots\leq|\lambda_n-\sigma|\end{aligned}$$ for a given target $\sigma\in\mathcal{C}$. We are interested in the eigenvalue $\lambda_1$ closest to the target $\sigma$ and/or the associated eigenvector ${\mathbf{x}}_1$. We denote $(\lambda_1,{\mathbf{x}}_1)$ by $(\lambda,{\mathbf{x}})$ for simplicity. A number of numerical methods [@bai2000templates; @parlett1998symmetric; @saad1992eigenvalue; @vandervorst2002eigenvalue; @stewart2001eigensystems] are available for solving this kind of problems. The Residual Arnoldi (RA) method and Shift-Invert Residual Arnoldi (SIRA) method are new ones that have their origins in the Jacobi–Davidson (JD) method [@sleijpen2000jacobi]. RA was initially proposed by van der Vorst and Stewart in 2001; see [@leestewart07]. The methods were then studied and developed by Lee [@lee2007residual] and Lee and Stewart [@leestewart07]. We briefly describe RA now.
Given a starting vector ${\mathbf{v}}_1$ with $\|{\mathbf{v}}_1\|=1$, suppose an orthonormal ${\mathbf{V}}_m=({\mathbf{v}}_1,\ldots,{\mathbf{v}}_m)$ has been constructed by the Arnoldi process. Then the columns of ${\mathbf{V}}_m$ form a basis of the $m$-dimensional Krylov subspace $\mathcal{K}_m({\mathbf{A}},{\mathbf{v}}_1)
=span\{{\mathbf{v}}_1,{\mathbf{A}}{\mathbf{v}}_1,\ldots,{\mathbf{A}}^{m-1}{\mathbf{v}}_1\}$, and the next basis vector ${\mathbf{v}}_{m+1}$ is obtained by orthogonalizing ${\mathbf{A}}{\mathbf{v}}_m$ against ${\mathbf{V}}_m$. Let $(\tilde\lambda,{\mathbf{y}})$ be the candidate Ritz pair of ${\mathbf{A}}$ for a desired eigenpair of ${\mathbf{A}}$ with respect to $\mathcal{K}_m({\mathbf{A}},{\mathbf{v}}_1)$, and define the residual ${\mathbf{r}}={\mathbf{A}}{\mathbf{y}}-\tilde\lambda{\mathbf{y}}$. Then the RA method orthogonalizes ${\mathbf{r}}$ against ${\mathbf{V}}_m$ to get the next basis vector, which, in exact arithmetic, is just ${\mathbf{v}}_{m+1}$ obtained by the Arnoldi process [@lee2007residual; @leestewart07]. So the Arnoldi method is mathematically equivalent to the RA method. However, van der Vorst and Stewart discovered a striking phenomenon that ${\mathbf{r}}$ in the RA method may allow much larger errors or perturbations than ${\mathbf{A}}{\mathbf{v}}_m$ in the Arnoldi method.
The Shift-Invert Arnoldi (SIA) method is the Arnoldi method applied to the shift-invert matrix ${\mathbf{B}}=({\mathbf{A}}-\sigma{\mathbf{I}})^{-1}$ and finds a few eigenvalues nearest to $\sigma$ and the associated eigenvectors. It computes ${\mathbf{v}}_{m+1}$ by orthogonalizing ${\mathbf{u}}={\mathbf{B}}{\mathbf{v}}_m$ against ${\mathbf{V}}_m$, whose columns now form a basis of $\mathcal{K}_m({\mathbf{B}},{\mathbf{v}}_1)$. So at step $m$ one has to solve the linear system $$\label{siainner}
({\mathbf{A}}-\sigma{\mathbf{I}}){\mathbf{u}}={\mathbf{v}}_m$$ for ${\mathbf{u}}$. The SIRA method [@lee2007residual; @leestewart07] is an alternative of the RA method applied to ${\mathbf{B}}$. At each step one has to solve the linear system $$\label{inneriteration}({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{u}}={\mathbf{r}}$$ for ${\mathbf{u}}$, where ${\mathbf{r}}={\mathbf{A}}{\mathbf{y}}-\nu{\mathbf{y}}$ is the residual of the current approximate eigenpair $(\nu,{\mathbf{y}})$ obtained by SIRA. Then the SIRA method computes the next basis vector ${\mathbf{v}}_{m+1}$ by orthogonalizing ${\mathbf{u}}$ against ${\mathbf{V}}_m$. A mathematical difference between SIA and SIRA is that the SIA method computes Ritz pairs of the shift-invert ${\mathbf{B}}$ with respect to $\mathcal{K}_m({\mathbf{B}},{\mathbf{v}}_1)$ and recovers an approximation to $(\lambda,{\mathbf{x}})$, while the SIRA method computes the Ritz pairs of the original ${\mathbf{A}}$ with respect to the same $\mathcal{K}_m({\mathbf{B}},{\mathbf{v}}_1)$ and gets an approximation to $(\lambda,{\mathbf{x}})$. So SIA and SIRA generally obtain different approximations to $(\lambda,{\mathbf{x}})$ with respect to the same subspace $\mathcal{K}_m({\mathbf{B}},{\mathbf{v}}_1)$.
However, for large (\[inneriteration\]), only iterative solvers are generally viable. This leads to the inexact SIRA, an inner-outer iterative method, built-up by outer iteration as the eigensolver and inner iteration as the solver of (\[inneriteration\]). Inexact eigensolvers have attracted much attention over the last two decades, and among them inexact SIA type methods [@simoncini2003ia; @simoncini2005ia; @spence2009ia; @xueelman10] are closely related to the work in the current paper. Central concerns on all inexact eigensolvers are how the accuracy of inner iterations ensures and affects the convergence of outer iterations and how to choose the accuracy requirements of inner iterations so that each inexact eigensolver mimics its corresponding exact counterpart very well in the sense that the two eigensolvers use almost the same or very comparable outer iterations to achieve the convergence.
The JD method with fixed or variable targets [@sleijpen2000jacobi] is a very popular inexact eigensolver, in which a correction equation (inner linear system) is solved iteratively at each outer iteration; see, e.g., [@bai2000templates; @vandervorst2002eigenvalue; @stewart2001eigensystems] and more recent [@hochstenbachnotay09; @notay02; @stathopoulos; @voss07]. Hitherto, however, there has been no result on the accuracy requirement of inner iterations involved in the standard JD method. Existing work only focuses on the simplified (or single-vector) JD method without subspace acceleration. One hopes that the results on the accuracy requirement of inner iterations developed for the simplified JD may help understand the standard JD. Nevertheless, such treatment may be too inaccurate and far from the essence of the standard JD. As is well known, the standard JD is much more complicated than the simplified JD, and the convergence of its outer iterations is much more involved; see [@jia2001analysis] and also [@bai2000templates; @vandervorst2002eigenvalue; @stewart2001eigensystems] for details. Therefore, the standard JD method lacks a general theory on inner iterations, and a rigorous and insightful analysis is necessary and very appealing.
For the inexact SIA method, Simoncini [@simoncini2003ia] has established a relaxation theory on the accuracy requirements of inner iterations of (\[siainner\]) as $m$ increases. She proved that the accuracy of approximate solution of (\[siainner\]) should be very high initially and is relaxed as the approximate eigenpairs start converging. Freitag and Spence [@spence2009ia] have extended Simoncini’s relaxation theory to the inexact implicitly restarted Arnoldi method. Xue and Elman [@xueelman10] have made a refined analysis on the relaxation strategy. So it may be very costly to implement the inexact SIA type methods.
For the SIRA method, it has been reported by Lee [@lee2007residual] and Lee and Stewart [@leestewart07] that when the accuracy of approximate solutions of (\[inneriteration\]) is low or modest at each step, the method may still work well. Lee and Stewart [@leestewart07] have made some analysis on the RA and SIRA methods but they did not derive any quantitative and explicit bounds for the accuracy requirements of inner iterations.
In this paper, we take a different approach from that in [@lee2007residual; @leestewart07] to giving a rigorous one-step analysis of the inexact SIRA method and establish a general and quantitative theory of the accuracy requirements of inner iterations. Our analysis approach applies to the JD method with the fixed target $\sigma$ as well. We first show that the exact SIRA and JD methods are mathematically equivalent. We then focus on a detailed quantitative analysis of the inexact SIRA and JD methods. Let $\varepsilon$ be the relative error of the approximate solution of the inner linear system. We prove that a modestly small $\varepsilon$, e.g., $\varepsilon \in [10^{-4},10^{-3}]$, is generally enough to make the inexact SIRA and JD use almost the same outer iterations as the exact ones to achieve the convergence. As a result, one only needs to solve all inner linear systems with low or modest accuracy in the inexact SIRA and the JD methods, and both methods are expected to be considerably more effective than the inexact SIA method. We should point out that our work is locally an one step analysis. A global analysis involving subspaces accumulating all previous perturbations is much harder and seems impossible. Actually, an one step local analysis is typical in the field of inexact eigensolvers, and it indeed sheds lights on the behavior of the inexact solvers.
The paper is organized as follows. In Section \[sec:sira\_vs\_jd\], we review the SIRA and JD methods and show the equivalence of two exact versions. In Section \[sec:eps\], we derive some relationships between $\varepsilon$ and subspace expansions and show that the inexact SIRA and methods behave very similar when their respective inner linear systems are solved with the same accuracy. In Section \[sec:teps\], we consider subspace improvement and the selection of $\varepsilon$ and prove that the inexact SIRA mimics the exact SIRA very well when $\varepsilon$ is modestly small at all steps. In Section \[issue\], we consider some practical issues and design practical stopping criteria for inner solves in the inexact SIRA and JD. In Section \[numer\], we report numerical experiments to confirm our theory and the considerable superiority of the inexact SIRA and JD algorithms to the inexact SIA algorithm. Meanwhile, we show that the inexact SIRA and JD are similarly effective. Finally, we conclude the paper and point out future work in Section \[concl\].
Throughout the paper, denote by $\|\cdot\|$ the 2-norm of a vector or matrix, by ${\mathbf{I}}$ the identity matrix with the order clear from the context, by the superscript $H$ the conjugate transpose of a vector or matrix, and by $\kappa({\mathbf{Q}})=\|{\mathbf{Q}}\|\|{\mathbf{Q}}^{-1}\|$ the condition number of a nonsingular matrix ${\mathbf{Q}}$. We measure the distance between a nonzero vector ${\mathbf{y}}$ and a subspace $\mathcal{V}$ by $$\sin\angle(\mathcal{V},{\mathbf{y}})=\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{y}}\|}{\|{\mathbf{y}}\|}
=\frac{\|{\mathbf{V}}_\perp^H{\mathbf{y}}\|}{\|{\mathbf{y}}\|}, \label{sinedef}$$ where ${\mathbf{P}}_{\mathbf{V}}$ is the orthogonal projector onto $\mathcal{V}$ and the columns of ${\mathbf{V}}_\perp$ form an orthonormal basis of the orthogonal complement of $\mathcal{V}$.
Equivalence of the exact SIRA and JD methods {#sec:sira_vs_jd}
============================================
Algorithms \[alg:sira\]–\[alg:jd\] describe the SIRA algorithm and the JD algorithm with the fixed target $\sigma$, respectively (for brevity we drop iteration subscript). Comparing them, we observe that the only seemingly differences between them are the linear systems to be solved (step 4) and the expansion vectors to be orthogonalized against the initial subspace $\mathcal{V}$. In fact, they are equivalent, as the following theorem shows.
\[equiva\] For the same initial $\mathcal{V}$, if $\sigma\not=\nu$, then the SIRA method and the JD method are mathematically equivalent when inner linear systems [(\[lssira\])]{} and [(\[ls\_jd\])]{} are solved exactly.
For the same initial ${\cal V}$, the two methods share the same ${\mathbf{H}}$, $\nu$ and ${\mathbf{y}}$, leading to the same ${\mathbf{r}}_S$ and ${\mathbf{r}}_J$. Let ${\mathbf{u}}_S$ and ${\mathbf{u}}_J$ be the exact solutions of (\[lssira\]) and (\[ls\_jd\]), respectively. Since ${\mathbf{B}}=({\mathbf{A}}-\sigma{\mathbf{I}})^{-1}$, we get $$\begin{aligned}
\label{u_sira}{\mathbf{u}}_S={\mathbf{B}}{\mathbf{r}}_S=(\sigma-\nu){\mathbf{B}}{\mathbf{y}}+{\mathbf{y}}.\end{aligned}$$ From (\[ls\_jd\]), we have $$\label{ls_jd_0}({\mathbf{A}}-\sigma{\mathbf{I}}){\mathbf{u}}_J=\left({\mathbf{y}}^H({\mathbf{A}}-\sigma{\mathbf{I}})
{\mathbf{u}}_J\right){\mathbf{y}}-{\mathbf{r}}_J=\gamma{\mathbf{y}}-({\mathbf{A}}-\sigma{\mathbf{I}}){\mathbf{y}},$$ where $\gamma={\mathbf{y}}^H({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{u}}_J-\sigma+\nu$. Premultiplying two sides of (\[ls\_jd\_0\]) by ${\mathbf{B}}$, we obtain $$\label{u_jd}{\mathbf{u}}_J=\gamma{\mathbf{B}}{\mathbf{y}}-{\mathbf{y}}.$$ Since ${\mathbf{u}}_J\perp {\mathbf{y}}$, we get $\gamma=\frac{1}{{\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}}$. Since ${\mathbf{y}}\in\mathcal{V}$, we have $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{y}}=\bf0$. So from (\[u\_sira\]) and (\[u\_jd\]), we get $$({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}=\frac{1}{\sigma-\nu}
({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}_S=\frac{1}{\gamma}({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}_J.
\label{expand}$$ Note that $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}_S$ and $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}_J$ (after normalization) are the subspace expansion vectors in SIRA and JD, respectively. The two methods generate the same subspace in the next iteration and $(\nu,{\mathbf{y}})$ obtained by them are thus identical.
From (\[ls\_jd\_0\]), define $$\label{rj'}{\mathbf{r}}_J'={\mathbf{A}}{\mathbf{y}}-(\sigma+\gamma){\mathbf{y}},$$ where $$\gamma={\mathbf{y}}^H({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{u}}_J-\sigma+\nu=\frac{1}{{\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}}.
\label{gamma}$$ Then (\[ls\_jd\_0\]) and thus (\[ls\_jd\]) become $$\label{ls_jd_1}({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{u}}={\mathbf{r}}_J', \label{mjd}$$ whose solution is $-{\mathbf{u}}_J$ and is the same as ${\mathbf{u}}_J$ up to the sign $-1$. So mathematically, hereafter we use (\[mjd\]) as the inner linear system in the JD method. Since ${\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}$ approximates the eigenvalue $\frac{1}{\lambda-\sigma}$ of ${\mathbf{B}}$, $\gamma+\sigma=\frac{1}
{{\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}}+\sigma$ approximates $\lambda$. So ${\mathbf{r}}_J'$ is a residual associated with the desired eigenpair $(\lambda,{\mathbf{x}})$, just like ${\mathbf{r}}_S$ in (\[lssira\]).
Relationships between the accuracy of inner iterations and subspace expansions {#sec:eps}
==============================================================================
We observe that (\[lssira\]) and (\[ls\_jd\_1\]) fall into the category of $$\label{ls_unified}({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{u}}=\alpha_1{\mathbf{y}}+\alpha_2
({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{y}},\label{alpha12}$$ where specifically $\alpha_1=\sigma-\nu$ and $\alpha_2=1$ in SIRA and $\alpha_1=-\frac{1}{{\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}}$ and $\alpha_2=1$ in JD. The exact solution ${\mathbf{u}}$ of (\[alpha12\]) is $$\label{u_unified}{\mathbf{u}}=\alpha_1{\mathbf{B}}{\mathbf{y}}+\alpha_2 {\mathbf{y}}.$$ Since $({\mathbf{I}}-{\mathbf{P}}_{{\mathbf{V}}}){\mathbf{y}}=\bf0$, the (unnormalized) subspace expansion vector is $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}=\alpha_1({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}$. Let $\tilde{{\mathbf{u}}}$ be an approximate solution of (\[ls\_unified\]), whose relative error is defined by $$\varepsilon=\frac{\|\tilde{{\mathbf{u}}}-{\mathbf{u}}\|}{\|{\mathbf{u}}\|}.
\label{errorsol}$$ Then we can write $$\tilde{{\mathbf{u}}}={\mathbf{u}}+\varepsilon\|{\mathbf{u}}\|{\mathbf{f}}\label{errorf}$$ with ${\mathbf{f}}$ the normalized error direction vector. So we get $$\label{IPtu}({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}=({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
{\mathbf{u}}+\varepsilon\|{\mathbf{u}}\|{\mathbf{f}}_\perp.$$ where $$\label{fperp} {\mathbf{f}}_\perp=({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{f}}.$$ Define $$\begin{aligned}
\label{tv_and_v}\tilde{{\mathbf{v}}}=\frac{({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}}
{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}\|},\ \ \ {\mathbf{v}}=\frac{({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}}
{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|},\end{aligned}$$ which are the normalized subspace expansion vectors in the inexact and exact methods, respectively. We measure the difference between $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}$ and $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}$ by the relative error $$\label{varepsilon}\tilde{\varepsilon}=\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
\tilde{{\mathbf{u}}}-({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}$$ or by $\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})$. Two quantities $\tilde{\varepsilon}$ and $\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})$ are two valid measures for the difference. Next we establish a relationship between $\tilde\varepsilon$ and $\sin\angle(\tilde{\mathbf{v}},{\mathbf{v}})$, which will be used in proving our final result in this paper.
\[lemma1\] With the notations defined above, it holds that $$\label{sin_tv_v_final}\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})
=\tilde{\varepsilon}\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{f}}_\perp).$$
Let ${\mathbf{U}}_\perp$ be an orthonormal basis of the orthogonal complement of $span\left\{({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}\right\}$ with respect to ${\cal C}^n$. Since ${\mathbf{U}}_\perp^H({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}=\mathbf{0}$, by definition (\[sinedef\]) we get $$\begin{aligned}
\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})
\nonumber&=&\sin\angle\left(({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}},({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
{\mathbf{u}}\right)\\
\nonumber&=&\frac{\left\|{\mathbf{U}}_\perp^H({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\right\|}
{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}\\
\nonumber&=&\frac{\left\|{\mathbf{U}}_\perp^H({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}-{\mathbf{U}}_\perp^H
({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\right\|}{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}\\
\label{sin_tv_v}&=&\frac{\left\|{\mathbf{U}}_\perp^H\left(({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
\tilde{{\mathbf{u}}}-({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\right)\right\|}{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}.\end{aligned}$$ From (\[IPtu\]) we have $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\tilde{{\mathbf{u}}}-({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
{\mathbf{u}}=\varepsilon\|{\mathbf{u}}\|{\mathbf{f}}_\perp$. Substituting it into (\[sin\_tv\_v\]) gives $$\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})
=\tilde{\varepsilon}\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{f}}_\perp).$$
In order to make the inexact SIRA method mimic the SIRA method well, we must require that $\tilde{{\mathbf{v}}}$ approximates ${\mathbf{v}}$ with certain accuracy, i.e., $\tilde\varepsilon$ suitably small, so that the two expanded subspaces have comparable quality. We will come back to this key point and estimate $\tilde\varepsilon$ quantitatively in Section \[sec:teps\].
In what follows we establish an important relationship between $\varepsilon$ and $\tilde{\varepsilon}$, and based on it we analyze how $\varepsilon$ varies with $\alpha_1$ and $\alpha_2$ for a given $\tilde\varepsilon$.
\[thm4\] Let ${\mathbf{y}}$ be the current approximate eigenvector and $\alpha=-\frac{\alpha_2}{\alpha_1}$ with $\alpha_1,\alpha_2$ in [(\[alpha12\])]{}. We have $$\label{eps_teps_relation_general}\varepsilon \leq \frac{2\|{\mathbf{B}}\|
\sin\angle({\mathbf{y}},{\mathbf{x}})}{\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon}.$$
By definition (\[fperp\]), we have $$\|{\mathbf{f}}_{\perp}\|=\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{f}}\|=\sin\angle(\mathcal{V},{\mathbf{f}}).$$ From (\[IPtu\]), we get $$\begin{aligned}
\varepsilon
&=&\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
\tilde{{\mathbf{u}}}-({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}{\|{\mathbf{u}}\|\|{\mathbf{f}}_{\perp}\|}\\
&=&\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}{\|{\mathbf{u}}\|\|{\mathbf{f}}_{\perp}\|}\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})
\tilde{{\mathbf{u}}}-({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}
\\
&=&\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}{\|{\mathbf{u}}\|\|{\mathbf{f}}_\perp\|}\tilde{\varepsilon}
=\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{u}}\|}{\|{\mathbf{u}}\|\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon}.\end{aligned}$$ By (\[u\_unified\]), we substitute ${\mathbf{u}}=\alpha_1{\mathbf{B}}{\mathbf{y}}+\alpha_2{\mathbf{y}}$ into the above, giving $$\begin{aligned}
\varepsilon
&=&\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})(\alpha_1{\mathbf{B}}{\mathbf{y}}+\alpha_2{\mathbf{y}})\|}
{\|\alpha_1{\mathbf{B}}{\mathbf{y}}+\alpha_2{\mathbf{y}}\|\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon} \nonumber\\
&=&\frac{\|\alpha_1({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}{\|\alpha_1{\mathbf{B}}{\mathbf{y}}+\alpha_2{\mathbf{y}}\|\sin\angle(\mathcal{V},{\mathbf{f}})}\tilde{\varepsilon} \nonumber\\
&=&\frac{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}{\left\|{\mathbf{B}}{\mathbf{y}}+
\frac{\alpha_2}{\alpha_1}{\mathbf{y}}\right\|\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon}. \label{eps_teps_relation}\end{aligned}$$ Decompose ${\mathbf{y}}$ into the orthogonal direct sum $$\label{decompose_y}{\mathbf{y}}=\cos\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{x}}+\sin\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{g}}$$ with ${\mathbf{g}}\perp {\mathbf{x}}$ and $\|{\mathbf{g}}\|=1$. Then we get $$\begin{aligned}
({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}&=&({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\left(\cos\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{B}}{\mathbf{x}}+\sin\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{B}}{\mathbf{g}}\right)\\
&=&({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\left(\frac{\cos\angle({\mathbf{y}},{\mathbf{x}})}{\lambda-\sigma}{\mathbf{x}}+
\sin\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{B}}{\mathbf{g}}\right)\\
&=&\frac{\cos\angle({\mathbf{y}},{\mathbf{x}})}{\lambda-\sigma}{\mathbf{x}_\perp}+\sin\angle({\mathbf{y}},{\mathbf{x}})
({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{g}},\end{aligned}$$ where ${\mathbf{x}_\perp}=({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{x}}$. Making use of $\|{\mathbf{x}_\perp}\|=\sin\angle(\mathcal{V},{\mathbf{x}})
\leq\sin\angle({\mathbf{y}},{\mathbf{x}})$ and $\frac{1}{|\lambda-\sigma|}\leq\|{\mathbf{B}}\|$, we obtain $$\begin{aligned}
\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|
&=&\left\|\frac{\cos\angle({\mathbf{y}},{\mathbf{x}})}{\lambda-\sigma}{\mathbf{x}_\perp}+\sin\angle({\mathbf{y}},{\mathbf{x}})
({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{g}}\right\|\nonumber\\
&\leq&\frac{|\cos\angle({\mathbf{y}},{\mathbf{x}})|}{|\lambda-\sigma|}\|{\mathbf{x}_\perp}\|+\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{g}}\|\sin\angle({\mathbf{y}},{\mathbf{x}})\nonumber\\
&\leq&\left(\frac{|\cos\angle({\mathbf{y}},{\mathbf{x}})|}{|\lambda-\sigma|}+
\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{g}}\|\right)\sin\angle({\mathbf{y}},{\mathbf{x}})\nonumber\\
&\leq&\left(\frac{1}{|\lambda-\sigma|}+\|{\mathbf{B}}\|\right)\sin\angle({\mathbf{y}},{\mathbf{x}})
\nonumber\\
&\leq&2\|{\mathbf{B}}\|\sin\angle({\mathbf{y}},{\mathbf{x}}). \label{Bybound}\end{aligned}$$ Therefore, combining the last relation with (\[eps\_teps\_relation\]) establishes (\[eps\_teps\_relation\_general\]).
Observe that the linear system $({\mathbf{A}}-\sigma{\mathbf{I}}){\mathbf{u}}={\mathbf{y}}$, which is also the one in the inverse power method at each step, falls into the form of (\[ls\_unified\]) by taking $\alpha_1=1$ and $\alpha_2=0$. For this case, from (\[eps\_teps\_relation\_general\]) we have $$\label{eps_teps_relation_original}\varepsilon\leq\frac{2\|{\mathbf{B}}\|
\sin\angle({\mathbf{y}},{\mathbf{x}})}{\left\|{\mathbf{B}}{\mathbf{y}}\right\|\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon}.$$ We comment that (i) $\sin\angle(\mathcal{V},{\mathbf{f}})$ is moderate as ${\mathbf{f}}$ is a general vector and (ii) $\|{\mathbf{B}}\|/\|{\mathbf{B}}{\mathbf{y}}\|=O(1)$ if ${\mathbf{y}}$ is a reasonably good approximation to ${\mathbf{x}}$ and in the worst case $\|{\mathbf{B}}\|/\|{\mathbf{B}}{\mathbf{y}}\|\leq\kappa({\mathbf{B}})$. In case that $\sin\angle(\mathcal{V},{\mathbf{f}})$ is small, $\varepsilon$ becomes big for a fixed small $\tilde{\varepsilon}$, that is, linear system (\[ls\_unified\]) is allowed to be solved with less accuracy. So a small $\sin\angle(\mathcal{V},{\mathbf{f}})$ is a lucky event.
We can use this theorem to further illustrate why it is bad to solve $({\mathbf{A}}-\sigma{\mathbf{I}}){\mathbf{u}}={\mathbf{y}}$ iteratively. For a fixed small $\tilde{\varepsilon}$, (\[eps\_teps\_relation\_original\]) tells us that $\varepsilon$ should become smaller as $\sin\angle({\mathbf{y}},{\mathbf{x}})\rightarrow 0$ as the algorithms converge. As a result, we have to solve inner linear systems with higher accuracy as ${\mathbf{y}}$ becomes more accurate. More generally, this is the case when $\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|$ is not small and typically of $O(\|{\mathbf{B}}\|)$. Therefore, for $\alpha=0$ and more general $\alpha$, the resulting method and SIA type methods are similar and no winner in theory. They are common in that they all require to solve inner linear systems accurately for some steps and they are different in that the former solves inner linear systems with poor accuracy initially and then with increasing accuracy as the algorithm converges, while the latter ones solve inner linear systems with high accuracy in some initial outer iterations and then with decreasing accuracy as the algorithms converge.
Based on (\[eps\_teps\_relation\_general\]), it is natural for us to maximize its upper bound with respect to $\alpha$ for a fixed $\tilde{\varepsilon}$. This will make $\varepsilon$ is as small as possible, so that we pay least computational efforts to solve (\[ls\_unified\]). This amounts to minimizing $\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|$. As is well known, the optimal $\alpha$ is $$\arg\min\limits_{\alpha\in\mathcal{C}}\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|
={\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}},$$ Such $\alpha=-\frac{\alpha_2}{\alpha_1}$ corresponds to the choice $\alpha_1=-\frac{1}{{\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}}$ and $\alpha_2=1$ in (\[ls\_unified\]), exactly leading to linear system (\[ls\_jd\_1\]) in the JD method. Therefore, in the sense of minimizing $\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|$, the JD method is the best. If we take $\alpha=\frac{1}{\nu-\sigma}$, which is the approximation to $\frac{1}{\lambda-\sigma}$ in SIRA, by letting $\alpha_1=\sigma-\nu$ and $\alpha_2=1$, then (\[ls\_unified\]) becomes $$\begin{aligned}
({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{u}}=({\mathbf{A}}-\sigma {\mathbf{I}}){\mathbf{y}}+(\sigma-\nu){\mathbf{y}}={\mathbf{r}}_S,\end{aligned}$$ which is exactly the linear system in the SIRA method. In each of JD and SIRA, $\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|$ is the residual norm of an approximate eigenpair $(\alpha,{\mathbf{y}})$ of ${\mathbf{B}}$.
In what follows, we denote $\varepsilon$ by $\varepsilon_S$ and $\varepsilon_J$ in the SIRA and JD methods, respectively. To derive our final and key relationships between $\varepsilon_S,\,\varepsilon_J$ and $\tilde{\varepsilon}$, we need the following lemma, which is direct from Theorem 6.1 of [@jia2001analysis] and establishes a close and compact relationship between $\sin\angle({\mathbf{y}},{\mathbf{x}})$ and the residual norm $\left\|{\mathbf{B}}{\mathbf{y}}-\alpha {\mathbf{y}}\right\|$.
\[lem:upper\_bound\_sin\_y\_x\] Suppose $\left(\frac{1}{\lambda-\sigma},{\mathbf{x}}\right)$ is a simple desired eigenpair of ${\mathbf{B}}\in\mathcal{C}^{n\times n}$ and let $({\mathbf{x}},{\mathbf{X}}_\perp)$ be unitary. Then $$\left[\begin{array}{c}{\mathbf{x}}^H \\ {\mathbf{X}}_\perp^H \end{array}\right]{\mathbf{B}}\left[\begin{array}{cc}{\mathbf{x}}& {\mathbf{X}}_\perp \end{array}\right]=
\left[\begin{array}{cc}\frac{1}{\lambda-\sigma} & {\mathbf{c}}^H \\
\mathbf{0} & {\mathbf{L}}\end{array}\right],$$ where ${\mathbf{c}}^H={\mathbf{x}}^H{\mathbf{B}}{\mathbf{X}}_\perp$ and ${\mathbf{L}}={\mathbf{X}}_\perp^H{\mathbf{B}}{\mathbf{X}}_\perp$. Let $(\alpha,{\mathbf{y}})$ be an approximation to $\left(\frac{1}{\lambda-\sigma},{\mathbf{x}}\right)$, assume that $\alpha$ is not an eigenvalue of ${\mathbf{L}}$ and define $${\mathrm{sep}}\left(\alpha,{\mathbf{L}}\right)=\|({\mathbf{L}}-\alpha{\mathbf{I}})^{-1}\|^{-1}>0.$$ Then $$\label{upper_bound_sin_y_x}\sin\angle({\mathbf{y}},{\mathbf{x}})\leq\frac{\|{\mathbf{B}}{\mathbf{y}}-
\alpha{\mathbf{y}}\|}{{\mathrm{sep}}\left(\alpha,{\mathbf{L}}\right)}.$$
Combining (\[upper\_bound\_sin\_y\_x\]) with Theorem \[thm4\], we obtain one of our main results.
\[thm5\] Assume that $\alpha$ is an approximation to $\frac{1}{\lambda-\sigma}$ and is not an eigenvalue of ${\mathbf{L}}$. Then $$\label{upper_bound_eps}\varepsilon\leq\frac{2\|{\mathbf{B}}\|}
{{\mathrm{sep}}\left(\alpha,{\mathbf{L}}\right)\sin\angle(\mathcal{V},{\mathbf{f}})}\tilde{\varepsilon}.$$ In particular, for $\alpha=\frac{1}{\nu-\sigma}$ and $\alpha={\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}$, which correspond to the SIRA and JD methods, respectively, assume that each of them is not an eigenvalue of ${\mathbf{L}}$. Then it holds that $$\label{eps_s}\varepsilon_S\leq\frac{2\|{\mathbf{B}}\|}{{\mathrm{sep}}\left(\frac{1}
{\nu-\sigma},{\mathbf{L}}\right)\sin\angle(\mathcal{V},{\mathbf{f}})}\tilde{\varepsilon},$$ and $$\label{eps_j}\varepsilon_J\leq\frac{2\|{\mathbf{B}}\|}
{{\mathrm{sep}}\left({\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}},{\mathbf{L}}\right)\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon}.$$
This theorem shows that once $\tilde\varepsilon$ is known we can a-priori determine the accuracy requirements $\varepsilon_S$ and $\varepsilon_J$ on approximate solutions of inner linear systems (\[lssira\]) and (\[ls\_jd\]).
It is important to observe from (\[upper\_bound\_eps\]) that $$\begin{aligned}
\varepsilon
\leq\frac{2\|{\mathbf{B}}\|}{{\mathrm{sep}}\left(\alpha,{\mathbf{L}}\right)\sin\angle(\mathcal{V},{\mathbf{f}})}
\tilde{\varepsilon}=\frac{2\|{\mathbf{B}}\|}{O(\|{\mathbf{B}}\|)}\tilde{\varepsilon}
=O(\tilde{\varepsilon})\end{aligned}$$ if $\alpha$ is well separated from the eigenvalues of ${\mathbf{B}}$ other than $\frac{1}{\lambda-\sigma}$ and ${\mathbf{B}}$ is normal or mildly non-normal and $\sin\angle(\mathcal{V},{\mathbf{f}})$ is not small. For $\sin\angle(\mathcal{V},{\mathbf{f}})$ small, noting that bound (\[upper\_bound\_eps\]) is compact, we are lucky to have a bigger $\varepsilon$, i.e., to solve the inner linear system with less accuracy. If ${\mathrm{sep}}\left(\alpha,{\mathbf{L}}\right)$ is considerably smaller than $\|{\mathbf{B}}\|$, then $\varepsilon$ may be bigger than $\tilde{\varepsilon}$ considerably and we are likely lucky to solve the inner linear system with less accuracy.
For the $\alpha$’s in the SIRA and JD methods, by continuity the corresponding two ${\mathrm{sep}}\left(\alpha,{\mathbf{L}}\right)$’s are close. Therefore, for a given $\tilde{\varepsilon}$, we have essentially the same upper bounds for $\varepsilon_S$ and $\varepsilon_J$. This means that we need to solve the corresponding inner linear systems (\[lssira\]) and (\[ls\_jd\]) in the SIRA and JD methods with essentially the same accuracy $\varepsilon$. In other words, the SIRA and JD methods behave very similar when (\[lssira\]) and (\[ls\_jd\]) are solved with the same accuracy.
Subspace improvement and selection of $\tilde{\varepsilon}$ and $\varepsilon$ {#sec:teps}
=============================================================================
In this section, we first focus on the fundamental problem of how to select $\tilde{\varepsilon}$ to make the inexact SIRA and JD mimic the exact SIRA very well from the current step to the next one. Then we show how to achieve our ultimate goal: the determination of $\varepsilon$.
Recall that the subspace expansion vectors are ${\mathbf{v}}$ and $\tilde{{\mathbf{v}}}$ for the exact SIRA and the inexact SIRA or JD; see (\[tv\_and\_v\]). Define ${\mathbf{V}}_+=\left[\begin{array}{cc}
{\mathbf{V}}& {\mathbf{v}}\end{array}\right]$, $\mathcal{V}_+=span\left\{{\mathbf{V}}_+\right\}$ and $\tilde{{\mathbf{V}}}_+
=\left[\begin{array}{cc}{\mathbf{V}}&\tilde{{\mathbf{v}}}\end{array}\right]$, $\tilde{\mathcal{V}}_+=span\{\tilde{{\mathbf{V}}}_+\}$. In order to make the inexact SIRA method mimic the exact SIRA method very well, we must require that the two expanded subspaces $\mathcal{V}_+$ and $\tilde{\mathcal{V}}_+$ have almost the same quality, namely, $\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})\approx\sin\angle(\mathcal{V}_+,{\mathbf{x}})$, whose quantitative meaning will be clear later.
\[thm:sin\_v\_xp\_div\_sin\_tv\_xp\] With the notations above, assume $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})\not=0$ with ${\mathbf{x}}_\perp=({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{x}}$.[^4] Then we have $$\begin{aligned}
\label{sin_v_xp} \sin\angle(\mathcal{V}_+,{\mathbf{x}})&=&
\sin\angle(\mathcal{V},{\mathbf{x}}) \sin\angle({\mathbf{v}},{\mathbf{x}_\perp}),\\
\label{sin_v_xp_div_sin_tv_xp}
\frac{\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})}{\sin\angle(\mathcal{V}_+,{\mathbf{x}})}
&=&\frac{\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{x}_\perp})}{\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})}.\end{aligned}$$ Suppose $\angle(\tilde{\mathbf{v}},{\mathbf{v}})$ is acute. If $\tau=\frac{2\tilde{\varepsilon}}{\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})}<1$, we have $$1-\tau\leq \label{sin_tV+_x_div_sin_V+_x}
\frac{\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})}{\sin\angle(\mathcal{V}_+,{\mathbf{x}})}
\leq 1+\tau. \label{tau}$$
Since $$\begin{aligned}
\sin^2\angle(\mathcal{V},{\mathbf{x}})-\sin^2\angle(\mathcal{V}_+,{\mathbf{x}})=
\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{x}}\|^2-\|({\mathbf{I}}-{\mathbf{P}}_{{\mathbf{V}}_+}){\mathbf{x}}\|^2=|{\mathbf{v}}^H{\mathbf{x}}|^2,\end{aligned}$$ by $\|{\mathbf{x}_\perp}\|=\sin\angle(\mathcal{V},{\mathbf{x}})$ we obtain $$\begin{aligned}
\frac{\sin\angle(\mathcal{V}_+,{\mathbf{x}})}{\sin\angle(\mathcal{V},{\mathbf{x}})}
\nonumber&=&\sqrt{1-\left(\frac{|{\mathbf{v}}^H{\mathbf{x}}|}{\sin\angle(\mathcal{V},{\mathbf{x}})}\right)^2}\\
\nonumber&=&\sqrt{1-\left(\frac{|{\mathbf{v}}^H{\mathbf{x}_\perp}|}{\sin\angle(\mathcal{V},{\mathbf{x}})}\right)^2}\\
\nonumber&=&\sqrt{1-\left(\frac{\|{\mathbf{x}_\perp}\|\cos\angle({\mathbf{v}},{\mathbf{x}_\perp})}
{\sin\angle(\mathcal{V},{\mathbf{x}})}\right)^2}\\
\nonumber&=&\sqrt{1-\cos^2\angle({\mathbf{v}},{\mathbf{x}_\perp})}\\
&=&\sin\angle({\mathbf{v}},{\mathbf{x}_\perp}),\end{aligned}$$ which proves (\[sin\_v\_xp\]). Similarly, we have $$\label{sin_tv_xp}\frac{\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})}
{\sin\angle(\mathcal{V},{\mathbf{x}})}
=\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{x}_\perp}).$$ Hence, from (\[sin\_v\_xp\]) and (\[sin\_tv\_xp\]), we get (\[sin\_v\_xp\_div\_sin\_tv\_xp\]).
Exploiting the trigonometric identity $$\sin\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})-\sin\angle({\mathbf{v}},{\mathbf{x}}_{\perp})
=2\cos\frac{\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})+\angle({\mathbf{v}},{\mathbf{x}}_{\perp})}{2}
\sin\frac{\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})-\angle({\mathbf{v}},{\mathbf{x}}_{\perp})}{2},$$ the angle triangle inequality $$|\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})-\angle({\mathbf{v}},{\mathbf{x}}_{\perp})|\leq\angle(\tilde{{\mathbf{v}}},{\mathbf{v}}).$$ and the monotonic increasing property of the $\sin$ function in the first quadrant, we get $$\begin{aligned}
|\sin\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})-\sin\angle({\mathbf{v}},{\mathbf{x}}_{\perp})|
&\leq&2\left|\sin\frac{\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})-\angle({\mathbf{v}},{\mathbf{x}}_{\perp})}{2}\right|
\nonumber\\
&=&2\sin\frac{|\angle(\tilde{\mathbf{v}},{\mathbf{x}}_{\perp})-\angle({\mathbf{v}},{\mathbf{x}}_{\perp})|}{2}\nonumber\\
&\leq&2\sin\frac{\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})}{2}\nonumber\\
&\leq&2\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}}). \label{triangle}\end{aligned}$$
From (\[sin\_v\_xp\_div\_sin\_tv\_xp\]), (\[triangle\]) and (\[sin\_tv\_v\_final\]), we obtain $$\begin{aligned}
\left|\frac{\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})}
{\sin\angle(\mathcal{V}_+,{\mathbf{x}})}-1\right|
&=&\left|\frac{\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{x}_\perp})}
{\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})}-1\right|\\
&=&\frac{\left|\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{x}_\perp})-
\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})\right|}{\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})}\\
&\leq&\frac{2\sin\angle(\tilde{{\mathbf{v}}},{\mathbf{v}})}
{\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})}\\
&\leq&\frac{2\tilde{\varepsilon}}{\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})}=\tau,\end{aligned}$$ from which it follows that (\[tau\]) holds.
From (\[sin\_v\_xp\]), we see that $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ is exactly one step subspace improvement when $\mathcal{V}$ is expanded to $\mathcal{V}_+$.
(\[tau\]) shows that, to make $\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})
\approx\sin\angle(\mathcal{V}_+,{\mathbf{x}})$, $\tau$ should be small. Meanwhile, (\[tau\]) also indicates that a very small $\tau$ cannot improve the bounds essentially. Actually, for our purpose, a fairly small $\tau$, e.g., $\tau=0.01$, is enough since we have $$0.99\leq\frac{\sin\angle(\tilde{\mathcal{V}}_+,{\mathbf{x}})}
{\sin\angle(\mathcal{V}_+,{\mathbf{x}})}\leq 1.01$$ and the lower and upper bounds are very near and differ marginally. Therefore, $\tilde{\mathcal{V}}_+$ and $\mathcal{V}_+$ are of almost the same quality for approximating ${\mathbf{x}}$. As a result, it is expected that the inexact SIRA or JD computes new approximation over $\tilde{\mathcal{V}}_+$ to the desired $(\lambda,{\mathbf{x}})$ that has almost the same accuracy as that obtained by the exact SIRA over $\mathcal{V}_+$. More precisely, the accuracy of the approximate eigenpair by the exact SIRA and that by the inexact SIRA or JD are generally the same within roughly a multiple $c\in [1-\tau,1+\tau]$ (this assertion can be justified from the results in [@jia05; @jia2001analysis]). So how near the constant $c$ is to one is insignificant, the inexact SIRA and JD generally mimic the exact SIRA very well when $\tau$ is fairly small. Concisely, we may well draw the conclusion that $\tau=0.01$ makes the inexact SIRA mimic the exact SIRA very well, that is, the exact and inexact SIRA methods use almost the same outer iterations to achieve the convergence.
Next we discuss the selection of $\tilde\varepsilon$. Once $\tilde\varepsilon$ is available, in principle we can exploit compact bounds (\[eps\_s\]) and (\[eps\_j\]) to determine the accuracy requirements $\varepsilon_S$ and $\varepsilon_J$ on inner iterations in the SIRA and JD.
From the definition of $\tau$, we have $$\label{teps}\tilde{\varepsilon}=\frac{\tau}{2}\sin\angle({\mathbf{v}},{\mathbf{x}_\perp}).$$ As Theorem \[thm:sin\_v\_xp\_div\_sin\_tv\_xp\] requires $\tau<1$, we must have $\tilde{\varepsilon}<\frac{1}{2}\sin\angle({\mathbf{v}},{\mathbf{x}}_{\perp})$. But ${\mathbf{x}_\perp}$ is not available and a-priori, so we can only use a reasonable estimate on $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ in (\[teps\]). In the following, we will look into $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ and show that it is actually independent of the quality of the approximate eigenvector ${\mathbf{y}}$, i.e., $\sin\angle({\mathbf{y}},{\mathbf{x}})$, and the subspace quality, i.e., $\sin\angle({\cal V},{\mathbf{x}})$. This means that $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ stays around some constant during outer iterations. Then we analyze its size, which is shown to be problem dependent and stay around some certain constant during outer iterations. Based on these results, we can propose a general practical selection of $\tilde\varepsilon$. Obviously, in order to achieve a given $\tau$, the smaller $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ is, the smaller $\tilde{\varepsilon}$ must be and the more accurately we need to solve the inner linear system.
We now investigate $|\cos\angle({\mathbf{v}},{\mathbf{x}}_\perp)|$ and show that it is bounded independently of $\sin\angle({\mathbf{y}},{\mathbf{x}})$ and $\sin\angle({\cal V},{\mathbf{x}})$, so is $\sin\angle({\mathbf{v}},{\mathbf{x}}_\perp)$. From (\[expand\]) and (\[tv\_and\_v\]), it is known that ${\mathbf{v}}$ and $({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}$ are in the same direction. Therefore, from decomposition (\[decompose\_y\]) of ${\mathbf{y}}$, we have $$\begin{aligned}
|\cos\angle({\mathbf{v}},{\mathbf{x}}_\perp)|
&=&\frac{|{\mathbf{x}}_\perp^H({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}|}
{\|{\mathbf{x}}_\perp\|\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}\\
&=&\frac{\left|{\mathbf{x}}_\perp^H({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}(\cos\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{x}}+\sin\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{g}})\right|}
{\|{\mathbf{x}}_\perp\|\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}\\
&=&\frac{\left|{\mathbf{x}}_\perp^H({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}})\left(\frac{\cos\angle({\mathbf{y}},{\mathbf{x}})}
{\lambda-\sigma}{\mathbf{x}}+\sin\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{B}}{\mathbf{g}}\right)\right|}
{\|{\mathbf{x}}_\perp\|\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}\\
&=&\frac{\left|\cos\angle({\mathbf{y}},{\mathbf{x}})\|{\mathbf{x}}_\perp\|^2+(\lambda-\sigma)
\sin\angle({\mathbf{y}},{\mathbf{x}}){\mathbf{x}}_\perp^H{\mathbf{B}}{\mathbf{g}}\right|}{|\lambda-\sigma|\|
{\mathbf{x}}_\perp\|\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}\\
&\leq&\frac{|\cos\angle({\mathbf{y}},{\mathbf{x}})|\|{\mathbf{x}}_\perp\|}{|\lambda-\sigma|
\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}+\frac{\sin\angle({\mathbf{y}},{\mathbf{x}})|
{\mathbf{x}}_\perp^H{\mathbf{B}}{\mathbf{g}}|}{\|{\mathbf{x}}_\perp\|\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}.\end{aligned}$$ Note that $|{\mathbf{x}}_\perp^H{\mathbf{B}}{\mathbf{g}}|\leq\|{\mathbf{x}}_\perp\|\|{\mathbf{B}}{\mathbf{g}}\|
\leq\|{\mathbf{x}}_\perp\|\|{\mathbf{B}}\|$ and $\|{\mathbf{x}}_\perp\|=\sin\angle(\mathcal{V},{\mathbf{x}})\leq\sin\angle({\mathbf{y}},{\mathbf{x}})$. So $$\begin{aligned}
|\cos\angle({\mathbf{v}},{\mathbf{x}}_\perp)|
\nonumber&\leq&\frac{|\cos\angle({\mathbf{y}},{\mathbf{x}})|\|{\mathbf{x}_\perp}\|}{|\lambda-\sigma|
\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}+\frac{\sin\angle({\mathbf{y}},{\mathbf{x}})
\|{\mathbf{B}}{\mathbf{g}}\|}{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}\\
\nonumber&\leq&\left(\frac{|\cos\angle({\mathbf{y}},{\mathbf{x}})|}
{|\lambda-\sigma|}+\|{\mathbf{B}}\|\right)\frac{\sin\angle({\mathbf{y}},{\mathbf{x}})}
{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}\\
\label{cosvxp}&\leq&\frac{2\|{\mathbf{B}}\|\sin\angle({\mathbf{y}},{\mathbf{x}})}
{\|({\mathbf{I}}-{\mathbf{P}}_{\mathbf{V}}){\mathbf{B}}{\mathbf{y}}\|}.\end{aligned}$$ Combining (\[cosvxp\]) and (\[Bybound\]), we have $$\begin{aligned}
|\cos\angle({\mathbf{v}},{\mathbf{x}}_\perp)|\leq\frac{O(\|{\mathbf{B}}\|)\sin\angle({\mathbf{y}},{\mathbf{x}})}
{O\left(\|{\mathbf{B}}\|\right)\sin\angle({\mathbf{y}},{\mathbf{x}})}=O(1),\end{aligned}$$ a seemingly trivial bound. However, the proof clearly shows that our derivation is general and does not miss anything essential. We are not able to make the bound essentially sharper and more elegant as the inequalities used in the proof cannot be sharpened generally. Nevertheless, this is enough for our purpose. A key implication is that the bound is independent of $\sin\angle({\mathbf{y}},{\mathbf{x}})$ and $\sin\angle({\cal V},{\mathbf{x}})$, so $|\cos\angle({\mathbf{v}},{\mathbf{x}}_\perp)|$ is expected to be around some constant during outer iterations, so is $\sin\angle({\mathbf{v}},{\mathbf{x}}_\perp)$.
It is possible to estimate $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ in some important cases. For the starting vector ${\mathbf{v}}_1$, it is known that the exact SIRA, SIA and JD methods work on the standard Krylov subspaces $\mathcal{V}=\mathcal{V}_m=
\mathcal{K}_m({\mathbf{B}},{\mathbf{v}}_1)$ and $\mathcal{V}_+=\mathcal{V}_{m+1}=\mathcal{K}_{m+1}({\mathbf{B}},{\mathbf{v}}_1)$. Here we have temporarily added iteration subscripts and assume that the current iteration step is $m$. It is direct from (\[sin\_v\_xp\_div\_sin\_tv\_xp\]) to get $$\label{sinprod}
\sin\angle(\mathcal{V}_{m+1},{\mathbf{x}})=\sin\angle({\mathbf{v}}_1,{\mathbf{x}})\prod_{i=2}^{m+1}
\sin\angle({\mathbf{v}}_i,{\mathbf{x}}_{i,\perp}),$$ where the ${\mathbf{v}}_i$ are exact subspace expansion vectors and ${\mathbf{x}}_{i,\perp}=({\mathbf{I}}-{\mathbf{P}}_{{{\mathbf{V}}_i}}){\mathbf{x}}$at steps $i=2,3,\ldots,m+1$.
For the Krylov subspaces ${\cal V}_m$ and ${\cal V}_{m+1}$, there have been some estimates on $\sin\angle(\mathcal{V}_{m+1},{\mathbf{x}})$ in [@jia95; @jia98; @saad1992eigenvalue]. For ${\mathbf{B}}$ is diagonalizable, suppose all the $\lambda_i,\ i=1,2,\ldots,n$ and $\sigma$ are real and $\frac{1}{\lambda-\sigma}$ is also the algebraically largest eigenvalue of ${\mathbf{B}}$, and define $$\eta=1+2\frac{\frac{1}{\lambda-\sigma}-\frac{1}{\lambda_2-\sigma}}
{\frac{1}{\lambda_2-\sigma}-\frac{1}{\lambda_n-\sigma}}
=1+2\frac{(\lambda_2-\lambda)(\lambda_n-\sigma)}
{(\lambda_n-\lambda_2)(\lambda-\sigma)}>1.$$ Then it is shown in [@jia98; @saad1992eigenvalue] that $$\sin\angle(\mathcal{V}_{m+1},{\mathbf{x}})=\sin\angle({\mathbf{v}}_1,{\mathbf{x}})\prod_{i=2}^{m+1}
\sin\angle({\mathbf{v}}_i,{\mathbf{x}}_{i,\perp})\leq C_{{\mathbf{v}}_1}\sin\angle({\mathbf{v}}_1,{\mathbf{x}})\left(\frac{1}
{\eta+\sqrt{\eta^2-1}}\right)^m,$$ where $C_{{\mathbf{v}}_1}$ is a certain constant only depending on ${\mathbf{v}}_1$ and the conditioning of the eigensystem of ${\mathbf{B}}$. So, ignoring the constant factor $C_{{\mathbf{v}}_1}$, we see the product $\prod_{i=2}^{m+1}
\sin\angle({\mathbf{v}}_i,{\mathbf{x}}_{i,\perp})$ converges to zero at least as rapidly as $$\left(\frac{1}
{\eta+\sqrt{\eta^2-1}}\right)^m.$$ As we have argued, all the $\sin\angle({\mathbf{v}}_i,{\mathbf{x}}_{i,\perp})$, $i=2,3,\ldots,m+1$, stay around a certain constant. So basically, each step subspace improvement $\sin\angle({\mathbf{v}}_i,{\mathbf{x}}_{i,\perp}),
\ i=2,3,\ldots,m+1$, behaves like and is no more than the factor $$\frac{1}{\eta+\sqrt{\eta^2-1}},$$ the average convergence factor for one step. Returning to our notation, we see the size of $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ crucially depends on the eigenvalue distribution. The better $\frac{1}{\lambda-\sigma}$ is separated from the other eigenvalues of ${\mathbf{B}}$, the smaller $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ is. Conversely, if $\frac{1}{\lambda-\sigma}$ is poorly separated from the others, $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ may be near to one. For more complicated complex eigenvalues and/or $\sigma$, quantitative results are obtained for $\sin\angle(\mathcal{V}_{m+1},{\mathbf{x}})$ and similar conclusions are drawn in [@jia95; @jia98]. However, we should point that these estimates may be conservative and also only predict linear convergence. In practice, a slightly superlinear convergence may occur sometimes, as has been observed in [@leestewart07].
For $\tau=0.01$, if $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})\in [0.02,0.2]$, then by (\[teps\]) we have $\tilde{\varepsilon}\in [10^{-4}, 10^{-3}]$. Such $\sin\angle({\mathbf{v}},{\mathbf{x}_\perp})$ means that $\frac{1}{\lambda-\sigma}$ is well separated from the other eigenvalues of ${\mathbf{B}}$ and the exact SIRA generally converges fast. In practice, however, for a given $\tilde{\varepsilon}$ we do not know the value of $\tau$ produced by $\tilde{\varepsilon}$ as $\sin\angle({\mathbf{v}},{\mathbf{x}}_{\perp})$ and its bound are not known. For a given $\tilde{\varepsilon}$, if we are unlucky to get a $\tau$ not small like $0.01$, the inexact SIRA may use more outer iterations than the exact SIRA. Suppose we select $\tilde\varepsilon=\frac{10^{-3}}{2}$. Then if each $\sin\angle({\mathbf{v}},{\mathbf{x}}_{\perp})=0.1$, we get $\tau=0.01$. For this case, we have a very good subspace $\mathcal{V}_m$ for $m=10$ since $\sin(\mathcal{V}_{10},{\mathbf{x}})\leq 10^{-9}$, so the exact SIRA generally converges very fast! For a real-world problem, however, one should not expect that $\frac{1}{\lambda-\sigma}$ is generally so well separated from the other eigenvalues that the convergence can be so rapid. Therefore, we generally expect that $\tilde\varepsilon\in [10^{-4},10^{-3}]$ makes $\tau\leq 0.01$, so that the inexact SIRA and JD mimic the exact SIRA very well.
Summarizing the above, we propose taking $$\tilde{\varepsilon}\in [10^{-4}, 10^{-3}].\label{tildeepsilon}$$
Our ultimate goal is to determine $\varepsilon_S$ and $\varepsilon_J$ for the inexact SIRA and JD. Compact bounds (\[eps\_s\]) and (\[eps\_j\]) show that they are generally of $O(\tilde{\varepsilon})$. However, it is impossible to compute the bounds cheaply and accurately. We will consider their practical estimates on $\varepsilon_S$ and $\varepsilon_J$ in Section \[issue\], where we demonstrate that these estimates are cheaply obtainable.
Restarted algorithms and practical stopping criteria for inner iterations {#issue}
=========================================================================
Due to the storage requirement and computational cost, Algorithms \[alg:sira\]–\[alg:jd\] will be impractical for large steps of outer iterations. To be practical, it is necessary to restart them for difficult problems. Let $\bf{\mathbf{M}}_{\max}$ be the maximum of outer iterations allowed. If the basic SIRA and JD algorithms do not converge, then we simply update ${\mathbf{v}}_1$ and restart them. We call the resulting restarted algorithms Algorithms 3–4, respectively.
In implementations, we adopt the following strategy to update ${\mathbf{v}}_1$. For outer iteration steps $i=1,2,
\ldots,{\mathbf{M}}_{\max}$ during the current cycle, suppose $(\nu_1^{(i)},{\mathbf{y}}_1^{(i)})$ is the candidate for approximating the desired eigenpair $(\lambda,x)$ of ${\mathbf{A}}$ at the $i$-th outer iteration. Then we take $$\label{revector}
{\mathbf{v}}_1={\mathbf{y}}=\arg\min_{i=1,2,\ldots,{\mathbf{M}}_{\max}}
\|({\mathbf{A}}-\nu_1^{(i)} {\mathbf{I}}){\mathbf{y}}_1^{(i)}\|$$ as the updated starting vector in the next cycle. Such a restarting strategy guarantees that we use the [*best*]{} candidate Ritz vector in the sense of (\[revector\]) to restart the algorithms.
In what follows we consider some practical issues and design practical stopping criteria for inner iterations in the (non-restarted and restarted) inexact SIRA and JD algorithms.
Given $\tilde{\varepsilon}$, since ${\mathbf{L}}$ is not available, it is impossible to compute ${\rm sep}(\frac{1}{\nu-\sigma},{\mathbf{L}})$ and ${\rm sep}({\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}},{\mathbf{L}})$ in (\[eps\_s\]) and (\[eps\_j\]). Also, we cannot compute $\sin\angle(\cal V,{\mathbf{f}})$ in (\[eps\_s\]) and (\[eps\_j\]). In practice, we simply replace the insignificant factor $\sin\angle(\cal V,{\mathbf{f}})$ by one, which makes $\varepsilon_S$ and $\varepsilon_J$ as small as possible, so that the inexact SIRA and JD algorithms are the safest to mimic the exact SIRA. We replace $\|{\mathbf{B}}\|$ by $\frac{1}{|\nu-\sigma|}$ in the inexact SIRA and JD, respectively. For ${\rm sep}(\frac{1}{\nu-\sigma},{\mathbf{L}})$, we can exploit the spectrum information of ${\mathbf{H}}$ to estimate it. Let $\nu_i,\,i=2,3,\ldots,m$ be the other eigenvalues (Ritz values) of ${\mathbf{H}}$ other than $\nu$. Then we use the estimate $$\label{sepL}
{\rm sep}\left(\frac{1}{\nu-\sigma},{\mathbf{L}}\right)\approx \min_{i=2,3,\ldots,m}
\left|\frac{1}{\nu-\sigma}-\frac{1}{\nu_i-\sigma}\right|.$$ Note that it is very expensive to compute ${\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}$ but ${\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}}\approx\frac{1}{\nu-\sigma}$. So we simply use $\frac{1}{\nu-\sigma}$ to estimate ${\rm sep}\left({\mathbf{y}}^H{\mathbf{B}}{\mathbf{y}},{\mathbf{L}}\right)$. With these estimates and taking the equalities in compact bounds (\[eps\_s\]) and (\[eps\_j\]), we get $$\label{epsilon}
\varepsilon_S=\varepsilon_J=\varepsilon
= 2\tilde{\varepsilon}\max\limits_{i=2,3,\ldots,m}
\left|\frac{\nu_i-\sigma}{\nu_i-\nu}\right|.$$ It might be possible to have $\varepsilon\geq 1$ for a given $\tilde\varepsilon$. This would make $\tilde{\mathbf{u}}$ no accuracy as an approximation to ${\mathbf{u}}$. As a remedy, from now on we set $$\label{epsmin}
\varepsilon=\min\{\varepsilon,0.1\}.$$ For $m=1$, we simply set $\varepsilon=\tilde\varepsilon$.
Note that $\frac{\|\tilde{{\mathbf{u}}}-{\mathbf{u}}\|}{\|{\mathbf{u}}\|}$ is a-priori and uncomputable. We are not able to determine whether it is below $\varepsilon$ or not. However, it is easy to verify that $$\frac{1}{\kappa({\mathbf{B}})}\frac{\|\tilde{{\mathbf{u}}}-{\mathbf{u}}\|}{\|{\mathbf{u}}\|}\leq
\frac{\|{\mathbf{r}}_S-({\mathbf{A}}-\sigma {\mathbf{I}})\tilde{{\mathbf{u}}}\|}{\|{\mathbf{r}}_S\|}\leq\kappa({\mathbf{B}})
\frac{\|\tilde{{\mathbf{u}}}-{\mathbf{u}}\|}{\|{\mathbf{u}}\|} \label{prioris}$$ and $$\frac{1}{\kappa({\mathbf{B}}')}\frac{\|\tilde{{\mathbf{u}}}-{\mathbf{u}}\|}{\|{\mathbf{u}}\|}\leq
\frac{\|-{\mathbf{r}}_J-({\mathbf{I}}-{\mathbf{y}}{\mathbf{y}}^H)({\mathbf{A}}-\sigma {\mathbf{I}})({\mathbf{I}}-{\mathbf{y}}{\mathbf{y}}^H)
\tilde{{\mathbf{u}}}\|}{\|{\mathbf{r}}_J\|}\leq\kappa({\mathbf{B}}')
\frac{\|\tilde{{\mathbf{u}}}-{\mathbf{u}}\|}{\|{\mathbf{u}}\|}, \label{priorij}$$ where $\tilde{\mathbf{u}}\perp{\mathbf{y}}$ and ${\mathbf{B}}'={\mathbf{B}}|_{{\mathbf{y}}^{\perp}}=({\mathbf{A}}-\sigma{\mathbf{I}})^{-1}|_{{\mathbf{y}}^{\perp}}$, the restriction of ${\mathbf{B}}$ to the orthogonal complement of $span\{{\mathbf{y}}\}$. Alternatively, based on the above two relations, in practice we require that inner solves stop when the a-posteriori computable relative residual norms $$\label{stopcrit}\frac{\|{\mathbf{r}}_S-({\mathbf{A}}-\sigma {\mathbf{I}})\tilde{{\mathbf{u}}}\|}{\|{\mathbf{r}}_S\|}
\leq\varepsilon$$ and $$\label{stopcritjd}\frac{\|-{\mathbf{r}}_J-({\mathbf{I}}-{\mathbf{y}}{\mathbf{y}}^H)({\mathbf{A}}-\sigma {\mathbf{I}})({\mathbf{I}}-{\mathbf{y}}{\mathbf{y}}^H)
\tilde{{\mathbf{u}}}\|}{\|{\mathbf{r}}_J\|}
\leq\varepsilon$$ for the inexact SIRA and JD, respectively.
[*Remark.*]{} In [@spence2009ia; @simoncini2005ia; @simoncini2003ia], a-priori accuracy requirements have been determined for inner iterations in SIA type methods. In computation, a-posteriori residuals are intuitive, and are probably the only practical way to approximate the a-priori residuals. Here, by the above lower and upper bounds (\[prioris\]) and (\[priorij\]) that relate the a-posteriori relative residuals to the a-priori errors of approximate solutions, we have simply demonstrated that (\[stopcrit\]) and (\[stopcritjd\]) are reasonable stopping criteria for inner solves. We see that the a-priori errors and the a-posteriori errors are definitely comparable once the linear systems are not ill conditioned.
Numerical experiments {#numer}
=====================
We report numerical experiments to confirm our theory. Our aims are mainly three-fold: (i) Regarding outer iterations, for fairly small $\tilde\varepsilon=10^{-3}$ and $10^{-4}$, the (non-restarted and restarted) inexact SIRA and JD behave very like the (non-restarted and restarted) exact SIRA. Even a bigger $\tilde\varepsilon=10^{-2}$ often works very well. (ii) Regarding inner iterations and overall efficiency, the inexact SIRA and JD algorithms are considerably more efficient than the inexact SIA. (iii) SIRA and JD are similarly effective.
All the numerical experiments were performed on an Intel (R) Core (TM)2 Quad CPU Q9400 $2.66$GHz with main memory 2 GB using Matlab 7.8.0 with the machine precision $\epsilon_{\rm mach}=2.22\times
10^{-16}$ under the Microsoft Windows XP operating system.
At the $m$th step of the inexact SIRA or JD method, we have ${\mathbf{H}}_m={\mathbf{V}}_m^H{\mathbf{A}}{\mathbf{V}}_m$. Let $(\nu_i^{(m)},{\mathbf{z}}_i^{(m)}),\ i=1,2,\ldots,m$ be the eigenpairs of ${\mathbf{H}}_m$, which are ordered as $$|\nu_1^{(m)}-\sigma|<|\nu_2^{(m)}-\sigma|\leq\cdots\leq |\nu_m^{(m)}-\sigma|.$$ We use the Ritz pair $(\nu_m,{\mathbf{y}}_m):=(\nu_1^{(m)},{\mathbf{V}}_m{\mathbf{z}}_1^{(m)})$ to approximate the desired eigenpair $(\lambda,x)$ of ${\mathbf{A}}$, and the associated residual is ${\mathbf{r}}_m={\mathbf{A}}{\mathbf{y}}_m-\nu_m{\mathbf{y}}_m$.
We stop the algorithms if $$\|{\mathbf{r}}_m\|\leq tol=\max\left\{\|{\mathbf{A}}\|_1,1\right\}\times10^{-10}.$$ In the inexact SIRA and JD, we stop inner solves when (\[stopcrit\]) and (\[stopcritjd\]) are satisfied, respectively, and denote by SIRA($\tilde\varepsilon$) and JD($\tilde\varepsilon$) the inexact SIRA and JD algorithms with the given parameter $\tilde\varepsilon$. We use the following stopping criteria for inner iterations in the exact SIRA and SIA algorithms and the inexact SIA algorithm.
- For the exact SIRA algorithm, we require the approximate solution $\tilde{{\mathbf{u}}}_{m+1}$ to satisfy $$\frac{\|{\mathbf{r}}_m-({\mathbf{A}}-\sigma{\mathbf{I}})\tilde{{\mathbf{u}}}_{m+1}\|}{\|{\mathbf{r}}_m\|}\leq10^{-14}.$$
- For the inexact SIA algorithm, we take the same outer iteration tolerance $tol=\max\left\{\|{\mathbf{A}}\|_1,1\right\}\times10^{-10}$, and use the stopping criterion (3.14) in [@spence2009ia] for inner solve, where $\varepsilon=tol$ and the steps $m$ suitably bigger than the number of outer iterations used by the exact SIRA so as to ensure the convergence of the inexact SIA with the same accuracy. For the restarted inexact SIA, we take $m$ the maximum outer iterations ${\mathbf{M}}_{\max}$ allowed for each cycle.
In the numerical experiments, we always take the zero vector as an initial approximate solution to each inner linear system and solve it by the right-preconditioned GMRES(30) method. Outer iterations start with the normalized vector $\frac{1}{\sqrt{n}}(1,1,\ldots,1)^H$. For the correction equation in the JD method, we use $$\tilde{{\mathbf{M}}}_m=({\mathbf{I}}-{\mathbf{y}}_m{\mathbf{y}}_m^H){\mathbf{M}}({\mathbf{I}}-{\mathbf{y}}_m{\mathbf{y}}_m^H),$$ the restriction of $M$ to the orthogonal complement of $span\{{\mathbf{y}}_m\}$, as a preconditioner, which is suggested in [@vandervorst2002eigenvalue]. $\tilde{{\mathbf{M}}}_m^{-1}|_{{\mathbf{y}}_m^{\perp}}$ means the inverse of $\tilde{{\mathbf{M}}}_m$ restricted to the orthogonal complement of $span\{{\mathbf{y}}_m\}$. Here ${\mathbf{M}}\approx{\mathbf{A}}-\sigma{\mathbf{I}}$ is some preconditioner used for all the inner linear systems involved in the algorithms tested except JD. We use the Matlab function $[L,U]=ilu(A-sigma*speye(n),setup)$ to compute the sparse incomplete LU factorization of $A-\sigma I$ with a given dropping tolerance $setup.droptol$. We then take $M=LU$. van der Vorst [@vandervorst2002eigenvalue] shows how to use $\tilde{{\mathbf{M}}}_m$ as a left preconditioner for (\[ls\_jd\]). It can also be used a right preconditioner for (\[ls\_jd\]) in the same spirit. Adapted from [@vandervorst2002eigenvalue p. 137-8], we briefly describe how to do so. Suppose that a Krylov solver for (\[ls\_jd\]) with right-preconditioning starts with zero vector as an initial guess to the solution. Then the starting vector for the Krylov solver is ${\mathbf{r}}_m$, which is in the subspace orthogonal to ${\mathbf{y}}_m$, and all iteration vectors for the Krylov solver are in that subspace. We compute $\tilde{{\mathbf{M}}}_m^{-1}|_{{\mathbf{y}}_m^{\perp}}{\mathbf{w}}$ for a vector ${\mathbf{w}}$ supplied by the Krylov solver at each inner iteration. Let ${\mathbf{z}}=\tilde{{\mathbf{M}}}_m^{-1}|_{{\mathbf{y}}_m^{\perp}}{\mathbf{w}}$ and note that ${\mathbf{z}}\perp{\mathbf{y}}_m$. Then it follows that $${\mathbf{w}}=\tilde{{\mathbf{M}}}_m{\mathbf{z}}=({\mathbf{I}}-{\mathbf{y}}_m{\mathbf{y}}_m^H){\mathbf{M}}{\mathbf{z}}={\mathbf{M}}{\mathbf{z}}-\beta{\mathbf{y}}_m,$$ where $\beta={\mathbf{y}}_m^H{\mathbf{M}}{\mathbf{z}}$. Equivalently, ${\mathbf{z}}={\mathbf{M}}^{-1}{\mathbf{w}}+\beta{\mathbf{M}}^{-1}{\mathbf{y}}_m$. Again, using ${\mathbf{z}}\perp{\mathbf{y}}_m$, we have ${\mathbf{y}}_m^H{\mathbf{M}}^{-1}{\mathbf{w}}+\beta{\mathbf{y}}_m^H{\mathbf{M}}^{-1}{\mathbf{y}}_m=0$, i.e., $\beta=-\frac{{\mathbf{y}}_m^H{\mathbf{M}}^{-1}{\mathbf{w}}}{{\mathbf{y}}_m^H{\mathbf{M}}^{-1}{\mathbf{y}}_m}$. Therefore, we can compute $\tilde{{\mathbf{M}}}_m^{-1}|_{{\mathbf{y}}_m^{\perp}}{\mathbf{w}}$ by $$\tilde{{\mathbf{M}}}_m^{-1}{\mathbf{w}}={\mathbf{M}}^{-1}{\mathbf{w}}-\left(\frac{{\mathbf{y}}_m^H{\mathbf{M}}^{-1}{\mathbf{w}}}{{\mathbf{y}}_m^H{\mathbf{M}}^{-1}{\mathbf{y}}_m}\right)
{\mathbf{M}}^{-1}{\mathbf{y}}_m.$$
In all the tables below, we denote by $I_{out}$ the number of outer iterations to achieve the convergence, by $I_{inn}$ the total number of inner iterations, i.e., the products of the matrix $A$ by vectors used by the Krylov solver, by $I_{0.1}$ the times of $\varepsilon=0.1$, by $T_1$ the total CPU time of solving the small eigenproblems, by $T_2$ the total CPU time of generating the orthonormal basis ${\mathbf{V}}$ and forming the projection matrix ${\mathbf{H}}$, by $T_3$ the time of constructing the preconditioner and by $T_4$ the total CPU time of the Krylov solver for solving right-preconditioned inner linear systems. We point out that the (inexact and exact) SIRA and JD methods must form the projection matrices explicitly while SIA does not and it gives its projection matrix as a byproduct when generating the orthonormal basis of ${\mathbf{V}}$. As a result, for the same dimension of subspace, $T_2$ for SIA is smaller than that for SIRA and JD. This will be confirmed clearly in later numerical experiments, and we will not mention this observation later. For Examples 1–3 we test Algorithms \[alg:sira\]–\[alg:jd\], the inexact SIA and exact SIRA; for Example 4 we test these algorithms and the restarted Algorithms 3–4 as well as the restarted inexact SIA.
**Example 1.** This problem is a large nonsymmetric standard eigenvalue problem of cry10000 of $n=10000$ that arises from the stability analysis of a crystal growth problem from [@matrixmarket]. We are interested in the eigenvalue nearest to $\sigma=7$. The computed eigenvalue is $\lambda\approx6.7741$. The preconditioner ${\mathbf{M}}$ is obtained by the sparse incomplete LU factorization of ${\mathbf{A}}-\sigma {\mathbf{I}}$ with $setup.droptol=0.001$. Table \[tab\_cry10000\] reports the results obtained, and the left and right parts of Figure \[fig\_cry10000\] depict the convergence curve of $\frac{\|{\mathbf{r}}_m\|}{\|{\mathbf{A}}\|_1}$ versus $I_{out}$ and the curve of $I_{inn}$ versus $I_{out}$ for the algorithms, respectively.
![*Example 1. cry10000 with $\sigma=7$. Left: relative outer residual norms versus outer iterations. Right: the numbers of inner iterations versus outer iterations.*[]{data-label="fig_cry10000"}](cry10000_norm.eps){height="5.8cm"}
![*Example 1. cry10000 with $\sigma=7$. Left: relative outer residual norms versus outer iterations. Right: the numbers of inner iterations versus outer iterations.*[]{data-label="fig_cry10000"}](cry10000_iter.eps){height="5.8cm"}
Algorithm $I_{inn}$ $I_{out}$ $I_{0.1}$ $T_1$ $T_2$ $T_3$ $T_4$
----------------- -- ----------- ----------- ----------- -- ------- ------- ------- --------
SIRA($10^{-2}$) $36$ $11$ $0$ $1$ $18$ $121$ $81$
JD($10^{-2}$) $38$ $12$ $0$ $2$ $21$ $121$ $103$
SIRA($10^{-3}$) $57$ $12$ $0$ $2$ $21$ $121$ $120$
JD($10^{-3}$) $57$ $12$ $0$ $2$ $21$ $121$ $136$
SIRA($10^{-4}$) $88$ $13$ $0$ $2$ $24$ $121$ $184$
JD($10^{-4}$) $78$ $12$ $0$ $2$ $21$ $121$ $176$
Inexact SIA $131$ $14$ $-$ $2$ $13$ $121$ $340$
Exact SIRA $277$ $11$ $-$ $1$ $18$ $121$ $1386$
: *Example 1. cry10000 with $\sigma=7$ (The unit of $T_1\sim T_4$ is $0.001$ second).*[]{data-label="tab_cry10000"}
We see from Table \[tab\_cry10000\] and Figure \[fig\_cry10000\] that for both $\tilde{\varepsilon}=10^{-2},10^{-3}$ the inexact SIRA and JD behaved like the exact SIRA very much and used almost the same outer iterations, while the inexact SIA had a small convergence delay. Clearly, smaller $\tilde{\varepsilon}$ is not necessary as it cannot reduce outer iterations anymore.
Regarding the overall efficiency, the exact SIRA was obviously the most expensive, as $I_{inn}$ and the dominant CPU time $T_3,\ T_4$ indicated. It used $27\sim29$ inner iterations per outer iteration. The inexact SIA was the second most expensive, in terms of the same measures. For it, the numbers of inner iterations were comparable and between $11\sim14$ at each of the first $7$ outer iterations where the accuracy of approximate eigenpairs was poor and the inner linear systems must be solved with high accuracy. As the approximate eigenpairs started converging, the relaxation strategy came into picture and the inner linear systems were solved with decreasing accuracy, leading to fewer inner iterations at subsequent outer iterations. Inner iterations used by the inexact SIA were only comparable to and finally below those used by the inexact SIRA and JD in the last very few iterations. In contrast, the figure indicates that, for the same $\tilde{\varepsilon}$, the inexact SIRA and JD solved the linear systems with almost the same inner iterations per outer iteration. Because of this, the inexact SIRA and JD were much more efficient than the inexact SIA and used much fewer inner iterations and computing time than the latter. Both the $I_{inn}$ and the total computing time in Table \[tab\_cry10000\] show that they were roughly one and a half to three times as fast as the inexact SIA, and SIRA and JD with $\tilde{\varepsilon}=10^{-2}$ were considerably more efficient than that with $\tilde{\varepsilon}=10^{-3}$, $10^{-4}$. Finally, we observe that the inexact SIRA and JD were equally effective, as indicated by the $I_{inn}$ and the computing time used for each $\tilde{\varepsilon}$.
In addition, we see from Table \[tab\_cry10000\] that $T_3$ is comparable to and can be more than $T_4$ when inner linear systems are solved with low accuracy, and it is less important for the inexact SIA, where the accuracy of inner inner iterations increases as outer iterations proceed, and especially for the exact SIRA, where inner linear systems are required to be solved exactly in finite precision arithmetic.
**Example 2.** We consider the unsymmetric sparse matrix sherman5 of $n=3312$ that has been used in [@spence2009ia; @simoncini2005ia] for testing the relaxation theory with $\sigma=0$. The computed eigenvalues is $\lambda\approx4.6925\times10^{-2}$. The preconditioner ${\mathbf{M}}$ is obtained by the sparse incomplete LU factorization of ${\mathbf{A}}-\sigma {\mathbf{I}}$ with $setup.droptol=0.001$. Table \[tab\_cry10000\] and Figure \[fig\_cry10000\] describe the results and convergence processes.
![*Example 2. sherman5 with $\sigma=0$. Left: relative outer residual norms versus outer iterations. Right: the numbers of inner iterations versus outer iterations.*[]{data-label="fig_sherman5"}](sherman5_norm.eps){height="5.8cm"}
![*Example 2. sherman5 with $\sigma=0$. Left: relative outer residual norms versus outer iterations. Right: the numbers of inner iterations versus outer iterations.*[]{data-label="fig_sherman5"}](sherman5_iter.eps){height="5.8cm"}
Algorithm $I_{inn}$ $I_{out}$ $I_{0.1}$ $T_1$ $T_2$ $T_3$ $T_4$
----------------- -- ----------- ----------- ----------- -- ------- ------- ------- --------
SIRA($10^{-2}$) $58$ $8$ $0$ $7$ $32$ $483$ $1713$
JD($10^{-2}$) $38$ $10$ $0$ $8$ $44$ $483$ $1425$
SIRA($10^{-3}$) $62$ $7$ $0$ $5$ $25$ $483$ $1820$
JD($10^{-3}$) $37$ $7$ $0$ $5$ $24$ $483$ $1259$
SIRA($10^{-4}$) $74$ $7$ $0$ $5$ $26$ $483$ $2174$
JD($10^{-4}$) $48$ $7$ $0$ $5$ $26$ $483$ $1567$
Inexact SIA $94$ $7$ $-$ $4$ $12$ $483$ $2821$
Exact SIRA $172$ $7$ $-$ $6$ $29$ $484$ $6583$
: *Example 2. sherman5 with $\sigma=0$ (The unit of $T_1\sim T_4$ is $0.0001$ second).*[]{data-label="tab_sherman5"}
We see from the left part of Figure \[fig\_sherman5\] that the inexact SIRA, JD and SIA behaved like the exact SIRA very much and used very comparable outer iterations. They mimic the exact SIRA better for $\tilde{\varepsilon}=10^{-3},10^{-4}$ than for $\tilde{\varepsilon}=10^{-2}$. The table also tells us that a smaller $\tilde{\varepsilon}<10^{-3}$ is definitely not necessary as it could not reduce the number of outer iterations and meanwhile consumed more inner iterations. The results confirm our theory and indicate that our selection of $\tilde{\varepsilon}$ and $\varepsilon$ worked very well. It is obvious that, as far as outer iterations are concerned, all the algorithms converged quickly and smoothly.
For the overall efficiency, the situation is very different. As is expected, we see from Table \[tab\_sherman5\] and Figure \[fig\_sherman5\] that the exact SIRA was the most expensive and the inexact SIA with was the second most expensive, as the $I_{inn}$ and the total computing time indicated. The exact SIRA used $28\sim29$ inner iterations per outer iteration, and the inexact SIA used $17$ inner iterations at each of the first $3$ outer iterations where the accuracy of approximate eigenpairs was poor and the inner linear systems must be solved with high accuracy. As the approximate eigenpairs started converging, the relaxation strategy took effect and the inner linear systems were solved with decreasing accuracy, so that the numbers of inner iterations became increasingly smaller as outer iterations proceeded. In contrast, the inexact SIRA and JD were much more efficient than the inexact SIA, they used much fewer inner iterations and computing time than the latter and were roughly one and a half to two times as fast as the inexact SIA. Furthermore, we observe that the inexact JD and SIRA used quite few and almost constant inner iterations per outer iteration for each $\tilde{\varepsilon}$, respectively, but the former was more effective than the latter. This may be due to the better conditioning of the coefficient matrix in the correction equation of JD.
Also, we observe from Table \[tab\_sherman5\] that the time $T_4$ of solving preconditioned inner linear systems dominates the total CPU time and on the other hand the construction of preconditioners is the second most expensive. So solving inner linear systems overwhelms is much more than the others, and both $I_{inn}$ and the sum of $T_4$ and $T_3$ reflect the overall efficiency of each algorithm very well.
**Example 3.** This problem arises from computational fluid dynamics and the test matrix af23560 of $n=23560$ is from transient stability analysis of Navier-Stokes solvers [@matrixmarket]. We want to find the eigenvalue nearest to $\sigma=0$. The computed eigenvalue is $\lambda\approx-0.2731$. The preconditioner ${\mathbf{M}}$ is obtained by the sparse incomplete LU factorization of ${\mathbf{A}}-\sigma {\mathbf{I}}$ with $setup.droptol=0.01$; see Table \[tab\_af23560\] and Figure \[fig\_af23560\] for the results.
![*Example 3. af23560 with $\sigma=0$. Left: outer residual norms versus outer iterations. Right: the numbers of inner iterations versus outer iterations.*[]{data-label="fig_af23560"}](af23560_norm.eps){height="5.8cm"}
![*Example 3. af23560 with $\sigma=0$. Left: outer residual norms versus outer iterations. Right: the numbers of inner iterations versus outer iterations.*[]{data-label="fig_af23560"}](af23560_iter.eps){height="5.8cm"}
Algorithm $I_{inn}$ $I_{out}$ $I_{0.1}$ $T_1$ $T_2$ $T_3$ $T_4$
----------------- -- ----------- ----------- ----------- -- ------- ------- ------- --------
SIRA($10^{-2}$) $258$ $32$ $19$ $1$ $68$ $89$ $1130$
JD($10^{-2}$) $250$ $31$ $23$ $1$ $63$ $89$ $1140$
SIRA($10^{-3}$) $283$ $24$ $0$ $1$ $37$ $89$ $1316$
JD($10^{-3}$) $324$ $25$ $0$ $1$ $35$ $89$ $1519$
SIRA($10^{-4}$) $429$ $23$ $0$ $1$ $35$ $89$ $2058$
JD($10^{-4}$) $400$ $23$ $0$ $1$ $32$ $89$ $1888$
Inexact SIA $1025$ $24$ $-$ $1$ $8$ $89$ $4232$
Exact SIRA $1967$ $24$ $-$ $1$ $31$ $89$ $8664$
: *Example 3. af23560 with $\sigma=0$ (The unit of $T_1\sim T_4$ is $0.01$ second).*[]{data-label="tab_af23560"}
Compared with Examples 1–2, we see from both Table \[tab\_af23560\] and Figure \[fig\_af23560\] that for this problem all the algorithms used considerably more outer iterations $I_{out}$ but $I_{inn}$ increases more rapidly than $I_{out}$ does. So this problem was considerably more difficult than the previous two ones. The difficulty is two-fold: the eigenvalue problem itself and the inner linear systems involved in the algorithms. The second difficulty means that $T_4$ is more dominant than it for Examples 1–2. Moreover, we see that $T_4$ is much more than the corresponding $T_3$, the setup time of the preconditioner. As as whole, $I_{inn}$ and the time of solving inner linear systems reflect the overall efficiency of an algorithm more accurately.
In this example, the case that $\varepsilon=0.1$ occurred at about $60\%$ and $75\%$ of outer iterations in SIRA($10^{-2}$) and JD($10^{-2}$), respectively. Regarding outer iterations, we observe from Figure \[fig\_af23560\] that for $\tilde{\varepsilon}=10^{-3}$ the inexact SIRA, JD and SIA behaved like the exact SIRA very much. For the bigger $\tilde{\varepsilon}=10^{-2}$, the inexact SIRA and SIA used more outer iterations and did not mimic the exact SIRA well. It is amazing that SIRA($10^{-4}$) and JD($10^{-4}$) used one less outer iteration than the exact SIRA. Again, the results confirmed our theory and showed that a low or modest accuracy $\tilde{\varepsilon}=10^{-3}$ is enough, a looser $\tilde{\varepsilon}=10^{-2}$ worked quite well and only a little bit more outer iterations were needed for it.
For the overall efficiency, the inexact SIA was better than the exact SIRA but much inferior to the inexact SIRA and JD. Actually, as $I_{inn}$ and $T_4$ show, the inexact SIRA and JD with $\tilde{\varepsilon}=10^{-2},10^{-3}$ were twice to almost four times as fast as the inexact SIA. Although SIRA($10^{-2}$) and JD($10^{-2}$) used more outer iterations than the others, they were the most efficient in terms of both $I_{inn}$ and $T_4$. The exact SIRA used roughly $85$ inner iterations per outer iteration. The inexact SIA used many inner iterations and needed to solve inner linear systems with high accuracy for most of the outer iterations. Even after the relaxation strategy played a role, it still used much more inner iterations than the inexact SIRA and JD with $\tilde{\varepsilon}=10^{-2},10^{-3}$ at each outer iteration. Although SIRA($10^{-4}$) and JD($10^{-4}$) behaved like the exact SIRA best and won all the others in terms of $I_{out}$, the overall efficiency of them was not as good as that of the the inexact methods with bigger $\tilde{\varepsilon}$. We find that, for the same accuracy $\tilde{\varepsilon}$, the inexact SIRA and JD solved the linear systems with slowly varying inner iterations at each outer iteration. This is expected as the accuracy requirements of inner iterations were almost the same. In terms of $I_{inn}$ and $T_4$, we also observe from Table \[tab\_af23560\] that the inexact SIRA and JD were equally effective and had very similar efficiency.
Still, similar to Examples 1–2, we see from $T_1\sim T_4$ that solving preconditioned inner linear systems is the most expensive and dominates the overall efficiency of each algorithm, while the construction of preconditioners overwhelms the solutions of small eigensystems as well as the generations of orthonormal basis and projected matrices.
**Example 4.** This unsymmetric eigenvalue problem dw8192 of $n=8192$ arises from dielectric channel waveguide problems [@matrixmarket]. We are interested in the eigenvalue nearest to the complex target $\sigma=0.01{\mathrm{i}}$. The computed eigenvalue is $\lambda\approx3.3552\times10^{-3}+1.1082\times10^{-3}{\mathrm{i}}$ The preconditioner ${\mathbf{M}}$ is obtained by the sparse incomplete LU factorization of ${\mathbf{A}}-\sigma {\mathbf{I}}$ with $setup.droptol=0.001$. Table \[tab\_dw8192\] displays the results.
Algorithm $I_{inn}$ $I_{out}$ $I_{0.1}$ $T_1$ $T_2$ $T_3$ $T_4$
----------------- -- ----------- ----------- ----------- -- ------- ------- ------- -------
SIRA($10^{-2}$) $312$ $99$ $82$ $12$ $53$ $3$ $129$
JD($10^{-2}$) $276$ $93$ $81$ $11$ $47$ $3$ $144$
SIRA($10^{-3}$) $386$ $87$ $0$ $8$ $41$ $3$ $144$
JD($10^{-3}$) $428$ $94$ $1$ $11$ $47$ $3$ $192$
SIRA($10^{-4}$) $466$ $71$ $0$ $4$ $26$ $3$ $171$
JD($10^{-4}$) $451$ $70$ $0$ $4$ $25$ $3$ $183$
Inexact SIA $1663$ $86$ $-$ $7$ $8$ $3$ $616$
Exact SIRA $1940$ $66$ $-$ $3$ $21$ $3$ $741$
: *Example 4. dw8192 with $\sigma=0.01{\mathrm{i}}$ (The unit of $T_1\sim T_4$ is $0.1$ second).*[]{data-label="tab_dw8192"}
As far as the eigenvalue problem is concerned, Table \[tab\_dw8192\] clearly indicates that this problem is much more difficult than Examples 1–3 since all the algorithms used much more outer iterations to achieve the convergence than those needed for Examples 1–3. But our inexact SIRA and JD algorithms still worked very well. The inexact SIRA and JD with $\tilde{\varepsilon}=10^{-4}$ behaved more like the exact SIRA than with $\tilde{\varepsilon}=10^{-3}$ and $\tilde{\varepsilon}=10^{-2}$. Therefore, we can infer that a smaller $\tilde{\varepsilon}<10^{-4}$ is not necessary and cannot improve the behavior of the inexact SIRA and JDl; it will make the inexact methods use almost the same outer iterations as the exact SIRA but consume more inner iterations. Furthermore, we have observed the inexact SIA did not mimic the exact SIRA very well as it used considerably more outer iterations than the exact SIRA.
For the overall efficiency, Table \[tab\_dw8192\] exhibited similar features to those in all the previous tables for Examples 1–3. The inexact SIRA and JD were similarly effective. Both of them were much more efficient than the inexact SIA and actually three to five times as fast as the latter, in terms of both $I_{inn}$ and the total computing time.
Since this problem is difficult, we turn to use restarted SIRA and JD algorithms, Algorithms 3–4, to solve it with the maximum ${\mathbf{M}}_{\max}=30$ outer iterations allowed during each cycle. We also test the implicitly restarted inexact SIA method [@spence2009ia; @xueelman10] with the same ${\mathbf{M}}_{\max}=30$ and make a comparison of all the restarted algorithms. Table \[tab\_dw8192\_restart\] lists the results obtained by the restarted inexact SIRA, JD and SIA as well as the restarted exact SIRA, where $I_{restart}$ denotes the number of restarts used, i.e., the number of the cycles of Algorithms 1–2 for the given ${\mathbf{M}}_{\max}$. Figure \[fig\_dw8192\_restart\] depicts the convergence curve of all the restarted algorithms and the curve of $I_{inn}$ versus $I_{restart}$, in which the zeroth restart in abscissa denotes the first cycle of Algorithms 3–4 and corresponds to the first restart in the left figure.
![*Example 4. Restarted algorithms with ${\mathbf{M}}_{\max}=30$. Left: outer residual norms versus outer iterations. Right: the numbers of inner iterations versus restarts.*[]{data-label="fig_dw8192_restart"}](dw8192_norm_restart.eps){height="5.8cm"}
![*Example 4. Restarted algorithms with ${\mathbf{M}}_{\max}=30$. Left: outer residual norms versus outer iterations. Right: the numbers of inner iterations versus restarts.*[]{data-label="fig_dw8192_restart"}](dw8192_iter_restart.eps){height="5.8cm"}
Algorithm $I_{inn}$ $I_{restart}$ $I_{out}$ $I_{0.1}$ $T_1$ $T_2$ $T_3$ $T_4$
----------------- -- ----------- --------------- ----------- ----------- -- ------- ------- ------- -------
SIRA($10^{-2}$) $876$ $8$ $250$ $142$ $2$ $32$ $3$ $346$
JD($10^{-2}$) $578$ $5$ $175$ $98$ $2$ $23$ $3$ $284$
SIRA($10^{-3}$) $518$ $3$ $115$ $0$ $1$ $15$ $3$ $203$
JD($10^{-3}$) $532$ $3$ $117$ $1$ $1$ $15$ $3$ $234$
SIRA($10^{-4}$) $601$ $3$ $100$ $0$ $1$ $12$ $3$ $222$
JD($10^{-4}$) $624$ $3$ $98$ $0$ $1$ $12$ $3$ $262$
Inexact SIA $1710$ $3$ $95$ $-$ $1$ $3$ $3$ $602$
Exact SIRA $2521$ $2$ $89$ $-$ $1$ $11$ $3$ $971$
: *Example 4. Restarted algorithms with ${\mathbf{M}}_{\max}=30$ (The unit of $T_1\sim T_4$ is $0.1$ second).*[]{data-label="tab_dw8192_restart"}
It is seen from Table \[tab\_dw8192\_restart\] and the left part of Figure \[fig\_dw8192\_restart\] that all the algorithms other than SIRA($10^{-2}$) and JD($10^{-2}$) solved the problem very successfully with no more than three restarts used and the convergence processes were very smooth. The restarted inexact SIA behaved like the restarted exact SIRA well but not so well as the restarted SIRA and JD with $\tilde{\varepsilon}=10^{-4}$, which behaved very like the restarted exact SIRA in the first two restarts and almost converged to our prescribed convergence accuracy at the second restart.
We also find that, compared with Table \[tab\_dw8192\_restart\], the restarted SIRA($10^{-4}$), JD($10^{-4}$) and exact SIRA performed excellently since $I_{out}$’s used by them were very near to the ones used by their corresponding non-restarted versions, respectively. For the restarted SIRA($10^{-2}$) and JD($10^{-2}$), the case that $\varepsilon=0.1$ occurred at $50\%$ of outer iterations. They did not mimic the exact SIRA well and used considerably more outer iterations than the inexact SIRA and JD with $\tilde{\varepsilon}=10^{-3}$ and $\tilde{\varepsilon}=10^{-4}$. So $\tilde{\varepsilon}=10^{-2}$ is not a good choice for the restarted inexact SIRA and JD for this example, though $I_{inn}$ and the total computing time are not so considerably more than those used by the algorithms with $\tilde\varepsilon=10^{-3},\ 10^{-4}$.
Regarding the overall performance, for given $\tilde{\varepsilon}=10^{-3}$ and $\tilde{\varepsilon}=10^{-4}$, the restarted SIRA and JD algorithms performed very similarly and were about more than twice as fast as the restarted inexact SIA, in terms of both $I_{inn}$ and the total computing time (actually $T_4$ now). During the last cycle, the restarted inexact SIRA($10^{-4}$) and JD($10^{-4}$) had already achieved the convergence at the tenth and eighth outer iteration, respectively. So we stopped the algorithm at that step and actually solved only about a third of twenty-nine inner linear systems needed to solve in each of the previous cycles. As a result, the number of inner iterations needed in the last circle was also about a third of that needed in each of the first three cycles. This is the reason why, in the right part of Figure \[fig\_dw8192\_restart\], the curves for the restarted SIRA($10^{-4}$) and JD($10^{-4}$) had a drastic decrease at last restart. As is expected, the restarted inexact SIRA and JD algorithms used almost constant inner iterations for the same $\tilde{\varepsilon}$ per restart, while the inexact SIA used fewer and fewer inner iterations as outer iterations converged. The figure clearly shows that the restarted inexact SIA used much more inner iterations than the restarted SIRA($10^{-4}$) and JD($10^{-4}$) at each of the first three cycles.
We see from Tables \[tab\_dw8192\]–\[tab\_dw8192\_restart\] that for this example the dominant cost is still paid to the solutions of preconditioned inner linear systems but unlike Examples 1–3 the construction of preconditioners is very cheap and negligible, compared with $T_4$.
In summary, it is seen from all the numerical experiments that both $I_inn$ and $T_4$ are reasonable measures of overall performance of SIRA, JD and SIA algorithms.
We have tested some other problems. We have also tested the algorithms when tuning is applied to our preconditioner ${\mathbf{M}}$ [@spence2009ia]. All of them have shown that the inexact SIRA and JD mimic the inexact SIA and the exact SIRA very well for $\tilde{\varepsilon}=10^{-3},10^{-4}$ and use much fewer inner iterations than the inexact SIA. As far as the overall efficiency is concerned, SIRA($10^{-2}$) and JD($10^{-2}$) may work well and often use comparably inner iterations than SIRA($10^{-3}$) and JD($10^{-3}$), but they are likely to need considerably more outer iterations and cannot mimic the exact SIRA well. Therefore, for the robust and general purpose, we propose using $\tilde{\varepsilon}\in[10^{-4},10^{-3}]$ in practice. We have found that the tuned preconditioning has no advantage over the usual preconditioning and is often inferior to the latter for the linear systems involved in the inexact SIRA, JD and SIA algorithms. For example, we have found that for Example 3 the tuned preconditioning used about three times more inner iterations than the usual preconditioning.
Conclusions and future work {#concl}
===========================
We have quantitatively analyzed the convergence of the SIRA and and JD methods over one step and proved that one only needs to solve all the inner linear systems involved in them with low or modest accuracy. Based on the theory established, we have designed practical stopping criteria for inner iterations of the inexact SIRA and JD. Numerical experiments have illustrated that our theory works very well and the non-restarted and restarted inexact SIRA and JD algorithms behave very like the non-restarted and restarted exact SIRA algorithms. Meanwhile, we have confirmed that the inexact SIRA and JD algorithms are similarly effective and both of them are much more efficient than the inexact SIA algorithms.
It is known that the (inexact) JD method with variable shifts is used more commonly. The analysis approach proposed in this paper may be extended to analyze the accuracy requirements of inner iterations in the JD method with variable shifts and a rigorous general theory may be expected. This work is in progress.
Since the harmonic projection may be more suitable to solve the interior eigenvalue problem, it is very significant to consider the harmonic version of SIRA. Moreover, it is known that the standard projection, i.e., the Rayleigh–Ritz method, and its harmonic version may have convergence problem when computing eigenvectors [@jia2001analysis; @jia05]. So it is worthwhile and appealing to use the refined Rayleigh–Ritz procedure [@jia97; @jia2001analysis] and the refined harmonic version [@jia2001analysis] for solving the large eigenproblem considered in this paper. These constitute our future work.
[**Acknowledgements**]{}. We thank the two referees for their comments and suggestions.
[10]{}
, [*Test matrix collection for non-Hermitian eigenvalue problems*]{}, http://math.nist.gov/MatrixMarket/.
, [ *[Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide]{}*]{}, SIAM, Philadelphia, PA, 2000.
, [*Shift-and-invert Arnoldi’s method with preconditioned iterative solvers*]{}, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 942–969.
, [*Controlling inner iterations in the Jacobi–Davidson method*]{}, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 460-477.
, [*The convergence of generalized Lanczos methods for large unsymmetric eigenproblems*]{}, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 843–862.
height 2pt depth -1.6pt width 23pt, [*Refined iterative algorithms based on Arnoldi’s process for unsymmmetric eigenproblems*]{}, Linear Algebra Appl., 259 (1997), pp. 1–23.
height 2pt depth -1.6pt width 23pt, [*Generalized block Lanczos methods for large unsymmetric eigenproblems*]{}, Numer. Math., 80 (1998), pp. 239–266.
height 2pt depth -1.6pt width 23pt, [*The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors*]{}, Math. Comput., 74 (2005), pp. 1441–1456.
, [*An analysis of the Rayleigh–Ritz method for approximating eigenspaces*]{}, Math. Comput., 70 (2001), pp. 637–648.
, [*Residual Arnoldi method: theory, package and experiments*]{}, Ph.D thesis, TR-4515, Department of Computer Science, University of Maryland at College Park, 2007.
, [*Analysis of the residual Arnoldi method*]{}, TR-4890, Department of Computer Science, University of Maryland at College Park, 2007.
, [*Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations*]{}, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1112–1135.
, [*Combination of Jacobi–Davidson and conjugate gradients for the partial symmetric eigenproblem*]{}, Numer. Linear Algebra Appl., 9 (2002), pp. 21–44.
, [*The Symmetric Eigenvalue Problem*]{}, SIAM, Philadelphia, PA, 1998.
, [*Numerical Methods for Large Eigenvalue Problems*]{}, Manchester University Press, UK, 1992.
, [*Variable accuracy of matrix-vector products in projection methods for eigencomputation*]{}, SIAM J. Numer. Anal., 43 (2005), pp. 1155–1174.
, [*Theory of inexact Krylov subspace methods and applications to scientific computing*]{}, SIAM J. Sci. Comput., 25 (2003), pp. 454–477.
, [*A Jacobi–Davidson iteration method for linear eigenvalue problems*]{}, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401–425. Reprinted in SIAM Review, (2000), pp. 267–293.
, [*Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: Seeking one eigenvalue*]{}, SIAM J. Sci. Comput., 29 (2007), pp. 2162–2188.
, [*Matrix Algorithms Vol II: Eigensystems*]{}, SIAM, Philadelphia, PA, 2001.
, [*Computational Methods for Large Eigenvalue Problems*]{}, Elsevier, North Hollands, 2002.
, [*A new justification of the Jacobi–Davidson method for large eigenproblems*]{}, Linear Algebra Appl., 424 (2007), pp. 448–455.
, [*Fast inexact implicitly restarted Arnoldi method for generalized eigenvalue problems with spectral transformation*]{}, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 433–459.
[^1]: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China, [jiazx@tsinghua.edu.cn]{}.
[^2]: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China, [licen07@mails.tsinghua.edu.cn]{}.
[^3]: Supported by National Basic Research Program of China 2011CB302400 and the National Science Foundation of China (No. 11071140).
[^4]: If it fails to hold, it is seen from (\[sin\_v\_xp\]) that $\sin\angle(\mathcal{V}_+,{\mathbf{x}})=0$ and the exact SIRA, SIA and JD methods terminate prematurely if $\dim(\mathcal{V}_+)<n$. In this case, $\mathcal{V}_+$ is an invariant subspace of ${\mathbf{A}}$ and we stop subspace expansion. We will exclude this rare case.
|
---
---
0 true cm
\
[**T Balehowsky**]{}[^1] and [**E Woolgar**]{}[^2] [$^\dag$ Dept of Mathematical and Statistical Sciences, University of Alberta,\
Edmonton, AB, Canada T6G 2G1.]{}
[**Abstract**]{}
It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant that depends on the area of the minimal sphere, we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound.
Introduction
============
A Ricci flow on a manifold $M$ is a family of Riemannian metrics $g_{ij}(t;x)$, $x\in M$, $t\in I\subseteq
{\mathbb R}$ the family parameter, and $I$ a connected interval, satisfying $$\frac{\partial g_{ij}}{\partial t} = -2R_{ij}\ , \label{eq1.1}$$ where $R_{ij}$ is the Ricci tensor of $g_{ij}(t,x)$. It is usually more convenient to study the [*Hamilton-DeTurck flow*]{} (or [*Ricci-DeTurck flow*]{}) $$\frac{\partial g_{ij}}{\partial t} = -2R_{ij}+\pounds_X g_{ij}
\ . \label{eq1.2}$$ If $g$ solves (\[eq1.2\]), then the pullback $\psi_t^*g$ solves (\[eq1.1\]) ([@CK], p 80), where $\psi_t$ is a family of time-dependent diffeomorphisms generated by the vector field $X$. From (\[eq1.2\]), the scalar curvature $R$ of $g_{ij}(t)$ evolves according to $$\frac{\partial R}{\partial t} = \Delta R + \nabla_X R +2R_{ij}R^{ij}
\ , \label{eq1.3}$$ where $\Delta:=g^{ij}\nabla_i \nabla_j$ is the ($t$-dependent) Laplacian.
Ricci flow of asymptotically flat manifolds arises in several physical and mathematical contexts, ranging from the physics of closed string tachyon condensation [@GHMS] to existence problems for static Einstein metrics [@GOW]. It is known that asymptotically flat initial data remain asymptotically flat and smooth when evolved by Ricci flow for some time interval $[0,T)$ [@DM; @OW]. For complete, rotationally symmetric initial data with no minimal sphere present (corresponding in static general relativity to the absence of black hole horizons), it has been shown that the flow exists for all future time and converges to flat space [@OW].
Then the question arises as to what happens when complete, rotationally symmetric initial data containing a minimal sphere are evolved. There are two cases, depending on whether the minimal sphere is topologically essential (representing a nontrivial class in $\pi_2(M)$) or, as will be the case for data on ${\mathbb R}^n$, inessential. Of these, the essential case is the easier one to study, and is the subject of the present paper.
It is well-known that a closed 3-manifold admitting an essential minimal 2-sphere, when evolved under the Ricci flow, will develop a singularity within finite time. Depending on the initial configuration, the singularity may be localized at or near the minimal sphere, or may be global in the sense that the manifold collapses everywhere at that time. It is expected that this result carries over to the noncompact case. Somewhat to the contrary though, asymptotically flat initial data are critical data with respect to the pinching theorem of M Simon [@Simon]. That is, Simon proves that warped products of a line with a positively curved closed manifold will evolve to form a neckpinch singularity in finite time, provided the initial data obey certain asymptotic conditions at the ends of the line, including a condition on asymptotic growth of the area of the closed manifold factor. Simon’s growth condition is written as a strict (i.e., open) inequality which is not satisfied by asymptotically flat manifolds, but such manifolds would lie in the closure. This suggests that such data may exhibit interesting evolutions, including perhaps critical phenomena of the sort observed in certain numerical evolutions in general relativity in a scenario sometimes called “critical collapse” [@Choptuik].
Garfinkle and Isenberg [@GI1] studied numerical Hamilton-DeTurck flow of a 3-sphere with “corsetted” initial metric, admitting an inessential minimal 2-sphere (the “waist”). For tight corsetting, meaning that the waist has very small area relative to the $2/3^{\rm
rds}$-power of the volume of the 3-sphere, the waist “pinches off” (a local singularity forms there), whereas for more gentle corsetting, the entire sphere shrinks to a point before a local singularity can form. The critical solution separating these two alternatives is a degenerate neckpinch singularity modelled by the Bryant soliton [@GI2].
The Garfinkle-Isenberg result shows that initial data for the Ricci flow with an inessential minimal sphere divides into two disjoint sets whose common boundary consists of points whose evolutions exhibit critical behaviour.
Husain and Seahra [@HS] then considered a numerical Hamilton-DeTurck flow of a sequence of initial metrics, each rotationally symmetric and reflection symmetric through an essential minimal 2-sphere, sometimes referred to as the “throat” or “bridge”. Evolution occurred on a bounded region, with boundary conditions imposed. Two alternatives were again found; for some initial data, the throat pinched off while, for other data, the throat expanded to infinity.
We study a related question. Consider simply connected, [*noncompact, complete*]{} manifolds with rotational symmetry. These are ${\mathbb R}^n$, ${\mathbb R}\times S^{n-1}$, and certain quotients thereof. As ${\mathbb R}^n$ has no essential minimal sphere, consider ${\mathbb R}\times S^{n-1}$, and endow it with an ${\rm
SO}(n)$-symmetric metric such that there is a minimal hypersphere located at, say, $r=0$ and an isometry corresponding to reflection in that hypersphere. One such metric is the $t=0$ slice of the $(n+1)$-dimensional Schwarzschild-Tangherlini metric. One can then identify points under the action ${\mathbb R}\times
S^{n-1}\ni(r,p)\mapsto (-r,-p)$, where $p\in S^{n-1}$ and $-p$ is the antipode of $p$. This produces a smooth metric on ${\mathbb
R}\times{\mathbb R}{\mathbb P}^{n-1}\cong{\mathbb R}{\mathbb P}^{n}
\setminus \{ {\rm pt} \}$. For $n=3$, this model is known in gravitational physics as the [*${\mathbb R}{\mathbb P}^{3}$ geon*]{}. We pose and will answer the question, “What is the Ricci flow evolution of the ${\mathbb R}{\mathbb P}^{3}$ geon?”[This was posed by DM Witt to the second author quite some time ago.]{}
In section 2, we adapt a standard Ricci flow argument from the setting of closed manifolds to that of asymptotically flat manifolds. This forms the basis for what follows, and shows that essential 2-spheres in asymptotically flat 3-manifolds collapse whenever the initial scalar curvature is bounded below by a nonpositive constant that can depend on the initial area of the minimal sphere. The result is valid for arbitrary Hamilton-DeTurck flow, including Ricci flow. As the covering space of the ${\mathbb
R}{\mathbb P}^{3}$ geon obeys this bound, this suffices to answer the question just posed.
This raises the possibility that, by choosing initial data for which the lower bound on initial scalar curvature is violated (e.g., by choosing a different initial metric on the geon manifold, with the same isometries), interesting dynamics might arise, such as observed in the studies cited above [@GI1; @GI2; @HS]. It also raises the question of the precise comparison, if any, of this result to the numerical work of [@HS], who found that for certain initial data with scalar curvature well above our lower bound, their numerical Hamilton-DeTurck evolution did not lead to collapse.
To understand these issues, we perform our own numerical simulations. In section 3, we lay the groundwork. We first discuss Hamilton-DeTurck flow with the “DeTurck trick” formulation which we use for our numerical evolutions in Section 4 (and which [@GI1] used). For comparison purposes, we then discuss the normal coordinate Hamilton-DeTurck formulation used in [@HS].
Section 4 contains our numerical results. We use the same form of initial data as Husain and Seahra [@HS] and, for practical purposes, we also now restrict our evolution to a bounded manifold, but we use different evolution equations[We use a different Hamilton-DeTurck system. Occurrence of collapse of the minimal surface should not depend on this choice; see section 2.]{} and different boundary conditions at the boundary at large $r$. Our numerical evolutions always exhibit collapse of the throat. Section 5 contains a brief discussion of the numerical evolution.
Our convention for the curvature tensor $R^a{}_{bcd}$ is that used in [@HE] and equals the quantity denoted by $R_{cdb}{}^a$ in [@CK]. We write $R_{abcd}:=g_{ae}R^e{}_{bcd}$. We denote the Laplacian by $\Delta:=g^{ab}\nabla_a\nabla_b$. For a definition of asymptotic flatness, see [@OW].
Time derivative of the area of a minimal sphere
===============================================
The maximum principle for scalar curvature
------------------------------------------
For Ricci flow on compact manifolds, it is standard that positive scalar curvature is associated to contraction and concentration of curvature under the Ricci flow. In this subsection is that scalar curvature, we recall that this is also true for complete manifolds with vanishing scalar curvature at infinity.
[**Proposition 2.1.**]{} [*Assume that (\[eq1.3\]) has a solution on some time interval $[0,T]$, $T>0$, on a complete manifold $M$ with one or more asymptotic ends, and assume that $R(t,x)\to 0$ as $x\to\infty$ (that is, as the point $x$ tends to an asymptotic end), for all $t\in [0,T]$. Let the scalar curvature $R_0(x)$ of the initial metric $g_0(x):=g(0,x)$ obey $$\inf_{x\in {\cal M}} \left \{ R_0(x) \right \}=:-a^2\ , \label{eq2.1}$$ for some $a\ge 0$. Then*]{} $$R(t,x)\ge \frac{-a^2}{1+\frac{2a^2}{n}t}\ . \label{eq2.2}$$
[**Remark 2.2.**]{} The existence assumption always holds for Ricci flow developing from asymptotically flat initial data ([@OW], [@DM]).
[**Proof.**]{} Equation (\[eq1.3\]) can be written as $$\frac{\partial R}{\partial t} = \Delta R + \frac{2}{n}R^2
+ 2\left \vert R_{ij}-\frac{1}{n}g_{ij}R \right \vert^2
+\nabla_X R\ . \label{eq2.3}$$ Exhaust $M$ by a sequence of closed, bounded sets $K_i$. An easy application of the maximum principle (e.g., [@Jost]) to the $i^{\rm th}$ subset yields that the minimum of $R$ occurs on the parabolic boundary of $[0,T]\times K_i$. Now take $i\to\infty$ and use that $R(t,x)\to 0$ as $x\to\infty$ to deduce that the minimum must either be zero or must occur on the initial data. This proves that $$R(t,x)\ge \inf_{x\in {\cal M}} \left \{ R_0(x) \right \}=-a^2,\ a\ge 0.
\label{eq2.4}$$ If $a=0$, this proves the theorem, so we now consider $a>0$, in which case the infimum is a minimum.
For $a>0$, let $Q := -\frac{1}{a^2} \left ( 1 +
\frac{2a^2}{n}t \right ) R$. Since $R=-a^2<0$ at its minimum, then $Q>0$ there. Hence the maximum of $Q$ is positive, and it then follows from the definition of $Q$ that, at the point where $Q$ achieves its maximum, $R$ is negative. As well, $Q(t,x)\to 0$ as $x\to \infty$.
Work on the intervalcompact set $[0,T]\times K_i$. From (\[eq2.3\]), we have $$\frac{\partial Q}{\partial t} \le \Delta Q +\nabla_X Q
+ \frac{2}{n} R \left ( Q-1 \right )\ . \label{eq2.5}$$ Then from the maximum principle (and since $R<0$ where $Q$ achieves its maximum), either the maximum of $Q$ occurs on the parabolic boundary of $[0,T]\times K_i$ or $Q-1\le 0$ at the maximum. If it occurs on the parabolic boundary, by taking $i\to\infty$ and using that $Q(t,x)\to 0$ as $x\to\infty$, we see that the maximum must occur on the initial boundary, where $Q(0,x)=-\frac{1}{a^2}R(0,x)\le
1$. Hence, $Q(t,x)\le 1$, and the result follows.
[**Remark 2.3.**]{} Obviously, these results also hold on $[0,T]\times K$, $K\subset M$ a compact set with boundary $\partial
K$, provided that $R(t,x)=0$ for $x\in\partial K$. But they do not hold if merely $R(t,x)\ge -const$ on $\partial K$, an observation which appears to be relevant in numerical studies (see section 3.2.2).
Evolution of hypersurfaces
--------------------------
In this subsection, we obtain the evolution equation for the area of a closed hypersurface $\Sigma$ in an $n$-manifold $({\cal M}^n,g)$, defined (locally at least) by an expression of the form $F(x^i)=0$, where $x^i$ are the coordinates held constant in the derivative $\frac{\partial}{\partial t}$ appearing in (\[eq1.2\]).[For hypersurfaces defined more generally by an equation of the form $F(t,x^i)=0$, the generalization of our result is easy to obtain, using the transport theorem.]{} Let $n^a$ be a smooth unit vector field normal to $\Sigma$. Let $h_{ab}$ be the induced metric and let $H_{ab}$ be the extrinsic curvature of $\Sigma$, with trace $H$ (thus $H$ is the mean curvature, taken here to mean the sum rather than the average of the principal curvatures at a point). If ${\tilde H}_{ab}$ denotes the trace-free part of $H_{ab}$, then $$H_{ab}={\tilde H}_{ab}+\frac{1}{n-1}h_{ab}H\ . \label{eq2.6}$$ If ${\cal R}$ denotes the scalar curvature of the induced metric $h_{ab}$ on $\Sigma$, then $$R - {\rm Ric}(n,n)= \frac12 \left [ {\cal R}
- \left ( \frac{n-2}{n-1}\right )
H^2 +R+\vert {\tilde H} \vert^2 \right ] \label{eq2.7}$$ by the Gauss equation. Finally, let the induced area element on $\Sigma$ be $d\mu$. From the Hamilton-DeTurck flow equation (\[eq1.2\]), the area $\vert \Sigma \vert:=\int_{\Sigma}d\mu$ of $\Sigma$ evolves as $$\begin{aligned}
\frac{d}{dt}\vert \Sigma \vert&=&\int_{\Sigma}\frac{\partial}{\partial t}d\mu
=\frac12\int_{\Sigma}h^{ij}\frac{\partial h_{ij}}{\partial t}d\mu
=\frac12\int_{\Sigma}h^{ij}\frac{\partial g_{ij}}{\partial t}d\mu
\nonumber\\
&=&-\int_{\Sigma}h^{ij}\left ( R_{ij}-\nabla_i X_j \right )d\mu
=-\int_{\Sigma}\left ( R-R_{ij}n^in^j-h^{ij}\nabla_i X_j\right ) d\mu
\nonumber\\
&=&-\frac12 \int_{\Sigma} \left ( {\cal R} - \left ( \frac{n-2}{n-1}\right )
H^2 +R+\vert {\tilde H} \vert^2-2Hn\cdot X\right ) d\mu
\ ,\label{eq2.8}\end{aligned}$$ using (\[eq2.7\]), the divergence theorem on $\Sigma$, and $H:=h^{ij}\nabla_i n_j$.
3-manifolds
-----------
Now fix the dimension to be $n=3$. Then we have $\int_{\Sigma} {\cal R} d\mu=4\pi\chi(\Sigma)$, where $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$. Then $$\frac{d}{dt}\vert \Sigma \vert=-2\pi\chi(\Sigma)+ W(\Sigma)
-\frac12 \int_{\Sigma} \left ( R+\vert {\tilde H} \vert^2 \right ) d\mu
+ \int_{\Sigma} Hn\cdot X d\mu\ , \label{eq2.9}$$ where $W(\Sigma):=\frac14 \int_{\Sigma} H^2 d\mu$ is the [*Willmore energy*]{} [@Willmore] of $\Sigma$.[When $\Sigma$ is a 2-sphere, the combination of Willmore energy and $\chi(\Sigma)$ that appears in (\[eq2.9\]) also appears in the definition of the Hawking quasi-local mass $m_H:=\frac{\vert \Sigma
\vert^{1/2}}{16\pi^{3/2}} \left [ 4\pi-W(\Sigma)\right ]$.]{} Using Proposition 2.1, (\[eq2.9\]) becomes $$\frac{d}{dt}\vert \Sigma \vert \le -2\pi\chi(\Sigma) + W(\Sigma)
+ \frac{3a^2\vert \Sigma \vert}{6+4t}
+ \int_{\Sigma}Hn\cdot X d\mu\ . \label{eq2.10}$$ The tightest bound occurs in the case of a minimal 2-sphere. Then $H\vert_{\Sigma}$ and the Willmore energy vanish and $\chi(S^2)=2$ so $$\frac{d}{dt}\vert \Sigma \vert \le -4\pi
+ \frac{3a^2}{6+4t}\vert \Sigma \vert \ . \label{eq2.11}$$ We note that the initial time derivative of $|\Sigma|$ will be negative whenever $$a^2<\frac{8\pi}{|\Sigma|_0}\ , \label{eq2.12}$$ where $|\Sigma|_0$ denotes the initial area of $\Sigma$. Even when not initially negative, the derivative can turn negative and remain so thereafter.
Now we can regard the left-hand side of (\[eq2.11\]) as the forward difference quotient of the functional whose value is the area of the smallest essential minimal surface present at $t$. Then (see Lemma 2.22 of [@MT]) an upper barrier for the area of the smallest minimal surface at any $t\ge 0$ is provided by the function $\Psi(t)$ that solves the initial value problem $$\begin{aligned}
\frac{d\Psi}{dt}-\frac{3a^2\Psi}{6+4t} &=& -4\pi\ , \label{eq2.13}\\
\Psi(0)=\vert \Sigma\vert_0\ . \nonumber\end{aligned}$$ The solution is $$\Psi(t) = \begin{cases} \frac{24\pi}{3a^2-4}(1+\frac23 t)
+\left ( |\Sigma|_0 - \frac{24\pi}{3a^2-4} \right )
\left ( 1+\frac23 t\right )^{3a^2/4} & \text{for $a^2\neq 4/3$,}\\
(1+\frac23 t)\left [ |\Sigma|_0 - 6\pi \log\left ( 1+\frac23 t\right )
\right ] & \text{for $a^2= 4/3$.} \label{eq2.14}
\end{cases}$$
[**Proposition 2.4.**]{} [*Say that the conditions of Proposition 2.1 hold and that $(M,g_0)$ contains a minimal sphere of area $4\pi\delta$ for some $\delta>0$. Let $3a^2-4<6/\delta$. Then the flow fails to exist in finite time.*]{}
[**Proof.**]{} We must show that if the flow lasts long enough, the minimal sphere collapses. If $a^2=4/3$, this is immediate from the second line of (\[eq2.14\]), since $\Psi$ has a zero at some $t>0$. Otherwise, let $\gamma:= 3a^2-4\neq 0$. The top line of (\[eq2.14\]) becomes $$\Psi(t)= 4\pi \left ( 1+\frac23 t \right ) \left [ \frac{6}{\gamma}
+ \left (\delta - \frac{6}{\gamma} \right ) \left ( 1 +\frac23 t
\right )^{\gamma/4} \right ] \ , \label{eq2.15}$$ so $\Psi(t)=0$ for some $t>0$ iff $$\left ( 1+\frac23 t \right )^{\gamma/4}=\frac{6}{6-\gamma\delta}\ . \label{eq2.16}$$ Now $\frac{6}{6-\gamma\delta}>1$ if $0<\gamma<6/\delta$ and $0<\frac{6}{6-\gamma\delta}<1$ if $\gamma<0$. The left-hand side equals 1 at $t=0$ and is otherwise monotonic, increasing without bound if $\gamma>0$, and decreasing with asymptote $0$ if $\gamma<0$. Hence there is always a root for $t>0$ if $\gamma<6/\delta$.
[**Corollary 2.5.**]{} For any $X$, consider the flow (\[eq1.3\]) developing from an initial metric which describes the ${\mathbb R}{\mathbb P}^3$-geon. Then the flow fails to exist in finite time.
[**Proof.**]{} The Riemannian double cover of the geon is the $\delta=1$, $a=0$ case above. Furthermore, since the initial data are asymptotically flat, then $R(t,r)\to 0$ as $r\to\infty$ for all $t$ during the flow ([@DM; @OW]). Then the result holds on the Riemannian double cover and, hence, on the manifold itself because the covering map is a local isometry.
Alternatively, the geon itself has $\chi(M)=1$ and $|\Sigma|=2\pi$, so equation (2.13) applies, with the $4\pi$ on the right-hand side of the differential equation replaced by $2\pi$ and $|\Sigma_0|=2\pi$. The solution (\[eq2.14\]) is then multiplied by an overall factor of $1/2$, which has no effect in the proposition.
2-dimensions
------------
The $n=2$ case is itself of some interest. Then (\[eq2.8\]) becomes $$\frac{d}{dt}\vert \Sigma \vert = -\frac12 \int_{\Sigma} \left ( R
-2Hn\cdot X \right ) d\mu\ , \label{eq2.17}$$ with $H$ the geodesic curvature of the closed curve $\Sigma$. Then $$\frac{d}{dt}\vert \Sigma \vert = -\frac12 \int_{\Sigma} Rd\mu \label{eq2.18}$$ if $\Sigma$ is a geodesic. If $R\to 0$ at infinity (or on any boundary), Proposition 2.1 yields that $R\ge -\frac{a^2}{1+a^2t}$, and so (2.19) becomes $$\frac{d}{dt}\vert \Sigma \vert \le \frac{a^2}{2(1+a^2t)}\vert \Sigma \vert
\ . \label{eq2.19}$$ Thus $$\vert \Sigma \vert(t) \le \vert \Sigma \vert (0) \sqrt{1+a^2t}\ . \label{eq2.20}$$
Rotational symmetry and Hamilton-DeTurck flow
=============================================
There are two scales in this problem, the initial area of the minimal surface and the minimum of the initial scalar curvature.[This presumes the scalar curvature has a negative minimum. In the initial data we study, the only exception will be the Schwarzschild-Tangherlini data, for which $R(0,p)=0$ for all points $p$.]{} The metric can be rescaled to fix one but not both of these. By choosing $\delta=1$, we fix the initial area. There is then the intriguing possibility that collapse of the minimal sphere will not occur if the geon initial data is replaced with initial data with scalar curvature sufficiently negative that $a^2>10/3$. In the next section we will study this question numerically. In this section, we formulate the equations we will need.
There are several versions of Hamilton-DeTurck system that are used in the study of rotationally symmetric Ricci flow. We discuss the formulation we will use for our numerical integrations in the first subsection. We then discuss, for comparison purposes, other formulations that have been used in related work.
DeTurck’s background connection method
--------------------------------------
### The flow equations with a background connection
The first Hamilton-DeTurck system arises from DeTurck’s original trick for proving short-time existence of Ricci flow, and results in a parabolic strictly system of two equations. The idea is that parabolicity fails only because there are families of flows whose members are distinguished from each other only by a continuous time-dependent deformation of the coordinates. Thus, parabolicity is restored by “breaking coordinate invariance”. This is done by fixing a $t$-independent background connection throughout the flow. That is, let $\Gamma^i_{jk}$ be the flowing connection (compatible with $g_{ij}(t)$) written in local coordinates, and let $\breve{\Gamma}^i_{jk}$ denote the chosen background connection in these coordinates. Define the vector field $$X^i:=g^{jk}\left ( \Gamma^i_{ij}-\breve{\Gamma}^i_{jk} \right )
\ . \label{eq3.1}$$ Then the system (\[eq1.2\]) is parabolic (for any choice of $t$-independent connection $\breve{\Gamma}$). For rotationally symmetric flow on ${\mathbb R}^n$, an obvious choice of coordinates is $$ds^2 = e^{2A(t,r)} dr^2 + r^2 e^{2B(t,r)}g(S^{n-1},{\rm can})\ ,
\label{eq3.2}$$ where $g(S^{n-1},{\rm can})$ denotes the metric of unit sectional curvature on $S^{n-1}$. Then it makes sense to choose a background connection to arise from a rotationally symmetric background metric. Since every rotationally symmetric metric is conformal to a flat metric, we write the background metric as $$\breve{g}=e^{2\psi(r)}\left [ dr^2 +r^2 g(S^{n-1},{\rm can})
\right ]\ , \label{eq3.3}$$ for some function $\psi(r)$ that must be specified. Choices of $\psi$ with rotational symmetry and asymptotic flatness include $$\psi = \begin{cases} 0 \ , & \text{flat background,} \\
\frac{2}{\alpha}\ln \left [ \left ( 1+
r^{-\alpha}\right ) \right ] \ , & \alpha=const>0. \label{eq3.4}
\end{cases}$$ The last choice listed is a family of metrics which were used as initial data for Ricci flow (with $1\le\alpha\le 2$ and $n=3$ dimensions) in [@HS] and which would, if $\alpha=n-2$, correspond to a time-symmetric slice of $(n+1)$-dimensional Schwarzschild spacetime.
Then the DeTurck vector field is $$\begin{aligned}
X=:V\frac{\partial}{\partial r} &=&\bigg \{ e^{-2A} \left [
\frac{\partial A}{\partial r} - \psi'(r) \right ]\label{eq3.5}\\
&&-(n-1) \left [ \frac{e^{-2A}-e^{-2B}}{r}
+e^{-2A} \frac{\partial B}{\partial r}
-e^{-2B}\psi'(r)\right ] \bigg \}\frac{\partial}{\partial r}\ . \nonumber\end{aligned}$$ The Hamilton-DeTurck system becomes $$\begin{aligned}
\frac{\partial A}{\partial t} &=& e^{-2A}\left \{
\frac{\partial^2 A}{\partial r^2} - \left (
\frac{\partial A}{\partial r} \right )^2 +(n-1)
\left ( \frac{\partial B}{\partial r} +\frac{1}{r} \right )^2-\psi''(r)
+\frac{\partial A}{\partial r}\psi'(r) \right \}
\nonumber\\
&&+(n-1)e^{-2B} \left \{ \frac{1}{r}\frac{\partial A}{\partial r}
-\frac{2}{r}\frac{\partial B}{\partial r}-\frac{1}{r^2} +\psi''(r)
+\frac{\partial A}{\partial r}\psi'(r)
-2\frac{\partial B}{\partial r}\psi'(r) \right \}\ ,\label{eq3.6}\\
\frac{\partial B}{\partial t} &=& e^{-2A}\left \{
\frac{\partial^2 B}{\partial r^2} -\frac{1}{r^2}
-\left ( \frac{\partial B}{\partial r}+\frac{1}{r}\right )
\psi'(r)\right \}\nonumber \\
&&+(n-1)e^{-2B}\left ( \frac{\partial B}{\partial r}+\frac{1}{r}\right )
\left ( \psi'(r)+\frac{1}{r}\right )-\frac{(n-2)}{r^2}e^{-2B}
\ .\label{eq3.7}\end{aligned}$$
If we define $$S:=V_{n-1}r^{n-1}e^{(n-1)B}\label{eq3.8}$$ to be the area of $r=const$ spheres, with $V_{n-1}$ being the volume of an $(n-1)$-sphere with unit sectional curvature, we can rewrite the above system with $B$ replaced by $S$.
In $n=3$ dimensions, this gives $$\begin{aligned}
\frac{\partial A}{\partial t} &=& e^{-2A}\left \{
\frac{\partial^2 A}{\partial r^2} - \left (
\frac{\partial A}{\partial r} \right )^2 +\frac{1}{2S^2}
\left ( \frac{\partial S}{\partial r}\right )^2-\psi''(r)
+\frac{\partial A}{\partial r}\psi'(r) \right \}
\nonumber\\
&&+\frac{8\pi r^2}{S} \bigg \{ \frac{1}{r}\frac{\partial A}{\partial r}
-\frac{1}{rS}\frac{\partial S}{\partial r}+\frac{1}{r^2}
+\psi''(r)+\left ( \frac{\partial A}{\partial r} +\frac{2}{r} \right )
\psi'(r) \nonumber\\
&&\qquad -\frac{1}{S}\frac{\partial S}{\partial r}\psi'(r) \bigg \}
\ ,\label{eq3.9}\\
\frac{\partial S}{\partial t} &=& e^{-2A}\left \{
\frac{\partial^2 S}{\partial r^2} -\frac{1}{S}\left (
\frac{\partial S}{\partial r} \right )^2
-\psi'(r)\frac{\partial S}{\partial r}\right \}
+\frac{8\pi r^2}{S}\frac{\partial S}{\partial r}
\left ( \psi'(r)+\frac{1}{r}\right ) -8\pi
\ .\label{eq3.10}\end{aligned}$$ Once a function $\psi$ is chosen, it is clear that this system is parabolic. This system was used in [@GI1] in their study of rotationally symmetric flow on “corsetted” 3-spheres.
### Minimal surface boundary conditions with background connection
Let us now consider the problem of the previous section, which is the evolution of an asymptotically flat metric on a manifold with inner boundary $\Sigma$ which is a minimal surface. At $\Sigma$, both the mean curvature and the vector field $X$ must vanish, so $$\begin{aligned}
V_{\Sigma}(t)&:=&V(t,r_0)=0\ , \label{eq3.11}\\
H_{\Sigma}(t)&:=&H(t,r_0)=0\ . \label{eq3.12}\end{aligned}$$ The second of these conditions is of course $\frac{\partial
S}{\partial r}\bigg \vert_{r_0}=0$ or, equivalently, $$\frac{\partial B}{\partial r}\bigg \vert_{r_0}+\frac{1}{r_0}=0
\ , \label{eq3.13}$$ while the first then yields $$e^{-2A(t,r_0)} \left ( \frac{\partial A}{\partial r}\bigg \vert_{r_0}
-\psi'(r_0)\right ) +(n-1)e^{-2B(t,r_0)} \left (\frac{1}{r_0}+\psi'(r_0)
\right ) =0\ . \label{eq3.14}$$
Polar coordinate gauge
----------------------
### The flow equations in Gaussian polar coordinates
This system was used by Husain and Seahra [@HS] and is not parabolic. The metric is $$ds^2 = dr^2+F^2(t,r)g(S^{n-1},{\rm can})\ , \label{eq3.15}$$ corresponding to $A=0$, $B=\log (F/r)$ in (\[eq3.2\]). The coordinate $r$ is distance from the minimal sphere at $r=0$. In this system, the Hamilton-DeTurck flow becomes a constrained system comprised of the differential equation $$\frac{\partial F}{\partial t} = \frac{\partial^2 F}{\partial r^2}
+ \frac{1}{F} \left ( \frac{\partial F}{\partial r}\right )^2
+V\frac{\partial F}{\partial r}-\frac{n-2}{F}\label{eq3.16}$$ and a differential constraint $$\frac{\partial V}{\partial r} = -\frac{(n-1)}{F}
\frac{\partial^2 F}{\partial r^2} \label{eq3.17}$$ which determines the generator $X=V\frac{\partial}{\partial r}$ of the diffeomorphism in (\[eq1.2\]). The mean curvature of a constant-$r$ sphere is $$H:=\frac{(n-1)}{F} \frac{\partial F}{\partial r} =\frac{1}{S}
\frac{\partial S}{\partial r}\ , \label{eq3.18}$$ where $S=V_{n-1}F^{n-1}$ is the area of an $r=const$ sphere (cf. (\[eq3.8\])), and so the solution of the constraint can be written as $$V(t,r)=C(t)+H(t,r_0)-H(t,r)-\frac{1}{(n-1)}\int\limits_{r_0}^r
H^2(t,r') dr' \ , \label{eq3.19}$$ where $C(t)$ is an arbitrary function of $t$, to be determined by boundary conditions at $r_0$. When the manifold contains an origin for the rotational symmetry (e.g., ${\mathbb R}^n$), this formulation suffers from the problem that $H$ diverges there.
In the case of $n=3$ dimensions, (\[eq3.16\]) leads to the remarkably simple equation $$\frac{\partial S}{\partial t}
=\frac{\partial^2 S}{\partial r^2}
+V\frac{\partial S}{\partial r}-8\pi\label{eq3.20}$$ for the area $S(t,r)$ of an orbit of the rotational symmetry. Indeed, this equation would be linear if not for the dependence of $V$ on $S$ via the constraint (\[eq3.17\]) (which takes a slightly more complicated form when $F$ is replaced by $S$).
### Minimal surface boundary conditions in polar coordinates
The condition that $V$ should vanish at the boundary $\Sigma$ is simply $C(t)=0$. The condition (\[eq3.12\]) that the boundary $\Sigma$ be a minimal surface is $H(t,r_0)=0$. These conditions, applied to (\[eq3.19\]), then imply that $$V(t,r)=-H(t,r)-\frac{1}{(n-1)}\int\limits_{r_0}^r H^2(t,r')dr' .
\label{eq3.21}$$ Since $H$ is not identically zero (for all $r$), then $V(t,r)<0$ on every convex surface $r>r_0$. Husain-Seahra [@HS] impose that $V\to 0$ at infinity and then approximate this for numerical purposes by imposing $V(t,r_c)=0$ on an outer boundary $r=r_c$ at all $t>0$.[In [@HS] the outer boundary is defined by $F(t,r_c)=const$ and thus $r_c$ can vary in time.]{} As a result, they have $$H(t,r_c)=-\frac{1}{(n-1)}\int\limits_{r_0}^{r_c} H^2(t,r)dr \le 0\ ,
\label{ew3.22}$$ and $=0$ only when $H(t,r)=0$ for all $r$, so the $r=r_c$ boundary is necessarily concave in the direction of increasing $r$ at all $t>0$. This has the potential to generate negative scalar curvature at $r=r_c$ since $$R=-2\frac{\partial H}{\partial r} - \frac{n}{(n-1)}H^2
+ (n-1)(n-2)\left ( S/V_{n-1} \right )^{-2/(n-1)}
\ . \label{ew3.23}$$ Then Proposition 2.1 and, concomitantly, Proposition 2.4 would no longer apply.
Area radius coordinates
-----------------------
This is also not a parabolic system, but can be reduced to a single parabolic equation. The technicque was used by [@OW] to prove existence and convergence of rotationally symmetric asymptoticall flat flows. Unfortunately the technique fails for initial data containing a minimal surface. For this reason, the results of [@OW] do not apply when minimal surfaces are present. In the rotationally symmetric setting, fix $X$ such that the components of the flow equation in directions tangent to the orbits of rotational symmetry are trivial; i.e., so that $\frac{\partial g_{ij}}{\partial t}=0$ for $i,j\neq 1$. This means that the metric coefficient $B$ above will be time-independent. We let $f=e^A$ and so we write the metric as $$ds^2 = f^2(t,r)dr^2+r^2g(S^{n-1},{\rm can})\ , \label{ew3.24}$$ and call $r$ the area radius. Then in local coordinates this fixes $$X=-\frac{R_{22}}{g_{11}\Gamma^1_{22}}\frac{\partial}{\partial r}
=-\frac{R_{33}}{g_{11}\Gamma^1_{33}}\frac{\partial}{\partial r}= \dots \ .
\label{ew3.25}$$ Then(\[eq1.2\]) becomes a constrained system in which the constraint can be solved, leading to the single parabolic equation $$\frac{\partial f}{\partial t}=\frac{1}{f^2}\frac{\partial^2
f}{\partial r^2} -\frac{2}{f^3}\left ( \frac{\partial f}{\partial r}
\right )^2 + \left ( \frac{(n-2)}{r}-\frac{1}{rf^2} \right )
\frac{\partial f}{\partial r}-\frac{(n-2)}{r^2f}\left ( f^2 -1 \right )
\ . \label{ew3.26}$$ Having a single parabolic equation instead of a system is an enormous advantage [@OW], and the maximum principle can be used to show that $f$ remains bounded whenever it is bounded on the initial data, yielding uniform parabolicity. However, unboundedness of $f$ on the initial data corresponds precisely to the presence of a minimal sphere, since the mean curvature of $r=const$ sphere is given in this system by $H=\frac{(n-1)}{rf}$. (It is shown in [@OW] that no solution of (\[ew3.26\]) can form a minimal sphere during the evolution if none is present initially.)
Conformal gauge: $n=2$ dimensions
---------------------------------
Every rotationally symmetric metric is conformally flat. However, for $n>2$, the conformal class varies throughout the flow. To see this, set $A=B$ in the metric (\[eq3.2\]). Then equations (\[eq3.6\], \[eq3.7\]) reduce to the single equation $$\frac{\partial A}{\partial t}= e^{-2A} \left \{
\frac{\partial ^2 A}{\partial r^2} + \frac{(n-1)}{r}
\frac{\partial A}{\partial r} +(n-2) \psi'(r) \left [
\frac{\partial A}{\partial r} + \frac{1}{r} \right ] \right \}
\ , \label{eq3.27}$$ together with a restriction on $\psi$: $$0=(n-2) \left [ \left ( \frac{\partial A}{\partial r} \right ) ^2
+\psi''(r) -2\frac{\partial A}{\partial r} \psi'(r) -\frac{1}{r}\psi'(r)
\right ] \ . \label{eq3.28}$$ When $n>2$, we obtain from (\[eq3.28\]) that $$\psi'(r)=re^{2A(t,r)}\left [ C(t) - \int \frac{e^{-2A(t,r')}}{r'}\left (
\frac{\partial A}{\partial r'} (t,r')\right )^2 dr'\right ] \ , \label{eq3.29}$$ for some function $C(t)$. Since the left-hand side is time-independent, so must be the right. Thus, the system (\[eq3.27\], \[eq3.28\]) is rarely solvable unless $n=2$. In $n=2$, however, (\[eq3.28\]) is trivial and then (\[eq3.27\]) can be solved.
Numerical results
=================
Initial data
------------
We begin with the class of metrics $$ds^2=\frac{d\rho^2}{1-(\rho_0/\rho)^{\alpha}}
+\rho^2 g(S^{n-1},{\rm can})\ , \label{eq4.1}$$ where $g(S^{n-1},{\rm can})$ is the metric with constant unit sectional curvature on the $(n-1)$-sphere. For $n=3$ and $\alpha\in
[2,1]$, these metrics were used as initial data by [@HS]. When $\alpha=n-2$, this is a static hypersurface in the Schwarzschild-Tangherlini metric. There is a minimal surface at $\rho=\rho_0$.
To obtain initial data for the functions $A(t,r)$ and $S(t,r)$ used in the evolution equations, we first transform to isotropic coordinates. We obtain $$\begin{aligned}
ds^2 &=&\beta^2(r) \left ( dr^2+r^2g(S^{n-1},{\rm can}) \right )
\ , \label{eq4.2}\\
\beta(r)&=&\left ( \frac{\rho_0}{2^{1/\alpha}r_0}\right )
\left (1+\frac{r_0^{\alpha}}{2r^{\alpha}} \right )^{2/\alpha}
\ , \label{eq4.3}\end{aligned}$$ where $r_0$ is a constant of integration arising in the coordinate transformation. Areas of constant-$r$ spheres are given by $$V_{n-1}r^{n-1}
\left ( \frac{\rho_0}{2^{1/\alpha}r_0}\right )^{\frac{2}{n-1}}
\left (1+\frac{r_0^{\alpha}}{2r^{\alpha}}
\right )^{\frac{4}{(n-1)\alpha}} \ , \label{eq4.4}$$ where $V_{n-1}$ is the volume of an $(n-1)$-sphere of constant unit sectional curvature. The minimal sphere now lies at $$r:=r_0/2^{1/\alpha}\ . \label{eq4.5}$$ We choose $r_0=2^{1/\alpha}$, so the minimal sphere occurs always at $r=1$. We then fix $\rho_0=1$, which fixes the area of the initial minimal sphere to be $V_{n-1}$. Then the initial data are $$\begin{aligned}
A(0,r)&=&\log \beta = \frac{2}{\alpha}\log
\left (1+\frac{1}{r^{\alpha}} \right )
-\frac{2}{\alpha} \log 2 \ , \label{eq4.6}\\
S(0,r)&=&V(r)r^{n-1}
\left ( \frac{1}{2}\right )^{\frac{4}{(n-1)\alpha}}
\left (1+\frac{1}{r^{\alpha}}
\right )^{\frac{4}{(n-1)\alpha}}\ . \label{eq4.7}\end{aligned}$$
The boundary data
-----------------
Our initial data have reflection symmetry in the minimal sphere, as well as rotational symmetry. The flow equations preserve isometries, so the subsequent evolution will share these symmetries.
We therefore choose to place a boundary at the location $r=1$ of the minimal sphere. The idea is then to take the Riemannian double $(D,g_D)$ (the topological double, with the Riemannian metric $g_D$ induced by pullback of $g(t)$ under the covering map) at any time $t$ along the evolution. For this, we must keep the mean curvature zero at $r=1$. Rotational symmetry then ensures the $r=1$ sphere is totally geodesic, so the double is a smooth manifold. In general, we do not expect $R(t,1)=0$ at the $r=1$ boundary. Nonetheless, the arguments of Section 2 apply to $(D,g_D(t))$ provided that $R(t,r)\to 0$ on the asymptotic ends ($D$, of course, has two of them).
As we wish to study the noncompact case, we would like to allow $r$ to range through all values $\ge 1$. However, for numerical purposes, we must either choose a finite cut-off or use a more sophisticated method (such as conformal compactification, which then would introduce the complication of dealing with a singular boundary value problem). For simplicity, we choose a finite cut-off, so $r\in
[1,r_c]$.
We now have two boundaries and a parabolic system of two PDEs, so we expect four boundary conditions, two at each boundary. Three are obvious. These are that the mean curvature should vanish at the $r=1$ boundary and the DeTurck vector field should vanish at both boundaries (to make it possible to infer the appropriate conclusions for Ricci flow from our results for the Hamilton-DeTurck flow on the same bounded manifold). For the remaining condition at the outer boundary, we would prefer to hold the scalar curvature constant (preferably zero) there. Instead, we must set a condition that contains no worse than first derivatives of $A$ and $S$. We choose to hold the mean curvature equal to a constant $\lambda$ on the outer boundary. For the case of a single boundary, Cortissoz [@Cortissoz] showed long-time existence of the flow with this boundary condition, so it would be interesting to see if any vestige of this result remains for the present case (specifically, noncollapse of the minimal surface when $\alpha$ is large).
The conditions become, for all $t>0$, $$\begin{aligned}
\text{At $r=1$:}&&\begin{cases} \frac{\partial S}{\partial r}=0 \\
\frac{\partial A}{\partial r} + 1 =0
\end{cases}\label{eq4.8}\\
\text{At $r=r_c$:}&&\begin{cases}e^{-A(t,r_c)}\frac{1}{S}
\frac{\partial S}{\partial r}
=\lambda = const\\
\frac{\partial A}{\partial r}=\psi'(r_c)+e^A\lambda
-(n-1)e^{2A} \left ( \frac{V_{n-1}}{S}
\right )^\frac{2}{n-1} r_c^2\left ( \frac{1}{r_c} + \psi'(r_c) \right )
\end{cases}\label{eq4.9}\end{aligned}$$ We take $\lambda$ to have the value that it would have in the background metric at $r=r_c$, so $$\lambda = (n-1)e^{-\psi(r_c)} \left ( \frac{1}{r_c}
+\psi'(r_c) \right ) \ . \label{eq4.10}$$
The numerical results
---------------------
For the initial data of section 4.1 and $n=3$ dimensions, the initial minimal surface is always of area $4\pi$, so $\delta=1$ and thus $a^2=10/3$ is the critical value in Proposition 2.4. This value is achieved for $\alpha=8/3$.
Our first numerical run probes the case of $n=3$ dimensions, with $\alpha$ taking values below or near the critical value. Figure 1 shows the results for $\alpha$ values $1$, $2$, $2.5$, and $3$. In every case, the minimal sphere collapses to zero area. Though we do not show it, we monitor the scalar curvature at the position of the minimal sphere and observe that it diverges to $+\infty$ at the collapse time.
In the second run, we probe larger values of $\alpha$, in the hope of seeing expansion of the minimal surface. The results are displayed in figure 2. We use $\alpha$ values all the way up to $\alpha=8$. In each case, collapse eventually commenced, after which there is no evidence for subsequent re-expansion.
However, at large $\alpha$, the expansion appears eventually to slow and halt. At such values, as the evolution proceeds, sectional curvature becomes highly concentrated at the minimal surface, manifested as a large value of the second spatial derivative of the area $S$. This may indicate that a singularity forms before collapse of the minimal surface occurs, but our numerics are not sufficiently reliable when derivatives become large.
Our third graph deals with the case of $n=2$ dimensions. For initial data derived from the form of (\[eq4.1\]) with $\alpha>0$, the initial scalar curvature will always be negative in 2 dimensions (as is the case in any dimension when $\alpha>n-2$, so each curve in figure 3 will initially expand. By (\[eq2.20\]), we do not have a collapsing upper barrier function for any $\alpha$, but it is still possible in principle that collapse could occur. We find no collapse, despite running for much longer times than for the higher-dimensional cases.
The numerical integrations were performed with MATLAB’s “pdepe” integration routine for partial differential equations. We performed several tests of the validity of the code. From equations (\[eq3.10\]), (\[eq4.6\]), and (\[eq4.7\]), we compute $\frac{\partial S}{\partial t} (0,1)
=4\pi(\alpha-2)$. This serves to define our unit of time, but also serves as a test of validity of the code, in the sense that the initial derivative should be linear in alpha and zero at $\alpha=2$. This is verified by figure 4, which presents a close-up of the evolution near the initial time.
For example, at $\alpha=2$ the slope is zero initially.
As additional verification, we vary the dimension, using initial data for which the scalar curvature obeys $R(0,r)=0$ in each case. The initial data may therefore be regarded as the metric on a moment of time symmetry in an $(n+1)$-dimensional Schwarzschild-Tangherlini exterior spacetime. We therefore use the evolution equations (\[eq3.6\]–\[eq3.8\]) with arbitrary $n\ge 3$ and set $\alpha=n-2$ in (\[eq4.6\], \[eq4.7\]). For these runs, we normalize the initial area of the minimal surface to be $2\pi^{n/2}/\Gamma(n/2)$. It is easy to verify, using equation (\[eq2.8\]), and the fact that $R(0,r)=0$ for such data (and thus $R(t,r)\ge 0$ by the maximum principle), that for asymptotically flat boundary conditions the area of the minimal surface will be bounded above at all times by $$S(t,0)\equiv\vert \Sigma \vert \le \frac{2\pi^{n/2}}{\Gamma(n/2)} \exp \left [
-\frac12 (n-1)(n-2)t \right ], \ n\ge 3. \label{eq4.11}$$ This should therefore give an indication of the effect of our boundary condition at finite $r=r_c$, which we typically pick to be at $r_c=10$ (where $r=1$ is the minimal sphere). The results are depicted in figure 5.
A further technical check concerns the choice of different Hamilton-DeTurck flow. For this, we return to (\[eq3.4\]) and this time choose the flat background, so $\psi=0$ in the evolution equations. The result is displayed in figure 6.
The evolutions take much longer to collapse, but collapse occurs just as before, and in fact the code seems more reliable close to the collapse time.[The difference appears due to numerical computation of second spatial derivatives, which grow to much larger magnitude when $\psi$ is the second (nonzero) option in (\[eq3.4\]) than when $\psi=0$.]{} We present graphs for small $\alpha$, but we carried out the calculation for larger $\alpha$ as well, with similar results.
Discussion
==========
The difference between our results and those of Husain-Seahra may have at least two origins. One is the diffuculty in dealing numerically with the nonparabolic system (\[eq3.16\], \[eq3.17\]). Another is the difficulty in numerically modelling fall-off conditions on a noncompact manifold using boundary conditions at a fixed boundary. Parabolic equations exhibit instantaneous action-at-a-distance, though it is typically exponentially suppressed. Our numerical runs appear to produce some positive scalar curvature at the large $r$ boundary, which must be chosen sufficiently distant to suppress the effect on the minimal surface. The effect is shown in figure 7. For the figure, we integrated over $r\in [1,100]$ but chose to display only the portion $r=[90,100]$. At early times we see a small pulse of positive scalar curvature concentrated at infinity, presumably attributable in some way to the boundary condition. As time progresses, the pulse dissipates. The figure shows three times, one shortly after the flow begins, one at around the midpoint in time of the flow, and one just before collapse of the minimal surface. We chose to display $\alpha=3$ but the behaviour is similar for all $\alpha$.
On the other hand, as discussed in subsection 3.2.2, the boundary conditions in [@HS] imply a concave boundary at $r=r_c$ and, concomitantly, appear to produce a source of negative curvature there. Then the maximum principle argument of section 2 would not apply. The critical behaviour observed in [@HS] may be a property of boundary conditions, rather than evidence for unstable regions in the space of initial data for asymptotically flat metrics.
Acknowledgements
================
We are grateful to Paul Mikula, whose numerical code for a related problem formed the basis from which our code was developed. We thank Viqar Husain and Sanjeev Seahra for discussions. TB was supported by an Undergraduate Summer Research Award from the Natural Sciences and Engineering Research Council of Canada (NSERC). This research was supported by an NSERC Discovery Grant to EW.
[99]{} M Choptuik, [*Universality and scaling in gravitational collapse of a massless scalar field*]{}, Phys Rev Lett 70 (1993) 9–12. B Chow and D Knopf, [*The Ricci Flow: An Introduction*]{} (AMS, Providence, 2004). JC Cortissoz, [*On the Ricci flow in rotationally symmetric manifolds with boundary*]{}, PhD thesis (Cornell, 2004, unpublished). X Dai and L Ma, [*Mass under the Ricci flow*]{}, Commun Math Phys 274 (2007) 65–80. D Garfinkle and J Isenberg, [*Critical behavior in Ricci flow*]{}, in [*Geometric Evolution Equations*]{}, Contemp Math 367 (2005) 103–114, eds S-C Chang, B Chow, S-C Chu, C-S Lin (AMS, Providence) \[arxiv.org:math/0306129\]. D Garfinkle and J Isenberg, [*The modelling of degenerate neck pinch singularities in Ricci flow by Bryant solitons*]{}, JMP 49 (2008) 073505 \[arxiv:0709.0514\]. L Gulcev, TA Oliynyk, and E Woolgar, [*On long-time existence for the flow of static metrics with rotational symmetry*]{}, preprint \[arxiv.org:0911.2037\]. M Gutperle, M Headrick, S Minwalla, and V Schomerus, [*Space-time energy decreases under world sheet RG flow*]{}, JHEP 0301 (2003) 073 \[arxiv:hep-th/0211063\]. SW Hawking and GFR Ellis, [*The large scale structure of space-time*]{} (Cambridge, 1973). V Husain and SS Seahra, [*Ricci flows, wormholes and critical phenomena*]{}, Class Quantum Gravit 25 (2008) 222002 \[arxiv:0808.0880\]. J Jost, [*Partial Differential Equations*]{}, Graduate Texts in Mathematics 214 (Springer, New York, 2002). J Morgan and G Tian, [*Ricci flow and the Poincaré conjecture*]{}, Clay Mathematics Monographs Vol 3 (AMS, Providence, 2007). TA Oliynyk and E Woolgar, [*Rotationally symmetric Ricci flow on asymptotically flat manifolds*]{}, Commun Anal Geom 15 (2007) 535–568 \[arxiv:math/0607438\]. M Simon, [*A class of Riemannian manifolds that pinch when evolved by Ricci flow*]{}, Manuscripta Math 101 (2000) 89–114. TJ Willmore, [*Total curvature in Riemannian geometry*]{} (Ellis Horwood, Chicester, 1982); [*Note on embedded surfaces*]{}, Ann St Univ Iasi, Section Ia Matematica 11B (1965) 493–496.
[^1]: balehows@ualberta.ca
[^2]: ewoolgar@math.ualberta.ca
|
---
abstract: 'We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules generates all graphs corresponding to a given stabilizer state; we do this by constructing an efficient procedure for testing the equality of any two stabilizer graphs.'
author:
- 'Matthew B. Elliott'
- Bryan Eastin
- 'Carlton M. Caves'
title: Graphical description of the action of Clifford operators on stabilizer states
---
Introduction
============
Stabilizer states are ubiquitous elements of quantum information theory, as a consequence both of their power and of their relative simplicity. The fields of quantum error correction, measurement-based quantum computation, and entanglement classification all make substantial use of stabilizer states and their transformations under Clifford operations [@gottesmanthesis; @oneway; @vandennest71]. Stabilizer states are distinctly quantum mechanical in that they can possess arbitrary amounts of entanglement, but the existence of a compact description that can be updated efficiently sets them apart from other highly entangled states. Their prominence, as well as their name, derives from this description, a formalism in which a state is identified by a set of Pauli operators generating the subgroup of the Pauli group that stabilizes it, i.e., the subgroup of which the state is the $+1$ eigenvector. In this paper we seek to augment the stabilizer formalism by developing a graphical representation both of the states themselves and of the transformations induced on them by Clifford operations. It is our hope that this representation will contribute to the understanding of this important class of states and to the ability to manipulate them efficiently.
The notion of representing states graphically is not new. Simple graphs are regularly used to represent *graph states*, i.e., states that can be constructed by applying a sequence of controlled-$Z$ gates to qubits each initially prepared in the state $({| 0 \rangle}+{| 1 \rangle})/\sqrt2$. The transformations of graph states under local Clifford operations were studied by Van den Nest [@vandennest69], who found that local complementation generated all graphs corresponding to graph states related by local Clifford operations. The results presented here constitute an extension of work by Van den Nest and others to arbitrary stabilizer states.
Our graphical depiction of stabilizer states is motivated by the equivalence of stabilizer states to graph states under local Clifford operations [@vandennest69]. Because of this equivalence, *stabilizer-state graphs* can be constructed by first drawing the graph for a locally equivalent graph state and then adding decorations, which correspond to local Clifford operations, to the nodes of the graph. Only three kinds of decoration are needed since it is possible to convert an arbitrary stabilizer state to some graph state by applying one of six local Clifford operations (including no operation) to each qubit. The standard form of the generator matrix for stabilizer states plays a crucial role in the development of this material, particularly in exploring the properties of *reduced graphs*, a subset of stabilizer graphs (which we introduce) that is sufficient for representing any stabilizer state. More generally, however, our stabilizer-state graphs are best understood in terms of a canonical circuit for creating the stabilizer state. This description also permits the use of circuit identities in proving various useful equalities. In this way, we establish a correspondence between Clifford operations on stabilizer states and graph operations on the corresponding stabilizer-state graphs. Ultimately, these rules allow us to simplify testing the equivalence of two stabilizer graphs to the point that the test becomes provably trivial.
This paper is organized as follows. Section \[sec:background\] contains background information on stabilizer states, Clifford operations, and quantum circuits. Stabilizer-state graphs are developed in Sec. \[sec:graphs\], and a graphical description of the action of local Clifford operations on these graphs is given in Sec. \[sec:transformations\]. The issue of the uniqueness of stabilizer graphs is taken up in Sec. \[sec:equiv\]. The appendix deals with the graph transformations associated with ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates.
Background {#sec:background}
==========
Stabilizer formalism\[subsec:stabilizer\]
-----------------------------------------
The *Pauli group* on $N$ qubits, $\mathcal{P}_N$, is defined to be the group, under matrix multiplication, of all $N$-fold tensor products of the identity, $I$, and the Pauli matrices, $X=\sigma_1$, $Y=\sigma_2$, and $Z=\sigma_3$, including overall phases $\pm 1$ and $\pm i$. A *stabilizer state* is defined to be the simultaneous $+1$ eigenstate of a set of $N$ commuting, Hermitian Pauli-group elements that are independent in the sense that none of them can be written as a product of the others. These elements are called *stabilizer generators* and are denoted here by $g_j$, while $g_{jk}$ is used to denote the $k$th Pauli matrix in the tensor-product decomposition of generator $g_j$. Stabilizer generator sets are not unique; replacing any generator with the product of itself and another generator yields an equivalent generating set. An arbitrary product of stabilizer generators, $g=g_1^{a_1}\cdots g_N^{a_N}$, where $a_j \in \{0,1\}$ is called a *stabilizer element*; the stabilizer elements make up a subgroup of the Pauli group known as the *stabilizer*.
A *graph state* is a special kind of stabilizer state whose generators can be written in terms of a simple graph as $$\label{eq:graphgenerators}
g_j = X_j \prod_{k \in {\mathcal{N}}(j)} Z_k\;,$$ where ${\mathcal{N}}(j)$ denotes the set of neighbors of node $j$ in the graph (see Sec. \[subsec:terminology\] and Ref. [@diestel] for graph terminology). *Simple graphs* and, hence, graph states can also be defined in terms of an *adjacency matrix* $\Gamma$, where $\Gamma_{\!jk}=1$ if $j \in {\mathcal{N}}(k)$ and $\Gamma_{\!jk}=0$ otherwise. In a simple graph, a node is never its own neighbor, i.e., there are no self-loops; thus the diagonal elements of the adjacency matrix of a simple graph are all equal to zero.
Binary representation of the Pauli group \[subsec:binary\]
----------------------------------------------------------
The binary representation of the Pauli group associates a two-dimensional binary vector $r(\sigma_j)$ with each Pauli matrix $\sigma_j$, where $r(I) = \left( \begin{array}{cc} 0 & 0
\end{array} \right)$, $r(X) = \left( \begin{array}{cc} 1 & 0
\end{array} \right)$, $r(Y) = \left( \begin{array}{cc} 1 & 1
\end{array} \right)$, and $r(Z) = \left( \begin{array}{cc} 0 & 1
\end{array} \right)$. This association is generalized to an arbitrary element $p\in\mathcal{P}_N$, whose $k$th Pauli matrix is $p_k$, by letting $r(p)$ be a $2N$-dimensional vector whose $k$th and $(N+k)$th entries are the entries of $r(p_{k})$, i.e., $r(p_{k})=\left(
\begin{array}{cc} [r(p)]_k & [r(p)]_{N+k}
\end{array} \right)$. The binary representation of a Pauli-group element specifies the element up to the overall phase of $\pm1$ or $\pm
i$; Hermitian Pauli-group elements are specified up to a sign.
The binary representation of the product of two Pauli-group elements, $p,q\in\mathcal{P}_N$, is the binary sum of their associated vectors, i.e., $r(p q) = r(p) + r(q)$. Two such elements commute if their *skew product*, $$r(p)\wedge r(q)=\sum_{j=1}^N[r(p)]_j[r(q)]_{N+j}+[r(p)]_{N+j}[r(q)]_j\;,$$ has value $0$; otherwise they anticommute.
Using binary notation, a full set of generators for a stabilizer state can be represented (up to a sign for each generator) by an $N
\times 2N$ *generator matrix* whose $j$th row is $r(g_j)$. Because the stabilizer generators commute, the rows of the generator matrix are orthogonal under the skew product. Similarly, the independence of the stabilizer generators under matrix multiplication implies that the rows of the generator matrix are linearly independent under addition. The freedom to take products of stabilizer generators without changing the stabilized state becomes, for the generator matrix, the freedom to add one row of the matrix to another. The exchange of any pair of qubits $j$ and $k$ of a stabilizer state corresponds to the exchange of columns $j$ and $k$ and columns $N+j$ and $N+k$ in the generator matrix. Since rows of the generator matrix are linearly independent and have vanishing skew product, these two operations are sufficient to allow us to transform any generator matrix to a *canonical form* [@nielsenchuang], $$\label{eq:canonicalstabilizer}
\left(\begin{array}{cc|cc}I & A & B & 0 \\ 0 & 0 & A^T & I \end{array}\right)\;,$$ where $B=B^T$ and the vertical line divides the matrix in half. The vanishing skew product of rows of the generator matrix implies that $B$ is a symmetric matrix and that $A$ and $A^T$ appear as indicated. The $I$s in Eq. (\[eq:canonicalstabilizer\]) denote a pair of identity matrices whose dimensions sum to $N$. The dimension of the upper left identity matrix is called the *left rank* of the generator matrix. Due to the freedom inherent in qubit exchange, the canonical form of the generator matrix is not unique.
For graph states, Eq. (\[eq:canonicalstabilizer\]) becomes $$\label{eq:canonicalgraph}
\left(\begin{array}{c|c}I & B \end{array}\right)\;,$$ where $B$ has only $0$s on the diagonal. Graph states thus have generator matrices of full left rank, with $B$ being the adjacency matrix of the graph state’s underlying graph. We denote generator matrices of this sort by the term *strict graph form*, whereas the term *graph form* is used for generator matrices of the form shown in Eq. (\[eq:canonicalgraph\]) where $B$ is any symmetric matrix.
The binary representation does not encode the sign of Pauli group elements, so the generator matrix really specifies a set of generators up to $2^N$ possible sign assignments and thus specifies not a single stabilizer state, but rather an orthonormal basis of simultaneous eigenstates of the generators, each member of which corresponds to one of the sign choices. Despite this, we continue, for convenience, to refer to *the* stabilizer state associated with a generator matrix.
Clifford operations\[subsec:clifford\]
--------------------------------------
The *Clifford group* is the normalizer of the Pauli group, i.e., the set of all unitary operators $U$ such that $UpU^{\dag} \in
\mathcal{P}_N$ for all $p \in \mathcal{P}_N$. Any local operation in the Clifford group can be obtained by repeated application of Hadamard and phase gates, which are written in the standard basis as $$\label{eq:HSdefinition}
\begin{array}{ccc}
\displaystyle{H = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1 & 1 \\1 & -1\end{array}\right)}
&
\mbox{and}
&
\displaystyle{S= \left(\begin{array}{cc}1 & 0 \\0 & i\end{array}\right)\;,}
\end{array}$$ respectively. Adding the two-qubit controlled-$Z$ (or controlled-sign) gate, $$\label{eq:CZdefinition}
\begin{array}{c}
{{{\vphantom{Z}}^{C}\!{Z}}}= \left(\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1\end{array}\right)\;,
\end{array}$$ completes the basis for Clifford operations [@gottesman; @aaronson].
Quantum circuits\[subsec:circuit\]
----------------------------------
Quantum-circuit notation [@nielsenchuang] is a pictorial method for representing the application of discrete operations to a quantum system. As with electrical circuits, repeated usage of a small number of simple, standard parts results in complex quantum circuits that are easier to understand and implement. A typical component gate set contains a basis for Clifford operations together with a single non-Clifford operation. For brevity, we employ more Clifford gates than are necessary to generate the Clifford group, augmenting the gates of Eqs. (\[eq:HSdefinition\]) and (\[eq:CZdefinition\]) by the Pauli matrices and $S^\dagger=S^3=ZS$.
Any state can be expressed in terms of a quantum circuit that prepares it from some fiducial state, traditionally taken to be the one in which each qubit is initially in the state ${| 0 \rangle}$. To prepare any $n$-qubit stabilizer state from ${| 0 \rangle}^{\otimes n}$, it is sufficient to apply Clifford gates, since, by definition of the Clifford group, there exists a Clifford operation that takes the stabilizer of the fiducial state to the stabilizer of the desired state under conjugation. Applying this operation to the fiducial state yields a $+1$ eigenstate of the stabilizers of the desired state, i.e., the desired stabilizer state.
Graph states can be written in a particularly simple form using quantum-circuit notation. Any graph state can be prepared using two layers of gates. In the first layer, $H$ is applied to each qubit. In the second layer, ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates are applied between all pairs of qubits corresponding to connected nodes on the graph. To prepare an arbitrary stabilizer state, it is sufficient to add a third layer that contains only $H$ and $S$ gates. This is because any stabilizer state is equivalent to some graph state under local Clifford operations [@vandennest69].
Stabilizer-state graphs\[sec:graphs\]
=====================================
Graphs from generator matrices\[subsec:gengraphs\]
--------------------------------------------------
Any stabilizer state with a generator matrix of canonical form can be converted by local Clifford operations to a state possessing a generator matrix of strict graph form. Applying $H$ to the last $N-r$ qubits of the stabilizer state, where $r$ is the initial left rank of the canonical-form generator matrix, exchanges columns $r+1$ through $N$ in the generator matrix with columns $N+r+1$ through $2N$, so that a generator matrix as in Eq. (\[eq:canonicalstabilizer\]) is transformed to $$\label{eq:intermediate}
\left(\begin{array}{cc|cc}I & 0 & B & A \\0 & I & A^T & 0\end{array}\right)\;.$$ The diagonal of $B$ in this graph-form generator matrix can then be stripped of $1$s without otherwise changing the generator matrix by applying $S$ to offending qubits. The resulting generator matrix has the form of a graph state and corresponds to a stabilizer state that differs from that represented by Eq. (\[eq:canonicalstabilizer\]) by at most a single $H$ or $S$ gate per qubit.
The close relationship between graph states and stabilizer states suggests the possibility of a graph-like representation of stabilizer states. One approach to such a representation is simply to transform the generator matrix of a stabilizer state into strict graph form, draw the graph thereby obtained, and add decorations to each node indicating whether an $H$ or $S$ was applied to the corresponding qubit in the process of reaching strict graph form. Thus are stabilizer graphs constructed in this paper, where we choose to signal the application of Hadamard gates by hollow (unfilled) nodes and the application of phase gates by loops. The decision to represent $S$ gates by loops is motivated by the standard graph convention that a $1$ on the diagonal of an adjacency matrix denotes a loop on that node.
![(a) A generator matrix for a stabilizer state. (b) A canonical-form generator matrix obtained from (a) by row and qubit swapping. (c) The adjacency matrix indicated by (b). (d) The adjacency matrix of (c) with the qubit swaps undone. (e) The reduced stabilizer graph associated with (a). In parts (a)-(d) the columns have been labeled by the corresponding qubit. In part (e) the nodes are labeled sequentially, beginning with the top node and moving clockwise. It is clear that the qubit swaps are not actually necessary, since we reverse them in the end. The generator matrix in (a) can be converted to graph form directly by exchanging columns $2$ and $3$ on the left with the matching columns on the right, an operation that corresponds to applying a Hadamard to qubits $2$ and $3$. In graph form, the adjacency matrix is just the right half of the generator matrix. Loops arise from $1$s on the diagonal of the adjacency matrix, and hollow nodes are used to indicate which columns were exchanged between the right and left halves of the generator matrix to get the adjacency matrix. Notice that there are no edges between hollow nodes, nor are there any loops on hollow nodes.\[fig:graph\]](Figure1){width="8.5cm"}
The following steps provide a recipe for translating an arbitrary stabilizer generator matrix into a *stabilizer graph*:
1. Through row reduction and qubit swapping, transform the generator matrix into canonical form, as in Eq. (\[eq:canonicalstabilizer\]), keeping track during the process of how columns of the generator matrix map to qubits.
2. Draw the graph corresponding to the adjacency matrix $$\label{eq:BAAT0}
\left(\begin{array}{cc}B&A\\A^T&0\end{array}\right)\;,$$ including loops for $1$s on the diagonal.
3. Make solid the nodes corresponding to the rows and columns of the submatrix $B$, and make hollow the nodes corresponding to the rows and columns of the submatrix $0$.
Notice that this procedure does not associate every combination of edges, loops, hollow nodes, and solid nodes with a stabilizer state. Because the submatrix $0$ in Eq. (\[eq:BAAT0\]) contains only $0$s, hollow nodes never have loops, and there are no edges between hollow nodes. We refer to stabilizer graphs having this property as *reduced*. An example of a generator matrix and an associated reduced stabilizer graph is given in Fig. \[fig:graph\].
It is important to note that the graphs in this paper are *labeled graphs* in that each node is associated with a particular qubit. Swapping two qubits is a physical operation that generally produces a different quantum state. In our graphs a *SWAP* gate can be described either by relabeling the corresponding nodes or by exchanging the nodes and all their decorations and connections while leaving the labeling the same. Since the process of bringing the generator matrix into canonical form can involve swapping qubits, we must keep track during this process of the correspondence between qubits and columns of the generator matrix and thus between these columns and the nodes of our graphs.
Just as a generator matrix contains no information about generator signs, so also are stabilizer graphs derived from generator matrices devoid of such information. It is for this reason that we can be cavalier about whether $S$ or $S^\dagger=ZS$ is used to convert a canonical generator matrix to strict graph form. In the absence of sign information, however, stabilizer graphs are best thought of as specifying an orthonormal basis rather than a single stabilizer state. Luckily, it is not hard to include sign information in the graph, as we show in the next subsection.
Graphs from quantum circuits {#subsec:circuitgraphs}
----------------------------
Having motivated a stabilizer-graph notation using generator matrices, we now turn to the quantum-circuit formalism to expand it. The binary representation of stabilizers lacks a convenient way to keep track of generator signs, applied gates, and qubit swaps. Since, for example, labeling is important, the column exchanges required to bring a generator matrix to canonical form must be tracked, perhaps by appending an extra row with qubit labels to the generator matrix. Details such as this are automatically dealt with when deriving stabilizer graphs from quantum circuits.
Consider a quantum circuit consisting of three layers of gates applied to $N$ qubits, each initially in the state ${| 0 \rangle}$. In the first layer, the Hadamard gate, $H$, is applied to each qubit. In the second, controlled-sign gates, ${{{\vphantom{Z}}^{C}\!{Z}}}$, are applied between various pairs of qubits. Finally, in the third layer, sign-flip gates, $Z$, phase gates, $S$, and Hadamard gates are applied (in that order) to various subsets of qubits. We refer to such circuits as having *graph form*.
The descriptor arises because a quantum circuit in this form can be depicted as a graph possessing three kinds of decoration. In such a graph, two nodes are linked if a ${{{\vphantom{Z}}^{C}\!{Z}}}$ gate is applied between the qubits of the circuit corresponding to those nodes; the various kinds of node decoration indicate the presence or absence of terminal $Z$, $S$, and $H$ gates on the corresponding qubits. It is the restrictions of such a representation that impel us to specify an order for the terminal gates, since $SH\neq HS$. The decoration corresponding to each gate is as follows: $Z$ gates are denoted by a minus sign in the node, $S$ gates by a loop, and $H$ gates by a hollow (as opposed to a solid) node. We refer to an arbitrary arrangement of solid and hollow nodes with loops, edges, and signs as a *stabilizer-state graph* or, more simply, as a *stabilizer graph*. An example graph-form circuit and the corresponding stabilizer graph are given in Fig. \[fig:stabgraph\].
![(a) A circuit in graph form. (b) A stabilizer graph corresponding to the circuit in (a). $\protect{{{\vphantom{Z}}^{C}\!{Z}}}$ gates between qubits are transformed into links between nodes, terminating $Z$ gates become negative signs, terminating $S$ gates result in loops, and terminating $H$ gates are denoted by hollow nodes. Nodes in (b) are labeled sequentially, beginning with the top node and moving clockwise. \[fig:stabgraph\]](Figure2){width="8.5cm"}
As might be guessed from our choice of decorations, the $Z$ gates in a graph-form circuit specify the signs of the stabilizer generators. The gates applied in the third layer of a graph-form circuit can be written as $$\begin{aligned}
\prod_{j=1}^N H_j^{c_j}S_j^{b_j}Z_j^{a_j}\;,
\label{eq:thirdLayer}\end{aligned}$$ where $a_j$, $b_j$, and $c_j$ are binary variables taking on the values $0$ or $1$. From Eq. (\[eq:thirdLayer\]) it follows that the third layer of gates transforms a set of graph-state generators as in Eq. (\[eq:graphgenerators\]) to the following stabilizer-state generators: $$\begin{aligned}
g_j=&(-1)^{a_j+b_jc_j}\nonumber\\
&\times X_j^{(b_j+1)(c_j+1)}Y_j^{b_j}Z_j^{(b_j+1)c_j}
\prod_{k\in\mathcal{N}(j)}Z_k^{c_k+1}X_k^{c_k}\;.
\label{eq:stabilizerStateGenerators}\end{aligned}$$ Equation (\[eq:stabilizerStateGenerators\]) shows that the exclusive effect of each terminal $Z$ operator is to flip the sign of a single stabilizer generator. This can also be seen through circuit identities, since pushing a $Z$ gate from the third layer of a graph-form quantum circuit to the beginning of the circuit merely transforms it to an $X$ gate; flipping an input bit is equivalent to flipping the sign of the associated stabilizer since the stabilizer of ${| 0 \rangle}$ is $Z$, which acquires a negative sign under conjugation by $X$. Using either of these methods, it is clear that terminal $Z$ gates, and hence the signs in stabilizer graphs, only impact the signs of the stabilizer generators, and can thus be omitted when these signs are thought to be unimportant.
Modulo generator signs, the definition of stabilizer graphs given in the previous subsection and the definition given in this one are compatible. The graph-form circuit and generator matrix associated with a stabilizer graph each specify the same stabilizer up to possible signs. The method of deriving stabilizer graphs from generator matrices described in Sec. \[subsec:gengraphs\], however, produces exclusively reduced stabilizer graphs, i.e., stabilizer graphs satisfying the restriction that hollow nodes never have loops and never be connected to other hollow nodes. In terms of graph-form quantum circuits, this is the restriction that lines with terminating Hadamard gates have no terminating $S$ gates and not be connected by ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates. The relative merits of stabilizer graphs and reduced stabilizer graphs are clarified in the following sections, particularly Sec. \[sec:equiv\]. Important roles are found for both.
{width="18cm"}
Graph transformations {#sec:transformations}
=====================
It is frequently useful to consider the way in which stabilizer states transform under the application of Clifford gates. Primarily, this is because Clifford gates take stabilizer states to stabilizer states, a property that follows from their preservation of the Pauli group. This same property implies that the action of a Clifford gate can be thought of as a transformation between the graphs representing the initial and final stabilizer states. In this section we consider the transformations induced by local Clifford gates. The transformations induced by ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates are discussed separately in the Appendix.
Terminology {#subsec:terminology}
-----------
We begin by introducing terminology for describing transformations on stabilizer graphs. We use a number of terms, some adopted from graph theory and others invented for the task at hand.
Among those terms common to graph theory are *neighbors*, *complement*, and *local complement*. The neighbors of a node $j$ are those nodes connected to $j$ by edges; we denote the set of neighbors of node $j$ by $\mathcal{N}(j)$. In the transformation rules that follow, a loop does not count as an edge, so a node is never its own neighbor. Complementing the edge between two nodes removes the edge if one is present and adds one otherwise. The local complement is performed by complementing a selection of edges, with the pattern of edges depending on whether local complementation is applied to a node or along an edge.
Performing local complementation on a node complements the edges between all of the node’s neighbors. Thus, local complementation at node $j$ transforms the adjacency matrix $\Gamma$ of a graph according to $$\label{eq:lc}
\Gamma_{lm}\rightarrow\Gamma'_{lm}=
\Gamma_{lm}+(1+\delta_{lm})\Gamma_{jl}\Gamma_{jm}\;;$$ i.e., it complements an edge if both nodes of the edge are neighbors of $j$.
Local complementation along an edge is equivalent to a sequence of local complementations on the *decision nodes*, i.e., the nodes defining the edge. The sequence is as follows: first perform local complementation on one of the decision nodes, then local complement on the other decision node, and finally local complement on the first decision node again. This sequence transforms the adjacency matrix of a simple graph in the following way: $$\begin{aligned}
\Gamma_{lm}\rightarrow\Gamma'_{lm}&=&\Gamma_{lm}
+(\Gamma_{jl}+\delta_{jl})(\Gamma_{km}+\delta_{km})\nonumber\\
&&\quad{}+(\Gamma_{jm}+\delta_{jm})(\Gamma_{kl}+\delta_{kl})\;,
\label{eq:threelc}\end{aligned}$$ where $j$ and $k$ are the decision nodes. Equation (\[eq:threelc\]) is symmetric in the two decision nodes, so it does not matter at which decision node local complementation is first performed. Additionally, since we do not consider self-loops to be edges, Eq. (\[eq:threelc\]) can be applied to adjacency matrices with nonzero diagonal entries simply by ignoring those entries. Notice that an edge one of whose nodes is a decision node transforms according to $\Gamma'_{lj}=\Gamma_{lk}$.
To describe the net effect of local complementation along an edge, it is helpful to define the *decision neighborhood* of a node as the intersection of its neighborhood with the decision nodes. Then local complementation along an edge can be summarized by three steps:
1. The edge between the decision nodes is left unchanged.
2. An edge one of whose nodes is a decision node is transferred from this decision node to the other.
3. An edge neither of whose nodes is a decision node is complemented if its end nodes have decision neighborhoods that are not empty and not identical.
To these terms we add *flip* and *advance*. Flip is used to describe the simple reversal of some binary property, such as the sign or the fill state (color) of a node. Advance refers specifically to an action on loops; advancing generates a loop on nodes where there was not previously one, and it removes the loop and flips the sign on nodes where there was a loop. Its action mirrors the application of the phase gate, since $S^2=Z$.
{width="18cm"}
Stabilizer-graph transformations\[subsec:stabgraphtrans\]
---------------------------------------------------------
An arbitrary local Clifford operation can be decomposed into a product of $H$, $S$, and $Z$ gates (the $Z$ is unnecessary, but convenient). The effect of a local Clifford operation on a stabilizer graph can thus be obtained by repeated application of the following six transformation rules.
1. Applying $H$ to a node flips its fill.
2. Applying $S$ to a solid node advances its loop.
3. Applying $S$ to a hollow node without a loop performs local complementation on the node and advances the loops of its neighbors.
If the node has a negative sign, flip the signs of its neighbors as well.
4. Applying $S$ to a hollow node with a loop flips its fill, removes its loop, performs local complementation on it, and advances the loops of its neighbors.
If the node does not have a negative sign, flip the signs of its neighbors as well.
5. Applying $Z$ to a solid node flips its sign.
6. Applying $Z$ to a hollow node flips the signs of all of its neighbors. If the node has a loop, its own sign is flipped as well.
These transformation rules can be derived from the circuit identities in Fig. \[fig:identproof\], which rely on the basic circuit identities given in Fig. \[fig:basics\]. Given an understanding of the relationship between circuits and graphs, transformation rules T1, T2, and T5 are trivial. Transformation rule T6 follows simply from Fig. \[fig:identproof\](a,b). Transformation rules T3 and T4 derive from Figs. \[fig:identproof\](c) and (d), respectively.
Reduced-stabilizer-graph transformations\[subsec:redstabgraphtrans\]
--------------------------------------------------------------------
The transformation rules T1–T6 do not generally take reduced stabilizer graphs to reduced stabilizer graphs. From Sec. \[subsec:gengraphs\], however, we know that there exists a reduced stabilizer graph corresponding to each stabilizer state, so it is always possible to represent the effect of a local Clifford operation as a mapping between reduced stabilizer graphs. The appropriate transformation rules for reduced stabilizer graphs are listed below, excepting those for $Z$ operations, which are identical to T5 and T6.
Applying $H$ to a solid node without a loop, which is only connected to other solid nodes, flips the fill of that node.
Applying $H$ to a solid node with a loop, which is only connected to other solid nodes, performs local complementation on the node and advances the loops of its neighbors.
Flip the node’s sign, and if it now has a negative sign, flip the signs of its neighbors as well.
Applying $H$ to a solid node without a loop, which is connected to a hollow node, flips the fill of the hollow node and performs local complementation along the edge connecting the nodes.
Flip the sign of nodes connected to both the solid and hollow nodes. If either of these two nodes has a negative sign, flip it and the signs of its current neighbors.
Applying $H$ to a solid node with a loop, which is connected to a hollow node, performs local complementation on the solid node and then on the hollow node. Then it removes the loop from the solid node, advances the loops of the solid node’s current neighbors, and flips the fill of the hollow node.
Flip the signs of nodes that were originally connected to both the solid and hollow nodes. If the originally solid node initially had a negative sign, flip it and the signs of its current neighbors, and if the originally hollow node initially had a negative sign, flip the signs of its current neighbors.
Applying $H$ to a hollow node flips its fill.
Applying $S$ to a solid node advances its loop.
Applying $S$ to a hollow node performs local complementation on that node and advances the loops of its neighbors.
If the node has a negative sign, flip the signs of its neighbors as well.
Of these transformation rules, T(i), T(v), and T(vi) are trivial, and T(vii) is a rewrite of T3. To prove the others requires results from Sec. \[sec:equiv\], in particular, the equivalence rules in Sec. \[subsec:circuitequiv\]. Specifically, T(ii) is obtained by applying equivalence rule E1, which gives an equivalent, but unreduced graph and then applying the Hadamard, via rule T1, which leaves a reduced graph. For T(iii), one first applies the Hadamard, via rule T1, and then uses equivalence rule E2 to convert to a reduced graph. In the case of T(iv), one applies the Hadamard, using rule T1, and then applies equivalence rule E1, first to the originally solid node and then to the hollow node. A key part of these transformations is the conversion of stabilizer graphs to reduced form, a process discussed in more detail in Sec. \[subsubsec:convert\].
It is not hard to check that, in using rules T(iii) and T(iv), any other hollow nodes that are connected to the originally solid node do not become connected and do not acquire loops, in accordance with the need to end up with a reduced graph.
Equivalent stabilizer graphs\[sec:equiv\]
=========================================
As we have defined it, the mapping between a graph-form circuit and its corresponding stabilizer graph is one-to-one. This does not imply, however, that each stabilizer state corresponds to a unique graph. On the contrary, an example of different graph-form circuits corresponding to the same stabilizer state can be found in Fig. \[fig:identproof\]. Applying an additional $S$ gate to the top qubit in Fig. \[fig:identproof\](d) makes the initial circuit identical to that in (b), but the final circuits are quite different. The two circuit identities thus define distinct transformation rules for applying a $Z$ gate to a hollow node with a loop, thereby demonstrating the existence of multiple, equivalent graph-form circuits. For every way of obtaining a particular stabilizer state from a circuit in graph form, there is an associated stabilizer graph. In this section we examine the resulting equivalence classes of stabilizer graphs, first by presenting equivalence rules for full and reduced stabilizer graphs and then by introducing simplified graph-form-circuit equalities which we use to show that the equivalence rules given here are complete.
{width="18cm"}
Stabilizer-graph equivalences {#subsec:circuitequiv}
-----------------------------
Applying either of the following two rules to a stabilizer graph yields an equivalent graph, i.e., one which represents the same stabilizer state.
1. Flip the fill of a node with a loop. Perform local complementation on the node, and advance the loops of its neighbors.
Flip the node’s sign, and if the node now has a negative sign, flip the signs of its neighbors as well.
2. Flip the fills of two connected nodes without loops, and local complement along the edge between them.
Flip the signs of nodes connected to both of the two original nodes. If either of the two original nodes has a negative sign, flip it and the signs of its current neighbors.
The first of these equivalence rules can be obtained by applying an additional $S$ gate to the top qubit in the identity of Fig. \[fig:identproof\](d) and equating the final circuit to the final circuit in Fig. \[fig:identproof\](b). For the second rule, we need yet another circuit identity. Figure \[fig:transIdent\](a) shows how Hadamards can be removed from a pair of connected qubits without $S$ gates. Figure \[fig:transIdent\](b) extends this identity to a demonstration of rule E2.
Reduced-stabilizer-graph equivalences {#subsec:redequiv}
-------------------------------------
The equivalence rules of the previous section can be reworked to yield equivalence rules for reduced stabilizer graphs. The resulting equivalence rules are
For a hollow node connected to a solid node with a loop, local complement on the solid node and then on the hollow node. Then remove the loop from the solid node, advance the loops of its current neighbors, and flip the fills of both nodes.
As for signs, follow the sequence in transformation rule T(iv).
For a hollow node connected to a solid node without a loop, local complement along the edge between them. Then flip the fills of both nodes.
As for signs, follow the sequence in equivalence rule E2.
There is a simple relationship between the two sets of equivalence rules. Equivalence rule E(ii) is identical to E2 for the case that the two connected nodes have opposite fill. Equivalence rule E(i) is simply rule E1 applied twice: first to the solid node with the loop and then once to the hollow node that has acquired a loop from the first application of E1. The second application of E1 is needed because the resulting graph is not reduced after one employment of the rule.
Both of these equivalence rules can also be derived by applying two Hadamards to a solid node, E(i) handling the case in which the solid node has a loop and E(ii) the case in which it does not. Thus equivalence rule E(i) is simply transformation rule T(iv) followed by use of rule T(i) to apply a second Hadamard to the originally solid node. Likewise, E(ii) is rule T(iii) followed by T(i) to apply a second Hadamard to the originally solid node. Notice that both rules preserve the number of hollow nodes.
{width="18cm"}
Constructive procedure for showing sufficiency of equivalence rules {#subsec:equivalenceproof}
-------------------------------------------------------------------
Having described a set of rules for converting between equivalent stabilizer graphs, we show in this section that, in each case, the aforementioned rules generate the entire equivalence class of stabilizer graphs. The proof is divided into three parts. The first part shows how to use rules E1 and E2 to convert an arbitrary stabilizer graph to an equivalent graph in reduced form. The second part explains how an equivalence test for a pair of reduced stabilizer graphs can be simplified, using rules E(i) and E(ii), to a special form. Finally, the third part proves that the graphs on the two sides of such a simplified equivalence test are equivalent only if they are trivially identical. Taken as a whole, this proves that the set of equivalence rules given in Secs. \[subsec:circuitequiv\] and \[subsec:redequiv\] is sufficient to convert (reversibly) between any two equivalent stabilizer graphs and thus that they generate all stabilizer graphs that are equivalent to the same stabilizer state. Similarly, considering only the final two parts of the proof shows that the rules given in Sec. \[subsec:redequiv\] are sufficient to generate all reduced stabilizer graphs.
### Converting stabilizer graphs to reduced form {#subsubsec:convert}
Two features identify a stabilizer graph as reduced. In a reduced graph, hollow nodes never have loops, and hollow nodes are never connected to each other. Equivalence rule E1 can be used to convert looped nodes from hollow to solid. Applying rule E1 in this sort of situation can cause hollow nodes to acquire loops, but each application fills one hollow node of the graph, so the procedure will terminate in at most a number of repetitions equal to the number of hollow nodes in the graph. Similarly, connected hollow nodes in the resulting graph can be converted to solid nodes using the appropriate case of rule E2. Once again, the process is guaranteed to terminate because the number of hollow nodes to which the rule might be applied decreases by two with each application of the rule. Concomitantly, the conversion of a stabilizer graph to an equivalent reduced graph never increases the number of hollow nodes.
### Simplifying reduced-graph equivalence testing {#subsubsec:simplify}
Equivalence testing for pairs of reduced graphs is facilitated by simplifying the equivalence such that nodes that are hollow only in the first graph never connect to nodes that are hollow only in the second. This simplification can be accomplished by iterating the following process. Choose a pair of connected (in either graph) nodes $a$ and $b$ such that $a$ is hollow in one graph and $b$ is hollow in the other, and to the graph in which they are connected, apply the relevant reduced equivalence rule to the selected nodes. Among other things, the equivalence operation reverses the fill of the two nodes it is applied to. Since one node is hollow and the other solid, this preserves the total number of hollow nodes while yielding a node that is hollow in both graphs. Because it is applied only to unpaired hollow nodes, subsequent uses of this equivalence rule do not disturb the newly paired hollow node. Consequently, this process also terminates in at most a number of repetitions equal to the number of hollow nodes in each of the graphs.
### Trivial evaluation of simplified reduced-graph\
equivalence tests {#subsubsec:evaluate}
The two reduced graphs composing a simplified reduced-graph equivalence test are equivalent, i.e., correspond to the same state, if and only if the graphs are identical. To see why this is so, we return to the graph-form quantum circuits discussed earlier. In addition to the standard restrictions for reduced graphs, the circuits corresponding to the graphs in a simplified equivalence test have the following property: if in one of the circuits, a qubit with a terminal $H$ participates in a ${{{\vphantom{Z}}^{C}\!{Z}}}$ gate with a second qubit (which cannot have a terminal $H$), then in the other circuit, it cannot be true that the second qubit has a terminal $H$ and the first does not. We prove the triviality of simplified reduced-graph equivalence testing by considering an arbitrary simplified reduced-graph equivalence test and showing that the two graphs must be identical if they are to correspond to the same state.
In terms of unitaries, an arbitrary graph-form circuit equality can be written as $$\begin{aligned}
\begin{split}
\prod_{g\in{\mathcal{H}}_l} &H_g
\prod_{j\in{\mathcal{S}}_l} S_j
\prod_{h\in{\mathcal{Z}}_l} Z_h
\prod_{\gamma\in{\mathcal{C}}_l} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma
\prod_{k} H_k {| 0 \rangle}^{\otimes n} \\
=&
\prod_{g\in{\mathcal{H}}_r} H_g
\prod_{j\in{\mathcal{S}}_r} S_j
\prod_{h\in{\mathcal{Z}}_r} Z_h
\prod_{\gamma\in{\mathcal{C}}_r} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma
\prod_{k} H_k {| 0 \rangle}^{\otimes n}\;,
\end{split}
\label{eq:arbSimpGraphFormCircEqual}\end{aligned}$$ where ${\mathcal{C}}$ lists the pairs of qubits participating in ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates and ${\mathcal{H}}$, ${\mathcal{Z}}$, and ${\mathcal{S}}$ are sets enumerating the qubits to which $H$, $Z$, and $S$ gates are applied respectively. The total number of qubits is denoted by $n$ and the subscripts $l$ and $r$ discriminate between the circuits on the left- and right-hand sides of the equation. In terms of these sets, a reduced-graph-form circuit satisfies the constraints ${\mathcal{H}}\cap{\mathcal{S}}=\varnothing$ and $\{a,b\}\not\in{\mathcal{C}}$ for all $a,b\in{\mathcal{H}}$. The circuits in simplified tests also satisfy $\{a,b\}\not\in{\mathcal{C}}_l,{\mathcal{C}}_r$ for all $a\in{\mathcal{H}_l\cap\overline{\mathcal{H}}_r}$ and $b\in{\overline{\mathcal{H}}_l\cap\mathcal{H}_r}$, where $\bar{{\mathcal{H}}}$ denotes the complement of ${\mathcal{H}}$, i.e., the set of qubits to which $H$ is not applied.
Suppose now that the two graphs have hollow nodes at different locations; i.e., at least one of the sets, ${\mathcal{H}_l\cap\overline{\mathcal{H}}_r}$ and ${\overline{\mathcal{H}}_l\cap\mathcal{H}_r}$, is not empty. For specificity, let’s say that ${\mathcal{H}_l\cap\overline{\mathcal{H}}_r}$ is not empty. In the language of circuits, this means that there exists a qubit $a$ that has a terminal $H$ on the left side of Eq. (\[eq:arbSimpGraphFormCircEqual\]), but not on the right side. Since qubit $a$ is part of a circuit for a reduced graph, it does not participate in ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates with other qubits that possess terminal $H$ gates. Consequently, on the left side of Eq. (\[eq:arbSimpGraphFormCircEqual\]), the ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates involving qubit $a$ can all be moved to the end of the circuit where they become ${{{\vphantom{X}}^{C}\!{X}}}$ gates with $a$ as the target. Doing this and transferring the ${{{\vphantom{X}}^{C}\!{X}}}$ gates to the other side yields,
$$\begin{aligned}
\begin{split}
\prod_{g\in{\mathcal{H}}_l} H_g
\prod_{j\in{\mathcal{S}}_l} S_j
\prod_{h\in{\mathcal{Z}}_l} Z_h
\prod_{\gamma\in{\mathcal{C}}_l\mbox{\scriptsize{\ s.t.\ }}a\not\in\gamma}\!\!\!\!\!{{{\vphantom{Z}}^{C}\!{Z}}}_\gamma\;\;\prod_{k} H_k
{| 0 \rangle}^{\otimes n} =\prod_{b\in{\mathcal{N}}_l(a)} {{{\vphantom{X}}^{C}\!{X}}}_{ba}
\prod_{g\in{\mathcal{H}}_r} H_g
\prod_{j\in{\mathcal{S}}_r} S_j
\prod_{h\in{\mathcal{Z}}_r} Z_h
\prod_{\gamma\in{\mathcal{C}}_r} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma
\prod_{k} H_k {| 0 \rangle}^{\otimes n}\;,
\end{split}\label{eq:modCircEqual}\end{aligned}$$
where ${\mathcal{N}}_l(a)$ denotes the set of qubits that participate in ${{{\vphantom{Z}}^{C}\!{Z}}}$ gates with qubit $a$ on the left-hand side of Eq. (\[eq:arbSimpGraphFormCircEqual\]).
Because the original graph equality (\[eq:arbSimpGraphFormCircEqual\]) was simplified, ${\mathcal{N}}_l(a)\cap{\mathcal{H}}_r=\varnothing$ and, by assumption, $a\not\in{\mathcal{H}}_r$, so the ${{{\vphantom{X}}^{C}\!{X}}}$ gates and the terminal Hadamards on the right side of Eq. (\[eq:modCircEqual\]) do not act on the same qubits. Thus we can commute the ${{{\vphantom{X}}^{C}\!{X}}}$ gates past the terminal Hadamards. Moreover, we can then move the ${{{\vphantom{X}}^{C}\!{X}}}$ gates to the beginning of the circuit where they have no effect and can therefore be dropped. During this migration, however, they generate a complicated menagerie of phases. The resulting expression for the right side of Eq. (\[eq:modCircEqual\]) is $$\begin{aligned}
\begin{split}
\prod_{g\in{\mathcal{H}}_r} H_g
\prod_{j\in{\mathcal{S}}_r} S_j
\raisebox{-.35em}{\Bigg(\Bigg.}
\prod_{d\in{\mathcal{N}}_{l}(a)}S_d {{{\vphantom{Z}}^{C}\!{Z}}}_{da}
\raisebox{-.35em}{\Bigg.\Bigg)}^{\!\!\mathbf{1}_{{\mathcal{S}}_r}(a)}
\prod_{h\in{\mathcal{Z}}_r} Z_h
\raisebox{-.35em}{\Bigg(\Bigg.}
\prod_{f\in{\mathcal{N}}_{l}(a)} Z_f
\raisebox{-.35em}{\Bigg.\Bigg)}^{\!\!\mathbf{1}_{{\mathcal{Z}}_r}(a)}
\prod_{\gamma\in{\mathcal{C}}_r}{{{\vphantom{Z}}^{C}\!{Z}}}_\gamma
\prod_{\delta\in{\mathcal{N}}(a)}{{{\vphantom{Z}}^{C}\!{Z}}}_\delta
\prod_{c\in{\mathcal{N}}_l(a)\cap{\mathcal{N}}_r(a)}\!\!\!Z_c\;\;\prod_{k} H_k{| 0 \rangle}^{\otimes n}\;,
\end{split}\label{eq:modCircEqualRight}\end{aligned}$$
where ${\mathcal{N}}_r(a)$ is defined similarly to ${\mathcal{N}}_l(a)$, $${\mathcal{N}}(a) = \{ \{ p, q \} | p\in{\mathcal{N}}_l(a), q\in{\mathcal{N}}_r(a),
p\neq q \}\;,$$ and $\mathbf{1}$ represents an indicator function, e.g., $\mathbf{1}_{{\mathcal{Z}}_r}(a)$ equals $1$ if $a\in{\mathcal{Z}}_r$ and $0$ otherwise.
{width="18cm"}
It might appear that we have made things substantially worse by this rearrangement, but in an important sense, Eq. (\[eq:modCircEqualRight\]) is now very simple with regard to qubit $a$: $H$ is applied to qubit $a$ followed by a sequence of unitary gates all of which are diagonal in the standard basis. This implies that there are an equal number of terms in the resultant state where qubit $a$ is in the state ${| 0 \rangle}$ and the state ${| 1 \rangle}$. On the left side of Eq. (\[eq:modCircEqual\]), however, the only gate remaining on qubit $a$ is the identity or an $X$, depending on whether $a\in{\mathcal{Z}}_l$; thus qubit $a$ is separable and is either in state ${| 0 \rangle}$ or ${| 1 \rangle}$ depending on whether $a\in{\mathcal{Z}}_l$. Consequently, our initial assumption that ${\mathcal{H}}_l \neq {\mathcal{H}}_r$ is incompatible with satisfying the equality.
The preceding discussion shows that two graphs composing a simplified equivalence test are equivalent only if they have hollow nodes in exactly the same locations. Given this constraint, the terminal Hadamards can be canceled from both sides of Eq. (\[eq:arbSimpGraphFormCircEqual\]), giving $$\begin{aligned}
\begin{split}
\prod_{h\in{\mathcal{Z}}_l}& Z_h \prod_{j\in{\mathcal{S}}_l} S_j \prod_{\gamma\in{\mathcal{C}}_l} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma
\prod_{k} H_k {| 0 \rangle}^{\otimes n} \\
=&\prod_{h\in{\mathcal{Z}}_r} Z_h \prod_{j\in{\mathcal{S}}_r} S_j \prod_{\gamma\in{\mathcal{C}}_r} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma
\prod_{k} H_k {| 0 \rangle}^{\otimes n}\;.\
\end{split}\label{eq:paredSimpGraphFormCircEqual}\end{aligned}$$ The state after the initial Hadamards is an equally weighted superposition of all the states in the standard basis. The subsequent unitaries are diagonal in the standard basis, so they put various phases in front of the terms in the equal superposition. Since a unitary is fully described by its action on a complete set of basis states, demanding equality term-by-term in Eq. (\[eq:paredSimpGraphFormCircEqual\]) amounts to requiring that $$\begin{aligned}
\prod_{h\in{\mathcal{Z}}_l}& Z_h
\prod_{j\in{\mathcal{S}}_l} S_j
\prod_{\gamma\in{\mathcal{C}}_l} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma=
\prod_{h\in{\mathcal{Z}}_r} Z_h
\prod_{j\in{\mathcal{S}}_r} S_j
\prod_{\gamma\in{\mathcal{C}}_r} {{{\vphantom{Z}}^{C}\!{Z}}}_\gamma\;,\end{aligned}$$ which is only satisfied when ${\mathcal{Z}}_l={\mathcal{Z}}_r$, ${\mathcal{S}}_l={\mathcal{S}}_r$, and ${\mathcal{C}}_l={\mathcal{C}}_r$. Thus, after simplification, the equivalence of pairs of reduced graphs is trivial to evaluate, since equivalence requires that the two graphs be identical.
An example of the entire process of testing graph equivalence is given in Fig. \[fig:equivalenceexample\]. An example which illustrates the circuit manipulations described algebraically in the text of this section is given in Fig. \[fig:partthreeexample\].
As mentioned above, our proof provides a constructive procedure for testing the equivalence of stabilizer graphs. Moreover, it shows that the set of equivalence rules given in Sec. \[subsec:circuitequiv\] is sufficient to convert between any equivalent stabilizer graphs and that the rules given in Sec. \[subsec:redequiv\] are sufficient to convert between any equivalent reduced stabilizer graphs. Since the conversion of an arbitrary stabilizer graph to reduced form never increases the number of hollow nodes and the rules E(i) and E(ii) that convert among reduced graphs preserve the number of hollow nodes, we conclude that the reduced stabilizer graphs for a stabilizer state are those that have the least number of hollow nodes.
Conclusion {#sec:conclude}
==========
Motivated by the relation between the graphs and the generator matrices associated with graph states, we extend the graph representation of states to encompass all stabilizer states. These *stabilizer graphs* differ from the graphs employed by [@hein] in that nodes can be either hollow or solid and can possess both loops and signs. The additional decorations identify the local Clifford operations that relate the desired stabilizer state to the graph state corresponding to the unadorned graph. Imposing the restriction that the number of hollow nodes be minimal yields a subset of the stabilizer graphs which we term reduced. Reduced graphs follow naturally from the binary representation of the stabilizer formalism, while generic stabilizer graphs are more closely related to the quantum-circuit formalism. For graph states, reduced stabilizer graphs are identical to the standard representation of these states in terms of graphs.
Using circuit identities, we derive a set of rules for transforming stabilizer graphs under the application of various local Clifford gates. From this list, we abstract a similar set for transforming reduced stabilizer graphs. Considering these transformation rules, particularly those for reduced stabilizer graphs, it becomes clear that the mapping between stabilizer states and stabilizer graphs is not one-to-one. Rules for converting between equivalent graphs are found, and we prove that the equivalence rules given are universal by developing a constructive procedure for testing the equivalence of any two stabilizer graphs.
This research was partly supported by Army Research Office Contract No. W911NF-04-1-0242 and National Science Foundation Grant No. PHY-0653596.
Controlled-$Z$ gates\[app:czgates\]
===================================
{width="18cm"}
The transformation rules given in Sec. \[sec:transformations\] suffice to describe the effect of any local Clifford operation on a stabilizer state. In order to complete the set of transformation rules for the Clifford group, we include transformation rules for the ${{{\vphantom{Z}}^{C}\!{Z}}}$ gate here. In the interest of brevity, we consider only reduced-stabilizer-graph transformations. Transformation rules for general stabilizer graphs can be derived by first using the equivalence rules in Sec. \[subsec:circuitequiv\] to convert to an equivalent reduced graph and then applying the transformation rules below.
Applying ${{{\vphantom{Z}}^{C}\!{Z}}}$ between two solid nodes complements the edge between them.
Applying ${{{\vphantom{Z}}^{C}\!{Z}}}$ between a hollow node and a solid node complements the edges between the solid node and the neighbors of the hollow node.\
------------------------------------------------------------------------
\
Flip the solid node’s sign if the two nodes were initially connected and the hollow node did not have a sign or if the two nodes were not connected and the hollow did have a sign.
Applying ${{{\vphantom{Z}}^{C}\!{Z}}}$ between two hollow nodes effects the third step of local complementation along the (unoccupied) edge between the two nodes.
Nodes that neighbor both decision nodes flip their signs. If a decision node initially had a sign, flip the signs of nodes connected to the other decision node.
Transformation rule T(viii) is trivial since the ${{{\vphantom{Z}}^{C}\!{Z}}}$ gate simply commutes into layer two of the reduced-graph-form circuit. The circuit identities needed to prove rules T(ix) and T(x) are given in Fig. \[fig:CZident\].
[cc]{}
D. Gottesman, PhD thesis, Caltech (1997).
R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. **86**, 5188 (2001).
M. Van den Nest, J. Dehaene, B. De Moor, Phys. Rev. A **71**, 062323 (2005).
M. Van den Nest, J. Dehaene, and B. De Moor, Phys.Rev. A **69**, 022316 (2004).
R. Diestel, *Graph Theory* (Springer-Verlag, New York, 2000).
M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information* (Cambridge University Press, Cambridge, England, 2000), Sec. 10.5.3.
D. Gottesman, Phys. Rev. A [**57**]{}, 127 (1998).
S. Aaronson and D. Gottesman, Phys. Rev. A [**70**]{}, 052328 (2004).
M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H.-J. Briegel, in *Entanglement in Graph Staes and its Applications*, Proceedings of the International School of Physics “Enrico Fermi,” Course 162, Varenna, July 5–15, 2005, edited by G. Casati, D. L. Shepelyansky, P. Zoller, and G. Benenti (IOS Press, Amsterdam, 2006), e-print arXiv:quant-ph/0602096.
|
---
abstract: |
This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti 1982 work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question – universes with zero total angular momentum, that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities – is capable of accommodating a wide range of classical physics phenomena. Furthermore, as I develop in Paper [**II**]{}, this relational particle model is a useful toy model for canonical general relativity.
I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I show how to access many examples of quantized relational particle models and then interpret these from a relational perspective. By these means, previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for some aspects of closed universe quantum cosmology. While, these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own.
---
[[**I. RECONCILIATION WITH STANDARD CLASSICAL AND QUANTUM THEORY**]{}]{}
[*Department of Physics, P-412 Avadh Bhatia Physics Laboratory,*]{}
[*University of Alberta, Edmonton, Canada, T6G2J1;*]{}
[*Peterhouse, Cambridge, U.K., CB21RD;*]{}
[*DAMTP, Centre for Mathemetical Sciences, Wilberforce Road, Cambridge, U.K., CB30WA.*]{}
[**PACS Numbers: 04.60-m, 04.60.Ds**]{}
Introduction
============
Newton’s formulation and conceptual framework for mechanics is based on his notions of absolute space and absolute time [@Newton]. However, Leibniz [@L], Berkeley [@B] and Mach [@M] envisaged that mechanics should rather be [*relational*]{} i.e feature solely relative distances, angles and times. Despite these occasional criticisms, no concrete realizations of such mechanical theories were achieved, and absolutism took hold. So while Lagrange [@Lag] and Jacobi [@Jac] invented notable techniques for the study of $N$ body Newtonian mechanics (NM): working in the barycentre frame, using relative particle separation coordinates, using the relative particle cluster separation Jacobi coordinates that diagonalize the relative formulation’s kinetic term, eliminating one angular momentum and using the Jacobi form for the action, they appear not to have assembled these techniques into a relationalist program [@Relper]. The first elements of that appear in Dziobek [@Dziobek] and Poincaré [@Poincare]. A concrete relational synthesis was attained by Barbour and Bertotti in 1982 (BB82) [@BB82] which I present in Sec 2. This has also been studied in [@BS; @Rovelli; @Smolin; @B94I; @B94II; @Buckets; @LB; @EOT; @Gergely; @GergelyMcKain; @RWR], reviewed in [@Kuchar92; @Kiefer; @Landerson] and philosophized about in [@Barbourphil; @Buckets; @EOT; @Pooley]. It consists of a classical relational reformulation of a [*portion of NM*]{}. BB82-type formulations may be perceived to have an undesirable indirectness in that while their derivation ends up free of absolute space, it proceeds indirectly via what may be viewed as absolutist notions (even if BB82 select these purely on grounds of convenience among various ways of conceptualizing in terms of configuration spaces). It is worth commenting that while it is not logically necessary for a relational viewpoint to lead to a recovery of NM, it is nevertheless interesting for a resolution of the absolute versus relative motion debate to have such a feature.
In this paper I examine and improve three key features [@Panderson] of BB82 relational particle mechanics (RPM). Firstly, it [*can*]{} also be cast in a [*directly relational form*]{}, which I attain in Jacobi action relative particle separation coordinate form in Sec 3 and furthermore recast in terms of [*relative Jacobi coordinates*]{} in Sec 4. Jacobi coordinates are particularly well-suited to RPM and are mathematically-central to this paper.
Secondly, by a ‘portion’ I mean the restriction to closed systems/universes of zero total angular momentum (AM) $\L = 0$, that are conservative and have a kinetic term that is homogeneous quadratic in the velocities. It is therefore important to investigate whether these are major restrictions. In Sec 5 I argue that this portion accommodates much, by tricks which counterbalance the [*sub*]{}system of interest with other subsystems elsewhere. This portion also admits a [*non*]{}homogeneous quadratic extension.
Thirdly, I begin to investigate the quantum implications of shifting from an absolutist to a relationalist viewpoint. This is particularly relevant given how GR and QM incorporate incompatible notions of time. This ‘Problem of Time’ and its conceptual ramifications are good reasons (see e.g. [@Battelle; @Kuchar80; @Kuchar92; @Isham93; @KucharPOTother; @B94II; @EOT]) why these two areas of theoretical physics have not been successfully combined insofar as we do not have a satisfactory account of quantum GR (nor a fully satisfactory account of any distinct theory of quantum gravity at all). In particular, the traditional framework of QM is based on Newton’s formulation of particle mechanics, absolute space and time and all, while GR, as well as being a successful reconciliation of SR and Newtonian gravity and a successful improvement of the theory of gravitation, is also Einstein’s attempt to free physics of absolute structure. Many theoretical physicists consider this feature of GR to be an important lesson about how the universe works, to the extent that quantum gravity schemes that, contrarily, have extraneous ‘background dependence’ [@bigcite; @Isham93; @Kiefer; @Dirac] can be regarded as not yet [*conceptually*]{} satisfactory. Indeed one major goal of nonperturbative M-theory is to attain this background independence.
Now, given that the relational reformulation of $\L = 0$ NM [*is itself a successful abolition of absolute structure*]{}, the following interesting question arises. Q1: Does this give a different sort of QM from that of the traditional absolutist approach? A further reason for this study is that the BB82 RPM approach to $\L = 0$ NM closely parallels [@BB82; @B94I; @RWR; @EOT; @Landerson] that to the geometrodynamical formulation of spatially compact without boundary GR [@BSW; @RWR; @Vanderson; @TP]. Indeed, [@BS; @Kuchar92; @B94I; @B94II; @EOT; @Kiefer] have considered BB82 RPM as a useful toy model arena for the investigation of quantum gravity and quantum cosmology issues such as the Problem of Time and difficulties with closed universes. Note however that there is a perceived obstruction to this program. While Barbour and others have been advocating BB82 RPM as a resolution of the absolute versus relative motion debate for some years, the Barbour–Smolin (BS) preprint [@BS] [*also*]{} claims that quantizing BB82 does not give a good semiclassical limit on two counts.
BS1: that small masses affect the QM spread of large masses.
BS2: Due to constituent subsystems being interlocked by additional restrictions.
This position is followed up by arguing against the applicability of standard quantization procedures, leading them instead to ‘seek a radical alternative’. [@B94II; @EOT] may be seen as the start of Barbour’s search for this – a particular [*consistent records*]{} program.
Nevertheless, in this paper I begin by addressing Q1. This subdivides into whether different formal mathematics becomes involved and whether there are interpretational differences. I principally consider simple 1-$d$ models, regarding extension to $d > 1$ models where possible as a useful bonus. I do so by use of relative Jacobi coordinates, noting that these map many relational quantum problems to problems whose formulation is known from the standard quantization of the absolutist formulation of NM. Thus the first steps of the study of these simplest RPM models do [*not*]{} involve formal mathematical differences, permitting one to benefit from separation and special function techniques. Thereby I can cover a wide range of RPM models for the minimally relationally-nontrivial case of 3 particles. These go well beyond BS’s (piecewise) constant potentials in 1-d to additionally solve a number of interparticle harmonic oscillator and Newton–Coulomb potential problems including those presented in Sec 7.
The first salient difference is interpretational. Spreads are now in terms of relational quantities (relative separations suffice in 1-d), from which perspective I show that objection BS1 is misplaced. The second is mathematical, but turns out not to spoil the above standard techniques by being treatable [*after*]{} deploying these techniques. This involves the collective of subsystems having interlocked energy and counterbalancing AM, and the subsystem eigenspectra sometimes exhibits truncations and gaps. These features are a further development of the mathematical observations which led to objection BS2, but I also develop further what the implications of these are. For universes that have a large particle number and a diverse content (such as free particles and of both positive- and negative-potential subsystems), gaps need no longer occur, truncations become acceptable, while interlocking and counterbalancing are obscured by the practicalities of experimentation involving only a small subsystem. As our own universe is large and diverse in content, the [*‘recovery of reality’*]{} is thereby not compromised by BS2. This opens the way for semiclassical exploration of these models, which I consider further, alongside the aforementioned consistent records formulation, in [**II**]{}.7–8. On the other hand, [*small universe models*]{} do exhibit these interesting theoretical effects whereby they are useful toy models for understanding subtle ramifications that arise from considering [*closed*]{}-universe quantum cosmologies.
Barbour–Bertotti 1982 formulation in terms of auxiliary variables
=================================================================
Let $\sq$ be the naïve $Nd$-dimensional configuration space in $d$ dimensions for $N$ particles whose positions are $q_{\alpha A}$. Jacobi actions are used, which implement temporal relationalism since they are [*reparametrization-invariant*]{} in the label-time $\lambda$. The actions considered in this article are for particle models whose kinetic term $\st$ which is homogeneous quadratic in its constituent velocities. These take the form \_ = 2 , \[Jac\] where $\sv$ is the potential term and $\se$ is the total energy, taken to be a fixed constant ‘energy of the universe’ $\un$. Note that the more usual particle mechanics action, $\sS_{\mbox{\scriptsize{Euler--Lagrange}}} = \int\d t (\st- \sv)$, can indeed be cast [@Lanczos] into the incipient Jacobi form (\[Jac\]) by adjoining the Newtonian time $t$ to $\sq$, then noting that $\dot{t} = \d t/\d\lambda$ alone features in the action and subsequently eliminating it by Routhian reduction.
The classical particle mechanics notion of spatial relationalism is implemented by passing to a suitable notion of arbitrary frame. This is achieved by the introduction of a translational auxiliary $d$-vector $\aa_{\alpha}(\lambda)$ and whichever rotational auxiliary corresponds to $d$: none for $d = 1$, a scalar b$(\lambda)$ for $d = 2$ or a 3-vector b$_{\alpha}(\lambda)$ for $d = 3$, so that the $A$th particle’s position $q_{\alpha A}$ is replaced by &\_[,]{}q\_[A]{} q\_[A]{} - å\_ - [\_]{}\^\_ q\_[A]{} . \[abcorr\] Whereas the 1- and 2-$d$ cases can be written in intrinsically 1- and 2-$d$ notation, I present all 3 cases together by use the extraneous 3-$d$ $\epsilon$ symbol, under the provisos that \_ = 0 d = 1 \_ = (0,0,\_3) , \_3 , \_ = (0, 0, [L]{}\_3) , \_3 d = 2. \[provisos\] While this may look like [*doubling*]{} the allusions to absolute space, $\sq \longrightarrow \sq \times \mbox{Eucl($d$)}$ \[for Eucl($d$) the Euclidean group of translations and whichever rotations exist in dimension $d$\], I reassure the reader that this doubling of redundancy leads promptly below to the removal of the redundancy in the fashion familiar in gauge theory. I proceed by requiring the action to be built as best as possible out of objects that transform well under $\lambda$-dependent Eucl($d$). In particular, defining the [*naïve*]{} or [*Lagrange relative coordinates*]{} (relative separations between particles) $r_{\alpha AB} \equiv q_{\alpha A} - q_{\alpha B}$, = (|r\_[AB]{}| ) , \[Vrel\] wherein the auxiliary corrections straightforwardly cancel each other out. The situation with the kinetic term is more complicated. $\frac{\pa}{\pa\lambda}$ is not a tensorial operation under $\lambda$-dependent Eucl(d) transformations. As explained in [@Stewart], $\frac{\pa}{\pa\lambda}$ should rather be seen as the Lie derivative $\pounds_{\frac{\pa}{\pa\lambda}}$ in a particular frame, which transforms to the Lie derivative with respect to ‘$\frac{\pa}{\pa\lambda}$ corrected additively by generators of translations and rotations.’ This gives a kinetic term of the form (\_I, \_J, , ) = \_[A = 1]{}\^[N]{}m\_A\^ ( &\_[, ]{}[q]{}\_[A]{} ) ( &\_[, ]{}[q]{}\_[A]{} ) . \[T\] So, finally, the proposed action is $\sS_{\mbox{\scriptsize{Jacobi}}}(\q_I, \dot{\q}_J, \dot{\fa}, \dot{\fb})$ of form (\[Jac\]) with (\[Vrel\]) and (\[T\]) substituted into it.
The momenta conjugate to the $q_{\alpha A}$ are p\^[A]{} = m\_A\^&\_[, ]{}\_[A]{} . By virtue of the particular reparametrization invariance of the Lagrangian, these obey the primary quadratic constraint \_[A = 1]{}\^[N]{}\_p\^[A]{}p\^[A]{} + = . The secondary linear constraints, \^ \_[A = 1]{}\^[N]{}p\^[A]{} = 0 \[ZM\] and whichever portion of \^ \_[A = 1]{}\^[N]{} [\^]{}\_ q\_[A]{} p\^[A]{} = 0 \[ZAM\] is relevant in the corresponding dimension, follow from variation of the cyclic auxiliary coordinates $\mbox{a}_{\alpha}$ and $\mbox{b}_{\alpha}$. These constraints obey the Poisson bracket algebra {[P]{}\^, [P]{}\^} = 0 , {[P]{}\^, } = 0 , {, } = 0 , {[P]{}\^, [L]{}\^ } = [\^]{}\_[P]{}\^ , {[L]{}\^, [L]{}\^} = [\^]{}\_[L]{}\_ , {[L]{}\^, } = 0 \[first6\] for $\Q = {\cal Q} - \se$. As this is closed, there are no further constraints. The constraints (\[ZM\], \[ZAM\]) then signify that the physics is not on the doubly redundant configuration space $\sq \times \mbox{Eucl($d$)}$ but on the quotient space $\sq/\mbox{Eucl($d$)}$, thus indeed rendering absolute space irrelevant.
It is worth counting to establish which are the minimal nontrivial dynamics examples. There are $dN$ degrees of freedom in the $d$-dimensional NM of $N$ particles. There are $d$ translations and $d(d - 1)/2$ rotations which make up $d(d + 1)/2$ irrelevant motions. So the $d$-dimensional $N$-particle BB82 RPM has $d(2N - d - 1)/2$ degrees of freedom. Furthermore, one wants to express one change in terms of another change rather than in terms of an arbitrarily-reparametrizable label-time. Thus dynamical nontriviality dictates $N$ and $d$ be such that $d(2N - d - 1)/2 > 1$ which gives $N \geq 3$ both in 3-$d$ and in 1-$d$. In this article I focus principally on the simplest nontrivial dynamics: $N = 3$.
Passage to direct formulation in terms of relative variables alone
==================================================================
In contrast to the above formulations, there is a distinct, non-Newtonian, failed and often rediscovered theory formulated directly in terms of the $r_{IJ}$ and without auxiliaries referring to absolute space. I should point out that this theory has a kinetic term that is exceptionally simple, = \_[IJ]{}\_[IK]{} , \[BB77T\] which is probably one reason why this theory has been rediscovered so many times. The question then arises whether BB82 RPM can [*also*]{} be cast as a presumably more complicated direct formulation of this kind, whereupon not even indirect or unphysical reference to absolute space is required. I address this below, partly foreshadowed by work of Lynden-Bell [@LB], Gergely [@Gergely] and Jacobi.
I use that the $d = 3$ working contains everything under the provisos (\[provisos\]). and then comment on each individual case $d = 1, 2, 3$ as these differ significantly. I begin with $\sS_{\mbox{\scriptsize Jacobi}}(\q_I, \dot{\q}_J, \dot{\fa}, \dot{\fb})$. I define $\tilde{m}$ as the total mass ${\sum_{I = 1}^{N} m_I}$. I eliminate $\dot{\aa}_{\alpha}$ from its own variational equation \_ = \_I m\_I ( \_[I]{} - [\_]{}\^\_q\_[I]{} ) \[the Lagrangian counterpart of the Hamiltonian expression (\[ZAM\])\]. This results in the (semi-)eliminated action \^\*(\_[IJ]{}, \_[KL]{}, ) = 2 \^\*(\_[KL]{}, \_[IJ]{}) = \_[I <]{}\_J |\_[IJ]{} - \_[IJ]{}|\^2. \[semi\] I next eliminate $\dot{\bb}_{\alpha}$ from its own variational equation \^ = (\^[-1]{}(\_[IJ]{}))\^ \_(\_[IJ]{}, \_[KL]{}) \[bvar\] where $\stackrel{\B}{I}_{\alpha\beta}$, $\stackrel{\B}{L}_{\alpha}$ are the barycentric inertia tensor and AM respectively, which are easily castable into relative separation coordinate form: \_(\_[IJ]{}) = \_[I <]{}\_J ( |\_[IJ]{}|\^2\_ - \_[IJ]{}\_[IJ]{} ) , \_(\_[IJ]{}, \_[KL]{}) = [\_]{}\^ \_[I <]{}\_J \_[IJ]{} \_[IJ]{} . \[relf\] This results in the eliminated action \^[\*\*]{}(\_[PQ]{}, \_[ST]{}) = 2 \[bmthgg\] for \^[\*\*]{}(\_[PQ]{}, \_[ST]{}) = \^[\*]{}(\_[PQ]{}, 0) - \_(\^[-1]{})\^\_ . \[Trel\]
(\[bmthgg\]) is the formulation of the stated portion of NM cast entirely in relative terms for $d = 3$. For $d = 1$, $\fb = 0$ so the working stops at (\[semi\]). For $d = 2$, (\[bvar\]) simplifies considerably since $\stackrel{\B}{I}_{\alpha\beta}$ becomes a mere scalar (\_[IJ]{}) \_[33]{}(\_[IJ]{}) \_[I <]{}\_[J]{}|\_[IJ]{}|\^2 , so one has trivial explicit invertibility and the second term in (\[Trel\]) simplifies considerably \[see also (\[Jenga\])\].
Coordinate improvements: Jacobi coordinates
===========================================
As written, my expressions depend on all of the $\r_{IJ}$’s rather than on an $(N - 1)$-member basis. I first redress this by noting that \_[IJ]{}, I = 1 N - 1, J = I + 1 \[basis\] and recasting the previous section’s symmetric but redundant ‘Lagrangian’ expressions asymmetrically yet nonredundantly in terms of these alone \[e.g using $\r_{13} = \r_{12} + \r_{13}$\].
Next I consider mapping the $N$ $\q_I$ position vector coordinates linearly to a maximal number $N - 1$ vector relative particle (cluster) separation coordinates $\urho_i$ and 1 vector absolute coordinate $\urho_{\mN}$. The relative coordinates therein can be recognized since then from (\[basis\]) \_I \_I = \_[J= 1]{}\^[N]{}\_[IJ]{}\_J \_[IJ]{} \_[J=1]{}\^N\_[IJ]{} = 0 . \[test\] If one has any such set of coordinates containing a maximal number of relative coordinates $\urho_i$, $i = 1$ to $N - 1$, the final coordinates $\urho_{\mN}$ being no true relative coordinates, the zero total momentum constraint of Sec 2 gives $$0 = \sum_{I = 1}^{N}\p^I
= \sum_{I = 1}^{N}\sum_{J = 1}^{N} \frac{\pa \urho_J}{\pa \q_I} \upi^I
= \sum_{I = 1}^{N}\sum_{J = 1}^{N} \uuA_{IJ}\upi^I
= 0 + \sum_{I = 1}^{N}\uuA_{\mN I}\upi^{\mN} \Rightarrow$$ \^ = 0 , \[irrel\] where $\upi^I$ are the momenta corresponding to the new set of coordinates, and using (\[test\]) in the last 2 steps. Thus the zero momentum constraint can be absorbed by passing to any such coordinate system, and this causes the absolute coordinates $\urho_{\mN}$ therein to drop out of all remaining equations, which thereby involve relative coordinates alone.
Furthermore, I wish $\st^*$ not only to be in terms of a basis of relative coordinates but also to be diagonal. This will be particularly useful in this paper when it comes to quantization, and is true of the [*Jacobi coordinates*]{} $\R_I$ [@LJR; @Marchal], which exist for all relevant $N$ (i.e $N \geq 3$). As among the ($N$ arbitrary) Jacobi coordinates, without loss of generality $\R_i$ $i = 1$ to $N - 1$ are relative, I set these to be the particular relative coordinates I use. Because of the existence of the zero momentum constraint, which may be cast in the form (\[irrel\]), it is irrelevant which single absolute coordinate vector I adjoin to these as the constraint will in any case wipe out this choice, so I can choose my last coordinate to be $\q_{\mN}$ rather than $\R_{\mN}$ for simplicity of computation. I then find that the relative Jacobi coordinates not only diagonalize $\st$ but have a large number of further properties which are particularly well-suited to the study of RPM. Namely, the form-preservation under the $\q_I$ [*space to*]{} $\R_i$ [*space map*]{} of $\st$ (both as a function of velocities and of momenta), the moment of inertia $J$, $\stackrel{\B}{I}_{\alpha\beta}$, $\L_{\alpha}$ and $\stackrel{\B}{L}_{\alpha}$; see also Sec [**II**]{}.4 for one more such preserved object and Secs 6 and [**II**]{}.6 for major applications of these properties.
As a first application of relative Jacobi coordinates, I recast the indirect formulation of Sec 2 as the action \^[\*]{}(, , ) = 2 \^\*(, \_k) = \_[i = 1]{}\^[N - 1]{}|\_i - \_i|\^2 . With $\underline{{\cal P}} = 0$ reducing to the absense of any additional absolute coordinates, the surviving constraints are = \_[i = 1]{}\^[N - 1]{} + - \[HamT\] and \_(\_i, \_j) = [\_]{}\^\_[i = 1]{}\^[N - 1]{}R\_[i]{} P\^i\_ = \_[i = 1]{}\^[N - 1]{}L\_[i]{} = 0 . \[DamT\] Note how employing relative Jacobi coordinates retains the separability of the zero AM constraint while preserving the form of the individual AM operators.
As a second application, I recast the direct formulation of Sec 3 dimension by dimension. For $d = 1$, \^[\*\*]{}\[R\_i\] = 2 \^[\*\*]{}() = \_[i = 1]{}\^[N - 1]{} \_i\^2 . So these coordinates are diagonal and also $d = 1$ ensures that this formulation is not just relative but also fully relational, because $\sS = \sS[|\R_{i}|]$ [*alone*]{}.
For $d = 2$, \^[\*\*]{}\[\_[i]{}\] = 2 for (\_i, ) = \_[i = 1]{}\^[N - 1]{} |\_i|\^2 - \_[i = 1]{}\^[N - 1]{}\_[j = 1]{}\^[N - 1]{} \^\^R\_[i]{}\_[i]{}R\_[j]{}\_[j]{} = \_[i = 1]{}\^[N - 1]{}[M\_i]{}|\_i|\^2 . \[Braz\] But \^\^R\_[i]{}\_[i]{}R\_[j]{}\_[j]{} = (\_1 \_2)(\_1 \_2) - (\_1 \_2)(\_2 \_1) , which is made out of relative angles and relative separations, and so the Jacobi coordinates serve to give a fully relational formulation. Alternatively \[to compare it with (\[BB77T\]), with [@Gergely] and for future reference\] (\[Braz\]) can be written in terms of an (inverse) configuration space metric (\_[i]{}, \_[j]{}) = \_[i = 1]{}\^[N - 1]{}\_[j = 1]{}\^[N - 1]{} (\^[-1]{})\^[ij]{}\_[i]{}\_[j]{} , (\^[-1]{})\^[ij]{} = M\_i\^\^[ij]{} - \^\^R\_\^[i]{}R\_\^[j]{} . \[Jenga\] .
For $d = 3$, \^[\*\*]{}(\_i, ) = \^[\*]{}(, 0) - \_ (\^[-1]{})\^ \_ , \_ = \_[i = 1]{}\^[N - 1]{}M\_i ( |\_i|\^2\_ - R\_[i]{}R\_[i]{} ) \_(\_i, \_j) = [\_]{}\^\_[i = 1]{}\^[N - 1]{}M\_iR\_[i]{}\_[i]{} . The inverse configuration space metric is now (\^[-1]{})\^[ij]{} = M\_i\^\^[ij]{} - M\_iM\_j [\_]{}\^(\^[-1]{})\^[\_]{}\^ R\_\^i R\_\^j . \[Janga\] Note the complication through the presence of the nontrivial inverse of the inertia tensor $I^{-1}$. I currently can do no better than express this as relational variables plus angles between Jacobi vectors and principal directions of the system’s inertia quadric. This underscores the approach in practice for papers [**I**]{} and [**II**]{}: work on techniques for the 1-$d$ problems since these will turn out to already exhibit many of the interesting features of RPM models; direct extensions of these methods to $d > 1$ are furthermore worth considering even though they will sometimes be a semi-relational ‘halfway house’ rather than fully relational methods. Moreover, some of the remaining interesting features appear in the scale-invariant 1-$d$ set-up (see paper [**II**]{}).
The above are the last formulations of the 1-$d$, 2-$d$ and 3-$d$ $\L = 0$ conservative homogeneous quadratic portion of NM in this paper. \[Sec [**II**]{}.4 contains yet further formulations and study of the configuration space metrics (\[Jenga\], \[Janga\])\]. I next address the question of whether focussing on this portion of NM is a significant restriction.
How restrictive is considering only this portion, at a classical level?
=======================================================================
BB82 RPM and $\L = 0$ NM give coincident physical predictions. If $\L \neq 0$ momentum physics is under study, one can always represent it as a [*subsystem*]{} of a $\L = 0$ universe [@BB82; @LB], so that the restriction to $\L = 0$ universes is by no means as severe as might be naïvely suggested.
One concern is that the viability of this rests on the $N$-body problem’s theoretical framework being such that initially distant subsystems do not fall together below any desired finite timescale internal to the subsystem under study. Reversely, such falling together requires the unboundedness of some velocities and hence of $\st$ and hence of $\sv$. This is termed a [*singularity*]{}. While it used to be thought that this could only occur in situations involving collisions, it has been shown that for 5 bodies it is possible to attain this by merely coming arbitrarily close to collisions [@Xia]. It is still mere conjecture however that singular solutions are of measure zero in the set of all solutions. In any case, sufficiently accurate physical modelling of the universe acts to prevent arbitrarily distant subsystems from falling together in finite time: astrophysics with realistic matter will experience short-range forces interfering with the potential being arbitrarily negative or GR (which respects a positive energy theorem) will take over, while SR will bound infall velocities.
Can one test whether $\L = 0$ or $\neq 0$ in our universe? It is true that $\L = 0$ and $\neq 0$ (sub)systems are capable of evolving qualitatively differently as exemplified by Hill’s work [@HillMoeckel]. That BB82 requires $\L = 0$ has been considered to be a prediction of the BB82 formulation as a separate theoretical entity from NM [@Pooley]; moreover $\L = 0$ appears to be true for the universe we are in. Also, the possibility of formulating $\L \neq 0$ NM relationally should not be dismissed. While this goes against the Poincaré principle that Barbour advocates, that at most positions and velocities should need to be specified in RPM, this principle might just be a simplicity; for sure, $\L \neq 0$ is known to be much more complicated than $\L = 0$ by the work of Dziobek [@Dziobek] and of Poincaré [@Poincare].
Finally, there are some robustness issues. Can branches of physics external to particle mechanics be incorporated? Doing so would remove interference by external torques violating AM conservation, and the possibility of absolute space or time being bestowed upon particle mechanics by other branches of physics. Electromagnetism can probably be incorporated (see also p. 14). Dissipative processes such as linear air resistance and diffusion I can accommodate by a counterbalancing trick [@MF] that adjoins nonphysical fields (which may be regarded as ‘elsewhere’) into the Lagrangian, but I cannot see any means of extending this to nonlinear dissipative processes such as air resistance quadratic in the velocity. In this paper, geared toward quantization, I stay clear of dissipative processes, which one would hope are in any case recastable in terms of more fundamental processes. What is excluded by considering only conservative dynamics? Is homogeneous quadraticity restrictive? While BS say “pretty well all dynamical systems in both normal mechanics and SR field theory can be cast in such forms", both Lanczos [@Lanczos] and I [@Vanderson] have argued contrarily. In mechanics, the direct study of systems cast with linear ‘gyroscopic’ terms would be precluded; linear terms also arise in the field-theoretic counterpart of this study for moving charges and for spin-1/2 fermions. While in principle nothing goes wrong if the treatment is extended to these, practical difficulties can arise if the ensuing complications thus brought into the actions are extended to their logical conclusion. However, as a compromise, as far as I know, nonhomogeneous quadraticity suffices to describe all established physics while retaining a form that is algebraically manageable, which gives Jacobi actions of the type = ( + \_ ) .
Setting up the quantization of relational particle models
=========================================================
I next address what kind of QM follows from such a formulation for the simplest models: $N = 3$ conservative, homogeneous quadratic $\L = 0$ models. In $d$ = 1 the configuration spaces are flat, so one can hope to employ standard mathematics. As discussed in Sec 4, I choose for the moment to continue to work with now only partly-reduced flat configuration spaces when passing to $d = 2$ and $3$. This is possible because of the good fortune that Jacobi coordinates preserve the separation of $\L = 0$.
Making use of the position representation $\hat{\q}_{A} = {\q}_{A}$ and $\hat{\p}_{A} = -i\hbar\pa/\pa{\q}_{A}$, and denoting $\pa^2/\pa \q_I^2$ by $\triangle_{q_I}$ the constraints of Sec 2 become the quantum constraints (\_A) \_[I=1]{}\^N (\_A) = 0 , \[bbqc1\] (\_A) ( \_[I=1]{}\^N-\_[q\_I]{} + (\_A) - ) (\_A) = 0 . \[bbqc2\] In 1-$d$, that is all, while for $d$ = 2 or 3 there is also (the portion relevant in dimension $d$ of) (\_A) \_[I = 1]{}\^N \_I (\_A) = 0 . Note that there are no operator ordering ambiguities in any of these constraints (only in the case of the zero AM constraint are there products of $p$’s and $q$’s, and even there the order is unambiguous by symmetry–antisymmetry). I am also ‘lucky’ in Dirac’s sense [@Dirac]: the full $d$-dimensional set of constraints quantum-closes; moreover this is in direct parallel with the classical closure (i.e $\{\mbox{ } , \mbox{ } \} \longrightarrow \frac{1}{i\hbar}[ \mbox{ } , \mbox{ } ]$), so I do not present it.
The kinetic term in (\[bbqc2\]) contains a sum of Laplacians on $\mbox{{\Large $\times$}}_{I = 1}^{N}\Re^{\md}$ in absolute, particle-position Cartesian coordinates $\q_I$. Toward addressing relational problems, I recast this in terms of relative coordinates. I wish the Laplacian to remain diagonal so as to exploit separability, so I employ relative Jacobi coordinates $\R_i$. The momentum constraint then becomes = 0 , \[37\] which expresses the translation-invariance of the wavefunction even more manifestly than (\[bbqc1\]). Alternatively, one could start in relative Jacobi coordinates, in which case (\[37\]) would have been built in automatically. By either route, the remaining constraints are = \_[i = 1]{}\^[N - 1]{}\_[R\_i]{}+ and (the portion relevant in dimension $d$ of) \_[i = 1]{}\^[N - 1]{} \_i = 0 . Thus I end up on the quotient space $Q/${Translations}= $\mbox{\Large $\times$}_{i = 1}^{N - 1}\Re^{\md}$ in Cartesian coordinates $\R_i$.
For $N = 3$, I employ the Jacobi coordinates \_1 = \_[12]{} = \_1 - \_2 , \[aljac1\] \_2 = [N]{} ( (m\_1 + m\_2)\_3 - [m\_1\_1 - m\_2\_2]{} ) , \[aljac2\] \_3 = \_3 ; \[aljac3\] (see Fig 1 for their meaning) with an as-yet unfixed normalization ${\cal N}$ for convenience.
\#1\#2[0.4\#1]{}
[The meaning of (the standard normalization of) the 3-body problem relative Jacobi coordinates $R_1$ and $R_2$. Among other things, Jacobi coordinates are well-suited to study the effect of an extra mass $m_3$ on a well-understood subsystem $m_1$, $m_2$.]{}
Then = - ( \_1 + \_2 ) (\_1, \_2) for $\mu_{12}$ the standard definition of reduced mass for a $m_1$, $m_2$ pair (I henceforth drop the $\mbox{}_{12}$ indices) and the choice of ${\cal N} = \sqrt{m_3/m_1m_2M}$. Thus one has 2$d$-dimensional flat space problems, which look like standard time-independent Schrödinger equations (TISE’s), except that it is now the relative coordinates $\R_1$ and $\R_2$ and not the particle positions which play the role of Cartesian coordinates on the reduced configuration space, (and, for $d > 1$, there is an additional equation restricting the total AM to be zero).
I am principally interested in physically-motivated examples such as free, (piecewise) constant, harmonic oscillator (HO) and Newton–Coulomb type potentials. Furthermore, as each potential involved is expressible as a function of the relative separations $|\r_{IJ}|$ alone, choosing to use these potentials is a relational input, as befits the current investigation. This inherent relationalism in many commonly studied potentials is an early indicator of some mathematical similarities with the standard absolutist QM. This is worth pointing out, since it means that, at least to start off with, one does [*not*]{} get the wrong sort of formal mathematics by quantizing absolute rather than relational particle mechanics. Interpretational and mathematical differences appear later in the study.
For such potentials, I demonstrate that passing to the relative Jacobi coordinates $\R_1, \R_2$ often permits separation (\_1, \_2) = \_1(\_1) \_2(\_2) \[sepans\] . Indeed the $\q_I \longrightarrow \R_i$ map I employ is [*form-preserving*]{} for many of these problems. I.e. it maps the 3$d$-dimensional configuration space quantum problem, for which the $\q_{I}$ ($I$ = 1 to 3) play the role of Cartesian coordinates and that is separable into 3 standard $d$-dimensional problems for each $\q_{I}$, to a corresponding $2d$-dimensional configuration space problem, for which $\R_i$ ($i$ = 1 to 2) play the role of Cartesian configuration space coordinates and that is separable into 2 of the [*same kind*]{} of standard $d$-dimensional problems for each $\R_i$. The normalization of the 1-d models’ wavefunctions is then standard and not required for this article’s applications.
Thus my strategy is to solve relational quantum problems by setting up $\q_I \longrightarrow \R_i$ maps to standard separable problems, and then exploiting the well-established mathematical machinery for these. Only then do I consider, from a careful relationalist perspective, what the interpretation of the relational QM models is, and explore the new mathematics which comes upon considering these as whole systems/universes.
Examples of $N$ = 3 relational QM
=================================
Example 1: 1-d free problem
---------------------------
The free problem straightforwardly separates to 2 copies of + \_i\_i = 0 , giving wavefunctions $
\psi_i = e^{\pm i \sqrt{2\mu\sse_i}R_i/\hbar}
$ for a positive continuous spectrum. The solution of the relational problem may then be reassembled as \_ = e\^[i( - ) r\_[12]{}/]{} e\^[i(m\_1 + m\_2)r\_[23]{}/]{} . Finally, the unusual relational feature is that $\se_1$ and $\se_2$ are not independent: $\se_1 + \se_2 = \un$.
Example 2: 1-d harmonic oscillators
-----------------------------------
For the single HO relational problem I exploit the following more widely applicable Move 1: as $\r_{12} = \R_1$, any potential depending at most on $\r_{12}$ is separable in $\R_1, \R_2$ coordinates, with identity map between the potential coefficients in $\q_I$-space and $\R_i$-space. Thus -(\_1 + \_2)+ (|\_1| )= -\_1\_1 + (|\_1|)\_1 = \_1\_1 , -\_2\_2 + = \_2\_2 \_1 + \_2 = . \[Etie\] Thus, let the HO be without loss of generality between particles 1 and 2 so that $R_1$ follows the separated-out 1-d problem - \_i + \_i\_i = 0 . This is solved by wavefunctions $
\psi_i(\nn_i) = \mbox{H}_{\mn_i}(Y_i)e^{-Y_i^2/2}
$ for $Y_i = (\mu H_i/\hbar^2)^{1/4}R_i$ and H$_{\mn_i}$ the $\nn_i$th Hermite polynomial and corresponding spectrum $\se_i = \sqrt{H_i/\mu}\hbar\left(\frac{1}{2} + \nn_i\right)$, $\nn_i\in N_0$. While, $R_2$ follows the separated-out free problem of Subsec 7.1. Thus \_ = ( e\^[(m\_1 + m\_2)r\_[23]{}/]{} ) ( e\^[ im\_1 r\_[12]{}/- r\_[12]{}\^2/2]{} \_ ( (H\_1/\^2)\^[1/4]{}r\_[12]{} ) ) . Finally, $\nn \equiv \nn_1$ and $\se_2$ aren’t independent: $\sqrt{ {H_1}/{\mu} }\hbar \left( \nn +\frac{1}{2} \right) + \se_2 = \un$.
For 2 or 3 HO potentials, map according to the following Move 2. Denoting the numerical coefficient in the $IJ$ pair’s potential by $h_{IJ}$ and the ith relative coordinate’s potential coefficient by $H_i$, for 2 HO’s, without loss of generality the potential is h\_[23]{}\_[12]{}\^2 + h\_[13]{}\_[13]{}\^2 = ( ( h\_[23]{}m\_1\^2 + h\_[13]{}m\_2\^2 ) \_1\^2 + \_2\^2 + 2\_1\_2 ) , so there is the separability restriction ${h_{13}}/{h_{23}} = {m_1}/{m_2}$, and the relations between the $\q_I$ and $\R_i$ HO’s Hooke coefficients are $H_1 = {h_{23}\mu}/{m_2}$, $H_2 = {h_{23}\mu M}/{m_2m_3}$. For 3 HO’s, the same separability restriction holds again and the relations are $H_1 = h_{12} + {h_{23}\mu}/{m_2}$, $H_2 = h_{23}\mu M/m_2m_3$. Thus one obtains two separated-out 1-$d$ HO problems, with coefficients in each case as given above. The desired solution in relative separation coordinates is then \_ = \_[\_1]{} ( (H\_1/\^2)\^[1/4]{}r\_[12]{} ) \_[\_2]{} ( (H\_1/\^2)\^[1/4]{}[N]{}(m\_1r\_[13]{} + m\_2r\_[23]{}) ) e\^[- r\_[12]{}\^2/+ ([N]{}\^2(m\_1 + m\_2)r\_[23]{} + m\_1r\_[12]{})\^2/]{} . Finally, $\nn_1$ and $\nn_2$ aren’t independent: $\sqrt{ {H_1}/{\mu} }\hbar\left(\nn_1 +\frac{1}{2}\right) +
\sqrt{ {H_2}/{\mu} }\hbar\left(\nn_2 +\frac{1}{2}\right) = \un$. Notice how the 2 and 3 HO solutions are twisted rather than trivial when expressed in terms of $r_{IJ}$ coordinates.
Example 3: a simple atomic model in 3-d
---------------------------------------
This is motivated 1) for more accurate real-world modelling purposes. 2) To tackle rotation/AM, which is the interesting and complicated part of relationalism, and the inclusion of which gives toy models that resemble geometrodynamics more closely (see e.g. [**II**]{}), such as the $N = 3$, $d = 3$ ‘Triangle Land’ that Barbour’s speculations [@EOT] are based on
As noted on p.7 AM doesn’t spoil separability. It is furthermore significant that the AM in $q_I$ space is mapped to its mathematical analogue in $R_i$ space, so that one can treat AM there as one usually does in $\q_I$ space. Full polar coordinates (i.e polars in 2-$d$ and spherical polars in 3-$d$) for each $\R_i$ are useful in this respect.
For a single Coulomb potential in 3-d (representing a hydrogen atom together with a free neutral particle), one immediately has separability by Move 1, into the 3-d Helmholtz equation and the simple atomic model. Additionally, the AM are tied between the 2 problems, so spherical polars $(\rho_i, \theta_i, \phi_i)$ are well-suited. Then both of these building blocks have the angular part $\YY_{\ml_i\mm_i}(\theta_i, \phi_i)$ (spherical harmonics).
The radial part of the separated-out free problem is solved by the spherical Bessel functions $
\psi_{\rho_i} = \jj_{\ml_i}(k_i \rho_i) \propto \JJ_{\ml_i + {1/2}}(k_i \rho_i)/{\sqrt{\rho_i}},
$ where the $k_i$ are continuous positive values related to $\se_i$ by $k= \sqrt{2\mu\se_i}/\hbar$. The radial part of the separated-out simple atomic problem is solved in terms of associated Laguerre polynomials
$
\psi_{\rho_i} = \rho_1^{\ml}\mbox{L}^{2\mn + 1}_{\mn - \ml - 1}
\left(
{\mu k}\rho_1/\hbar^2
\right)
e^{- \mu k \rho_1/\hbar}.
$ The corresponding energy eigenspectrum of bound states is $\se_i = - \mu k/2\hbar^2\nn^2$ for an attractive potential $\sv = - k/\rho_1$. There is also a continuum of positive energies representing ionized states. Thus overall this simple relational atom model is solved by \_ = \_(\_1, \_1) \_1\^\^[2+ 1]{}\_[- - 1]{}([k]{}\_1/\^2) e\^[-k \_1/\^2]{} \_[-]{}(\_2, \_2)j\_ (\_2/) where $\se_1$ and $\se_2$ aren’t independent: $\se_1 + \se_2 = \un$ and the angular momentum conterbalancing explained on p.13 is in use. I don’t present this here re-expressed in terms of $\R_i$ and then $r_{IJ}$ as that becomes messy.
As regards further examples, Move 1 is widely applicable, while Move 2 also works for 2 or 3 isotropic HO’s in 2- or 3-d (see [@v2] for these).
Interpretation of N = 3 relational QM
=====================================
Decent semiclassicality
-----------------------
While one common requirement is for the wavelength $\lambda_{\mbox{\scriptsize Q}}$ to be smaller than some characteristic scale $l_{\mbox{\scriptsize C}}$ of the problem, there is no universal rigorous notion of semiclassical limit for quantum theories. Here are various procedures that one might consider in connection with such an investigation, all of which play some role in papers [**I**]{} or [**II**]{}.
1\) Consider the spread (i.e width) of the wavefunctions \[this is essentially what BS do\].
2\) Furthermore investigate how localized wavepackets are as a whole (as opposed to the spread of each wavefunction in their summand/integrand).
3\) Consider what happens to the system for large quantum numbers whereupon the wavefunction becomes wiggly on scales much shorter than $l_{\mbox{\scriptsize C}}$.
4\) Consider a WKB ansatz for the wavefunction and expand in powers of $\lambda_{\mbox{\scriptsize Q}}/l_{\mbox{\scriptsize C}}$.
[**Spread of the wavefunctions**]{}
Barbour and Smolin’s first objection BS1 was to the sensitivity of the spread of the wavefunctions of large-mass particles to the values of the mass of small-mass particles. This was in connection with piecewise-constant potential models with two small masses and one large one. I clarify why this is not a problem as follows. First, bear in mind that in the absolutist quantization, one thinks primarily of particle wavepackets, but in a 1-d relationalist quantization one should think of [*relative distance wavepackets*]{}. Then, intuitively, once one rephrases one’s standard quantum intuition about small masses being more spread out in relational terms, it is clear that the position uncertainty of the small mass dominates the relational formulation’s relative separation uncertainty between that small mass and a large mass (Fig 2).
\#1\#2[0.4\#1]{}
[For two particles, position spread a) versus relative separation spread b). For 3 particles in $d > 1$, c) and d) are position spread versus a choice to represent relative spread in Jacobi coordinates \[$\theta$ = arccos($\R_1\cdot\R_2/|\R_1||\R_2|$)\].]{}
BS’s specific example is akin to my Example 1, except that they have a constant $U$ and set $\un = 0$ while I have no $U$ but a constant $\un$. Converting, the wavefunction may then be rewritten as = e\^[ i ( r\_[12]{} ( - ) + r\_[23]{} ( + ) )/]{} for $n_a = 1/m_a$, $\nu_{ab} = 1/\mu_{ab}$, $\chi = n_3^2\mu_{12} + n_1^2\mu_{23} + 2n_1n_2n_3$ and where the mass-independent constants present are parametrized by an angle $\theta$. This has much nicer 1,3 symmetry than 1,2 symmetry, so in setting two of the masses to be equal to match BS’s example, I opt for $m_1, m_3 = m << M = m_2$ Then the wavefunction goes as \~ e\^[iJr\_[12]{}/]{}e\^[iKr\_[23]{}/]{} (for $J$, $K$ mass-independent constants). Thus, indeed as BS claim, the small masses dominate all the uncertainties. But by my above interpretation, these uncertainties are actually in the separations between a big mass and a small mass, so this situation conforms to standard quantum intuitions rather than constituting an impasse. The [*truly*]{} relevant test to establish whether there is a semiclassical limit problem is rather to check what happens if $m_1, m_3 = M >> m = m_2$, for then there is a big mass–big mass relative separation, and it is [*this*]{} which one would not expect to be influenced much by a small mass somewhere else. And indeed, upon performing the new approximation, and isolating the big mass–big mass separation $r_{13}$ as the variable whose spread is of relevance, I find that \~ e\^[ i |[J]{} r\_[13]{}/]{}e\^[-i|[K]{}r\_[12]{}/]{} (for $\bar{J}$, $\bar{K}$ mass-independent constants) so that indeed only the big masses contribute significantly to the spread in the big mass-big mass separation. This basic conclusion is unaffected by having the two identical masses replaced by merely similar masses, and holds widely throughout the models presented in this paper when suitable pairs of quantities are set to be relatively large and small.
[**Wavepackets**]{}
Consider the 1-$d$ problems which separate out of the relational problems of this section, piecemeal and as formal pieces of mathematics in which an eigenspectrum and wavefunctions are obtained from a differential equation and boundary conditions that depend on some abstract set of variables and parameters. Then the wavepackets built up by summing and/or integrating the wavefunctions (generally with some weighting) over the eigenspectrum may also be considered as formal pieces of mathematics. My first point is that, at the level of the separated-out pieces of the relational problems considered in this section, this formal mathematics is the same as in the usual absolutist quantization. Thus the piecemeal construction of wavepackets for the 1-d quantum problems does not care whether these arise from separation in relational problems or in absolutist ones, so both behave equally well. The best-known examples of these wavepackets are the fixed-size one for the free particle and the pulsating one for the harmonic oscillator (see e.g. [@Schiff; @Robinett]).
A first difference between wavepackets in the absolutist and relational QM schemes arises in their interpretation. This parallels the above situation with the spreads. A second difference is that there are limitations building composite wavepackets in the relational case. Unlike in the absolutist case, composition of subsystem wavepackets cannot be extended to include the whole system. This is due to the energy of the universe being a fixed quantity.
Also note that the application of polar coordinates for $d > 1$ brings out that the standard QM interpretation is close to being relational. This is most familiar in the study of the hydrogen atom, for which a simple standard approach is to treat the proton as fixed and then consider the spread of the radial separation $\rho$ between the proton (or more accurately the atom’s barycentre) and the electron. All that is missing as regards obtaining a fully relational perspective is to consider the position of the barycentre not only to be uninteresting but also to be meaningless. Then one considers the spread in $|\R_1| (= |\r_{12}| \equiv \rho_1)$. The ready availability of this familiar picture is one reason why it is unfortunate that BS restricted their study to 1-$d$ examples. I should add that interpreting $d > 1$ RPM’s furthermore requires an additional notion of [*spread in relative angle*]{} \[Fig 2 d)\].
**Interesting features of the RPM examples as closed universe systems**
-----------------------------------------------------------------------
[**Subsystem energy interlocking and truncation, gaps in the energy spectrum**]{}
1\) As mentioned above, there is energy interlocking between constituent subsystems. E.g. this requires the above naïvely free problem to have a line segment rather than a quadrant as its overall eigenspectrum, and the single HO and free particle system to have a set of points rather than an infinity of lines as its overall eigenspectrum, and the coupled HO’s have a small set of points rather than a regular lattice as its eigenspectrum (Fig 3).
\#1\#2[0.4\#1]{}
[Effect on spectra of $\un$ taking a forever fixed value.]{}
By energy interlocking the small universe models built up from individual problems’ wavepackets additionally contain a delta function (\_ \_ - ) acting inside the sums and integrals required to build it up, which causes it to differ mathematically from e.g the direct product of subsystem wavepackets in a fully separable universe.
Moreover, unlike in the usual interpretation of few-particle QM, the energy here is the energy of the universe, which is not only fixed, but is also a separate attribute of the universe so that the fixed value it takes need bear no relation to the eigenspectra of the universe’s contents. This leads to the following effects.
2\) In my multiple HO example, all the energies \[$\se_1(\nn_1)$ and $\se_2(\nn_2)$\] are positive. Then $\un < \se_1(0) + \se_2$ so no wavefunctions exist. Universes failing to meet the zero point energy of their content fail to have a wavefunction.
3\) Such universes could rather fail to meet the energy required for just some of the states which lie above a given energy that is greater than the zero point energy. then one would obtain not the conventional eigenspectrum but rather a [*truncation*]{} of it.
4\) In my 2 or 3 HO examples, one is required to solve $k_1\nn_1 + k_2\nn_2 = q$ for $\nn_i \in |N_0$, $k_i = \sqrt{ \frac{H_i}{\mu}}\hbar $ and $q = \un - \frac{\bar{h}}{2}\frac{\sqrt{H_1} + \sqrt{H_2}}{\sqrt{\mu}}$. Then if $k_i$ and $q$ mismatch through some being rational and some irrational, or even if all are integers but the highest common factor of $k_1$ and $k_2$ does not divide $q$, no solutions exist. This can also be set up to give [*gaps*]{} in what would otherwise look like a truncation of the conventional eigenspectrum. Similar effects can be achieved by more complicated matches and mismatches in the other examples’ $\se_i$ dependences on $\nn_i$.
[**How energy interlocking does not affect the ‘recovery of reality’ in practice for large universes** ]{}
Objection BS2 is due to a less developed version of the above material. However, I develop the following conditional counterstatements.
5\) In universes which furthermore contain free particles, because these have continuous spectra, the missing out of some conventional states by 4) cannot occur. This is the case e.g for the 1 HO example above, while ‘tensoring’ free particles with the 2 or 3 HO setting above alleviates this ‘missing state’ problem in the setting of a universe with a slightly larger particle number.
6\) If negative energies are possible, then subsystems can attain energies higher than $\un$. So if one tensors a negative potential example such as hydrogen with an independent HO pair, one can have less truncation of the HO. One will still have many, or all, states missing, depending on how the coefficients of the two independent subsystems’ potentials are related. But if one then tensors in free particles one can have everything up to the truncation. Thus truncation can be displaced at least for some universes, by a modest increase in constituent particle number. It should be noted that there is a mismatch between e.g. the HO which has excited states unbounded from above in conventional QM and hydrogen which has negative energy states bounded from below. I get out of this difficulty by pointing out that in any case very high positive energy states are unlikely to be physically meaningful; at the very least the physical validity of the model would break down due to e.g pair production and ultimately the breakdown of spacetime. To ensure one’s model can attain high enough (but finite) positive energies, wells that are deep and/or numerous enough can be brought in.
7\) Unlike 2)–4), energy interlocking [*does not*]{} go away with particle number increase or the accommodation of a variety of potentials within one’s universe model. However, the large particle number and high quantum number aspects of semiclassicality are relevant here. Subsystems remain well-behaved and all experimental studies in practice involve subsystems. But subsystems may be taken to have conventional wavepackets insofar as these [*are*]{} products of their constituent separated-out problems’ wavepackets. It is still true that these may be truncated rather than built out of arbitrarily many eigenfunctions along the lines of 1) to 3) above, but this will be alleviable in practice by ensuring that sufficient additional free particles or particles whose mutual potentials are of an opposing sign to the original subsystem’s. In such a framework, the correlation of a quantity within a subsystem with another in the rest of the universe is overwhelmingly likely to evade detection provided that the rest of the universe contains plenty of other particles. And of course the real universe is indeed well-populated with particles.
Thus, BS’s second bad semiclassicality objection should be replaced by: 1–4) lead to simple small relational particle universes constituting good toy models for gaining a deeper closed universes and of the Problem of Time, while 5–7) ensure that thinking of the (diverse, large particle number) real universe in terms of relational particles is [*not*]{} in practice compromised by any manifest bad semiclassicality. I should also caution that solving TISE’s is not as complete as setting up the full machinery of QM so more detailed contentions may emerge in further work.
Whether the WKB ansatz may additionally be applied in to whole universes is an issue relevant to the semiclassical approach to quantum cosmology in the general (rather than just RPM) context. I touch on this issue in Sec 8.3, but mostly leave its discussion to [**II**]{}.8 and [@Soland]. For the moment I show that increasing the dimension away from 1 reveals further such ‘small closed universe’ effects.
[**Angular momentum counterbalancing**]{}
2-$d$ RPM solutions depend on 2 and not 4 quantum numbers via the AM constraint as follows: $L_i\psi_i = \nm_i\psi_i$ $i = 1, 2$ arise in each separated problem, but these are ‘joined together’ by $0 = \hat{L}\Psi = \hat{L}_1 \otimes \hat{1}_2 \Psi + \hat{1}_1 \otimes \hat{L}_2\Psi
= \nm_1\Psi + \nm_2\Psi $, so $\nm_1 = - \nm_2 \equiv m$. 3-$d$ RPM solutions such as Example 3 depend on 3 and not 6 quantum numbers via a [*more elaborate*]{} angular momentum counterbalancing as follows: $\nm_1 = - \nm_2 \equiv \nm$ as for the 2-d case using $0 = \hat{L}_z\Psi$ in place of $0 = \hat{L}\Psi$. But now, also, $0 = \hat{L}_x\Psi$ and $0 = \hat{L}_y\Psi$, so $\hat{L}_{x1}\Psi = - \hat{L}_{x2}\Psi$ and $\hat{L}_{x1}\Psi = - \hat{L}_{x2}\Psi$. Thus, using $\hat{L}_1^2\psi_i = \nl_i(\nl_i+1)\psi_i
\mbox{ } \Rightarrow \mbox{ }
(\hat{L}_{xi}^2 + \hat{L}_{yi}^2)\psi_i = (\nl_i(\nl_i + 1) - \nm_i^2)\psi_i$ for each of $i = 1, \mbox{ } 2$, so $\nl_1(\nl_1 + 1) - \nm_1^2 = \nl_2(\nl_2 + 1) -\nm_2^2$. But $\nm_1 = -\nm_2 \equiv \nm$ and $\nl_1, \nl_2 \in {\cal N}_0$ so there is no choice but $\nl_1 = \nl_2 \equiv \nl$. Thus whole universe models have a variety of additional naïvely unexpected correlations between their ordinary-looking constituent subsystems due to limited energy resources and due to having to balance out each others’ angular momentum. A similar argument to that presented for energy interlocking leads to one expecting angular momentum counterbalancing not to be noticed in the study of subsystems within a large universe.
[**Further types of question about the ‘recovery of reality’**]{}
Conceptualizing in terms of an atom and some other subsystem, lends itself to raising further types of questions. It is all very well that these subsystems can be constructed to have the right sort of spectra in the sense of energy levels (mathematical sets of eigenvalues). But are [*transitions*]{} between these, most obviously manifested by spectra (in the distinct, practical sense of emissive/absorptive frequency patterns) or chemical reactions, possible? After all, the universes in question are governed by TISE’s which possess solely stationary solutions? One answer, developed in [**II**]{}.7 and [@Soland], is that this set-up [*is*]{} nevertheless capable of allowing [*subsystems*]{} to take on [*the appearance*]{} of dynamics. This is a [*semiclassical approach*]{} which relies on the WKB ansatz and on (perhaps small) terms that cause nonseparability. One situation which has nonseparability is the multi-Coulomb potential (in some cases through multiple charged particles, while all models will have small gravitational interactions). Consistent records schemes such as Barbour’s concern a different answer (see [**II**]{}.8. for further comparison of these). N.B. that, while this discussion of apparent dynamics from TISE’s may appear unusual at first sight, it [*is*]{} a simplified discussion about the plausible situation that our own universe, which certainly possesses subsystems which appear to be dynamical, may (at some level) be describable [*overall*]{} by a GR TISE: the so-called Wheeler–DeWitt equation (See [**II**]{} for references).
It is subsequently relevant to ask about [*mechanism*]{}. Insisting on interpreting such models as closed (i.e self-contained) amounts to [*not*]{} assuming a ‘surrounding photon sea’. Does the absense of this preclude all dynamical processes? The answer is no, insofar that further particles of the same species can also mediate energy at some effective level (this holds as far as Fermi theory, overcoming the non-renormalizability of which leads to quantum electrodynamics, in which photons [*are*]{} then necessary for interactions). For atoms, it is well known that a grazing free particle can result in electron excitation, in ionization or in the (4-particle) Auger effect [@Mott]. If one must furthermore deploy a counterbalancing particle/subsystem in order to recover standard results, at least 4 or 5 body problems are required to model such situations.
A separate point is that, while photon-free transitions as above are known, many [*more*]{} effects require interaction by photons or taking into account the atomic and fields. This is one reason why it would be interesting to extend the present work so as to ‘build in’ electromagnetism by considering closed relational Newton–Maxwell universes. This would also serve both as a robustness check for the results of paper [**II**]{} and as a toy model for closed Einstein–Maxwell universes. Inclusion of intrinsic spin would also be useful.
Conclusion
==========
As regards the classical absolute versus relative motion debate, I have provided a concrete synthesis of directly-formulated relational particle mechanics (RPM) which is equivalent to a reformulation of the portion of Newtonian mechanics that is conservative, has $\L = 0$ and whose kinetic term is homogeneous quadratic in its velocities. I have presented this in relative Lagrange variables and in relative Jacobi variables. I have furthermore presented improved (counterbalancing subsystem) arguments that adopting this portion or the straightforward nonhomogeneous enlargement of it is not for many purposes a major restriction at the classical level.
As regards QM implications of adopting the relational stance, I have been able to go further than in previous relational studies by bringing in Jacobi variables. Thus I have been able to work with more complicated, more realistic and more relationally motivated quantum RPM models than Barbour and Smolin (BS) or any other paper in the RPM literature to date. This is based on how many simple problems are separable in Jacobi coordinates and share much formal mathematical structure with the conventional absolutist approach. Thus at the very simplest level, the answer to my question about whether absolutism has misled QM is [*no*]{}, in particular if one continues to adhere with conventional QM’s emphasis (see e.g. [@DiracQM]) on formal structures and their manipulation. However, differences in the [*interpretation*]{} of that mathematics are readily manifest due to the one employing particle positions and the other employing interparticle (cluster) separations. Indeed the bona fide relational interpretation that I provide enables me to reject BS’s spread sensitivity objection to RPM models having a good semiclassical limit.
Moreover the answer is [*yes as regards whole closed universes*]{}. The models of this paper exhibit several interesting features, which suffice to show that insisting on modelling a closed universe as a whole courts difficulties well beyond those encountered in studying its constituent subsystems. (Of course, there is more to modelling quantum cosmology than just this.) These features are energy interlocking, AM counterbalancing, eigenspectrum gaps due to the universe and its contents being mismatched, and eigenspectrum truncation due to finite resources. While these mathematical observations are an extension of BS’s second objection to RPM models having a good semiclassical limit, moreover these effects do not affect the ‘recovery of reality’ for a large universe with sufficiently varied contents. For, the presence of free particles and of potentials of both signs serve to overcome gaps and noticeable truncation, while the practicalities of experimentation involving only small subsystems means that interlocking and counterbalancing would be likely to go unnoticed.
[**Acknowledgments**]{}
I thank my Father, who taught and encouraged me prior to his death in 1995. I thank also my other early teachers, lecturers, supervisors and professors and the Royal Society of Chemistry for illusioning me. I thank Professor Malcolm MacCallum, Dr Julian Barbour and Professor Don Page for more recent guidance toward this project. I thank Professor Claus Kiefer for sending me the BS preprint while Julian was ill. I thank Professor Bruno Bertotti, Dr Martin Bojowald, Dr Harvey Brown, Mr. Brendan Foster, Dr Laszlo Gergely, Professor Gary Gibbons, Ms. Isabelle Herbauts, Dr Bryan Kelleher, Dr Adrian Kent, Professor Jacek Klinowski, Professor Niall Ó Murchadha, Dr Jonathan Oppenheim, Dr Oliver Pooley, Professor Reza Tavakol, Professor Lee Smolin, Dr Vardarajan Suneeta and Dr Eric Woolgar for discussions. I thank the Barbour family, Peterhouse and DAMTP for hospitality during which some of this work was done. I acknowledge funding at various stages from Peterhouse and the Killam Foundation.
[99]{}
I. Newton, [*Philosophiae Naturalis Principia Mathematica*]{} (1686). For an English translation, see e.g I.B. Cohen and A. Whitman (University of California Press, Berkeley, 1999). In particular, see the Scholium on absolute motion therein.
See *The Leibnitz–Clark Correspondence, ed. H.G. Alexander (Manchester 1956).*
Bishop G. Berkeley, *The Principles of Human Knowledge (1710); *Concerning Motion (De Motu) (1721).**
E. Mach, [*Die Mechanik in ihrer Entwickelung, Historisch-kritisch dargestellt*]{} (J.A. Barth, Leipzig, 1883). The English translation is [*The Science of Mechanics: A Critical and Historical Account of its Development*]{} (Open Court, La Salle, Ill. 1960).
J.-L. Lagrange, [*Le probleme des trois corps, Oeuvres*]{}, Vol 6, 229 (1772).
C.G.J. Jacobi, [*Sur l’elimination des noeuds dans le probleme des trois corps*]{}, Math Werke Vol. 1 30 (1843).
Relevant aspects of Lagrange and Jacobi’s work are also discussed in e.g A. Chenciner, Ravello Summer School Notes (unpublished, from 1997); J. Barbour, material for unfinished volume II of [*Discovery of Dynamics*]{}.
O. Dziobek, [*Die Mathematischen theorien der planeten-bewegungen* ]{} (Barth, Leipzig 1888), available in English as [*Mathematical Theories of Planetary Motions*]{} (1892), now available as (Dover, New York 1962).
H. Poincaré, [*Science et Hypotheses*]{} (1902), English translation (Science Press, New York 1905).
J.B. Barbour and B. Bertotti, Proc. Roy. Soc. Lond. **A382 295 (1982).**
J.B. Barbour and L. Smolin, unpublished preprint dating from 1989.
C. Rovelli, p 292 in [*Conceptual Problems of Quantum Gravity*]{} ed. A. Ashtekar and J. Stachel (Birkhäuser, Boston, 1991).
L. Smolin, in [*Conceptual Problems of Quantum Gravity*]{} ed. A. Ashtekar and J. Stachel (Birkhäuser, Boston, 1991). J.B. Barbour, Class. Quantum Grav. **11 2853 (1994).**
J.B. Barbour, Class. Quantum Grav. **11 2875 (1994).**
J.B. Barbour, [*GR as a perfectly Machian theory*]{}, in [*Mach’s principle: From Newton’s Bucket to Quantum Gravity*]{} ed. J.B. Barbour and H. Pfister (Birkhäuser, Boston 1995).
D. Lynden-Bell, in *Mach’s principle: From Newton’s Bucket to Quantum Gravity, ed. J.B. Barbour and H. Pfister (Birkhäuser, Boston 1995).*
J.B. Barbour, [*The End of Time*]{} (Oxford University Press, New York 1999).
L.Á Gergely, Class. Quantum Grav. [**17**]{} 1949 (2000), gr-qc/0003064.
L.Á Gergely and M. McKain, Class. Quantum Grav. [**17**]{} 1963 (2000), gr-qc/0003065.
J.B. Barbour, B.Z. Foster and N. Ó Murchadha, Class. Quantum Grav. **19 3217 (2002), gr-qc/0012089.**
K.V. Kuchař, in *Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, ed. G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992).*
C. Kiefer, [*Quantum Gravity*]{} (Clarendon, Oxford 2004).
E. Anderson, in [*General Relativity Research Trends, Horizons in World Physics*]{} [**249**]{} Ed. A. Reimer (Nova, New York 2005), gr-qc/0405022.
J.B. Barbour, in [*Quantum concepts in space and time*]{} ed. R. Penrose and C.J. Isham (Oxford University Press, Oxford, 1986). J. Earman, *World Enough and Space-Time: Absolute versus Relational Theories of Space and Time (MIT Press, Cambridge MA, 1989). L. Smolin, in [*Time and the Instant*]{} ed. R. Durie (Clinamen Press, Manchester 2000), gr-qc/0104097. J.N. Butterfield, Brit. J. Phil. Sci. [**53** ]{} 289 (2002), gr-qc/0103055; J.B. Barbour, in [*Decoherence and Entropy in Complex Systems (Proceedings of the Conference DICE, Piombino 2002*]{} ed. H.-T. Elze (Springer Lecture Notes in Physics 2003), gr-qc/0309089.*
These results were announced in outline in E. Anderson AIP Conf. Proc. [**861**]{} 285 (2006), gr-qc/0509054.
O. Pooley and H.R. Brown, Brit. J. Phil. Sci. **53 183 (2002); O. Pooley, Chronos (Proceedings of the Philosophy of Time Society 2003-4).**
E. Anderson Class. Quant. Grav. [**23**]{} 2491 (2006), gr-qc/0511069. A. Einstein, Sitz. Preuss. Ak. Wiss. 142 (1917); [*The Meaning of Relativity*]{} (Princeton University Press, Princeton 1955).
J.A. Wheeler, in *[Battelle rencontres: 1967 lectures in mathematics and physics]{} ed. C. DeWitt and J.A. Wheeler (Benjamin, New York 1968).*
K.V. Kuchař, in [*Quantum Gravity 2: A Second Oxford Symposium*]{} ed. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, Oxford 1981).
C.J. Isham, in [*Integrable systems, quantum groups and quantum field theories*]{} ed. L.A. Ibort and M.A. Rodríguez (Kluwer, Dordrecht 1993), gr-qc/9210011.
K.V. Kuchař, in [*Conceptual Problems of Quantum Gravity*]{} ed. A. Ashtekar and J. Stachel (Birkhäuser, Boston, 1991); K.V. Kuchař, in *The Arguments of Time ed. J. Butterfield (Oxford University Press, Oxford 1999).*
C.J. Isham, in Lect. Notes Phys. [**434**]{} (1994) ,gr-qc/9310031; writeup of GR14 plenary lecture, gr-qc/9510063; C. Rovelli, Living Rev. Rel. [**1**]{} 1 (1998), gr-qc/9710008; J. Butterfield and C.J Isham, in [*Physics meets Philosophy at the Planck Scale*]{}, ed. C. Callender and N. Huggett, Cambridge University Press (2000), gr-qc/9903072; S. Carlip, Rept. Prog. Phys. [**64**]{} 885 (2001), gr-qc/0108040; T. Thiemann, Lect. Notes Phys. [**631**]{} 41 (2003), gr-qc/0210094; L. Smolin, hep-th/0303185; hep-th/0408048; A. Ashtekar and J. Lewandowski Class. Quant. Grav. [**21**]{} R53 (2004), gr-qc/0404018; O. Dreyer, hep-th/0409048.
P.A.M. Dirac, *Lectures on Quantum Mechanics (Yeshiva University, NY, 1964).*
R.F. Baierlain, D. Sharp and J.A. Wheeler, Phys. Rev. **126 1864 (1962).**
E. Anderson, Phys. Rev. [**D68**]{} 104001 (2003), gr-qc/0302035.
E. Anderson, “Geometrodynamics: spacetime or space?" (Ph.D. Thesis, University of London 2004), gr-qc/0409123; E. Anderson, gr-qc/0511070.
J.B. Barbour, Class. Quantum Grav. **20**, 1543 (2003), gr-qc/0211021.
E. Anderson, J.B. Barbour, B.Z. Foster, B. Kelleher and N Ó Murchadha, Class. Quantum Grav. [**22**]{} 1795 (2005), gr-qc/0407104.
See e.g. R.G. Littlejohn and M. Reinsch, Rev. Mod. Phys. [**69**]{} 213 (1997).
C. Lanczos, *The Variational Principles of Mechanics (University of Toronto Press, Toronto 1949; Dover, New York 1986).*
J. Stewart, [*Advanced General Relativity*]{} (Cambridge University Press, Cambridge, 1991).
J.B. Barbour and B. Bertotti, Nuovo Cim. **B38 1 (1977).**
V.W. Hughes, H.G. Robinson and V. Beltran–Lopez, Phys. Rev. Lett. [**4**]{} 342 (1960); R.W.P. Drever, Phil. Mag. [**6**]{} 683 (1961).
See e.g. C. Marchal, [*Celestial Mechanics*]{} (Elsevier, Tokyo 1990).
This was conjectured by P. Painlevé, [*Leçons sur la théorie analytique des équations différentielles*]{} (Hermann, Paris 1897); one example which has now been built is Z. Xia, Ann. Math [**135**]{} 411 (1992); I mention this one in particular as it is for $\L = 0$; for other examples and a simple introduction see F.N. Diacu, [*Singularities of the $N$-body problem*]{} (Les Publications CRM, Montréal, 1992).
See e.g. R. Moeckel, Contemp. Math. [**81**]{} 1 (1988) and references therein.
E. Anderson, v1 of the preprint gr-qc/0511068.
P.M. Morse and H. Feshbach, [*Methods of Theoretical Physics Pert I*]{} (McGraw–Hill, New York 1953). R.W. Robinett, [*Quantum mechanics: classical results, modern systems, and visualized examples*]{} (Oxford University Press, New York 1997).
L.D. Landau and E.M. Lifschitz, [*Quantum Mechanics. Non-relativistic theory*]{} (English translation: Pergamon, London, 1958).
L.I. Schiff, [*Quantum mechanics*]{} (McGraw-Hill, New York 1968).
E. Anderson, [**24**]{} 2935 (2007), gr-qc/0611007; E. Anderson, [**24**]{} 2971 (2007), gr-qc/0611008; E. Anderson 2 further forthcoming papers.
J. Schwinger [*Quantum Mechanics*]{} ed. B.-G. Englert (Springer, Berlin 2001).
N.F. Mott, [*Elements of wave mechanics*]{} (Cambridge University Press, Cambridge 1952); N.F. Mott and H.S.W. Massey, [*The theory of atomic collisions*]{} (Clarendon Press, Oxford 1949).
See e.g. P.A.M. Dirac, [*Principles of Quantum Mechanics*]{} 4th edition (Clarendon Press, Oxford 1983).
|
---
abstract: 'Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gröbner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gröbner basis of a one-level quasi-cyclic product code is derived.'
author:
- '[^1]'
-
bibliography:
- 'qccsccextract.bib'
title: 'Construction of Quasi-Cyclic Product Codes'
---
Cyclic code, Gröbner basis, module minimization, product code, quasi-cyclic code, submodule
Introduction
============
A linear block code of length $\QCCl \QCCm $ over a finite field $\Fq$ is a quasi-cyclic code if every cyclic shift of a codeword by $\QCCl$ positions, for some integer $\QCCl$ between one and $\QCCl \QCCm$, results in another codeword. Quasi-cyclic codes are a natural generalization of cyclic codes (where $\QCCl = 1$), and have a closely linked algebraic structure. In contrast to cyclic codes, quasi-cyclic codes are known to be asymptotically good (see Chen–Peterson–Weldon [@chen_results_1969]). Several such codes have been discovered with the highest minimum distance for a given length and dimension (see Gulliver–Bhargava [@gulliver_best_1991] as well as Chen’s and Grassl’s databases [@chen_database_2014; @Grassl_Codetables]). Several good LDPC codes are quasi-cyclic (see e.g. [@butler_bounds_2013]) and the connection to convolutional codes was investigated among others in [@solomon_connection_1979; @esmaeili_link_1998; @lally_algebraic_2006].
Recent papers of Barbier *et al.* [@barbier_quasi-cyclic_2012; @barbier_decoding_2013], Lally–Fitzpatrick [@lally_algebraic_2001; @lally_quasicyclic_2003; @lally_algebraic_2006], Ling–Solé [@ling_algebraic_2001; @ling_algebraic_2003; @ling_algebraic_2005], Semenov–Trifonov [@semenov_spectral_2012], Güneri–Özbudak [@guneri_bound_2012] and ours [@zeh_decoding_2014] discuss different aspects of the algebraic structure of quasi-cyclic codes including lower bounds on the minimum Hamming distance and efficient decoding algorithms.
The focus of this paper is on a simple method to combine two given quasi-cyclic codes into a product code. More specifically, we give a description of a quasi-cyclic product code when one component code is quasi-cyclic and the second one is cyclic.
The work of Wasan [@wasan_quasi_1977] first considers quasi-cyclic product codes while investigating the mathematical properties of the wider class of quasi-abelian codes. Some more results were published in a short note by Wasan and Dass [@dass_note_1983]. Koshy proposed a so-called “circle” quasi-cyclic product codes in [@koshy_quasi-cyclic_1972].
Our work considers quasi-cyclic product codes that generalize the results of Burton–Weldon [@burton_cyclic_1965] and Lin–Weldon [@lin_further_1970] (see also [@macwilliams_theory_1988 Chapter 18]) based on the reduced Gröbner basis representation of Lally–Fitzpatrick [@lally_algebraic_2001] of the quasi-cyclic component code. We derive a representation of the generating set of a quasi-cyclic product code, where one component code is quasi-cyclic and the other is cyclic (in Thm. \[theo\_QCCTimesCYC\]) and we give a reduced Gröbner basis for the special class of one-level quasi-cyclic product codes (in Thm. \[theo\_OneLevelQC\]).
The paper is structured as follows. In Section \[sec\_Preliminaries\], we give necessary preliminaries on quasi-cyclic codes over finite fields. We outline relevant basics of the reduced Gröbner basis representation of Lally–Fitzpatrick [@lally_algebraic_2001]. Furthermore, the special class of $r$-level quasi-cyclic codes is defined in this section. Section \[sec\_ProductQCCQCC\] contains the main result on quasi-cyclic product codes, where the row-code is quasi-cyclic and the column-code is cyclic. Moreover, an explicit expression of the basis of a 1-level quasi-cyclic product code is derived in Section \[sec\_ProductQCCQCC\]. For illustration, we explicitly give an example of a binary $2$-quasi-cyclic product code in Section \[sec\_Example\]. Section \[sec\_Conclusion\] concludes this paper.
Preliminaries {#sec_Preliminaries}
=============
Let $\Fq$ denote the finite field of order $q$ and $\Fqx$ the polynomial ring over $\Fq$ with indeterminate $X$. Let $a, b$ with $b > a$ be two positive integers and denote by $\interval{a,b}$ the set of integers $\{a,a+1,\dots,b-1\}$ and by $\interval{b}=\interval{0,b}$. A vector of length $n$ is denoted by a lowercase bold letter as $\vec{v} = (v_0 \ v_1 \ \cdots \ v_{n-1})$ and an $m \times n$ matrix is denoted by a capital bold letter as $\M{M}=(m_{i,j})_{i \in \interval{m}}^{j \in \interval{n}}$.
A linear code $\QCC$ of length $\QCCl \QCCm$, dimension $\QCCk$ and minimum Hamming distance $\QCCd$ over $\Fq$ is $\QCCcyc$-quasi-cyclic if every cyclic shift by $\QCCcyc$ of a codeword is again a codeword of $\QCC$, more explicitly if:\
=.4pt
--------------------------------------------------- ---------------------------------- ----- --------------------------------------
$ (c_{0,0} \cdots c_{\QCCcyc-1,0}$ $c_{0,1} \cdots c_{\QCCcyc-1,1}$ ... $c_{\QCCcyc-1,\QCClen-1}) \in \QCC $
$ \Rightarrow $
$(c_{0,\QCClen-1} \cdots c_{\QCCcyc-1,\QCClen-1}$ $c_{0,0} \cdots c_{\QCCcyc-1,0}$ ... $c_{\QCCcyc-1,\QCClen-2}) \in \QCC$.
--------------------------------------------------- ---------------------------------- ----- --------------------------------------
\
We can represent a codeword of an $\QCCcyc$-quasi-cyclic code as $\mathbf{c}(X) = (c_0(X) \ c_1(X) \ \cdots \ c_{\QCCcyc-1}(X)) \in \Fqx^{\ell} $, where $$\label{eq_UnivariatePolyCodeword}
c_i(X) \defeq \sum_{j=0}^{\QCClen-1} c_{i,j} X^{j}, \quad \forall i \in \interval{\QCCcyc}.$$ Then, the defining property of $\QCC$ is that each component $c_i(X)$ of $\mathbf{c}(X)$ is closed under multiplication by $X$ and reduction modulo $X^{\QCClen}-1$.
\[lem\_VectorToUnivariatePoly\] Let $(c_0(X) \ c_1(X) \ \cdots \ c_{\QCCcyc-1}(X))$ be a codeword of an $\QCCcyc$-quasi-cyclic code $\QCC$ of length $\QCClen \QCCcyc$, where the components are defined as in . Then a codeword in $\QCC$ represented as one univariate polynomial of degree smaller than $\QCClen \QCCcyc$ is $$\label{eq_VectorToUnivariatePoly}
c(X) = \sum_{i=0}^{\QCCcyc-1} c_i(X^{\QCCcyc})X^i.$$
Substitute into : $$\begin{aligned}
c(X) & = \sum_{i=0}^{\QCCcyc-1} c_i(X^{\QCCcyc})X^i = \sum_{i=0}^{\QCCcyc-1} \sum_{j=0}^{\QCClen-1} c_{i,j} X^{j\QCCcyc+i}.\end{aligned}$$
Lally and Fitzpatrick [@lally_construction_1999; @lally_algebraic_2001] showed that this enables us to see a quasi-cyclic code as an $R$-submodule of the algebra $R^{\QCCcyc}$, where $R = \Fqx/\langle X^{\QCClen}-1 \rangle$. The code $\QCC$ is the image of an $\Fqx$-submodule $\tilde{\QCC}$ of $\Fqx^{\QCCl}$ containing $\basis = \langle (X^{\QCCm}-1)\mathbf{e}_j, j \in \interval{\QCCl} \rangle$ (where $\mathbf{e}_j$ is the standard basis vector with one in position $j$ and zero elsewhere) under the natural homomorphism $$\begin{aligned}
\phi: \; \Fqx^{\QCCl} & \rightarrow R^{\QCCcyc} \\
(c_0(X) \ \cdots \ c_{\QCCcyc - 1}(X)) & \mapsto (c_0(X) + \langle X^{\QCClen} \minus 1 \rangle \ \cdots \\
& \qquad \qquad c_{\QCCcyc \minus 1}(X) +\langle X^{\QCClen} \minus 1 \rangle ).\end{aligned}$$ It has a generating set of the form $\{ \mathbf{a}_i, i \in \interval{z}, (X^{\QCCm}-1)\mathbf{e}_j, j \in \interval{\QCCl} \}$, where $\mathbf{a}_i \in \Fqx^{\QCCl}$ and $z \leq \QCCl$ (see e.g. [@cox_using_1998 Chapter 5] for further information). Therefore, its generating set can be represented as a matrix with entries in $\Fqx$: $$\label{eq_GeneratorWithBasis}
\arraycolsep=1pt
\mathbf{M}(X) =
\begin{pmatrix}
a_{0,0}(X) & a_{0,1}(X) & \cdots & a_{0,\QCCl-1}(X) \\
a_{1,0}(X) & a_{1,1}(X) & \cdots & a_{1,\QCCl-1}(X) \\
\vdots & \vdots & \ddots & \vdots \\
a_{z-1,0}(X) & a_{z-1,1}(X) & \cdots & a_{z-1,\QCCl-1}(X) \\
X^{\QCCm}-1 & & \\
& X^{\QCCm}-1 & \multicolumn{2}{c}{\bigzero} \\
\multicolumn{2}{c}{\bigzero} & \ddots \\
& & & X^{\QCCm}-1
\end{pmatrix}.$$ Every matrix $\mathbf{M}(X)$ as in of the preimage $\tilde{\QCC}$ can be transformed into a reduced Gröbner basis (RGB) with respect to the position-over-term order (POT) in $\Fqx^{\QCCcyc}$ (see [@lally_construction_1999; @lally_algebraic_2001]). This basis can be represented in the form of an upper-triangular $\ell \times \ell$ matrix with entries in $\Fqx$ as follows: $$\label{def_GroebBasisMatrix}
\mathbf{G}(X) =
\begin{pmatrix}
g_{0,0}(X) & g_{0,1}(X) & \cdots & g_{0,\QCCcyc-1}(X) \\
& g_{1,1}(X) & \cdots & g_{1,\QCCcyc-1}(X) \\
\multicolumn{2}{c}{\bigzero}& \ddots & \vdots \\
& & & g_{\QCCcyc-1,\QCCcyc-1}(X)
\end{pmatrix},$$ where the following conditions must be fulfilled:\
---- ------------------- ------------------------------------- ---------------------------------------------
1) $g_{i,j}(X)$ $= 0,$ $\forall 0 \leq j < i < \QCCcyc$,
2) $\deg g_{j,i}(X)$ $ < \deg g_{i,i}(X),$ $ \forall j < i, i \in \interval{\QCCcyc}$,
3) $g_{i,i}(X)$ $| \hspace{.2cm} (X^{\QCClen}-1),$ $\forall i \in \interval{\QCCcyc}$,
4) if $g_{i,i}(X)$ $=X^{\QCClen}-1$ then
$g_{i,j}(X)$ $=0,$ $ \forall j \in \interval{i+1,\QCCcyc}$.
---- ------------------- ------------------------------------- ---------------------------------------------
\
The rows of $\mathbf{G}(X)$ with $g_{i,i}(X) \neq X^{\QCClen}-1$ (i.e., the rows that do not map to zero under $\phi$) are called the reduced generating set of the quasi-cyclic code $\QCC$. A codeword of $\QCC$ can be represented as $\mathbf{c}(X) = \mathbf{i}(X) \mathbf{G}(X)$ and it follows that $\QCCk = \QCClen \QCCcyc - \sum_{i=0}^{\QCCcyc-1} \deg g_{i,i}(X)$. Let us recall the following definition (see also [@lally_construction_1999 Thm. 3.2]).
\[def\_LevelQC\] We call an $\QCCl$-quasi-cyclic code $\QCC$ of length $\QCCl \QCCm$ an $\level$-level quasi-cyclic code if there is an index $\level \in \interval{\QCCl}$ for which the RGB/POT matrix as defined in is such that $g_{\level-1,\level-1}(X) \neq X^{\QCCm}-1$ and $g_{\level,\level}(X) = \dots = g_{\QCCl-1,\QCCl-1}(X) = X^{\QCCm}-1$.
We recall [@lally_construction_1999 Corollary 3.3] for the case of a $1$-level quasi-cyclic code in the following.
\[cor\_OneLevelQC\] The generator matrix in RGB/POT form of a $1$-level $\QCCl$-quasi-cyclic code $\QCC$ of length $\QCCl \QCCm$ is: $$\mathbf{G}(X) =
\begin{pmatrix}
g(X) & g(X) f_{1}(X) & \cdots & g(X)f_{\QCCl-1}(X)
\end{pmatrix},$$ where $g(X) | (X^{\QCCm}-1)$ and $f_{1}(X), \dots, f_{\QCCl-1}(X) \in \Fqx$.
To describe quasi-cyclic codes explicitly, we need to recall the following facts of *cyclic* codes. A $q$-cyclotomic coset $\coset{i}{\QCClen}$ is defined as: $ \coset{i}{\QCClen} \defeq \big\{ iq^j \mod \QCClen \, \vert \, j \in \interval{a} \big\}$, where $a$ is the smallest positive integer such that $iq^{a} \equiv i \bmod \QCClen$. The minimal polynomial in $\Fqx$ of the element $\alpha^i \in \F{q^{\QCCext}}$ is given by $$\label{eq_MinPoly}
\minpoly{i}{\QCClen} = \prod_{j \in \coset{i}{\QCClen} } (X-\alpha^j).$$ The following fact is used in Section \[sec\_ProductQCCQCC\].
\[fact\_ModuloBlaBla\] Let four nonzero integers $y, a, \ell, m$ be such that $$y \equiv a \ell \mod m \ell$$ holds. Then $\ell \mid y$ and $y/\ell \equiv a \mod m$.
Quasi-Cyclic Product Code {#sec_ProductQCCQCC}
=========================
Throughout this section we consider a linear product code $\QCCa \otimes \QCCb$, where $\QCCa$ is the row-code and $\QCCb$ the column-code, respectively. Furthermore, w.l.o.g. let $\QCCa$ be an ${\ensuremath{\ell}}$-quasi-cyclic code with reduced Gröbner basis in POT form as defined in : $$\label{eq_GroebMatrixCodeA}
\genmat[A] =
\begin{pmatrix}
\gen[A][0][0] & \gen[A][0][1] & \cdots & \gen[A][0][{\ensuremath{\ell}}-1]\\
& \gen[A][1][1] & \cdots & \gen[A][1][{\ensuremath{\ell}}-1] \\
\multicolumn{2}{c}{\bigzero}& \ddots & \vdots \\
& & & \gen[A][{\ensuremath{\ell}}-1][{\ensuremath{\ell}}-1]
\end{pmatrix},$$ and let $\QCCb$ be an cyclic code with generator polynomial $\gen[B]$ of degree $\QCCbm - \QCCbk $.
Throughout the paper, we assume that $\gcd({\ensuremath{\ell}}\QCCam, \QCCbm) = 1$ and we furthermore assume that the two integers $\inta$ and $\intb$ are such that $$\label{eq_BEzoutRel}
\inta {\ensuremath{\ell}}\QCCam + \intb \QCCbm = 1.$$ We recall the lemma of Wasan [@wasan_quasi_1977], that generalizes the result of Burton–Weldon [@burton_cyclic_1965 Theorem I] for cyclic product codes to the case of an ${\ensuremath{\ell}}$-quasi-cyclic product code of an ${\ensuremath{\ell}}$-quasi-cyclic code $\QCCa$ and a cyclic code $\QCCb$. A codeword of $\QCCa \otimes \QCCb$ represented as univariate polynomial $c(X)$ can then be obtained from the matrix representation $(m_{i,j})_{i \in \interval{\QCCbm}}^{j \in \interval{{\ensuremath{\ell}}\QCCam}}$ as follows: $$\label{eq_OneUnivariatePolyProduct}
c(X) \equiv \sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{{\ensuremath{\ell}}\QCCam-1} m_{i,j} X^{\map{i}{j}} \mod X^{{\ensuremath{\ell}}\QCCam \QCCbm}-1,$$ where $$\label{def_MappingMatrixPolyQCCQCC}
\map{i}{j} \defeq i \inta {\ensuremath{\ell}}\QCCam {\ensuremath{\ell}}+ j \intb \QCCbm \mod {\ensuremath{\ell}}\QCCam \QCCbm.$$
\[lem\_MappingToUnivariatePolyQCC\] Let $\QCCa$ be an ${\ensuremath{\ell}}$-quasi-cyclic code of length ${\ensuremath{\ell}}\QCCam$ and let $\QCCb$ be a cyclic code of length $\QCCbm$. The product code $\QCCa \otimes \QCCb$ is an ${\ensuremath{\ell}}$-quasi-cyclic code of length ${\ensuremath{\ell}}\QCCam \QCCbm$ if $\gcd({\ensuremath{\ell}}\QCCam, \QCCbm) = 1$.
Let $(m_{i,j})_{i \in \interval{\QCCbm}}^{j \in \interval{{\ensuremath{\ell}}\QCCam}}$ be a codeword of the product code $\QCCa \otimes \QCCb$, where each row is a codeword of $\QCCa$ and each column is a codeword of $\QCCb$. The entry $m_{i,j}$ is the coefficient $c_{\map{i}{j}}$ of the codeword $\sum_{i} c_i X^i$ as in . In order to prove that $\QCCa \otimes \QCCb$ is ${\ensuremath{\ell}}$-quasi-cyclic it is sufficient to show that a shift by ${\ensuremath{\ell}}$ positions of a codeword serialized to a univariate polynomial by of $\QCCa \otimes \QCCb$ is again a codeword of $\QCCa \otimes \QCCb$.
A shift by ${\ensuremath{\ell}}$ in each row and a shift by one each column clearly gives a codeword in $\QCCa \otimes \QCCb$, because $\QCCa$ is ${\ensuremath{\ell}}$-quasi-cyclic and $\QCCb$ is cyclic. With $$\begin{aligned}
& \map{i+1}{j+{\ensuremath{\ell}}} \\
& \equiv (i+1) \inta {\ensuremath{\ell}}\QCCam {\ensuremath{\ell}}+ (j+{\ensuremath{\ell}}) \intb \QCCbm \mod {\ensuremath{\ell}}\QCCam \QCCbm \\
& \equiv i \inta {\ensuremath{\ell}}\QCCam {\ensuremath{\ell}}+ j \intb \QCCbm + {\ensuremath{\ell}}(\inta {\ensuremath{\ell}}\QCCam + \intb \QCCbm) \mod {\ensuremath{\ell}}\QCCam \QCCbm \\
& \equiv \map{i}{j} + {\ensuremath{\ell}}\mod {\ensuremath{\ell}}\QCCam \QCCbm, \end{aligned}$$ we obtain an ${\ensuremath{\ell}}$-quasi-cyclic shift of the univariate codeword obtained by and .
Instead of representing a codeword of $\QCCa \otimes \QCCb$ as one univariate polynomial as in , we want to represent it as ${\ensuremath{\ell}}$ univariate polynomials as defined in .
\[lem\_MappingUnivariateQCCQCC\] Let $\QCCa$ be an ${\ensuremath{\ell}}$-quasi-cyclic code of length ${\ensuremath{\ell}}\QCCam$ and let $\QCCb$ be a cyclic code of length $\QCCbm$. Let the matrix $(m_{i,j})_{i \in \interval{\QCCbm}}^{j \in \interval{{\ensuremath{\ell}}\QCCam}}$ be a codeword of $\QCCa \otimes \QCCb$, where each row is in $\QCCa$ and each column is in $\QCCb$. The ${\ensuremath{\ell}}$ univariate polynomials of the corresponding codeword $(c_{0}(X) \ c_{1}(X) \ \cdots \ c_{{\ensuremath{\ell}}-1}(X)) $, where each component is defined as in , are given by: $$\label{eq_ComponentExpression}
\begin{split}
c_{h}(X) & \equiv X^{h(-\inta \QCCam)} \cdot \sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{\QCCam-1} m_{i,j{\ensuremath{\ell}}+h} X^{ \mapb{i}{j}} \\
& \qquad \qquad \mod X^{\QCCam \QCCbm} - 1, \quad \forall h \in \interval{{\ensuremath{\ell}}},
\end{split}$$ where $$\label{eq_MappingSubCodewordQCCQCC}
\mapb{i}{j} \equiv i \inta {\ensuremath{\ell}}\QCCam + j \intb \QCCbm \mod \QCCam \QCCbm.$$
From Fact \[fact\_ModuloBlaBla\] we have for the exponents in : $$\begin{aligned}
& \mapb{i}{j} + h ( -\inta \QCCam) \equiv i \inta {\ensuremath{\ell}}\QCCam + j \intb \QCCbm \mod \QCCam \QCCbm \nonumber \\
& \Leftrightarrow \nonumber \\
& {\ensuremath{\ell}}\big( \mapb{i}{j} + h ( -\inta \QCCam ) \big) \nonumber \\
& \equiv {\ensuremath{\ell}}(i \inta {\ensuremath{\ell}}\QCCam + j \intb \QCCbm + h ( -\inta \QCCam ) ) \mod {\ensuremath{\ell}}\QCCam \QCCbm. \label{eq_FinalBigToSmallMatrix}\end{aligned}$$ With $ -\inta {\ensuremath{\ell}}\QCCam = \intb \QCCbm-1 $, we can rewrite : $$\begin{aligned}
{\ensuremath{\ell}}\big( \mapb{i}{j} + h ( -\inta \QCCam ) \big) & = {\ensuremath{\ell}}\mapb{i}{j} + {\ensuremath{\ell}}h ( -\inta \QCCam ) \nonumber \\
& = {\ensuremath{\ell}}\mapb{i}{j} + h \intb \QCCbm -h , \nonumber \end{aligned}$$ and this gives with $\mapb{i}{j}$ as in and $\map{i}{j}$ as in : $$\begin{aligned}
& {\ensuremath{\ell}}\mapb{i}{j} + h \intb \QCCbm - h \nonumber \\
& \equiv {\ensuremath{\ell}}(i \inta {\ensuremath{\ell}}\QCCam + j \intb \QCCbm) + h \intb \QCCbm - h \nonumber \\
& \equiv {\ensuremath{\ell}}i \inta {\ensuremath{\ell}}\QCCam + (j {\ensuremath{\ell}}+h) \intb \QCCbm - h \mod {\ensuremath{\ell}}\QCCam \QCCbm \nonumber \\
& = \map{i}{j{\ensuremath{\ell}}+h} - h. \label{eq_ExpressionForComponennt}\end{aligned}$$ Inserting in of Lemma \[lem\_VectorToUnivariatePoly\] leads to: $$\begin{aligned}
c(X) & = \sum_{h=0}^{{\ensuremath{\ell}}-1} c_h(X^{{\ensuremath{\ell}}}) X^h \nonumber \\
& = \sum_{h=0}^{{\ensuremath{\ell}}-1} \sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{\QCCam-1} m_{i,j{\ensuremath{\ell}}+h} X^{\map{i}{j{\ensuremath{\ell}}+h}} \nonumber \\
& = \sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{{\ensuremath{\ell}}\QCCam-1} m_{i,j} X^{\map{i}{j}},\end{aligned}$$ which equals .
The mapping $\mapb{i}{j}$ from of the ${\ensuremath{\ell}}$ submatrices $(m_{i,j{\ensuremath{\ell}}})_{i \in \interval{\QCCbm}}^{j \in \interval{\QCCam}}, (m_{i,j{\ensuremath{\ell}}+1})_{i \in \interval{\QCCbm}}^{j \in \interval{\QCCam}}, \dots, (m_{i,j{\ensuremath{\ell}}+{\ensuremath{\ell}}-1})_{i \in \interval{\QCCbm}}^{j \in \interval{\QCCam}}$ to the ${\ensuremath{\ell}}$ univariate polynomials $c_0(X), c_1(X), \dots, c_{{\ensuremath{\ell}}-1}(X) $ is the same as the one used to map the codeword of a cyclic product code from its matrix representation to a polynomial representation (see [@burton_cyclic_1965 Thm. 1]).
In Fig. \[fig\_CyclicQCC\], we illustrate the $\map{i}{j}$ as in for $\inta=1$, ${\ensuremath{\ell}}= 2$, $\QCCam = 17$ and $\intb=-11$, $\QCCbm=3$. Subfigure \[fig\_QCCCYCfull\] shows the values of $\map{i}{j}$. The two submatrices $(m_{i,j2})$ and $(m_{i,j2+1})$ for $i \in \interval{3}$ and $j \in \interval{17}$ are shown in Subfigure \[fig\_QCCCYCb\]. Subfigure \[fig\_QCCCYCc\] contains the coefficients of the two univariate polynomials $c_0(X)$ and $c_1(X)$, where $(c_0(X) \ c_1(X))$ is a codeword of the $2$-quasi-cyclic product code of length $102$.
\
\
\[fig\_CyclicQCC\]
The following theorem gives the basis representation of a quasi-cyclic product code, where the row-code is quasi-cyclic and the column-code is cyclic.
\[theo\_QCCTimesCYC\] Let $\QCCa$ be an $\LINQCC{{\ensuremath{\ell}}}{\QCCam}{\QCCak}{\QCCad}{q}$ ${\ensuremath{\ell}}$-quasi-cyclic code with generator matrix $\genmat[A] \in \Fqx^{{\ensuremath{\ell}}\times {\ensuremath{\ell}}}$ as in and let $\QCCb$ be an $\LIN{\QCCbm}{\QCCbk}{\QCCbd}{q}$ cyclic code with generator polynomial $\gen[B] \in \Fqx$.
Then the ${\ensuremath{\ell}}$-quasi-cyclic product code $\QCCa \otimes \QCCb$ has a generating matrix of the following (unreduced) form: $$\label{eq_UnReducedBasis}
\genmat =
\begin{pmatrix}
\genmat[0] \\
\genmat[1] \\
\end{pmatrix},$$ where $$\label{eq_UnReducedBasisPart1}
\begin{split}
& \genmat[0] = \genarg{B}{X^{\inta {\ensuremath{\ell}}\QCCam }} \cdot \\
& \begin{pmatrix}
\genarg{A}[0][0]{X^{\intb \QCCbm}} & \genarg{A}[0][1]{X^{\intb \QCCbm}} & \cdots & \genarg{A}[0][{\ensuremath{\ell}}-1]{X^{\intb \QCCbm}} \\
& \genarg{A}[1][1]{X^{\intb \QCCbm}} & \cdots & \genarg{A}[1][{\ensuremath{\ell}}-1]{X^{\intb \QCCbm}} \\
\multicolumn{2}{c}{\bigzerob} & \ddots & \vdots \\
& & & \genarg{A}[{\ensuremath{\ell}}-1][{\ensuremath{\ell}}-1]{X^{\intb \QCCbm}} \\
\end{pmatrix},\\
& \cdot \diag \begin{pmatrix}
1, X^{-\inta \QCCam}, X^{-2 \inta \QCCam}, \dots, X^{- ({\ensuremath{\ell}}-1) \inta \QCCam}
\end{pmatrix}
\end{split}$$ and $$\label{eq_UnReducedBasisPart2}
\genmat[1] = (X^{\QCCam \QCCbm}-1) \mathbf{I}_{{\ensuremath{\ell}}},$$ where $\mathbf{I}_{{\ensuremath{\ell}}}$ is the ${\ensuremath{\ell}}\times {\ensuremath{\ell}}$ identity matrix.
We first give an explicit expression for each component of a codeword $(c_{0}(X) \ c_{1}(X) \ \cdots \ c_{{\ensuremath{\ell}}-1}(X))$ in $\QCCa \otimes \QCCb $ depending on the components of a codeword $(a_{0}(X) \ a_{1}(X) \ \cdots \ a_{{\ensuremath{\ell}}-1}(X) )$ of the row-code $\QCCa$ and depending the column-code $\QCCb$ based on the expression of Lemma \[lem\_MappingUnivariateQCCQCC\]. Let the $\QCCbm \times {\ensuremath{\ell}}\QCCam $ matrix $(m_{i,j})$ be a codeword of the ${\ensuremath{\ell}}$-quasi-cyclic product code $\QCCa \otimes \QCCb$ and let the polynomial $$\label{eq_ColumnComponentQCCQCC}
a_{i,h}(X) \defeq \sum_{j=0}^{\QCCam-1} m_{i,j{\ensuremath{\ell}}+h} X^j, \quad \forall h \in \interval{{\ensuremath{\ell}}}, i \in \interval{\QCCbm}$$ denote the $h$th component of a codeword $(a_{i,0}(X) \ a_{i,1}(X) \ \cdots \ a_{i, {\ensuremath{\ell}}-1}(X))$ in $\QCCa$ in the $i$th row of the matrix $(m_{i,j})$. Denote a codeword $b_{j}(X)$ of $\QCCb$ in the $j$th column by $$\label{eq_RowComponentQCCQCC}
b_{j}(X) = \sum_{i=0}^{\QCCbm-1} m_{i,j} X^i, \quad \forall j \in \interval{{\ensuremath{\ell}}\QCCam},$$ respectively. From , we have for the $h$th component of a codeword of the product code $ \QCCa \otimes \QCCb$: $$\label{eq_ProductComponentQCC}
\begin{split}
c_{h}(X) & \equiv X^{h(-\inta \QCCam)} \sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{\QCCam-1} m_{i,j{\ensuremath{\ell}}+h} X^{ \mapb{i}{j} } \\
& \qquad \qquad \mod X^{\QCCam \QCCbm} - 1, \quad \forall h \in \interval{{\ensuremath{\ell}}},
\end{split}$$ and with $\mapb{i}{j}$ as in of Lemma \[lem\_MappingUnivariateQCCQCC\] we can write explicitly: $$\begin{aligned}
& c_{h}(X) \equiv X^{h(-\inta \QCCam)} \sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{\QCCam-1} m_{i,j{\ensuremath{\ell}}+h} X^{i \inta {\ensuremath{\ell}}\QCCam + j\intb \QCCbm} \nonumber \\
& \qquad \qquad \qquad \mod X^{\QCCam \QCCbm} - 1, \quad \forall h \in \interval{{\ensuremath{\ell}}}. \label{eq_ExplicitComponentProductQCC} \end{aligned}$$ We define a shifted component: $$\label{eq_ExplicitComponentProductQCCShifted}
\tilde{c}_{h}(X) \equiv c_{h}(X) X^{h(\inta \QCCam)} \mod X^{\QCCam \QCCbm} \minus 1, \: \forall h \in \interval{{\ensuremath{\ell}}}.$$ Since $$\begin{aligned}
\sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{\QCCam-1} & m_{i,j{\ensuremath{\ell}}+h} X^{i \inta {\ensuremath{\ell}}\QCCam + j \intb \QCCbm} \\
& = \sum_{i=0}^{\QCCbm-1} X^{i \inta {\ensuremath{\ell}}\QCCam} \sum_{j=0}^{\QCCam-1} m_{i,j{\ensuremath{\ell}}+h} X^{j \intb \QCCbm} \\
& = \sum_{i=0}^{\QCCbm-1} X^{i \inta {\ensuremath{\ell}}\QCCam} a_{i,h}(X^{\intb \QCCbm}), \quad \forall h \in \interval{{\ensuremath{\ell}}},\end{aligned}$$ and from and in terms of the components of the row-code as defined in , we obtain: $$\label{eq_ExplicitComponentWithRemainderQCCQCC}
\begin{split}
& \tilde{c}_{h}(X) = q_{h}(X) (X^{\QCCam \QCCbm}-1) + \\
& \qquad \qquad \sum_{i=0}^{\QCCbm-1} X^{i \inta {\ensuremath{\ell}}\QCCam} a_{i,h}(X^{\intb \QCCbm}), \quad \forall h \in \interval{{\ensuremath{\ell}}},
\end{split}$$ for some $q_{h}(X) \in \Fqx$. Therefore $\tilde{c}_{h}(X)$ is a multiple of $\sum_{i=0}^h \epsilon_{i}(X) \genarg{A}[i][h]{X^{\intb \QCCbm}}$ for some $\epsilon_i(X) \in \Fqx$. A codeword $b_{j}(X)$ in $\QCCb$ in the $j$th column of $(m_{i,j})$ is a multiple of $\genarg{B}{X}$ and we obtain: $$\begin{aligned}
\sum_{i=0}^{\QCCbm-1} \sum_{j=0}^{{\ensuremath{\ell}}\QCCam-1} & m_{i,j} X^{i \inta {\ensuremath{\ell}}\QCCam+j\intb\QCCbm} \\
& = \sum_{j=0}^{{\ensuremath{\ell}}\QCCam-1} X^{j \intb \QCCbm} \sum_{i=0}^{\QCCbm-1} m_{i,j} X^{i \inta {\ensuremath{\ell}}\QCCam} \\
& = \sum_{j=0}^{{\ensuremath{\ell}}\QCCam-1} X^{j \intb \QCCbm} b_{j}(X^{\inta {\ensuremath{\ell}}\QCCam}),\end{aligned}$$ and therefore $\tilde{c}_h(X)$ is a multiple of $\genarg{B}{X^{\inta {\ensuremath{\ell}}\QCCam}}$ modulo $X^{\QCCam \QCCbm} - 1$.
Similar to the proof of [@burton_cyclic_1965 Thm. III], it can be shown that every shifted component $\tilde{c}_h(X)$ is a multiple of the product of $\genarg{B}{X^{\inta {\ensuremath{\ell}}\QCCam}}$ and $\sum_{i=0}^h \epsilon_{i} \genarg{A}[i][h]{X^{\intb \QCCbm}} $ modulo $(X^{\QCCam \QCCbm}-1)$. Therefore, we can represent each codeword in $\QCCa \otimes \QCCb$ as: $$\begin{split}
& \big(c_0(X) \ c_1(X) \ \cdots \ c_{{\ensuremath{\ell}}-1}(X) \big) \\
& \qquad = \big(i_0(X) \ i_1(X) \ \cdots \ i_{{\ensuremath{\ell}}-1}(X) \big) \genmat,
\end{split}$$ where $\genmat$ is as in .
The following theorem gives the reduced Gröbner basis (as defined in ) representation of the quasi-cyclic product code from Thm. \[theo\_QCCTimesCYC\], where the row-code is a 1-level quasi-cyclic code.
\[theo\_OneLevelQC\] Let $\QCCa$ be an $\LINQCC{{\ensuremath{\ell}}}{\QCCam}{\QCCak}{\QCCad}{q}$ 1-level ${\ensuremath{\ell}}$-quasi-cyclic code with generator matrix in RGB/POT form: $$\begin{aligned}
& \genmat[A] \nonumber \\
& = \begin{pmatrix}
\gen[A][0][0] & \hspace{.2cm} \gen[A][0][1] & \hspace{.55cm} \cdots & \hspace{.3cm} \gen[A][0][{\ensuremath{\ell}}-1]
\end{pmatrix} \nonumber \\
& = \begin{pmatrix}
\gen[A] & \gen[A] f_{1}^{A}(X) & \cdots & \gen[A] f_{{\ensuremath{\ell}}-1}^{A}(X)
\end{pmatrix} \label{eq_GenMatrixQCCOneLevel}\end{aligned}$$ as shown in Corollary \[cor\_OneLevelQC\]. Let $\QCCb$ be an $\LIN{\QCCbm}{\QCCbk}{\QCCbd}{q}$ cyclic code with generator polynomial $\gen[B] \in \Fqx$.
Then a generator matrix of the $1$-level ${\ensuremath{\ell}}$-quasi-cyclic product code in RGB/POT form is:
$$\begin{split}
\genmat = & \begin{pmatrix}
\gen & \gen f_{1}^{A}(X^{\intb \QCCbm}) & \cdots & \gen f_{{\ensuremath{\ell}}-1}^{A}(X^{\intb \QCCbm })
\end{pmatrix}\\
& \cdot \diag \begin{pmatrix}
1, X^{-\inta \QCCam}, X^{-2 \inta \QCCam}, \dots, X^{- ({\ensuremath{\ell}}-1) \inta \QCCam}
\end{pmatrix},
\end{split}$$
where $$\label{eq_GCDOneLevel}
\gen = \gcd \big( X^{\QCCam \QCCbm}-1, \genarg{A}{X^{\intb \QCCbm}} \genarg{B}{X^{\inta {\ensuremath{\ell}}\QCCam}} \big) .$$
Let two polynomials $u_0(X), v_0(X) \in \Fqx$ be such that: $$\label{eq_BezoutProductDiag}
\begin{split}
\gen & = u_0(X) \genarg{A}{X^{\intb \QCCbm}} \genarg{B}{X^{\inta {\ensuremath{\ell}}\QCCam }}\\
& \qquad \qquad + v_0(X) (X^{\QCCam \QCCbm}-1).
\end{split}$$ We show now how to reduce the basis representation to the RGB/POT form. We denote a new Row $i$ by $\rowop{i}'$. For ease of notation, we omit the term $\diag(1, X^{-\inta \QCCam}, X^{-2 \inta \QCCam}, \dots ,X^{- ({\ensuremath{\ell}}-1) \inta \QCCam} )$ and denote by $Y = X^{\intb \QCCbm}$ and $Z = X^{\inta {\ensuremath{\ell}}\QCCam}$.
We write the basis of the submodule in unreduced form (as in ): $$\begin{aligned}
& \begin{pmatrix}
\genarg{A}{Y} \genarg{B}{Z} & \genarg{A}{Y} f_{1}^{A}(Y) \genarg{B}{Z} & \cdots & \\ X^{\QCCam \QCCbm}-1 & \\
& X^{\QCCam \QCCbm}-1 & \\
\multicolumn{1}{c}{\bigzero} & & \ddots & \\
\end{pmatrix} \label{eq_StartMatrix} \\[2ex]
& \rightarrow \rowop{0}' = u_0(X)\rowop{0} + v_0(X) \rowop{1} + v_0(X) f_{1}^{A}(Y) \rowop{2} \nonumber \\
& \qquad \qquad \qquad + \dots + v_0(X) f_{{\ensuremath{\ell}}-1}^{A}(Y) \rowop{{\ensuremath{\ell}}} \nonumber \\[2ex]
& \begin{pmatrix}
\gen & \gen f_{1}^{A}(Y) & \cdots & \\
\genarg{A}{Y} \genarg{B}{Z} & \genarg{A}{Y} f_{1}^{A}(Y) \genarg{B}{Z} & \cdots & \\
X^{\QCCam \QCCbm}-1 & \\
& X^{\QCCam \QCCbm}-1 & \\
\multicolumn{1}{c}{\bigzero} & & \ddots & \\
\end{pmatrix}, \label{eq_MatrixFirstMerge}\end{aligned}$$ where the $i$th entry in new row 0 was obtained using: $$\begin{aligned}
& u_0(X) \genarg{A}{Y} f_{i}^{A}(Y) \genarg{B}{Z} + v_0(X) f_{i}^{A}(Y) (X^{\QCCam \QCCbm} -1) \nonumber \\
& = f_{i}^{A}(Y) \big( u_0(X) \genarg{A}{Y} \genarg{B}{Z} \nonumber \\
& \qquad \qquad + v_0(X) (X^{\QCCam \QCCbm} - 1) \big), \label{eq_PreGCDForm}\end{aligned}$$ and with we obtain from $$\begin{aligned}
& f_{i}^{A}(Y) \big( u_0(X) \genarg{A}{Y} \genarg{B}{Z} + v_0(X) (X^{\QCCam \QCCbm}-1) \big) \\
& = f_{i}^{A}(Y) \gen.\end{aligned}$$ Clearly, $\gen$ divides $\genarg{A}{Y} \genarg{B}{Z}$ and it is easy to check that Row 1 of the matrix in can be obtained from Row 0 by multiplying by $\genarg{A}{Y} \genarg{B}{Z}/\gen$. Therefore, we can omit the linearly dependent Row $1$ in and write the reduced basis as: $$\begin{aligned}
& \begin{pmatrix}
\gen \hspace*{.3cm} & \gen f_{1}^{A}(X^{\intb \QCCbm}) & \cdots & \gen f_{{\ensuremath{\ell}}-1}^{A}(X^{\intb \QCCbm})
\end{pmatrix}, \end{aligned}$$ where we omitted the matrix $\diag (1, X^{-\inta \QCCam}, X^{-2 \inta \QCCam}, \dots, X^{- ({\ensuremath{\ell}}-1) \inta \QCCam})$ for the first row during the proof, but it will only influence the row-operations by a factor.
Note that is exactly the generator polynomial of a cyclic product code. A $1$-level ${\ensuremath{\ell}}$-quasi-cyclic product has rate greater than $({\ensuremath{\ell}}-1)/{\ensuremath{\ell}}$ and is therefore of high practical relevance. The explicit RGB/POT form of the $1$-level quasi-cyclic product code as in Thm. \[theo\_OneLevelQC\] allows statements on the minimum distance and to develop decoding algorithms.
Example {#sec_Example}
=======
We consider a $2$-quasi-cyclic product code with the same parameters as the one illustrated in Fig. \[fig\_CyclicQCC\]. In this section we investigate a more explicit example to be able to calculate the basis as given in Thm. \[theo\_OneLevelQC\].
Let $\QCCa$ be a binary $2$-quasi-cyclic code of length ${\ensuremath{\ell}}\QCCam = 2\cdot 17 = 34$ and let $\QCCb$ be a cyclic code of length $\QCCbm=3$. We have $X^{17}-1=\minpoly{17}{0} \minpoly{17}{1} \minpoly{17}{3} $, where the minimal polynomials are as defined in . Let the generator matrix of $\QCCa$ in RGB/POT form as in be $ \genmat[A] = \begin{pmatrix} \gen[A][0][0] & \gen[A][0][1] \end{pmatrix}$ where $$\begin{aligned}
\gen[A][0][0] & = \minpoly{17}{1}\\
& = X^8 + X^7 + X^6 + X^4 + X^2 + X + 1, \\
\gen[A][0][1] & = \minpoly{17}{1} \cdot m_0(X)^3 \cdot (X^3+X^2+1) \\
& = X^{14} + X^{13} + X^{12} + X^{11} + X^8 + 1,\end{aligned}$$ and $\QCCa$ is a $\LINQCC{17}{2}{9}{11}{2}$ $2$-quasi-cyclic code. Let $\alpha$ be a $17$th root of unity in $\Fxsub{2^{8}} \cong \Fxsub{2}/(X^8 + X^4 + X^3 + X^2 + 1)$. Let $\gen[B] = \minpoly{3}{0} = X+1$ be the generator polynomial of the $\LIN{3}{2}{2}{2}$ cyclic code $\CYCb$ and let $\inta = 1$ and $\intb = -11$ be such that holds. We have $$\begin{aligned}
X^{51}-1 & = \minpoly{51}{0}\minpoly{51}{1}\minpoly{51}{3}\minpoly{51}{5}\minpoly{51}{9}\\
& \qquad \minpoly{51}{11}\minpoly{51}{17}\minpoly{51}{19}.\end{aligned}$$ According to Thm. \[theo\_OneLevelQC\], we calculate $$\begin{aligned}
f_{1}^{A}(X^{-11 \cdot 3}) & \equiv f_{0,1}^{A}(X^{18}) = m_0(X^{18})^3 \cdot (X^{54}+X^{36}+1)\\
& = (X^{18}+1)^3 \cdot (X^{54}+X^{36}+1) \\
& = X^{108} + X^{54} + X^{18} + 1 \\
& \equiv X^{18} + X^6 + X^3 + 1 \mod (X^{51}+1), \end{aligned}$$ and we obtain the generator matrix $\genmat = (\gen[0][0] \ \gen[0][1])$ of $\QCCa \otimes \QCCb$, where: $$\begin{aligned}
\gen[0][0] & = \minpoly{51}{0}\minpoly{51}{1}\minpoly{51}{3}\minpoly{51}{9}\minpoly{51}{19}\\
& = X^{33} + X^{32} + X^{30} + X^{27} + X^{25} + X^{23} + X^{20} \\
& \quad + X^{18} + X^{17} + X^{16} + X^{15} + X^{13} + X^{10} + X^{8} \\
& \quad + X^{6} + X^{3} + X + 1.\end{aligned}$$ With Thm. \[theo\_OneLevelQC\], we obtain: $$\begin{aligned}
\gen[0][1] & \equiv a_{1}^{A}(X^{-11 \cdot 3}) \gen[0][0] \\
& \equiv X^{50} + X^{48} + X^{45} + X^{43} + X^{41} + X^{39} + X^{36}\\
& \quad + X^{34} + X^{32} + X^{29} + X^{27} + X^{26} + X^{25} + X^{23}\\
& \quad + X^{22} + X^{21} + X^{19} + X^{18} + X^{17} + X^{16} + X^{15}\\
& \quad + X^{14} + X^{12} + X^{11} + X^{10} + X^{8} + X^{7} + X^{6}\\
& \quad + X^{4} + X \mod (X^{51}+1).\end{aligned}$$
Conclusion and Outlook {#sec_Conclusion}
======================
Based on the RGB/POT representation of an ${\ensuremath{\ell}}$-quasi-cyclic code $\QCCa$ and the generator polynomial of a cyclic code $\QCCb$, a basis representation of the ${\ensuremath{\ell}}$-quasi-cyclic product code $\QCCa \otimes \QCCb$ was proven. The reduced basis representation of the special case of a $1$-generator quasi-cyclic product code was derived.
The general case of the basis representation of an $\QCCl_A \QCCbl$-quasi cyclic product code from an $\QCCl_A$-quasi-cyclic code $\QCCa$ and an $\QCCbl$-quasi-cyclic code $\QCCb$ as well as the reduction of the basis remains an open future work. Furthermore, a technique to bound the minimum distance of a given quasi-cyclic code by embedding it into a product code similar to [@zeh_decoding_2012] seems to be realizable.
[^1]: A. Zeh has been supported by the German research council (Deutsche Forschungsgemeinschaft, DFG) under grant Ze1016/1-1. S. Ling has been supported by NTU Research Grant M4080456.
|
---
abstract: 'We study moduli stabilization in combination with inflation in heterotic orbifold compactifications in the light of a large Hubble scale and the favored tensor-to-scalar ratio $r \approx 0.05$. To account for a trans-Planckian field range we implement aligned natural inflation. Although there is only one universal axion in heterotic constructions, further axions from the geometric moduli can be used for alignment and inflation. We argue that such an alignment is rather generic on orbifolds, since all non-perturbative terms are determined by modular weights of the involved fields and the Dedekind $\eta$ function. We present two setups inspired by the mini-landscape models of the $\mathbb Z_{6-\text{II}}$ orbifold which realize aligned inflation and stabilization of the relevant moduli. One has a supersymmetric vacuum after inflation, while the other includes a gaugino condensate which breaks supersymmetry at a high scale.'
---
DESY-15-040
[ ]{}\
\
*Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany*
Introduction {#sec:Introduction}
============
Precision measurements of the CMB radiation increasingly favor the paradigm that the very early universe can be described by a phase of single-field slow-roll inflation [@Planck:2015xua; @Ade:2015lrj]. In particular, recent observations of polarization fluctuations in the CMB indicate the possibility of substantial tensor modes among the primordial perturbations [@Ade:2014xna; @Ade:2015tva]. This necessitates large-field models of inflation, i.e., the inflaton field must traverse a trans-Planckian field range during the last 60 $e$-folds of inflation [@Lyth:1996im]. Since large-field inflation is potentially susceptible to an infinite series of Planck-suppressed operators, this requires an understanding of possible quantum gravity effects. Thus, there has been renewed interest in obtaining inflation models from string theory.
In this context natural inflation, first proposed in [@Freese:1990rb], is among the most promising candidates. Here the flatness of the inflaton potential is guaranteed by an axionic shift symmetry which is exact in perturbation theory, but potentially broken by non-perturbative effects [@Wen:1985jz; @Dine:1986vd]. Nevertheless, discrete symmetries may survive which protect the potential even at trans-Planckian field values. However, while axions are abundant in string theory compactifications we still face a problem: trans-Planckian inflaton values require an axion decay constant which is larger than the Planck scale. However, in string theory one generically expects the decay constant to be smaller than the string scale [@Banks:2003sx; @Svrcek:2006yi].
Different paths have been proposed to address this problem. In N-flation [@Dimopoulos:2005ac; @Kim:2006ys], for example, many axions with sub-Planckian decay constants contribute to the trans-Planckian field range of the inflaton, which is a linear combination of axions. However, this typically requires a very large number of axions which might be challenging to realize explicitly while maintaining control over the models. Another option was considered in [@Abe:2014pwa], where the authors obtain trans-Planckian axions by choosing large gauge groups and by stabilizing the Kähler moduli at values much below the Planck scale. In that case, in principle one has to worry about perturbative control of the supergravity approximation, i.e., stringy corrections may be important. Furthermore, there is axion monodromy inflation [@Silverstein:2008sg; @McAllister:2008hb] which uses a single sub-Planckian axion with a multi-valued potential to create an effectively trans-Planckian field range during inflation.
Another way of obtaining a large effective axion decay constant from a few number of axions is by alignment as proposed in [@Kim:2004rp] and further developed in [@Kappl:2014lra; @Kappl:2015pxa], or by kinetic alignment [@Bachlechner:2014hsa].[^1] In the minimal setup of [@Kim:2004rp] there are two axions which appear as a linear combination in multiple non-perturbative contributions to the superpotential. If the axion decay constants are almost aligned one obtains an effective axion with a large decay constant, although the individual decay constants were small. In this paper we focus on the KNP alignment mechanism and its realization in ${\ensuremath{\text{E}_{8}}\xspace}\times{\ensuremath{\text{E}_{8}}\xspace}$ heterotic string theory [@Gross:1984dd] on orbifolds [@Dixon:1985jw; @Dixon:1986jc]. Progress in this direction has recently been made in [@Ali:2014mra], where the authors embedded aligned natural inflation in a supergravity model motivated by heterotic string compactifications on smooth Calabi-Yau manifolds with vector bundles. However, the authors did not specify the mechanism of moduli stabilization or an underlying reason for the alignment of the non-perturbative terms. The authors of [@Ben-Dayan:2014zsa] proposed a related model of hierarchical axion inflation and how it could be embedded in type IIB string theory. For other attempts to embed aligned natural inflation in type IIB string theory see [@Long:2014dta; @Gao:2014uha; @Abe:2014xja; @Bachlechner:2014gfa; @Shiu:2015uva; @Shiu:2015xda], and [@Grimm:2007hs] for a related analysis.
We study whether alignment of heterotic axions may be achieved by considering world-sheet instantons or a combination of the latter with gaugino condensates. Since the contributions arise from completely different mechanisms a natural question arises: why should the two effects be aligned? We attempt to answer this question, focusing our discussion on heterotic orbifolds where the moduli dependence of both effects, the condensing gauge group and the world-sheet instantons, can be computed using methods of conformal field theory. We argue that an alignment of the two terms is not as unnatural as one may think, essentially because the moduli dependence of both effects is determined by modular weights and Dedekind $\eta$ functions.
Furthermore, we address the issue of consistent moduli stabilization. Whenever inflation is discussed in string theory one desires a hierarchy of the form $$\begin{aligned}
\label{eq:hierarchy}
M_\text s, \, M_\text{KK} > M_\text{moduli} > H\,,\end{aligned}$$ where $M_\text s$ denotes the string scale, $M_\text{KK}$ the Kaluza-Klein scale, and $H$ is the Hubble scale during inflation. This hierarchy is essential to ensure that inflation can be described by an effective four-dimensional supergravity theory where the inflaton is the only dynamical degree of freedom. In addition, in case of metastable vacua the barriers protecting the minima of the moduli must be larger than $H^2$. This is to avoid moduli destabilization during inflation as pointed out in [@Buchmuller:2004xr; @Kallosh:2004yh]. Using the terms needed for successful inflation and other contributions to the superpotential we provide such a hierarchy explicitly for the complex dilaton field and the two Kähler moduli whose axions combine to form the effective inflaton. In a similar way this has recently been discussed for two Kähler moduli in aligned inflation in [@Kappl:2015pxa].
While finalizing this project, concerns about large-field inflation models, among them aligned axion inflation, were raised by the authors of [@Rudelius:2015xta; @Montero:2015ofa; @Brown:2015iha]. They argue that most models involving trans-Planckian axions are generically challenged by potential contributions from gravitational instantons. In [@Montero:2015ofa] the authors outline examples involving Euclidean D-brane instantons in type IIB string theory. To fully assess the implications of these analyses it would be interesting to study the potentially dangerous gravitational instantons on heterotic orbifolds in order to explicitly check whether these instantons do arise or are forbidden by the orbifold symmetries. This is, however, left for future investigation.
This paper is organized as follows. In Section \[sec:Orbifolds\] we briefly review important properties of orbifold spaces for reference in later sections. Afterwards, in Section \[sec:Inflation\] we review the axion alignment mechanism and explain how to compute the various contributions to the superpotential. We discuss how this can be combined with moduli stabilization without spoiling the alignment or the dynamics of inflation. In Section \[sec:Example\] we present two toy examples of our mechanism based on the ${\ensuremath{\mathbb{Z}_{6-\text{II}}}\xspace}$ orbifold of the mini-landscape models [@Lebedev:2007hv; @Lebedev:2008un]. Section \[sec:Conclusion\] contains our conclusions and an outlook.
Properties of orbifolds {#sec:Orbifolds}
=======================
In this section we briefly review those properties of heterotic orbifolds relevant for our discussion. A good and detailed review can, for example, be found in [@Bailin:1999nk]. References [@Dundee:2010sb; @Parameswaran:2010ec] discuss the relevance of these ingredients for moduli stabilization.
In the construction of Abelian heterotic toroidal orbifolds one starts with a six-torus $T^6$ parameterized by three complex coordinates $z_{1,2,3}$ and mods out a discrete ${\ensuremath{\mathbb{Z}_{N}}\xspace}$ symmetry[^2] $\theta$, $$\begin{aligned}
\theta:~(z_1,z_2,z_3)\mapsto(e^{2\pi{\text{i}}n_1/N}z_1,e^{2\pi{\text{i}}n_2/N}z_2,e^{2\pi{\text{i}}n_3/N}z_3)=(e^{2\pi{\text{i}}v_1}z_1,e^{2\pi{\text{i}}v_2}z_2,e^{2\pi{\text{i}}v_3}z_3)\;,\end{aligned}$$ where we have defined the twist vector $v=(v_1,v_2,v_3)$. Requiring that the resulting singular space is Calabi-Yau imposes $v_1+v_2+v_3\in{\ensuremath{\mathbb{Z}_{}}\xspace}$. The ${\ensuremath{\mathbb{Z}_{6-\text{II}}}\xspace}$ orbifold, for example, has ${v=(1/3,-1/2,1/6)}$, i.e., it acts with an order-three rotation on the first, with an order-two rotation on the second, and with an order-six rotation on the third torus. Hence, each orbifold has $N$ twisted sectors $\theta^k$, $k=0,\ldots,N-1$. To ensure modular invariance of the one-loop string partition functions, these twists have to be accompanied by a shift in the ${\ensuremath{\text{E}_{8}}\xspace}\times{\ensuremath{\text{E}_{8}}\xspace}$ gauge degrees of freedom. This shift is parameterized by the shift-vector $V$. In addition, depending on the geometry one can allow for up to six independent Wilson lines $W_i$ on the torus.
The massless string spectrum is given in terms of the twist $v$, the shift $V$, and the Wilson lines $W_i$. In addition to the usual untwisted strings in the $\theta^0$ sector, which close already on the torus, it contains new string states in twisted sectors $\theta^k$ which are called twisted strings. These only close under the orbifold action and are thus forced to localize at orbifold fixed points. Depending on the orbifold action and the toroidal lattice, the amount of untwisted Kähler moduli $T_i$, $i=1,\ldots,h^{1,1}$ and complex structure moduli $U_j$, $j=1,\ldots,h^{2,1}$ may vary. In the ${\ensuremath{\mathbb{Z}_{6-\text{II}}}\xspace}$ case, for example, one has $h^{1,1}=3$ and $h^{2,1}=1$. The $T_i$ parameterize the size of the three $T^2$ sub-tori while $U$ parameterizes the shape of the $T^2$ on which the orbifold has a ${\ensuremath{\mathbb{Z}_{2}}\xspace}$ action.
Modular transformations {#subsec:ModularTrafos}
-----------------------
The Kähler and complex structure moduli have an $\text{SL}(2,{\ensuremath{\mathbb{Z}_{}}\xspace})$ symmetry under which the $T_i$ transform as $$\begin{aligned}
\label{eq:TDuality}
T_i~\rightarrow~\frac{a_i T_i -{\text{i}}\, b_i}{{\text{i}}\, c_i T_i+d_i} \,,\end{aligned}$$ and likewise for the moduli $U_j$. Here, $a_i, b_i, c_i, d_i \in \mathbb{Z}$ and $a_i d_i-b_i c_i =1$.
At zeroth order the Kähler potential of the moduli reads $$\begin{aligned}
K_\text{moduli}=-\sum_{i=1}^{h^{1,1}}\ln(T_i+\overline{T}_i)-\sum_{j=1}^{h^{2,1}}\ln(U_j+\overline{U}_j)\,.\end{aligned}$$ It is readily checked that under the transformation the Kähler potential transforms as $$\begin{aligned}
K_\text{moduli} \rightarrow K_\text{moduli}+\sum_{i=1}^{h^{1,1}}\ln|{\text{i}}\, c_iT^i+d_i|^2+\sum_{j=1}^{h^{2,1}}\ln|{\text{i}}\, c_j U^j+d_j|^2\,.\end{aligned}$$ Hence the shift symmetry of the moduli in the Kähler potential is protected by the modular symmetry. Since $G=K_\text{moduli} + K_\text{matter} +\ln |W|^2$, which appears in the supergravity Lagrangian, has to be invariant we find that the superpotential has to transform with modular weight $-1$, $$\begin{aligned}
W \rightarrow W\,\prod_{i=1}^{h^{1,1}}({\text{i}}\, c_iT^i+d_i)^{-1}\,\prod_{j=1}^{h^{2,1}}({\text{i}}\, c_j U^j+d_j)^{-1}\,.\end{aligned}$$ In addition to the Kähler and the superpotential, also the chiral fields have non-trivial modular transformations, $$\begin{aligned}
\Phi_\alpha~\rightarrow~\Phi_\alpha\,\prod_{i=1}^{h^{1,1}} \left({\text{i}}\,c_i T_i+d_i\right)^{m^i_\alpha}\,\prod_{j=1}^{h^{2,1}} \left({\text{i}}\,c_j U_j+d_j\right)^{\ell^j_\alpha}\,.\end{aligned}$$ The modular weights $m^i_\alpha$ and $\ell^j_\alpha$ depend on the orbifold twisted sector $k$ and oscillator numbers. Defining $w_i(k)=kv_i\text{ mod }1$, they are given by [@Dixon:1989fj; @Louis:1991vh; @Ibanez:1992hc] $$\begin{aligned}
\begin{alignedat}{1}
\label{eq:ModularWeights}
m^i=\left\{
\begin{array}{ll}
0\;, \quad & \text{if}\: w_i=0\;,\\
w_i-1-\widetilde{N}^i+\widetilde{N}^{i\,*}\;, \quad & \text{if}\: w_i\neq0\;.
\end{array}
\right.\\[4mm]
\ell^j=\left\{
\begin{array}{ll}
0\;, \quad & \text{if}\: w_j=0\;,\\
w_i-1+\widetilde{N}^j-\widetilde{N}^{j\,*}\;, \quad & \text{if}\: w_j\neq0\;.
\end{array}
\right.
\end{alignedat}\end{aligned}$$ Here, the $\widetilde{N}^i$ and $\widetilde{N}^{i\,*}$ are integer oscillation numbers. In the $p^\text{th}$ complex plane of the untwisted sector we have $m_p^i=-\delta_p^i$, $\ell_p^j=-\delta_p^j$. From this we find for the Kähler potential for the matter fields at lowest order $$\begin{aligned}
\label{eq:Kmatter}
K_\text{matter} = \sum_\alpha\prod_{i=1}^{h^{1,1}} \left(T_i + \overline{T_i}\right)^{m_\alpha^i}\,\prod_{j=1}^{h^{2,1}}\left(U_j + \overline{U}_j\right)^{\ell_\alpha^j}\, |\Phi_\alpha|^2\,.\end{aligned}$$ Since the matter fields transform non-trivially and the superpotential has to have modular weight $-1$, the coupling “constants” $y_{\alpha_1\ldots\alpha_L}$ of the $L$-point correlator $$\begin{aligned}
\label{eq:NPWSSchematic}
W\supset y_{\alpha_1\ldots\alpha_L}\Phi_{\alpha_1}\ldots\Phi_{\alpha_L}\end{aligned}$$ have to be appropriate modular functions such that the overall modular weight is $-1$. Specifically, $$\begin{aligned}
y_{\alpha_1\ldots\alpha_L}\Phi_{\alpha_1}\ldots\Phi_{\alpha_L} \propto \prod_{i=1}^{h^{1,1}}~\prod_{j=1}^{h^{2,1}}\eta(T_i)^{2 r_i}~\eta(U_j)^{2 s_j}~\Phi_{\alpha_1}\ldots\Phi_{\alpha_L}\,,\end{aligned}$$ where $\eta$ denotes the Dedekind $\eta$ function[^3] defined by $$\begin{aligned}
\eta(T)=e^{-\frac{\pi T}{12}}\prod_{\rho=1}^\infty \left(1-e^{-2\pi \rho T}\right)\,,\end{aligned}$$ and the constant parameters $r_i$ and $s_j$ are determined by the modular weights, $$\begin{aligned}
r_i=-1-\sum_\alpha m_\alpha^i\,,\qquad s_j=-1-\sum_\alpha \ell_\alpha^j\,.\end{aligned}$$ The Dedekind $\eta$ function transforms under modular transformations up to a phase, $$\begin{aligned}
\eta(T)\rightarrow ({\text{i}}\, c T+d)^{1/2}~\eta(T)\,.\end{aligned}$$ For $T > 1$ in Planck units we use the approximation $$\begin{aligned}
\label{eq:EtaApprox}
\eta(T)=e^{-\frac{\pi T}{12}}\,.\end{aligned}$$ As a result the non-perturbative superpotential terms are of the schematic form $$\begin{aligned}
\label{eq:NPWS}
W_\text{NP}^{\text{WS}}=A(\Phi_\alpha)\;e^{-\frac{2\pi}{12}\left(\sum_i r_i T_i+\sum_j s_j U_j\right)}\,.\end{aligned}$$ Note that, if the fields $\Phi_\alpha$ are charged under an anomalous 1 symmetry, $S$ may appear in the exponent as well. In particular, this is the case when the model-independent axion contained in $S$ cancels the anomalies, as explained in more detail below.
Anomalous U(1) and FI terms {#subsec:AnomalousU1}
---------------------------
In orbifold models with shift embeddings, the primordial ${\ensuremath{\text{E}_{8}}\xspace}\times{\ensuremath{\text{E}_{8}}\xspace}$ gauge symmetry is broken rank-preservingly into Abelian and non-Abelian gauge factors. Generically one 1 is anomalous, henceforth denoted by 1. This anomaly is canceled via a Green-Schwarz (GS) mechanism [@Green:1984sg]. More precisely, the dilaton $S$ transforms under such an anomalous gauge variation as $S\rightarrow S-{\text{i}}\Lambda \delta_\text{GS}$, where $\Lambda$ is the superfield gauge parameter and $\delta_\text{GS}$ is a real constant. As a consequence the combination $S+\overline{S}-\delta_\text{GS}V_A$ is gauge-invariant, where $V_A$ is the vector multiplet associated with 1.
The non-trivial 1 transformation of $S$ has two important consequences. First, we observe that GS anomaly cancellation results in a field-dependent Fayet-Iliopoulos (FI) term[^4] of the form $$\begin{aligned}
\label{eq:FITerm}
\xi=\frac{\delta_\text{GS}}{(S+\overline{S})}\;.\end{aligned}$$ In order to preserve $D$-flatness, this means that some chiral orbifold fields $\Phi_\alpha$ with appropriate charge must get a vacuum expectation value (VEV) to cancel $\xi$. The VEV of these fields can, at the same time, break unwanted extra gauge groups and lift vector-like exotics and other extra hidden fields in a Higgs-like mechanism. Generically, the primordial ${\ensuremath{\text{E}_{8}}\xspace}\times{\ensuremath{\text{E}_{8}}\xspace}$ is broken to many U(1) factors under which the orbifold fields are charged simultaneously. Hence, $D$-flatness of the other U(1) symmetries requires that many fields obtain a non-vanishing VEV. Second, superpotential terms involving the dilaton in the exponent have to be such that the whole correlator is gauge-invariant.
Moreover, $S$ has a non-trivial modular transformation to ensure anomaly cancellation in the underlying sigma-model [@Dixon:1990pc; @Derendinger:1991hq]: $$\begin{aligned}
\label{eq:DilatonTrafo}
S\rightarrow S+\frac{1}{8\pi^2}\sum_{i=1}^{h^{1,1}}\delta^i\ln({\text{i}}c_i T_i+d_i)+\sum_{j=1}^{h^{2,1}}\delta^j\ln({\text{i}}c_j U_j+d_j) \,,\end{aligned}$$ where $\delta^i$ and $\delta^j$ are real constants of order 1 that can be computed from the sigma-model anomaly cancellation condition. As a consequence, the modular invariant Kähler potential of the dilaton reads $$\begin{aligned}
\label{eq:KDilatonExact}
K_\text{dilaton}=-\ln(Y)=-\ln\left(S+\overline{S}+\frac{1}{8\pi^2}\left[\sum_{i=1}^{h^{1,1}}\delta^i\ln(T_i+\overline{T}_i)+\sum_{j=1}^{h^{2,1}}\delta^j\ln(U_j+\overline{U}_j)\right]\right)\,.\end{aligned}$$ Due to the loop suppression factor $8 \pi^2$, these corrections are small as long as the $T_i$ are not stabilized at substantially larger field values than $S$. This is not the case in the models we study.
Gauge kinetic function and gaugino condensation {#subsec:GaugeKinFunction}
-----------------------------------------------
The one-loop gauge kinetic function of a gauge group $G_a$ at Kač–Moody level 1 is given by [@Dixon:1989fj; @Dixon:1990pc; @Lust:1991yi] $$\begin{aligned}
\label{eq:GKF}
f_a(S,T,U)=S+\frac{1}{8\pi^2}\sum_{i=1}^{h^{1,1}}b_a^i(m) \frac{g_i}{N}\ln(\eta(T^i))^2+\frac{1}{8\pi^2}\sum_{j=1}^{h^{2,1}}b_a^j(\ell) \frac{g_j}{N}\ln(\eta(U^j))^2\;,\end{aligned}$$ where $b_a^i$ are the $\beta$-function coefficients in the $i^\text{th}$ torus of the gauge group $G_a$. They are non-vanishing in the $\mathcal{N}=2$ twisted sub-sectors of the theory and depend on the Dynkin indices and on the modular weights of the states charged under $G_a$. Furthermore, the $g_i$ are the order of the little group of the orbifold action in the $i^\text{th}$ torus, i.e., the order of the group that leaves the $i^\text{th}$ torus fixed. Depending on the lattice and the presence of Wilson lines, the modular symmetry group SL(2,[$\mathbb{Z}_{}$]{}) might be reduced such that only a subgroup $\Gamma^0(N/g_i)$ or $\Gamma_0(N/g_i)$ is realized [@Bailin:1993fm; @Bailin:1993ri; @Bailin:2014nna]. In the example of the factorized ${\ensuremath{\mathbb{Z}_{6-\text{II}}}\xspace}$ orbifold the $\mathcal{N}=2$ twisted sectors are $\theta^k$ with $k=2,3,4$, $N=6$, $g_1=2$, $g_2=3$ and the modular group is not reduced.
The gauginos of $G_a$ may condense at a scale $\Lambda_a^\text{GC}$ which depends on the low-energy effective $\mathcal{N}=1$ $\beta$-function, given by $$\begin{aligned}
\label{eq:EffectiveBetaFunction}
\beta_a = \frac{11}{3}C_2(\textbf{Ad}_a)-\frac{2}{3}\left(C_2(\textbf{Ad}_a)+\sum_{\psi_{\mathbf{R}_a}}C_2(\mathbf{R}_a)\right)-\frac{1}{3}\sum_{\phi_{\mathbf{R}_a}}C_2(\mathbf{R}_a)\,,\end{aligned}$$ where $C_2(\mathbf{R}_a)$ is the quadratic Casimir operator of the irreducible representation $\mathbf{R}_a$. $\Lambda_a^\text{GC}$ can then be written in terms of the gauge kinetic function as [@Taylor:1982bp; @Affleck:1983mk], $$\begin{aligned}
\label{eq:CondensationScale}
\Lambda_a^\text{GC} = e^{-\frac{8\pi^2}{\beta_a}f_a(S,T,U)}\,.\end{aligned}$$ As discussed above, all extra fields become massive. If their mass is larger than the condensation scale they can be integrated out. The $\beta$-function is then simply $3\check{c}$, where $\check{c}$ denotes the dual Coxeter number. The effective superpotential term generated by gaugino condensation is $\propto (\Lambda_a^\text{GC})^3$. In addition, it depends on the fields charged under the condensing gauge group and on the fields that get a VEV and give an effective mass term to those fields. The final expression involves, in addition to the $\mathcal{N}=2$ beta function of the condensing gauge group, the modular weights of the fields that enter in the condensate. To obtain the final expression, we insert into , and include a field-dependent pre-factor from integrating out the heavy fields [@Binetruy:1996uv]. Using the transformation behavior of the dilaton and requiring that the result has again modular weight $-1$, we find $$\begin{aligned}
\label{eq:NPGC}
W_\text{NP}^{\text{GC}}=B(\Phi_\rho)\;e^{-\frac{8\pi^2}{\check{c}}S+\sum_{i}(-2+\frac{2\delta^i}{\check{c}})\ln\eta(T_i)+\sum_{j}(-2+\frac{2\delta^j}{\check{c}})\ln\eta(U_j)}\,.\end{aligned}$$ Hence, we observe that both the non-perturbative world-sheet instanton contributions and the non-perturbative gaugino condensation terms depend on the modular weights and on the Dedekind $\eta$ function. The combined superpotential, using and , has the schematic form $$\begin{aligned}
\label{eq:SPotFull}
W \supset \prod_\alpha \Phi_\alpha\;e^{-\sum_\alpha \frac{q_\alpha}{\delta_\text{GS}} Y - \frac{2\pi}{12}\left( \sum_{i} r_i T_i + \sum_{j} s_j U_j\right)}+ B(\Phi_\rho)e^{-\frac{8\pi^2}{\check{c}}S+\frac{2\pi}{12}(\sum_{i} b_i T_i+\sum_{j} b_j U_j)}\,,\end{aligned}$$ where $q_\alpha$ are the 1 charges of the fields $\Phi_\alpha$. Note that the modular weights $m$ and $\ell$ are negative and such that $r_i,s_j\geq0$ for most couplings. The constants $b_i$ and $b_j$ also depend on the modular weights and in addition on the $\mathcal{N}=2$ beta function coefficients, $$\begin{aligned}
b_i=1-\sum_i \frac{\delta^i}{\check{c}}\,,\qquad b_j=1-\sum_j \frac{\delta^j}{\check{c}}\,.\end{aligned}$$ As mentioned before, the $\delta^i$ and $\delta^j$ are typically of order 1 so that $b_i, b_j\approx1$, especially for large gauge groups. Note that in many couplings at least some of these constants are zero and hence the corresponding modulus does not appear in those superpotential terms.
Inflation in heterotic orbifolds {#sec:Inflation}
================================
Let us now discuss how inflation can be realized in heterotic orbifold compactifications. We briefly review the alignment mechanism proposed in [@Kim:2004rp; @Kappl:2014lra] and subsequently put the ingredients of Section \[sec:Orbifolds\] together to build an aligned axion inflation model with all moduli stabilized at a high scale.
The alignment mechanism {#subsec:AlignmentMechanism}
-----------------------
Remember that alignment means, on the level of the effective potential for two axions $\tau_{1,2}$, $$\begin{aligned}
\label{eq:effpot}
V=\kappa_1\left(1-\cos(\beta_1\tau_1+\beta_2\tau_2)\right)+\kappa_2\left(1-\cos(n_1\tau_1+n_2\tau_2)\right)\,,\end{aligned}$$ that there is a flat direction if $$\begin{aligned}
\label{eq:ratios}
\frac{\beta_1}{n_1}=\frac{\beta_2}{n_2}\,.\end{aligned}$$ Notice that the coefficients $\beta_i$ and $n_i$ are the inverse of the axion decay constants. To slightly lift this flat direction one can introduce a small misalignment parameterized by [@Kappl:2014lra] $$\begin{aligned}
\label{eq:AlignmentParameter}
k:=\frac{1}{n_2}-\frac{\beta_1}{\beta_2}\frac{1}{n_1}\,,\end{aligned}$$ which vanishes for perfect alignment. After rotating to a convenient field basis, ${(\tau_1,\tau_2)\mapsto(\varphi_1,\varphi_2)}$ and canonically normalizing the kinetic terms, we obtain for the almost flat direction $\varphi_1$ an effective decay constant $f_\text{eff}$ which reads [@Kappl:2014lra; @Ali:2014mra] $$\begin{aligned}
f_\text{eff}=\frac{\beta_1^2 \sqrt{(\beta_1^{-2}+\beta_2^{-2})(\beta_1^{-2}+n_1^{-2})}}{k n_1 \beta_2}\,.\end{aligned}$$ It is arbitrarily large for arbitrarily small $k$ and hence closely aligned axions $\tau_i$. A sizeable tensor-to-scalar ratio $r\approx0.05$ requires a misalignment of $k\approx0.2$.
Alignment and moduli stabilization on orbifolds {#subsec:ModuliStabilization}
-----------------------------------------------
A complete treatment of stabilizing all moduli while keeping three MSSM generations of particles and one pair of Higgs fields with realistic Yukawa couplings, decoupling extra vector-like exotics, and breaking additional U(1) symmetries generically present in these models is beyond the scope of this paper. Moduli stabilization in similar setups without considering inflation has been investigated in [@Dundee:2010sb; @Parameswaran:2010ec]. However, the mechanisms used there typically yield masses below the currently favored large Hubble scale and are thus incompatible with single-field inflation.
From the discussion in Section \[sec:Orbifolds\] it should be clear that the effective potential is sourced by a superpotential with two non-perturbative terms, both of which contain two Kähler moduli $T_1$ and $T_2$. In particular, we mostly focus on the two Kähler moduli which correspond to the tori that have an $\mathcal{N}=2$ sub-sector.[^5] In fact, all orbifolds have at least three untwisted Kähler moduli and up to three untwisted complex structure moduli. Concerning their stabilization, note that those Kähler moduli which correspond to tori that have fixed points in all twisted sectors $\theta^k$ do not enter in the gauge kinetic function and thus can only be stabilized via world-sheet instantons. Whether they appear in a world-sheet instanton coupling depends on the modular weights as discussed above. For the sake of simplicity we assume that the moduli not involved in the stabilization or alignment mechanism, as well as other potentially present fields, have been stabilized at a scale above $H$ and consequently decouple from inflation.
The real parts of the $T_i$ govern the size of the compactification manifold. The imaginary parts, albeit not involved in the anomaly cancellation except for the small one-loop contribution, enjoy an axionic shift symmetry inherited from the SL(2,[$\mathbb{Z}_{}$]{}) symmetry. They yield a cosine-potential as in and can consequently be used as inflaton candidates. The real part of the complex dilaton field determines the gauge coupling strength while its imaginary part is the so-called universal axion which is responsible for Green-Schwarz anomaly cancellation, cf. Section \[subsec:AnomalousU1\]. For a suitable choice of the superpotential, comprised of the terms generically available in orbifold compactifications, the effective potential after integrating out all moduli and the universal axion takes the form . In the following we discuss which parts of the superpotential may achieve this while ensuring consistent stabilization of the aforementioned relevant moduli.
### Inflation with world-sheet instantons only {#inflation-with-world-sheet-instantons-only .unnumbered}
A first option is to employ only world-sheet instanton contributions. For two aligned Kähler moduli this has recently been discussed in [@Kappl:2015pxa], based on the mechanism proposed in [@Wieck:2014xxa]. We extend this to include dilaton stabilization by considering the part of the orbifold superpotential which has the form $$\begin{aligned}
\label{eq:WWSInstantons}
\begin{split}
W &= \chi_1 \left[A_1(\phi_\alpha,\chi_\beta) e^{-n_1 T_1 - n_2 T_2 } - P_1(\chi_\gamma) \right] + \chi_2 \left[ A_2(\phi_\mu,\chi_\nu) e^{-n_3 T_1 - n_4 T_2 } - P_2(\chi_\rho) \right] \\
&+ \chi_3 \left[A_3(\chi_\sigma) e^{-\frac{q}{\delta_\text{GS}} S} - P_3(\chi_\lambda) \right]\,,
\end{split}\end{aligned}$$ where the $\chi_i$ and $\phi_i$ are untwisted and twisted chiral superfields, respectively, and $n_i=\frac{\pi}{6}r_i$ in the notation of . Note that we have neglected the loop contributions to $S$. To obtain the correct $T_i$ dependence in the various terms twisted fields necessarily enter the functions $A_i$, while the $P_i$ are functions of untwisted moduli. Since untwisted fields have modular weight $-1$ they do not induce a moduli dependence of the couplings. Likewise, the moduli dependence in the couplings of $A_1$ and $A_2$ arises from the twisted fields. The discussion has again been tailored to the ${\ensuremath{\mathbb{Z}_{6-\text{II}}}\xspace}$ orbifold. For other orbifolds, especially for ${\ensuremath{\mathbb{Z}_{M}}\xspace}\times{\ensuremath{\mathbb{Z}_{N}}\xspace}$ orbifolds, there also exist couplings that involve only twisted states which nevertheless have modular weight $-1$, so that no extra $T_i$ occur in these terms.
In the above parameterization we assume that the fields entering $A_i$ and $P_i$ obtain non-vanishing vacuum expectation values via $F$- and $D$-terms. In our supergravity analysis we treat them as numerical constants given by the VEVs of these fields. Those VEVs are generically of the order of the string scale, $M_\text s \lesssim 0.1$. We assume that the other fields we have not made explicit obtain a mass in a similar way from couplings to fields that get a VEV.
The effective theory defined by and the Kähler potential discussed in Section \[sec:Orbifolds\] has a supersymmetric Minkowski vacuum at $\langle \chi_1 \rangle = \langle \chi_2 \rangle = \langle \chi_3 \rangle = 0$. The auxiliary fields of the $\chi_i$ stabilize the complex scalars $S$, $T_1$, and $T_2$ at mass scales determined by the $A_i$ and $P_i$. In the heterotic mini-landscape models of [@Lebedev:2007hv; @Lebedev:2008un] there are many examples in which the coefficients $n_i$ are such that sufficient alignment is possible. There is then a light linear combination of $T_1$ and $T_2$ whose imaginary part is the inflaton field. All other degrees of freedom can be sufficiently stabilized in many examples. More details and an explicit example which realizes the hierarchy are given in Section \[sec:Example\].
After inflation has ended supersymmetry must be broken to avoid phenomenological problems. As pointed out in [@Kappl:2015pxa] the above scheme can accommodate low-energy supersymmetry breaking, for example, via the $F$-term of a Polonyi field. A more generic situation on orbifolds is supersymmetry breaking via gaugino condensates. As is well-known, these can also lead to a suppression of the supersymmetry-breaking scale compared to the Hubble scale. From the perspective of aligned inflation this is desirable in the above setup, since the gaugino condensate must not interfere with the alignment of $T_1$ and $T_2$. No matter how supersymmetry is broken, the resulting vacuum will have a positive cosmological constant which is determined by the scale of supersymmetry breaking. This must be cancelled by a fine-tuned constant contribution to the superpotential to high accuracy.
Since many non-Abelian gauge groups arise from the breaking of ${\ensuremath{\text{E}_{8}}\xspace}\times{\ensuremath{\text{E}_{8}}\xspace}$ the appearance of gaugino condensates is quite generic in orbifolds. In the following we discuss an example in which a gaugino condensate participates in the alignment mechanism.
### Inflation with world-sheet instantons and gaugino condensates {#inflation-with-world-sheet-instantons-and-gaugino-condensates .unnumbered}
A second option to achieve alignment and moduli stabilization is to use a combination of gaugino condensates and world-sheet instantons. In this case supersymmetry is necessarily broken at a high scale in order to stabilize all fields above the Hubble scale. In many models we find superpotentials of the form , or more specifically $$\begin{aligned}
\label{eq:WGCAndWSInstantons}
W = \chi_1 \left[B_1 e^{-\frac{8 \pi^2}{\check c} S + \beta_1 T_1 + \beta_2 T_2 } - P_1 \right] + \chi_2 \left[A_1 e^{- n_1 T_1 - n_2 T_2} - P_2 \right]+ \chi_3 \left[A_2 e^{-\frac{q}{\delta_\text{GS}} S} - P_3 \right]\,,\end{aligned}$$ with $\beta_i = \frac{\pi}{6}b_i$. The notation and the field dependence of the $A_i$, $P_i$, $n_i$ is as in the previous example, and again we assume them to be constants arising from other fields that obtain a VEV. As explained in Section \[subsec:GaugeKinFunction\] the $\beta_i$ depend on the particle content of the $\mathcal{N}=2$ sub-sector and the modular weights of $\chi_1$ and the fields entering $B_1$. In the effective theory of inflation $B_1$ is assumed to be constant as well. It is in general a non-analytic function of mesonic degrees of freedom which are integrated out above the scale of gaugino condensation. As explained in more detail in [@Dundee:2010sb], $B1 \langle \chi_1 \rangle$ determines the meson mass in the vacuum, which must be larger than $H$ and the condensation scale to ensure decoupling in the effective theory and during inflation. This means that $\langle \chi_1 \rangle \neq 0$ in such setups. This is typically guaranteed by $D$-terms associated with 1 or, as in the above case, other U(1) symmetries. This means that the superpotential in yields a type of racetrack potential for the moduli, which are stabilized by their own $F$-terms and those of the $\chi_i$.
Moduli stabilization via the superpotential in generically yields dS vacua since supersymmetry is broken by the gaugino condensate. The scale of supersymmetry breaking is proportional to $B_1 \langle \chi_1 \rangle$ and necessarily lies, as explained above, close to the inflationary Hubble scale. However, to avoid a potentially destructive back-reaction of the auxiliary fields responsible for supersymmetry breaking, cf. the discussion in [@Buchmuller:2014pla], one must find examples in which the gravitino mass is not substantially larger than $H$. We demonstrate that this is possible in a second benchmark model in Section \[sec:Example\].
Two benchmark models {#sec:Example}
====================
Let us now turn to two examples. We chose to discuss inflation in the context of the ${\ensuremath{\mathbb{Z}_{6-\text{II}}}\xspace}$ mini-landscape models because these are the most-discussed models in the literature. However, the mechanisms discussed here apply to most orbifolds in a similar vein.
Example 1: World-sheet instantons only
--------------------------------------
Let us start with the situation described in Section \[subsec:ModuliStabilization\], where we stabilize the moduli via world-sheet instantons only. We assume that some untwisted and twisted fields $\Phi_\alpha$ have obtained a string-scale VEV from $D$-terms which we do not include explicitly here. As explained above, we take the $\chi_i$ to be untwisted and the $\phi_i$ to be twisted matter fields. Furthermore, we consider $\chi_3$ to carry 1 charge $q=1$.
The Kähler potential in this case reads $$\begin{aligned}
K = -\ln{\left( S + \overline S \right)} -\ln{\left( T_1 + \overline T_1 - |\chi_1|^2 \right)} -\ln{\left( T_2 + \overline T_2 - |\chi_2|^2 \right)} + |\chi_3|^2\,,\end{aligned}$$ where we have neglected the loop contributions to $S$. The contributions of the twisted fields do not affect the results of our discussion as long as all VEVs of the matter fields are of the order of the string scale or below. We consider the part of the full superpotential given in . The possible values for the modular weights $n_i$ are taken from the `orbifolder` [@Nilles:2011aj], in this case $n_1 = \pi/6$, $n_2 = \pi/6$, $n_3 = \pi/3$, and $n_4 = \pi/2$. In typical models $\delta_\text{GS}\sim\mathcal{O}(0.1)$ and the 1 charges of the fields entering $A_2$ are $\mathcal{O}(1)$, such that we obtain an overall prefactor of $S$ of order $1$. The VEVs of the fields entering $A_2$ cancel the $D$-term induced by $\delta_\text{GS}$. The remaining input parameters for this example are summarized in Table \[tab:Ex1Params\].
$A_1$ $A_2$ $A_3$ $P_1$ $P_2$ $P_3$
--------------------- --------------------- --------------------- --------------------- -------------------- ---------------------
$3.2 \cdot 10^{-4}$ $6.8 \cdot 10^{-4}$ $1.6 \cdot 10^{-3}$ $9.7 \cdot 10^{-5}$ $3.2 \cdot10^{-5}$ $2.6 \cdot 10^{-4}$
: Input parameters for the constants used in Example 1. The $A_i$ and $P_i$ arise from $3$- and $4$-point couplings. \[tab:Ex1Params\]
The resulting theory has a supersymmetric vacuum at $\langle \chi_1 \rangle = \langle \chi_2 \rangle = \langle \chi_3 \rangle = \langle \text{Im}\, S \rangle = 0$ and $\langle \text{Re}\, S \rangle \approx 1.8$. The lightest eigenvalue in the mass matrix corresponds to the aligned linear combination of $T_1$ and $T_2$. A convenient field basis is therefore $$\begin{aligned}
T_1 \to \tilde T_1 = a T_1 + b T_2\,, \qquad
T_2 \to \tilde T_2 = -b T_1 + a T_2\,,\end{aligned}$$ with $a \approx -0.64$ and $b \approx -0.77$ in this case. $\tilde T_2$ is the lightest direction and $\text{Im} \, \tilde T_2$ is the inflaton. In the vacuum its real part is as heavy as the inflaton because supersymmetry is unbroken. Thus, one may worry that it contributes quantum fluctuations to the system, yielding a multi-field inflation model. However, during inflation $\text{Re} \, \tilde T_2$ receives a soft mass term of the same order as the Hubble scale. Indeed, a numerical analysis of the coupled equations of motion, similar to the one carried out in [@Kappl:2015pxa], reveals that all fields except the inflaton are sufficiently stabilized during inflation. For 60 $e$-folds of slow-roll inflation we summarize the predictions for the CMB observables and other relevant parameters in Table \[tab:Ex1CMB\].
$\langle T_1 \rangle = \langle \overline T_1 \rangle$ $\langle T_2 \rangle = \langle \overline T_2 \rangle$ $f_\text{eff}$ $n_\text s$ $r$
------------------------------------------------------- ------------------------------------------------------- ---------------- ------------- ----------
$ 1.06 $ $ 1.24 $ $ 5.7 $ $ 0.96 $ $ 0.03 $
: CMB observables and other relevant parameters for 60 $e$-folds of inflation in Example 1. \[tab:Ex1CMB\]
Apparently, successful inflation in line with recent observations is possible in this setup. However, since we have chosen to only employ a portion of the total superpotential of such orbifold models, one may worry about additional terms which can interfere with moduli stabilization or the alignment mechanism. In particular, there may be terms of the form $$\begin{aligned}
W \supset C(\Phi_\alpha) e^{-f(T_1,T_2)}\,,\end{aligned}$$ where the function $f$ contains some linear combination of the two moduli. On the one hand, this term clearly breaks supersymmetry if $ C(\Phi_\alpha) \neq 0$. The effects on inflation, however, are not significant as long as the resulting gravitino mass is not much larger than $H$, which is generically fulfilled. On the other hand, the additional dependence on the moduli may interfere with the alignment of the effective inflaton field. We have verified that this is negligible as long as $C < A_i$. This means that additional terms of this type must be suppressed up to slightly higher order than the ones in the part of the superpotential we consider.
Example 2: World-sheet instantons and gaugino condensates
---------------------------------------------------------
The setup which includes a gaugino condensate is slightly more generic, but also more complicated. Similar to the previous example the Kähler potential reads $$\begin{aligned}
K = -\ln{\left( S + \overline S \right)} -\ln{\left( T_1 + \overline T_1 - |\chi_1|^2 \right)} -\ln{\left( T_2 + \overline T_2 - |\chi_2|^2 \right)} + |\chi_3|^2\,,\end{aligned}$$ where we have once more neglected the loop-suppressed correction to the dilaton Kähler potential and the contribution of the twisted matter fields. The superpotential is this time given by with $n_1 = \pi/2$, $n_2 = \pi/3$, $\beta_1 = \pi/6$, $\beta_2 = \pi/6$ and $q/\delta_\text{GS}=1$. As in the previous example, the FI term of 1 is canceled by the VEVs of the fields entering $A_2$. Note that $\chi_3$ cannot cancel this FI term since we assume in our example that its charge has the wrong sign. Nevertheless, on orbifolds fields are typically charged under many U(1) factors simultaneously. To account for this, we include a $D$-term $\zeta$ originating from another U(1) under which $\chi_{3}$ has charge $-1$, $$\begin{aligned}
V_D = \frac{1}{S+ \overline S}\left( \chi_3 K_{\chi_3} -\zeta \right)^2\,,\end{aligned}$$ with $\zeta = 10^{-3}$. This $D$-term is canceled by $\langle \chi_3\rangle\neq0$, which results in a non-vanishing VEV of the other fields, $\langle \chi_{1,2} \rangle \neq 0$. All other input parameters are summarized in Table \[tab:Ex2Params\].
$B_1$ $A_1$ $A_2$ $P_1$ $P_2$ $P_3$
---------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
$ 14.4 $ $ 6.8 \cdot 10^{-3} $ $ 3.6 \cdot 10^{-3} $ $ 2.1 \cdot 10^{-4} $ $ 9.1 \cdot 10^{-5} $ $ 7.1 \cdot 10^{-4} $
: Input parameters for the constants used in Example 2. The $A_i$ and $P_i$ arise from $3$- and $4$-point couplings. \[tab:Ex2Params\]
The resulting scalar potential has a dS vacuum specified in Table \[tab:Ex2VEVs\].
$ \langle S \rangle = \langle \overline S \rangle $ $\langle T_1 \rangle = \langle \overline T_1 \rangle$ $\langle T_2 \rangle = \langle \overline T_2 \rangle$ $\langle \chi_1 \rangle $ $ \langle \chi_2 \rangle $ $ \langle \chi_3 \rangle $
----------------------------------------------------- ------------------------------------------------------- ------------------------------------------------------- --------------------------- ---------------------------- ----------------------------
$ 1.6 $ $ 1.97 $ $ 1.16 $ $ 9.6 \cdot 10^{-3} $ $ -7.9 \cdot 10^{-2} $ $ -2.2 \cdot 10^{-2} $
: Vacuum expectation values of all relevant fields in the dS minimum with $\langle V \rangle \approx 2 \cdot 10^{-14}$. In addition, the imaginary parts of the $\chi_i$ obtain VEVs much smaller than 1. \[tab:Ex2VEVs\]
The positive vacuum energy can be cancelled by a fine-tuned constant contribution to $W$, and the gravitino mass in the near-Minkowski vacuum is $m_{3/2} \approx 6.2 \cdot 10^{-7}$. There is again a lightest direction in the mass matrix which is $\tilde T_2$ with $a \approx -0.82$ and $b \approx -0.56$, and its imaginary part is the inflaton. Once more we solve the coupled equations of motion to ensure that all other degrees of freedom are sufficiently stable during inflation. The CMB predictions for this second case are summarized in Table \[tab:Ex2CMB\].
$f_\text{eff}$ $n_\text s$ $r$
---------------- ------------- ----------
$ 5.7 $ $ 0.96 $ $ 0.04 $
: CMB observables for 60 $e$-folds of inflation and the effective axion decay constant in Example 2. \[tab:Ex2CMB\]
Further contributions to the superpotential must satisfy the same constraints as in Example 1 to not interfere with inflation.
Conclusions {#sec:Conclusion}
===========
We have analyzed the feasibility of natural inflation with consistent moduli stabilization in heterotic orbifold compactifications. To allow for the trans-Planckian axion field range favored by recent observations of the CMB polarization, we implement aligned natural inflation with two axions. Generic properties of orbifolds naturally permit sufficient alignment for 60 $e$-folds of slow-roll inflation with a detectable tensor-to-scalar ratio and a scalar spectral index of $n_\text s \approx 0.96$, and at the same time provide a mechanism to stabilize the relevant moduli and the dilaton.
The alignment is produced by two non-perturbative terms in the superpotential of the orbifold. They may either be sourced by two world-sheet instantons which couple to twisted and untwisted matter fields, or by a world-sheet instanton and a gaugino condensate of a non-Abelian gauge group in the hidden sector of the primordial ${\ensuremath{\text{E}_{8}}\xspace} \times {\ensuremath{\text{E}_{8}}\xspace}$. The axions which mix are the imaginary parts of two complex untwisted Kähler moduli, governing the size of two tori. A crucial observation is that both possible non-perturbative effects are determined by the modular weights of the fields involved in the coupling and the Dedekind $\eta$ function. This leads to many instantonic couplings with similar coefficients in the exponential, corresponding to the individual axion decay constants, which in turn allows for aligned inflation. Since any embedding of inflation in string theory must address moduli stabilization, we demonstrate how both Kähler moduli and the dilaton can be stabilized at a high scale. This can happen through the terms needed for inflation and additional terms involving the VEVs of twisted and untwisted matter fields.
In the case of two world-sheet instantons all axion coefficients are determined by sums of modular weights and the Dedekind $\eta$ function. Thus, the more fields are involved in the correlator, the larger the coefficients of the moduli in the instantonic terms. This way, couplings generated at fourth or higher order generically have coefficients which allow for just the right amount of alignment. The case in which inflation is driven by a world-sheet instanton and a gaugino condensate is more constrained, and thus more predictive. The coefficients in the gaugino condensate are fixed by symmetry arguments and the Dedekind $\eta$ function. Alignment can occur when the world-sheet instanton coupling is introduced at sufficiently high order. In both cases, additional terms in the superpotential do not interfere with inflation or moduli stabilization, as long as their magnitude is below the inflationary Hubble scale.
We provide benchmark models for both cases to illustrate our findings. In the first case we find a supersymmetric Minkowski vacuum in which the flattest direction is a linear combination of the two Kähler moduli, the imaginary part of which is the aligned inflaton. All other degrees of freedom are stabilized at a higher scale and decouple from inflation. During inflation the real part of the aligned modulus receives a Hubble-scale soft mass and is sufficiently stable as well. This situation is similar in the second case, although in the vacuum supersymmetry is spontaneously broken by the gaugino condensate with $m_{3/2} \lesssim H$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Wilfried Buchmüller, Emilian Dudas, Rolf Kappl, Hans Peter Nilles, Martin Winkler, and Alexander Westphal for useful discussions. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Center (SFB) 676 “Particles, Strings and the Early Universe”. The work of C.W. has been supported by a scholarship of the Joachim Herz Foundation.
[10]{}
Collaboration P. Ade [*et al.*]{} “[P]{}lanck 2015 results. [XIII]{}. [C]{}osmological parameters,” [\[[1502.01589]{}\]](http://www.arXiv.org/abs/1502.01589). Collaboration P. Ade [*et al.*]{} “[P]{}lanck 2015. [XX]{}. [C]{}onstraints on inflation,” [\[[1502.02114]{}\]](http://www.arXiv.org/abs/1502.02114). Collaboration P. Ade [*et al.*]{} “[D]{}etection of ${B}$-[M]{}ode [P]{}olarization at [D]{}egree [A]{}ngular [S]{}cales by [BICEP]{}2,” [*Phys.Rev.Lett.*]{} [**112**]{} (2014) no. 24, 241101 [\[[1403.3985]{}\]](http://www.arXiv.org/abs/1403.3985). Collaboration P. Ade [*et al.*]{} “[A]{} [J]{}oint [A]{}nalysis of [BICEP]{}2/[K]{}eck [A]{}rray and [P]{}lanck [D]{}ata,” [*Phys.Rev.Lett.*]{} (2015) [\[[1502.00612]{}\]](http://www.arXiv.org/abs/1502.00612). D. H. Lyth “[W]{}hat would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy?,” [*Phys.Rev.Lett.*]{} [**78**]{} (1997) 1861–1863 [\[[hep-ph/9606387]{}\]](http://www.arXiv.org/abs/hep-ph/9606387). K. Freese, J. A. Frieman, and A. V. Olinto “[N]{}atural inflation with pseudo - [N]{}ambu-[G]{}oldstone bosons,” [*Phys.Rev.Lett.*]{} [**65**]{} (1990) 3233–3236. X. Wen and E. Witten “[W]{}orld [S]{}heet [I]{}nstantons and the [Peccei-Quinn]{} [S]{}ymmetry,” [*Phys.Lett.*]{} [**B166**]{} (1986) 397. M. Dine and N. Seiberg “[N]{}onrenormalization [T]{}heorems in [S]{}uperstring [T]{}heory,” [*Phys.Rev.Lett.*]{} [**57**]{} (1986) 2625. T. Banks, M. Dine, P. J. Fox, and E. Gorbatov “[O]{}n the possibility of large axion decay constants,” [*JCAP*]{} [**0306**]{} (2003) 001 [\[[hep-th/0303252]{}\]](http://www.arXiv.org/abs/hep-th/0303252). P. Svrcek and E. Witten “[A]{}xions [I]{}n [S]{}tring [T]{}heory,” [*JHEP*]{} [ **0606**]{} (2006) 051 [\[[hep-th/0605206]{}\]](http://www.arXiv.org/abs/hep-th/0605206). S. Dimopoulos, S. Kachru, J. McGreevy, and J. G. Wacker “[N]{}-flation,” [ *JCAP*]{} [**0808**]{} (2008) 003 [\[[hep-th/0507205]{}\]](http://www.arXiv.org/abs/hep-th/0507205). S. A. Kim and A. R. Liddle “[N]{}flation: multi-field inflationary dynamics and perturbations,” [*Phys.Rev.*]{} [**D74**]{} (2006) 023513 [\[[astro-ph/0605604]{}\]](http://www.arXiv.org/abs/astro-ph/0605604). H. Abe, T. Kobayashi, and H. Otsuka “[T]{}owards natural inflation from weakly coupled heterotic string theory,” [\[[1409.8436]{}\]](http://www.arXiv.org/abs/1409.8436). E. Silverstein and A. Westphal “[M]{}onodromy in the [CMB]{}: [G]{}ravity [W]{}aves and [S]{}tring [I]{}nflation,” [*Phys.Rev.*]{} [**D78**]{} (2008) 106003 [\[[0803.3085]{}\]](http://www.arXiv.org/abs/0803.3085). L. McAllister, E. Silverstein, and A. Westphal “[G]{}ravity [W]{}aves and [L]{}inear [I]{}nflation from [A]{}xion [M]{}onodromy,” [*Phys.Rev.*]{} [**D82**]{} (2010) 046003 [\[[0808.0706]{}\]](http://www.arXiv.org/abs/0808.0706). J. E. Kim, H. P. Nilles, and M. Peloso “[C]{}ompleting natural inflation,” [ *JCAP*]{} [**0501**]{} (2005) 005 [\[[hep-ph/0409138]{}\]](http://www.arXiv.org/abs/hep-ph/0409138). R. Kappl, S. Krippendorf, and H. P. Nilles “[A]{}ligned [N]{}atural [I]{}nflation: [M]{}onodromies of two [A]{}xions,” [*Phys.Lett.*]{} [**B737**]{} (2014) 124–128 [\[[1404.7127]{}\]](http://www.arXiv.org/abs/1404.7127). R. Kappl, H. P. Nilles, and M. W. Winkler “[N]{}atural [I]{}nflation and [L]{}ow [E]{}nergy [S]{}upersymmetry,” [\[[1503.01777]{}\]](http://www.arXiv.org/abs/1503.01777). T. C. Bachlechner, M. Dias, J. Frazer, and L. McAllister “[C]{}haotic inflation with kinetic alignment of axion fields,” [*Phys.Rev.*]{} [**D91**]{} (2015) no. 2, 023520 [\[[1404.7496]{}\]](http://www.arXiv.org/abs/1404.7496). S. H. H. Tye and S. S. C. Wong “[H]{}elical [I]{}nflation and [C]{}osmic [S]{}trings,” [\[[1404.6988]{}\]](http://www.arXiv.org/abs/1404.6988). K. Choi, H. Kim, and S. Yun “[N]{}atural inflation with multiple sub-[P]{}lanckian axions,” [*Phys.Rev.*]{} [**D90**]{} (2014) no. 2, 023545 [\[[1404.6209]{}\]](http://www.arXiv.org/abs/1404.6209). D. J. Gross, J. A. Harvey, E. J. Martinec, and R. Rohm “[T]{}he [H]{}eterotic [S]{}tring,” [*Phys. Rev. Lett.*]{} [**54**]{} (1985) 502–505. L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten “[S]{}trings on [O]{}rbifolds,” [*Nucl. Phys.*]{} [**B261**]{} (1985) 678–686. L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten “[S]{}trings on [O]{}rbifolds. 2,” [*Nucl. Phys.*]{} [**B274**]{} (1986) 285–314. T. Ali, S. S. Haque, and V. Jejjala “[N]{}atural [I]{}nflation from [N]{}ear [A]{}lignment in [H]{}eterotic [S]{}tring [T]{}heory,” [\[[1410.4660]{}\]](http://www.arXiv.org/abs/1410.4660). I. Ben-Dayan, F. G. Pedro, and A. Westphal “[H]{}ierarchical [A]{}xion [I]{}nflation,” [*Phys.Rev.Lett.*]{} [**113**]{} (2014) no. 26, 261301 [\[[1404.7773]{}\]](http://www.arXiv.org/abs/1404.7773). C. Long, L. McAllister, and P. McGuirk “[A]{}ligned [N]{}atural [I]{}nflation in [S]{}tring [T]{}heory,” [*Phys.Rev.*]{} [**D90**]{} (2014) no. 2, 023501 [\[[1404.7852]{}\]](http://www.arXiv.org/abs/1404.7852). X. Gao, T. Li, and P. Shukla “[Combining Universal and Odd RR Axions for Aligned Natural Inflation]{},” [*JCAP*]{} [**1410**]{} (2014) no. 10, 048 [\[[1406.0341]{}\]](http://www.arXiv.org/abs/1406.0341). H. Abe, T. Kobayashi, and H. Otsuka “[Natural inflation with and without modulations in type IIB string theory]{},” [\[[1411.4768]{}\]](http://www.arXiv.org/abs/1411.4768). T. C. Bachlechner, C. Long, and L. McAllister “[P]{}lanckian [A]{}xions in [S]{}tring [T]{}heory,” [\[[1412.1093]{}\]](http://www.arXiv.org/abs/1412.1093). G. Shiu, W. Staessens, and F. Ye “[W]{}idening the [A]{}xion [W]{}indow via [K]{}inetic and [S]{}tückelberg [M]{}ixings,” [\[[1503.01015]{}\]](http://www.arXiv.org/abs/1503.01015). G. Shiu, W. Staessens, and F. Ye “[L]{}arge [F]{}ield [I]{}nflation from [A]{}xion [M]{}ixing,” [\[[1503.02965]{}\]](http://www.arXiv.org/abs/1503.02965). T. W. Grimm “[Axion inflation in type II string theory]{},” [*Phys.Rev.*]{} [**D77**]{} (2008) 126007 [\[[0710.3883]{}\]](http://www.arXiv.org/abs/0710.3883). W. Buchmuller, K. Hamaguchi, O. Lebedev, and M. Ratz “[D]{}ilaton destabilization at high temperature,” [*Nucl.Phys.*]{} [**B699**]{} (2004) 292–308 [\[[hep-th/0404168]{}\]](http://www.arXiv.org/abs/hep-th/0404168). R. Kallosh and A. D. Linde “[L]{}andscape, the scale of [SUSY]{} breaking, and inflation,” [*JHEP*]{} [**0412**]{} (2004) 004 [\[[hep-th/0411011]{}\]](http://www.arXiv.org/abs/hep-th/0411011). T. Rudelius “[C]{}onstraints on [A]{}xion [I]{}nflation from the [W]{}eak [G]{}ravity [C]{}onjecture,” [\[[1503.00795]{}\]](http://www.arXiv.org/abs/1503.00795). M. Montero, A. M. Uranga, and I. Valenzuela “[T]{}ransplanckian axions !?,” [\[[1503.03886]{}\]](http://www.arXiv.org/abs/1503.03886). J. Brown, W. Cottrell, G. Shiu, and P. Soler “[F]{}encing in the [S]{}wampland: [Q]{}uantum [G]{}ravity [C]{}onstraints on [L]{}arge [F]{}ield [I]{}nflation,” [\[[1503.04783]{}\]](http://www.arXiv.org/abs/1503.04783). O. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, [*et al.*]{} “[T]{}he [H]{}eterotic [R]{}oad to the [MSSM]{} with [R]{} parity,” [*Phys.Rev.*]{} [**D77**]{} (2008) 046013 [\[[0708.2691]{}\]](http://www.arXiv.org/abs/0708.2691). O. Lebedev, H. P. Nilles, S. Ramos-Sanchez, M. Ratz, and P. K. S. Vaudrevange “[H]{}eterotic mini-landscape ([II]{}): completing the search for [MSSM]{} vacua in a ${Z}_6$ orbifold,” [*Phys. Lett.*]{} [**B668**]{} (2008) 331–335 [\[[0807.4384]{}\]](http://www.arXiv.org/abs/0807.4384). D. Bailin and A. Love “[O]{}rbifold compactifications of string theory,” [ *Phys.Rept.*]{} [**315**]{} (1999) 285–408. B. Dundee, S. Raby, and A. Westphal “[M]{}oduli stabilization and [SUSY]{} breaking in heterotic orbifold string models,” [*Phys.Rev.*]{} [**D82**]{} (2010) 126002 [\[[1002.1081]{}\]](http://www.arXiv.org/abs/1002.1081). S. L. Parameswaran, S. Ramos-Sanchez, and I. Zavala “[O]{}n [M]{}oduli [S]{}tabilisation and de [S]{}itter [V]{}acua in [MSSM]{} [H]{}eterotic [O]{}rbifolds,” [*JHEP*]{} [**1101**]{} (2011) 071 [\[[1009.3931]{}\]](http://www.arXiv.org/abs/1009.3931). L. J. Dixon, V. Kaplunovsky, and J. Louis “[O]{}n [E]{}ffective [F]{}ield [T]{}heories [D]{}escribing (2,2) [V]{}acua of the [H]{}eterotic [S]{}tring,” [*Nucl.Phys.*]{} [**B329**]{} (1990) 27–82. J. Louis “[N]{}onharmonic gauge coupling constants in supersymmetry and superstring theory,”. L. E. Ibanez and D. Lust “[D]{}uality anomaly cancellation, minimal string unification and the effective low-energy [L]{}agrangian of 4-[D]{} strings,” [*Nucl.Phys.*]{} [**B382**]{} (1992) 305–364 [\[[hep-th/9202046]{}\]](http://arXiv.org/abs/hep-th/9202046). J. Lauer, J. Mas, and H. P. Nilles “[D]{}uality and the [R]{}ole of [N]{}onperturbative [E]{}ffects on the [W]{}orld [S]{}heet,” [*Phys.Lett.*]{} [ **B226**]{} (1989) 251. S. Ferrara, D. Lust, and S. Theisen “[T]{}arget [S]{}pace [M]{}odular [I]{}nvariance and [L]{}ow-[E]{}nergy [C]{}ouplings in [O]{}rbifold [C]{}ompactifications,” [ *Phys.Lett.*]{} [**B233**]{} (1989) 147. J. Lauer, J. Mas, and H. P. Nilles “[T]{}wisted sector representations of discrete background symmetries for two-dimensional orbifolds,” [ *Nucl.Phys.*]{} [**B351**]{} (1991) 353–424. S. Stieberger, D. Jungnickel, J. Lauer, and M. Spalinski “[Y]{}ukawa couplings for bosonic [Z]{}([N]{}) orbifolds: [T]{}heir moduli and twisted sector dependence,” [*Mod.Phys.Lett.*]{} [**A7**]{} (1992) 3059–3070 [\[[hep-th/9204037]{}\]](http://www.arXiv.org/abs/hep-th/9204037). M. B. Green and J. H. Schwarz “[A]{}nomaly [C]{}ancellation in [S]{}upersymmetric [D]{}=10 [G]{}auge [T]{}heory and [S]{}uperstring [T]{}heory,” [*Phys. Lett.*]{} [ **B149**]{} (1984) 117–122. P. Fayet and J. Iliopoulos “[S]{}pontaneously [B]{}roken [S]{}upergauge [S]{}ymmetries and [G]{}oldstone [S]{}pinors,” [*Phys.Lett.*]{} [**B51**]{} (1974) 461–464. M. Dine, N. Seiberg, and E. Witten “[Fayet-Iliopoulos Terms in String Theory]{},” [*Nucl.Phys.*]{} [**B289**]{} (1987) 589. L. J. Dixon, V. Kaplunovsky, and J. Louis “[M]{}oduli dependence of string loop corrections to gauge coupling constants,” [*Nucl.Phys.*]{} [**B355**]{} (1991) 649–688. J. Derendinger, S. Ferrara, C. Kounnas, and F. Zwirner “[O]{}n loop corrections to string effective field theories: [F]{}ield dependent gauge couplings and sigma model anomalies,” [*Nucl.Phys.*]{} [**B372**]{} (1992) 145–188. D. Lust and C. Munoz “[D]{}uality invariant gaugino condensation and one loop corrected [K]{}ahler potentials in string theory,” [*Phys.Lett.*]{} [**B279**]{} (1992) 272–280 [\[[hep-th/9201047]{}\]](http://www.arXiv.org/abs/hep-th/9201047). D. Bailin, A. Love, W. Sabra, and S. Thomas “[String loop threshold corrections for Z(N) Coxeter orbifolds]{},” [*Mod.Phys.Lett.*]{} [**A9**]{} (1994) 67–80 [\[[hep-th/9310008]{}\]](http://www.arXiv.org/abs/hep-th/9310008). D. Bailin, A. Love, W. Sabra, and S. Thomas “[Modular symmetries in Z(N) orbifold compactified string theories with Wilson lines]{},” [ *Mod.Phys.Lett.*]{} [**A9**]{} (1994) 1229–1238 [\[[hep-th/9312122]{}\]](http://www.arXiv.org/abs/hep-th/9312122). D. Bailin and A. Love “[R]{}educed modular symmetries of threshold corrections and gauge coupling unification,” [\[[1412.7327]{}\]](http://www.arXiv.org/abs/1412.7327). T. Taylor, G. Veneziano, and S. Yankielowicz “[S]{}upersymmetric [QCD]{} and [I]{}ts [M]{}assless [L]{}imit: [A]{}n [E]{}ffective [L]{}agrangian [A]{}nalysis,” [ *Nucl.Phys.*]{} [**B218**]{} (1983) 493. I. Affleck, M. Dine, and N. Seiberg “[D]{}ynamical [S]{}upersymmetry [B]{}reaking in [S]{}upersymmetric [QCD]{},” [*Nucl.Phys.*]{} [**B241**]{} (1984) 493–534. P. Binetruy and E. Dudas “[Gaugino condensation and the anomalous U(1)]{},” [*Phys.Lett.*]{} [**B389**]{} (1996) 503–509 [\[[hep-th/9607172]{}\]](http://www.arXiv.org/abs/hep-th/9607172). C. Wieck and M. W. Winkler “[I]{}nflation with [F]{}ayet-[I]{}liopoulos [T]{}erms,” [*Phys.Rev.*]{} [**D90**]{} (2014) no. 10, 103507 [\[[1408.2826]{}\]](http://www.arXiv.org/abs/1408.2826). W. Buchmuller, E. Dudas, L. Heurtier, and C. Wieck “[L]{}arge-[F]{}ield [I]{}nflation and [S]{}upersymmetry [B]{}reaking,” [*JHEP*]{} [**1409**]{} (2014) 053 [\[[1407.0253]{}\]](http://www.arXiv.org/abs/1407.0253). H. P. Nilles, S. Ramos-Sanchez, P. K. Vaudrevange, and A. Wingerter “[The Orbifolder: A Tool to study the Low Energy Effective Theory of Heterotic Orbifolds]{},” [*Comput.Phys.Commun.*]{} [**183**]{} (2012) 1363–1380 [\[[1110.5229]{}\]](http://www.arXiv.org/abs/1110.5229).
[^1]: See [@Tye:2014tja; @Choi:2014rja] for related alignment mechanisms.
[^2]: A similar discussion applies to the case of ${\ensuremath{\mathbb{Z}_{M}}\xspace}\times{\ensuremath{\mathbb{Z}_{N}}\xspace}$ orbifolds.
[^3]: In general, other modular functions can appear as well [@Lauer:1989ax; @Ferrara:1989qb; @Lauer:1990tm; @Stieberger:1992bj].
[^4]: Notice that this commonly used terminology is slightly misleading. A field-dependent FI term is usually the $D$-term of a complex field with a logarithmic Kähler potential, which, if integrated out at a high scale, may mimic a constant FI term as the one introduced in [@Fayet:1974jb]. We refer to the original discussion in [@Dine:1987xk] for more details.
[^5]: In the prime orbifolds ${\ensuremath{\mathbb{Z}_{3}}\xspace}$ and ${\ensuremath{\mathbb{Z}_{7}}\xspace}$ no torus has an $\mathcal{N}=2$ sub-sector while in the ${\ensuremath{\mathbb{Z}_{M}}\xspace}\times{\ensuremath{\mathbb{Z}_{N}}\xspace}$ orbifolds all three tori do.
|
---
abstract: 'Cost functions for non-hierarchical pairwise clustering are introduced, in the probabilistic autoencoder framework, by the request of maximal average similarity between the input and the output of the autoencoder. The partition provided by these cost functions identifies clusters with dense connected regions in data space; differences and similarities with respect to a well known cost function for pairwise clustering are outlined.'
address:
- 'Dipartimento Interateneo di Fisica, Università di Bari and I.N.F.N., Bari, Italy'
- 'Dipartimento dell’Emergenza e dei Trapianti di Organi, Sezione Fisica Medica, Università di Bari and I.N.F.N., Bari, Italy'
author:
- 'Leonardo Angelini, Mario Pellicoro, Sebastiano Stramaglia'
- Luigi Nitti
title: |
Cost functions for pairwise data clustering \
\
---
7.3in -0.5in 8.8in -0.3in
Clustering methods aim at partitioning a set of data-points in classes such that points that belong to the same class are more alike than points that belong to different classes [@ripley]. These classes are called clusters and their number may be preassigned or can be a parameter to be determined by the algorithm. There exist applications of clustering in such diverse fields as pattern recognition [@duda], astrophysics [@west], communications [@linde], biology [@alon], business [@mantegna] and many others. Two main approaches to clustering can be identified: parametric and non-parametric clustering.
Non-parametric approaches make few assumptions about about the data structure and, typically, follow some local criterion for the construction of clusters. Typical examples of the non-parametric approach are the agglomerative and divisive algorithms that produce dendrograms. In the last years non-parametric clustering algorithms have been introduced employing the statistical properties of physical systems. The Super-Paramagnetic approach by Domany and coworkers [@domany] exploits the analogy to a model granular magnet: the spin-spin correlation of a Potts model, living on the data-points lattice and with pair-couplings decreasing with the distance, is used to partition points in clusters. The synchronization properties of a system of coupled chaotic maps are used in [@angelini] to produce hierarchical clustering.
Parametric methods make some assumptions about the underlying data structure. Generative mixture models [@bishop] treat clustering as a problem of density estimation: data are viewed as coming from a mixture of probability distributions, each representing a different cluster, and the parameters of these distributions are adjusted to achieve a good match with the distribution of the input data. This can be obtained by maximizing the data likelihood (ML) or the posterior (MAP) if additional prior information on the parameters is available [@utsugi].
Many parametric clustering methods are based on a cost function: the best partition of points in clusters is assumed to be the one with minimum cost. Often cost functions incorporate the loss of information incurred by the clustering procedure when trying to reconstruct the original data from the compressed cluster representation: the most popular algorithm to optimize a cost function is $K$-means [@bishop]. Starting from a statistical [*ansatz*]{} and invoking maximum likelihood leads to a cost function which has been observed to work for clustering financial time series [@marsili].
It is important to stress the difference between [*central*]{} clustering, where it is assumed that each cluster can be represented by a prototype [@rose], and [*pairwise*]{} clustering where data are indirectly characterized by pairwise comparison instead of explicit coordinates [@hofman]; pairwise algorithms require as input only the matrix of dissimilarities. Obviously the choice of the measure of dissimilarity is not unique and it is crucial for the performance of any pairwise clustering method. It is worth remarking that it often happens that the dissimilarity matrix violates the requirements of a distance measure, i.e. the triangular inequality does not necessarily holds.
Folded Markov chains are used in the Probabilistic Autoencoder Framework to derive cost functions for clustering [@luttrel]. Some examples of two-stage folded Markov chains, and the corresponding algorithms for clustering and topographic mapping [@bishop1], are thoroughly analyzed in [@graepel], where it is also shown that the cost function for pairwise clustering, introduced in [@hofman], may be seen as a consequence of Bayes’ theorem and the requirement of minimal average distorsion in a probabilistic autoencoder.
It is the purpose of this work to introduce a new class of cost functions for pairwise clustering which can be obtained, in the autoencoder frame, by requiring [*maximal similarity*]{} instead of minimal distorsion. We show that the cost functions here introduced provide a non-hierarchical clustering of points where dense connected regions of points in the data space are recognized as clusters.
Let us now discuss autoencoders described by one-stage folded Markov chains. Let us consider a point $x$, in a data space, sampled with probability distribution $P_0 \left(x\right)$; a code index $\alpha \in \{1,\ldots,q\}$ is assigned to $x$ according to conditional probabilities $P\left( \alpha |x\right)$. A reconstructed version of the input, $x'$, is then obtained by use of the Bayesian decoder: $$P\left( x'|\alpha\right)={P\left( \alpha |x'\right) P_0
\left(x'\right)\over P\left( \alpha\right)}. \label{eq:bayes}$$ The joint distribution of $x$, $x'$ and $\alpha$, describing this encoding-decoding process, is $$P\left( x,x',\alpha\right)=P_0 \left( x\right)P\left( \alpha
|x\right) P\left(x'|\alpha\right); \label{eq:bayes1}$$ owing to (\[eq:bayes\]), the joint distribution reads: $$P\left( x,x',\alpha\right)={P_0 \left(x\right)P_0 \left(x'\right)
P\left( \alpha |x\right) P\left( \alpha |x'\right) \over P\left(
\alpha\right)}. \label{eq:bayes2}$$ The conditional probabilities $\{ P\left( \alpha |x\right)\}$ are the free parameters that must be adjusted to force the autoencoder to emulate the identity map on the data space.
Let $d(x,x')$ be a measure of the distorsion between input and output of the autoencoder. The average distorsion is then given by: $${\cal D}=\sum_{\alpha=1}^q \int dx \int dx'{P_0 \left(x\right)P_0
\left(x'\right) P\left( \alpha |x\right) P\left( \alpha |x'\right)
\over P\left( \alpha\right)} d(x,x'). \label{eq:aved}$$ Moreover, let s(x,x’) be a measure of the similarity between input and output; the average similarity is then given by $${\cal S}=\sum_{\alpha=1}^q \int dx \int dx'{P_0 \left(x\right)P_0
\left(x'\right) P\left( \alpha |x\right) P\left( \alpha |x'\right)
\over P\left( \alpha\right)} s(x,x'). \label{eq:aves}$$ It is natural to postulate a one-to-one mapping between values of distorsion and similarity, $s=F(d)$, with $F$ a strictly decreasing function. A good autoencoder is obviously characterized by a low value of ${\cal D}$ and high value of ${\cal S}$. However we remark that the two requirements $Min({\cal D})$ and $Max({\cal
S})$, for reasonable choices of $F$, are not generically equivalent.
Now we turn back to the clustering problem. Given a data-set $\{x_i\}$ of cardinality $N$, partitioning these points in $q$ classes corresponds, in this frame, to design an autoencoder, with $q$ code indexes, acting on data space. We choose the encoder to be deterministic: $$P\left( \alpha |x\right)=\delta_{\alpha\;\sigma(x)},
\label{eq:delta}$$ $\sigma (x)\in \{1,\ldots,q\}$ being the code index associated to $x$. The estimate for the average distorsion (\[eq:aved\]), based on the data-set at hand, is given by $\hat{{\cal D}}=N H_d
[\sigma]$, where we introduce the hamiltonian $H_d$ for the Potts variables $\{\sigma_i\}$: $$H_d [\sigma]=\sum_{\alpha=1}^q {\sum_{i,j=1}^N \delta_{\alpha
\sigma_i}\delta_{\alpha \sigma_j}d_{ij}\over \sum_{k=1}^N
\delta_{\alpha \sigma_k}}, \label{hamd}$$ where $\sigma_i =\sigma (x_i)$, $d_{ij}=d(x_i,x_j)$. It turns out that $H_d$ is equivalent to the cost function for pairwise clustering, influential in the clustering literature, introduced in [@hofman].
The estimate for the average similarity is, similarly, given by $\hat{{\cal S}}=-N H_s [\sigma]$, where we introduce the hamiltonian $H_s$: $$H_s [\sigma]=- \sum_{\alpha=1}^q {\sum_{i,j=1}^N \delta_{\alpha
\sigma_i}\delta_{\alpha \sigma_j}s_{ij}\over \sum_{k=1}^N
\delta_{\alpha \sigma_k}}. \label{hams}$$
If we choose the autoencoder by minimizing the average distorsion, then the best partition of the data-set in $q$ classes corresponds to the ground state of $H_d$. If we choose it by maximizing the average similarity, then the ground state of $H_s$ must be sought for, instead. Since both $\{d_{ij}\}$ and $\{s_{ij}\}$ may be taken positive, it follows that $H_d$ is characterized by antiferromagnetic couplings between the Potts variables, while $H_s$ is made of ferromagnetic couplings. Denominators in both $H_d$ and $H_s$ serve to enforce the coherence among the $q$ clusters. In particular, without the denominator the ground state of $H_s$ would correspond to a single big cluster.
The form of the function $F$, determining the relation between $s$ and $d$, has to be specified. In what follows we consider two forms of this relation. A scale-free relation $$s_{ij}=F_\gamma (d_{ij})=\left({d_{ij}\over \langle
d\rangle}\right) ^{-\gamma}, \label{dd}$$ depending on the exponent $\gamma$, and a scale-dependent relation $$s_{ij}=F_a (d_{ij})=\exp\left(-{1\over 2a^2}\left({d_{ij}\over
\langle d\rangle}\right)^2\right), \label{ff}$$ dependent on the scale $a$. In the formulas above, $\langle d
\rangle$ is the average dissimilarity over all the pairs of data-set points. The exponent $\gamma$ will be restricted to assume small values so as to characterize the corresponding Potts model by long-range ferromagnetic couplings; the scale parameter $a$ will be bounded in $[0,1]$.
At this point it is worth stressing that minimization of the distorsion and maximization of the similarity yield, in the autoencoder frame, different cost functions. The hamiltonian $H_d$ embodies the requirement that pairs of distant points (large $d_{ij}$) should belong to different clusters. On the other hand, the hamiltonian $H_s$, for reasonable choices of $F$, concentrates on pairs of close points (small $d$) and forces them to belong to the same cluster. In other words, $H_s$ may be seen to implement the idea that clusters should be searched for as dense connected regions in the data space.
We describe now the application of the variational criterions for clustering, described above, to some artificial and real data-sets. We consider two optimization algorithms to find the configuration of minimum cost: simulated annealing [@gelatt] and mean-field annealing [@yuille]. Both approaches associate a Gibbs probability distribution to the functional to be optimized. Simulated annealing is a Monte-carlo technique which samples the Gibbs distribution as the temperature is reduced to zero, while mean-field annealing attempts to track an approximation, to the mean of the distribution, known as [*mean field*]{} approximation [@parisi]. We remark that an efficient mean-field annealing algorithm for cost function (\[hamd\]), based on the EM scheme [@demp], is described in [@hofman]: the generalization of that algorithm to (\[hams\]) is straightforward.
{width="8cm"}
In many cases cost functions $H_d$ and $H_s$ have very close global minima. For example in Fig.1a we depict an artificial data-set generated by two overlapping isotropic Gaussian distributions. In this case the natural measure of dissimilarity is Euclidean metrics, and we use $q=2$. In Fig.1b the corresponding ground state of $H_d$ [@gr] is depicted: it is very close to the Bayesian solution, i.e. the solution obtained drawing the symmetry plane for the centers of the two Gaussians. A similar partition is obtained minimizing, by simulated annealing, $H_s$. As a measure of the difference between two partitions $\{\sigma_i\}$ and $\{\eta_i\}$, we evaluate the following quantity: $$\epsilon = {1\over N(N-1)}\sum_{i=1}^N \sum_{j=1,j\ne i}^N
\left(\delta_{\sigma_i \sigma_j}-\delta_{\eta_i \eta_j}\right)^2
\label{ep}$$ which counts the number of pairs of points upon which the two partitions disagree. Using the scale-dependent $F_a$, we find the ground state of $H_s$ to differ from those of $H_d$ by $\epsilon < 0.01$ varying $a$ in $[0.05,1]$. Analogously, using the scale-free $F_\gamma$, with $\gamma\in [0.1,1.5]$, we find $\epsilon < 0.02$ when we compare the ground state of $H_s$ with those of $H_d$. Hence, on this data set, the cost functions introduced above work similarly within wide ranges of $\gamma$ and $a$ values.
We find a similar behaviour with respect to the famous IRIS data of Anderson [@anderson]. This data set has often been used as a standard for testing clustering algorithms: it consists of three clusters (Virginica, Versicolor and Setosa) and there are $50$ objects in ${\mathbf R}^4$ per cluster. Two clusters (Verginica, Versicolor) are very overlapping. The clustering result, with $q=3$ and minimizing $H_d$, consists of three clusters of $61$, $39$ and $50$ points respectively, with $90\%$ of correct classification percentage. We obtain exactly the same partition by minimizing $H_s$ using a scale-free $F$ (with $\gamma \in
[0.15,1.45]$), and using a scale-dependent $F$ (with $a\in
[0.25,1]$). For $a\in[0.1,0.25]$ we obtain, in the scale-dependent case, a slightly different partition with clusters’ sizes $58$, $42$, $50$ and correct classification percentage $93.3\%$. These results show that also in the IRIS case the pairwise clustering procedures by distorsion minimization and similarity maximization are almost equivalent.
{width="8cm"}
A typical situation resulting in different answers from $H_d$ and $H_s$ is depicted in Fig.2a. This two-dimensional data-set is made of an elongated cluster and a Gaussian distributed circular one. It is evident that two dense connected regions are present, and that the farthest pairs of points belong to the same connected region. This is the type of data-set such that minimizing the distorsion is not equivalent to maximizing the similarity. In fig.2b the partition we obtain minimizing $H_d$ is depicted: it fails to recognize the structure in the data-set. Let us now consider the ground state of $H_s$ with the scale-dependent $F$. For $a<0.7$ the ground state, depicted in Fig.2c, recognizes with $99\%$ accuracy the data structure. At $a\sim 0.7$ a transition phenomenon occurs: the configuration depicted in Fig.2c ceases to be the global minimum, the new ground state (Fig.2d) being very close to the solution by $H_d$.
{width="8cm"}
In Fig.3a we depict the efficiency of the classification versus the resolution parameter $a$, for the scale dependent $F$, while in Fig.3b we consider a sequence of $a$-values and we plot the $\epsilon$ between partitions corresponding to adjacent values of $a$. The peak at $a=0.7$ is the indicator of the transition between global minima. Finally, in Fig. 3c the size of the two clusters, versus $a$, is depicted. Concerning the scale-free $F$, in Fig.4 the same plots as in Fig.3 are depicted, showing that the [*good*]{} minimum is stable for a wide range of $\gamma$.
{width="8cm"}
The choice of the optimization algorithm deserves a comment. All the results described above are obtained by simulated annealing; we also apply the mean-field annealing scheme, described in [@hofman], and we always find a configuration very close to the one from simulated annealing, while spending less computational time. This confirms that optimization algorithms rooted on mean-field theory yield quickly a good solution on these problems [@yuille].
In summary, we address non-hierarchical pairwise clustering and, working in the probabilistic autoencoder frame, we introduce a class of cost functions arising from the request of maximal average similarity between the input and the output of the autoencoder. Our simulations show that the partition provided by these new cost functions corresponds to extract dense connected regions in data space, and that a relevant discrepancy with the partition provided by the cost function introduced in [@hofman] is to be expected in case of non-trivial geometry of clusters. We note that the approach to clustering here described has some similarities with the method in [@domany]: indeed in both cases clustering is mapped onto a ferromagnetic Potts model with couplings decreasing with the distance. In the superparamagnetic approach, however, $q$ is not related to the number of classes present in the data-set and one obtains hierarchical clustering as the temperature of the Potts model is varied. In the present case $q$ is the number of classes, which is supposed to be known (non-hierarchical clustering), and the denominators in the hamiltonian, ensuring clusters’s coherence, leads to a non-trivial ground state which reflects data structure. We consider two classes of cost function. Scale-free cost functions depend on the exponent $\gamma$, while scale-dependent ones depend on the scale-parameter $a$. Varying $a$, i.e. changing the resolution at which the data-set is processed, may give rise to transitions between different partitions; in the scale-free case, the clustering output is fairly stable, with respect to $\gamma$, in a wide range.
Further work will be devoted to test these new cost functions on other real applications and to study related issues, such as the introduction of an [*adaptive*]{} relation between distorsion and similarity, i.e. the function $s=F(d)$ might be depending on the properties of the data-set in a neighbourhood of the pair of points under consideration. It will be also important to develop cluster-validity criterions to provide a means to choose an optimal $q$ value in situations where the number of classes is ambiguous.
[1]{}
B.D. Rypley, [*Pattern Recognition and neural networks*]{}. Cambridge University Press, Cambridge U.K., 1996.
R.O. Duda, P.E. Hart, [*Pattern Recognition and scene analysis*]{}. Wiley, New York, 1973.
A. Dekel, M.J. West, Astrophys. J. [**228**]{}, p. 411 (1985).
Y. Linde, A. Buzo, R.M. Gray, IEEE Trans. on Communications [**28**]{}, p. 84 (1980).
U. Alon, N. Barkai, D.A. Notterman, K. Gish, S. Ybarra, D. Mack, A.J. Levine, Proc. Natl. Acad. Sci. USA [**96**]{}, p. 6745 (1999).
L. Kullmann, J. Kertesz, R.N. Mantegna, Physica [**A 287**]{}, p. 412 (2000).
M. Blatt, S. Wiseman, E. Domany, Phys. Rev. Lett. [**76**]{}, pp. 3251-3255 (1996).
L. Angelini, F. De Carlo, C. Marangi, M. Pellicoro, S. Stramaglia, Phys. Rev. Lett. [**85**]{}, pp. 554-557 (2000).
C.M. Bishop, [*Neural networks for pattern recognition*]{}. Clarendon Press, Oxford, 1995.
A. Utsugi, Network [**7**]{}, p. 727 (1996).
L. Giada, M. Marsili, ’Data clustering and noise undressing of correlation matrices’, preprint cond-mat/0101237.
K. Rose, E. Gurewitz, G. Fox, Phys. Rev. Lett. [**65**]{}, pp. 945-948 (1990).
T. Hofman, J.M. Buhmann, IEEE Trans. P.A.M.I. [**19**]{}, pp.1-14 (1997).
S.P. Luttrel, Neural Computation [**6**]{}, p. 767, 1994.
C.M. Bishop, M. Svensen, C.K.I. Williams, Neural Computation [**10**]{}, p.215 (1997).
T. Graepel, [*Statistical Physics of clustering algorithms*]{}, Diplomarbeit, Technique Universitat, FB Physik, Institut fur Theoretishe Physik, Berlin, April 1998.
S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Science [**220**]{}, p.671 (1983).
See, e.g., A.L. Yuille, J.J. Kosowsky, Neural Computation [**6**]{}, pp. 341-356 (1994), and references therein.
G. Parisi, [*Statistical Field Theory*]{}, Addison Wesley, California 1988.
A.P. Dempster, N.M. Laird, D.B. Rubin, Jour. Royal Stat. Soc. [**39**]{}, p.1, (1977).
In the text we use an operational definition of [*ground state*]{} as the best output over a number (10-50) of simulated annealing runs. The true ground state might be found only by an unpractical exhaustive search.
E. Anderson, Bull. Amer. Iris Soc. [**59**]{}, p.2 (1935).
|
---
abstract: |
The field of automatic image inpainting has progressed rapidly in recent years, but no one has yet proposed a standard method of evaluating algorithms. This absence is due to the problem’s challenging nature: image-inpainting algorithms strive for realism in the resulting images, but realism is a subjective concept intrinsic to human perception. Existing objective image-quality metrics provide a poor approximation of what humans consider more or less realistic.
To improve the situation and to better organize both prior and future research in this field, we conducted a subjective comparison of nine state-of-the-art inpainting algorithms and propose objective quality metrics that exhibit high correlation with the results of our comparison.
author:
- |
Ivan Molodetskikh, Mikhail Erofeev, Dmitry Vatolin\
Lomonosov Moscow State University\
Moscow, Russia\
[ivan.molodetskikh@graphics.cs.msu.ru]{}
bibliography:
- 'references.bib'
title: Perceptually Motivated Method for Image Inpainting Comparison
---
{width="\linewidth"}
Introduction
============
Image inpainting, or hole filling, is the task of filling in missing parts of an image. Given an incomplete image and a hole mask, an inpainting algorithm must generate the missing parts so that the result looks realistic. Inpainting is a widely researched topic. Many classical algorithms have been proposed [@telea2004image; @criminisi2004region], but over the past few years most research has focused on using deep neural networks to solve this problem [@Pathak_2016_CVPR; @song2017image; @liu2017semantically; @li2018context; @Yu_2018_CVPR; @Liu_2018_ECCV; @yu2018free].
Naturally, because of the many avenues of research in this field, the need to evaluate algorithms emerges. The specifics of image inpainting mean this problem has no simple solution. The goal of an inpainting algorithm is to make the final image as realistic as possible, but image realism is a concept intrinsic to humans. Therefore, the most accurate way to evaluate an algorithm’s performance is a subjective experiment where many participants compare the outcomes of different algorithms and choose the one they consider the most realistic.
Unfortunately, conducting a subjective experiment involves considerable time and resources, so many authors resort to evaluating their proposed methods using traditional objective image-similarity metrics such as PSNR, SSIM and mean $l_2$ loss relative to the ground-truth image. This strategy, however, is inadequate. One reason is that evaluation by measuring similarity to the ground-truth image assumes that only a single, best inpainting result exists—a false assumption in most cases. As a trivial example, consider that inpainting is frequently used to erase unwanted objects from photographs. Therefore, at least two realistic outcomes are possible: the original photograph with the object and the desired photograph without the object. Furthermore, we show that the popular SSIM image-quality estimation metric correlates poorly with human responses and that the inpainting result can seem more realistic to a human than the original image does, limiting the applicability of full-reference metrics to this task.
Moreover, owing to the lack of a clear and objective way to evaluate inpainting algorithms, different authors present results of different metrics on different data sets when considering their proposed algorithms. Comparing these algorithms is therefore even harder.
Thus, a perceptually motivated objective metric for inpainting-quality assessment is desirable. The objective metric should approximate the notion of image realism and yield results similar to those of a subjective study when comparing outputs from different algorithms.
We conducted a subjective evaluation of nine state-of-the-art classical and deep-learning-based approaches to image inpainting. Using the results, we examine different methods of objective inpainting-quality evaluation, including both full-reference methods (taking both the resulting image and the ground-truth image as an input) and no-reference methods (taking just the resulting image as an input).
It is important to note that we are *not* proposing objective quality-evaluation models trained on a database of subjective scores, as obtaining sufficiently large and diverse databases of this nature is impractical. We do, however, evaluate the proposed models by comparing their correlations to human responses.
Related work
============
Little work has been done on objective image inpainting-quality evaluation or on inpainting detection in general. The somewhat related field of manipulated-image detection has seen moderate research, including both classical and deep-learning-based approaches. This field focuses on detecting altered image regions, usually involving a set of common manipulations: copy-move (copying an image fragment and pasting it elsewhere in the same image), splicing (pasting a fragment from another image), fragment removal (deleting an image fragment and then performing either a copy-move or inpainting to fill in the missing area), various effects such as Gaussian blur and median filtering, and recompression (usually indicating that the image was handled in a photo editor). Among these manipulations, the most interesting for this work is fragment removal with inpainting.
The classical approaches to image-manipulation detection include [@pun2015image; @li2017image], and the deep-learning-based approaches include [@Bappy_2017_ICCV; @zhu2018deep; @salloum2018image; @Zhou_2018_CVPR]. These algorithms aim to locate the manipulated image regions by outputting a mask or a set of bounding boxes enclosing suspicious regions. Unfortunately, they are not directly applicable to inpainting-quality estimation because they have a different goal: whereas an objective quality-estimation metric should strive to accurately compare realistically inpainted images similar to the originals, a forgery-detection algorithm should strive to accurately tell one apart from the other.
Inpainting subjective evaluation {#sec:subjective-evaluation}
================================
The gold standard for evaluating image-inpainting algorithms is human perception, since each algorithm strives to produce images that look the most realistic to humans. Thus, to obtain a baseline for creating an objective inpainting-quality metric, we conducted a subjective evaluation of multiple state-of-the-art algorithms, including both classical and deep-learning-based ones. To assess the overall quality and applicability of the current approaches and to see how they compare with manual photo editing, we also included human-produced images. We asked several professional photo editors to fill in missing regions of the test photos just like an automatic algorithm would.
Test data set
-------------
Since human photo editors were to perform inpainting, our data set excluded publicly available images: we wanted to ensure that finding the original photos online and achieving perfect results would be impossible. We therefore created our own private set of test images by taking photographs of various outdoor scenes, which are the most likely target for inpainting.
Each test image was $512\times512$ pixels with a square hole in the middle measuring $180\times180$ pixels. We chose a square instead of a free-form shape because one algorithm in our comparison [@Yang_2017_CVPR] lacks the ability to fill in free-form holes. The data set comprised 33 images in total. Figure \[fig:subjective-comparison-photos\] shows examples.
Inpainting methods
------------------
[ m[2ex]{} m[0.3]{} m[0.3]{} m[0.3]{} ]{} & & &\
& ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/city_flowers_0/fl_1.png "fig:"){width="\linewidth"} & ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/city_flowers_0/fl_2.jpg "fig:"){width="\linewidth"} & ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/city_flowers_0/fl_3.jpg "fig:"){width="\linewidth"}\
& ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/splashing_sea_0/fl_1.png "fig:"){width="\linewidth"} & ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/splashing_sea_0/fl_2.jpg "fig:"){width="\linewidth"} & ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/splashing_sea_0/fl_3.jpg "fig:"){width="\linewidth"}\
& ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/forest_0/fl_1.png "fig:"){width="\linewidth"} & ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/forest_0/fl_2.jpg "fig:"){width="\linewidth"} & ![Three images from our test set, inpainted by three human artists.[]{data-label="fig:artist-results"}](media/human/forest_0/fl_3.jpg "fig:"){width="\linewidth"}\
We evaluated nine classical and deep-learning-based approaches:
- Exemplar-based inpainting [@criminisi2004region] is a well-known classical algorithm that finds the image patches that are most similar to the region being filled and copies them into that region in a particular order to correctly preserve image structures.
- Statistics of patch offsets [@he2012statistics] is a more recent classical algorithm that employs distributions of statistics for the relative positions of similar image patches.
- Content-aware fill is a tool in the popular photo-editing software Adobe Photoshop. We picked it because, being part of a popular image editor, it is highly likely to be used for inpainting. Thus, comparing it with other state-of-the-art approaches is valuable. We tested the content-aware fill in Adobe Photoshop CS5; its implementation preceded deep learning’s explosion in popularity.
- Deep image prior [@Ulyanov_2018_CVPR] is an unconventional deep-learning-based method. It relies on deep generator networks converging to a realistic image from a random initial state rather than by learning realistic image priors from a large training set.
- Globally and locally consistent image completion [@iizuka2017globally] is a deep-learning-based approach that uses global and local discriminators, thereby improving both the coherence of the resulting image as a whole and the local consistencies of image patches.
- High-resolution image inpainting [@Yang_2017_CVPR], another deep-learning-based method, employs multiscale neural patch synthesis to preserve both contextual structures and high-frequency details in high-resolution images.
- Shift-Net [@Yan_2018_ECCV] is a U-Net architecture that implements a special shift-connection layer to improve the inpainting quality. The shift-connection layer offsets encoder features of known regions to estimate the missing ones.
- Generative image inpainting with contextual attention [@Yu_2018_CVPR] uses a two-part convolutional neural network to first predict the structural information and then restore the fine details. It also has a special contextual-attention layer that finds the most similar patches from the known image areas to aid generation of fine details.
- Image inpainting for irregular holes using partial convolutions [@Liu_2018_ECCV] follows the idea that regular convolutional layers in deep inpainting networks are suboptimal because they unconditionally use both valid (known) and invalid (unknown) pixels from the input image. This method implements a partial convolution layer that masks out the unknown pixels.
Additionally, we hired three professional photo/restoration and photo-retouching artists to manually inpaint three randomly selected images from our test data set. To encourage them to produce the best possible results, we offered a 50% honorarium bonus for the ones that outranked the competitors. Although we imposed no strict time limit, all three artists completed their work within 90 minutes. Figure \[fig:artist-results\] shows their results.
![Subjective-comparison results across three images inpainted by human artists.[]{data-label="fig:human-results-overall"}](media/subjective-comparison-humans-overall.pdf){width="\linewidth"}
{width="\linewidth"}
Test method
-----------
The subjective evaluation took place through the <http://subjectify.us> platform. Human observers were shown pairs of images and asked to pick from each pair the one they found most realistic. Each image pair consisted of two different inpainting results for the same picture (the set also contained the original image). Also included were two verification questions that asked the participants to compare the result of exemplar-based inpainting [@criminisi2004region] with the ground-truth image. The final results excluded all responses from participants who failed to correctly answer one or both verification questions. In total, 6,945 valid pairwise judgements were collected from 215 participants.
The judgements were then used to fit a Bradley-Terry model [@bradley1952rank]. The resulting subjective scores maximize likelihood given the pairwise judgements.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Artist \#1 Statistics of Patch Offsets [@he2012statistics]
![Comparison of inpainting results from Artist \#1 and statistics of patch offsets [@he2012statistics] (preferred in the subjective comparison).[]{data-label="fig:artist-1-patch-shift"}](media/human/city_flowers_0/fl_1.png "fig:"){width="0.49\linewidth"} ![Comparison of inpainting results from Artist \#1 and statistics of patch offsets [@he2012statistics] (preferred in the subjective comparison).[]{data-label="fig:artist-1-patch-shift"}](media/city-flowers-patch-shift-stats.png "fig:"){width="0.49\linewidth"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Results
-------
**Algorithms vs. artists.** Figure \[fig:human-results-overall\] shows the results for the three images inpainted by the human artists. The artists outperformed all state-of-the-art automatic algorithms, and out of the deep-learning-based methods, only generative image inpainting [@Yu_2018_CVPR] trained on the Places 2 data set outperformed the classical inpainting methods.
The individual results for each of these three images appear in Figure \[fig:human-results-individual\]. In only one case did an algorithm beat an artist: statistics of patch offsets [@he2012statistics] scored higher than one artist on the “Urban Flowers” photo. Figure \[fig:artist-1-patch-shift\] shows the respective inpainting results. Additionally, for the “Splashing Sea” photo, two artists actually “outperformed” the original image: their results turned out to be more realistic. This outcome illustrates that the ideal quality-estimation metric should be a no-reference one.
**Algorithms vs. algorithms.** We additionally performed a subjective comparison of various inpainting algorithms among the entire 33-image test set, collecting 3,969 valid pairwise judgements across 147 participants. The overall results appear in Figure \[fig:algorithm-results-overall\]. They confirm our observations from the first comparison: among the deep-learning-based approaches we evaluated, generative image inpainting [@Yu_2018_CVPR] seems to be the only one that can outperform the classical methods.
![Subjective-comparison results for 33 images inpainted using automatic methods.[]{data-label="fig:algorithm-results-overall"}](media/subjective-comparison-algorithms-overall.pdf){width="\linewidth"}
**Conclusion.** The subjective evaluation allows us to draw the following conclusions:
- Manual inpainting by human artists remains the only viable way to achieve quality close to that of the original images.
- Automatic algorithms can approach manual-inpainting results only for certain images.
- Classical inpainting methods continue to maintain a strong position, even though a deep-learning-based algorithm took the lead.
Objective inpainting-quality estimation
=======================================
Using the results we obtained from the subjective comparison, we evaluated several approaches to objective inpainting-quality estimation. Using these objective metrics, we estimated the inpainting quality of the images from our test set and then compared them with the subjective results. For each of the 33 images, we applied every tested metric to every inpainting result (as well as to the ground-truth image) and computed the Pearson and Spearman correlation coefficients with the subjective result. The final value was an average of the correlations over all 33 test images.
Below is an overview of each metric we evaluated.
Full-reference metrics
----------------------
Full-reference metrics take both the ground-truth (original) image and the inpainting result as an input.
To construct a full-reference metric that encourages semantic similarity rather than per-pixel similarity, as in [@johnson2016perceptual], we evaluated metrics that compute the difference between the ground-truth and inpainted-image feature maps produced by an image-classification neural network. We selected five of the most popular deep architectures: VGG [@simonyan2014very] (16- and 19-layer deep variants), ResNet-V1-50 [@He_2016_CVPR], Inception-V3 [@Szegedy_2016_CVPR], Inception-ResNet-V2 [@szegedy2017inception] and Xception [@Chollet_2017_CVPR]. We used the models pretrained on the ImageNet [@deng2009imagenet] data set. The mean squared error between the feature maps was the metric result.
For VGG we tested the output from several deep layers (convolutional and pooling layers from the last block) and found that the deepest layer has the highest correlation with the subjective-study results. For each of the other architectures, therefore, we tested only the deepest layer.
We additionally included the structural-similarity (SSIM) index [@wang2004image] as a full-reference metric. SSIM is widely used to compare image quality, but it falls short when applied to inpainting-quality estimation.
The best correlations among the full-reference metrics emerged when using the last layer of the VGG-16 model.
No-reference metrics
--------------------
No-reference metrics take only the target (possibly inpainted) image as an input, so they apply to a much wider range of problems.
We used a deep-learning approach to constructing no-reference metrics. We picked several popular image-classification neural-network architectures and trained them to differentiate “clean” (realistic, original) images without any inpainting from partially inpainted images. The architectures included VGG [@simonyan2014very] (16- and 19-layer deep), ResNet-V1-50 [@He_2016_CVPR], ResNet-V2-50 [@he2016identity], Inception-V3 [@Szegedy_2016_CVPR], Inception-V4 [@szegedy2017inception] and PNASNet-Large [@Liu_2018_ECCV_pnasnet].
**Data set.** For training, we used clean and inpainted images based on the COCO [@lin2014microsoft] data set. To create the inpainted images, we cropped the input images to a square aspect ratio, resized them to $512\times512$ pixels and masked out a square of $180\times180$ pixels from the middle (the same procedure we used when creating our subjective-evaluation data set). We then inpainted the masked images using five inpainting algorithms [@criminisi2004region; @iizuka2017globally; @he2012statistics; @Yan_2018_ECCV; @Yu_2018_CVPR] in eight total configurations. The total number of images in the data set appears in Table \[tab:training-data set-image-counts\].
------------------------------------------------------- -------
Exemplar-Based (patch size 9) [@criminisi2004region] 16777
Exemplar-Based (patch size 13) [@criminisi2004region] 16777
Globally and Locally Consistent [@iizuka2017globally] 13870
Statistics of Patch Offsets [@he2012statistics] 7955
Shift-Net [@Yan_2018_ECCV] 8928
Generative Inpainting (Places 2) [@Yu_2018_CVPR] 8928
Generative Inpainting (CelebA) [@Yu_2018_CVPR] 8928
Generative Inpainting (ImageNet) [@Yu_2018_CVPR] 8927
Total inpainted 91090
Total original (clean) 81520
------------------------------------------------------- -------
: Total number of images in the training data set by inpainting algorithm.[]{data-label="tab:training-data set-image-counts"}
![Inpainting quality estimated by VGG-16 for one image at the peak Pearson correlation and after further training. The score distribution at the correlation peak is similar to a subjective-score distribution; after further training, however, the network starts to heavily underscore most inpainting algorithms.[]{data-label="fig:overfitting"}](media/overfitting.pdf){width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
**Training.** The network architectures take a square image as an input and output the score—a single number where 0 means the image contains inpainted regions and 1 means the image is “clean.” The loss function was mean squared error. Some network architectures were additionally trained to output the predicted class using one-hot encoding (similar to binary classification); the loss function for this case was softmax cross-entropy.
The network architectures were identical to the ones used for image classification, with one difference: we altered the number of outputs from the last fully connected layer. This change allowed us to initialize the weights of all previous layers from the models pretrained on ImageNet, greatly improving the results compared with training from random initialization.
For some experiments we additionally tried using the RGB noise features described in [@Zhou_2018_CVPR] and the spectral weight normalization described in [@miyato2018spectral].
In addition to the typical validation on part of the data set, we also monitored correlation of network predictions with the subjective scores collected in Section \[sec:subjective-evaluation\]. We used the networks to estimate the inpainting quality of the 33-image test set, then computed correlations with subjective results in the same way as the final comparison. The training of each network was stopped once the correlation of the network predictions with the subjective scores peaked and started to decrease (possibly because the networks were overfitting to the inpainting results of the algorithms we used to create the training data set). Figure \[fig:overfitting\] compares the network scores for one test image at the correlation peak and after further training.
Results
-------
We evaluated several objective quality-estimation approaches, both full reference and no reference as well as both classical and deep-learning based. Figure \[fig:correlations\] shows the overall results. We additionally did a comparison that excluded ground-truth scores from the correlation, because the full-reference approaches always yield the highest score when comparing the ground-truth image with itself. The overall results for that comparison appear in Figure \[fig:correlations-no-gt\].
As Figures \[fig:correlations\] and \[fig:correlations-no-gt\] show, the no-reference methods achieve slightly weaker correlation with the subjective-evaluation responses than do the best full-reference methods. But the results of most no-reference methods are still considerably better than those of the full-reference SSIM. The best correlations among the no-reference methods came from the Inception-V4 model trained to output one score, with spectral weight normalization.
It is important to emphasize that we did *not* train the networks to maximize correlation with human responses or with the subjective-evaluation data set in general. We trained them to distinguish “clean” images from inpainted images, yet their output showed good correlation with human responses.
Conclusion
==========
We have proposed a number of perceptually motivated no-reference and full-reference objective metrics for evaluating image-inpainting quality. We evaluated the metrics by correlating them with human responses from a subjective comparison of state-of-the-art image-inpainting algorithms.
The results of the subjective comparison indicate that although a deep-learning-based approach to image inpainting holds the lead, classical algorithms remain among the best in the field.
The highest mean Pearson correlation with the human responses from the subjective study was achieved by the block5\_conv3 layer of the VGG-16 model trained on ImageNet (full reference) and by the Inception-V4 model with spectral normalization (no reference).
We achieved good correlation with the subjective/comparison results without specifically training our proposed objective quality-evaluation metrics on the subjective-comparison response data set.
|
---
abstract: 'Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maaß cusp forms of weight $0$ or $1$ for the congruence subgroups $\Gamma_0(q)$, $\Gamma_1(q)$, and $\Gamma(q)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.'
address: 'Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA'
author:
- Peter Humphries
title: Density Theorems for Exceptional Eigenvalues for Congruence Subgroups
---
Introduction
============
Let $\kappa \in \{0,1\}$, let $\Gamma$ be a congruence subgroup of ${\mathrm{SL}}_2({\mathbb{Z}})$, and let $\chi$ be a congruence character of $\Gamma$ satisfying $\chi(-I) = (-1)^{\kappa}$ should $-I$ be a member of $\Gamma$. Denote by ${\mathcal{A}}_{\kappa}(\Gamma,\chi)$ the space spanned by Maaß cusp forms of weight $\kappa$, level $\Gamma$, and nebentypus $\chi$, namely the $L^2$-closure of the space of smooth functions $f : {\mathbb{H}}\to {\mathbb{C}}$ satisfying
- $f(\gamma z) = \chi(\gamma) j_{\gamma}(z)^{\kappa} f(z)$ for all $\gamma \in \Gamma$ and $z \in {\mathbb{H}}$, where for $\gamma = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \Gamma$, $$j_{\gamma}(z) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{cz + d}{|cz + d|},$$
- $f$ is an eigenfunction of the weight $\kappa$ Laplacian $$\Delta_{\kappa} {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}-y^2 \left(\frac{{\partial}^2}{{\partial}x^2} + \frac{{\partial}^2}{{\partial}y^2}\right) + i\kappa y \frac{{\partial}}{{\partial}x},$$
- $f$ is of moderate growth, and
- the constant term is zero in the Fourier expansion of $f$ at every cusp ${\mathfrak{a}}$ of $\Gamma \backslash {\mathbb{H}}$ that is singular with respect to $\chi$.
We may choose a basis ${\mathcal{B}}_{\kappa}(\Gamma,\chi)$ of the complex vector space ${\mathcal{A}}_{\kappa}(\Gamma,\chi)$ consisting of Hecke eigenforms. For $f \in {\mathcal{B}}_{\kappa}(\Gamma,\chi)$, we let $\lambda_f = 1/4 + t_f^2$ denote the eigenvalue of the weight $\kappa$ Laplacian, where either $t_f \in [0,\infty)$ or $it_f \in (0,1/2)$. Similarly, we let $\lambda_f(p)$ denote the eigenvalue of the Hecke operator $T_p$ at a prime $p$, so that $|\lambda_f(p)| < p^{1/2} + p^{-1/2}$. The generalised Ramanujan conjecture states that $t_f$ is real and that $|\lambda_f(p)| \leq 2$ for every prime $p$. Exceptions to this conjecture are called exceptional eigenvalues. It is known that exceptional Laplacian eigenvalues cannot occur if $\kappa = 1$, while for $\kappa = 0$ there are no exceptional Laplacian eigenvalues for Maaß cusp forms of squarefree conductor less than $857$ [@BS Theorem 1]. The best current bounds towards the generalised Ramanujan conjecture are due to Kim and Sarnak [@Kim Proposition 2 of Appendix 2]; they show that $$\lambda_f \geq \frac{1}{4} - \left(\frac{7}{64}\right)^2, \qquad \left|\lambda_f(p)\right| \leq p^{7/64} + p^{-7/64}.$$
Results
-------
In this paper, we use the Kuznetsov formula to prove density results for exceptional eigenvalues for the congruence subgroups $$\begin{aligned}
\Gamma_0(q) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\mathrm{SL}}_2({\mathbb{Z}}) : c \equiv 0 \hspace{-.2cm} \pmod{q}\right\}, \\
\Gamma_1(q) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\mathrm{SL}}_2({\mathbb{Z}}) : a,d \equiv 1 \hspace{-.2cm} \pmod{q}, \ c \equiv 0 \hspace{-.2cm} \pmod{q}\right\}, \\
\Gamma(q) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\mathrm{SL}}_2({\mathbb{Z}}) : a,d \equiv 1 \hspace{-.2cm} \pmod{q}, \ b,c \equiv 0 \hspace{-.2cm} \pmod{q}\right\},\end{aligned}$$ with $\chi$ equal to the trivial character for the latter two congruence subgroups. Recall that $$\operatorname{vol}(\Gamma \backslash {\mathbb{H}}) = \frac{\pi}{3} \left[{\mathrm{SL}}_2({\mathbb{Z}}) : \Gamma\right] = \begin{dcases*}
\frac{\pi}{3} q \prod_{p \mid q} \left(1 + \frac{1}{p}\right) & if $\Gamma = \Gamma_0(q)$, \\
\frac{\pi}{3} q^2 \prod_{p \mid q} \left(1 - \frac{1}{p^2}\right) & if $\Gamma = \Gamma_1(q)$, \\
\frac{\pi}{3} q^3 \prod_{p \mid q} \left(1 - \frac{1}{p^2}\right) & if $\Gamma = \Gamma(q)$.
\end{dcases*}$$ When $\chi$ is the trivial character, we write ${\mathcal{B}}_{\kappa}(\Gamma)$ in place of ${\mathcal{B}}_{\kappa}(\Gamma,\chi)$, while when $\Gamma = \Gamma_0(q)$, we write this as ${\mathcal{B}}_{\kappa}(q,\chi)$. Given positive integers $q$ and $q_{\chi}$ with $q_{\chi} \mid q$, we factorise $q = \prod_{p^{\alpha} \parallel q} p^{\alpha}$ and $q_{\chi} = \prod_{p^{\gamma} \parallel q_{\chi}} p^{\gamma}$, and define $$\dot{Q} = \dot{Q}(q,q_{\chi}) = \prod_{\substack{p^{\alpha} \parallel q \\ p^{\gamma} \parallel q_{\chi}}} \dot{Q}(p^{\alpha},p^{\gamma}), \qquad \ddot{Q} = \ddot{Q}(q,q_{\chi}) = \prod_{\substack{p^{\alpha} \parallel q \\ p^{\gamma} \parallel q_{\chi}}} \ddot{Q}(p^{\alpha},p^{\gamma})$$ with $$\begin{aligned}
\dot{Q}(p^{\alpha},p^{\gamma}) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\begin{dcases*}
p^{\lfloor \frac{3\alpha + 1}{4} \rfloor - \frac{\alpha}{2}} & if $p$ is odd and $\alpha = \gamma \geq 3$, \\
2^{\lfloor \frac{3\alpha + 1}{4} \rfloor - \frac{\alpha}{2}} & if $p = 2$ and $\gamma + 1 \geq \alpha \geq \geq 3$, \\
1 & otherwise,
\end{dcases*} \\
\ddot{Q}(p^{\alpha},p^{\gamma}) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\begin{dcases*}
p & if $p$ is odd and $\alpha = \gamma \geq 3$, \\
4 & if $p = 2$ and $\alpha = \gamma \geq 3$, \\
2 & if $p = 2$ and $\alpha = \gamma + 1 \geq 3$, \\
1 & otherwise.
\end{dcases*}\end{aligned}$$
\[Sarnakthm\] For any fixed finite collection of primes ${\mathcal{P}}$ not dividing $q$, any $\alpha_p \in (2, p^{1/2} + p^{-1/2})$ and $0 \leq \mu_p \leq 1$ for all $p \in {\mathcal{P}}$ with $\sum_{p \in {\mathcal{P}}} \mu_p = 1$, we have that $$\begin{gathered}
\label{SarnakGamma1}
\#\left\{f \in {\mathcal{B}}_{\kappa}(\Gamma_1(q)) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma_1(q) \backslash {\mathbb{H}})^{1 - 3 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \left(T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e},\end{gathered}$$ $$\begin{gathered}
\label{SarnakGamma}
\#\left\{f \in {\mathcal{B}}_{\kappa}(\Gamma(q)) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma(q) \backslash {\mathbb{H}})^{1 - \frac{8}{3} \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \left(T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e},\end{gathered}$$ $$\begin{gathered}
\label{SarnakGamma0}
\#\left\{f \in {\mathcal{B}}_{\kappa}(q,\chi) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \left(T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \\
\times \min\left\{\dot{Q}^{4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}}, \ddot{Q}^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}}\right\}.\end{gathered}$$
should be compared to the Weyl law, which states that $$\#\left\{f \in {\mathcal{B}}_{\kappa}(\Gamma,\chi) : t_f \in [0,T]\right\} \sim \frac{\operatorname{vol}\left(\Gamma \backslash {\mathbb{H}}\right)}{4\pi} T^2.$$
For $\Gamma = {\mathrm{SL}}_2({\mathbb{Z}})$, so that $\chi$ is the trivial character, and ${\mathcal{P}}$ consisting of a single prime $p$, is a result of Blomer, Buttcane, and Raulf [@BBR Proposition 1], improving on a slightly weaker result of Sarnak [@Sarnak Theorem 1.1], who uses the Selberg trace formula in place of the Kuznetsov formula and obtains instead (see [@BBR Footnote 1]) $$\#\left\{f \in {\mathcal{B}}_0\left({\mathrm{SL}}_2({\mathbb{Z}})\right) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha\right\} \ll \left(T^2\right)^{1 - 2 \frac{\log \alpha / 2}{\log p}}.$$
\[Huxleythm\] For any fixed finite (possibly empty) collection of primes ${\mathcal{P}}$ not dividing $q$, any $\alpha_0 \in (0,1/2)$, $\alpha_p \in (2, p^{1/2} + p^{-1/2})$, and $0 \leq \mu_0,\mu_p \leq 1$ for all $p \in {\mathcal{P}}$ with $\mu_0 + \sum_{p \in {\mathcal{P}}} \mu_p = 1$, we have that $$\begin{gathered}
\label{HuxleyGamma1}
\#\left\{f \in {\mathcal{B}}_0(\Gamma_1(q)) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma_1(q) \backslash {\mathbb{H}})^{1 - 3 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right) + \e}\end{gathered}$$ $$\begin{gathered}
\label{HuxleyGamma}
\#\left\{f \in {\mathcal{B}}_0(\Gamma(q)) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma(q) \backslash {\mathbb{H}})^{1 - \frac{8}{3} \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right) + \e}.\end{gathered}$$ $$\begin{gathered}
\label{HuxleyGamma0}
\#\left\{f \in {\mathcal{B}}_0(q,\chi) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{1 - 4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right) + \e} \\
\times \min\left\{\dot{Q}^{4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right)}, \ddot{Q}^{1 - 4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right)}\right\}.\end{gathered}$$
When ${\mathcal{P}}$ is empty and $\chi$ is the trivial congruence character, improves upon a result of Huxley [@Hux], who uses the Selberg trace formula in place of the Kuznetsov formula and obtains instead this result with the exponent $2$ for each of the three congruence subgroups instead of $3$, $8/3$, and $4$ respectively. When ${\mathcal{P}}$ is empty and $\chi$ is the trivial congruence character, is a result of Iwaniec [@Iwa Theorem 11.7] ; see also [@IK (16.61)].
Since $$\lfloor \frac{3\alpha + 1}{4} \rfloor - \frac{\alpha}{2} \leq \frac{3 \alpha}{10},$$ so that $\dot{Q} \ll \operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{3/10}$, the right-hand side of is bounded by $$\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{1 - \frac{14}{5} \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \left(T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e},$$ while the right-hand side of is bounded by $$\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{1 - \frac{14}{5} \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right) + \e}.$$ On the other hand, taking ${\mathcal{P}}$ to consist of a single prime in recovers the Selberg bound $\lambda_f(p) \ll_{\e} p^{1/4 + \e}$ for an individual element $f \in {\mathcal{B}}_{\kappa}(q,\chi)$ by taking $T$ sufficiently large, while taking ${\mathcal{P}}$ to be empty in recovers the Selberg bound $\lambda_f \geq 3/16$ by embedding $f$ in ${\mathcal{B}}_{\kappa}(qQ,\chi)$ and taking $Q$ sufficiently large.
Idea of Proof
-------------
By Rankin’s trick, it suffices to find bounds for $$\sum_{\substack{f \in {\mathcal{B}}_{\kappa}\left(\Gamma, \chi\right) \\ t_f \in [0,T]}} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}, \qquad \sum_{\substack{f \in {\mathcal{B}}_0\left(\Gamma, \chi\right) \\ it_f \in (0,1/2)}} X^{2it_f} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}$$ for nonnegative integers $\ell_p$ and a positive real number $X \geq 1$ to be chosen. To bound these quantities, we begin with Kuznetsov formula for ${\mathcal{B}}_{\kappa}(q,\chi)$; we then use the Atkin–Lehner decomposition to turn this into a Kuznetsov formula for ${\mathcal{B}}_{\kappa}(\Gamma,\chi)$. We take a test function in the Kuznetsov formula that localises the spectral sum to cusp forms with $t_f \in [0,T]$ in the case of and to cusp forms with $it_f \in (0,1/2)$ in the case of . We use the Hecke relations to introduce powers of the Hecke eigenvalues into the Kuznetsov formula. By positivity, we discard the contribution of the continuous spectrum, and we are left with bounding the right-hand side of the Kuznetsov formula.
The chief novelty of the proof is the bounds for sum of Kloosterman sums in the Kuznetsov formula for each congruence subgroup. As well as the usual Weil bound, we use character orthogonality for $\Gamma_1(q)$ and $\Gamma(q)$, at which point we only use the trivial bound for the resulting sum of Kloosterman sums. For $\Gamma_0(q)$ and $\chi$ the principal character, we may also use the Weil bound, but for $\chi$ nonprincipal, additional difficulties arise in bounding the Kloosterman sum, with the bound possibly depending on the conductor of $\chi$; it is for this reason that the bounds and involve $\dot{Q}$, for $\dot{Q}$ arises when only weaker bounds than the Weil bound are possible for the Kloosterman sums involved.
It is worth mentioning that the results in this paper ought to generalise naturally to cusp forms on ${\mathrm{GL}}_2$ over arbitrary number fields $F$. In [@BrMia], Bruggeman and Miatello prove a form of the Kuznetsov formula for ${\mathrm{GL}}_2$ over a totally real field and use this to prove weighted Weyl law for cusp forms. Similarly, in [@Mag], Maga proves a semi-adèlic version of the Kuznetsov formula for ${\mathrm{GL}}_2$ over an arbitrary number field. In the former case, this formula is valid for congruence subgroups of the form $\Gamma_0({\mathfrak{q}})$ for a nonzero integral ideal ${\mathfrak{q}}$ of the ring of integers ${\mathcal{O}}_F$ of $F$ and arbitrary congruence characters $\chi$ modulo ${\mathfrak{q}}$, while the latter only treats the case of trivial congruence character but should easily be able to be generalised to arbitrary congruence character; this is precisely what is required for density theorems for the congruence subgroups $\Gamma_0({\mathfrak{q}})$, $\Gamma_1({\mathfrak{q}})$, and $\Gamma({\mathfrak{q}})$.
The Kuznetsov Formula
=====================
The background on automorphic forms and notation in this section largely follows [@DFI]; see [@DFI Section 4] for more details. Let $\kappa \in \{0,1\}$, and let $\chi$ be a primitive Dirichlet character modulo $q_{\chi}$, where $q_{\chi}$ divides $q$, satisfying $\chi(-1) = (-1)^{\kappa}$; this defines a congruence character of $\Gamma_0(q)$ via $\chi(\gamma) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\chi(d)$ for $\gamma = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \Gamma_0(q)$. We denote by $L^2\left(\Gamma_0(q) \backslash {\mathbb{H}},\kappa,\chi\right)$ the $L^2$-completion of the space of all smooth functions $f : {\mathbb{H}}\to {\mathbb{C}}$ that are of moderate growth and satisfy $f(\gamma z) = \chi(\gamma) j_{\gamma}(z)^{\kappa} f(z)$. This space has the spectral decomposition $$L^2\left(\Gamma_0(q) \backslash {\mathbb{H}},\kappa,\chi\right) = {\mathcal{A}}_{\kappa}(q,\chi) \oplus {\mathcal{E}}_{\kappa}(q,\chi)$$ with respect to the weight $\kappa$ Laplacian, where ${\mathcal{A}}_{\kappa}(q,\chi) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}{\mathcal{A}}_{\kappa}\left(\Gamma_0(q),\chi\right)$ is the space spanned by Maaß cusp forms of weight $\kappa$, level $q$, and nebentypus $\chi$, and ${\mathcal{E}}_{\kappa}(q,\chi)$ is the space spanned by incomplete Eisenstein series parametrised by the cusps ${\mathfrak{a}}$ of $\Gamma_0(q) \backslash {\mathbb{H}}$ that are singular with respect to $\chi$.
We denote by ${\mathcal{B}}_{\kappa}(q,\chi)$ an orthonormal basis of Maaß cusp forms $f \in {\mathcal{A}}_{\kappa}(q,\chi)$ normalised to have $L^2$-norm $1$: $$\langle f, f \rangle_q {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\int_{\Gamma_0(q) \backslash {\mathbb{H}}} |f(z)|^2 \, d\mu(z) = 1,$$ where $d\mu(z) = \dfrac{dx \, dy}{y^2}$ is the ${\mathrm{SL}}_2({\mathbb{R}})$-invariant measure on ${\mathbb{H}}$. Later we will use the Atkin–Lehner decomposition of ${\mathcal{A}}_{\kappa}(q,\chi)$ in order to specify that ${\mathcal{B}}_{\kappa}(q,\chi)$ can be chosen to consist of linear combinations of Hecke eigenforms. The Fourier expansion of $f \in {\mathcal{B}}_{\kappa}(q,\chi)$ is $$f(z) = \sum_{\substack{n = -\infty \\ n \neq 0}}^{\infty} \rho_f(n) W_{\operatorname{sgn}(n) \frac{\kappa}{2}, it_f}(4\pi|n|y) e(nx),$$ where $W_{\alpha,\beta}$ is the Whittaker function and $$\rho_f(n) W_{\operatorname{sgn}(n) \frac{\kappa}{2}, it_f}(4\pi|n|y) = \int_{0}^{1} f(z) e(-nx) \, dx.$$
For a singular cusp ${\mathfrak{a}}$, we define the Eisenstein series $$E_{{\mathfrak{a}}}(z,s,\chi) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\sum_{\gamma \in \Gamma_{{\mathfrak{a}}} \backslash \Gamma_0(q)} \overline{\chi}(\gamma) j_{\sigma_{{\mathfrak{a}}}^{-1} \gamma}(z)^{-\kappa} \Im\left(\sigma_{{\mathfrak{a}}}^{-1} \gamma z\right)^s,$$ which is absolutely convergent for $\Re(s) > 1$ and extends meromorphically to ${\mathbb{C}}$, with the Fourier expansion $$\delta_{{\mathfrak{a}},\infty} y^{1/2 + it} + \varphi_{{\mathfrak{a}},\infty}\left(\frac{1}{2} + it,\chi\right) y^{1/2 - it} + \sum_{\substack{n = -\infty \\ n \neq 0}}^{\infty} \rho_{{\mathfrak{a}}}(n,t,\chi) W_{\operatorname{sgn}(n) \frac{\kappa}{2}, it}(4\pi|n|y) e(nx)$$ for $s = 1/2 + it$ with $t \in {\mathbb{R}}\setminus \{0\}$, where $$\begin{aligned}
\delta_{{\mathfrak{a}},\infty} y^{1/2 + it} + \varphi_{{\mathfrak{a}},\infty}\left(\frac{1}{2} + it,\chi\right) y^{1/2 - it} & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\int_{0}^{1} E_{{\mathfrak{a}}}\left(z, \frac{1}{2} + it, \chi\right) \, dx, \\
\rho_{{\mathfrak{a}}}(n,t,\chi) W_{\operatorname{sgn}(n) \frac{\kappa}{2}, it}(4\pi|n|y) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\int_{0}^{1} E_{{\mathfrak{a}}}\left(z, \frac{1}{2} + it, \chi\right) e(-nx) \, dx.\end{aligned}$$ The subspace ${\mathcal{E}}_{\kappa}(q,\chi)$ consists of functions $g \in L^2\left(\Gamma_0(q) \backslash {\mathbb{H}}, \kappa, \chi\right)$ that are orthogonal to every Maaß cusp form $f \in {\mathcal{A}}_{\kappa}(q,\chi)$; it is the $L^2$-closure of the space spanned by incomplete Eisenstein series, which are functions of the form $$\label{incompleteEiseneq}
E_{{\mathfrak{a}}}(z,\psi,\chi) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{1}{2\pi i} \int_{\sigma - i\infty}^{\sigma + i\infty} E_{{\mathfrak{a}}}(z,s,\chi) \widehat{\psi}(s) \, ds$$ for some singular cusp ${\mathfrak{a}}$ and some smooth function of compact support $\psi : {\mathbb{R}}^+ \to {\mathbb{C}}$, where $\sigma > 1$ and $$\widehat{\psi}(s) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\int_{0}^{\infty} \psi(x) x^{-s} \, \frac{dx}{x}.$$
For $m,n \geq 1$ and $r \in {\mathbb{R}}$, $$\begin{gathered}
\sum_{f \in {\mathcal{B}}_{\kappa}(q,\chi)} \frac{4\pi \sqrt{mn} \overline{\rho_f}(m) \rho_f(n)}{\cosh \pi (r - t_f) \cosh \pi (r + t_f)} + \sum_{{\mathfrak{a}}} \int_{-\infty}^{\infty} \frac{\sqrt{mn} \overline{\rho_{{\mathfrak{a}}}}(m,t,\chi) \rho_{{\mathfrak{a}}}(n,t,\chi)}{\cosh \pi (r - t) \cosh \pi (r + t)} \, dt \\
= \frac{\left|\Gamma\left(1 - \frac{\kappa}{2} - ir\right)\right|^2}{\pi^2} \left(\delta_{m,n} + \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{S_{\chi}(m,n;c)}{c} I_{\kappa}\left(\frac{4\pi \sqrt{mn}}{c},r\right)\right),\end{gathered}$$ where $$\begin{aligned}
S_{\chi}(m,n;c) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\sum_{d \in ({\mathbb{Z}}/c{\mathbb{Z}})^{\times}} \chi(d) e\left(\frac{md + n\overline{d}}{c}\right), \\
I_{\kappa}(t,r) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}-2t \int_{-i}^{i} (-i\zeta)^{\kappa - 1} K_{2ir}(\zeta t) \, d\zeta,\end{aligned}$$ with the latter integral being over the semicircle $|z| = 1$, $\Re(z) > 0$.
By the reflection formula for the gamma function, we have that for $r \in {\mathbb{R}}$, $$\left|\Gamma\left(1 - \frac{\kappa}{2} - ir\right)\right|^2 = \begin{dcases*}
\frac{\pi r}{\sinh \pi r} & if $\kappa = 0$, \\
\frac{\pi}{\cosh \pi r} & if $\kappa = 1$.
\end{dcases*}$$
Given a sufficiently well-behaved function $h$, we may multiply both sides of the pre-Kuznetsov formula for $\kappa = 0$ by $$\frac{1}{2} \left(h\left(r + \frac{i}{2}\right) + h\left(r - \frac{i}{2}\right)\right) \cosh \pi r$$ and then integrate both sides from $-\infty$ to $\infty$ with respect to $r$. This yields the following Kuznetsov formula.
Let $\delta > 0$, and let $h$ be a function that is even, holomorphic in the vertical strip $|\Im(t)| \leq 1/2 + \delta$, and satisfies $h(t) \ll (|t| + 1)^{-2 - \delta}$. Then $$\begin{gathered}
\sum_{f \in {\mathcal{B}}_0(q,\chi)} 4\pi \sqrt{mn} \overline{\rho_f}(m) \rho_f(n) \frac{h(t_f)}{\cosh \pi t_f} \\
+ \sum_{{\mathfrak{a}}} \int_{-\infty}^{\infty} \sqrt{mn} \overline{\rho_{{\mathfrak{a}}}}(m,t,\chi) \rho_{{\mathfrak{a}}}(n,t,\chi) \frac{h(t)}{\cosh \pi t} \, dt \\
= \delta_{mn} g_0 + \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{S_{\chi}(m,n;c)}{c} g_0\left(\frac{4\pi \sqrt{mn}}{c}\right),\end{gathered}$$ where $$\begin{aligned}
g_0 & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{1}{\pi} \int_{-\infty}^{\infty} r h(r) \tanh \pi r \, dr, \\
g_0(x) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}2i \int_{-\infty}^{\infty} J_{2ir}(x) \frac{r h(r)}{\cosh \pi r} \, dr.\end{aligned}$$
The left-hand side of the Kuznetsov formula is called the spectral side; the first term is the contribution from the discrete spectrum, while the second term is the contribution from the continuous spectrum. The right-hand side of the Kuznetsov formula is called the geometric side; the first term is the delta term and the second term is the Kloosterman term.
Decomposition of Spaces of Modular Forms
========================================
Eisenstein Series and Hecke Operators
-------------------------------------
The space ${\mathcal{E}}_{\kappa}(q,\chi)$ is spanned by incomplete Eisenstein series of the form , which are obtained by integrating test functions against Eisenstein series indexed by singular cusps ${\mathfrak{a}}$; in this sense, the Eisenstein series $E_{{\mathfrak{a}}}(z,s,\chi)$ are a spanning set for ${\mathcal{E}}_{\kappa}(q,\chi)$. We may instead choose a different spanning set of Eisenstein series for ${\mathcal{E}}_{\kappa}(q,\chi)$; in place of the set of Eisenstein series $E_{{\mathfrak{a}}}(z,s,\chi)$ with ${\mathfrak{a}}$ a singular cusp, we may instead choose a spanning set of Eisenstein series of the form $E(z,s,f)$ with Fourier expansion $$c_{1,f}(t) y^{1/2 + it} + c_{2,f}(t) y^{1/2 - it} + \sum_{\substack{n = -\infty \\ n \neq 0}}^{\infty} \rho_f(n,t,\chi) W_{\operatorname{sgn}(n) \frac{\kappa}{2}, it}(4\pi|n|y) e(nx)$$ for $s = 1/2 + it$ with $t \in {\mathbb{R}}\setminus \{0\}$, where ${\mathcal{B}}(\chi_1,\chi_2) \ni f$ with $\chi_1 \chi_2 = \chi$ is some finite set depending on $\chi_1,\chi_2$ corresponding to an orthonormal basis in the space of the induced representation constructed out of the pair $(\chi_1,\chi_2)$; see [@BHM Section 2.1.1] or [@KL Chapter 5]. For our purposes, we need not be more specific about ${\mathcal{B}}(\chi_1,\chi_2)$, other than noting that for each $f \in {\mathcal{B}}(\chi_1,\chi_2)$, the Eisenstein series $E(z,1/2 + it,f)$ is an eigenfunction of the Hecke operators $T_n$ for $(n,q) = 1$ with Hecke eigenvalues $$\lambda_f(n,t) = \sum_{ab = n} \chi_1(a) a^{it} \chi_2(b) b^{-it},$$ where for $g : {\mathbb{H}}\to {\mathbb{C}}$ a periodic function of period one, $$(T_n g)(z) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{1}{\sqrt{n}} \sum_{ad = n} \chi(a) \sum_{b \hspace{-.25cm} \pmod{d}} g\left(\frac{az + b}{d}\right).$$ So for $f \in {\mathcal{B}}(\chi_1,\chi_2)$, $$\begin{aligned}
\lambda_f(m,t) \lambda_f(n,t) & = \sum_{d \mid (m,n)} \chi(d) \lambda_f\left(\frac{mn}{d^2},t\right), \label{Eisensteinmult}\\
\overline{\lambda_f}(n,t) & = \overline{\chi}(n) \lambda_f(n,t), \label{Eisensteinconj}\\
\rho_f(1,t) \lambda_f(n) & = \sqrt{n} \rho_f(n,t) \label{Eisensteinrholambda}\end{aligned}$$ whenever $m,n \geq 1$ with $(mn,q) = 1$ and $s = 1/2 + it$.
For any prime $p \nmid q$ and positive integer $\ell$, we have that $$\label{EisensteinHecke2ell}
\left|\lambda_f(p,t)\right|^{2\ell} = \sum_{j = 0}^{\ell} \alpha_{2j,2\ell} \overline{\chi}(p)^j \lambda_f\left(p^{2j},t\right)$$ for any $f \in {\mathcal{B}}(\chi_1,\chi_2)$ and $s = 1/2 + it$, where $$\label{Lobbnumber}
\alpha_{2j,2\ell} = \frac{2j + 1}{\ell + j + 1} \binom{2\ell}{\ell + j} = \begin{dcases*}
\binom{2\ell}{\ell - j} - \binom{2\ell}{\ell - j - 1} & if $0 \leq j \leq \ell - 1$, \\
1 & if $j = \ell$,
\end{dcases*}$$ so that each $\alpha_{2j,2\ell}$ is positive and satisfies $$\label{sumalphabound}
\sum_{j = 0}^{\ell} \alpha_{2j,2\ell} = \binom{2\ell}{\ell} \leq 2^{2\ell}.$$
That follows from is clear. For , we have that $$\overline{\chi}(p)^{j/2} \lambda_f\left(p^j,t\right) = U_j\left(\frac{\overline{\chi}(p)^{1/2} \lambda_f(p,t)}{2}\right),$$ where $U_j$ is the $j$-th Chebyshev polynomial of the second kind, because $U_j$ satisfies $U_0(x/2) = 1$, $U_1(x/2) = x$, and the recurrence relation $$U_{j + 1}\left(\frac{x}{2}\right) = x U_j\left(\frac{x}{2}\right) - U_{j - 1}\left(\frac{x}{2}\right)$$ for all $j \geq 1$, and $\overline{\chi}(p)^{j/2} \lambda_f\left(p^j,t\right)$ satisfies the same recurrence relation from . Since $$\frac{2}{\pi} \int_{-1}^{1} U_j(x) U_k(x) \sqrt{1 - x^2} \, dx = \delta_{j,k},$$ we have that $$x^{2\ell} = \sum_{j = 0}^{2\ell} \alpha_{j,2\ell} U_j\left(\frac{x}{2}\right),$$ where $$\alpha_{j,2\ell} = \frac{2^{2\ell + 1}}{\pi} \int_{-1}^{1} x^{2\ell} U_j(x) \sqrt{1 - x^2} \, dx.$$ This vanishes if $j$ is odd as $U_j(-x) = (-1)^j U_j(x)$, while for $j$ even we have the identity from [@GR 7.311.2]. Combined with , this proves .
Atkin–Lehner Decomposition for 04000(q)
---------------------------------------
Similarly, we may choose a basis of ${\mathcal{A}}_{\kappa}(q,\chi)$ consisting of linear combinations of Hecke eigenforms. Let ${\mathcal{B}}_{\kappa}^{\ast}(q,\chi)$ denote the set of newforms of weight $\kappa$, level $q$, and nebentypus $\chi$, and let ${\mathcal{A}}_{\kappa}^{\ast}(q,\chi)$ denote the subspace of ${\mathcal{A}}_{\kappa}(q,\chi)$ spanned by such newforms. Recall that a newform $f \in {\mathcal{B}}_{\kappa}^{\ast}(q,\chi)$ is an eigenfunction of the weight $\kappa$ Laplacian $\Delta_{\kappa}$ with eigenvalue $1/4 + t_f^2$ and of every Hecke operator $T_n$, $n \geq 1$, with eigenvalue $\lambda_f(n)$, as well as the operator $Q_{1/2 + it_f,\kappa}$ as defined in [@DFI Section 4], with eigenvalue $\epsilon_f \in \{-1,1\}$; we say that $f$ is even if $\epsilon_f = 1$ and $f$ is odd if $\epsilon_f = -1$. In particular, $$\begin{aligned}
\lambda_f(m) \lambda_f(n) & = \sum_{\substack{d \mid (m,n) \\ (d,q) = 1}} \chi(d) \lambda_f\left(\frac{mn}{d^2}\right), \label{cuspmult}\\
\rho_f(1) \lambda_f(n) & = \sqrt{n} \rho_f(n) \label{cusprholambda}\end{aligned}$$ whenever $m,n \geq 1$, and $$\label{cuspconj}
\overline{\lambda_f}(n) = \overline{\chi}(n) \lambda_f(n)$$ for $n \geq 1$ with $(n,q) = 1$. Using and , we have the following.
For any prime $p \nmid q$ and positive integer $\ell$, we have that $$\label{cuspHecke2ell}
\left|\lambda_f(p)\right|^{2\ell} = \sum_{j = 0}^{\ell} \alpha_{2j,2\ell} \overline{\chi}(p)^j \lambda_f\left(p^{2j}\right)$$ for any $f \in {\mathcal{B}}_{\kappa}^{\ast}(q,\chi)$, where once again $\alpha_{2j,2\ell}$ is given by .
The Atkin–Lehner decomposition states that $${\mathcal{A}}_{\kappa}(q,\chi) = \bigoplus_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \bigoplus_{f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)} \bigoplus_{d \mid q_2} {\mathbb{C}}\cdot \iota_{d,q_1,q} f,$$ where $\iota_{d,q_1,q} : {\mathcal{A}}_{\kappa}\left(q_1,\chi\right) \to {\mathcal{A}}_{\kappa}(q,\chi)$ is the map $\iota_{d,q_1,q} f(z) = f(dz)$. The map $\iota_{d,q_1,q}$ commutes with the weight $k$ Laplacian $\Delta_{\kappa}$ and the Hecke operators $T_n$ whenever $n \geq 1$ and $(n,q) = 1$. It follows that if $g = \iota_{d,q_1,q} f$ for some $f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right)$, then $t_g = t_f$ and $\lambda_g(n) = \lambda_f(n)$ whenever $n \geq 1$ and $(n,q) = 1$. Note, however, that $\rho_g(1) = 0$ unless $d = 1$, in which case $\rho_g(1) = \rho_f(1)$.
Unfortunately, the inner Atkin–Lehner decomposition $$\bigoplus_{d \mid q_2} {\mathbb{C}}\cdot \iota_{d,q_1,q} f$$ is not an orthogonal decomposition. Nonetheless, one may make use of this decomposition in determining an orthonormal basis of ${\mathcal{A}}_{\kappa}(q,\chi)$. For squarefree $q$ and principal nebentypus, this is a result of Iwaniec, Luo, and Sarnak [@ILS Lemma 2.4], while Blomer and Milićević have generalised this to nonsquarefree $q$ [@BlMil Lemma 9]. Here we generalise this further to nonprincipal nebentypus; this has also independently been derived by Ian Petrow (personal communication).
\[innerproductlemma\] Suppose that $\chi$ has conductor $q_{\chi} \mid q$, and suppose that $q_1 q_2 = q$ with $q_1 \equiv 0 \pmod{q_{\chi}}$. For $f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)$ and $\ell_1, \ell_2 \mid q_2$, we have that $$\frac{\left\langle \iota_{\ell_1,q_1,q} f, \iota_{\ell_2,q_1,q} f\right\rangle_q}{\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f\rangle_q} = A_f\left(\frac{\ell_2}{(\ell_1,\ell_2)}\right) \overline{A_f}\left(\frac{\ell_1}{(\ell_1,\ell_2)}\right),$$ where $A_f(n)$ is the multiplicative function defined on prime powers by $$A_f(p^t) = \begin{dcases*}
\frac{\lambda_f(p)}{\sqrt{p}(1 + \chi_{0(q_1)}(p) p^{-1})} & if $t = 1$, \\
\frac{\lambda_f(p^t) - \chi_{(q_1)}(p) \lambda_f(p^{t - 2}) p^{-1}}{p^{t/2} (1 + \chi_{0(q_1)}(p) p^{-1})} & if $t \geq 2$,
\end{dcases*}$$ where $\chi_{0(q_1)}$ denotes the principal character modulo $q_1$ and $\chi_{(q_1)} {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\chi \chi_{0(q_1)}$ denotes the Dirichlet character modulo $q_1$ induced from $\chi$.
For $\Re(s) > 1$, consider the integral $$F(s) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\int_{\Gamma_0(q) \backslash {\mathbb{H}}} f(\ell_1 z) \overline{f}(\ell_2 z) E(z,s) \, d\mu(z),$$ where $$E(z,s) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\sum_{\gamma \in \Gamma_{\infty} \backslash \Gamma_0(q)} \Im(\gamma z)^s.$$ Unfolding the integral and using Parseval’s identity, $$F(s) = \int_{0}^{\infty} y^{s - 1} \underset{\ell_1 n_1 = \ell_2 n_2}{\sum_{\substack{n_1 = -\infty \\ n_1 \neq 0}}^{\infty} \sum_{\substack{n_2 = -\infty \\ n_2 \neq 0}}^{\infty}} \rho_f(n_1) \overline{\rho_f}(n_2) W_{\operatorname{sgn}(n_1) \frac{\kappa}{2}, it_f}(4\pi \ell_1 |n_1|y)^2 \, \frac{dy}{y}.$$ From and the fact from [@DFI Equation (4.70)] that $$\rho_f(-n) = \epsilon_f \frac{\Gamma\left(\frac{1 + \kappa}{2} + it_f\right)}{\Gamma\left(\frac{1 - \kappa}{2} + it_f\right)} \rho_f(n)$$ for $n \geq 1$, where $\epsilon_f \in \{-1,1\}$, we find that $$\begin{gathered}
F(s) = \frac{|\rho_f(1)|^2}{(4\pi [\ell_1, \ell_2])^{s - 1} \sqrt{\ell' \ell''}} \sum_{n = 1}^{\infty} \frac{\lambda_f(\ell'' n) \overline{\lambda_f}(\ell' n)}{n^s} \\
\times \int_{0}^{\infty} y^{s - 1} \left(W_{\frac{\kappa}{2}, it_f}(y)^2 + \left|\frac{\Gamma\left(\frac{1 + \kappa}{2} + it_f\right)}{\Gamma\left(\frac{1 - \kappa}{2} + it_f\right)}\right|^2 W_{-\frac{\kappa}{2}, it_f}(y)^2\right) \, \frac{dy}{y},\end{gathered}$$ where we have written $n_1 = \ell'' n$, $n_2 = \ell' n$, with $\ell' = \ell_1/(\ell_1,\ell_2)$ and $\ell'' = \ell_2/(\ell_1,\ell_2)$.
Next, by the multiplicativity of the Hecke eigenvalues of $f$ together with the fact that $(\ell',\ell'') = 1$, the sum over $n$ is equal to $$\sum_{\substack{n = 1 \\ (n,\ell' \ell'') = 1}}^{\infty} \frac{|\lambda_f(n)|^2}{n^s} \prod_{p^t \parallel \ell''} \sum_{r = 0}^{\infty} \frac{\lambda_f(p^{r + t}) \overline{\lambda_f}(p^r)}{p^{rs}} \prod_{p^t \parallel \ell'} \sum_{r = 0}^{\infty} \frac{\lambda_f(p^r) \overline{\lambda_f}(p^{r + t})}{p^{rs}}.$$ From and , we find that $$\begin{aligned}
\sum_{r = 0}^{\infty} \frac{\lambda_f(p^{r + t}) \overline{\lambda_f}(p^r)}{p^{rs}} & = B_f(p^t;s) \sum_{r = 0}^{\infty} \frac{|\lambda_f(p^r)|^2}{p^{rs}}, \\
\sum_{r = 0}^{\infty} \frac{\lambda_f(p^r) \overline{\lambda_f}(p^{r + t})}{p^{rs}} & = \overline{B_f}(p^t;\overline{s}) \sum_{r = 0}^{\infty} \frac{|\lambda_f(p^r)|^2}{p^{rs}},\end{aligned}$$ where $B_f(n;s)$ is defined to be the multiplicative function $$B_f(p^t;s) = \begin{dcases*}
\frac{\lambda_f(p)}{1 + \chi_{0(q_1)}(p) p^{-s}} & if $t = 1$, \\
\frac{\lambda_f(p^t) - \chi_{(q_1)}(p) \lambda_f(p^{t - 2}) p^{-s}}{1 + \chi_{0(q_1)}(p) p^{-s}} & if $t \geq 2$,
\end{dcases*}$$ so that $A_f(n) = n^{-1/2} B_f(n;1)$. We surmise that $F(s)$ is equal to $$\begin{gathered}
\label{F(s)}
\frac{|\rho_f(1)|^2}{(4\pi [\ell_1, \ell_2])^{s - 1} \sqrt{\ell' \ell''}} B_f(\ell'';s) \overline{B_f}(\ell';\overline{s}) \sum_{n = 1}^{\infty} \frac{|\lambda_f(n)|^2}{n^s} \\
\times \int_{0}^{\infty} y^{s - 1} \left(W_{\frac{\kappa}{2}, it_f}(y)^2 + \left|\frac{\Gamma\left(\frac{1 + \kappa}{2} + it_f\right)}{\Gamma\left(\frac{1 - \kappa}{2} + it_f\right)}\right|^2 W_{-\frac{\kappa}{2}, it_f}(y)^2\right) \, \frac{dy}{y}.\end{gathered}$$ The result follows by taking the residue at $s = 1$, noting that $E(z,s)$ has residue equal to $1/\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})$ at $s = 1$ independently of $z \in \Gamma_0(q) \backslash {\mathbb{H}}$, and comparing to the case $\ell_1 = \ell_2 = 1$.
An orthonormal basis of ${\mathcal{A}}_{\kappa}(q,\chi)$ is given by $$\label{basisqchi}
{\mathcal{B}}_{\kappa}(q,\chi) = \bigsqcup_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \bigsqcup_{f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)} \bigsqcup_{d \mid q_2} \left\{f_d = \sum_{\ell \mid d} \xi_f(\ell,d) \iota_{\ell,q_1,q} f\right\},$$ where each $f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right)$ is normalised such that $\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f \rangle_q = 1$, and the function $\xi_f(\ell,d)$ is jointly multiplicative. For $0 \leq r \leq t$, $\xi_f(p^r,p^t)$ is equal to $$\begin{dcases*}
1 & if $r = t = 0$, \\
-\frac{\overline{A_f}(p)}{\sqrt{1 - |A_f(p)|^2}} & if $r = 0$ and $t = 1$, \\
\frac{1}{\sqrt{1 - |A_f(p)|^2}} & if $r = t = 1$, \\
\frac{\overline{\chi}_{(q_1)}(p)}{p} \frac{1}{\sqrt{(1 - \chi_{0(q_1)}(p) p^{-2}) (1 - |A_f(p)|^2)}} & if $r = t - 2$ and $t \geq 2$, \\
-\frac{\overline{\lambda_f}(p)}{\sqrt{p}} \frac{1}{\sqrt{(1 - \chi_{0(q_1)}(p) p^{-2}) (1 - |A_f(p)|^2)}} & if $r = t - 1$ and $t \geq 2$, \\
\frac{1}{\sqrt{(1 - \chi_{0(q_1)}(p) p^{-2}) (1 - |A_f(p)|^2)}} & if $r = t$ and $t \geq 2$, \\
0 & if $0 \leq r \leq t - 3$ and $t \geq 3$.
\end{dcases*}$$
The key point is that the coefficients $\xi_f(\ell,d)$ are chosen such that the ratio of inner products $$\delta_f(d_1,d_2) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{\langle f_{d_1}, f_{d_2}\rangle_q}{\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f\rangle_q} = \sum_{\ell_1 \mid d_1} \sum_{\ell_2 \mid d_2} \xi_f(\ell_1,d_1) \overline{\xi_f}(\ell_2,d_2) \frac{\left\langle \iota_{\ell_1,q_1,q} f, \iota_{\ell_2,q_1,q} f\right\rangle_q}{\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f\rangle_q}$$ is equal to $1$ if $d_1 = d_2$ and $0$ otherwise.
The proof follows the same lines as [@BlMil Proof of Lemma 9]; we omit the details.
Explicit Kuznetsov Formula
--------------------------
We may use the explicit basis together with and to rewrite the discrete part of the Kuznetsov formula, noting that for $f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)$, $d \mid q_2$, and $n \geq 1$ coprime to $q$, $$\rho_{f_d}(n) = \xi_f(1,d) \rho_f(1) \frac{\lambda_f(n)}{\sqrt{n}}.$$ Similarly, the continuous part can be rewritten in terms of the Eisenstein spanning set ${\mathcal{B}}(\chi_1,\chi_2)$ with $\chi_1 \chi_2 = \chi$ together with and . This yields the following explicit versions of the pre-Kuznetsov and Kuznetsov formulæ.
When $m,n \geq 1$ with $(mn,q) = 1$, the pre-Kuznetsov formula has the form $$\begin{gathered}
\label{pre-Kuznetsov}
\sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right)} 4 \pi \xi_f \left|\rho_f(1)\right|^2 \frac{\overline{\chi}(m) \lambda_f(m) \lambda_f(n)}{\cosh \pi(r - t_f) \cosh \pi (r + t_f)} \\
+ \sum_{\substack{\chi_1,\chi_2 \hspace{-.25cm} \pmod{q} \\ \chi_1 \chi_2 = \chi}} \sum_{f \in {\mathcal{B}}(\chi_1,\chi_2)} \int_{-\infty}^{\infty} \left|\rho_f(1,t)\right|^2 \frac{\overline{\chi}(m) \lambda_f(m,t) \lambda_f(n,t)}{\cosh \pi(r - t) \cosh \pi (r + t)} \, dt \\
= \frac{\left|\Gamma\left(1 - \frac{\kappa}{2} - ir\right)\right|^2}{\pi^2} \left(\delta_{mn} + \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{S_{\chi}(m,n;c)}{c} I_{\kappa}\left(\frac{4\pi \sqrt{mn}}{c},r\right)\right)\end{gathered}$$ for $\kappa \in \{0,1\}$, where we define $$\xi_f {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\sum_{d \mid q_2} \left|\xi_f(1,d)\right|^2,$$ while the Kuznetsov formula for $\kappa = 0$ has the form $$\begin{gathered}
\label{Kuznetsov}
\sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{f \in {\mathcal{B}}_0^{\ast}\left(q_1,\chi\right)} \frac{4\pi \xi_f \left|\rho_f(1)\right|^2}{\cosh \pi t_f} \overline{\chi}(m) \lambda_f(m) \lambda_f(n) h(t_f) \\
+ \sum_{\substack{\chi_1,\chi_2 \hspace{-.25cm} \pmod{q} \\ \chi_1 \chi_2 = \chi}} \sum_{f \in {\mathcal{B}}(\chi_1,\chi_2)} \int_{-\infty}^{\infty} \frac{\left|\rho_f(1,t)\right|^2}{\cosh \pi t} \overline{\chi}(m) \lambda_f(m,t) \lambda_f(n,t) h(t) \, dt \\
= \delta_{mn} g_0 + \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{S_{\chi}(m,n;c)}{c} g_0\left(\frac{4\pi \sqrt{mn}}{c}\right).\end{gathered}$$ In both formulæ, each $f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right)$ is normalised such that $\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f \rangle_q = 1$.
Atkin–Lehner Decomposition for 04001(q)
---------------------------------------
We recall the decomposition $${\mathcal{A}}_{\kappa}\left(\Gamma_1(q)\right) = \bigoplus_{\substack{\chi \hspace{-.25cm}\pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} {\mathcal{A}}_{\kappa}(q,\chi),$$ which follows from the fact that $\Gamma_1(q)$ is a normal subgroup of $\Gamma_0(q)$ with quotient group isomorphic to $({\mathbb{Z}}/q{\mathbb{Z}})^{\times}$, noting that ${\mathcal{A}}_{\kappa}(q,\chi) = \{0\}$ if $\chi(-1) \neq (-1)^{\kappa}$. From this, we obtain the natural basis of ${\mathcal{A}}_{\kappa}\left(\Gamma_1(q)\right)$ given by $$\label{basisGamma1(q)}
{\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) = \bigsqcup_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \bigsqcup_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \bigsqcup_{f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)} \bigsqcup_{d \mid q_2} \left\{f_d = \sum_{\ell \mid d} \xi_f(\ell,d) \iota_{\ell,q_1,q} f\right\}.$$ This allows us to use the pre-Kuznetsov and Kuznetsov formulæ and for ${\mathcal{B}}_{\kappa}(\Gamma_1(q))$ and ${\mathcal{B}}_0(\Gamma_1(q))$, even though ostensibly these two formulæ are only set up for ${\mathcal{B}}_{\kappa}(q,\chi)$ and ${\mathcal{B}}_0(q,\chi)$.
Atkin–Lehner Decomposition for (q)
----------------------------------
A similar decomposition also holds for ${\mathcal{A}}_{\kappa}(\Gamma(q))$. In this case, the fact that $$\begin{aligned}
\Gamma_0\left(q^2\right) \cap \Gamma_1(q) & = \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\mathrm{SL}}_2({\mathbb{Z}}) : a,d \equiv 1 \hspace{-.25cm} \pmod{q}, \ c \equiv 0 \hspace{-.25cm} \pmod{q^2}\right\} \\
& = \begin{pmatrix} q^{-1} & 0 \\ 0 & 1 \end{pmatrix} \Gamma(q) \begin{pmatrix} q & 0 \\ 0 & 1 \end{pmatrix}\end{aligned}$$ implies that $${\mathcal{A}}_{\kappa}(\Gamma(q)) = \iota_{q^{-1}} {\mathcal{A}}_{\kappa}\left(\Gamma_0\left(q^2\right) \cap \Gamma_1(q)\right),$$ where $\iota_{q^{-1}} : {\mathcal{A}}_{\kappa}\left(\Gamma_0\left(q^2\right) \cap \Gamma_1(q)\right) \to {\mathcal{A}}_{\kappa}(\Gamma(q))$ is the map $\iota_{q^{-1}} f(z) = f\left(q^{-1}z\right)$. As $\Gamma_0\left(q^2\right) \cap \Gamma_1(q)$ is a normal subgroup of $\Gamma_0\left(q^2\right)$ with quotient group isomorphic to $({\mathbb{Z}}/q{\mathbb{Z}})^{\times}$, we obtain the decomposition $${\mathcal{A}}_{\kappa}\left(\Gamma(q)\right) = \bigoplus_{\substack{\chi \hspace{-.25cm}\pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \iota_{q^{-1}} {\mathcal{A}}_{\kappa}\left(q^2,\chi\right),$$ thereby allowing us to choose an explicit basis ${\mathcal{B}}_{\kappa}(\Gamma(q))$ of ${\mathcal{A}}_{\kappa}(\Gamma(q))$ of the form $$\label{basisGamma(q)}
\bigsqcup_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \bigsqcup_{\substack{q_1 q_2 = q^2 \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \bigsqcup_{f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)} \bigsqcup_{d \mid q_2} \left\{\iota_{q^{-1}} f_d = \sum_{\ell \mid d} \xi_f(\ell,d)\iota_{q^{-1}} \iota_{\ell,q_1,q} f \right\}.$$ Once again, this allows us to make use of the pre-Kuznetsov and Kuznetsov formulæ and for ${\mathcal{B}}_{\kappa}(\Gamma(q))$ and ${\mathcal{B}}_0(\Gamma(q))$.
Bounds for Fourier Coefficients of Newforms
===========================================
In the Kuznetsov formula , the Fourier coefficients $|\rho_f(1)|^2$ and the normalisation factor $\xi_f$ both appear naturally. To remove these weights, we obtain lower bounds for $|\rho_f(1)|^2$ and $\xi_f$. For the former, such bounds are well-known, appearing in some generality in [@DFI Equation (7.16)]; nevertheless, we take this opportunity to correct some of the minor numerical errors in this proof, as well as greatly streamline the proof via the recent work of Li [@Li] on obtaining upper bounds for $L$-functions at the edge of the critical strip.
\[xif(1)lemma\] For $f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)$, we have that $$\xi_f = \sum_{n \mid q_2^{\infty}} \frac{|\lambda_f(n)|^2}{n} \prod_{p \parallel q_2} \left(1 - \frac{\chi_{0(q_1)}(p)}{p^2}\right).$$ In particular, $\xi_f \gg 1$.
By multiplicativity, $$\sum_{d \mid q_2} |\xi_f(1,d)|^2 = \prod_{p^t \parallel q_2} \sum_{r = 0}^{t} |\xi_f(1,p^r)|^2.$$ We have that $$\sum_{r = 0}^{t} |\xi_f(1,p^r)|^2 = \begin{dcases*}
1 & if $t = 0$, \\
\frac{1}{1 - |A_f(p)|^2} & if $t = 1$, \\
\frac{1}{(1 - \chi_{0(q_1)}(p) p^{-2}) (1 - |A_f(p)|^2)} & if $t \geq 2$.
\end{dcases*}$$ The result then follows from the fact that $$\frac{1}{1 - |A_f(p)|^2} = \left(1 - \frac{\chi_{0(q_1)}(p)}{p^2}\right) \sum_{k = 0}^{\infty} \frac{|\lambda_f(p^k)|^2}{p^k}.\qedhere$$
For $f \in {\mathcal{B}}_{\kappa}(q,\chi)$, we define $$\nu_f {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\Gamma\left(\frac{1 + \kappa}{2} + it_f\right) \Gamma\left(\frac{1 + \kappa}{2} - it_f\right) |\rho_f(1)|^2.$$ Note that $$\Gamma\left(\frac{1 + \kappa}{2} + it\right) \Gamma\left(\frac{1 + \kappa}{2} - it\right) = \begin{dcases*}
\frac{\pi}{\cosh \pi t} & if $\kappa = 0$, \\
\frac{\pi t}{\sinh \pi t} & if $\kappa = 1$.
\end{dcases*}$$
\[L2idlemma\] Suppose that $f \in {\mathcal{B}}_{\kappa}^{\ast}(q_1,\chi)$ for some $q_1 \mid q$. Then $$\frac{\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f \rangle_q}{\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)} = \nu_f \operatorname*{Res}_{s = 1} \sum_{n = 1}^{\infty} \frac{|\lambda_f(n)|^2}{n^s}.$$
We let $\ell_1 = \ell_2 = 1$ in and take the residue at $s = 1$, yielding $$\begin{gathered}
\frac{\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f \rangle_q}{\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)} = |\rho_f(1)|^2 \operatorname*{Res}_{s = 1} \sum_{n = 1}^{\infty} \frac{|\lambda_f(n)|^2}{n^s} \\
\times \int_{0}^{\infty} \left(W_{\frac{\kappa}{2}, it_f}(y)^2 + \left|\frac{\Gamma\left(\frac{1 + \kappa}{2} + it_f\right)}{\Gamma\left(\frac{1 - \kappa}{2} + it_f\right)}\right|^2 W_{-\frac{\kappa}{2}, it_f}(y)^2\right) \, \frac{dy}{y},\end{gathered}$$ since the residue of $E(z,s)$ at $s = 1$ is $1/\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)$. We have by [@GR 7.611.4] that for $\kappa \in {\mathbb{C}}$ and $-1/2 < \Re(it) < 1/2$, $$\int_{0}^{\infty} W_{\frac{\kappa}{2}, it}(y)^2 \, \frac{dy}{y} = \frac{\pi}{\sin 2\pi it} \frac{\psi\left(\frac{1 - \kappa}{2} + it\right) - \psi\left(\frac{1 - \kappa}{2} - it\right)}{\Gamma\left(\frac{1 - \kappa}{2} + it\right) \Gamma\left(\frac{1 - \kappa}{2} - it\right)},$$ where $\psi$ is the digamma function; note that a slightly erroneous version of this appears in [@DFI Equation (19.6)]. By the gamma and digamma reflection formulæ, we find that $$\begin{gathered}
\int_{0}^{\infty} \left(W_{\frac{\kappa}{2}, it_f}(y)^2 + \left|\frac{\Gamma\left(\frac{1 + \kappa}{2} + it_f\right)}{\Gamma\left(\frac{1 - \kappa}{2} + it_f\right)}\right|^2 W_{-\frac{\kappa}{2}, it_f}(y)^2\right) \, \frac{dy}{y} \\
= \Gamma\left(\frac{1 + \kappa}{2} + it_f\right) \Gamma\left(\frac{1 + \kappa}{2} - it_f\right)\end{gathered}$$ assuming that $t_f \in [0,\infty)$ if $\kappa = 1$ and $t_f \in [0,\infty)$ or $it_f \in (0,1/2)$ if $\kappa = 0$.
Suppose that $f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right)$ for some $q_1 \mid q$. Then $$\label{rho1lowerbound}
\nu_f \gg_{\e} \frac{\langle \iota_{1,q_1,q} f, \iota_{1,q_1,q} f \rangle_q}{\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)} \left(q \left(3 + t_f^2\right)\right)^{-\e}.$$
It is known that $$\sum_{n = 1}^{\infty} \frac{|\lambda_f(n)|^2}{n^s} = \frac{\zeta(s) L(s, \operatorname{ad}f)}{\zeta(2s)} \prod_{p \mid q} P_{f,p}(p^{-s}),$$ where for each prime $p$ dividing $q$, $P_{f,p}(z)$ is a rational function satisfying $p^{-\e} \ll_{\e} P_{f,p}(p^{-1}) \leq 1$. The work of Li [@Li Theorem 2] then shows that $$L(1, \operatorname{ad}f) \ll \exp\left(C \frac{\log \left(q \left(3 + t_f^2\right)\right)}{\log \log \left(q \left(3 + t_f^2\right)\right)}\right)$$ for some absolute constant $C > 0$, thereby yielding the result.
Bounds for Sums of Kloosterman Sums
===================================
We denote by $$S(m,n;c) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\sum_{d \in ({\mathbb{Z}}/c{\mathbb{Z}})^{\times}} e\left(\frac{md + n\overline{d}}{c}\right)$$ the usual Kloosterman sum with trivial character, for which the Weil bound holds: $$\label{Weilbound}
|S(m,n;c)|\leq \tau(c) \sqrt{(m,n,c) c}.$$ We also require bounds for Kloosterman sums with nontrivial character. For $c \equiv 0 \pmod{q}$, $m,n \geq 1$, and $(a,q) = 1$, we have that $$\begin{gathered}
\sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \overline{\chi}(a) S_{\chi}(m,n;c) \\
= \frac{1}{2} \sum_{d \in ({\mathbb{Z}}/c{\mathbb{Z}})^{\times}} \sum_{\chi \hspace{-.25cm} \pmod{q}} \overline{\chi}(a) \left(\chi(d) + (-1)^{\kappa} \chi(-d)\right) e\left(\frac{md + n\overline{d}}{c}\right).\end{gathered}$$ We break this up into two sums. In the second sum, we can replace $d$ with $-d$ and $\chi$ with $\overline{\chi}$ and use character orthogonality to see that $$\label{sumKloosterman}
\sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \overline{\chi}(a) S_{\chi}(m,n;c) =
\begin{dcases*}
\varphi(q) \Re\left(S_{a(q)}(m,n;c)\right) & if $\kappa = 0$, \\
i \varphi(q) \Im\left(S_{a(q)}(m,n;c)\right) & if $\kappa = 1$,
\end{dcases*}$$ where we set $$S_{a(q)}(m,n;c) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\sum_{\substack{d \in ({\mathbb{Z}}/c{\mathbb{Z}})^{\times} \\ d \equiv a \hspace{-.25cm} \pmod{q}}} e\left(\frac{md + n\overline{d}}{c}\right).$$
If $c = c_1 c_2$ with $(c_1,c_2) = 1$ and $c_1 c_2 \equiv 0 \pmod{q}$, then we let $d = c_2 \overline{c_2} d_1 + c_1 \overline{c_1} d_2$, where $d_1 \in ({\mathbb{Z}}/c_1 {\mathbb{Z}})^{\times}$, $d_2 \in ({\mathbb{Z}}/c_2 {\mathbb{Z}})^{\times}$, and $c_2 \overline{c_2} \equiv 1 \pmod{c_1}$, $c_1 \overline{c_1} \equiv 1 \pmod{c_2}$. By the Chinese remainder theorem, $$S_{a(q)}(m,n;c) = S_{a((q,c_1))}\left(m\overline{c_2}, n\overline{c_2}; c_1\right) S_{a((q,c_2))}\left(m\overline{c_1}, n\overline{c_1}; c_2\right).$$ To bound $S_{a(q)}(m,n;c)$, it therefore suffices to find bounds for $S_{a(p^{\alpha})}\left(m,n;p^{\beta}\right)$ for any prime $p$ and any $\beta \geq \alpha \geq 1$. The trivial bound is merely $$\label{trivKloostbound}
\left|S_{a(p^{\alpha})}\left(m,n;p^{\beta}\right)\right| \leq p^{\beta - \alpha}.$$ Somewhat surprisingly, this is sufficient for our needs. Indeed, we cannot do better than this when $\beta = \alpha$, and in our applications, this will be the dominant contribution.
We also require bounds for $S_{\chi}(m,n;c)$. Unfortunately, it is not necessarily the case that this is bounded by $\tau(c) \sqrt{(m,n,c) c}$, which can be observed numerically at [@LMFDB]; see also [@KL Example 9.9].
\[weakWeillemma\] Let $p$ be an odd prime, let $\chi_{p^{\gamma}}$ be a Dirichlet character of conductor $p^{\gamma}$, and suppose that $(mn,p) = 1$. Then for any $c \geq 1$ with $(c,p) = 1$ and $\beta \geq \gamma \geq 0$, we have that $$\left|S_{\chi_{p^{\gamma}}}(mc,nc;p^{\beta})\right| \leq 2 p^{\beta/2}$$ unless $\beta = \gamma \geq 3$, in which case we only have that $$\left|S_{\chi_{p^{\gamma}}}(mc,nc;p^{\beta})\right| \leq 2 p^{\lfloor \frac{3\beta + 1}{4} \rfloor}.$$
Similarly, let $\chi_{2^{\gamma}}$ be a Dirichlet character of conductor $2^{\gamma}$, and suppose that $(mn,2) = 1$. Then for any $c \geq 1$ with $(c,2) = 1$ and $\beta \geq \gamma \geq 0$, we have that $$\left|S_{\chi_{2^{\gamma}}}(mc,nc;2^{\beta})\right| \leq 8 \cdot 2^{\beta/2}$$ unless $\gamma + 1 \geq \beta \geq 3$, in which case we only have that $$\left|S_{\chi_{2^{\gamma}}}(mc,nc;2^{\beta})\right| \leq 4 \cdot 2^{\lfloor \frac{3\beta + 1}{4} \rfloor}.$$
This follows from [@KL Propositions 9.4, 9.7, 9.8, and Lemmata 9.6].
When $(m,n) = 1$, we have that $$\begin{aligned}
\sum_{\substack{c \leq 4\pi \sqrt{mn} \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^{3/2}} & \ll \frac{(\log (mn + 1))^2}{q^{3/2}} \prod_{p \mid q} \frac{1}{1 - p^{-1/2}}, \label{Gamma1(q)cleq}\\
\sum_{\substack{c \leq 4\pi \sqrt{mn} \\ c \equiv 0 \hspace{-.25cm} \pmod{q^2}}} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^{3/2}} & \ll \frac{(\log (mn + 1))^2}{q^2} \prod_{p \mid q} \frac{1}{1 - p^{-1/2}}. \label{Gamma(q)cleq}\end{aligned}$$ If we additionally assume that $(mn,q) = 1$, then given a Dirichlet character $\chi$ modulo $q$, we have that $$\label{Gamma0(q)cleq}
\sum_{\substack{c \leq 4\pi \sqrt{mn} \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}} \frac{\left|S_{\chi}(m,n;c)\right|}{c^{3/2}} \ll (\log (mn + 1))^2 \frac{2^{\omega(q)} \dot{Q}}{\varphi(q)}.$$
We write $q = p_1^{\alpha_1} \cdots p_{\ell}^{\alpha_{\ell}}$, so that the left-hand side of is $$\begin{gathered}
\sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{\left(p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}\right)^{3/2}} \sum_{\substack{c \leq 4\pi \sqrt{mn} p_1^{-\beta_1} \cdots p_{\ell}^{-\beta_{\ell}} \\ (c,q) = 1}} \frac{1}{c^{3/2}} \\
\times \left|S\left(m \overline{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}}, n \overline{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}}; c\right)\right| \left|S_{a(q)}\left(m \overline{c}, n \overline{c}; p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}\right)\right|.\end{gathered}$$ Using the Weil bound for the first Kloosterman sum and the trivial bound for the second, we find that this is bounded by $$\frac{1}{q} \sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{\sqrt{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}}} \sum_{\substack{c \leq 4\pi \sqrt{mn} \\ (c,q) = 1}} \frac{\tau(c) \sqrt{(m,n,c)}}{c}.$$ If $(m,n) = 1$, the inner sum is bounded by a constant multiple of $(\log(mn + 1))^2$, and so the sum is bounded by a constant multiple of $$\frac{(\log (mn + 1))^2}{q} \sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{\sqrt{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}}} ,$$ which yields upon evaluating these geometric series. follows similarly. Finally, follows via the same method but using to bound the Kloosterman sums, yielding the bound $$8 \cdot 2^{\omega(q)} \dot{Q} \sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}} \sum_{\substack{c \leq 4\pi \sqrt{mn} \\ (c,q) = 1}} \frac{\tau(c)}{c}$$ for the left-hand side of , from which the result easily follows.
When $(m,n) = 1$, we have that $$\begin{aligned}
\sum_{\substack{c > 4\pi \sqrt{mn} \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^2} \left(1 + \log \frac{c}{4\pi\sqrt{mn}}\right) & \ll \frac{(\log (mn + 1))^2}{(mn)^{1/4}} \frac{1}{q^{3/2}} \prod_{p \mid q} \frac{1}{1 - p^{-1/2}}, \label{Gamma1(q)cgeq}\\
\sum_{\substack{c > 4\pi \sqrt{mn} \\ c \equiv 0 \hspace{-.25cm} \pmod{q^2}}} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^2} \left(1 + \log \frac{c}{4\pi\sqrt{mn}}\right) & \ll \frac{(\log (mn + 1))^2}{(mn)^{1/4}} \frac{1}{q^2} \prod_{p \mid q} \frac{1}{1 - p^{-1/2}}. \label{Gamma(q)cgeq}\end{aligned}$$ If we additionally assume that $(mn,q) = 1$, then given a Dirichlet character $\chi$ modulo $q$, we have that $$\label{Gamma0(q)cgeq}
\sum_{\substack{c > 4\pi \sqrt{mn} \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}} \frac{\left|S_{\chi}(m,n;c)\right|}{c^2} \left(1 + \log \frac{c}{4\pi\sqrt{mn}}\right) \ll \frac{(\log (mn + 1))^2}{(mn)^{1/4}} \frac{2^{\omega(q)} \dot{Q}}{\varphi(q)}.$$
As before, with $q = p_1^{\alpha_1} \cdots p_{\ell}^{\alpha_{\ell}}$, the left-hand side of is bounded by $$\frac{1}{q} \sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}} \sum_{\substack{c > 4\pi \sqrt{mn} p_1^{-\beta_1} \cdots p_{\ell}^{-\beta_{\ell}} \\ (c,q) = 1}} \frac{\tau(c) \sqrt{(m,n,c)} \log c}{c^{3/2}}.$$ If $(m,n) = 1$, then the inner sum is bounded by a constant multiple of $$\frac{(\log (mn + 1))^2}{(mn)^{1/4}} \sqrt{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}}.$$ It follows that the sum is bounded by a constant multiple of $$\frac{(\log (mn + 1))^2}{(mn)^{1/4}} \frac{1}{q} \sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{\sqrt{p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}}},$$ which gives . The proof of is analogous, while again follows upon using to bound the Kloosterman sums.
For all $1/2 < \sigma < 1$, $$\begin{aligned}
\sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^{1 + \sigma}} & \leq \frac{18 \tau((m,n))}{(2\sigma - 1)^2} \frac{1}{q^{1 + \sigma}} \prod_{p \mid q} \frac{1}{1 - p^{-\sigma}}, \label{Gamma1(q)csigma}\\
\sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q^2}}}^{\infty} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^{1 + \sigma}} & \leq \frac{18 \tau((m,n))}{(2\sigma - 1)^2} \frac{1}{q^{1 + 2\sigma}} \prod_{p \mid q} \frac{1}{1 - p^{-\sigma}}. \label{Gamma(q)csigma}\end{aligned}$$ If we additionally assume that $(m,n) = (mn,q) = 1$, then given a Dirichlet character $\chi$ modulo $q$, we have that $$\label{Gamma0(q)csigma}
\sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{\left|S_{\chi}(m,n;c)\right|}{c^{1 + \sigma}} \leq \frac{72}{(2\sigma - 1)^2} \frac{2^{\omega(q)} \dot{Q}}{\varphi(q) q^{\sigma - 1/2}}.$$
Once again writing $q = p_1^{\alpha_1} \cdots p_{\ell}^{\alpha_{\ell}}$ and bounding the Kloosterman sums, we have that $$\begin{aligned}
\sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{\left|S_{a(q)}(m,n;c)\right|}{c^{1 + \sigma}} & \leq \sum_{\substack{c = 1 \\ (c,q) = 1}}^{\infty} \frac{\tau(c) \sqrt{(m,n,c)}}{c^{1/2 + \sigma}} \frac{1}{q} \sum_{\beta_1 = \alpha_1}^{\infty} \cdots \sum_{\beta_{\ell} = \alpha_{\ell}}^{\infty} \frac{1}{\left(p_1^{\beta_1} \cdots p_{\ell}^{\beta_{\ell}}\right)^{\sigma}} \\
& = \sum_{\substack{c = 1 \\ (c,q) = 1}}^{\infty} \frac{\tau(c) \sqrt{(m,n,c)}}{c^{1/2 + \sigma}} \frac{1}{q^{1 + \sigma}} \prod_{p \mid q} \frac{1}{1 - p^{-\sigma}} \\
& \leq \zeta\left(\sigma + \frac{1}{2}\right)^2 \sum_{d \mid (m,n)} \frac{\tau(d)}{d^{\sigma}} \frac{1}{q^{1 + \sigma}} \prod_{p \mid q} \frac{1}{1 - p^{-\sigma}} \\
& \leq \frac{18 \tau((m,n))}{(2\sigma - 1)^2} \frac{1}{q^{1 + \sigma}} \prod_{p \mid q} \frac{1}{1 - p^{-\sigma}}.\end{aligned}$$ This proves . The inequality follows by a similar argument, as does once the Kloosterman sums are bounded via .
Bounds for Test Functions
=========================
We require bounds for the test function that we will obtain by multiplying the pre-Kuznetsov formula by a function dependent on $r$ and then integrating both sides over $r \in [0,T]$.
\[hkTlemma\] For $T \geq 1$, let $$\begin{aligned}
h_{\kappa,T}(t) & {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{\pi^2}{\Gamma\left(\frac{1 + \kappa}{2} + it\right) \Gamma\left(\frac{1 + \kappa}{2} - it\right)} \int_{0}^{T} \frac{r \left|\Gamma\left(1 - \frac{\kappa}{2} + ir\right)\right|^{-2}}{\cosh \pi (r - t) \cosh \pi (r + t)} \, dr \\
& = \begin{dcases*}
\cosh \pi t \int_{0}^{T} \frac{\sinh \pi r}{\cosh \pi (r - t) \cosh \pi (r + t)} \, dr & if $\kappa = 0$, \\
\frac{\sinh \pi t}{t} \int_{0}^{T} \frac{r \cosh \pi r}{\cosh \pi (r - t) \cosh \pi (r + t)} \, dr & if $\kappa = 1$.
\end{dcases*}\end{aligned}$$ Then $h_{\kappa,T}(t)$ is positive for all $t \in {\mathbb{R}}$ and additionally, should $\kappa$ be equal to $0$, for $it \in (-1/2,1/2)$. Furthermore, $h_{\kappa,T}(t) \gg 1$ for $t \in [0,T]$.
Using the fact that $$\cosh \pi (r - t) \cosh \pi (r + t) = \cosh^2 \pi t + \sinh^2 \pi r = \sinh^2 \pi t + \cosh^2 \pi r,$$ it is clear that $h_{\kappa,T}(t)$ is positive for all $t \in {\mathbb{R}}$ and additionally, should $\kappa$ be equal to $0$, if $it \in (-1/2,1/2)$.
For $\kappa = 0$, we have that $$\begin{aligned}
h_{0,T}(t) & = \frac{\cosh \pi t}{\pi} \int_{1}^{\cosh \pi T} \frac{1}{x^2 + \sinh^2 \pi t} \, dx \\
& = \frac{\coth \pi t}{\pi} \arctan \frac{\sinh \pi t \left(\cosh \pi T - 1\right)}{\sinh^2 \pi t + \cosh\pi T},\end{aligned}$$ where the second line follows from the arctangent subtraction formula. The first expression shows that $h_{0,T}(t) \gg 1$ when $t$ is small, while when $t$ is large, the argument of $\arctan$ is essentially $$\frac{e^{\pi(T + t)} - e^{\pi t}}{e^{2\pi t} + e^{\pi T}},$$ and this is bounded from below provided that $t \leq T$, so that again $h_{0,T}(t) \gg 1$.
For $\kappa = 1$, we can similarly show via integration by parts that $$\begin{aligned}
h_{1,T}(t) & = \frac{\sinh \pi t}{\pi^2 t} \int_{0}^{\sinh \pi T} \frac{\operatorname{arsinh}x}{x^2 + \cosh^2 \pi t} \, dx \\
& = \frac{\tanh \pi t}{\pi^2 t} \int_{0}^{\sinh \pi T} \frac{ \arctan \frac{\sinh \pi T}{\cosh \pi t} - \arctan \frac{x}{\cosh \pi t}}{\sqrt{x^2 + 1}} \, dx.\end{aligned}$$ The first expression shows that $h_{1,T}(t) \gg 1$ when $t$ is small, while when $t$ is large, we break up the second expression into two integrals: one from $0$ to $\sinh \frac{\pi t}{2}$ and one from $\sinh \frac{\pi t}{2}$ to $\sinh \pi T$. Trivially bounding the numerator in each integral, we find that $$\begin{aligned}
h_{1,T}(t) & \geq \frac{\tanh \pi t}{2\pi} \left(\arctan \frac{\sinh \pi T}{\cosh \pi t} - \arctan \frac{\sinh \frac{\pi t}{2}}{\cosh \pi t}\right) \\
& = \frac{\tanh \pi t}{2\pi} \arctan \frac{\cosh \pi t \left(\sinh \pi T - \sinh \frac{\pi t}{2}\right)}{\cosh^2 \pi t + \sinh \pi T \sinh \frac{\pi t}{2}}.\end{aligned}$$ The argument of $\arctan$ is essentially $$\frac{e^{\pi(T + t)} - e^{3\pi t/2}}{e^{2\pi t} + e^{\pi(T + t/2)}},$$ and this is bounded from below provided that $t \leq T$, while $\tanh \pi t$ is bounded from below provided that $t$ is larger than some fixed constant. It follows again that $h_{1,T}(t) \gg 1$.
We also require the following bound, which arises from the Kloosterman term in the pre-Kuznetsov formula .
For $\kappa \in \{0,1\}$ and $T > 0$, we have the bound $$\label{uniformIkint}
\int_{0}^{T} r I_{\kappa}(a,r) \, dr \ll \begin{dcases*}
\sqrt{a} & if $a \geq 1$, \\
a\left(1 + \log \frac{1}{a}\right) & if $0 < a < 1$
\end{dcases*}$$ uniformly in $T$.
From [@Kuz Equation (5.13)], we have that $$\int_{0}^{T} r I_0(a,r) \, dr = a \int_{0}^{\infty} \frac{\tanh \xi}{\xi} (1 - \cos 2T \xi) \sin(a \cosh \xi) \, d\xi.$$ Similarly, using the fact that $$K_{2ir}(\zeta) = \int_{0}^{\infty} e^{-\zeta \cosh \xi} \cos 2r \xi \, d\xi$$ for $r \in {\mathbb{R}}$ and $\Re(\zeta) > 0$ from [@GR 8.432.1], we have that $$\int_{0}^{T} r I_1(a,r) \, dr = -2a \int_{0}^{\infty} \int_{0}^{T} r \cos 2r \xi \, dr \int_{-i}^{i} e^{-\zeta a \cosh \xi} \, d\zeta \, d\xi.$$ Evaluating each of the inner integrals and then integrating by parts, we find that $$\begin{gathered}
\int_{0}^{T} r I_1(a,r) \, dr = i a \int_{0}^{\infty} \frac{\tanh \xi}{\xi} (1 - \cos 2T \xi) \cos(a \cosh \xi) \, d\xi \\
- i \int_{0}^{\infty} \frac{\tanh \xi}{\xi} (1 - \cos 2T \xi) \frac{\sin(a \cosh \xi)}{\cosh \xi} \, d\xi.\end{gathered}$$ From here, one can show via stationary phase on subintervals of $(0,\infty)$ that $\int_{0}^{T} r I_0(a,r) \, dr$ and the first term in the above expression for $\int_{0}^{T} r I_1(a,r) \, dr$ both are bounded by a constant multiple of $$\begin{dcases*}
\sqrt{a} & if $a \geq 1$, \\
a\left(1 + \log \frac{1}{a}\right) & if $0 < a < 1$;
\end{dcases*}$$ see [@Kuz Equation (5.14)]. The second term in the expression for $\int_{0}^{T} r I_1(a,r) \, dr$ is uniformly bounded for $a \geq 1$, so we need only consider when $0 < a < 1$. In this case, the fact that $|\sin x| \leq \min\{1,|x|\}$ for $x \in {\mathbb{R}}$ implies that this is bounded by $$2a \int_{0}^{\log \frac{1}{a}} \frac{\tanh \xi}{\xi} \, d\xi + 2 \int_{\log \frac{1}{a}}^{\infty} \frac{\tanh \xi}{\xi} \frac{1}{\cosh \xi} \, d\xi \\
\ll a \left(1 + \log \frac{1}{a}\right).\qedhere$$
Sarnak’s Density Theorem for Exceptional Hecke Eigenvalues
==========================================================
We are now in a position to prove .
By Rankin’s trick, $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\leq \prod_{p \in {\mathcal{P}}} \alpha_p^{-2\ell_p} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) \\ t_f \in [0,T]}} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}\end{gathered}$$ for any nonnegative integers $\ell_p$ to be chosen. Using the explicit basis of ${\mathcal{A}}_{\kappa}\left(\Gamma_1(q)\right)$ together with the lower bound for $\nu_f$, $$\begin{aligned}
\hspace{2cm} & \hspace{-2cm} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) \\ t_f \in [0,T]}} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\
& = \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right) \\ t_f \in [0,T]}} \tau(q_2) \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\
& \ll_{\e} q^{1 + \e} T^{\e} \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right) \\ t_f \in [0,T]}} \xi_f \nu_f \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}.\end{aligned}$$
We take $m = 1$ and $n = \prod_{p \in {\mathcal{P}}} p^{2j_p}$ in the pre-Kuznetsov formula , multiply both sides by $\prod_{p \in {\mathcal{P}}} \alpha_{2j_p,2\ell_p} \overline{\chi}(p)^{j_p}$, and sum over all $0 \leq j_p \leq \ell_p$, over all $p \in {\mathcal{P}}$, and over all Dirichlet characters $\chi$ modulo $q$ satisfying $\chi(-1) = (-1)^{\kappa}$. We then multiply both sides by $\pi^2 r \left|\Gamma\left(1 - \frac{\kappa}{2} + ir\right)\right|^{-2}$ and integrate both sides with respect to $r$ from $0$ to $T$.
On the spectral side, , , and allow us to use positivity to discard the contribution from the continuous spectrum, while we may discard the contribution of the discrete spectrum with $t \notin [0,T]$ via , , and , so that the spectral side is bounded from below by a constant multiple of $$\sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right) \\ t_f \in [0,T]}} \xi_f \nu_f \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}.$$ On the geometric side, we only pick up the delta term when $j_p = 0$ for all $p \in {\mathcal{P}}$, in which case the term is bounded by a constant multiple of $q T^2 \prod_{p \in {\mathcal{P}}} \alpha_{0,2\ell_p}$. For $\kappa = 0$, we use to write the Kloosterman term in the form $$\begin{gathered}
\frac{\varphi(q)}{\pi} \sum_{\substack{j_p = 0 \\ p \in {\mathcal{P}}}}^{\ell_p} \prod_{p \in {\mathcal{P}}} \alpha_{2j_p,2\ell_p} \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{\Re\left(S_{\prod_{p \in {\mathcal{P}}} p^{j_p} (q)}\left(1,\prod_{p \in {\mathcal{P}}} p^{2j_p};c\right)\right)}{c} \\
\times \int_{0}^{T} r I_0\left(\frac{4\pi \prod_{p \in {\mathcal{P}}} p^{j_p}}{c},r\right) \, dr.\end{gathered}$$ For $\kappa = 1$, the Kloosterman term is the same except with $i\Im$ in place of $\Re$ and $I_1$ in place of $I_0$. In either case, we bound the integral via , which allows us to use and to bound the summation over $c$, so that the Kloosterman term is bounded by a constant multiple of $$\frac{1}{\sqrt{q}} \prod_{p' \mid q} \frac{1}{1 - {p'}^{-1/2}} \sum_{\substack{j_p = 0 \\ p \in {\mathcal{P}}}}^{\ell_p} \prod_{p \in {\mathcal{P}}} \alpha_{2j_p,2\ell_p} p^{j_p/2} \left(\log \left(\prod_{p \in {\mathcal{P}}} p^{2j_p} + 1\right)\right)^2.$$ We bound the summation over $j_p$ and over $p \in {\mathcal{P}}$ via , thereby obtaining $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} q^{1 + \e} T^{\e} \prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{-2\ell_p} \left(qT^2 + \frac{\prod_{p \in {\mathcal{P}}} p^{\ell_p/2} \left(\log \prod_{p \in {\mathcal{P}}} p^{\ell_p/2}\right)^2}{\sqrt{q}} \prod_{p' \mid q} \frac{1}{1 - {p'}^{-1/2}}\right).\end{gathered}$$ It remains to take $$\ell_p = \left\lfloor \frac{\mu_p \log \left(\operatorname{vol}\left(\Gamma_1(q) \backslash {\mathbb{H}}\right)^{3/2} T^4\right)}{\log p} \right\rfloor.\qedhere$$
We use , , and in place of , , and , thereby finding that $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}\left(\Gamma(q)\right) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\leq \prod_{p \in {\mathcal{P}}} \alpha_p^{-2\ell_p} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}\left(\Gamma(q)\right) \\ t_f \in [0,T]}} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p},\end{gathered}$$ with $$\begin{aligned}
& \sum_{\substack{f \in {\mathcal{B}}_{\kappa}\left(\Gamma(q)\right) \\ t_f \in [0,T]}} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\
& = \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \sum_{\substack{q_1 q_2 = q^2 \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_{\kappa}^{\ast}\left(q_1,\chi\right) \\ t_f \in [0,T]}} \tau(q_2) \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\
& \ll_{\e} q^{2 + \e} T^{\e} \prod_{p \in {\mathcal{P}}} 2^{2\ell_p} \left(qT^2 + \frac{\prod_{p \in {\mathcal{P}}} p^{\ell_p/2} \left(\log \prod_{p \in {\mathcal{P}}} p^{\ell_p/2}\right)^2}{q} \prod_{p' \mid q} \frac{1}{1 - {p'}^{-1/2}}\right).\end{aligned}$$ Taking $$\ell_p = \left\lfloor \frac{\mu_p \log \left(\operatorname{vol}\left(\Gamma(q) \backslash {\mathbb{H}}\right)^{4/3} T^4\right)}{\log p} \right\rfloor$$ completes the proof.
Using , , and in place of , , and , $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}(q,\chi) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} q^{1 + \e} T^{\e} \prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{-2\ell_p} \left(T^2 + \prod_{p \in {\mathcal{P}}} p^{\ell_p/2} \left(\log \prod_{p \in {\mathcal{P}}} p^{\ell_p/2}\right)^2 \frac{2^{\omega(q)} \dot{Q}}{\varphi(q)}\right).\end{gathered}$$ Upon taking $$\ell_p = \left\lfloor \frac{\mu_p \log \left(\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)^2 T^4 \dot{Q}^{-2} \right)}{\log p} \right\rfloor,$$ we conclude that $$\begin{gathered}
\label{SarnakGamma0(q)dotq}
\#\left\{f \in {\mathcal{B}}_{\kappa}(q,\chi) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \left(\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}}) T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \dot{Q}^{4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}}.\end{gathered}$$
On the other hand, by the inclusion ${\mathcal{A}}_{\kappa}(q,\chi) \subset {\mathcal{A}}_{\kappa}(q \ddot{Q},\chi)$, $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}(q,\chi) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\leq \#\left\{f \in {\mathcal{B}}_{\kappa}(q \ddot{Q},\chi) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\}.\end{gathered}$$ Since $q_{\chi \psi^2} \mid q_{\chi}$, we have that $\dot{Q}(q \ddot{Q}, q_{\chi \psi^2}) = 1$. Consequently, yields the bound $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}(q,\chi) : t_f \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \left(\operatorname{vol}(\Gamma_0(q \ddot{Q}) \backslash {\mathbb{H}}) T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \\
\ll_{\e} \left(\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}}) T^2\right)^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \ddot{Q}^{1 - 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}}.\end{gathered}$$ This completes the proof.
Should we wish to improve to be uniform in ${\mathcal{P}}$, then one needs to take into account the fact that $$\begin{gathered}
\prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{-2\ell_p} = \left(\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}}) T^2\right)^{- 4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p} + \e} \dot{Q}^{4 \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}} \\
\times \prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{2 \left\{\frac{\mu_p \log \left(\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)^2 T^4 \dot{Q}^{-2} \right)}{\log p}\right\}},\end{gathered}$$ and the last term need not necessarily be $\ll_{\e} \left(\operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}}) T^2\right)^{\e}$. For this reason, [@BBR Proposition 1] is not correct in the generality in which it is stated, namely the claim that the result is uniform for $T > p$. Instead, one requires that $p \ll_{\e} T^{\e}$.
Huxley’s Density Theorem for Exceptional Laplacian Eigenvalues
==============================================================
is proved similarly to , though we use the Kuznetsov formula with a carefully chosen test function in place of the pre-Kuznetsov formula , and we require different methods to bound the Kloosterman term.
We again use Rankin’s trick with nonnegative integers $\ell_p$ and a positive real number $X \geq 1$ to be chosen: $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_0\left(\Gamma_1(q)\right) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\leq X^{-2\alpha_0} \prod_{p \in {\mathcal{P}}} \alpha_p^{-2\ell_p} \sum_{\substack{f \in {\mathcal{B}}_0\left(\Gamma_1(q)\right) \\ it_f \in (0,1/2)}} X^{2it_f} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}.\end{gathered}$$ Again using and , $$\begin{gathered}
\sum_{\substack{f \in {\mathcal{B}}_0\left(\Gamma_1(q)\right) \\ it_f \in (0,1/2)}} X^{2it_f} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\
= \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = 1}} \sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_0^{\ast}\left(q_1,\chi\right) \\ it_f \in (0,1/2)}} \tau(q_2) X^{2it_f} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\
\ll_{\e} q^{1 + \e} \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = 1}} \sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_0^{\ast}\left(q_1,\chi\right) \\ it_f \in (0,1/2)}} \xi_f \nu_f X^{2it_f} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p} \\\end{gathered}$$
We take $m = 1$, $n = \prod_{p \in {\mathcal{P}}} p^{2j_p}$, and $$h(t) = h_X(t) = \left(\frac{X^{it} + X^{-it}}{t^2 + 1}\right)^2$$ in the Kuznetsov formula , multiply both sides by $\prod_{p \in {\mathcal{P}}} \alpha_{2j_p,2\ell_p} \overline{\chi}(p)^{j_p}$, and sum over all $0 \leq j_p \leq \ell_p$, over all $p \in {\mathcal{P}}$, and over all even Dirichlet characters modulo $q$. On the spectral side, we discard all but the discrete spectrum for which $it_f \in (0,1/2)$ via positivity, so that the spectral side is bounded from below by a constant multiple of $$\sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = 1}} \sum_{\substack{q_1 q_2 = q \\ q_1 \equiv 0 \hspace{-.25cm} \pmod{q_{\chi}}}} \sum_{\substack{f \in {\mathcal{B}}_0^{\ast}\left(q_1,\chi\right) \\ it_f \in (0,1/2)}} \xi_f \nu_f X^{2it_f} \prod_{p \in {\mathcal{P}}} \left|\lambda_f(p)\right|^{2\ell_p}.$$ We only pick up the delta term on the geometric side when $j_p = 0$ for all $p \in {\mathcal{P}}$, in which case the term is bounded by a constant multiple of $q \prod_{p \in {\mathcal{P}}} 2^{2\ell_p}$. We write the Kloosterman term in the form $$\begin{gathered}
\frac{\varphi(q)}{2\pi i} \sum_{\substack{j_p = 0 \\ p \in {\mathcal{P}}}}^{\ell_p} \prod_{p \in {\mathcal{P}}} \alpha_{2j_p,2\ell_p} \int_{\sigma - i\infty}^{\sigma + i\infty} \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{\Re\left(S_{\prod_{p \in {\mathcal{P}}} p^{j_p}(q)}\left(1,\prod_{p \in {\mathcal{P}}} p^{2j_p};c\right)\right)}{c} \\
\times J_s\left(\frac{4\pi \prod_{p \in {\mathcal{P}}} p^{j_p}}{c}\right) \frac{s h_X\left(\frac{is}{2}\right)}{\cos \frac{\pi s}{2}} \, ds\end{gathered}$$ for any $1/2 < \sigma < 1$. We have, via [@GR 8.411.4], the bound $$J_s(x) \ll \frac{x^{\sigma}}{\left|\Gamma\left(s + \frac{1}{2}\right)\right|} \ll e^{\pi|s|/2} \left(\frac{x}{|s|}\right)^{\sigma},$$ and so the integral in the Kloosterman term is bounded by a constant multiple of $$\prod_{p \in {\mathcal{P}}} p^{j_p \sigma} \sum_{\substack{c = 1 \\ c \equiv 0 \hspace{-.25cm} \pmod{q}}}^{\infty} \frac{\left|S_{\prod_{p \in {\mathcal{P}}} p^{j_p} (q)}\left(1,\prod_{p \in {\mathcal{P}}} p^{2j_p};c\right)\right|}{c^{1 + \sigma}} \int_{\sigma/2 - i\infty}^{\sigma/2 + i\infty} \left|r^{3/4} h_X(ir)\right| \, dr.$$ We take $$\sigma = \frac{1}{2} + \frac{1}{\log \left(X \prod_{p \in {\mathcal{P}}} p^{\ell_p}\right)},$$ so that the integral is bounded by a constant multiple of $\sqrt{X}$, and use to bound the summation over $c$ and to bound the summation over $j_p$ and $p \in {\mathcal{P}}$ in order to find that $$\begin{aligned}
& \#\left\{f \in {\mathcal{B}}_0\left(\Gamma_1(q)\right) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
& \qquad\ll_{\e} q^{1 + \e} X^{-2\alpha_0} \prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{-2\ell_p} \\
& \hspace{2.5cm} \times \left(q + \sqrt{X} \prod_{p \in {\mathcal{P}}} p^{\ell_p/2} \left(\log \left(X \prod_{p \in {\mathcal{P}}} p^{\ell_p}\right)\right)^2 \frac{1}{\sqrt{q}} \prod_{p' \mid q} \frac{1}{1 - {p'}^{-1/2}}\right).\end{aligned}$$ The result follows upon taking $$X = \operatorname{vol}\left(\Gamma_1(q) \backslash {\mathbb{H}}\right)^{3\mu_0/2}, \qquad \ell_p = \left\lfloor \frac{\mu_p \log \operatorname{vol}\left(\Gamma_1(q) \backslash {\mathbb{H}}\right)^{3/2}}{\log p} \right\rfloor.\qedhere$$
By using and in place of and , we obtain $$\begin{aligned}
& \#\left\{f \in {\mathcal{B}}_0\left(\Gamma(q)\right) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
& \qquad\ll_{\e} q^{2 + \e} X^{-2\alpha_0} \prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{-2\ell_p} \\
& \hspace{2.5cm} \times \left(q + \sqrt{X} \prod_{p \in {\mathcal{P}}} p^{\ell_p/2} \left(\log \left(X \prod_{p \in {\mathcal{P}}} p^{\ell_p}\right)\right)^2 \frac{1}{q} \prod_{p' \mid q} \frac{1}{1 - {p'}^{-1/2}}\right),\end{aligned}$$ and it remains to take $$X = \operatorname{vol}\left(\Gamma(q) \backslash {\mathbb{H}}\right)^{4\mu_0/3}, \qquad \ell_p = \left\lfloor \frac{\mu_p \log \operatorname{vol}\left(\Gamma(q) \backslash {\mathbb{H}}\right)^{4/3}}{\log p} \right\rfloor.\qedhere$$
We use and in place of and , so that $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_0(q,\chi) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} q^{1 + \e} X^{-2\alpha_0} \prod_{p \in {\mathcal{P}}} \left(\frac{\alpha_p}{2}\right)^{-2\ell_p} \left(1 + \sqrt{X} \prod_{p \in {\mathcal{P}}} p^{\ell_p/2} \left(\log \left(X \prod_{p \in {\mathcal{P}}} p^{\ell_p}\right)\right)^2 \frac{2^{\omega(q)} \dot{Q}}{\varphi(q)}\right).\end{gathered}$$ We find that $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_0(q,\chi) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{1 - 4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right) + \e} \dot{Q}^{4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right)}.\end{gathered}$$ by taking $$X = \operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)^{2\mu_0} \dot{Q}^{-2\mu_0}, \qquad \ell_p = \left\lfloor \frac{\mu_p \log \left(\operatorname{vol}\left(\Gamma_0(q) \backslash {\mathbb{H}}\right)^2 \dot{Q}^{-2}\right)}{\log p} \right\rfloor.$$ Again, we also have that $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_0(q,\chi) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\leq \#\left\{f \in {\mathcal{B}}_0(q \ddot{Q},\chi \psi^2) : it_f \in (\alpha_0,1/2), \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\}\end{gathered}$$ for any primitive character $\psi$ modulo $\ddot{Q}$, which implies that $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_0(q,\chi) : it_f \in (\alpha_0,1/2) \in [0,T], \ \left|\lambda_f(p)\right| \geq \alpha_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\ll_{\e} \operatorname{vol}(\Gamma_0(q) \backslash {\mathbb{H}})^{1 - 4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right) + \e} \ddot{Q}^{1 - 4 \left(\mu_0 \alpha_0 + \sum_{p \in {\mathcal{P}}} \mu_p \frac{\log \alpha_p/2}{\log p}\right)}.
\qedhere\end{gathered}$$
Improving Theorems \[Sarnakthm\] and \[Huxleythm\] for 323 04001(q) via Twisting
================================================================================
If we take any primitive character $\psi$ modulo $q_{\psi}$ with $q_{\psi} \mid q$, then the twisted cusp form $$(f \otimes \psi)(z) {\mathrel{\rlap{ \raisebox{0.3ex}{$\m@th\cdot$}} \raisebox{-0.3ex}{$\m@th\cdot$}} =}\frac{1}{\tau\left(\overline{\psi}\right)} \sum_{d \hspace{-.25cm} \pmod{q_{\psi}}} \overline{\psi}(d) f\left(z + \frac{d}{q_{\psi}}\right)$$ of $f \in {\mathcal{B}}_{\kappa}(q,\chi)$ is a nontrivial element of ${\mathcal{A}}_{\kappa}\left(q^2, \chi \psi^2\right)$ by [@AL Proposition 3.1], where $\tau\left(\overline{\psi}\right)$ denotes the Gauss sum of $\overline{\psi}$. Note that twisting by a Dirichlet character preserves the Laplacian eigenvalue $\lambda_f = 1/4 + t_f^2$ and the absolute value $|\lambda_f(n)|$ of the Hecke eigenvalues of $f$ for all $(n,q) = 1$. Moreover, if $f_1 \in {\mathcal{B}}_{\kappa}\left(q,\chi_1\right)$, $f_2 \in {\mathcal{B}}_{\kappa}\left(q,\chi_2\right)$ are such that there exist primitive Dirichlet characters $\psi_1$ modulo $q_{\psi_1}$ and $\psi_2$ modulo $q_{\psi_2}$ with $q_{\psi_1}, q_{\psi_2} \mid q$ such that $$f_1 \otimes \psi_1 = f_2 \otimes \psi_2,$$ then $f_2 = f_1 \otimes \psi_1 \overline{\psi_2}$.
If $q$ is squarefree, $\psi$ is a primitive Dirichlet modulo $q_{\psi}$, where $q_{\psi} \mid q$, and $f \in {\mathcal{B}}_{\kappa}(q,\chi)$, then $f_1 \otimes \psi \in {\mathcal{A}}_{\kappa}\left(\Gamma_1(q)\right)$ if and only if $\overline{\psi}$ divides $\chi$, in the sense that $\psi \chi$ has conductor dividing $q_{\chi}$.
This follows via the methods of [@Hum]. For $p \mid q$, let $\pi_p$ be the local component of the cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_2({\mathbb{A}}_{{\mathbb{Q}}})$ associated to $f$, so that the central character $\omega_p$ of $\pi_p$ is the local component of the Hecke character $\omega$ that is the idèlic lift of $\chi$. As $q$ is squarefree, $\pi_p$ is either a principal series representation or a special representation.
In the former case, $\pi_p = \omega_{p,1} \boxplus \omega_{p,2}$ with central character $\omega_p = \omega_{p,1} \omega_{p,2}$, where $\omega_{p,1},\omega_{p,2}$ are characters of ${\mathbb{Q}}_p^{\times}$ with conductor exponents $c(\omega_{p,1}), c(\omega_{p,2}) \in \{0,1\}$ such that the conductor exponent $c(\pi_p)$ of $\pi_p$ is $c(\omega_{p,1}) + c(\omega_{p,2}) \in \{0,1\}$. The twist $\pi_p \otimes \omega_p'$ of $\pi_p$ by a character $\omega_p'$ of ${\mathbb{Q}}_p^{\times}$ of conductor exponent $c(\omega_p') \in \{0,1\}$ is $\omega_{p,1} \omega_p' \boxplus \omega_{p,2} \omega_p'$ with corresponding conductor exponent $c(\pi_p \otimes \omega_p') = c(\omega_{p,1} \omega_p') + c(\omega_{p,2} \omega_p')$. For this to be at most $1$, either $\omega_p'$ is unramified, or one of $c(\omega_{p,1} \omega_p'), c(\omega_{p,2} \omega_p')$ must be equal to $0$, so that $\overline{\omega_p'}$ is equal to $\omega_{p,1}$ or $\omega_{p,2}$ up to multiplication by an unramified character.
In the latter case, $\pi_p = \omega_{p,1} {\mathrm{St}}$ with central character $\omega_p = \omega_{p,1}^2$ such that $c(\omega_{p,1}) = 0$, so that $c(\pi_p) = 1$. The twist of $\pi_p$ by $\omega_p'$ is $\omega_{p,1} \omega_p' {\mathrm{St}}$, with corresponding conductor exponent $c(\pi_p \otimes \omega_p') = \max\{1, 2c(\omega_{p,1} \omega_p')\}$. For this to be at most $1$, $\omega_p'$ must be unramified.
It follows that if the Hecke character $\omega'$ is the idèlic lift of $\psi$, then the conductor of $\pi \otimes \omega'$ divides $q$ if and only if the conductor of $\omega' \omega$ divides the conductor of $\omega$.
From this, we have the following.
There are at most $\tau(q)$ nontrivial members of ${\mathcal{B}}_{\kappa}(q,\chi)$ that can be twisted by a Dirichlet character of conductor dividing $q$ to give the same member of ${\mathcal{A}}_{\kappa}\left(\Gamma_1(q)\right)$.
Let $q$ be squarefree, let ${\mathcal{P}}$ be a finite collection of primes not dividing $q$, let $E_0$ be a measurable subset of $[0,\infty) \cup i(0,1/2)$, and let $E_p$ be a measurable subset of $[0,\infty)$ for each $p \in {\mathcal{P}}$. Then $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) : t_f \in E_0, \ |\lambda_f(p)| \in E_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
\leq \frac{\tau(q)}{\varphi(q)}
\#\left\{f \in {\mathcal{B}}_{\kappa}\left(\Gamma(q)\right) : t_f \in E_0, \ |\lambda_f(p)| \in E_p \text{ for all $p \in {\mathcal{P}}$}\right\}.\end{gathered}$$
From , we have that $$\begin{gathered}
\#\left\{f \in {\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right) : t_f \in E_0, \ |\lambda_f(p)| \in E_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
= \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \#\left\{f \in {\mathcal{B}}_{\kappa}(q,\chi) : t_f \in E_0, \ |\lambda_f(p)| \in E_p \text{ for all $p \in {\mathcal{P}}$}\right\} \\
= \frac{1}{\varphi(q)} \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \sum_{\psi \hspace{-.25cm} \pmod{q}} \#\left\{f \otimes \psi \in {\mathcal{A}}_{\kappa}(q^2,\chi \psi^2) : f \in {\mathcal{B}}_{\kappa}(q,\chi), \ t_f \in E_0, \ \right. \\
\left. |\lambda_f(p)| \in E_p \text{ for all $p \in {\mathcal{P}}$}\right\}.\end{gathered}$$ Note that $g = f \otimes \psi$ has level dividing $q^2$ and nebentypus of conductor dividing $q$, and there are at most $\tau(q)$ elements of ${\mathcal{B}}_{\kappa}\left(\Gamma_1(q)\right)$ that can be twisted by a Dirichlet character of conductor dividing $q$ that yield $g$. So the above quantity is bounded by $$\frac{\tau(q)}{\varphi(q)} \sum_{\substack{\chi \hspace{-.25cm} \pmod{q} \\ \chi(-1) = (-1)^{\kappa}}} \#\left\{f \in {\mathcal{B}}_{\kappa}(q^2,\chi) : t_f \in E_0, \ |\lambda_f(p)| \in E_p \text{ for all $p \in {\mathcal{P}}$}\right\},$$ which, upon recalling the explicit basis of ${\mathcal{B}}_{\kappa}(\Gamma(q))$, yields the result.
Combining this with the fact that $\operatorname{vol}\left(\Gamma(q) \backslash {\mathbb{H}}\right) = q \operatorname{vol}\left(\Gamma_1(q) \backslash {\mathbb{H}}\right)$, we can improve and \[Huxleythm\] for $\Gamma_1(q)$ with $q$ squarefree.
When $q$ is squarefree, and hold with the exponent $3$ replaced by $4$.
It is likely that a more careful analysis could obtain this same result even when $q$ is not squarefree via the methods in [@Hum].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author thanks Peter Sarnak for many helpful discussions on this topic.
[BrMia09]{}
A. O. L. Atkin and Wen-Ch’ing Winnie Li, “Twists of Newforms and Pseudo-Eigenvalues of $W$-Operators”, *Inventiones Mathematicae* **48**:3 (1978), 221–243. <span style="font-variant:small-caps;">doi</span>:[10.1007/BF01390245](https://doi.org/10.1007/BF01390245)
Valentin Blomer, Jack Buttcane, and Nicole Raulf, “A Sato–Tate Law for ${\mathrm{GL}}(3)$”, *Commentarii Mathematici Helvetici* **89**:4 (2014), 895–919. <span style="font-variant:small-caps;">doi</span>:[10.4171/CMH/337](https://doi.org/10.4171/CMH/337)
Valentin Blomer, Gergely Harcos, and Philippe Michel, “Bounds for Modular $L$-Functions in the Level Aspect”, *Annales Scientifiques de l’École Normale Supérieure, $4^{\mathrm{e}}$ série* **40**:5 (2007), 697–740. <span style="font-variant:small-caps;">doi</span>:[10.1016/j.ansens.2007.05.003](https://doi.org/10.1016/j.ansens.2007.05.003)
Valentin Blomer and Djordje Milićević, “The Second Moment of Twisted Modular $L$-Functions”, *Geometric and Functional Analysis* **25**:2 (2015), 453–516. <span style="font-variant:small-caps;">doi</span>:[10.1007/s00039-015-0318-7](https://doi.org/10.1007/s00039-015-0318-7)
Andrew R. Booker and Andreas Strömbergsson, “Numerical Computations with the Trace Formula and the Selberg Eigenvalue Conjecture”, *Journal für die reine und angewandte Mathematik* **607** (2007), 113–161. <span style="font-variant:small-caps;">doi</span>:[10.1515/CRELLE.2007.047](https://doi.org/10.1515/CRELLE.2007.047)
Roelof W. Bruggeman and Roberto J. Miatello, *Sum Formula for ${\mathrm{SL}}_2$ over a Totally Real Number Field*, Memoirs of the American Mathematical Society **197**:919, American Mathematical Society, Providence, 2009. <span style="font-variant:small-caps;">doi</span>:[10.1090/memo/0919](https://doi.org/10.1090/memo/0919)
J. B. Conrey, W. Duke, and D. W. Farmer, “The Distribution of the Eigenvalues of Hecke Operators”, *Acta Arithmetica* **78**:4 (1997), 405–409. <https://eudml.org/doc/206959>
W. Duke, J. B. Friedlander, and H. Iwaniec, “The Subconvexity Problem for Artin $L$-Functions”, *Inventiones Mathematicae* **149**:3 (2002), 489–577. <span style="font-variant:small-caps;">doi</span>:[10.1007/s002220200223](https://doi.org/10.1007/s002220200223)
I. S. Gradshteyn and I. M. Ryzhik, *Table of Integrals, Series, and Products, Seventh Edition*, editors Alan Jeffrey and Daniel Zwillinger, Academic Press, Burlington, 2007.
C. P. Hughes and Steven J. Miller, “Low-Lying Zeros of $L$-Functions with Orthogonal Symmetry”, *Duke Mathematical Journal* **136**:1 (2007), 115–172. <span style="font-variant:small-caps;">doi</span>:[10.1215/S0012-7094-07-13614-7](https://doi.org/10.1215/S0012-7094-07-13614-7)
Peter Humphries, “Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level”, preprint (2015), 26 pages. arXiv:[math.NT/1502.06885](https://arxiv.org/abs/1502.06885)
M. N. Huxley, “Exceptional Eigenvalues and Congruence Subgroups”, in *The Selberg Trace Formula and Related Topics*, editors Dennis A. Hejhal, Peter Sarnak and Audrey Anne Terras, Contemporary Mathematics **53**, American Mathematical Society, Providence, 1986, 341–349. <span style="font-variant:small-caps;">doi</span>:[10.1090/conm/053/853564](https://doi.org/10.1090/conm/053/853564)
Henryk Iwaniec, *Spectral Methods of Automorphic Forms, Second Edition*, Graduate Studies in Mathematics **53**, American Mathematical Society, Providence, 2002. <span style="font-variant:small-caps;">doi</span>:[10.1090/gsm/053](https://doi.org/10.1090/gsm/053)
Henryk Iwaniec and Emmanuel Kowalski, *Analytic Number Theory*, American Mathematical Society Colloquium Publications **53**, American Mathematical Society, Providence, 2004. <span style="font-variant:small-caps;">doi</span>:[10.1090/coll/053](https://doi.org/10.1090/coll/053)
Henryk Iwaniec, Wenzhi Luo, and Peter Sarnak, “Low Lying Zeros of Families of $L$-Functions”, *Publications Mathématiques de l’Institut des Hautes Études Scientifiques* **91**:1 (2000), 55–131. <span style="font-variant:small-caps;">doi</span>:[10.1007/BF02698741](https://doi.org/10.1007/BF02698741)
Henry H. Kim, “Functoriality for the Exterior Square of ${\mathrm{GL}}_4$ and the Symmetric Fourth of ${\mathrm{GL}}_2$”, Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Henry H. Kim and Peter Sarnak, *Journal of the American Mathematical Society* **16**:1 (2003), 139–183. <span style="font-variant:small-caps;">doi</span>:[10.1090/S0894-0347-02-00410-1](https://doi.org/10.1090/S0894-0347-02-00410-1)
A. Knightly and C. Li, *Kuznetsov’s Trace Formula and the Hecke Eigenvalues of Maass Forms*, Memoirs of the American Mathematical Society **224**:1055, American Mathematical Society, Providence, 2013. <span style="font-variant:small-caps;">doi</span>:[10.1090/S0065-9266-2012-00673-3](https://doi.org/10.1090/S0065-9266-2012-00673-3)
N. V. Kuznecov, “The Petersson Conjecture for Cusp Forms of Weight Zero and the Linnik Conjecture. Sums of Kloosterman Sums”, *Mathematics of the USSR. Sbornik* **39** (1981), 299–342. <span style="font-variant:small-caps;">doi</span>:[10.1070/SM1981v039n03ABEH001518](https://doi.org/10.1070/SM1981v039n03ABEH001518)
Xiannan Li, “Upper Bounds for $L$-Functions at the Edge of the Critical Strip”, *International Mathematics Research Notices* **2010**:4 (2010), 727–755. <span style="font-variant:small-caps;">doi</span>:[10.1093/imrn/rnp148](https://doi.org/10.1093/imrn/rnp148)
The LMFDB Collaboration, *The $L$-Functions and Modular Forms Database*, <http://www.lmfdb.org>, 2013, \[Online; accessed 30 January 2017\].
Péter Maga, “A Semi-Adelic Kuznetsov Formula over Number Fields”, *International Journal of Number Theory* **9**:7 (2013), 1649–1681. <span style="font-variant:small-caps;">doi</span>:[10.1142/S1793042113500498](https://doi.org/10.1142/S1793042113500498)
Ian Petrow and Matthew P. Young, “A Generalized Cubic Moment and the Petersson Formula for Newforms”, preprint (2016), 53 pages. arXiv:[math.NT/1608.06854](https://arxiv.org/abs/1608.06854)
Peter Sarnak, “Statistical Properties of Eigenvalues of the Hecke Operators”, in *Analytic Number Theory and Diophantine Problems. Proceedings of a Conference at Oklahoma State University, 1984*, editors A. C. Adolphson, J. B. Conrey, A. Ghosh, and R. I. Yager, Progress in Mathematics **70**, Birkhäuser, Boston, 1987, 321–331. <span style="font-variant:small-caps;">doi</span>:[10.1007/978-1-4612-4816-3\_19](https://doi.org/10.1007/978-1-4612-4816-3_19)
|
---
abstract: '[ We investigate the energy relaxation of hot carriers produced by photoexcitation of graphene through coupling to both intrinsic and remote (substrate) surface polar phonons using the Boltzmann equation approach. We find that the energy relaxation of hot photocarriers in graphene on commonly used polar substrates, under most conditions, is dominated by remote surface polar phonons. We also calculate key characteristics of the energy relaxation process, such as the transient cooling time and steady state carrier temperatures and photocarriers densities, which determine the thermoelectric and photovoltaic photoresponse, respectively. Substrate engineering can be a promising route to efficient optoelectronic devices driven by hot carrier dynamics. ]{}'
author:
- 'Tony Low$^1$, Vasili Perebeinos$^1$, Raseong Kim$^2\footnote{Present address: Components Research, Intel Corporation, Hillsboro, OR 97124, USA}$, Marcus Freitag$^1$ and Phaedon Avouris$^1$'
title: Cooling of photoexcited carriers in graphene by internal and substrate phonons
---
Introduction
============
Upon fast excitation of graphene carriers with light or other means, the dynamics of the resulting non-equilibrium carrier distribution evolve on a fast time scale and has been extensively studied both experimentally[@SWDL08; @MSCRS08; @KPSFW05; @BRE09; @Newson_2009; @Choi_APL_2009; @Wang_APL_2008; @George_NL_2008; @Kumar_2009; @Ishioka_2008; @Seibert_1990; @Winnerl_2011] and theoretically[@Winzer_2010; @KPA11; @Malic_PRB_2011]. The relaxation involves an initial fast evolution towards quasi-thermal distribution on a femtosecond timescale via electron-electron collisions[@Hertel_2000; @Heinz_PRL_2010; @Winzer_2010; @KPA11], followed by energy transfer to phonons on a longer picosecond timescale. The conversion of the excess energy of these photoexcited carriers into electrical current before they lose this energy to the phonon baths represents one of the key challenges to efficient optoelectronic device.\
In this paper, we study the energy relaxation pathways of the photoexcited carriers via different inelastic scattering channels. Energy relaxation processes in graphene due to intrinsic optical and acoustic phonons have already been studied[@BM09; @TS09; @KPA11]. High energy optical phonon emission by hot carriers is responsible for the subpicosecond fast cooling process[@SWDL08; @MSCRS08; @SAJ12], followed by cooling via the acoustic modes. The latter is a slow process, that creates an electron-phonon cooling bottleneck[@BM09]. Here, we focus on an extrinsic mechanism for cooling of photoexcited carriers in graphene via the remote surface polar phonon modes (SPP) of the substrate and compare their efficiency under different conditions with those of the internal phonon modes.\
In polar substrates such as SiO$_2$, a non-vanishing fluctuating electric field is generated by the propagating surface phonon modes[@AM76]. The interactions of these SPP modes with charged carriers in the conduction channel was first explored in the context of inversion layer of semiconductor-oxide interface[@WM72; @HV79; @FNC01]. They have also been studied in other material systems such as carbon nanotubes[@PR06; @PRPA09; @Bhupesh_2011], where close proximity between charged carriers and the underlying substrate renders the SPP-phonon scattering more prominent. Similarly, in graphene, SPP was found to limit electronic transport properties[@MHY08; @CJXIF08; @FG08; @PA10; @KFJ10; @Li_SPP; @PHS12; @SSG12; @ZHKZ10; @DZJ10]. Recently, the SPP coupling with graphene plasmons was also probed experimentally through infrared spectroscopy[@LW10; @FAB11; @CBGTH12]. In this work, we found irrespectively of the mechanism i.e. thermoelectric or photovoltaic, that SPP limits the overall strength of the steady state photoresponse on common substrates, and our results suggest that elimination of the SPP cooling channel can lead to an order of magnitude enhancement in the photoresponse.\
In Sec.\[sec:theory\], we present the general theory, where details of the models for the electron cooling power via the different phonon baths are presented in the Appendix. We present the results of our calculation of the cooling powers in Sec.\[sec:results\]A and discuss their relative contribution in detail, as a function of doping and electronic/lattice temperatures. In Sec.\[sec:results\]B, we apply the above models to the study of the cooling dynamics of hot carriers due to continuous or pulsed light excitations. We calculate key experimental observables such as the transient cooling time, and steady state quantities such as the non-equilibrium electronic temperatures, excess photocarriers density and the out-of-plane thermal conductivity for graphene on common substrates.
\[sec:theory\] Theory and Models
================================
Transition probability for emission and absorption of phonons with a particular phonon bath $\alpha$ is described by the Fermi’s golden rule, $$\begin{aligned}
S_{\alpha}(\bold{k},\bold{k}')=\frac{2\pi}{\hbar}\sum_{\bold{q}}\frac{1}{A}\left|M^{\alpha}_{\bold{k},\bold{k}'}\right|^2\left\{ N_{\omega_q}
\delta_{\bold{k}'-\bold{k}-\bold{q}}\delta(E_\bold{k}'-E_\bold{k}-\hbar\omega_{q})+
(N_{\omega_q}+1)
\delta_{\bold{k}'-\bold{k}+\bold{q}}\delta(E_\bold{k}'-E_\bold{k}+\hbar\omega_{q})
\right\}\end{aligned}$$ where $\bold{q}$ is the phonon momentum, $N_{\omega_q}=[\mbox{exp}(\hbar\omega_{q}/k_{B}T_L)-1]^{-1}$ is the Bose-Einstein distribution and $M_{\bold{k},\bold{k}'}^{\alpha}$ are the transition matrix elements related to the coupling with phonon bath $\alpha$, to be defined below. For brevity, summation $\sum_{\bold{q}}\delta_{\bold{k}'-\bold{k}\pm\bold{q}}$ shall be implicit hereafter. The cooling power is computed numerically via, $$\begin{aligned}
\nonumber
{\cal P}^{\alpha}_{ss'} &=& \frac{g_s g_v}{A}\sum_{\bold{k}}\sum_{\bold{k}'} S_{\alpha}(\bold{k},\bold{k}')
(E_{\bold{k}}-E_{\bold{k}'})f_{\bold{k}}(1-f_{\bold{k}'})\\
\nonumber
&=& \frac{2\pi g_s g_v}{\hbar A^2} \sum_{\bold{k},\bold{k}'} \left|M^{\alpha}_{\bold{k},\bold{k}'}\right|^2
\delta(E_\bold{k}'-E_\bold{k}-\hbar\omega_{q})(E_{\bold{k}'}-E_{\bold{k}}) {\cal F}(k,k')\\
&=& \frac{g_s g_v }{(2\pi)^2\hbar } \int_{0}^{\infty}k dk \int_{0}^{\infty}k' dk' \int_{0}^{2\pi} d\theta
\left|M^{\alpha}_{\bold{k},\bold{k}'}\right|^2
\delta(k'-k-\omega_{q}v_{F}^{-1})(k'-k) {\cal F}(k,k')
\label{cpower}\end{aligned}$$ where $T_{L}$ and $T_{E}$ are the lattice and electron temperatures respectively, and the electron distribution function is described by $f_{\bold{k}}$, and we define a composite Fermi-Boson distribution function as, $$\begin{aligned}
{\cal F}(k,k')&\equiv& (N_{\omega_q}+1)f_{\bold{k}'}(1-f_{\bold{k}})-N_{\omega_q}f_{\bold{k}}(1-f_{\bold{k}'})
%\\
%&=& \frac{\mbox{exp}(\frac{E_{k}-\mu}{k_{B}T_{E}})\left(\mbox{exp}(\frac{\hbar\omega_{q}}{k_{B}T_{L}})-\mbox{exp}(\frac{\hbar\omega_{q}}{k_{B}T_{E}})\right)}
%{\left(\mbox{exp}(\frac{\hbar\omega_{q}}{k_{B}T_{L}})-1\right)
%\left(\mbox{exp}(\frac{E_{k}-\mu}{k_{B}T_{E}})+1\right)\left(\mbox{exp}(\frac{E_{k'}-\mu}{k_{B}T_{E}})+1\right)}
\label{comF}\end{aligned}$$ As indicated by experiments[@SWDL08; @MSCRS08; @Heinz_PRL_2010], the electronic system is thermalized by the electron-electron interactions which occur at much faster timescale than the electron-phonon processes we are calculating here. Hence, it is appropriate to simply assume that $f_{\bold{k}}$ follows the Fermi Dirac distribution function, i.e. $[1+\mbox{exp}(\beta(E_{\bold{k}}-\mu))]^{\mbox{-}1}$ where $\beta=1/k_B T_E$ and $\mu$ is the chemical potential. It is apparent that the composite electron-phonon distribution function ${\cal F}(k,k')$ becomes zero when $T_{E}=T_{L}$, hence zero cooling power.\
In this work, we are interested in the energy exchange of electrons with the different phonon baths i.e. intrinsic acoustic phonons (AP), optical phonons (OP) and surface phonon polaritons (SPP). Their transition matrix elements, $M_{\bold{k},\bold{k}'}^{\alpha}$, are well-known in the literatures[@HS08; @LPM05; @ANDO06; @FG08; @FNC01; @PA10]. We therefore defer their discussions to the Appendix, while focusing on the key results in what follows.
\[sec:results\] Results and discussions
=======================================
Competing cooling pathways
--------------------------
We begin with a simple illustration of the possible cooling pathways for photoexcited carriers in graphene in Fig.\[fig1\]. Each thermal bath can be characterized by their respective temperatures $T_{\alpha}$, and are in general different from the ambient temperature $T_0$. The heat exchange between these thermal baths can be described by the thermal conductivity, $\kappa$, defined as the ratio between the power exchange per unit temperature difference i.e. $\delta P/\delta T$. In general, the different phonon baths can each establish a different temperature upon interactions with the electrons (see also discussion in Sec.\[sec:cooldy\]). In this work, we shall assume a common temperature for all these phonon baths, denoted simply as the lattice temperature $T_L$.\
For a typical SiO$_2$ substrate thickness of $h=50-300\,$nm, $\kappa_0=\kappa/h$ varies in a range $\approx 5\,$MW/Km$^2 - 10\,$MW/Km$^2$, where SiO$_2$ film thermal conductivity is $\kappa=0.5-1.4\,$W/mK[@YNKT02]. The interface thermal conductance of graphene on SiO$_2$ substrate has been measured using various experimental techniques[@Balan11; @Chen_APL_2009; @Freitag_NL_2009; @Heinz_APL_2010; @Pop_NL_2010], with values ranging from $\approx 25\,$MW/Km$^2 - 180\,$MW/Km$^2$. On the theory front, several approaches have been employed to estimate this interface thermal conductance[@Pop_PRB_2010; @PU10; @Persson_2010; @Volokitin_2011; @Rotkin_unpub], which varies from $\approx 1\,$MW/Km$^2 - 100\,$MW/Km$^2$. As illustrated in Fig.\[fig1\], energy transfered from electrons to the internal phonon baths is conducted to the underlying substrate through a phonon-limited $\kappa_{TB}$. $\kappa_{TB}$ between carbon surface and SiO$_2$ substrate has been estimated from molecular dynamics and is $\approx 60\,$MW/Km$^2$[@Pop_PRB_2010], and can depend also on the surface roughness. Alternatively, energy can be transfered directly to the substrate via the SPP phonons, i.e. $\kappa_{SPP}$, and can depend sensitively on doping. For an undoped graphene, $\kappa_{SPP}$ is on the order of $1\,$MW/Km$^2$ while $\kappa_{LAT}$ is even smaller, as we will see later in the discussion. In this section, we discuss how these cooling pathways depend on the various experimental conditions.\
Detailed balance condition of in- and out-scattering processes requires that ${\cal P}^{\alpha}$ vanishes under equilibrium condition i.e. $\delta T\equiv T_E - T_L=0$. In the theory, this is ensured by the composite Fermi-Boson distribution function ${\cal F}(T_E,T_L)$. The energy exchange efficiency with these various phonon baths depends upon, among other factors, the doping and electronic/lattice temperatures. Using the models described in Sec.\[sec:theory\], we calculate ${\cal P}^{\alpha}(T_E)$ due to the various phonon baths for intrinsic/doped graphene under cold/hot (defined at $T_L=10,$ $300\,$K respectively) lattice temperature as shown in Fig.\[fig2\].\
First, we discuss results on intrinsic graphene, see Fig.\[fig2\](a,c,e), which can be understood on the basis of scattering phase space arguments. For cold neutral graphene, Pauli blocking limits the electronic transitions involved to mainly interband processes. Hence, under near equilibrium condition i.e. $\delta T$ being small, we observed that the cooling power is mainly dominated by interband processes by optical and SPP modes. Increasing the electronic temperature alleviates Pauli blocking, and allows for intraband processes to take place. As $\delta T$ increases further, we observe that intraband cooling begins to dominate over the interband counterpart. The efficiency of energy exchange can be explained by the electron-phonon occupation number, quantified by the composite distribution function ${\cal F}(T_E,T_L)$ defined in Eq.\[comF\]. For inelastic processes, one can show that ${\cal F}(T_E,T_L)$ is independent of $T_L$ when $\delta T\rightarrow\infty$. This is as reflected in Fig.\[fig2\] for ${\cal P}^{OP,K}$ and ${\cal P}^{SPP,H}$. On the other hand, for quasi-elastic acoustic phonons, the cooling power is proportional $\delta T/T_E$ instead.\
The results on doped graphene are shown in Fig.\[fig2\](b,d,f). Contrary to the intrinsic case, Pauli blocking promotes intraband electronic transitions over interband processes in doped graphene. In addition, ${\cal P}^{\alpha}_{cc}\neq {\cal P}^{\alpha}_{vv}$, with larger cooling power for the majority carriers. At moderate doping of $\mu = 0.2\,$eV, their cooling power differs by more than an order of magnitude. The reduced electron-hole symmetry upon doping also leads to smaller interband cooling power. Quasi-elasticity of acoustic phonon scattering results in a phase space restriction in the scattering, with a Bloch-Gr$\ddot{u}$neisen temperature determined by the doping[@HS08; @EK10], i.e. $T_{BG}=2\hbar v_{S}k_F /k_B$, in contrast to normal metals. This increase in phase space in conjunction with Pauli blocking greatly enhances the cooling power due to AP over the optical phonon baths. In fact, for moderate $T_E\lesssim 100\,$K, ${\cal P}^{AP}$ dominates over all other mechanisms for cold graphene.\
The lattice temperature, $T_L$, also plays an important role in the competing cooling pathways. Fig.\[fig3\] compares the fractional cooling powers ${\cal P}^\alpha$/${\cal P}^{T}$ for intrinsic graphene, where ${\cal P}^{T}=\sum_{\alpha}{\cal P}^\alpha$. To obtain a quantitative estimate, we include in-plane screening of the SPP scattering potential in graphene. The screening is incorporated through a standard procedure[@FG97] $\left|M_{\bold{k},\bold{k}'}\right|$$\rightarrow$$\left|M_{\bold{k},\bold{k}'}\right|/\epsilon_{2D}(\bold{q},\omega)$. For simplicity, we employed the static screening dielectric function $\epsilon_{2D}(\bold{q},0)$, which in the long-wavelength limit assumes a simple form[@Ando06b] $\epsilon_{2D}(\bold{q},0)$$\approx$$ 1+q_s/q$, where $q_s$=$ e^2/2\epsilon_{0}\kappa \int \tfrac{\partial f}{\partial \epsilon} {\cal D}(\epsilon)d\epsilon$ and ${\cal D}$ is graphene density-of-states. $f$ is the Fermi distribution function and is a function of the electronic temperature.\
We analyze the results in two non-equilibrium temperature limits, namely “near equilibrium" ($T_E-T_L=10\,$K) and “far from equilibrium" ($T_E-T_L=100\,$K) conditions. Fig.\[fig3\]a considers the condition of “near equilibrium". At low $T_L$, AP dominates cooling. Increasing $T_L$ populates the low-energy SPP mode, which begins to overtake the cooling power at a temperature of $\sim 20\,$K. This transition temperature increases with doping e.g. is $\sim 50\,$K at a doping of $0.1\,$eV. The low energy SPP mode is overtaken by its high-energy mode at $\sim 170\,$K. A downturn in the high-energy SPP cooling power is observed, due to larger screening at higher temperatures. Eventually, the optical phonons overtake the SPP for temperatures larger than $1000\,$K. Fig.\[fig3\]b considers the condition of “far from equilibrium". In this case, the SPP dominates the cooling power for all $T_L$, except at temperature $>$$1000\,$K where optical phonons begin to overtake it.
\[sec:cooldy\] Cooling dynamics
-------------------------------
We are interested in the role played by these various phonon baths on the cooling dynamics of photoexcited carriers, more specifically, the temporal evolution of $T_E$. The acoustic and optical phonon baths can each establish a different temperature upon interactions with the electrons, but processes such as anharmonic phonon-phonon scattering serve to thermalize them on a picosecond time scale[@KPSFW05; @BLMM07; @Heinz_PRL_2008; @YSMC09]. In this work, we shall assume a common lattice temperature $T_L$, but acknowledge that in experiments with ultrafast pump-probe, this will not hold true. On the other hand, under continuous light excitation, coupling to the heat sink via the supporting substrate substantially cools the lattice temperature to within a few degrees Kelvin of the ambient temperature $T_0$ under usual photoexcitation conditions[@FLXA12]. Typically, $T_E-T_L$$\gg 1 K$ under low/moderate excitation power levels used in our studies. In this regard, the relatively small differences among the various phonon baths can be safely ignored.\
Hot carriers dynamics can be probed through optical measurements[@SHAH92; @SWDL08; @MSCRS08; @KPSFW05]. Following a pulsed light excitation, the temporal evolution of carrier relaxation, quantified by its electronic temperature $T_E$, can be measured using differential transmission spectroscopy. The electron dynamics are usually described by $\Delta T_E\propto \mbox{exp}(-t/\tau_E)$, and can be estimated with[@SRL11], $$\begin{aligned}
\tau_E = {\cal C} \left(\frac{d{\cal P}^T}{dT_E}\right)^{-1}
\label{tauE}\end{aligned}$$ where ${\cal C}=d{\cal E}/dT_E$ is the electron specific heat and ${\cal E}$ is the energy density of graphene. In this work, ${\cal C}$ is computed numerically. However, we note that for $T_E\ll \mu/k_B$, ${\cal C}$ increases linearly with $T_E$, i.e. ${\cal C}\approx \tfrac{2\pi^2}{3}{\cal D}(\mu)k_B^2 T_E$. Having computed the total cooling power ${\cal P}^T$ in Sec.\[sec:results\], $\tau_E$ can be obtained directly from Eq.\[tauE\].\
Fig.\[fig4\] plots the cooling time, $\tau_E$, for neutral graphene at $T_L=10\,$K. It is calculated for common substrates such as SiO$_2$, BN and non-polar substrate such as diamond. At very hot electron temperatures, i.e. $T_E$$>$$500\,$K, $\tau_E$ is given by a relatively constant sub-picosecond cooling time. This is in agreement with experiments[@SAJ12]. The constancy of $\tau_E$ suggests the $\mbox{exp}(-t/\tau_E)$ decay characteristics typical during the initial fast cooling process. As $T_E$ cools down, the cooling bottleneck due to AP begins to set in, leading to much slower cooling times. The transition temperature into this slow cooling regime varies with the choice of substrate as indicated in Fig.\[fig4\]. This transition temperature is dictated by the lowest frequency SPP mode of the substrate. In BN, the SPP modes are located at energies $\hbar\omega_{L,H}=101.7\,$meV and $195.7\,$meV respectively, with Froehlich coupling $F_{L}^{2}=0.258\,$meV and $F_{H}^{2}=0.52\,$meV respectively[@PA10; @BN_par], see also Appendix for details about the model. Unscreened results, which overestimate the SPP cooling power, yield much shorter cooling lifetimes than experimentally reported[@SAJ12]. We also note that inclusion of disorder assisted cooling[@SRL11], might enhance the decay rate, especially in the slow cooling regime.\
The optoelectronic response in graphene, photovoltaic[@XMLGA09; @LBW08; @PAR09] or thermoelectric[@GSM11; @XGA09; @SAJ12], is also a measure of the energy transport of these hot carriers. These experiments are usually performed under a continuous light illumination of an electrostatic junction. Their relative contribution depends on the electrostatic junction characteristics, doping, and even extrinsic factors such as electron-hole puddles[@SRML11]. Nevertheless, at steady state, the photovoltaic current is proportional to the photo-generated excess carrier density, $\delta n$, via $e\delta n\mu_n \xi$ where $\mu_n$ is the carrier mobility and $\xi$ the local electric field. The thermoelectric response, on the other hand, is proportional to the local elevated temperature, $\delta T$=$T_E-T_L$, via $\sigma(S_1-S_2)\delta T$ where $S_{1,2}$ is the Seebeck coefficient of the two junction and $\sigma$ is the device conductivity. Here, we discuss estimates of $\delta n$ and $\delta T$.\
Under steady state condition, $$\begin{aligned}
{\cal P}^{0}=\sum_{\alpha} {\cal P}^{\alpha}+{\cal P}^{M}
\label{heatcons}\end{aligned}$$ where ${\cal P}^{0}$ is the laser power absorbed by graphene, and ${\cal P}^{M}$ is the heat dissipation via the metallic contacts, if any. In the absence of contacts, all heat dissipation is via the supporting substrate. At steady state, ${\cal P}^{AP}+{\cal P}^{OP}+{\cal P}^{SPP}\approx (T_L-T_0)\kappa_{0}$. Eq.\[heatcons\] is then solved self-consistently in conjunction with charge conservation i.e. $\delta n=n_{e}(T_E,\mu)-n_{e}(T_0,\mu_0)$ =$n_{h}(T_E,\mu)-n_{h}(T_0,\mu_0)$, arriving at steady state values for $T_E$ and $\mu$. The photoexcited carrier density, $\delta n$, and the elevated temperature, $\delta T$ are plotted in Fig.\[fig5\](a-b) respectively, assuming typical experimental values of ${\cal P}^0 = 1\times 10^7\,$W/m$^{2}$ and $\kappa_0 = 10\,$MW/Km$^{2}$. Both $\delta n$ and $\delta T$ decrease with increasing ambient temperature $T_0$, due to more efficient cooling as phonon occupation increases. For SiO$_2$, $\delta n\approx 10^{10}\,$cm$^{-2}$ and $\delta T\approx 10\,$K under room temperature condition, of the same order typically seen in measurement[@FLXA12], and these values decrease with doping.\
We also estimate the heat dissipation via contacts phenomenologically with ${\cal P}^M$$\approx$ $\int(f-f_0)(E-\mu){\cal D}/\tau_M dE$ where $f_0$ is the distribution function before light excitation. First, we consider the simple ballistic limit where $\tau_M$ is just the device lifetime given by $L/v_F$, where $L$ is the length of the device and $v_F\approx 10^6$ m/s is the Fermi velocity. Here, we assume a typical $\tau_M=1\,$ps. We found that including ${\cal P}^M$ only leads to few-fold decrease in the quantitative results presented in Fig.\[fig5\] (not shown). In the realistic case where the carrier transport is in the diffusive dominated regime, ${\cal P}^M$ would be even smaller, by a factor of $\lambda/L$, where $\lambda$ is the carrier’s mean free path.\
Fig.\[fig5\] also suggests an order of magnitude enhancement in the optoelectronic response of graphene, by suppressing the SPP heat dissipation through a non-polar substrate, such as diamond-like carbon[@WLBJ11], or by suspending graphene. In fact, the amount of heat transfer to the substrate via the electron coupling with the SPP can be quantified by an out-of-plane thermal conductance $\kappa_{SPP}$, defined as $\kappa_{SPP}={\cal P}^{SPP}/\delta T$. This quantity sets the lower limit on the interfacial thermal conductance and it is plotted in Fig.\[fig6\]. At room temperature, $\kappa_{SPP}\approx 1$MW/Km$^2$ for undoped graphene, and can increases with doping to order of $10$MW/Km$^2$, see also Ref.[@Rotkin_unpub]. For typical photocurrent experiments, $L$ is typically $\gg \lambda$, and transport is in the diffusive regime. Here, the in-plane electronic thermal conductivity, $\kappa_e$, can be estimated from the Wiedemann Franz relation. We found that our estimated value of $\kappa_{SPP}$ is significantly larger than $\kappa_e/L^2$ for typical experimental situations. This suggests that out-of-plane heat dissipation via SPP dominates over the in-plane electronic heat conduction. The former leads to an increased temperature of the graphene lattice. This result reconciles with recent experiment[@FLXA12], which reveals significant lattice heating upon laser excitation. From the experiment[@FLXA12], we can estimate an out-of-plane thermal conductance of $\kappa_{exp}=P^0/\delta T\approx 10\,$MW/Km$^2$. This value is consistent with our estimated $\kappa_{SPP}$. In fact, $\kappa_{LAT}$ alone is orders of magnitude smaller than the experiment as shown in Fig.\[fig6\].
\[conclude\] Conclusions
========================
Our results point to the limiting role played by remote substrate phonons in the energy relaxation of hot photocarriers. Substrate engineering therefore presents a promising route to efficient optoelectronic devices driven by hot carrier dynamics.\
*Acknowledgements:* TL acknowledges use of a computing cluster provided by Network for Computational Nanotechnology, partial funding from INDEX-NRI and in part by the NSF under Grant No. NSF PHY05-51164 (KITP). We thank F. Xia, H. Yan, F. Guinea, E. Hwang and X. Xu for useful discussions.
Acoustic phonon
===============
We consider first the energy exchange with the acoustic phonon (AP) bath. The total matrix element for electron-acoustic phonon scattering due to the two acoustic phonon modes, i.e. $\Gamma_{LA}$ and $\Gamma_{TA}$, is given by[@HS08; @PA10], $$\begin{aligned}
\left|M^{AP}_{\bold{k},\bold{k}'}\right|^2=\frac{ D_{ac}^2 \hbar q}{2\rho_{m}v_S}\end{aligned}$$ where $D_{ac}$ is the acoustic deformation potential, taken to be $7.1\,$eV in our calulations, which is very similar to the recent ab-initio calculations of $6.8\,$eV[@Kaasbjerg_2012]. We note that the electron-phonon matrix element for these two acoustic modes have different angular dependencies with transition matrix elements[@PSZR80; @Kaasbjerg_2012], which became negated after summing them[@PSZR80]. $v_S$ is graphene effective sound velocity defined as[@PA10] $2v_{S}^{-2}=v_{LA}^{-2}+v_{TA}^{-2}$, where $v_S=17\,$km/s, $v_{LA}=24\,$km/s and $v_{TA}=14\,$km/s. $\rho_{m}$ is graphene mass density taken to be $7.6\times 10^{-7}\,$kg/m$^2$. The acoustic phonon is then described by an effective Debye linear dispersion $\omega_{q}=v_{S}q$. Since $v_S \ll v_F$, $\hbar \omega_q$ is typically much smaller than other energy scale in the problem. The acoustic phonon scattering is thus approximated to be elastic[@BM09] i.e. $k'\approx k$.\
The cooling power can then be written as, $$\begin{aligned}
\nonumber
{\cal P}^{AP}_{cc}&=&\frac{g_s g_v }{(2\pi)^2\hbar } \int_{0}^{\infty}k dk \int_{0}^{\infty}k' dk' \int_{0}^{2\pi} d\theta
\frac{D_{ac}^{2}\hbar q}{2\rho_{m}v_S}
\delta(k'-k-\omega_{q}v_{F}^{-1})(k'-k) {\cal F}(k,k')\\
&\approx&\frac{g_s g_v }{(2\pi)^2\hbar } \frac{D_{ac}^{2}\hbar }{2\rho_{m}v_S}\int_{0}^{\infty} \int_{0}^{2\pi} k^2 q^2 \frac{v_S}{v_F} {\cal F}(k,k) dk d\theta\end{aligned}$$ Under the assumption $\hbar\omega_{q}\ll T_E, T_L$, we have $$\begin{aligned}
{\cal F}(k,k)\approx (1-f_{k})f_{k}\frac{T_E-T_L}{T_E}\end{aligned}$$ Making use of the relations $q^2\approx 2k^2(1-\mbox{cos}\theta)$, we then obtain a simplified form for the cooling power, $$\begin{aligned}
{\cal P}^{AP}_{cc}\approx \frac{g_s g_v D_{ac}^{2}}{2\pi \rho_{m}v_S}
\frac{T_E-T_L}{T_E}
\int_{0}^{\infty}k^4 (1-f_{k})f_{k} dk\end{aligned}$$ Contributions from interband processes, ${\cal P}^{AP}_{cv,vc}$, are forbidden due to energy-momentum conservations.
Optical phonons
===============
Next, we discuss energy exchanges with high energy dispersionless optical phonon (OP) modes i.e. $\Gamma_{LO}$, $\Gamma_{TO}$ and $\mbox{K}_{TO}$. We consider first the electron-phonon coupling of long-wavelength optical phonon modes, $\Gamma_{LO}$ and $\Gamma_{TO}$. Their sum is expressed as[@LPM05; @ANDO06], $$\begin{aligned}
\left|M^{OP,\Gamma}_{\bold{k},\bold{k}'}\right|^2=\frac{ D_{OP,\Gamma}^2 \hbar }{2\rho_{m}\omega_{o}}\end{aligned}$$ where $D_{OP,\Gamma}=3\sqrt{2}g/2\approx 11\,$eV$\AA^{-1}$ is the optical-phonon deformation potential with a coupling constant of $g=5.3\,$eV$\AA^{-1}$, and $\hbar\omega_{o}=197\,$meV. We note that the electron-phonon matrix element for these two optical modes have different angular dependencies with transition matrix elements[@LPM05; @ANDO06] i.e. $ 1\pm ss'\mbox{cos}(\theta_{\bold{k}}-\theta_{\bold{k}'})$ where $s=\pm 1$ denotes conduction/valence bands, which again became negated after summing them.\
We consider first the intraband cooling power, written as, $$\begin{aligned}
\nonumber
{\cal P}^{OP,\Gamma}_{cc}&=&\frac{g_s g_v }{(2\pi)^2\hbar } \int_{0}^{\infty}k dk \int_{0}^{\infty}k' dk' \int_{0}^{2\pi} d\theta
\frac{D_{OP,\Gamma}^{2}\hbar }{2\rho_{m}\omega_0}
\delta(k'-k-\omega_{0}v_{F}^{-1})(k'-k) {\cal F}(k,k')\\
\nonumber
&=&\frac{g_s g_v D_{OP,\Gamma}^{2}}{4\pi \rho_{m}v_{F}}\int_{0}^{\infty}k
(k+\omega_{0}v_{F}^{-1}){\cal F}(k,k+\omega_{0}v_{F}^{-1})dk\\
&=&\frac{g_s g_v D_{OP,\Gamma}^{2}}{4\pi \rho_{m}v_{F}}N_{\omega_0}
\left[\mbox{exp}(\frac{\hbar\omega_0}{k_{B}T_{L}})-\mbox{exp}(\frac{\hbar\omega_0}{k_{B}T_{E}})\right]
\int_{0}^{\infty}k(k+\omega_{0}v_{F}^{-1})(1-f_k)f_{k+\omega_{0}v_{F}^{-1}}dk\end{aligned}$$ In similar fashion, the interband cooling power is written as, $$\begin{aligned}
{\cal P}^{OP,\Gamma}_{cv}&=&\frac{g_s g_v D_{OP,\Gamma}^{2}}{4\pi \rho_{m}v_{F}}N_{\mbox{-}\omega_0}
\left[\mbox{exp}(\frac{-\hbar\omega_0}{k_{B}T_{L}})-\mbox{exp}(\frac{-\hbar\omega_0}{k_{B}T_{E}})\right]
\int_{0}^{\infty}k(\omega_{0}v_{F}^{-1}-k)H_v[\omega_{0}v_{F}^{-1}-k](1-f_k)f_{k-\omega_{0}v_{F}^{-1}}dk\end{aligned}$$ where $H_v$ is the Heaviside function.\
For zone edge phonon modes, only the transverse $\mbox{K}_{TO}$ contributes to carrier scattering, and the matrix element is[@LPM05], $$\begin{aligned}
\left|M^{OP,K}_{\bold{k},\bold{k}'}\right|^2=\frac{ D_{OP,K}^2 \hbar }{2\rho_{m}\omega_{o}}\frac{1-ss'\mbox{cos}\theta}{2}\end{aligned}$$ where $D_{OP,K}=3g \approx 16\,$eV$\AA^{-1}$ and $\hbar\omega_0=157\,$meV. The cooling power is similar to the $\Gamma$ phonons case, except a factor of $\tfrac{1}{2}$ smaller due to the angular dependence.
Surface polar phonons
=====================
The surface polar phonons coupling is given by[@FG08; @FNC01; @PA10], $$\begin{aligned}
\left|M^{SPP}_{\bold{k},\bold{k}'}\right|^2=\frac{\pi e^2}{\epsilon_0} F_{j}^{2}\frac{\mbox{exp}(-2qz_0)}{q}\frac{1+ss'\mbox{cos}\theta}{2}\end{aligned}$$ where $\epsilon_0$ is the free space permittivity and $z_0$ is the separation between graphene and the substrate. The magnitude of the polarization field is given by the Frohlich coupling parameter, $F_{j}^{2}$. In common SiO$_2$ dielectrics, there are two dominant surface optical phonon modes having energies $\hbar \omega_{1}=58.9\,$meV and $\hbar \omega_{2}=156.4\,$meV, with Frohlich coupling $F_{1}^{2}=0.237\,$meV and $F_{2}^{2}=1.612\,$meV respectively[@PA10].\
We consider first the intraband cooling power, written as, $$\begin{aligned}
\nonumber
{\cal P}^{SPP}_{cc}&=&\frac{g_s g_v }{(2\pi)^2\hbar } \int_{0}^{\infty}k dk \int_{0}^{\infty}k' dk' \int_{0}^{2\pi} d\theta
\frac{\pi e^2}{\epsilon_0} F_{j}^{2}\frac{\mbox{exp}(-2qz_0)}{q}\frac{1+ss'\mbox{cos}\theta}{2}
\delta(k'-k-\omega_{j}v_{F}^{-1})(k'-k) {\cal F}(k,k')\\
&=&\frac{g_s g_v }{(2\pi)^2\hbar }\frac{\omega_j}{v_F}\frac{\pi e^2}{\epsilon_0} F_{j}^{2} \int_{0}^{\infty} k(k+\omega_{j}v_{F}^{-1}) {\cal F}(k,k+\omega_{j}v_{F}^{-1}) \int_{0}^{2\pi}
\frac{\mbox{exp}(-2qz_0)}{q}\frac{1+\mbox{cos}\theta}{2} d\theta dk\end{aligned}$$ The phonon momentum $q$ has the constraint $q^2=k'^2+k^2-2k'k\mbox{cos}\theta$. Under typical conditions, the factor $\mbox{exp}(-2qz_0)\approx 1$. Linearizing $\mbox{exp}(-2qz_0)$, the intraband cooling power then becomes, $$\begin{aligned}
{\cal P}^{SPP}_{cc}&=&\frac{g_s g_v \omega_j\pi e^2 F_{j}^{2}}{(2\pi)^2\hbar v_F \epsilon_0} N_{\omega_j}
\left[\mbox{exp}(\frac{\hbar\omega_j}{k_{B}T_{L}})-\mbox{exp}(\frac{\hbar\omega_j}{k_{B}T_{E}})\right] \int_{0}^{\infty} kk' (1-f_k)f_{k'} \Theta(k,k') \mbox{exp}\left(\frac{-z_0}{\Theta(k,k')}\right) dk\end{aligned}$$ where $k'\equiv k+\omega_{0}v_{F}^{-1}$ and $$\begin{aligned}
\nonumber
\Theta(k,k')&\equiv&\int_{0}^{2\pi}
\frac{1+\mbox{cos}\theta}{2\sqrt{k'^2+k^2-2k'k\mbox{cos}\theta}} d\theta \\
&=& \frac{k+k'}{kk'}
\left[I_{K}\left(\frac{2\sqrt{kk'}}{k+k'}\right)-I_{E}\left(\frac{2\sqrt{kk'}}{k+k'}\right)\right]\end{aligned}$$ where $I_{K,E}$ are the complete elliptic integrals of first and second kind. In similar fashion, the interband cooling power is written as, $$\begin{aligned}
\nonumber
{\cal P}^{SPP}_{cv}&=&\frac{g_s g_v \omega_j\pi e^2 F_{j}^{2}}{(2\pi)^2\hbar v_F \epsilon_0} N_{\mbox{-}\omega_j}
\left[\mbox{exp}(\frac{-\hbar\omega_j}{k_{B}T_{L}})-\mbox{exp}(\frac{-\hbar\omega_j}{k_{B}T_{E}})\right]\times\\
&& \int_{0}^{\infty} kk' H_v[k'](1-f_k)f_{-k'}
\Theta(k,-k')\mbox{exp}\left(\frac{-z_0}{\Theta(k,-k')}\right) dk\end{aligned}$$ where $k'\equiv \omega_{0}v_{F}^{-1}-k$.
\[0.55\][![ (Color online) Cartoon illustrating the typical cooling pathways of hot electrons produced by continuous photoexcitation of graphene with detail descriptions in the main text. Heat can also be dissipated through metallic contacts attached to graphene (not shown), as discussed in Sec.\[sec:cooldy\]. []{data-label="fig1"}](fig1.pdf "fig:")]{}
\[1.0\][![ (Color online) **(a-b)** Electron cooling power due to acoustic phonons bath, ${\cal P}^{AP}$, as function of electron temperature $T_{E}$ for different lattice temperature $T_{L}$ calculated for neutral and doped graphene respectively. Intraband ($cc$, $vv$) and interband ($cv$, $vc$) processes are indicated. **(c-d)** Similar, except for $K$ optical phonons modes, ${\cal P}^{OP,K}$ and **(e-f)** for high energy *unscreened* SPP mode, ${\cal P}^{SPP,H}$. $\Gamma$ optical and the low energy SPP phonons show similar characteristic (not shown). []{data-label="fig2"}](fig2.pdf "fig:")]{}
\[1.0\][![ (Color online) $\bold{(a)}$ Fractional cooling power ${\cal P}^{\alpha}/{\cal P}^{T}$ where ${\cal P}^{T}=\sum _{\alpha}{\cal P}^{\alpha}$, calculated at “near equilibrium"’ condition of $T_E-T_L=10\,$K for neutral graphene, including 2D screening $\epsilon_{2D}(q)$ described in text. For SPP and OP, the dashed (solid) line represents the low (high) energy mode. $\bold{(b)}$ Same as (a), except calculated for “far from equilibrium"’ condition of $T_E-T_L=100\,$K. []{data-label="fig3"}](fig3.pdf "fig:")]{}
\[1.3\][![ (Color online) Carrier’s cooling time $\tau_E$ after photoexcitation plotted as function of $T_E$ calculated for cold neutral graphene. Various substrates are considered, namely SiO$_2$, BN and non-polar, calculated for screened and unscreened SPP scattering potentials. Dotted line distinguishing fast/slow cooling is rather arbitrary, and serve only as guide to the eye. []{data-label="fig4"}](fig4.pdf "fig:")]{}
\[1.1\][![ $\bold{(a)}$ (Color online) Steady-state excess carrier density, $\delta n$, upon continuous photoexcitation as function of the ambient temperature for neutral graphene. Various substrates are considered, namely SiO$_2$, BN and non-polar, calculated for screened and unscreened SPP scattering potentials. $\bold{(b)}$ Elevated temperatures, $T_E-T_L$, calculated for same conditions in (a). All calculations assumed ${\cal P}^0 = 1\times 10^7\,$W/m$^{2}$ and $\kappa_0 = 10\,$MW/Km$^{2}$. []{data-label="fig5"}](fig5.pdf "fig:")]{}
\[1.3\][![ (Color online) Out-of-plane thermal conductance, $\kappa_{SPP}$, defined as $\kappa_{SPP}={\cal P^{SPP}}/\delta T$ calculated for SiO$_2$ for different conditions such as (i) screened and unscreened SPP scattering potentials and (ii) different doping $\mu$. $\kappa_{LAT}$ for undoped graphene on a non-polar substrate is plotted as reference. []{data-label="fig6"}](fig6.pdf "fig:")]{}
[73]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , , , , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , ****, ().
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, , , , , , , ****, ().
, p. ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , , , , ****, ().
, , , , , , ****, ().
, , , , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , , ().
, , , ().
, , , , ****, ().
, , , , ****, ().
, ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ().
, ****, ().
, , , , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , , , , ****, ().
, , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ().
, ****, ().
, ****, ().
, ().
, ****, ().
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , ****, ().
, , , , ().
, ().
, , , ().
, , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , , , , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , , , , , , ****, ().
, , , ****, ().
, , , , ****, ().
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.